THE
MECHIIANICAL PRINCIPLES
ENGINEERING
AND
ARCHITECTURE.
BY
HENRY MOSELEY, M.A. F.1P.S.
CHAPLAIN IN ORDINARY TO THE QUEEN, CANON OF BRISTOL, VICAR OF OLVESTON;
CORRESPONDING MEMBER OF THE INSTITUTE OF FRANCE, AND FORMERLY PROFESSOR
OF NATURAL PHILOSOPHY AND ASTRONOMY IN KING'S COLLEGE, LONDON.
First America from Second London Editions.
WITH ADDITIONS BY
D. H. MAHAN, LL.D
U. S. MILITARY ACADEMY.
WITH ILLUSTRATIONS ON WVOOD.
NEW YORK:
WILEY & HALSTED.
1856.




ENTERED according to Act of Congress, in the year 1856, by
WILEY & HALSTED,
In the Clerk's Office of the District Court of the United States, for the Southern District
of New York.
W. H. TINSON, Stereotyper.                             R. CRAIGHEAD, Printer.




EDITOR'S  PREFACE.
TiE high place that Professor Moseley occupies in the
scientific world, as an original investigator, and the clearness and elegance of the methods he has employed in this
work have made it a standard text book on the subjects it
treats of. In undertaking its revision for the press, at the
request of the publishers of this edition, it has been deemed
advisable, in view of the class of students into whose hands
it may fall, to make some slight addition to the original.
This has been done in the way of Notes thrown into an
Appendix, the matter of which has been gathered from
various authorities; but chiefly from notes taken by the
editor, whilst a pupil at the French military school at Metz,
of lectures delivered by General Poncelet, at that time, 1829,
professor in that school. It is a source of great pleasure to
the editor to have this opportunity of publicly acknowledging his obligations to the teachings of this eminent saqvan,
who is distinguished not more for his high scientific attainment, and the advancement he has given to mechanical
science, than for having brought these to minister to the
wants of the industrial classes, the intelligent success of
whose operations depends so much upon mechanical science,
by presenting it in a form to render it attainable by the most
ordinary capacities.
ii




iv                  EDITOReS PREFACE.
The editor would remark that he has carefully refrained
from making any alterations in the text revised, except corrections of typographical errors, and in one instance where,
from a repetition of apparently one of these, he apprehended
some difficulty might be offered to the student if allowed
to remain exactly as printed in the original.
UNITED STATES MILITARY ACADEMY,
West Point 3March 8, 1856.




PREFACE TO THE SECOND  EDITION.
I HAVE added in this Edition articles:-first, "On the
Dynamical Stability of Floating Bodies;" secondly, "On
the Rolling of a Cylinder;" thirdly, " On the descent of a
body upon an inclined plane, when subjected to -variations of
temperature, which would otherwise rest upon it;" fourthly,' On the state bordering upon motion of a body moveable
about a cylindrical axis of finite dimensions, when acted
upon by any number of pressures."
The conditions of the dynamical stability of floating
bodies include those of the rolling and pitching motion of
ships. The discussion of the rolling motion of a cylinder
includes that of the rocking motion to which a locomotive
engine is subject, when its driving wheels are falsely
balanced, and that of the slip of the wheel due to the same
cause. The descent of a body upon an inclined plane
when subjected to variations in temperature, which otherwise would rest upon it, appears to explain satisfactorily the
descent of glaciers.
The numerous corrections made in the text, I owe chiefly
to my old pupils at King's College, to whom the lectures
of which it contains the substance, were addressed. For




vi          PREFACE TO THE SECOND EDITION.
several important ones I am, however, indebted to Mr.
Robinson, Master of the School for Shipwrights' Apprentices,
in Her Majesty's Dockyard, Portsea; to whom I have also to
express my warm  acknowledgments for the care with
which he has corrected the proof sheets whilst going through,the press.
May, 1855




PRE FA CE.
Is the following work, I have proposed to myself to apply
the principles of mechanics to the discussion of the most
important and obvious of those questions which present
themselves in the practice of the engineer and the architect;
and I have sought to include in that discussion all the
circumstances on which the practical solution of such questions may be assumed to depend. It includes the substance
of a course of lectures delivered to the students of King's
College in the department of engineering and architecture,
during the years 1840, 1841, 1842.*
In the first part I have treated of those portions of the
science of STATICs, which have their application in the theory
of machines and the theory of construction.
In: the second, of the science of DYNnsaCs, and, under this
head, particularly of that union of a continued pressure with
a continued motion which has received from English writers
the various names of " dynamical effect," "efficiency," "work
done," "labouring force," " work," &c.; and "moment
d'activit6," "quantit6  d'action," "puissance me'canique,"
"travail," from French writers.
Among the latter this variety of terms has at length given
place to the most intelligible and the simplest of them,
* The first 170 pages of the work were printed for the use of my pupils in the
year 1840. Copies of them were about the same time in the possession of
several of my friends in the Universities.
vii




Viii                      PREFACE.
i" travail."  The English word " work" is the obvious translation of "travail," and the use of it appears to be recommended by the same considerations. The work of overcoming
a pressure of one pound through a space of one foot has, in
this country, been taken as the unit, in terms of which any
other amount of work is estimated; and in France, the work
of overcoming a pressure of one kilogramme through a space
of one metre.  M. Dupin has proposed the application of the
term dyname to this unit.
I have gladly sheltered myself from the charge of having
contributed to increase the vocabulary of scientific words,
by assuming the obvious term " unit of work" to represent
concisely and conveniently enough the idea which is attached
to it.
The work of any pressure operating through any space is
evidently measured in terms of such units, by multiplying
the number of pounds in the pressure by the number of feet
in the space, if the direction of the pressure be continually
that in which the space is described. If not, it follows, by
a simple geometrical deduction, that it is measured by the
product of the number of pounds in the pressure, by the
number of feet in the projection of the space described,*
upon the direction of the pressure; that is, by the product
of the pressure by its virtual velocity. Thus, then, we
conclude at once, by the principle of virtual velocities, that
if a macFine work under a constant equilibrium of the
pressures applied to it, or if it work uniformly, then is the
aggregate work of those pressures which tend to accelerate
its motion equal to the aggregate work of those which tend
to retard it; and, by the principle of vis vava, that if the
machine do not work under an equilibrium  of the forces
impressed upon it, then is the aggregate work of those which
tend to accelerate the motion of the machine greater or less
* If the direction of the pressure remain always parallel to itself, the space
described may be any finite space; if it do not, the space is understood to be
so small, that the direction of the pressure may be supposed to remain parallel
to itself whilst that space is described.




PREFACG.                      ix
than the aggregate work of those which tend to retard its
motion by one half the aggregate of the vires vivce acquired
or lost by the moving parts of the system, whilst the work is
being done upon it. In no respect' have the labours of the
illustrious president of tile Academy of Sciences more contributed to the development of the theory of machines than
in the application which he has so successfully made to it of
this principle of vis viva.* In the elementary discussion of
this principle, which is given by M. Poncelet, in the introduction to his Mecaniquge industrielle, he has revived the
term  vis inertice (vis inertice, vis insita, Newton), and,
associating with it the definitive idea of a force of resistance
opposed to the acceleration or the retardation of a body's
motion, he has shown (Arts. 66. and 122.) the work expended
in overcoming this resistance through any space, to be
measured by one half the vis viva accumulated through the
space; so that throwing into the consideration of the forces
under which a machine works, the vires inertice of its moving
elements, and observing that one half of their aggregate vis
viva is equal to the aggregate work of their vires inertice, it
follows, by the principle of virtual velocities, that the difference between the aggregate work of those forces impressed
upon a machine, which tend to accelerate its motion, and
the aggregate work of those which tend to retard the motion,
is equal to the aggregate work of the vires inertice of the
moving parts of the machine: under which form the principle of vis vi;va resolves itself into the principle of virtual
velocities. So many difficulties, however, oppose themselves
to the introduction of the term vis inertice, associated with
the definitive idea of a force, into the discussion of questions
of mechanics, and especially of practical and elementary
mechanics, that I have thought it desirable to avoid it. It
is with this view that I have given a new interpretation to
that function of the velocity of a moving body which is
known as its vie viva.  One half that function I have interpreted to represent the number of units of work accumulated
* See Poncelet, Mecanique Industrielle, troisieme partie.




OXr                   PREFACE.
in the body so long as its motion is continued. This number
of units of work it is capable of reproducing upon any resistance opposed to its motion. A very simple investigation
(Art. 66.) establishes the truth of this interpretation, and
gives to the principle of vis viva the following more simple
enunciation:-" The difference between the aggregate work
done upon the machine, during any time, by those forces
which tend to accelerate the motion, and the aggregate
work, during the same time, of those which tend to retard
the motion, is equal to the aggregate number of units of
work accumulated in the moving parts of the machine
during that time if the former aggregate exceed the latterj:and lost from them during that time if the former aggregate
fall short of the latter." Thus, then, if the aggregate work
of the forces which tend to accelerate the motion of a
machine exceeds that of the forces which tend to retard it,
then is the surplus work (that, done upon the driving points,
above that expended upon the prejudicial resistances and
upon the working points) continually accumulated in the
moving elements of the machine, and their motion is thereby
continually accelerated. And if the former aggregate be
less than the latter, then is the deficiency supplied from the
work already accumulated in the moving elements, so that
their motion is in this case continually retarded.
The moving power divides itself whilst it operates in a,machine, first, into that which overcomes the prejudicial
resistances of the machine, or those which are opposed by
friction and other causes, uselessly absorbing the work in its
transmission.  Secondly, into that which accelerates the
amotion of the various moving parts of the machine, and which
accumulates in them so long as the work done by the moving
power upon it exceeds that expended upon the various
resistances opposed to the motion of the machine. Thirdly,
into that which overcomes the useful resistances, or those
which are opposed: to the motion of the machine at the
working point, or points, by the useful work which is done
by it.




PREFACE.                     Xi
Between these three elements there obtains in every
machine an mathematical relation, which I have called its
MODULUS. The general form of this modulus I have discussed
in'a memoir on the'Theory of Machines" published in the
Philosophical Transactions for the year 1841. The determination of the particular moduli of those elements of
machinery which are most commonly in use, is the subject
of the third part of the following work. From a combination
of the moduli of any such elements there results at once the
modulus of the machine compounded of them.
When a machine has acquired a state of uniform motion,
work ceases to accumulate in its moving elements, and its
modulus assumes the fortl of a. direct relation between the
work done by the motive power upon its driving point and
that yielded at its working points. I have determined by a
general method * the modulus in this case, from that statical
relation between the driving and working pressures upon
the' machine which obtains in the state bordering upon its
motion, and which may be deduced from the known conditions of equilibrium and the established laws of friction. In
making this deduction I have, in every case, availed nmyself
of the following principle, first published in my paper on the
theory of the arch, read before the Cambridge Philosophical
Society in Dec. 1833, and printed in their Trlansactions of
the following year: —"In the state bordering upon motion
of one body upon the'surface of another, the resultant
pressure upon their common surface of contact is inclined
to the normal, at an angle whose tangent is equal to the
coefficient of friction."
This angle I have called the limiting angle of resistance.
Its values calculated, in respect to a great variety of surfaces
of contact, are given in a table at the conclusion of the
second part, from the admirable experiments of M. Morin,t
into the mechanical details of which precautions have been
introduced hitherto unknown to experiments of this class,
* Art. 152. See Phil. Trans., 1841, p. 290.
t Nowuelles EIxpriences sur le Frottement, Paris, 18338.




Xli                        PREFACE.
and which have given to our knowledge of the laws of
friction a precision and a certainty hitherto unhoped for.
Of the various elements of machinery those which rotate
about cylindrical axes are of the most frequent occurrence
and the most useful application; I have, therefore, in the
first place sought to establish the general relation of the
state bordering upon motion between the driving and the
working pressures upon such a machine, reference being
had to the weight of the machine.*  This relation points out
the existence of a particular direction in which the driving
pressure should be applied to any such machine, that the
amount of work expended upon the friction of the axis may
be the least possible. This direction of the driving pressure
always presents itself on the same side of the axis with that
of the working pressure, and when the latter is vertical it
becomes parallel to it; a principle of the economy of power
in machinery which has received its application in the
parallel motion of the marine engines known as the Gorgon
Engines.
I have devoted a considerable space in this portion of my
work to the determination of the modulus of a system of
toothed wheels; this determination I have, moreover,
extended to bevil wheels, and have included in it, with the
influence of the friction of the teeth of the wheels, that of
their axes and their weights.  An approximate form  of this
modulus applies to any shape of the teeth under which they
may be made to work correctly; and when in this approximate form of the modulus the terms which represent the
influence of the friction of the axis and the weight of the
wheel are neglected, it resolves itself into a well known
theorem of M. Poncelet, reproduced by M. Xavier and the
Rev. Dr. Whewell.t  In respect to wheels having epicy* In my memoir on the " Theory of Machines " (Phil. Trans. 1841), I have
extended this relation to the case in which the number of the pressures and
their directions are any whatever. The theorem which expresses it is given in
the Appendix of this work.
f In the discussion of the friction of the teeth of wheels, the direction of the
mutual pressures of the teeth is determined by a method first applied by me to




PREFACE.                        Xiii
cloidal and involute teeth, the modulus assumes a character
of mathematical exactitude and precision, and at once
establishes the conclusion (so often disputed) that the loss of
power is greater before the teeth pass the line of centres
than at corresponding points afterwards; that the contact
should, nevertheless, in all cases take place partly before
and partly after the line of centres has been passed. In the
case of involute teeth, the proportion in which the arc of
contact should thus be divided by the line of centres is
determined by a simple formula; as also are the best
dimensions of the base of the involute, with- a view to the
most perfect economy of power in the working of the
wheels.
The greater portion of the discussions in the third part of
my work I believe to be new to science. In the fourth part
I have treated of " the theory of the stability of structures,"
referring its conditions, so far as they are dependent upon
the rotation of the parts of a structure upon one another, to
the properties of a certain line which may be conceived to
traverse every structure, passing through those points in it
where its surfaces of contact are intersected by the resultant
pressures upon them. To this line, whose properties I first
discussed in a memoir upon i" the Stability of a System of
Bodies in Contact," printed in the sixth volume of the Camb.
Phil. Trans., I have given the name of the line of resistance; it differs essentially in its properties from a line
referred to by preceding writers under the name of the
curve of equilibrium or the line of pressure.
The distance of the line of resistance from the extrados of
a structure, at the point where it most nearly approaches it.
I have taken as a measure of the stability of a structure,* and
that purpose in a popular treatise, entitled Mechanics applied to the Arts,
published in 1834.
* This idea was suggested to me by a rule for the stability of revetement
walls attributed to Vauban, to the effect, that the resultant pressure should
intersect the base of such a wall at a point whose distance from its extrados is
4ths the distance between the extrados at the base and the vertical through
the centre of gravity




Xiv                       PREFACE.
have called it the modulus of stability; conceiving this
measure of the stability to be of more obvious and easier
application than the coefficient of stability used by the
French writers.
That structure in respect to:every independent element
of which the modulus of stability is the same, is evidently
the structure of the greatest stability having a given quantity
of material employed in its construction; or of the greatest
economy of material having a given stability.
The application of these principles of construction to the
theory of piers, walls supported by counterforts and shores,
buttresses, walls supporting the thrust of roofs, and the
weights of the floors of dwellings, and Gothic structures,
has suggested to me a class of problems never, I believe,
before:treated mathematically.
I have applied the well known principle of Coulomb to
the determination of the pressure of earth upon revetement
walls, and a modification of that principle, suggested by M.
Poncelet, to the determination of the resistance opposed to
the overthrow of a wall backed by earth.  This determination has an obvious application to the theory of foundations.
In the application of the principle of Coulomb I have
availed myself, with great advantages of the properties of
the limiting angle of resistance.  All my results have thus
received a new and a simplified form.
The theory of the arch I have discussed upon principles
first laid down in my memoir on " the Theory of the Stability
of a System  of Bodies in Contact," before referred to, and
subsequently in a memoir printed in the "Treatise  on
Bridges"  by Professor Hosking and Mr. Hann.*  They
differ essentially from those on which the theory of Coulomb
is founded;t when, nevertheless, applied to the case treated
* I have made extensive use of the memoir above referred to in the following
work, by the obliging permission of the publisher, Mr. Weale.
* The theory of Coulomb was unknown to me at the time of the publication
of my memoirs printed in the Camb. Phil. Trans. For a comparison of the
two methods see Mr. Hann's treatise.




PREFACE.                        XV
by the French mathematicians, they lead to identical results.
I-have inserted at the -conclusion of my work the tables of
the thrust of circular arches, calculated by M. Garidel from
formulae founded on the theory of Coulomb.
The fifth part of the work treats of the "strength of
materials," and applies a new method to the determination
of the defiexion of a beam under given pressures.
In the case of a beam  loaded uniformly over its whole
length, and supported at four different points, I have determined the several pressures upon the points of support by a
method applied by M. Navier to a similar determination in
respect to a beam loaded at given points.*
In treating of rupture by elongation I have been led to a
discussion of the theory of the suspension bridge. This
question, so complicated when reference is had to'the weight
of the roadway and the weights of the suspending rods, and
when the suspending chains are assumed to be of uniform
thickness, becomes comparatively easy when the section of
the chain is assumed so to vary its dimensions as to be every
where of the same strength. A suspension bridge thus
constructed is obviously that which, being of a given
strength, can be constructed with the least quantity of
materials; or, which is of the greatest strength having a
given quantity of materials used in its construction.t
The theory of rupture by transverse strain has suggested
a new class of problems, having reference to the forms of
girders having wide flanges connected by slender ribs or by
open frame work: the consideration of their strongest forms
leads to results of practical importance.
In discussing the conditions of the strength of breastsummers, my attention has been directed to the best positions
of the columns destined to support them, and to a comparison
* As in fig. p. 487. of the following work.
f That particular case of this problem, in which the weights of the suspending
rods are neglected, has been treated by Mr. Hodgkinson in the fourth vol. of
Manchester Transactions, with his usual ability. He has not, however, succeeded in effecting its complete solution.




XVi                         PREFACE.
of the strength of a beam carrying a uniform load and supported freely at its extremities, with that of a beam similarly
loaded  but having  its  extremities  firmly  imbedded  in
masonry.
In treating of the strength of columns I have gladly
replaced the mathematical speculations upon this subject,
which are so obviously founded upon false data, by the
invaluable  experimental results of Mr. E. Hodgkinson,
detailed  in  his well known  paper in  the Philosophical
Transactions for 1840.
The sixth and last part of my work treats on "impact;"
and the Appendix includes, together with tables of the
mechanical properties of the materials of construction, the
angles of rupture and the thrusts of arches, and complete
elliptic functions, a demonstration of the admirable theorem
of M. Poncelet for determining an approximate value of the
square root of the sum or difference of two squares.
In respect to the following articles of my work I have to
acknowledge my obligations to the work of M. Poncelet,
entitled ifeceangiqe Industrielle. The mode of demonstration
is in some, perhaps, so far varied as that their origin might
with difficulty be traced; the principle, however, of each
demonstration-all that constitutes its novelty or its valuebelongs to that distinguished author.
30,* 38, 40, 45, 46, 47, 52, 58, 62, 75, 108,t 123, 202,
267,4 268, 269, 270, 349, 354, 365.~
* The enunciation only of this theorem is given in the  iec. Ind., 2me partie,
Art. 38.
f Some important elements of the demonstration of this theorem are taken
from the Mec. Ind., Art. 79. 2me partie. The principle of the demonstration
is not, however, the same as in that work.: In this and the three following articles I have developed the theory of the
fly-wheel, under a different form from that adopted by M. Poncelet (Mec. Ind.,
Art. 56. 3me partie). The principle of the whole calculation is, however,
taken from his work. It probably constitutes one of the most valuable of his
contributions to practical science.
~ The idea of determining the work necessary to produce a given deflection
of a beam from that expended the compression and the elongation of its component fibres was suggested by an observation in the Mec. Ind., Art. 75. Sme
partie.




OO N T E N T S
PART I.
STATICS.
Page
The Parallelogram of Pressures.                                          3
The Principle of the Equality of Moments.                  6
The Polygon of Pressures.                                             10
The Parallelopipedon of Pressures..... 14
Of Parallel Pressures.16
The Centre of Gravity..20.The Properties of Guldinus.                                          36. 
PART II.
DYNAMICS.
Motion                          4...........
Velocity.................48
WOR.......48
Work of Pressures applied in, different Directions to a Body moveable
about affixed Axis........... 5
Accumulation of Work.......58
Angular Velocity.........                                             65
The Moment of. Inertia.70
THE.ACCELERATION OF MOTION BY GIVEN MOVING FORCES.. 
The Descent of. a Body upon a Curve.....                               83
The Simple Pendulum....... 85
Impulsive Force................86
The Parallelogram of Motion.....                                  86
The Polygon of. Motion..............88
The Principle of D'Alembert.,..                          89
Motion of.Translation.............
Motion of.Rotation about a fixed Axis..                               91
The Centre of Percussion.............. 96
The Centre of Oscillation...                                 96
Projectiles.....99
Centrifugal Force.....106
*




XViii                          CONTENTS.
Page
The Principle of virtual Velocities....  112
The Principle of Vis Viva..116
Dynamical Stability......  121
FRICTION......... 124
Summary of the Laws of Friction.. 130
The limiting Angle of Resistance..131
The Cone of Resistance..133
The two States bordering upon Motion.                133
THE RIGIDITY OF CORDS..142
PART III.
THE THEORY OF MACHINES.
The Transmission of Work by Machines... 146
The Modulus of a Machine moving with a uniform or periodical Motion. 148
The Modulus of a Machine moving with an accelerated or a retarded
Motion...150
The Velocity of a Machine moving with a variable Motion... 151
To determine the Co-efficients of the Modulus of a Machine...153
General Condition of the State bordering upon Motion in a Body acted
upon by Pressures in the same Plane, and moveable about a cylindrical
Axis.........154
The Wheel and Axle.........  155
The Pulley....... 160
System of one fixed and one moveable Pulley... 161
A System of one fixed and any Number of moveable Pulleys.. 163
A Tackle of any Number of Sheaves..166
The Modulus of a compound Machine... 169
The Capstan.                                                          194
The Chinese Capstan. 199
The Horse Capstan, or the Whim Gin..202
The Friction of Cords..                                              207
The Friction Break. 213
The Band... 215
The modulus of the Band....217
The Teeth of Wheels..22
Involute Teeth......... 234
Epicycloidal and Hypocycloidal Teeth.                                 236
To set out the Teeth of Wheels..239
A Train of Wheels...........241
The Strength of Teeth...243
To describe EpicycloidalTeeth.                                      245
To describe involute Teeth.                                           261
The Teeth of a Rack and Pinion....253




CONTENTS.                              xix
Page
The Teet'h of a Wheel working with a Lantern or Trundle. 2 51
The driving and working Pressures on Spur Wheels.                    259
The Modulus of a System of two Spur Wheels.. 268
The Modulus of a Rack and Pinion..                                   2S8
Conical or Bevil Whleels..2S4
The Modulus of a System of two Bevil Wheels.2 88
The Modulus of a Train of Wheels....301
The Train of least Resistance.31
The Inclined Plane.. 319
The Wedge driven by Pressure... 321
The Wedge driven by Impact.. 32'3
The mean Pressure of Impact.. 5325
The Screw....326
Applications of the Screw.2..9
The Differential Screw...... 31
itunter's Screw....                    382
The Theory of the Screw with a Square Thread in reference to the variable Inclination of the Thread at different Distances from the Axis. 333
The Beam of the Steam Engine..337
The Crank............ 341
The Dead Points in the Crank.3. 45
The Double Crank......
The Crank Guide...                                                    51.
The Fly-wheel....5
The Friction of the Fly-wheel.. 36 2
The Modulus of the Crank and Fly-wheel.63. 
The Governor.. 364
The Carriage-wheel.. 68
On the State of the accelerated or retarded Motion of a Machine. 3. 3
PART IV.
THE THEORY OF TIHE STABILITY OF S'T'RUCTUtERS.
General Conditions of the Stability of a Structure of Unceinented Stones 371
The Line of Resistance..
The Line of Pressure.                                             379
The Stability of a Solid Body...8:)
The Stability of a Structure.....
The Wall or Pier..82
The Line of Resistance in a Pier...9 383p
The Stability of a Wall supported by Shores.....
The Gothic Buttress......39
The Stability of Walls sustaining Roofs.                397
The Plate Band.......    4');
The sloping Buttress




XX                            CONTEXNTS.
Page
The Stability of a Wall sustaining the Pressure of a Fluid. 408
Earth Works. 41'
Revrtement Walls...41()
The Arch..  429
The Angle of Rupture.                                               437
The Line of Resistance in a circular arch whose Voussoirs are equal, and
whose Load is distributed over different Points of its Extrados. 440
A segmental Arch whose Extrados is horizontal.                     441
A Gothic Arch, the Extrados of each Semi-Arch being a straight Line
inclined at any given Angle to the Horizon, and the Material of the
Loading different from that of the Arch..                        4142
A circular Arch having equal Voussoirs and sustaining the Pressure of
Water.......444
The Equilibrium of an Arch, the Contact of whose Voussoirs is geometrically accurate.                                                   446;
Applications of the Theory of the Arch.. 4-48
Tables of the Thrust of Arches..45-4
PART V.
THE STRENGTHt OF MATERIALS.
Elasticity............
Elongation...........               45
The Moduli of Resilience and Fragility.4(; 2
Deflection.. 4.67
The Deflexion of Beams loaded uniformly..481
The Deflexion of Breast Summers.                                     4...;
Rupture...
Tenacitv..2.....
The Suspension Bridge... 505
Tile Catenary..506
The Suspension Bridge of greatest Strength.                         510
Rupture by Compression.51
The Section of Rupture in a Beam..     52m 1
General Conditions of the Rupture of a Beam.. 521
The Beam of greatest Strength..                                    527
The Strength of Breast Summers.54()
The best Positions of their Points of Support..        54-12
Formule representing the absolute Strength of a Cylindrical Column to
sustain a Pressure in the Direction of its Length.. 545
Torsion.                                                             46




CONTENTS.                             xxi
PART VI.
IMIPACT.
Page
The Impact of two Bodies whose centres of Gravity move in the same
right Line........... 553
Greatest Compression of the Surface of the Bodies...          55.5
Velocity of two elastic Bodies after Impact..                 556
The Pile Iriver.                            564
AADDITIONS BY TlE r~AMERICAN EDITOR...                               571
APPEND IX.
Note A...631
Note B.-Poncelet's Theorems..   632
Note C.-On the Rolling of Ships..637
Note D.......653
Note E.-On the Rolling Motion of a Cylinder..                    655
Note F.-On the Descent upon an Inclined Plane of a Body subject to
Variations of Temperature, and on the Motion of Glaciers.     6'i5
Note G. -The best Dimensions of a Buttress... 683
Note H.-Dimensions of the Teeth of Wheels...               684
Note I.-Experiments of M. Morin oil the Traction of Carriages. 685
Note K. —On the Strength of Columns..686
Table I.-The Numerical Values of complete Elliptic Functions of the.first and second Orders for Values of the Modulus k corresponding to
each Deglree of the Angle sin.-lkc.                               687
Table II.-Showing the Angle of Rupture t of an Arch whose Loading
is of the same Material with its Voussoirs, and whose Extrados is
inclined at a given Angle to the Horizon.                        688
Table III.-Showing the HIorizontal Thrust of an Arch, the Radius of
whose Intrados is Unity, and the Weight of each Cubic Foot of its
Material and that of its Loadinag, Unity..691
Table IV.-Mechanical Properties of the Materials of Construction.   694
Table V.-Useful Numbers.                                            698








THE
MECHANICAL: PRINCIPLES
CIVIL ENGINEERING.
PART I.
STATICS.
1. FORCE is that which tends to cause or to destroy
motion, or which actually causes or destroys it.
Tile direction of a force is that straight line in which it
tends to cause motion in the point to which it is applied, or
in which it tends to destroy the motion in it.*
When more forces than one are applied to a body, and
their respective tendencies to communicate motion to it
counteract one another, so that the body remains at rest,
these forces are said to be in EQUILIBRIUM, and are called
PRESSURES.
It is found by experiment t that the effect of a pressure,
when applied to a solid body, is the same at whatever point
in the line of its direction it is applied; so that the conditions of the equilibrium of that pressure, in respect to other
pressures applied to the same body, are not altered, if, without altering the direction of the pressure, we remove its
point of application, provided only the point to which we
remove it be in the straight line in the direction of which it
acts.
The science of STATICS is that which treats of the equilibrium of pressures. When two pressures only are applied to
* Note (a) Ed. Appendix.  t Note (b) Ed. Appendix.




2                 THE UNIT OF PRESSURE.
a body, and hold it at rest, it is found by experiment that
these pressures act in opposite directions, and have their
directions always in the same straight line. Two such pressures are said to be equal.
If, instead of applying two pressures which are thus equal
in opposite directions, we apply them  both in the same
direction, the single pressure which must be applied in a
direction opposite to the two to sustain them, is said to be
double of either of them.  If we take a third pressure, equal
to either of the two first, and apply the three in the same
direction, the single pressure, which must be applied in a
direction opposite to the three to sustain them, is said to be
triple of either of them: and so of any number of pressures.
Thus, fixing upon any one pressure, and ascertaining how
many pressures equal to this are necessary, when applied in
an opposite direction, to sustain any other greater pressure,
we arrive at a true conception of the amount of that greater
pressure in terms of the first.
That single pressure, in terms of which the amount of any
other greater pressure is thus ascertained, is called an UNIT
of pressure.
Pressures, the amount of which are determined in terms
of some known unit of pressure, are said to be measured.
Different pressures, the amounts.of which can be determined in terms of the same unit, are said to be commensurable.
The units of pressure which it is found most convenient to
use, are the weights of certain portions of matter, or the
pressures with which they tend towards the centre of the
earth. The units of pressure are different in different countries. With us, the unit of pressure from which all the rest
are derived is the weight of 22'815 * cubic inches of distilled
water. This weight is one pound troy; being divided into
5760 equal parts, the weight of each is a grain troy, and
7000 such grains constitute the pound avoirdupois.
If straight lines be taken in the directions of any number
of pressures, and have their lengths proportional to the
numbers of units in those pressures respectively, then these
lines having to one another the same proportion in length
that the pressures have in magnitude, and being moreover
drawn in the directions in which those pressures respectively
act, are said to represent them in magnitude and direction.
* This standard was fixed by Act of Parliament, in 1824. The temperature
of the water is supposed to be 620 Fahrenheit, the weight to be taken in air,
and the barometer to stand at 30 inches.




THE PARALLELOGRAM OF PRESSURES.             3
A system of pressures being in equilibrium, let any number of them be imagined to be taken away and replaced by
a single pressure, and let this single pressure be such that
the equilibrium which before existed may remain, then this
single pressure, producing the same effect in respect to the
equilibrium that the pressures which it replaces produced, is
said to be the RESULTANT.
The pressures which it replaces are said to be the COMPONENTS of this single pressure; and the act of replacing them
by such a single pressure, is called the COMPOSITION of
pressures.
If, a single pressure being removed from a system in equilibrium, it be replaced by any number of other pressures,
such, that whatever effect was produced by that which they
replace singly, the same effect (in respect to the conditions of
the equilibrium) may be produced by those pressures conjointly, then is that single pressure said to have been RESOLVED into these, and the act of making this substitution
of two or more pressures for one, is called the RESOLUTION
of pressures.
THE PARALLELOGRAM OF PRESSURES.
2. The resultacnt of any two pressures applied to a point,
is represented in direction by the octagonal of a parallelogram, whose adjacent sides represent those pressures in
magnitude and direction.*
(Duchayla's Method.t)
To the demonstration of this proposition, after the excellent method of Duchayla, it is necessary in the first place
to show, that if there be any two pressures P2 and P2 whose
directions are in the same straight line, and a third pressure
P1 in any other direction, and if the proposition be true in
respect to P1 and P2, and also in respect to P1 and P., then
it will be true in respect to P, and P2+P,.
Let P1, P,O and P,, form part of any system of pressures in
C. D  equilibrium, and let them be applied to the point
A; take AB and AC to represent, in magnitude
i`; A and direction, the pressures P, and P2, and CD
RX`  the pressure P., and complete the parallelograms
CB and DF. Suppose the proposition to be true with regard
* This proposition constitutes the foundation of the entire science of Statics.
t Note (c) Ed. App.




4                 THE PARALLELOGRAM
to P- and Pa, the resultant of P, and P2 will then be in the
direction of the diagonal AF of the parallelogram BC, whose
adjacent sides AC and AB represent P, and P2 in magnitude
and direction. Let P, and P2 be replaced by this resultant.
It matters not to the, equilibrium where in the line AF it is
applied; let it then be applied at F. But thus applied at
F it may, without affecting the conditions of the equilibrium,
be in its turn replaced by (or resolved into) two other pressures
acting in CF and BF, and these will manifestly be equal to
P, and P2, of which P, may be transferred without altering
the conditions to C, and P2 to E. Let this be done, and let
PS be transferred from A to C, we shall then have P1 and
P3 acting in the directions CF and CD at C and P2, in the
direction FE at E, and the conditions of the equilibrium will
not have been affected by the transfer of them to these
points. Now suppose that the proposition is also true in
respect to P1 and P2 as well as P1 and P2. Then since CF
and CD represent P, and P3 in magnitude and direction,
therefore their resultant is in the direction of the diagonal
CE. Let them be replaced by this resultant, and let it be
transferred to E, and let it then be resolved into two other
pressures acting in the directions DE and FE; these will
evidently be P1 and P.. We have now then transferred all
the three pressures P1, P2, P,, from A to E, and they act at E
in directions parallel to the directions in which they acted at
A, and this has been done without affecting the conditions of
the equilibrium; or, in other words, it has been shown that
the pressures PF, P2, P., produce the same effect as it respects the conditions of the' equilibrium, whether they be
applied at A or E. The resultant of PI, P2, P,, must therefore produce the same effect as it regards the conditions of
the equilibrium, whether it be applied at A or E. But in
order that this resultant may thus produce the same effect
when acting at A or E, it must act in the straight line AE,
because a pressure produces the same effect when applied at
two different points only when both those points are in the
line of its direction. On the supposition made, therefore,
the resultant of P1, P2, and P., or of P, and P2 + P3
acts in the direction of the diagonal AE of the parallelogramn BD, whose adjacent sides AD and AB represent
P2 + P1 and P, in magnitude and direction; and it has been
shown, that if the proposition be true in respect to P, and
P2, and also in respect to P1 and P,, then it is true in respect
to P1 and P2 + P.. Now this being the case for all values
of P1, P2, P, it is the case when P,, P2, and FP, are equal




OF PRESSURES.                   5
to one another. But if P1 be equal to P2 their resultant
will manifestly have its direction as much towards one of
these pressures as the other; that is, it will have its direction midway between them, and it will bisect the angle BAC:
but the diagonal AF in this case also bisects the angle BAC,
since P1 being equal to P,, AC is equal to AB; so that in
this particular case the direction of the resultant is the
direction of the diagonal, and the proposition is true, and
similarly it is true of P1 and P,, since these pressures are
equal. Since then it is true of P1 and P2 when they are
equal, and also of P1 and P,, therefore it is true in this case
of P, and P2 + P,, that is of P, and 2 P1. And since it is
true of P1 and P1, and also of P1 and 2 P1, therefore it is true
of P, and P1 + 2 P1, that is of P1 and 3 P1; and so of P1 and
m P1, if m be any whole number; and similarly since it is
true of m P1 and P1, therefore it is true of m, P and 2 P,- &c.,
and of m P1 and n P1 where n is any whole number. Therefore the proposition is true of any two pressures m P, and
n P, which are corlnmensurable.
It is moreover true when the pressures are ian-::.. commensurable.  For let AC and AB represent:: —-::  any two such pressures P, and P1 in magnitude
and direction, and complete the parallelogram
ABDC, then will the direction of the resultant of P1 and
P2 be in AD; for if not, let its direction be AE, and draw
EG parallel to CD. Divide AB into equal parts, each less
than GC, and set off on AC parts equal to those from A
towards C. One of the divisions of these will manifestly
fall in GC. Let it be II, and complete the parallelogram
AHFB.  Then the pressure P. being conceived to be
divided into as many equal units of pressure as there are
equal parts in the line AB, AH may be taken to represent a
pressure Pa containing as many of these units of pressure
as there are equal parts in AH, and these pressures P1 and
P1 will be conzmensurable, being measured in terms of the
same unit. Their resultant is therefore in the direction AF,
and this resultant of P1 and P1 has its direction nearer to
AC than the resultant AEL of P1 and P1 has; which is
absurd, since P, is greater than P1.
Therefore AE is not in the direction of the resultant of
P1 and P,; and in the same manner it may be shown that no
other than AD' is in that direction. Therefore, &c.




6               THE PRINCIPLES OF THE
3. The resultant of two pressures applied in any directions
to apoint, is represented in magnitude as well as in direction by the diagonal of the pcarallelogram, whose adjacent
sides represent those pressures in magnitude and in direction.
Let BA and CA represent, in magnitude and
ri".,,   direction, any two pressures applied to the point
A. Complete the parallelogram BC. Then by
"I.   the last proposition AD will represent the resultant of these pressures in direction. It will also
represent it in magnitude; for, produce DA to G, and conceive a pressure to be applied in GA equal to the resultant
of BA and CA, and opposite to it, and let this pressure be
represented in magnitude by the line GA. Then will the
pressures represented by the lines BA, CA, and GA, manifestly be pressures in equilibrium. Complete the parallelogram BG, then is the resultant of GA and BA in the
direction FA; also since GA and BA are in equilibrium
with CA, therefore this resultant is in equilibrium with CA,
but when two pressures are in equilibrium, their directions
are in the same straight line; therefore FAC is a straight
line. But AC is parallel to BD, therefore FA is parallel to
BD, and FB is, by construction, parallel to GD, therefore
AFBD is a parallelogram, and AD is equal to FB and
therefore to AG. But AG represents the resultant of CA
and BA in magnitude, AD therefore represents it in magnitude. Therefore, &c.*
THE PRINCIPLE OF THE EQUALTY  OF MOMENTS.
4. DEFINITION. If any number of pressures act in the
same plane, and any point be taken in that plane, and perpendiculars be drawn from it upon the directions of all these
pressures, produced if necessary, and if the number of units
in each pressure be then multiplied by the number of units
in the corresponding perpendicular, then this product is
called the moment of that pressure about the point from
which the perpendiculars are drawn, and these moments are
said to be measured from that point.
* Note (d) Ed. App.




EQIJALITY OF MOMENTS.             7
5. If three pressures be in equilibrium, and their moments
be taken about any point in the plane in which they act,
then the sume of the mnoents of those two pressures which
tend to turn the plane in one direction about the -point
feomn which the moments are measured, is equal to the
momeJnt of that pressure which tends to turn it in the
opposite cdirection.'.-._..M.  _ Let P, P., Po, acting in the directions
N ag- I2  P1O, P2,0  PG, be any three pressures in
- -D%..-::2:   equilibrium. Take any point A in the plane
A —-....in which they act, and measure their moments
from A, then will the sumn of the moments of P2 and P,,
which tend to turn the plane in one direction about A, equal
the moment of P,, which tends to turn it in the opposite
direction.
Through A draw DAB parallel to OP1, and produce OP2
to meet it in D. Take OD to represent P,, and take DB
such a length that OD may have the same proportion to
DB that P, has to P,. Complete the parallelogram ODBC,
then will OD and OC represent P, and P, in magnitude and
direction.  Therefore OB will represent P, in magnitude.
and direction.
Draw AM, AN, AL, perpendiculars on OC, OD, OB,
and join AO, AC. Now the triangle OBC is equal to the
triangle OAC, since these triangles are upon the same base
and between the same parallels.
Also,  A ODA+ AG OAB = A OBD = a OBC,.*  ODA +  AB =  OAC, -_ AO7
ODxAN+OB x AL  GCxA, a 2
-P, xAN+PxAL=PxAM.                S' 
Now P, x AM, P2 x AN, P, x AL, are the moments of P,,
P,, P3, about A (Art. 4.).. mt P2 + mt P  mt P.(1).
Therefore, &c. &c.
6. If IR be the resultant of P2 and PF, then since R is
equal to P, and acts in the same straight line, mtR - mtP,,.. mltP2 + ntP -= mtR. (8)
The sum of the moments therefore, about any point, of
two pressures, P, and P, in the same plane, which tend to




8                THE PRtINCIPLEL OF THE
turn it in the samne direction about that point, is equal to
the moment of their resultant about that point.
If they had tended to turn it in opposite directions, then
the d~ifrenece of their moments would have equalled the
moment of their resultant. For let R be the resultant of
P1 and P3, which tend to turn the plane in opposite directions about A, &c. Then is R equal to P,, and in the same
straight line with it, therefore -moment R  is equal to
moment P.  But by equation (1) mtP — mtP,   mtP,;
*itpimnt~P  -- MtR
Generally, therefore, mt P1 + qnt P1 -mt R.      (2)
the moment, therefore, of the resultant of any two pressures
in the same _plane is equal to the sum or difference of the
moments of its cormponents, accordiqng  as they act to turn the
plane in the same direction about the point from which the
enoments are mecasured, or in opposite directions.*'
7T. If any number of pressures in the same cplane be in eqilibrium. and cany point be taken, in that pclane, from
which their momnents are measured, then the Su2m of the
mnoments of those pressures which tend to turn the plane
in one direction about that _point is equal to the su7 of the
moments of those which tend to turn it in the opposite
directiown.
Let P,, P, P,..... P be any number of pressures in
the same plane which are in equi-'~2y~R~   ~      librium, and A any point in the
_l..     _ 7:''; plane from  which their moments
}-<T    t J2,-1    are measured, then will the sum of
the moments of those pressures
which tend to turn the plane in one direction about A equal
the sum  of the momrients of those which tend to turn it in
the opposite direction.
Let IR be the resultant of P, and P,,
tt~.......IR, and P,,
&R......  &c.
ARZ3 * * * o * * * R2 nd P 0
R,2i  * O * * *   Rn_2 and P,.
Therefore, by the last proposition, it being understood
that the moments of those of the pressures P,, P2, which
tend to turn the plane to the left of A, are to be taken negatively, we have
Note (c) Ed. App.




EQUALITY OF MOMENTS.               9
mt R, = mt P, + mt P,.
mt R2 = mt R  + mt P,
mt R3  nmt R2 + mt P4,
&c.     &.       &c.
mt iRn1  = mt R-z2 + mt Pn.
Adding these equations together, and striking out the
terms common to both sides, we have
mt R     -1 mt P1 +- mt P2 + mt P3 +..... + mt Pn
(3), where R-   is the resultant of all the pressures P,,
But these pressures- are in equilibrium; they have, therefore, no resultant.. R,  =  O.. mt R,_1   0,.  m  P + mt P2 + mt,..... mt P, = 0.... (4).
Now, in this equation the moments of those pressures which
tend to turn the system to the left hand are to be taken
negatively. 1Moreover, the sum of the negative terms must
equal the sum of the positive terms, otherwise the whole
sum could not equal zero. It follows, therefore, that the
sum of the moments of those pressures which tend to turn
the system to the right must equal the sumn of the moments
of those which tend to turn it to the left. Therefore, &c. &c.
8. af any number qf presswres acting in the same plane be in
eqazilbrium, and'they be imagined to be moved pcrallel to
t7heir existing directiolns, and all a plied to the same point,
so as all to act upon that point Zn directions parallel to
those in which they before acted upon different points, then
will they be in equilibrium  about that point.
For (see the preceding figure) the pressure R1 at whateverpoint in its direction it be conceived to be applied, may be
resolved at that point into two pressures parallel and equal
to P1 and P2: similarly, R2 may be resolved, at any point in
its direction, into two pressures parallel and equal to R, and
P,, of which RI may be resolved into two, parallel and equal
to P1 and P2, so that R, may be resolved at any point of its
direction into three pressures parallel and equal to P1, P,, P;:
and, in like manner, R, may be resolved into two pressures
parallel and equal to R, and P,, and therefore into four pressures parallel and equal to P1, P,, P3, P,, and so of the rest.




10                  THE POLYGON
Therefore Rnl  may, at any point of its direction be resolved
into n pressures parallel and equal to P1, P,, P,..... P;
if, therefore, n such pressures were applied to that point,
they would just be held in equilibrium by a pressure equal
and opposite to R,-1. But Rnl = 0; these n pressures
would, therefore, be in equilibrium with one another if
applied to this point.
Now it is evident, that if, being thus applied to this point,
they would be in equilibrium, they would be in equilibrium
if similarly applied to any other point. Therefore, &c.
THE POLYGON OF PRESSURES.
9. ihe conditions of the egquilibrium of any number ofpresssures avpplied to a point.
Let OPT, OP, OP, &c., represent in magnitude and direction pressures P1, P2, &c.,
%  ^.//~/  \applied to the same point O. Complete the
parallelogram OP, AP,, and draw its diagonal OA; then will OA represent in magnitude and direction the resultant of P2 and
P2.  Complete the parallelogram  OABPS, then will OB
represent in magnitude and direction the resultant of OA
and P.; but OA is the resultant of P1 and P,, therefore OB
is the resultant of P1, P2, PS; similarly, if the parallelogram
OBCP4 be completed, its diagonal OC represents the resultant of OB and P4, that is, of P1, P2, P,, P4, and in like
manner OD, the diagonal of the parallelogram OCDP5,
represents the resultant of P1, P2, P,, 4 P, P.
Now let it be observed, that AP1 is equal and parallel to
OP2, AB to OP., BC to OP4, CD to OP,, so that PA, AB,
BC, CD, represent P, P, P, P, respectively in magnitude,
and are parallel to their directions. Moreover, OP1 is in the
direction of P1 and represents it in magnitude, so that the
sides OP2, PA, AB, BC, CD, of the polygon OP1, ABCDO
represent the pressures P,, P,, Ps, P,, P, respectively in
magnitude, and are parallel to their directions; whilst the:side OD, which completes that polygon, represents the
resultant of those pressures in magnitude and direction.
If, therefore, the pressures P,, P, P, P4, P,, be in equilibrium, so that they have no resultant, then the side OD of
the polygon must vanish, and the point D coincide with O.'Thus, then, if any number of pressures be applied to a point




OF PRESSURES.                   11
and lines be drawn parallel to the directions of those pressures, and representing them in magnitude, so as to form
sides of a polygon (care being taken to draw each line from
the point where it nnites with the preceding, towards the
direction in which the corresponding pressure acts), then the
line thus drawn parallel to the last pressure, and representing
it in magnitude, will pass through the point from which the
polygon commenced, and will just complete it if the pressures be in equilibrium; and if they be not in equilibrium,
then this last line will not complete the polygon, and if a
line be drawn completing it, that line will represent the
resultant of all the pressures in magnitude and direction.
This principle is that of the POLYGON OF PRESSURES; it
obtains in respect to pressures applied to the same point,
whether they be in the same plane or not.
10. If any number of pressures in the same plcne be in eguilibriuwr, and each be resolved in directions parallel to any
two rectangular axes, then the sum of all those resolved
pressures, whose tendency is to communicate motion in one
direction along either anxis, is efucal to the sum of those
whose tendency is in the opposite direction.
Let the polygon of pressures be formed in respect to any
number of pressures, P, P25, PS, P4, in the same plane and in.~            equilibrium (Arts. 8, 9), and let the sides of
this polygon be projected on any straight line
A:.,i!    Ax in the same plane. Now it is evident,
i,G'?-q.',i ~' that the sum of the projections of those sides..:_;1: 1. -. of the polygon which form that side of the
figure which is nearest to Ax, is equal to the sum of the projections of those sides which form the opposite side of the
polygon: moreover, that the former are those sides of the
polygon which represent pressures tending to communicate
motion from A towards x, or from left to right in respect to
the line Ax; and the latter, those which tend to communicate motion in the opposite direction. Now each projection
is equal to the corresponding side of the polygon, multiplied
by the cosine of its inclination to Ax. The sum of all those
sides of the polygon which represent pressures tending to
communicate motion from A towards x, multiplied each by
the cosine of its inclination to Ax, is equal, therefore, to the
sum of all the sides representing pressures whose tendency
is in the opposite direction, each being similarly multiplied
by the cosine of its inclination to Ax. Now the sides of the




12                   THE RESOLUTION
polygon represent, the pressures in magnitude, and are
inclined at the same.angles to Ax. Therefore, each pressure
being multiplied by the cosine of its inclination to Ax, the
sum -of all these products, in respect to those which tend to
communicate motion in one direction, equals the sum similarly taken in respect to those which tend to communicate
motion in the opposite direction; or, if in taking this sum it
be understood that each term into which there enters a pressure, whose tendency is from A towards x, is to be taken
positively, whilst each into which there enters a pressure
which tends from x towards A is to be taken negatively,
then the sum of all these terms will equal zero; that is,
calling the inclinations of the directions of P1, P2, P3.  P4
to As, a,, a,, a..... a, respectively,
P1 cos. a, 4- P2 cos.a2 + Pcos.a, +.... + P, cos. a, =0... (5),
in which expression all those terms are to be taken negatively which include pressures, whose tendency is from x
towards A.
This proposition being true in respect to any axis, Ax is
true in respect to another axis, to which the inclinations of
the directions of the pressures are represented by Pi, 12, 2,,....   so     that,
P, cos. 13  + P2 cos.  32+.... + Pn cos. n =0.
Let this second axis be at right angles to the first:
then  3,  -    al.', cos. /1= sin. a1, P- 2-  a2,.. cos. 2
= sin. a_ &c. - &c.'. Pi sin. a1 + P2 sin. a +....  -   P sin. a =0.... (6);
those terms in this equation, involving pressures which tend
to communicate motion in one direction, in respect to the
axis Ay being taken with the positive sign, and those which
tend in the opposite direction with the negative sign.
If the pressures P,, Pt, &c. be each of them  resolved
into two others, one of which is parallel to the axis Ax, and
the other to the axis Ay, it is evident that the pressures
thus resolved parallel to Ax, will be represented by P, cos. a,,
P2 cos. a2, &c., and those resolved parallel to Ay, by
P1 sin. a1 P2 sin. a2, &c. Thus then it follows, that if
any system  of pressures in equilibrium  be thus resolved
parallel to two rectangular axes, the sum of those resolved
pressures, whose tendency is in one direction along either




OF PRESSURES.                   13
axis, is equal to the sum of those whose tendency is in the
opposite direction.*
This condition, and that of the equality of moments, are
necessary to the equilibrium of any number of pressures in
the same plane, and they are together suflicient to that equilibrium.
11. To determine the resultant of any number of pressures
in the scame plane.
If the pressures P1 P2.... P be not in
I,i'~-~.  equilibrium, and have a resultant, then one
side is wanting to complete the polygon of:,21.    pressures, and that side represents the res-----!' ultant of all the pressures in magnitude,
and is parallel to its direction (Art. 9).
Moreover it is evident, that in this case the sum of the projections on Ax (Art. 10) of those lines which form  one
side of the polygon, will be deficient of the sum of those of
the lines which form the other side of the polygon, by the
projection of this last deficient side; and therefore, that the
sum of the resolved pressures acting in one direction along
the line Ax, will be less than the sum of the resolved pressures in the opposite direction, by the resolved part of the
resultant along this line. Now if R represent this resultant,
and 0 its inclination to Ax, then R cos. 0 is the resolved part
of R in the direction of Ax. Therefore the signs of the terms
being understood as before, we have
R cos. 0=P, cos. al+P, cos. a,~+.   +. +Pcos. a,.. (7).
And reasoning similarly in respect to the axis Ay, we have
R sin. 0 — P1 sin. a+ -P2 sin. a+...  +P sin. an... (8).
Squaring these equations and adding them, and observing
that R2 sin.2 0 + R cos.2 0  R= (sin.20 + cos.0 ) - R2, we have
R   =(zP sin. a)2 + (IP cos. a)2......... (9),
where YP sin. a is taken to represent the sum P1 sin. a, +
P2 sin. a2 + P sin. a2 + &c., and IP cos. a to represent the
sum P, cos. a+ P2 cos. a+ P, cos. a2+ &c.
Dividing equation (8) by equation (7),
tan.          a.(10).
cP cos. a
Thus then by equation (9) the magnitude of the resultant
* Note (f) Ed. App.




14              TIHE PARALLELOPIPEDON
R is known, and by equation (10) its inclination 0 to the axis
Ax is known. In order completely to determine it, we have
yet to find the perpendicular distance at which it acts from
the given point A.  For this we must have recourse to the
condition of the equality of moments (Art. 7).
If the sum of the moments of those of the pressures, P1,
P2... Pn, which tend to turn the system in one direction about A, do not equal the sum of the moments of those
which tend to turn it the other way, then a pressure being
applied to the system, equal and opposite to the resultant R,
will bring about the equality of these two sums, so that the
moment of R must be equal to the difference of these sums.
Let then p equal the perpendicular distance of the direction
of R from A. Therefore
Rp=mt P~+mt P2+mt P       ~+.... +mt Pn... (11),
in the second member of which equation the moments of
those pressures are to be taken negatively, which tend to
communicate motion round A towards the left.
Dividing both sides by R we have
mt Pi1mt P2...    mtP             (12).
Thus then by equations (9), (10), (12), the magnitude of
the resultant R., its inclination to the given axis Ax, and the
perpendicular distance of its direction from the point A, are
known; and thus the resultant pressure is completely determined in magnitude and direction.
THE PARALLELOPrPEDON OF PRESSURES.
12. Three pressures, P1, P2, P,, being apltied to the same
point A, in directions xA, yA, zA,, hich are not in the
same plane, it is required to deterrine their resultant.
Take the lines P, A, P2 A, P, A, to represent the pressures
P1, P2, P,, in magnitude and direction.
_-:      Complete the parallelopipedon RPP,PP,,:....:   ofwhichAP, AP, AP, are adj acent edges,
and draw its diagonal RA; then will RA
—..'  represent the resultant of P,, P, P,, in
direction and  magnitude.  For since
PSPA is a parallelogram, whose adjacent
sides P1 A, P2 A, represent the presures
P, and P in magnitude and direction, therefore its diagonal




OF THREE PRESSURES.               15
SA represents the resultant of these two pressures.  And
similarly RA, the diagonal of the parallelogram RSAP,, represents in magnitude and direction the resultant of SA and
3, that is, of P, P, and P, since SA is the resultant of
P, and P2.
It is evident that the fourth pressure necessary to produce
an equilibrium with P., P2, P, being equal and opposite to
their resultant, is represented in magnitude and direction
by AR.
13. Three pressures, P,, P2, P3, being in equilibrium, it is
required to determine the third P, in terms of the other
two; a/nd their inclination to one another.
Let AP, and AP2 represent the pressures P, and P2 in
magnitude and direction, and let the inclination
p' P, AP2 of P, to P. be represented by 10. Com-?  plete the parallelogram AP, RP,, and draw its..::  diagonal AR.  Then does AR represent the
resultant of P1 and P2 in magnitude and direction. But this resultant is in equilibrium with P,, since P,
and P2 are in equilibrium with P,. It acts, therefore, in the
same straight line with P,, but in an opposite direction, and
is equal to it. Since then AR represents this resultant in
magnitude and direction, therefore RA represents P3 in magnitude and direction.
Now, AR2'AP 2-2AP,. PR. cos. APR+PpRR;
also, APR- r-PAP2-= -,102, PR=AP2, and AP,, AP2,
AR, represent P, P2, P,, in magnitude.
P. 32=P12 —2PP2 cos. ( —, 02) +P2).
Now cos. (r —,02) = -cos.,0,. P=P +s- 2PP1 cos. 102+ P,22.' P,=  4/P,2+2P,P2cos. 02 + P22.....  (13).
14. If three pressures, P,, P2, P,, be in equilibrium, any two
of them are to one another inversely as the sines of their'tnclinations to the third.
Let the inclination of P, to P3 be represented by 10,, and
that of P, to P3 by,03.
Now PAR=-i-PAP=r —, 0,.'. sin. PAR=sin. 101;
P,RA_=P2AR=7-PAP3 =r-20,.' sin. P,RA=sin. 20,,.




16             OF PARALLEL PRESSURES.
Also,       AP,  AP1  sin. PIRA
AP2 - PR - sin. P1AP'
P1  sin. 20s
0P2  sin.0 O03' *,  *..  (14).
P, — sin., 0
That is, P1 is to P2 inversely, as the sine of the inclination of P1 to P2 is to the sine of the inclination of P2 to P3.
Therefore, &c. &c.                           [Q. E. D.]
OF PARALLEL PRESSURES.
15. The principle of the equality of moments obtains in
respect to pressures in the same plane whatever may be
their inclinations to one another, and therefore if their
inclinations be infinitely small, or if they be parallel.
In this case of parallel pressures, the same line AB, which:T T ~   is drawn from a given point A, perpendicular
to one of these pressures, is also perpendicular:-2. f\ ~   to all the rest, so that the perpendiculars are
ff\~;   here the parts of this line AM1, AM2, &c.
intercepted between the point A and the directions of the pressures respectively. The principle is not however in this case true only in respect to the intercepted parts
of this perpendicular line AB, but in respect to the intercepted parts of any line AC, drawn through the point A
across the directions of the pressures, since the intercepted
parts Am,, Am2, Am., &c. of this second line are proportional
to those, AM1, AM2, &c. of the first.
Thus taking the case represented in the figure, since by
the principle of the equality of moments we have,
AM,. P1 + AM.. PA       + P2+AM3P3 + AMP,;
dividing both sides by AM,,
AM,    AM[. AM       AM,
AM * 1+ AM    A 4          P- AM    + AM  P+P1;
AM2 Am1 AM Amw
But by similar triangles, AM-1 Am1 AM   Am1 A &c.-c
Am1       Am        Am2      Am3'A       AP, +A'.P4=A.P+.P,+m P.
Therefore multiplying by Amn,,
Am1. P1+ Am,. P, =Am,. P2 +Am. P + Amr. P.
Therefore, &c.                                [Q.E.D.]




OF PARALLEL PRESSURES.             17
16. To find the resultant of any number of parallel pressures
in the same plane.
It is evident that if a pressure equal and opposite to the
resultant were added to the system, the whole would be in
equilibrium. And being in equilibrium it has been shown
(Art. 8.), that if the pressures were all moved from their
present points of application, so as to remain parallel to their
existing directions, and applied to the same point, they are
such as would be in equilibrium about that point.  But
being thus moved, these parallel pressures would all have
their directions in the same straight line. Acting therefore all
in the same straight line, and being in equilibrium, the suni
of those pressures whose tendency is in one direction along
that line must equal the sum of those whose tendency is in
the opposite direction. Now one of these sums includes the
resultant R. It is evident then that before ZR was introduced
the two sums must have been unequal, and that R equals the
excess of the greater sum over the less; and generally that if
IP represent the sum of any number of parallel pressures,
those whose tendency is in one direction being taken with
the positive sign, and -those whose tendency is in the opposite
direction, with the negative sign; then
-R     P.......(15).
the sign of R indicating whether it act in the direction of
those pressures which are taken positively, or those which are
taken negatively.
Moreover since these pressures, including R, are in equilibrium, therefore the sum of the moments about any point,
of those whose tendency is to communicate motion in one
direction, must equal the sum of the moments of the restthese moments being measured on any line, as AC; but one
of these sums includes the moment of R; these
two sums must therefore, before the introduc~..: "~' tion of R, have been unequal, and the moment
\ 4agE'  of R must be equal to the excess of the greater
sum over the less, so that, representing the
sum of the moments of the pressures (PR not being included)
bly I mt P, those whose tendency is to communicate motion
in one direction, having the positive sign, and the rest the
negative; and representing by x the distance from A, measured along the line AC, at which R intersects that line, we
have, since xR is the moment of R, xZR = z mt P, where the
2




18             OF PARALLEL PRESSURES.
sign of xR indicates the direction in which R tends to turn
the system about A, but R =-  P,
I:mt. 1
~  2 mt.P.          (16).
zp
Equations (15) and (16) determine completely the magnitude and the direction of the resultant of a system of parallel
pressures in the same plane.
17. To determine the resultant of any number of parallel
pressures not'n the same plane.
Let P1 and P2 be the points of application of any two of
these pressures, and let the pressures themselves
be represented by P, and P2. Also let their
if~..resultant R1 intersect the line joining the points
P1 and P2 in the point R1; produce the line
P, P2, to intersect any plane given in position,
in the point L. Through the points P1, P,, and R1, draw
P,1 M, PlM2,  and R1N perpendicularly to this plane: these
lines will be in the same plane with one another and with
P,L; let the intersection of this last mentioned plane with
the first be LM1, then will PAM1, P31M2, and R1N, be perpendiculars to LM,; moreover by the last proposition,
Q LP1 + P LP2 - R- LR1;
pLP1         LP 
LR     2 LR1
But by similar triangles,
IP, P1M,        LP  PAM2
LRJZR N1       LRJZR N,
PM, PM.P.    RN   2,   =R1
Let now the resultant, R,, of R1 and P,
3bt (;-l i.intersect the line joining the points R1 and
P in the point R,, and similarly let the
-)<'i. — ~.. - resultant, R1, of R, and P4 intersect the
ine joining the points R, and P, in the...... a'"i'" point R., and so on: then by the last equation.




OF PARALLEL PRESSURES.             19
P,. P1 1+P2. P2M2  IR, It1,N
Similarly,   tI. RRN, + P. PM,  - R, R,N,,
R,.,Nt +P  PM. P  RR  N,
&C.  +   &C.         &C.
RLa_ 2 i_ t     N_-+P,. RPR-M2=R1,_.   URn- N-_,.
Adding these equations, and striking out terms common to
both sides,
P,. PII + P2 P   +. +      + P,. PM-, R-,.* R,, N1,, (17).
Now,  R-,=P,+P,,        R, Rl+P,=Pl+P,+P,,
R. -R2+P4=P,+P2+P,+P4. &.&c.
R_ 1-=  P +P2+P3 +.....  +P.,;
R_1 N,,1. Pl+2+P,+&c.+P=P,. P,M, ++P1,
P~  +.....    + Pn. PnM};
P, PM,+P2 P,2M+... +P,. PnMn.P1++P2+P3...  +Pn
in which expression those of the parallel pressures P,, P,,
&c. which tend in one direction, are to be taken positively,
whilst those which tend in the opposite direction are to be
taken negatively.
The line R,_1 N,,_ represents the perpendicular distance
from the given plane of a point through which the resultant
of all the pressures P,, P2....P,, passes. In the same
manner may be determined the distance of this point from
any other plane. Let this distance be thus determined in
respect to three given planes at right angles to one another.
Its actual position in space will then be known. Thus then
we shall know a point through which the resultant of all the
pressures passes, also the direction of that resultant, for it is
parallel to the common direction of all the pressures, and we
shall know its amount, for it is equal to the sum of all the
pressures with their proper signs. Thus then the resultant
pressure will be completely known. The point R,,1 is called
the CENTRE OF PARALLEL PRESSURES.
18. The product of any pressure by its perpendicular distance from a plane (or rather the product of the number of
units in the pressure by the number of units in the perpendicular), is called the moment of the pressure, in resjpect to
that plane. Whence it follows from equation (17) that the
sum of the moments of any number of parallel pressures in




20             iTHE CENTRE OF GRAVITY.
respect to a given plane is equal to the mowment of their
resultant in respect to that plane.
19. It is evident, from  equation (18), that the distance
R1- N-1 of the centre of pressure of any number of
parallel pressures from a given plane, is independent of the
directions of these parallel pressures, and is dependent
wholly upon their amounts and the perpendicular distances
P1M, P2M2, &c. of their points of application from the
given plane.
So that if the directions of the pressures were changed,
provided'hat their amounts and points of application
remained the same, their centre of pressure, determined as
above, would remain unchanged; that is, the resultant,
although it would alter its direction with the directions of
the component pressures, would, nevertheless, always pass
through the same point.
The weights of any number of different bodies or different
parts of the same body, constitute a system of parallel pressures; the direction, therefore, through this system of the
resultant weight may be determined by the preceding proposition; their centre of pressure is their centre of gravity.
TmI  CENTRE OF GRAVITY.
20. The resultant of the weights of any number of bodies
or parts of the same body united into a system qf isnva
riable forr  passes through the same point in it, into whatever position it may be turned.
For the effect of turning it into different positions is to
cause the directions of the weights of its parts to traverse
the heavy body or system in different directions, at one time
lengthwise for instance, at another across, at another
obliquely; and the effect upon the direction of the resultant
weight through the body, produced by thus turning it into
different positions, and thereby changing the directions in
which the weights of its component parts traverse its mass,
is manifestly the same as would be produced, if without altering the position of the body, the direction of gravity could
be changed so as, for instance, to make it at one time traverse that body longitudinally, at another obliquely, at a
third transversely. But by Article 19, this last mentioned
change, altering the common direction of the parallel pres



THE CENTRE OF GRAVITY.                     21
sures through the body without altering their amounts or
their points of application, would not alter the position of
their centre of pressure in the body; therefore, neither would
the first mentioned change. Whence it follows that the
centre of pressure of the weights of the parts of a heavy
body, or of a system of invariable form, does not alter its
position in the body, whatever may be the position into
which the body is turned; or in other words, that the
resultant of the weights of its parts passes always through
the same point in the body or system in whatever position
it may be placed.
This point, through which the resultant of the weights of
the parts of a body, or system  of bodies of invariable form,
passes, in whatever position it is placed; or, if it be a body
or system of variable form, through which the resultant
would pass, in whatever position it were placed, if it became
rigid or invariable in its form,, is called the CENTRE OF
GRAVITY.
21. Since the weights of the parts of a body act in
parallel directions, and all tend in the same direction, therefore their resultant is equal to their sum.  Now, the resultant of the weights of the parts of the body would produce,
singly, the same effect as it regards the conditions of the
equilibrium of the body, that the weights of its parts
actually do collectively, and this weight is equal to the sum
of the weights of the parts, that is, to the whole weight of
the body, and in every position it acts vertically downwards
through the same point in the body, viz. the centre of
gravity.  Thus then it follows, that in every position of the
body and under every circumstance, the weights of its parts
produce the same efect in respect to the conditions of its
eguilibrium, as though they were all collected in and acted
through that one point of it —its centre of gravity.*
* That the resultant of the weights of all the parts of a rigid body passes
in all the positions of that body through the same point in it is a property of
many and most important uses in the mechanism of the universe, as well as in
the practice of the arts; another proof of it is therefore subjoined, which
may be more satisfactory to some readers than that given in the text. The
system being rigid, the distance P1, P2, of the points of
application of any two of the pressures remains the
Mfs;;'l.Z    same, into whatever position the body may be turned:
the only difference produced in the circumstance under
which they are applied is an alteration in the inclina-',.   i ~, \ tions of these pressures to the line P1, P2: now being
weights, the directions of these pressures always remain
parallel to one another, whatever may be their inclination; thus then it follows by the principle of the equa



22                  TIEE CEINTRE OF GRAVITY.
22. To determine the positio;.f the centre off gravity of
two weights, P, and P2, for ling part of ac rigid  system.
Let it be represented by G.  Then since the resultant of
P, and P2 passes through  G, we have by equaC   --- 2 tion (16), taking P, as the point from  which the
moments are measured,
P1 + P2  P1G- P2. P1P2,
*PG P.  P1P2
P, + P2
whence the position of G is known.
23. ft is reqgirec7 to determinze the centre of gravity of three
weights P,, P, P, not in the same straight line, and forming part of a rigid systenm.
Find the centre of gravity G1, of P1 and P2, as in the last
proposition. Suppose the weights P1 and P2 to
6 be collected in G1, and find as before the com0, mon centre of gravity G2 of this weight P1 + P2,
~1   3     0 ~  so collected in GO, and the third weight P1.  It
i is evident that this point G2 is the centre of
gravity  required.   Since  G2 is the  centre  of
gravity of P, and P1 + P2 collected in G1, we have by the
last proposition
GG2. P1+P2+P,3-G P, P,
G 1P3. P 
2 — P1+P2+P3.
lity of moments (Art. 15), that P1+P2.PR1=P2. P2P2, so that for every
such inclination of the pressures to P1 P2, the line P1R1 is of the same length,
and the point R1 therefore the same point; therefore, the line P3R1 is always
the same line in the body; and R, which equals P1+P2, is always the same
pressure, as also is P3, and these pressures always remain parallel, therefore,
for the same reason as before, R2 is always the same point in the body in
whatever position it may be turned, and so of R2, R......  and R;,:-. That
is, in every position of the body, the resultant of the weights of its parts
passes through the same point R,,-, in it. Since the resultant of the weights
of the parts of a body always passes through its centre of gravity, it is
evident, that a single force applied at that point equal and opposite to this
resultant, that is, equal in amount to the whole weight of the body, and in a
direction vertically upwards, would in every position of the body sustain it.
This property of the centre of gravity, viz. that it is a point in the body where
a single force would support it is sometimes taken as the definition of it.




OF A TRIANGLE.                23
If P1) P2, P3, be all equal, then
G1G2=~  G1P3.
Moreover in this case,
PG1=l P1P2.
24. To find the centre of gravity of four weights not in the
same plane.
Let P1, P2, PS, P4, represent these weights; find the
centre of gravity G, of the weights P1, P2,
P,, as in the last proposition; suppose these
three weights to be collected in G2, and then
/':find the centre of gravity G, of the weight
/'I  thus collected in G2 and P,. G3 will be the
-— e: —h'-rs  centre of gravity required, and since G, is
the centre of gravity of P4 acting at the
point P4, and of P, +P2 +P3 collected at G2,. G2G3. P,~-P2+PS+P4=G2P4. P,,
G2P4 P4:.  $ -Pl+P2+PS+P4  -
If all these weights be equal, then by the above equation,
G2G,=- G2P4,
also,    GG2 =  G1P,
and      GP —=I PP2.
25. THE CENTRE OF GRAVITY OF A TRIANGLE.
Let the sides AB and BC of the triangular lamina ABC
be bisected in E and D, and the lines CE and
AD drawn to the opposite angles, then is the
intersection G of these lines the centre of gravity
of the triangle: for the triangle may be supposed
to be made up of exceedingly narrow rectangular
~ strips or bands, parallel to BC, each of which will
be bisected by the line AD; for by similar triangles
PR' DB:: AR: AD:: RQ: DC, therefore, alternando,
PR: RQ:: DB: DC; but DB-DC; therefore PR-RQ.
Therefore, each of the elementary bands, or rectangles
parallel to BC, which compose the triangle ABC, would
separately balance on the line AD; therefore, all of them




24               TIHE CENTRE OF GRAVITY
joined together would balance on the line AD, therefore the
centre of gravity of the triangle is in AD.
In the same manner it may be shown that the centre of
gravity of the triangle is in the line CE; therefore, the centre of gravity is at the intersection G of these lines.
Now DG=~  DA: for imagine the triangle to be without
weight, and three equal weights to be placed at the angles
A, B, and C, then it is evident that these three weights will
balance upon AD; for AD being supported, the weight A
will be supported, since it is in that line; moreover, B and
C will be supported since they are equidistant from that
line.
Since, then, all three of the weights will balance upon
AD, their centre of gravity is in AD. In like manner it
may be shown that the centre of gravity of all three weights
is in CE; therefore it is in G, and coincides with the centre
of gravity of the triangle.
Now, suppose theweights B and C to be collected in their
centre of gravity D, and suppose each weight to be represented in amount by A, a weight equal to 2A will then be
collected in D, and a weight equal to A at A, and the centre
of gravity of these is in G; therefore DA x A=DG x
(2A + A),. DA = 3 DG, or i)G = ~    DA.*     [Q.E.D.]
26. THE CENTRE OF GRAVITY OF THIE PYRAMID.
Let ABC be a pyramid, and suppose it to be
made up of elementary laminae bed, parallel to
the base BCD. Take G, the centre of gravity'//i   of the base BCD, and join AG; then A will
c pass through the centre of gravity g of the
laimina bcd,t therefore each of the laminae will separately
balance on the straight line AGC; therefore the laminae when
combined will balance upon this line; therefore the whole
figure will balance on AG, and the centre of gravity of
the whole is in AG. In like manner if the centre of gravity
H of the face ABD be taken, and CH be joined, then it may
be shown that the centre of gravity of the whole is in CH;
* Note (g) Ed. App.'[ For produce the plane ABG to intersect the plane ADO in AM, then by
similar triangles DM: MC:: dmi: me, but DM 5 MC; therefore dm = me. Also
by similar triangles GM: BM:: gmi: bm, but GM —  BM; therefore gm=j
bin. Since then dmin   de and gm - i bm, therefore g is the centre of gravity
of the triangle bdc.




OF A PYRAMID.                 25
therefore the lines AG and CII intersect, and the centre of
gravity is at their intersection K.
Now GK is one-fourth of GA; for suppose equal weights
to be placed at the angles A, B, C, and D of the pyramid
(the pyramid itself being imagined without weight), then
will these four weights balance upon the line AG, fobr one
of them, A, is in that line, and the line passes through the
centre of gravity G of the other three.
Since then, the equal weights A, B, C, and D balance
upon the line AG, their centre of gravity is in AG; in the
same manner it may be shown that the centre of gravity of
the four weights is in CIH, therefore it is in K, and coincides
with the centre of gravity of the pyramid.
Now let the number of units in each weight be represented by A, and let the three weights B, C, and D be
supposed to be collected in their centre of gravity G; the
four weights will then be reduced to two, viz. 3A at G, and
A at A, whose common centre of gravity is K,.. GKx3A+A= GA x A,.. 4-GK   GA or GK0-  - GA.*     [Q.E.D.]
27. The centre of gravity of a pyramid with a polygonal base
is situated at a vertical heigAt frnom the base, equal to one
fourth the whole height of the pyramid.
For any such pyramid ABCDEF may be supposed to
be made up of triangular pyramids ABCF,
ACDF, and ADEF, whose centres of gravity
G, I G, and K, are situated in lines AL, AM,
and AN, drawn to the centres of gravity L, M,
~-., Sia\  and N of their bases; LG being one-fourth of
LA, MH one-fourth of MA, and NK one-fourth
of NA. The points G0, IH, and K, are therefore in a plane
parallel to the base of the pyramid, and whose vertical distance from the base equals one-fourth the vertical height of
the pyramid.
Since then the centres of gravity G0, H, and K of the eleinentary triangular pyramids which compose the whole polygonal pyramid are in this plane, therefore the centre of gravity
of the whole is in this plane, i. e. the centre of gravity of the
whole polygonal pyramid is situated at a vertical height from
the base, equal to one fourth the vertical height of the whole
* Note (h) Ed. App.




,26             THE CENTRE OF GRAVITY
pyramid, or at a vertical depth from the vertex, equal to three
fourths of the whole. Now the above proportion is true,
whatever be the number of the sides of the polygonal base,
and therefore if they be infinite in number; and therefore it
is true of the cone, which may be considered a pyramid having a polygonal base, of an infinite number of sides; and it
is true whether the cone orpyranmid be an oblique or a right
cone or pyramid.
28. If a body be of a prismatic form, and symmetrical
about a certain plane, then its whole weight may be supposed to be collected in the surface of that plane, and uniformly distributed through it. For let
ACBEFD represent such a prismatic
body, and abc a plane about which it is
t —- -;-,  symmetrical: take m, an element of uniX..  -------— " form thickness whose sides are parallel to
" the sides of the prism, and which is
terminated by the faces ACB and DFE of the prism;
it is evident that this element m will be bisected by the
plane abc, and that its centre of gravity will therefore
lie in that plane, so that its whole weight may be supposed collected in that plane; and this being true of
every other similar element, and all these elements being equal, it follows that the whole weight of the body
may be supposed to be collected in and uniformly distributed through that plane. It is in this sense only that we
can speak with accuracy of the weight and the centre of gravity of a plane, whereas a plane being a surface only, and
having no thickness, can have no weight, and therefore no
centre of gravity. In like manner when we speak of the
centre of gravity of a curved surface, we mean the centre of
gravity of a body, the weights of all whose parts may be supposed to be collected and uniformly distributed throughout
that curved surface. It is evident that this condition is
approached to whenever the body being hollow, its material
is exceedingly thin. Its whole weight may then be conceived
to be collected in a surface equidistant from its two external
surfaces. In like manner an exceedingly thin uniform curved
rod may be imagined to have its weight collected uniformly
in a line passing along the centre of its thickness, and in this
sense we may speak of the centre of gravity of a line,
although a line having no breadth or thickness can have no
weight, and therefore no centre of gravity.




OF ANY QUADRILATERAL FIGURE.         27
29. THE CENTRE OF GRIAVITY OF A TRAPEZOID.
Let AD and BC be the parallel sides of the trapezoid, of
C 0         which AD is the less. Let AD be represented
\      J   by a, BC by b, and the perpendicular distance
o;.'~.'?~NL of th'e two sides by h. Draw DE parallel
_\si      to AB.  Let G, be the intersection of
u-~?~?+-    the diagonals of the parallelogram ABED,
then will G~ be the centre of gravity of that parallelogram. Bisect CE in L, join DL, and take DG2,-  DL,
then will G2 be the centre of gravity of the triangle DEC.
Draw G M, and G2M, perpendiculars to AD; then since
AG1=_-  AE, therefore GMl= —M1 FE-= A.  And since
DG2 —  DL, therefore G2M2 -= NL -  h. Suppose the
whole parallelogram to be collected in its centre of gravity
G,, and the whole triangle in its centre of gravity G. Let
G be the centre of gravity of the whole trapezoid, and draw
GM perpendicular to AD. Then would the whole be supported by a single force equal to the weight of the trapezoid
acting upwards at G.. Therefore (Art. 17),
MG. ABCD-=GIM,. ABED+G2M2. CED
Now,       ABCD = ~ h (a+ b), ABED = ha,
CED =- h (b- a), G1M-   A -, GM =- h,.MG..h (a+b) =  1h. ha +- h. i A (b-a),... MG (a+b)=ha+  h (b-a) =I h(a+2b),..MG- -         F aA+b.Y Q.  (19).
30. THE CENTRE OF GRAVITY OF ANY QUADRILATERAL FIGURE.
Draw the diagonals AC and BD of any quadrilateral figure
ABCD, and let them intersect in E,
and from the greater of the two parts,
BE and DE, of either diagonal BD set
off a part BF equal to the less part.
Bisect the other diagonal AC in H, join
As —-I    ""IDIHF and take HG equal to one third of
HF; then will G be the centre of gravity of the whole
figure.
For if not, let y be the centre of gravity, join HB and HD
and take HG1, =  HB and HG2=   HD, then will G, and
G be the centres of gravity of the triangles ABC and ADC




28             THE CENTRE OF GRAVITY.
respectively (Art. 25). Suppose these triangles to be collected in their centres of gravity GI, G2; it is evident that
the centre of gravity g, of the whole figure, will be in the
straight line joining the points G, G,: let this line intersect
AC in 1K; then since a pressure equal to the weight of the
whole figure acting upwards at y, will be in equilibrium with
the weights of the triangles collected in G, and G,, we have,
by the principle of the equality of moments (Art. 15),
Kg7. ABCD-KG,. ABlC-KG2. ADO.
Now since HG1 _  i HB, and HG- =g  liD, therefore G, G2
is parallel to DB, therefore KG1 =-  BE, and KG2 —  DE.
Now let the angle AED   BEC =    T. Therefore the perpendicular from B upon AC   BE sin. i, and that fiom D = DE
sin.,, therefore area of triangle ABC = AC. BE sin. i,
and area of triangle ADC _=  AC. DE sin. i therefore area
of quadrilateral ABCD =- AC. 3E sin. +~  AC. A  DE
sin. i -- (BE+DE) AC sin. ~. Substituting these values in
the preceding equation,
Kg. - (BE+DE) AC sin. — =  BE. - AC. BE sin. -
_  DE. ~ AC. DE sin.,
Kg. (BE+DE)=  (BE2-DE2),
BE 2DE2.-. K =  1- (BE- DE) i (BEBF)  --  FE.
BE+DE
But since HG = i EF,..KG =     E,.'. - Kg= KG; that
is, the true centre of gravity g coincides with the point G.
Therefore, &c.                               [Q. E. D.]
*31. In the examples hitherto given, the centre of pressure
of a system of weights, or their centre of gravity, has been
determined by methods which are indirect as compared with
the direct and general method indicated in Article 17. That
method supposes, however, a determination of the sum of the
moments of the weights of all the various elements of the
body in respect to three given planes. Now in a continzuous
body these elements are inftni;te in number, each being infinitely small; this determination supposes, therefore, the summation of an infinite number of infinitely small quantities,
and requires an application of the principles of the intregal
calculus.
Let aM be taken to represent any small element of the




THE CENTRE OF GRAVITY.                29
volume M  of a continuous body, and x its perpendicular
distance from a given plane.  Then will x:J zAM  represent
the moment of the weight of this element about that plane,
t representing the weight of each unit of the volume M.
Let Vpx AM represent the sum of all such moments, taken in
respect to all the small elements, such as AM, which make
up the volume of the body.  Then if G, represent the distance of the centre of gravity of the body from the given
plane; since FRxAM represents the sum of the moments of a
system of parallel pressures about that plane, FiM the sum of
those pressures, and G1 the distance of their centre of pressure from the plane (Art. 19), it follows by equation (18) that
_     AM. AM    Xm. AM
1 M               M(20).
Now it is proved in the theory of the integral calculus,*
that a sum, such as is represented -by the above expression
EXAM, whose terms are infinite in number, and each the product of a finite quantity x, and an infinitely small quantity
AM, and in which M is, as in this case, a function of x (and
therefore x a function of M), is equal to the definite integral
f xdiM. Therefore, generally,
02
Z13
Similarly            G                          (21).
Gs 
In the two last of which equations y and z are taken to represent, respectively, the distances of the element AM  of the
* Poisson, Journal de l'Ecole Polytechnique, 18me cahier, p. 320, or Art. 2,
in the Treatise on Definite Integrals in the Encyclopaedia Metropolitana by the
author of this work. See Appendix, note A.




30              THE CENTRE OF GRAVITY.
body from two other planes, as x represents its distance from
the first plane; and G1 and G2 to represent the distances of
its centre of gravity from those planes. The distances G,,
G2, G., of the centre of gravity from three different planes
being thus known, its actual position in space is fully determined. These three planes are usually taken at right angles
to one another, and are.then called rectangular co-ordinate
planes, and their common intersections rectangular co-ordinate axes.
If the centre of gravity of the body be known to lie in a
certain plane, and one of the co-ordinate planes spoken of
above, as for instance that from which G3 is measured, be
taken to coincide with this plane in which the centre of gravity is known to lie, then G,  O0, and the position of the centre of gravity is determined by the two first only of the above
three equations. This case occurs when the body, whose
centre of gravity is to be determined, is symmetrical about a
certain plane, since then its centre of gravity evidently lies
in its plane of symmetry. If the centre of gravity of the
body be known to lie in a certain line, and two of the co-ordinate planes, those for instance from which G2 and G3 are
measured, be taken so as to intersect one another in that line,
then the centre of gravity will be in both those planes; therefore G, -0  and G = 0, and its position is determined by the
first of the preceding equations alone. This case occurs
when the body is symmetrical about a given line; its centre
of gravity is then manifestly in that line.
*32. THE CENTRE OF GRAVITY OF A CURVED LINE WHICH LIES
WHOLLY IN THE SAME PLANE.
Taking M  to represent the length S of such a line, we
have, by equations (21),
L       )      2   s...- (22),
EXAMPLE.-Let it be required to determine the centre of
gravity of a circular are EF.
The centre of gravity of such an arc is evidently in the
radius CA, which bisects it; since the arc
is symmetrical about that radius. Take a
i..... ~::-  plane Cy perpendicular to this radius, and
C.-. —. —--—.  passing through the centre, to measure the
moments from.  Let x represent the distance PM of any point P in this are from




OF A CURVED LINE.              31
this plane; also let s represent the are PA, and S the arn
EAF, a the radius CA, and C the chord EF..x  - PM  CP cos. CPM = CP cos. ACP = a cos. a
2S            2S
f.xdS=afcos. a s =a2fos. --         - 2a sidn.2  -
— is          — s
the integral being taken between the limits IS and — S,
because these are the values of s which correspond to the
extreme points F and E of the arc.
Now 2 sin. ia (-)   chord of EAF =,..fdds = aC,
aC. G    S....  (23).
The distance of the centre of gravity of a circular arc from
the centre of the circle is therefore a fourth proportional to
the length of the arc, the length of the chord, and the radius
of the arc.
X33. THE CENTRE OF GRAVITY OF A CURVILINEAR AREA
WHICH LIES WHOLLY IN THE SAME PLANE.
Let BAC represent such an area.' If x and y represent
the perpendicular distances PN and PM of any
point P in the curve AB from planes AC and
~-"~ C.    AD, perpendicular to the plane of the given area
and to one another, and M represent the area
a.X }   __ PAM, then, considering this area to be made up
of rectangles parallel to PM, the width of each
of which is represented by the exceedingly small quantity
Ax, the area AM of each such rectangle will be represented
by yex, and its moment about AD by ptxyAx.
1dx!XxyAX xyd
Therefore by equation (20), G,    M   -M    (24).
A similar expression determines the value of G2; but one
more convenient for calculation is obtained, if we consider
the weight of each of the rectangles, whose length is y, to
be collected in its centre of gravity, whose distance from AC




.32             TTE CENTRE OF GRAVITY.
is y.  Thle moment of the weight of each rectangle about
AC will then be represented,uy2ax; whence it follows that
2  M
G2-         - _'- 2:.... (25).
ExAMPLE.-Suppose the curve APB to be a parabola, whose
axis is AC.
D    By the equation to the parabola y    4ax, if a
be the distance of the focus from the vertex.
".,   YaMoreover, the limits between which the integral
is to be taken are 0 and x, and 0 and y,, since at
A, xA 0,   y- 0, and at C, x = x,, y    y,,
ml           VC1    4                   1
thereforexydx= 2 /    3af dx=2  / ax, 2; also,M Afydx
C2        0                         0
ml     4                     3
2 4       I d     -ax, therefore  =  y.
3      2
ACB, then A=AC, IIG-4   B.
5            8*            88
34. TUE CENTRE OF GRAVITY OF A SURFACE OF REVOLUTION.
Any surface of revolution BAC is evidently symmetrical
~  about its axis of revolution AD, its centre of
gravity is therefore in that axis. Let the mof   ments be measured from a plane passing through
A and perpendicular to the axis AD, and let x
I and y be co-ordinates of any point P in the
generating curve APB of the surface, and s the
length of the curve AP. Then M being taken to represent
the area of the surface, and being supposed to be made up
of bands parallel to PQ, the area AM of each such band is
represented (see Art. 40.) * by 2iryas, and its moment by
2rxy            Church's Diff. Calculus,
* Church's Dill. Calculus, Art. 91.




OF A suIRFACE.                 33
Si
27r, xyds: G- 2xyA          8 s.    (26).
M           M(
EXAMPLE.- To determine the centre of gravity of the surface of any zone or segment of a sphere.
T    Let B AC0 represent the surface of a sphere,
whose centre is D, and whose radius DP is represented by a, and the arc AP by s. Then x = DM
= DP cos. PDM        a cos., y= PM =DP sin.
PDM=a sin.,.* 2xy = 2a sin. - cos.  = a2 sin..
a                a     a        a
S,. S,
2.rf xyds =  a fsin.2s - s
a
s,         S,
=   ra3 f cos. 2S  cos.  2   S
a               a)
c COS.2 - Cos2  -   cos.2 S,...  (27).
ca        a
where Si and S2 are the values of s at the points
^  Bi and B,, where the zone is supposed to terminate.
cl                             ^S
Also, since   - =    2ry,. M = 2  ryds
ds
S,
S1
= 27ra sin. dS =- 2ra2   cos. A2 - COs.j1
4/     a          [     a       a:   a {cos. -   + coS. - 
3




34              THE CENTRE OF GRAVITY
DE2+ DE,      =DE.....(28),
if E be the bisection of EE,.
If S2 = 0, or the zone commence from A, then
Gi    a     +cos.      a cos.....(29).
2     (aa               2a
*35. THE CENTRE OF GRAVITY OF A SOLID OF REVOLUTION.
Any solid of revolution BAC is evidently symmetrical
about its axis of revolution AD, its centre of
~-/r   gravity is therefore in that line; and taking a
/.. plane passing through A and perpendicular to
that axis as the plane from which the moments
are measured, we have only to determine the
distance AG of the centre of gravity, from
that plane.
Now, if x and y represent the co-ordinates of any point P
in the generating curve, and M the volume of the portion
PAQ of this solid, then, conceiving it to be made up of
cylindrical laminae parallel to PQ, the thickness of each of
which is ax, the volume of each is represented by ~yAx, and
its moment by axy2^x.
~fxy'dx
*MG,=       =   2 M...... (30).
- M.
EXAMPLE.-To determine the centre of gravity of any solid
segment of a sphere.
Let B1AC1 represent any such segment of a
i3.   sphere whose centre is D and its radius a. Let x
A    -D and y represent the co-ordinates AM and MP of
1  any point P, x being measured from A; then by
the equation to the circle y2=2ax- 2,
C4
x,        x1.r;fxy2dx==J'x (2 ax-a) dx=^ (ax-      4).
X,        X,
x2        0
Also, M=qrfy 2dX =, j  (2ax_2) dx=qr (aX1-jxi8).
X0        0




OF TIIE SEGMfIENT OF AN ARCHI.       35.. G,-  i  --  i  *         *...*  (31).'4 1  \3a-x /..
If the segment become a hemisphere, -a=,.~ G,= -;a.
Let CAB represent such a sector; conceive the arc ADB
to be a polygon of an infinite number of sides: a.' and lines, to be drawn from all the angles of the
c.^ —-l polygon to the centre C of the circle, these will
divide the. sector into as many triangles. Now
the centre of gravity of each triangle will be at
a distance from C equal to   - the line drawn from the vertex
C of that triangle to the bisection of its base, that is equal
to 2 the radius of the circle, so that the centres of gravity of
all the triangles will lie in a circular arc FE, whose centre is
C and its radius CF equal to -CA, and the weights of the
triangles may be supposed to be collected in this arc FE,
and to be uniformly distributed through it, so that the centre of gravity G of the whole sector CAB is the centre of
gravity of the circular arc FE. Therefore by equation (23),
if S1, C1, and a', represent the arc FE, its chord FE, and its
radius CF, and S, C, a, the similar arc, chord, and radius of
ADB, then CG =; but since the arcs AB and FE are
Si
similar, and that a' = -a,.. C' =  C and S = -S. Substituting these values in the last equation, we have
CG=    a.          (32).
S
37. The centre of gravity of any portion of a circular ring
or of an arch of equal voussoirs.
Let B,CiCB2 represent any such portion of a circular ring
c, whose centre is A.  Let a, represent the
C    \ radius, and Cl the chord of the arc B~,C, and
A< -~ —-  S, its length, and let a, C similarly represent' — J the radius and chord of the arc B2C,, and S,' the length of that arc.
Also let G1 represent the centre of gravity of the sector
ABC,, G, that of the sector ABC,, and G the centre of
gravity of the ring. Then
AG, x sect. ABC +AG x ring BCB2C,-AG x sect. AB1,A.
Now (by equation 32), AG,  -a ai AG,=-| 2 _;
b,  7     -  3'




36                  THE PROPERTIES
also sector AB,C,= Slal, sector ABC,= -Sa,,:. ring B1C,CB,= sect. ABC,- sect. ABC,= iSal — iSa,,
S2'     a C....S, Sa,+AG 2 S1A-S)a,)=..S,... (Sa,- Sa2) =- (CSa,'- Ca2),. AG- =-  CAala2...  (33).
Sa,- Sa''
38. THE PROPERTIES OF GILDINUS.
If NL represent any plane area, and AB be any awis, in the
same plane, about which the area is made to
iA.     revolve, so tha; NL is by this revolution made to
generate a solid of revolution, then is the volume,....)   of this solid equal to that of a prism whose base.M' " I4s NL, and whose height is equal to the length
3  -r.  of the path which the centre of gra/vity G of the
area NL is made to describe.
For take any rectangular area PRSQ in NL, whose sides
are respectively parallel and perpendicular to AB, and let
MT be the mean distance of the points P and Q; or R and
S, from AB. Now it is evident that in the revolution of
NL about AB, PQ will describe a superficial ring.
Suppose this to be represented by QFPK, let M be the
centre of the ring, and let the arc subtended by
77I3  the angle QMF at distance unity from AM be represented by 6, then the area FQPK equals the sector'   FQM-the sector KPM=MQ2 x 6-i MP' x 6
i6 (QMQ-MP2)=(MQ +MP)x(MQ-MP)=(MT x PQ).
Now the solid ring generated by PRSQ is evidently equal
to the superficial ring generated by PQ, multiplied by the
distance PR. This solid ring equals therefore 6 (MT x PQ
x PR) or x MT xPRSQ. Now suppose the area PRSQ
to be exceedingly small, and the whole area NL to be made
up of such exceedingly small areas, and let them be represented by a,, a,,,, &c. and their mean distances MT by x,,
x,2 x, &c. then the solid annuli generated by these areas
respectively will (as we have shown), be represented by
sxa,, Oxea, 6xa,, &c. &c.; and the sum of these annuli,




OF GULDINUS.                  37
or the whole solid, will be represented by lx,a, + Ox-t, 
Oxa +&c., or by (la,+x,+a,+xa,+&c.). Now if v. represent the weight of any superficial element of the plane NL,
xraj=the moment of the weight of a, about the axis AB,
taji=that of the area a, about the same axis AB, and so
on, therefore the sum (x,a, + xa, + xa- + &c.) - =the moment
of the whole area NL about AB; but if G be the centre of
gravity of NL, and GI its distance from AB, enthen the
moment of NL about AB-GIxNLP;.l    ~      therefore the whole solid=O. GI. NL;
but 6. GI equals the length of the circuT >  ( j lar path described by G; therefore the
n    1\g*   volume of the solid equals NL multiplied by the length of the path deL, scribed by G, i. e. it equals a prism NM,
_1___-   _ *  whose base is NL, and whose height GH:
is the length of the path described by
7- -G; which is the first property of GULDINUS.
39. The above proposition is applicable to finding the
solid contents of the thread of a screw of variable diameter, or of the material in a spiral staircase: for it is
evident that the thread of a screw may be supposed to be
made up of an infinite number of small solids of revolution,
arranged one above another like the steps of a staircase, all
of which (contained in one turn of the thread) might be
made to slide along the axis, so that their surfaces should all
lie in the same plane; in which case they would manifestly
form one solid of revolution, such as that whose volume has
been investigated. The principle is moreover applicable to
determine the volume of any solid (however irregular may
be its form otherwise), provided only that it may be conceived to be generated by the motion of a given plane area,
perpendicular to a given curved line, which always passes
through the same point in the plane. For it
<^^ ^  ~is evident that whatever point in this curved
line the plane may at any instant be traversing, it may at that instant be conceived to be revolving
about a certain fixed axis, passing through the centre of
curvature of the curve at that point; and thus revolving
about a fixed axis, it is generating for an instant a solid of
revolution about that axis, the volume of which elementary
solid of revolution is equal to the area of the plane multi



38                   TITE PROPERTIES
plied by the length of the path described by its centre of
gravity; and this being true of all such elementary solids,
each being equal to the product of the plane by the corresponding elementary path of the centre of gravity, it follows
that the whole volume of the solid is equal to the product
of the area by the whole length of the path.
40. If AB represent any curved line made to revolve about
the axis AD so as to generate the surface of revolution BAC, and G be the...ce..ntre of gravity of this curved line,.-..:: t    —...i  then is the area of this surface equal
to the product of the length of the
curved line AB, by the length of the
path described by the point G, during
the revolution of the curve about AD.  This is the second
property of Guldinus.
Let PQ be any small element of the generating curve,
and PQFK a zone of the surface generated by this element,
this zone may be considered as a portion of the surface of a
cone whose apex is M, where the tangents to the curve at T
and V, which are the middle points of PQ and FK, meet
when produced. Let this band PQFK of the cone QIMF be
developed*, and let PQFK represent its develop-'~oj~' ment; this figure PQFK will evidently be a circular ring, whose centre is M; since the developM   ment of the whole cone is evidently a circular
sector MQF whose centre M     corresponds to the
apex of the cone, and its radius MQ to the side MQ of the
cone.
Now, as was shown in the last proposition, the area of
this circular ring when thus developed, and therefore of the
conical band before it was developed, is represented by
6. MT. PQ, where 6 represents the arc subtended by QMF
at distance unity. Now the arc whose radius is MT is
represented by 0. MT; but this arc, before it was developed
from the cone, formed a complete circle whose radius was
NT, and therefore its circumference 2irNT; since then the
circle has not altered its length by its development, we
have
* If the cone be supposed covered with a flexible sheet, and a band such
as PQFK be imagined to be cut upon it, and then unwrapped from the cone
and laid upon a plane, it is called the development of the band.




OF GULDINUS.                  39
O MT=2~rNT.
Substituting this value of OIMT in the expression for the area
of the band we have
area of zone PQFK=2t. NT. PQ.
Let the surface be conceived to be divided into an infinite
number of such elementary bands, and let the lengths of
the corresponding elements of the curve AB be represented
by 81s s), s3, &c. and the corresponding values of NT by y,
yO, yS, &c. Then will the areas of the corresponding zones
be represented by 2irys, 2ry2s2, 2ry33,s, &c. and the area of
the whole surface BAC by 2;ryls, + 2tY-22 + 2ty383 +.... or
by 2t(y1s, +Y282 +y33+s....). But since G is the centre of
gravity of the curved line AB, therefore AB. GITI represents the moment of the weight of a uniform thread or wire
of the form of that line about AD,, being the weight of
each unit in the length of the line: moreover, this moment
equals the sum of the moments of the weights sy s, s2  s,,
&c. of the elements of the line... AB. GHlu = (ys  + ys2+ ys 3+..).-.AB. GIH=-y  +y2sY2+YS38+..
Therefore  area  of surface BAC-=2rAB        GH=AB
(2~GH).
But 2IGH equals the length of the circular path described
by G in its revolution about AD. Therefore, &c.
This proposition, like the last, is true not only in respect to
a surface'of revolution, but of any surface generated by a
plane curve, which traverses perpendicularly another curve
of any form whatever, and is always intersected by it in the
same point. It is evident, indeed, that the same demonstration applies to both propositions. It must, however, be observed, that neither proposition applies unless the motion of
the generating plane or curve be such, that no two of its consecutive positions intersect or cross one another.
41. The volume of any truncated prismatic or cylindrical
body ABCD, of which one —extremity CD is perpendicular
to the sides of the prism, and the other AB inclined to
them, is equal to that of an upright prism ABEF, having
for its base the plane AB, and for its height the perpendicular height GN of the centre of gravity G of the plane
DC, above the plane of AB.
For let I represent the inclination of the plane DC to AB;




40           THE PROPERTIES OF GULDINUS.
take m, any small element of the plane
~=Z —-  CD, and let mr be a prism whose base is m
/.  and whose sides are parallel to AD and
[~      ] 3'I BC; of elementary prisms similar to which
the whole solid ABCD may be supposed
to be made up. Now the volume of this prisrm, whose base
is in and its height mr, equals mr x m = sec. J x (mr. cos. i)
x in = sec. I x (mr. sin. mrn) nm = sec. i x mn x m.
Therefore the whole solid equals the sum of all such products as mn x m, each such product being multiplied by the
constant quantity sec. J, or it is equal to the sum just spoken
of, that sum being divided by cos. I. Let this sum be represented by Imn x in, therefore the volume of the solid is remnnm x i
presented by Zmnxm. Now suppose CD to represent a
thin lamina of uniform thickness, the weight of each square
unit of which is a, then will the weight of the element ma be
represented by a x i, and its moment about the plane ABN
by i x mn x m, and Zmin x m will represent the sum of the
moments of all the elements of the lamina similar to m about
that plane. Now by Art. 15. this sum equals the moment of
the whole weight of the lamina Pi x CD supposed to be collected in G, about that plane. Therefore
P x CD x NG=Pmn x nm,
CD xNG = mn x m.
Substituting this value of:mn x n, we have
volume of solid = sec.' x CD x NG.
But the plane CD is the projection of AB, therefore CD
= AB cos. I,.. CD xsec. i = AB;. vol. of solid ABCD = AB x NG = vol. of prism ABEF.
Therefore, &c.
[Q. E. D.]




MOTION.                     41
PAIR T II.
DYNAMICS.
42. MOTION is change of place.
The science of DYNAMICs is that which treats of the laws
which govern the motions of material bodies, and of their
relation to the forces whence those motions result.
The SPACES described by a moving body are the distances
between the positions which it occupies at different successive periods of time.
UNIFORM MOTION is that in which equal spaces are described in equal successive intervals of time.
The VELOCITY of uniform motion is the space which a
body moving uniformly describes in each second of time.
Thus if a body move uniformly with a velocity represented
by V, and during a time represented in seconds by T, then
the space S described by it in those T seconds is represented
S        S;
by TV, or S=TV. Whence it follows that V = -and T=V
so that if a body move uniformly, the space described by it
is equal to the velocity multiplied by the time in seconds,
the velocity is equal to the space divided by the time, and
the time is equal to the space divided by the velocity.
43. It is a law of motion, established from constant observation upon the motions of the planets, and by experiment
upon the motions of the bodies around us, that when once
communicated to a body, it remains in that body, unaffected
by the lapse of time, carrying it forward for ever with the
same velocity and in the same direction in which it first began to move, unless some force act afterwards in a contrary
direction to destroy it.*
* This is the first LAW OF MOTION. For numerous illustrations of this fundamental law of motion, the reader is referred to the author's work, entitled,
ILLUSTRATIONS OF MECHANICS, Art. 193.




42                   VELOCITY.
The velocity, at any instant, of a body moving with a
VARIABLE MOTION, is the space which it would describe in.
one second of time if its motion were from that instant to
become rNIFORM.
An ACCELERATING FORCE is that which acting continually
upon a body in the direction of its motion, produces in it a
continually increasing velocity of motion.
A RETARDING FORCE is that which acting upon a body in
a direction opposite to that of its motion produces in it a
continually diminishing velocity.
An IMPULSIVE FORCE is that which having communicated
motion to a body, ceases to act upon it after an exceedingly
small time from the commencement of the motion.
44. A UNIFORMLY accelerating or retarding force is that
which produces equal increments or decrements of velocity
in equal successive intervals of time.  If f represent the
additional velocity communicated to a body by a uniformly
accelerating force in each successive second of time, and T
the number of seconds during which it moves, then since by
the first law of motion it retains all these increments of velocity (if its motion be unopposed), it follows that after T
seconds, an additional velocity represented by fT, will have
been communicated to it; and if at the commencement of
this T seconds its velocity in the same direction was V, then
this initial velocity having been retained (by the first law of
motion), its whole velocity will have become V+fT.
If, on the contrary, f represents the velocity continually
taken away from a body in each successive second of time,
by a uniformly retarding force, and V the velocity with
which it began to move in a direction opposite to that in
which this retarding force acts, then will its remaining velocity after T seconds be represented by V-f T; so that generally the velocity V of a body acted upon by a uniformly
accelerating or retarding force is represented, after T seconds,
by the formula
= V+fT....        (34).
The force of gravity is, in respect to the descent of bodies
near the earth's surface, a constantly accelerating force,
increasing the velocity of their descent by 32, feet in each
successive second, and if they be projected upwards it is a
constantly retarding force, diminishing their velocity by that
quantity in each second. The symbol g is commonly used to




VELOCITY.                   43
represent this number 32-; so that in respect to gravity the
above formula becomes v=VY~T, the sign ~ being taken
according as the body is projected downwards or upwards.
A VARIABLE acceleratintg force is that which communicates
unequal increments of velocity in equal successive intervals
of time; and a variable retarding force that which takes
away unequal decrements of velocity.*
45. To DETERMINE THE RELATION BETWEEN THE VELOCITY AND
THE SPACE, AND THE SPACE AND TIME OF A BODY'S MOTION.
Let AM1, M1MI, MAM1, &c. represent the exceeding small
successive periods of a body's motion, and
AP the velocity with which it began to
C  move, MlP, the velocity at the expiration
\ \ |       ^of the first interval of time, MP2 that at
Ir —— B    the expiration of the second, MP1 of the
third interval of time, and so on; and
instead of the body varying the velocity of its motion continually throughout the period. AMN, suppose it to move
through that interval with a velocity which is a mean
between the velocity AP at A, and that MJP1 at M1, or with
a velocity equal to I (AP + MA1P).
Since on this supposition it moves with a uniform motion,
the space it describes during the period AMI equals the
product of that velocity by that period of time, or it equals
2-(AP+MPP1)AIM,. Now this product represents the area
of the trapezoid AM1PP. The space described then in the
interval AM,, on the supposition that the body moves during
that interval with a velocity which is the mean between
those actually acquired at the commencement and termination of the interval, is represented by the trapezoidal
area AMPP.
Similarly the areas PM,, PM,, &c. represent the spaces
the body is made to describe in the successive intervals
M1M,, M2MS, &c.; and therefore the whole polygonal area
APCB represents the whole space the body is made to
describe in the whole time AB, on the supposition that it
moves in each successively exceeding small interval of time
with the mean velocity of that interval. Now the less the
intervals are, the more nearly does this mean velocity of each
interval approach the actual velocity of that interval; and
if they be infinitely small, and therefore infinitely great in
* Note (i) Ed. App.




W44               MOTION UNIFORMLY
number, t thethe mean velocity coincides with the actual
velocity of each interval, and in this case the polygonal area
passes into the curvilinear area APCB.
Generally, therefore, if we represent by the abscissac of a
curve the times through which a, body has moved, and by
the corresponding ordinates of that curve the velocities which
it has acquired after those times, then the area of that curve
will represent the space through which the body has moved;
or in other words, if a curve PC be taken such that the number of equal parts in any one of its abscissae AMI being taken
to represent the number of seconds during which a body has
moved, the number of those equal parts in the corresponding
ordinate M3P3 will represent the number of feet in the velocity then acquired; then the space which the body has
described will be represented by the number of these equal
parts squared which are contained in the area of that curve.
46. To DETERMINE THE SPACE DESCRIBED IN A GIVEN TIME BY
A BODY WHICH IS PROJECTED WITH A GIVEN VELOCITY, AND
WHOSE MOTION IS UNIFORMLY ACCELERATED) OR UNIFORMLY
RETARDED.
Take any straight line AB to represent the whole time T,. in seconds, of the body's motion, and draw AD
Hi~   ~.:perpendicular to it, representing on the same
Kx ~I!   scale its velocity at the commencement of its
A''M  motion. Draw DE parallel to AB, and according as the motion is accelerated or retarded
draw DC or DF inclined to DE, at an angle whose tangent
equalsf, the constant increment or decrement of the body's
velocity. Then if any abscissa AM be taken to represent a
number of seconds t during which the body has moved, the
corresponding ordinate MP or MQ will represent the velocity
then acquired by it, according as its motion is accelerated or
retarded. For PR = RQ = DR tan. PDE=AM tan. PDE;
but AM=t, and tan. PDE=f: therefore PR     RQ==ft.
Also RM=AD=V, therefore MP= RM +PR =V +t, and
MQ=RM-RQ=V-ft; therefore by equation (34), MP or
MQ represents the velocity after the time AM according as
the motion is accelerated or retarded. The same being true
of every other time, it follows, by the last proposition, that
the whole space described in the time T or AB is represented
by the area ABCD if the motion be accelerated, and by the
area ABFD if it be retarded.




ACCELERATED OR RETARDED.              45
Now area ABCD=JAB(AD+BC), but AB=T, AD=V,
BC=V+/T,. area ABCD=2T(V+V+fT)=VT+I T.
Also area ABFD=iAB (AD+BF), where AB and AD
have the same values as before, and BF=V-fT,
area ABFD=T(V+V-f/T)=VT —f/T.
Therefore generally, if S represent the space described after
T seconds,
S=VT ~    fT2..... (35);
in which formula the sign ~- is to be taken according as the
motion is accelerated or retarded.
47. To DETERMINE A RELATION BETWEEN THE SPACE DESCRIBED
AND THE VELOCITY ACQUIRED BY A BODY WHICH IS PROJECTED
WITH A GIVEN VELOCITY, AND WHOSE MOTION IS UNIFORMLY
ACCELERATED OR RETARDED.
Let v be the velocity acquired after T seconds, then by
equation (34), v =V ~tfT,.. T=     v-  ).
p    c    Now area ABCD - =AB (AD + BO), where
D      p-..   AB=T=(-V   ) AD-=V,BC=v,. areaABOCD=     --  (             -V+)(,' —
area ABFD = JAB (AD +BF), where AB= T -(,
AD =V, BF=v.. areaABFD=        (-V) (v+V)     -( —y).
f AF2f
Therefore generally, if S represent the space through which
the velocity v is acquired, then S +  ( vV,.*.V'-v2=~2..... (36);
in which formula the ~ sign is to be taken according as
the motion is accelerated or retarded.
If the body's motion be retarded, its velocity v will eventually be destroyed. Let S, be the space which will have been




46               THE UNIT OF WORK.
described when v thus vanishes, then by the last equation
o —=_ - 2/SI..'. V2=2fS1....    (3S)
where V is the velocity with which the body is projected
in a direction opposite to the force, and S, the whole space
which by this velocity of projection it can be made to
describe.
If the body's motion be accelerated and it fall from rest,
or have no velocity of projection, then 2 — 0 -  2fS,
*. =2f.......       (8).
Let S2 be the space through which it must in this case
move to acquire a velocity V equal to that with which it
rwas projected in the last case, therefore'2'S 2fS. Whence
it follows that S,-8,, or that the whole space S, throulgh
which a body will move when projected with a given veloci ty V, and uniformly retcrded by any force, is equLal to the
space 8,, through which it must move to acquire that velocity when uniformly accelerated by the same force.
In the case of bodies moving freely, and acted upon by
gravity, f equals 32- feet and is represented by g; and the
space S,, through which any given velocity V is acquired, is
then said to be that due to that velocity.
WORK.
48. WoRK is the union of a continued pressure with a
continued motion. And a mechanical agent is thus said to
woRx when a pressure is continually overcome, and a point
(to which that pressure is applied) contin:ally moved by it.
Neither pressure nor motion alone is sufficient to constitute
work; so that a man who merely supports a load upon his
shoulders, without moving it, no more workcs, in the sense in
which that term is here used, than does a column which sustains a heavy weight upon its summit; and a stone, as it falls
freely in vaczto, no more works than do the planets as they
wheel unresisted through space.*
49. THE UNIT OF WORK.-The unit of work used in this
country, in terms of which to estimate every other amount
*Note (j) Ed. App.




VARIABLE WORK.                 47
of work, is the work necessary to overcome a pressure of one
pound through a distance of one foot, in a direction opposite
to that in which a pressure acts. Thus, for instance, if a
pound weight be raised through a vertical height of one foot,
one unit of work is done; for a pressure of one pound is
overcome through a distance of one foot, in a direction opposite to that in which the pressure acts.
50. The6 number of units of work necessary to overcome a
presscure of M pournds through a distance of N feet, is
equal to the product MN.
For since, to overcome a pressure of one pound through
one foot requires one unit of work, it is evident that to overcome a pressure of M pounds through the same distance of
one foot, will require M units. Since, then, M units of work
are required to overcome this pressure through one foot, it
it evident that N times as many units (i. e. NM) are required
to overcome it through N feet. Thus, if we take U to represent the number of units of work done in overcoming a constant pressure of M pounds through N feet, we have
U=MN........ (39).
51. To ESTIMATE THE WORK DONE UNDER A VARIABLE
PRESSURE.
Let PC be a curved line and AB its axis, such that any
one of its abscissae AM,, containing as many
- -.c equal parts as there are units in the space
r<i'',     through which any portion of the work has..-..M M, been done, the corresponding ordinate MP,
may contain as many of those equal parts,
as there are in the pressure under which it is then being
done. Divide AB into exceedingly small equal parts, AM,
MMa, &c.,. and draw the ordinates MP, M,P,, &c.;:ten if
we conceive the work done through the space AM, (which
is in reality done under pressures varying from AP to MIP,),
to be done uniformly under a pressure, which is the arithmetic mean between AP and MP,, it is evident that the
number of units in the work done through that small space
will equal the number of square units in the trapezoid
APPM, (see Art. 45.), and similarly with the other trapeNote (k) Ed. App.




48                TTHE RESOLUTION
zoids; so that the number of units in the.whole work done
through the space AB will equal the number of square units
in the whole polygonal area APP1P2P1, &c., CB.
But since AM1, MI1M, &c., are exceedingly small, this
polygonal area passes into the curvilinear area APCB; the
whole work done is therefore represented by the number of
square equal parts in this area.
Now, generally, the area of any curve is represented by
the integralJfydx, where y represents the ordinate, and x
the corresponding abscissa. But in this case the variable
pressure P is represented by the ordinate, and the space S
described under this variable pressure by the abscissa. If
therefore U represent the work done between the values S,
and S2 of S, we have.
S2
fPdS......(40).
S1
iMfean pressure is that under which the same work would
be done over the same space, provided that pressure, instead
of varying throughout that space, remained
-B...._____...- D the same: thus, the mean pressure in respect to an amount of work represented by
the curvilinear area AEFC, is that under
Al           c which an amount of work would be done
represented by the rectilineal area ABDC, the area ABDC
being equal to the curvilinear area AEFC; the mean pressure in this case is represented by AB. Thus, to determine
the mean pressure in any case of variable pressure, we have
only to find a curvilinear area representing the work done
under that variable pressure, and then to describe a rectangular parallelogram on the same base AC, which shall have
an area equal to the curvilinear area.
If S represent the space described under a variable pressure, U the work done, and p the mean pressure, then
pS = U, therefore p.*
52. To estimate the work of a pressure, whose direction is not
that in which itspoint of application is made to move.
Hitherto the work of a force has been estimated only on
* Note (I) Ed. App.




OF WORK.                    49
the supposition that the point of applicat'   tion of that force is moved in the direction
-,7' \  in which the force operates, or in the oppo/\  8.^  site direction. Let PQ be the direction of
a pressure, whose point of application Q
is made to move in the direction of the
straight line AB. Suppose the pressure P to remain constant, and its direction to continue parallel to itself. It is
required to estimate the work done, whilst the point of
application has been moved from A to Q.
Resolve P into R and S, of which R is parallel and S perpendicular to AB. Then since no motion takes place in the
direction of SQ, the pressure S does no work, and the whole
work is done by R; therefore the work= R. AQ.
Now R=P. cos. PQR, therefore the work =P. AQ cos.
PQR. From the point A draw AM perpendicular to PQ,
then AQ cos. PQR=-QM; therefore work=P. QM. Therefore the work of any pressure as above, not acting in the
direction of the motion of the point of application of that
pressure, is the same as it would have been if the point of
application had been made to move in the direction of the
pressure, provided that the space through which it was so
moved had been the projection of the space through which
it actually moves. The product P. QM may be called the
work of P resolved in the direction of P.
The above proposition which has been proved, whatever
may be the distance through which the point of application
is moved, in that particular case only in which the pressure
remains the same in amount and always parallel to itself, is
evidently true for exceedingly small spaces of motion, even
if the pressure be variable both in amount and direction;
since for such exceedingly small variations in the positions
of the points of application, the variations of the pressures
themselves, both in amount and direction, arising from these
variations of position, must be exceedingly small, and therefore the resulting variations in the work exceedingly small
as compared with the whole work.*
* Note (m) Ed. App.
4




50                   THE WORK OF
53. If any number of pressures P,, P,, P,, be applied to the
same point A, and remain constant and parallel to themselves, whilst the point A is made to move through the
straight line AB, then the whole work done is equal to the
sum of the works of the different pressures resolved in the
directions of those pressures, each being taken negatively
whose point of application is made to move in an opposite
direction to the pressure upon it.
Let al, a2 "O, &c. represent the inclinations of the pressures P,, P, &c. to the line AB, then will
Ya -       the resolved parts of these pressures in the
r; -'direction of that line be P1 cos. a, P, cos.',.\      %2 P3 cos. %3 &c. and they will be equivav/'i/  >J   lent to a single pressure in the direction
3%-^  ~of that line represented by P, cos. c, + P2
cos. a,+P, cos. %, &c. in which sum all
those terms are to be taken negatively which involve pressures whose direction is from B towards A (since the single
pressure from A towards B is manifestly equal to the difference between the sum of those resolved pressures which act
in that direction, and those in the opposite direction). Therefore the whole work is equal to {P, cos. ac + P, cos. c, + P,
cos og+.....      AB-=P. AB cos. c +P,. AB cos. a;
+PSAB cos. a,+..     P,. BM,+P2. BM+ P3. BMS,....; in which expression the successive terms are the
works of the different pressures resolved in the several
directions of those pressures, each being taken positively or
negatively, according as the direction of the corresponding
pressure is towards the direction of the motion or opposite
to it.
Thus if U represent the whole work and U, and U, the
sums of those done in opposite directions, then
U=U,-u.... (41).
54. If any number of pressures applied to a point be in equilibrium, and their point of application be moved, the whole
work done by these pressures in the direction of the motion
will equal the whole work done in the opposite direction.
For if the pressures P,, P,, P,, &c, (Art. 53) be in equilibrium, then the sums of their resolved pressures in opposite
* Note (n) Ed. App.




CENTRAL FORCES.                    51
directions along AB will be equal (Art. 10); therefore the
whole work U along AB, which by the last proposition is
equal to the work of a pressure represented by the difference
of these sums, will equal nothing, therefore 0 —U — U,,
therefore U1=U2, that is, the whole work done in one direction along AB, by the pressures P    2, P, &c. is equal to the
whole work done in the opposite direction.
55. If a body be acted upon by a force whose direction is
always towards a certain point S, called a centre of force,
and be made to describe any given curve PA in a direction
opposed to the action bf that force, and Sp be measured on
SA equal to SP, then will the work done in moving the
body through the curve PA be equal to that which would
be necessary to move it in a straight line from p to A.
For suppose the curve PA to be a portion of a polygon of
A  an infinite number of sides, PP,, PiP21 &c.
/   Through the points P, P1, P2, &c. describe circup2ai,   lar arcs with the radii SP, SP, SP,, &c. and let:%;      them intersect SA in p, p,, p2, &c. Then since
PP1 is exceedingly small, the force may be considered to act throughout this space always in a
direction parallel to SP,; therefore the work done
through PP, is equal to the work which must be
done to move the body through the distance mP1 (Art. 52.),
since mP1 is the projection of PP, upon the direction SP, of
the force. But mP,=pp,; therefore the work done through
PP, is equal to that which would be required to move the
body along the line SA through the distance pp; and similarly the work done through P1P2 is equal to that which
must be done to move the body through plp,, so that
the work through PP, is equal to that through ppa, and so
of all other points in the curve. Therefore the work through
PA is equal to that through pA.*   Therefore, &c.  [Q.E.D.]
* Of course it is in this proposition supposed that the force, if it be not
constant, is dependant for its amount only on the distance of the point at
which it acts from the centre of force S; so that the distances of p and P
from S being the same, the force at p is equal to that at P; similarly the
force at p, is equal to that at Pi, the force at p2 equal to that at P,, &c.




52                   THE WORK OF
56. If S 7he at an exceedingly great distance as compared
with AP, then all the lines drawn from S to AP may be considered parallel. This is the case with the force of gravity
at the surface of the earth, which tends towards a point, the
earth's centre, situated at an exceedingly great distance, as
compared with any of the distances through which the work
of mechanical agents is usually estimated.
Thus then it follows that the work necessary to move a
heavy body up any curve PA, or inclined plane, is the same
as would be necessary to raise it in a vertical line pA to the
same height.
The dimensions of the body are here supposed to be exceeding small.  If it be of considerable dimensions, then
whatever be the height through which its centre of gravity
is raised along the curve, the work expended is the same
(Art 60.) as though the centre of gravity were raised vertically to that height."
57. In the preceding propositions the work has been estimated on the supposition that the body is made to move so
as to increase its distance from the centre S, or in a direction
opposed to that of the force impelling it towards S. It is
evident, nevertheless that the work would have been precisely
the same, if instead of the body moving from P to A it had
moved from A to P, provided only that in this last case
there were applied to it at every point such a force as would
prevent its motion from being accelerated by the force continually impelling it towards S; for it is evident that to prevent this acceleration, there must continually be applied to
the body a force in a directionfrom S equal to that by which
it is attracted towards it; and the work of such a force is
manifestly the same, provided the path be the same, whether
the body move in one direction or the other along that path,
being in the two cases the work of the same force over the
same space, but in opposite directions.
* The only force acting upon the body is in this proposition supposed to be
that acting towards S. No account is taken of friction or any other forces
which oppose themselves to its motion.




PARALLEL FORCES.                 53
58. If there be any number of parallel pressures, P,, P2, P,,
&c. whose points of application are transferred, each
through any given distance from one position to another,
then is the work which would be necessary to transfer their
resultant through a space equal to that by which their
centre of pressure is displaced in this change of position,
equal to the difference between the aggregate work of those
pressures whose points of application have been moved in
the directions in which thepressures applied to them act,
and those whose points of application have been moved in
the opposite directions to their pressures.
For (Art. 17.), if y,, y,, y,, &c. represent the distances of
the points of application of these pressures from any given
plane in their first position, and h the distance of their centre
of pressure from that plane, and if Y,, Y2, Y, &c. and H represent the corresponding distances in the second position,
and if P, P,, P &, &c. be taken positively or negatively according as their directions are from, or towards the given
plane, h {P,+P2+P+....  =Py+Py,+P3y....
and  I{P,+P2+P,+.... -       PY,+PY,+P3Y,+..... (H-h) {P1~+P.S+P,+.   } =P (Y,-y,)+P2 (Y,-Y,)
+P (Ys-y)+.           (42);
in the second member of which equation the several terms
are evidently positive or negative, according as the pressure
P corresponding to each, and the difference Y-y of its distances from the plane in its two positions, have the same or
contrary signs. Now by supposition P is positive or negative
according as it acts from or towards the plane; also Y-y is
evidently positive or negative according as the point of application of P is moved from or towards the plane; each term
is therefore positive or negative, according as the corresponding point of application is transferred in a direction towards
that in which its applied pressure acts, or in the opposite
direction.
Now the plane from which the distances of the points of
application are measured may be any plane whatever.  Let
it be a plane perpendicular to the directions of the pressures.




64                   THE WORK OF
Let Axy represent this plane, and let P
z         VP' represent the two positions of the point
of application of the pressure P (the path. i,   described by it between these two positions
|Z  I   having been any whatever). Let MP and
siA I!t,4    M'P' represent the perpendicular disy... / ^ /   tances of the points P and P' from  the
plane, and draw Pm from P perpendicular
to M'P'. Then P (Y —y)=P(M'P'-MP)=P. mP'; but by
Art. 55., P. mP' equals the work of P as its point of application is transferred from P to P'. Thus each term of the second
member of equation (42) represents the work of the,corresponding pressure, so that if u1,, represent the aggregate
work of those pressures whose points of application are transferred towards the directions in which the pressures act, and
lu2 the work of those whose points of application are moved
opposite to the directions in which they severally act, then
the second member of the equation is represented by zu —
u2. Moreover the first member of the equation is evidently
the work necessary to transfer the resultant pressure Pi +
P2-P,P  &c. through the distance H-h, which is that by
which the centre of pressure is removedfrom or towards the
given plane, so that if U represent the quantity of work
necessary to make this transfer of the centre of pressure,
TU=Zmu-:s...... (43).
59. If the sum of those parallel pressures whose tendency
is in one direction equal the sum of those whose tendency
is in the opposite direction, then P + P+P3+.... 
In this case, therefore, U=O, therefore, — Yzu,=0, therefore 2u= lu2,; so that when in any system of parallel pressures the sum of those whose tendency is in one direction
equals the sum of those whose tendency is in the opposite direction, then the aggregate work of those whose points of application are moved in the directions of the pressure severally
applied to them is equal to the aggregate work of those whose
points of application are moved in the opposite directions.
This case manifestly obtains when the parallel pressures
are in EQUILIBRIUM, the sum of those whose tendency is in
one direction then equalling the sum of those whose tendency
is in the opposite direction, since otherwise, when applied to
a point, these pressures could not be in equilibrium about
that point (Art. 8.).




PARALLEL FORCES.                           55
60. The preceding proposition is manifestly true in respect
to a system of weights, these being pressures whose directions
are always parallel, wherever their points of application may
be moved.      Now    the centre    of pressure of a system         of
weights is its centre of gravity (Art. 19).         Thus then it follows, that if the weights composing such a system be separately moved in any directions whatever, and through any
distances whatever, then the difference between the aggregate work done upwards in making this change of relative
position   and that done downwards is equal to             the work
necessary to raise the sum     of all the weights through a height
equal to that through which their centre of gravity is raised
or depressed.*      Moreover that if such a system        of weights
be supported in equilibrium        by the resistance of any fixed
point or points, and be put in motion, then (since the work
of the resistance of each such point is nothing) the aggregate
* This proposition has numerous applications. If, for instance, it be required
to determine the aggregate expenditure of work in raising the different elements of a structure, its stone, cement, &c., to the different positions they
occupy in it, we make this calculation by determining the work requisite to
raise the whole weight of material at once to the height of the centre of gravity of the structure. If these materials have been carried up by labourers, and
we are desirous to include the whole of their labour in the calculation, we
ascertain the probable amount of each load, and conceive the weight of a labourer to be added to each load, and then all these at once to be raised to the
height of the centre of gravity.
Again, if it be required to determine the expenditure of work made in raising the material excavated from a well, or in pumping the water out of it, we
know that (neglecting the effect of friction, and the weight and rigidity of the
cord) this expenditure of work is the same as though the whole material had
been raised at one lift from the centre of gravity of the shaft to the surface.
Let us take another application of this principle which offers so many practical
results. The material of a railway excavation of considerable length is to be
removed so as to form an embankment across a valley at some distance, and it
is required to determine the expenditure of work made in this transfer of the
material. Here each load of material is made to traverse a different distance,
a resistance from the friction, &c., of the road being continually opposed to its
motion. These resistances on the different loads constitute a system of parallel pressures, each of whose points of application is separately transferred from
one given point to another given point, the directions of transfer being also
parallel. Now by the preceding proposition, the expenditure of work in all
these separate transfers is the same as it would have been had a pressure equal
to the sum of all these pressures been at once transferred from the centre of
resistance of the excavation to the centre of resistance of the embankment.
Now the resistances of the parts of the mass moved are the frictions of its elements upon the road, and these frictions are proportional to the weights of the
elements; their centre of resistance coincides therefore with the centre of gravity of the mass, and it follows that the expenditure of work is the same as
though all the material had been moved at once from the centre of gravity of
the excavation to that of the embankment. To allow for the weight of the
carriages, as many times the weight of a carriage must be added to the weight
of the material as there are journeys made.




56             STABILITY OF TIE CENTRE.
work of those weights which are made to descend, is equal
to that of those which are made to ascend.
61. If a plane be taken peendicular to the directions of any
number of parallel pressures and there be two different positions of the points of application of certain of these pressures in which they are at different distances from the
plane, whilst the points of application of the rest of these
pressures remain at the same distance from that plane,
and if in both positions the system be 6in equilibrium, then
the centre of pressure of the first mentioned pressures will
be at the same distance from the plane in bothpositions.
For since in both positions the system is in equilibrium,
therefore in both positions P1+P2 +P+... =0,
~ (-yJ)P + (Y, -yy( 2 )P2+ (Y3 -y3)P,+ +  + P(Y,-y)=0.
Now let P. be any one of the pressures whose points of application is at the same distance from the given plane in both
positions,.   YE=y, and Y —y- = 0,
(Y1- y)P, + (Y2 -y2)P2 +.. + (Yn_,-y )P,, =0.Y1,P+Y2,P +... +Y,_lPn-_ =y1P1++y2P,+     +y. _lPI,
Y1P,+Y2P2+. -.+P..     1     P+y2P2+ *,+...+ yP,_,? -I P2+ ~ ~ ~ +Pz-_     PI +PV+    ~ ~ ~ +Pn-I.. Hn-l =An-lI
where HIT_ represents the distance of the centre of pressure
of P,, P2... PF-, from the given plane in the first position,
and hAn- its distance in the second position. Its distance in
the first position is therefore the same as in the second.
Therefore, &c.
From this proposition, it follows that if a system of weights
be supported by the resistances of one or more fixed points,
and if there be any two positions whatever of the weights in
both of which they are in equilibrium with the resistances
of those points, then the height of the common centre of
gravity of the weights is the same in both positions. And
that if there be a series of positions in all of which the
weights are in equilibrium about such a resisting point or
points, then the centre of gravity remains continually at the
same height as the system passes through this series of positions.
If all these positions of equilibrium be infinitely near to




WORK OF PRESSURES.               57
one another, then it is only during an infinitely small motion
of the points of application that the centre of gravity ceases
to ascend or descend; and, conversely, if for an infinitely
small motion of the points of application the centre of
gravity ceases to ascend or descend, then in two or more
positions of the points of application of the system, infinitely near to one another, it is in equilibrium.
WORK OF PRESSURES APPLIED IN DIFFERENT DIRECTIONS TO
A BODY MOVEABLE ABOUT A FIXED AXIS.
62. The work of a pressure applied to a body moveable about
a ixed axis is the same at whatever point in its proper
direction that pressure may be applied.
For let AB represent the direction of a pressure applied
to a body moveable about a fixed axis
D    "'' —' O 0; the work done by this pressure
~_r_~;y/-j      - 7 will be the same whether it be ap\ //5///          plied at A or B. For conceive the
I/f'             body to revolve about 0, through an
~~o    ~      exceedingly small angle AOC, or
BOD, so that the points A and D may describe circular arcs
AC and BD. Draw Cm, Dn, and OE, perpendiculars to
AB, then if P represent the pressure applied to AB, P. Am
will represent the work done by P when applied at A (Art.
52.), and P. Bn will represent the work done by P when
applied at B; therefore the work done by P at A is the same
as that done by P at B, if Am is equal to Bn.
Now AC and BD being exceedingly small, they may be
conceived to be straight lines.  Since BD and BE are
respectively perpendicular to OB and OE, therefore L DBE
= /BOE; and because AC and AE are perpendicular to
OA and OE, therefore Z CAE = / AOE.       Now Am=
CA
CA. cos. CAE -=CA. cos. AOE =         OA. cos. AOE
OA
CA
= A x OE. Similarly Bn = DB cos. DBE= DB. cos.
BID                  BD
BOE = OB       OB cos. BOE-B     x OE, i.e. Am= OE.
GOB                  GB    GE
* It is a well-known principle of Geometry, that if two lines be inclined at
any angle, and any two others be drawn perpendicular to these, then the inclination of the last two to one another shall equal that of the first two.




58            THE ACCUMULATION OF WORK.
CA               BD       CA BD
A' and Bn = OE OB. But OA-B' since the ZAOC=
Z BOD, therefore Am= Bn.*
63. If any number of pressures oe zn equilibrium about a
fixed axis, then the whole work of those which tend to move
the system in one direction about that axis is equal to the
whole work of those which tend to move it in the opposite
direction about the same axis. For let P be any one of such
a system of pressures, and 0 a fixed axis, and OM perpendicular to the direction of P, then whatever may be the
point of application of P, the work of that pressure is the
same as though it were applied at M. Suppose the whole
system to be moved through an exceeding small' /   angle 6 about the point 0, and let OM be represented by p, then will p6 represent the space.....  described by the point M, which will be actually.o in the direction of the force P, therefore the work
of P=P. p. 8. Now let P1, P,, P3, &c. represent those
pressures which act in the direction of the motion, and P',,
P', &c. those which act in the opposite direction, and let
p1, p, p2, &c. be the perpendiculars on the first, andp',, p',,
p' &c. be the perpendiculars on the second; therefore by
the principle of the equality of moments POp, +P^p + Pp
+ &c. = P'P' + P'2', + P'sp', + &c.; therefore multiplying
both sides by 0, PPp, + Pp,o=              + P''  P', 2  +
P'p'3, +&c.; but PpA, P'1p6', &c. are the works of the
forces P,, P',, &c.; therefore the aggregate work of those
which tend to move the system in one direction is equal to
the aggregate of those which tend to move it in the opposite
direction.
64. THE ACCUMULATION OF WORK IN A MOVING BODY.
In every moving body there is accumulated, by the action
of the forces whence its motion has resulted, a certain
amount of power which it reproduces upon any resistance
opposed to its motion, and which is measured by the work
done by it upon that obstacle. Not to multiply terms, we
shall speak of this accumulated power of working, thus
measured by the work it is capable of producing, as ACCUMULATED WORK. It is in this sense that in a ball fired from
* Note (o) Ed. App.




THE ACCUMULATION OF WORK.           59
a cannon there is understood to be accumulated the work it
reproduces upon the obstacles which it encounters in its
flight; that in the water which flows through the channel
of a mill is accumulated the work which it yields up to the
wheel; * and that in the carriage which is allowed rapidly
to descend a hill is accumulated the work which carries it a
considerable distance up the next hill. It is when the pressure under which any work is done, exceeds the resistance
opposed to it, that the work is thus accumulated in a moving
body; and it will subsequently be shown (Art. 69.) that in
every case the work accumulated is precisely equal to the
work done upon the body beyond that necessary to overcome the resistances opposed to its motion, a principle
which might almost indeed be assumed as in itself evident.
65. The amount of work thus accumulated in a body
moving with a given velocity, is evidently the same, whatever may have been the circumstances under which its
velocity has been acquired. Whether the velocity of a ball
has been communicated by projection from a steam gun, or
explosion from a cannon, or by being allowed to fall freely
from a sufficient height, it matters not to the result; provided the same velocity be communicated to it in all three
cases, and it be of the same weight, the work accumulated
in it, estimated by the effect it is capable of producing, is
evidently the same.
In like manner, the whole amount of work which it is
capable of yielding to overcome any resistance is the same,
whatever may be the nature of that resistance.
66. To ESTIMATE THE NUMBER OF UNITS OF WORK ACCUMTULATED IN A BODY MOVING WITH A GIVEN VELOCITY.
Let w be the weight of the body in pounds, and v its
velocity in feet.
Now suppose the body to be projected with the velocity v
in a direction opposite to gravity, it will ascend to the height
A from which it must have fallen, to acquire that same velocity v (Art. 47.); there must then at the instant of projection
have been accumulated in it an amount of work sufficient to
raise it to this height h; but the number of units of work
* This remark applies more particularly to the under-shot wheel, which is
carried round by the rush of the water.




60            TIIE ACCUMULATION OF WORK.
requisite to raise a weight w to a height h, is represented by
wh; this then is the number of units of work accumulated
in the body at the instant of projection. But since h is the
height through which the body must fall to acquire the velocity v, therefore v2=2gh (Art. 47.); therefore h=- g; whence
it follows that if U represent the number of units of work
accumulated,
UT=-..... (44).
Moreover it appears by the last article that this expression
represents the work accumulated in a body weighing w
pounds, and moving with a velocity of  feet, whatever may
have been the circumstances under which that velocity was
accumulated.
The product (-) 2 is called the vis VIVA of the body, so
that the accumulated work is represented by half the vis
viva, the quotient (-) is called the MAss of the body.*
67. To estimate the work accumulated in a body, or lost by
it, as it passesfrom one velocity to another.
In a body whose weight is w, and which moves with a
velocity v there is accumulated a number of units of work
represented (Art. 66.) by the formula g-v2  After it has
passed from this velocity to another V, there will be accumulated in it a number of units of work, represented by — Y2,
so that if its last velocity be greater than the first, there
will have been added to the work accumulated in it a number of units represented by i-VY2 —-2; or if the second
velocity be less than the first, there will have been taken
from the work accumulated in it a number of units represented by - — 2_  V'. So that generally if U represent
the work accumulated or lost by the body, in passing from
the velocity v to the velocity V, then
* Note (p) Ed. App.




TlE ACCUMULATION OF WORK.           61
Lu=~. {i2_^......(45),
where the ~ sign is to be taken according as the motion is
accelerated or retarded.
68. lhe work accumulated in a body, whose motion is accelerated through any given space by given forces is equal to
the work which it would be necessary to do upon the body
to cause it to move back again through the same space
when acted upon by the same forces.
For it is evident that if with the velocity which a body
has acquired through any space AB by the
action of any forces whose direction is from A
towards B, it be projected back again from B
towards A, then as it returns through each
successive small part or element of its path, it
will be retarded by precisely the same forces as those by
which it was accelerated when it before passed through it;
so that it will, in returning through each such element, lose
the same portion of its velocity as before it gained there;
and when at length it has traversed the whole distance BA,
and reached the point A, it will have lost between B and A
a velocity, and therefore an amount of work (Art. 67.),
precisely equal to that which before it gained between A
and B. Now the work lost between B and A is the work
necessary to overcome the resistances opposed to the motion
through BA. The work accumulated from A to B is therefore equal to the work which would be necessary to overcome the resistances between B and A, or which would be
necessary to move the body from a state of rest, and with a
uniform motion, in opposition to these resistances, through
BA. Let this work be represented by U; also let v be the
velocity with which the body started from A, and V that
W
which it has acquired at B. Then will -1  (V2 -v) represent the work accumulated between A and B,
*   (V _ v2)=U *,   Y -v=2gU
If the body, instead of being accelerated, had been
retarded, then the work lost being that expended in overcoming the retarding forces, is evidently that necessary to




62            THE ACCUMUTLATION OF WORK.
move the body uniformly in opposition to these retarding
forces through-AB; so thatfif this force be represented by
W
U, then, since W- (v'-VY) is in this case the work lost, we
shall have v2-V2 2gU    Therefore, generally,
-- W-. Therefore, generally,
2gU
W2-v2=.. (46),
where the sign ~ is to be taken according as the motion is
accelerated or retarded.
69. The work accumulated in a body which has moved
through any spe acted upon by any force, is equal to the
excess of the work which has been done uyoon it by those
forces which tend to accelerate its motion above that which
has been done upon it by those which tend to retard its
motion.
For let R be the single force which would at any point P
(see last fig.) be necessary to move the body back again
through an exceeding small element of the same path (the
other forces impressed upon it remaining as before); then it
follows by Art. 54. that the work of R over this element of
the path is equal to the excess of the work over that
element of the forces which are impressed upon the body in
the direction of its motion above the work of those
impressed in the opposite direction. Now this is true at
every point of the path; therefore the whole work of the
force R necessary to move the body back again from B to A
is equal to the excess of the work done upon it, by the
impressed forces in the direction of its motion, above the
work done upon it by them in a direction opposed to its
motion; whence also it follows, by the last proposition, that
the accumulated work is equal to this excess.   Therefore, &c.
*70. If P represent the force in the direction of the
motion which at a given distance S, measured along the
path, acts to accelerate the motion of the body, this force
being understood not to be counteracted by any other, or to
be the surplus force in the direction of the motion over and




THE ACCUMULATION OF WORK.           63
above any resistance opposed to it, then will  PdS be the
0
work which must be done in an opposite direction to overs
come this force through the space S, or UT —fPdS,
0
2gfPdS. by equation (46), V2-v2= —~...   (47).
71. If the force P tends at first towards the direction in
which the body moves, so as to accelerate the motion, and
if after a certain space has been described it changes its
direction so as to retard the motion, and U1 represent the
value of U in respect to the former motion, and V, the
velocity acquired when that motion has terminated, whilst
U2 is the value of U in respect to the second or retarded
motion, and if v be the initial and V the ultimate or actual
velocity, then
V 2 2 22gU,
v - -V2              (48).
V _a    W...        ( 
As UT increases, the actual velocity V of the body continually diminishes; and when at length UT-,=U  that is
when the whole work done (above the resistances) in a
direction opposite to the motion, comes to equal that done,
before, in the direction of the motion, then V —v, or the
velocity of the body returns again to that which it had
when the force P began to act upon it. This is that general case of reciprocating motion which is so frequently presented in the combinations of machinery, and of which the
crank motion is a remarkable example.
*72. If the force which accelerates the body's motion act
always towards the same centre S, and Sb be taken equal to




64           THE ACCULTULATION OF WORK.
SB, it has been shown (Art 55.) that the work
necessary to move the body along the curve from
B to A, is equal to that which would be necessary.*""7  to move it through the straight line bA. The
accumulated work is therefore equal to that necessary to move the body through the difference bA
of the two distances SA and SB (Art. 68.). If these
8  distances be represented by R1 and R2, and P
represent the pressure with which the body's motion along
hA would be resisted at any distance R from the point S,
RI
thenjPdR will represent this work. Moreover the work
R2
accumulated in the body between A and B is represented
by i_ (VY2 —v), if V represent the velocity at B and v that
at A,                            ~
i (Vs 7        PdR)
a V _  2     R
~-   V-FdR.....(49).
R2
73. The work accumulated in the body while it descends
the curve AB, is the same as that which it would acquire in
falling directly towards S through the distance AB,, for both
of these are equal to the work which would be necessary to
raise the body from b to A. Since then the work accumulated by the body through AB is equal to that which it would
accumulate if it fell through Ab, it follows that velocity
acquired by it in falling, from rest, through AB is equal to
that which it would acquire in falling through Ab. For if
V represent the velocity acquired in the one case, and V,
that in the other, then the accumulated work in the first case
W                             W
is represented by,i -V2, and that in the second case by i- V2',
W       W
therefore i  V2 = -- -Y, therefore VY=1.
g       a
From this it follows, that if a body descend, being projected obliquely into free space, or sliding from rest upon
any curved surface or inclined plane, and be acted upon only
by the force of gravity (that is, subject to no friction or
resistance of the air or other retarding cause), then the velo



THE ACCUMIULATION OF WOKr.            65
city acquired by it in its descent is precisely the same as
though it had fallen vertically through the same height.
74. DEFINITION. The ANGULAR VELOCITY of a body which
rotates about a fixed axis is the arc which every particle of
the body situated at a distance unity from the axis describes
in a second of time, if the body revolves uniformly; or, if
the body moves with a variable motion, it is the arc which it
would describe in a second of time if (from the instant when
its angular velocity is measured) its revolution were to
become uniform.
75. THE ACCU]MULATION OF WORK IN A BODY WHICH
ROTATES ABOUT A FIXED AXIS.
Propositions 68 and 69 apply to every case of the motion
of a heavy body. In every such case the work accumulated
or lost by the action of any moving force or pressure, whilst
the body passes from any one position to another, is equal
to the work which must be done in an opposite direction, to
cause it to pass back from the second position into the first.
Let us suppose U to represent this work in respect to a body
of any given dimensions, which has rotated about a fixed.
axis from one given position into another, by the action of
given forces.
Let ca be taken to represent the ANGULAR VELOCITY OF the.
body after it has passed from one of these positions into.
another. Then since a is the actual velocity of a particle at
distance unity from the axis, therefore the velocity of a particle at any other distance p, from the axis is cap. Let [J.
represent the weight of each unit of the volume of the body,
and me the volume of any particle whose distance from the
axis is p1, then will the weight of that particle be %m,; also
its velocity has been shown to be ap1, therefore the amount
of work accumulated in that particle is represented by
I 2   2      1 2, 2o
if,-mepAl, or by ^  ~^p2 *qp.
Similarly the different amounts of work accumulated in
tile other particles or elements of the body whose distances
from the axis are represented by p,, p,,... and their
volumes by m, mn,,.... are represented by i~2  M2 P2,
^3~pa p2 &c.; so that the whole work accumulated is repreg                     5




66               ANGULAR VELOCITY.
rented by the sum      p -m12 2 m+  2  p M2+i2P  +......., or by 1-    p12+ Wp, +  p32 +......
The sum m,1p2 +   +m2  p2 +...., or?mp2 taken in
respect to all the particles or elements which compose the
body, is called its MOMENT OF INERTIA in respect to the
particular axis about which the rotation takes place. Let it
be represented by I; then will a. -. I, represent the
whole amount of work accumulated in the body whilst it has
been made to acquire the angular velocity a from rest. If
therefore U represent the work which must be done in an
opposite direction to cause the body to pass back from its
last position into its first,
Ia2( )I=U,
a=22(g)I           (50).
If instead of the body's first position being one of rest, it
had in its first position been moving with an angular velocity
a. which had passed, in its second position, into a velocity
a; and if U represent, as before, the work which must be
done in an opposite direction, to bring this body back from
its second into its first position, then is a2 () -  - i2 (~) 
or i (- ) (a2 -  ) I, the work accumulated between the first
and second positions; therefore
i  (;al2)I= ~t    )
a=2-2(....     (51),
where the sign ~ is to be taken according as the motion is
accelerated or retarded between the first and second positions, since in the one case the angular velocity increases
during the motion, so that a2 is greater than cc12, whilst in the
latter case it diminishes, so that a2 is less than ol2.
76. If during one part of the motion, the work of the




ANGULAR VELOCITY.               67
impressed forces tends to accelerate, and during another to
retard it, and the work in the former case be represented by
TU, and in the latter by U2, then
2    2+ 2=) I,.....2  ( 
*,2+ 2  (U)l- _U.....  (52).
From this equation it follows that when U —=U, or when
the work U2 done by the forces which tend to resist the
motion at length, equals that done by the forces which tend
to accelerate the motion, then oa=al or the revolving body
then returns again to the angular velocity from which it set
out. Whilst, if U, never becomes equal to U1 in the course
of a revolution, then the angular velocity a does not return
to its original value, but is increased at each revolution;
and on the other hand, if U2 becomes at each revolution
greater than U1, then the angular velocity is at each revolution diminished.
The greater the moment of inertia I of the revolving
mass, and the greater the weight PL of its unit of volume
(that is, the heavier the material of which it is formed), the
less is the variation produced in the angular velocity a by
any given variation of U or U1T-U  at different periods of
the same revolution, or from revolution to revolution; that
is, the more steady is the motion produced by any variable
action of the impelling force. It is on this principle that
the fly-wheel is used to equalize the motion of machinery
under a variable operation of the moving power, or of the
resistance. It is simply a contrivance for increasing the
moment of inertia of the revolving mass, and thereby
giving steadiness to its revolution, under the operation of
variable impelling forces, on the principles stated above.
This great moment of inertia is given to the fly-wheel, by
collecting the greater part of its material on the rim, or
about the circumference of the wheel, so that the distance
p of each particle which composes it, from the axis about
which it revolves, may be the greatest possible, and thus
the sum -mp2, or I, may be the greatest possible. At the
same time the greatest value is given to the quantity a, by
constructing the wheel of the heaviest material applicable
to the purpose.
What has here been said will best be understood in its
application to the CRlANK.




68                ANGUTLAR VELOCITY.
77. If we conceive a constant pressure Q to act upon the
~                 iB tedarm CB of the crank
-.. —                 arm G    f in the direction AB of
f/  yy. ^^^^SSthe crank rod, and a
|.c.y t  i                    constant resistance R
to be opposed to the.". —-.." revolution of the axis
C always at the same perpendicular distance from that axis,
it is evident that since the perpendicular distance at which
Q acts from the axis is continually varying (being at one
time nothing, and at, another equal to the whole length CB
of the arm of the   rank)he effective pressure upon the
arm CB must at certain periods of each revolution exceed the
constant resistance opposed to the motion of that arm, and
at other periods fall short of it; so that the resultant of
this pressure and this resistance, or the unbalanced pressure
P upon the arm, must at one period of each revolution have
its direction in the direction of the motion, and at another
time opposite to it. Representing the work done upon the
arm in the one case by U, and in the other by U2, it follows
that if U1-TTU the arm will return in the course of each
revolution, from the velocity which it had when the work
U, began to be done, to that velocity again when the work
UL is completed. If on the contrary UT exceed U2, then the
velocity will increase at each revolution; and if U1 be less
than TT2, it will diminish. It is evident from equation (52),
that the greater the moment of inertia I of the body put
in motion, and the greater the weight v of its unit of
volume, the less is the variation in the value of a, produced
by any given variation in the value of U, -UI; the less
therefore is the variation in the rotation of the arm of the
crank, and of the machine to which it gives motion, produced by the varying action of the forces impressed upon it.
Now the fly-wheel bing fixed upon the same axis with the
crank arm, and revolving with it, adds its own moment of
inertia to that of the rest of the revolving mass, thereby
increasing greatly the value of I, and therefore, on the principles stated above, equalizing the motion, whilst it does not
otherwise increase the resistance to be overcome, than by
the friction of its axis, and the resistance which the air
opposes to its revolution.*
* We shall hereafter treat fully of the crank and fly-wheel.




ANGULAR VELOCITY.               69
78. The rotation of a body about a fixed axis when acted
upon by no other moving force than its weight.
Let U represent the work necessary to raise it from its
second position into the first if it be descending, or from its
first into its second position if it be ascending, and let ac be
its angular velocity in the first position, and a in the second;
then by equation (51),.2'\! + (U).
Now it has been shown (Art. 60.), that the work necessary
to raise the body from its second position into the first if it
be descending, or from its first into its second if it be
ascending (its weight being the only force to be overcome),
is the same as would be necessary to raise its whole weight
collected in its centre of gravity from the one position into
the other position of its centre of gravity. Let CA represent the one, and CA, the other position of..      -   the body, and G and G, the two correspond-':..-'.."...: ing positions of the centre of gravity, then
^   will the work necessary to raise the body
~"~,:.from its position CA to its position CA1, be
Di "'J'    equal to that which is necessary to raise its
whole weight W, supposed collected in G,
from that point to G; which by Article 56, is the same as
that necessary to raise it through the vertical height GM.
Let now CG=-CG=Ah, let CD be a vertical line through
C, let GOCD=-   and GCD-=, in the case in which the
body descends, and conversely when it ascends; therefore
GM=NN=-CN-CN1=h cos. 6-h cos. 1 when the body
descends, or =h cos. 01-h cos. 0 when it ascends from the
position AC to AC1, since in this last case GCD=01 and,GCD=-. Therefore GM= ~hA (cos. -cos. 8), the sign ~
being taken according as the body ascends or descends.
Now UT   W. GM= ~W      (cos. — cos. 80),:. by equation (51) ac2=ci2 +(2 ^ ) (cos. 8-cos. 80).
If M represent the volume of the revolving body MP=W,
~ -- a 12 +     (cos. 6-cos. 8)...(53).
When the body has descended into the vertical position,




'~~7~0       MOMENT OF INERTIA.
8=0, so that (cos. - cos. )1)=-1-cos. 8-=2 sin.2. When
it has ascended into that position =-r, so that (cos. — cos.
8))= - (1 + cos. 8)= - -2 cos.'2yl.
In the first case, therefore,
+2 124       )hin. S..... (54).
In the second case,
( 4gthM 
a2 -l2^-(4g\I/cos....... (55).
When the body has descended or ascended into the horizontal position 8=2, so that (cos. — cos. 1)= —cos. 08. But
it is to be observed, that if the body have descended into
the horizontal position, B, must have been greater than -
and therefore cos. 8, must be negative and equal to -cos.
BCG1; so that if we suppose 1 to be measured from CB or
CD, according as the body descends or ascends, then (cos.
— cos. 81)= ~cos. 81, and we have for this case of descent
or ascent to a horizontal position
2ghM
o2 C    ~ i   cos. 01.... (56.)
If the body descend from a state of rest, a,=0..'. by equation (53) c2-= I (cos. O-cos. )... (57).
Thus the angular velocity acquired from rest is less as the
moment of inertia I is greater as compared with the volume
M, or as the mass of the body is collected farther from its
axis.
THE MOMENT OF INERTIA.
79. Having given the moment of inertia of a body, or system
of bodies, about an axis passing through its centre of
gravity, to fnd its moment of inertia about an axis, parallel to the first, passing through any other point in the
body or system.
Let m, be any element of the body or system, m,AG a




MOMENT OF INERTIA.              71
plane perpendicular to the axis, about
/NG       which the moments are to be measured, A:-t..-...(  the point where this plane is intersected......... by that axis, and G the point where it is'........,intersected by the parallel axis passing
through the centre of gravity of the body.  Join AG,
Aml, Gi,, and draw mM, perpendicular to AG.     Let
Am,1=p, Gm1=r,, GMI=x,, AG=h.
Now (Euclid, 2-12.), Am12 = AG2+Gm 2+ 2AG. GM1,
or p12 h +r12+ 2hx,.
If therefore the volume of the element be represented by
m,, and both sides of the above equation be multiplied by it,
plm = h2m, + r12m1 + 2hxm,.
Nnd if mi, n,, m,, &c. represent the volumes of any other
elements, and p,, r, x,; p,, r,, $,, &c. be similarly taken in
respect to those elements, then,
p22m 2= h2n2 + r22m2 + 2hx2m2,
p,32, =k 2m, + r,2m + 2Axm,,
&c.=&c.
Adding these equations we have, p,2m, + p2nm + pm2, +...
=h (miM +m     -... )+ (r,2mS1+22m2+r,3m,+....)+
2h (xm, +  m,  m3+ m +. * ),
or zp2m=h2= m + 2rm + 2hA2zm.
NTow z2m is the sum of the moments of all the elements
of the body about a plane perpendicular to AG, and passing
through the centre of gravity G of the body. Therefore
(Art. 17.) 2-zm=0,:. Zp2m=/2am+ xm.
Also p2in is the moment of inertia of the body about the
given axis passing through A, and.r.m is the moment of
inertia about an axis parallel to this, passing through the
centre of gravity of the body. Let the former moment be
represented by II; and the latter by I; and let the volume
of the body zm be represented by M,:. I = h +I...... (58).
From which relation the moment of inertia (I,) about any
axis may be found, that (I) about an axis parallel to it, and
passing through the centre of gravity of the body being
known.
80. TuE RADIUS OF GYRATION. If we suppose k1 to be the
distance from the axis passing through A, at which distance,




72                MOMENT OF INERTIA.
if the whole mass of the body were collected, the moment of
inertia would remain the same, so that k12M=I, then ki is
called the RADIUS OF GYRATION, in respect to that axis.
If k be the radius of gyration, similarly taken in respect
to the axis passing through G1, so that k2M=I, then, substituting in the preceding equation, and dividing by M,
ik' h=+Pk...... (t59).
The following are examples of the determination of the
moments of inertia of bodies of some of the more common
geometrical forms, about the axes passing through their centres of gravity: they may thence be found about any other
axes parallel to these, by equation (59).
*81. The moment of inertia of a thin uniform rod about an
axis perpendicular to its length and passing through its
middle point.
Let m represent an element of the rod contained between
A  ~ ~ two plane sections perpendicular to its
e       \ --— ~:i  faces, the area of each of which is c, and
Al      -     whose distance from one another is Ap,
I              and let X and Ap be so small that every
point in this element may be considered to be at the same
distance p from the axis A, about which the rod revolves.
Then is the volume of the element represented by Lap, and
its moment of inertia about A by cp'Ap. So that the whole
moment of inertia I of the bar is represented by?cp'Ap, or,
since Xi is the same throughout (the bar being uniform), by
czp2Ap; or since Ap is infinitely small, it is represented by
the definite integral Kc p'dp, where 1 is the whole length
of the bar,
*.I=-C i(l)?-(-(i)- }
or I=   l3..... v (60).
*82. The moment of inertia of a thin rectangular lamina
about an axis, passing through its centre of gravity, and
parallel to one of its sides.
It is evident that such a lamina may be conceived to be




MOMENT OF INERTIA.              73
made up of an infinite number of slender
rectangular rods of equal length, each of
-: —---------- EE which will be bisected by the axis AB,
a...d t.hat the n moment of inertia of the
whole lamina is equal to the sum of the
moments of inertia of these rods. Now if i be the section
of any rod, and Z the length of the lamina, then the moment
of inertia of that rod is, by the last proposition, represented
by 1-td3; so that if the section of each rod be the same, and
they be n in number, then the whole moment of inertia of
the lamina is  niclt3. Now nK is the area of the transverse
section of the lamina, which may be represented by K, so
that the moment of inertia of the lamina about the axis AB
is represented by the formula
I=   3 2....  (61).
*83. The moment of inertia of a rectangular arallelo2ppedon about an axis, passing through its centre of gravity,
and parallel to either of its edges.
Let CD be a rectangular parallelopipedon, and AB an.     axis passing through its centre of gravity and
parallel to either of its edges; also let ab be...../  an axis parallel to the first, passing through
b ---:   the centre of gravity of a lamina contained
a  b I/  by planes parallel to either of the faces of the
D **.   parallelopiped.  Let a, b, c, represent the
three edges ED, EF, EG, of the parallelopiped, then will the moment of inertia of the lamina about
the axis ab be represented by -1Kb', where K is the transverse section of the lamina (equation 61). Now let the
perpendicular distance between the two axes AB and ab be
represented by x. Then (by equation 58) the moment of
inertia of the lamina about the axis AB is represented by
the formula x2M+ KKb3, where M represents the volume of
the lamina. Let the thickness of the lamina be represented
by Ax;.. -M =abAx, K = aAx,;. mt ina of lama = abxax +
-2ab'Ax;.. whole mt ina of parallelopiped = abx,2Ax +
~ab'zAx; or taking Ax infinitely small, and representing the
moment of inertia of the parallelopiped by I.
I= abJ f2dx + -Lab 9  dx;
-Fc        -fo




74~:        MMOMENT OF INERTIA.
or I=ab i(2c)- ( —)c)  + 2Tal Ic)-( —c)1
=12abc + -2ab'c,.I= -ab(b' + c2)..... (62).
*84. The moment of inertia of an upright triangular prism
about a vertical axis passing through its centre of gravity.
Let AB be a vertical axis passing through the centre of
gravity of a prism, whose horizontal section is
D a, W  an isosceles triangle having the equal sides ED,   and EF.
Let two planes be drawn parallel to the face
I^ |  1  DF of the prism, and containing between them
a thin lamina pq of its volume. Let Cm, the
perpendicular distance of an axis passing through
the centre of gravity of this lamina from the
axis AB, be represented by x; also let Lx represent the
thickness of the lamina.
Let DF=_ a, DG = b, and let the perpendicular from the
vertex E to the base DF of the triangle DEF be represented
by c,
EC =   c, Em=-c,- $; also  == E
DF     c,'pq = - (c-w$); also transverse section K of lamina = bax.
ab. volume M of lamina- =  (c —x)Ax.  Therefore by equations (58) and (61),
mt inm of lama about AB- a^(c-x)2Ax+-jLb-c       )A-;. mt in of prism about
ab   +tc          ba'   +io
AB =        c-w   dx+    f     C-fdx.
Gc -fic          Gc"V' -4c
Performing the integrations here indicated, and representing the inertia of the prism about AB by I, we have
I=   abe( a2 + c2)...... (63).




MOMENT OF INERTIA,              75
*85. The moment of inertia of a solid cylinder about its
axis of symmetry.
Let AB be the axis of such a cylinder, whose radius AC
is represented by a, and its height by b. Con-, ceive the cylinder to be made up of cylindrical
~:~C. rings having the same axis; let AP= p be the,'1 i, jinternal radius of one of these, and let its thickl::I I ness PQ be represented by Ap, so that p Q+?- is
_-..jl. D the exteral radius AQ of the ring. Then will
-   the volume of the ring be represented by,b(p + Ap)2-rbp2, or by.6[2pap + (Ap)2]; or if Ap
be taken exceedingly small, so that (Ap)2 may vanish as compared with 2pap, then is the volume of the ring represented
by 2,bpap.
Now this being the case, the ring may be considered as an
element AM of the volume of the solid, every part of which
element is at the same distance p from the axis AB, so that
the whole moment of inertia zp'AM   of the cylinder=
jp ( 2,rbpp ) —  2,rbz p 3ap,. I= 2Srb f p3cp=ba4....   (64).
0
*86. The moment of inertia of a hollow cylinder about its
axis of symmetry.
Let a1 be the external radius AC, and a2 the internal
radius AP, and b the height of the cylinder;
Hi,?~   then by the last proposition the moment of inertia of the cylinder CD, if it were solid, would'i ia i Sbe iv&ba,4; also the moment of inertia of the
cylinder PE, which is taken from this solid to
form the hollow cylinder, would be 42,ba,4. Now
ia   let I represent the moment of inertia of the hollow cylinder CP, therefore I+irba24=Itba,14.. I=-rb(a4-a'24)=-fb(a2 2-2)(al + a22) = b(a -a2)
(a1 + a)(a12 +a).
Let the thickness a- -a2 of the hollow cylinder be represented by c, and its mean radius i(a, +a) by R, therefore
a=ER+-Ic, u2=R-Xc.




76                MOMENT OF INEERTIA.
Substituting these values in the preceding equation, we obtain
I=S2bceR R2+.....  (65).
*87. The moment of inertia of a cylinder about an axis
passing through its centre of gravity, and perpendicular
to its axis of symmetry.
Let AB be such an axis, and let PQ represent a lamina
contained between planes perpendicular to'r         this axis, and exceedingly near to each other.
Let CD, the axis of the cylinder, be repre1 4      -    sented by b, its radius by a, and let CM=a.
k Take Ax to represent the thickness of the
lamina, and let MP-y. Now this lamina
aL   ~may be considered a rectangular parallelopiped traversed through its centre of gravity by the axis AB;
therefore by equation (62) its moment of inertia about that axis
is represented by -2(Ax)b(2y) Ib2+(2y)21}  b b2y+ 4y'} Ax.
Now the whole moment of inertia I of the cylinder about
AB is evidently equal to the sum of the moments of inertia
of all such laminse;
I I= 6b {62y +~ 4y AX     (b2y+4y')dx.
-a
Also, since x and y are the co-ordinates of a point in a
circle from its centre, therefore y= (a2-x-2).  Substituting
this value of y, and integrating according to the well known
rules of the integral calculus,* we have
I={bae2(a2 +~b2)..... (66).
*88. The moment of inertia of a cone about its axis of
symmetry.
The cone may be supposed to be made up of laminae, such
as PQ, contained by planes perpendicular to
the axis of symmetry AB, and each having its
centre of gravity in that axis. Let BP-x, and
let Ax represent the thickness of the lamina,
\':.c  and y its radius PR. Then, since it may be
considered a cylinder of very small height, its
moment of inertia about AB (equation 64) is
represented by 2y4ax. Now the moment of
* Church's Diff. and Integ. Calculus, Arts. 148, 149.




MOIMENT OF INTERTIA.            77
inertia I of the whole cone is equal to the sum of the moments of all such elements,. I,=zIy4Ax.
Let the radius of the base of the cone be represented by
xb               4
a, and its height by b; therefore-=-, thereforeax= -Ay;
y  a             a... l b   <y 4Ay==gYJ y'dy;
o
0.*.I=  ba..... (67).
89. The moment of inertia of a sphere about one of its
diameters.
Let C be the centre of the sphere and AB the diameter
about which its moment is to be determined.
~?~~     Let PQ be any lamina contained by planes
/  ____  perpendicular to AB; let CM=-, and let Ax.... iP'  represent the thickness of the lamina, and y its../   radius; also let CA-=a; then since this lamina,
being exceedingly thin, may be considered a
cylinder, its moment of inertia about the axis AB is (equation 64) fry4/'x; and the moment of inertia I of the whole
sphere is the sum of the moments of all such laminae,
flea.1. I=2r'sy4aX=  l fydx.
-a
Now by the equation to the circle y'=a'-x', therefore
y4=a-4-2cx2 +x4. If this value be substituted for y4, and
the integration be completed according to the common
methods, we shall obtain the equation,
I_==a...... (68).
90. The momrent of inertia of a cone about an axis passing
through its centre of grawity and perpendicular to its axis
of symmetry.
Let CD be an axis passing through the centre of gravity




78                MOMENT OF INERTIA.
G of the cone, and perpendicular to its axis of
-     symmetry, and let GP the distance of the lamina
c \ —     from G, measured along the axis, be represented
by x; also let the thickness of the lamina be represented by Ax. Now this lamina may be considered a cylinder of exceedingly small thickness.  If its radius be represented by y, its moment of inertia about an axis parallel to CD passing through
its,centre, is therefore (equation 66) represented by
iyz 2 y + t(A)2} AX, or if Ax be assumed exceedingly small,
it is represented by ty'ax. Now this being the moment of
the lamina about an axis parallel to CD, passing through its
centre of gravity, and the distance of this axis from CD being x, and also the volume of the lamina being ny'Ax, it follows (equation 58), that the moment of the lamina about CD
is represented by 7y2x~2ax+ y4Ax=: {y2x2 +y4} Ax.
Now the moment I of the whole cone about CD equals
the sum of the moments of all such elements,
* I=-=(y2x  +-ay4)Aa.
Now if a be the radius of the base of the cone and b its
height, then since BG=,b,
-5b —_        b                 b
*' __;- x= —(4a-y) andAx= —Ay; -
Y    a ~     C:a               a..I=_-b    {     _A (-y)2y2+k  4  Ay,.A.I=-^   f {2 -j a^ y)2y2+^y  dy,
a
=:.I lwra2b {a2+b2..... (69).
91. The moment of inertia of a segment of a sphere about
a diameter parallel to the plane of section.
Let ADBE represent any such portion of a sphere, and
D      AB a diameter parallel to the plane of section......    Let CD=a, CE-b, and let PQ be any lamina
ip  -.  contained by planes parallel to the plane of
section: let the distance of the lamina from
C=x, and let its thickness be Ax and its radius
y. Then considering it a cylinder of exceeding small thick



MOVING FORCES.                79
ness, its moment of inertia about an axis passing through its
centre of gravity and parallel to AB, is represented (equation 66) by ty2 y2+ -(Ax)2 Alx, or (neglecting powers of Ax
above the first- by Hy4ax. Hence, therefore, the moment of
this lamina about the axis AB is represented (equation
58) by -y()(Ax)x2++ry4x, or by r\y2x2+4}\ Ax; now the
whole moment I of inertia of ADBE about AB is evidently
equal to the sum of the moments of all such laminae,
a.. I=2 {y2X2 +  4 A f= (y22X +ty4)d.
-b
Now y- =a2-x, therefore y2x+ ~y4=1 2a2x2 —3x4 + a4.
Substituting this value in the integral and integrating, we
have
I —- l16a+15 a4b+o12b3-..b... ((70)
THE ACCELERATION        OF MOTION     BY GIVEN
MOVING FORCES.
92. IF the forces applied to a moving body in the direction of its motion exceed those applied to it in the opposite
direction (both sets of forces being resolved in the direction
of a tangent to its path), the motion of the body will be accelerated; if they fall short of those applied in the opposite
direction, the motion will be retarded. In either case the
excess of the one set of forces above the other is called the
MOVING FORCE upon the body: it is measured by that single
pressure which being applied to the body in a direction opposite to the greater force, would just balance it; or which,
had it been applied to the body (together with the other
forces impressed upon it) when in a state of rest, would have
maintained it in that state; and which, therefore, if applied
when its motion had commenced, would have caused it to
pass from a state of variable to one of uniform motion. Thus
the moving force upon a body which descends freely by gravity, is measured by its weight, that is, by the single force
which, being applied to the body before its motion had commenced in a direction opposite to gravity, would just have
supported it, and which being applied to it at any instant of
* Note (q) Ed. App.




80                   RELATIONS OF
its descent, would have caused its motion at that instant to
pass from a state of variable to a state of uniform motion.
If the resistance of the air upon its descent be taken into
account, then the moving force upon the body at any instant
is measured by that single pressure which, being applied upwards, would, together with the resistance of the air at that
instant, just balance the weight of the body.
A moving force being thus understood to be measured by
a pressure, being in fact the unbalanced pressure upon the
moving body, the following relations between the amount of
a moving force thus measured, and the degree of acceleration
produced by it will become intelligible. These are laws of
motion which have become known by experiment upon the
motions of the bodies immediately around us, and by observation upon those of the planets.
93. When the moving force upon a body remains constantly the same in amount (as measured by the equivalent
pressure) throughout the motion, or is a uniform moving
force, it communicates to it equal additions of velocity in
equal successive intervals of time. Thus the moving force
upon a body descending freely by gravity (measured by its
weight) being constantly the same in amount throughout its
descent (the resistance of the air being neglected), the body
receives from it equal additions of velocity in equal successive intervals of time, viz. 32- feet in each successive second
of time (Art. 44.).
94. The increments of velocity communicated to equal
bodies by unequal moving forces (supposed uniform as above)
are to one another as the amounts of those moving forces
(measured by their equivalent pressures).
Thus let P and P1 be any two unequal moving forces upon
two equal bodies, and let them act in the directions in which
the bodies respectively move; let them be the only forces
tending to communicate motion to those bodies, and remain
constantly the same in amount throughout the motion. Also
let f and f, represent the additional velocities which these
two forces respectively communicate to those two equal
bodies in each successive second of time; then it is a law of
the motion of bodies, determined by observation and experiment, that P: P::f:f/
* Pressure and moving force are indeed but different modes of the operation
of the same principle of force.




PRESSURE AND MOTION.              81
If one of the moving forces, as for instance P1, be the
weight W  of the body moved, then the value f, of the
increment of velocity per second corresponding to that
moving force is 321 (Art. 44.) represented by g,
P.P: W::: g,
w
p= f.....().
95. If the amount or magnitude of the moving force does
not remain the same throughout the motion, or if it be a
variable moving force, then the increments of velocity communicated by it in equal successive intervals of time are not
equal; they increase continually if the moving force
increases, and they diminish if it diminishes.
If two unequal moving forces, one or both of them, thus
variable in magnitude, become the moving forces of two
equal bodies, the additional velocities which they would
communicate in the same interval of time to those bodies,
if at any period of the motion from variable they become
uniform, are to one another (Art. 94.) as the respective
moving forces at that period of the motion.
Thus letf andf, represent the additional velocities which
would thus be communicated to two equal bodies in one
second of time, if at any instant the pressures P and P,,
which are at that instant the moving forces of those bodies,
were from  variable to become constant pressures, then
(Art. 94.),
P: P,::/:f.
This being true of any two moving forces, is evidently true,
if one of them become a constant force. Let P, represent
the weight W  of the body, then will f, be represented
by g,
-.P: W::f: g.
Let the moving force P be supposed to remain constant
during a number of seconds or parts of a second, represented by At, and let aV be the increment of velocity in
the time At on this supposition.  Now f represents the
increment of velocity in each second, and aV the increment
of velocity in At seconds: moreover the force P is supposed
constant during At, so that the motion is uniformly accelerated during that time (Art. 44.).
6




82                  RELATIONS OF
0..fAtAA,.f../=V
At
-Now this is true (if the supposition, that P remains constant
during the time At, on which it is founded, be true), however small the time At may be. But if this time be
infinitely small, the supposition on which it is founded is in
all cases true, for P may in all cases be considered to remain
the same during an infinitely small period of time, although
it does not remain the same during any time which is not
AV   dV
infinitely small.  Now when at is infinitely small — _dt;:At dt
generally therefore f=- d.
If V increase as the time t increases, or if the motion be
dV o
accelerated, then  t is necessarily a positive quantity. If,
on the contrary, V diminishes as the time increases, then
dV
is negative; so that, generally,
f= t.(72),
the sign ~ being taken according as the motion is accelerated or retarded. Substituting this value of f in the last
proportion we have in the case, in which P represents a
variable pressure,
W dV
g dt.....      73).
The principles stated above constitute the fundamental relations of pressure and motion.
96. The velocity V at any instant of a body moving with
a variable motion, being the space which it would describe
in a second of time, if at that instant its motion were to
become uniform, it follows, that if we represent by At any
number of seconds or parts of a second, beginning from that
instant, and by AS, the space which the body would describe
* Note (r) Ed. App.




PRESSURE AND MOTION.               83
in the time At, if its motion continued uniform from the commencement of that time, then,
AS
VAt=AS,     V==At.
Now this is true if the motion remain uniform during the
time At, however small that time may be, and therefore if it
be infinitely small. But if the time At be infinitely small,
the motion does remain uniform during that time, however
variable may be the moving force; also when at is infinitely small, AtS -d  Therefore, generally,
dS
Th1dt.   (74).
V-.. *. (d)t
The equations (73) and (74) are the fundamental equations
of dynamics: they involve those dynamical results which
have been discussed on other principles in the preceding
parts of this work.*
THE DESCENT OF A BODY UPON A CURVE.
* 97. If the moving force P upon a body varies directly as its
distance at any t&mefrom a given point towards which it
falls, then the whole time of the body's falling to that
point will be the same, whatever may be the distance from
which it falls.
Let A be the point from which the body falls, and B a
~A ~     point towards which it falls along the path
APB, which may be either curved or straight;
also let the body be acted upon at each
point P of its path, by a force in the direction of its path at that point which varies as
* Thus if the latter equation be inverted, and multiplied by the former, we
obtain the equation
d.   W    J(dV.  W(dV2
dt -    * via d) =~  d )
dV2   2g
— =WfS1
which is ideial with equation ().S
which is identical with equation (47).




84                   RELATIONS OF
its distance BP, measured along the path from B; the time
of falling to B will be the same, whatever may be the distance of the point A from which the body falls.
For let BP=S, and let the force impelling the body
towards B be represented by cS, where c is a constant quantity; suppose the body, instead of falling from A towards
B, to be projected with any velocity from B towards A, and
let v be the velocity acquired at P, and V that at A, and
let BA=Si then by equation (4i),
_ 2~:-          _9 2/;       _ =,_ ~.
V    2-jSdS          -  W(SI   S2).
Suppose now the velocity of projection from B to have
been such as would only just carry the body to A, so that
V=O,
=WIoSa-S2)..... (75).
Now by equation (74),
dt 1         dS
^       dS
t' J^   ((S-S2t)
and if 4-T represent the whole time in seconds occupied in
the ascent of the body from B to A,
/W    f  dS         W10 S) -o
iT=       -,2   S2)     — (t    cos.-S -- OS-  
i. iT=   2.h
It is clear that the time required for the body's descent
from A to B is equal to that necessary for the ascent from
B to A, so that the whole time required to complete the
ascent and descent is equal to T, and is represented by the
formula
T=()..... (76).
09r 




PRIESSUC-E AND MOTION.            85
Now this expression does not contain S,, i. e. the distance
from which the body falls to B; the time T is the same
therefore, whatever that distance may be.
THE SIMPLE PENDULUM.
98. If a heavy particle P be imagined to be suspended from a
point C by a thread without weight, and allowed to oscillate
freely, but so as to deviate but little on either side of the
vertical, then will its oscillations, so long as they are thus
small, be performed in the same time whatever their amplitudes may be.
For let the inclination POB of CP to the vertical be represented by 8, and let the weight w of the particle
/P P, which acts in the direction of the vertical VP,
v / Ibe resolved into two others, one'of which is in the
direction CP, and the other perpendicular to that
g   direction: the former will be wholly counteracted
by the tension of the thread CP, and the latter will
be represented by w sin. VPC=w sin. 0; and, acting in the direction in which the particle P moves, this will
be the whole impressed moving force upon it (Art. 92.) Now
so long as the arc 8 is small, this arc does not differ sensibly
from its sine, so that for small oscillations the impressed moving force upon P is represented by w8, or by (, or by w,
if I represent the length CP of the suspending thread, and S
the length of the arc BP.  Now in this expression w and I
are constant throughout the oscillation, the moving force varies therefore as S. Hence by the last proposition, the small
oscillations on either side of CB are isochronous, since so long
as they are thus small, the impressed moving force in the
direction of the motion varies as the length of the path BP
from the lowest point B. Since in the last proposition the
moving force was assumed equal to cS, and that here it is
w                         w
represented by - S, therefore in this case = —.  SubstitutL                        I
ing this value in equation (76),
T=(    i...... (77).
A single particle thus suspended by a thread without




86          THE PARALLELOGRAM OF MOTION.
weight, is that which is meant by a SIMPLE PENDULUM. It is
evident that the time of oscillation increases with the length
I of the pendulum.
IMPULSrVE FORCE.
99. If any number of different moving forces be applied
to as many equal bodies, the velocities communicated to
them in the same exceedingly small interval of time, will be
to one another as the moving forces.  For let P1, P2, represent the moving forces, and, f2, the additional velocities
they would communicate per second if each moving force
remained continually of the same magnitude (Art. 93.), then
would tf, tf, be the whole velocities communicated on this
supposition in t seconds; let these be represented by V1, V,;
therefore by Art. 94.
P: P2f,: f,::tf: tf2:: V,: V,.
The proposition is therefore true on the supposition that P,
and P2 remain constant during the interval of time t; but
if t be exceedingly small, then whatever the pressures P,
and P2 may be, they may be considered to remain the same
during that time. Therefore the proposition is true generally,
when, as above, the moving forces are supposed to act on
equal bodies, or successively on the same body, through
equal exceedingly small intervals of time.
Moving forces thus acting through exceedingly small intervals of time only, are called IMPULTrVE FORCES.
THE PARALLELOGRAM OF MOTION.
100. If two impulsive forces P1, P2, whose directions are AB
and AC, be impressed at the same time upon.. -      a body at A, which if made to act upon it
// _/B     separately would cause it to move through
AB and AC in the same given time, then
will the body be made, by the simultaneous action of these
impulsive forces, to describe in that time the diagonal AD
of the parallelogram, of which AB and AC are adjacent
sides.
For the moving forces P and P2 acting separately upon




INDEPENDENCE OF SIMULTANEOUi MOTIONS.    8,
the same body through equal infinitely small times, communicate to it velocities which are (Art. 99.) as those forces,
therefore the spaces AB and AC described with these velocities in any given time are also as those forces. Since then
AB and AC are to one another as the pressures P1 and P.,
therefore by the principle (Art. 2.) of the parallelogram of
pressures, the resultant R of Pi and P2 is in the direction of
the diagonal AD, and bears the same proportion to P, and
P, that AD does to AB and AC.
Therefore the velocity which the resultant R of P1 and P,
would communicate to the body in any exceedingly small
time is to the velocities which P, and P2 would separately
communicate to it in the same time as AD to AB and AC
(Art. 99.), and therefore the spaces which the body would
describe uniformly with these three velocities in any equal
times are in the ratio of these three lines. But AB and AC
are the spaces actually described in the equal times by reason of the impulses of P, and P,. Therefore AD is the space
described in that time by reason of the impulse of R, that is,
by reason of the simultaneous impulses of P, and P2.
101. THE INDEPENDENCE OF SIMULTANEOUS MOTIONS.
It is evident that if the body starting from A had been
made to describe AB in a given time, and then'::2-     had been made in an equal time to describe
la.,   BBD, it would have arrived precisely at the same
point D to which the simultaneous motions
AC and AB have brought it, so that the body is made to
move by these simultaneous motions precisely to the same
point to which it would have been brought by those motions,
communicated to it successively, but in half the time.  The
following may be taken as an illustration of this principle of
the independence of simultaneous motions. Let a canal-boat, _     be imagined to extend across the whole
lr-~~7 ^H,,   width of the canal, and let it be supposed..... w that a person standing on the one bank at
A is desirous to pass to a point D on the
opposite bank, and that for this purpose, as the boat passes
him, he steps into it, and walks across it in the direction
AB, arriving at the point B in the boat precisely at the instant when the motion of the boat has carried it through
BD; it is clear that he will be brought, by the joint effect




88             TiHE POLYGON OF MOTION.
of his own motion across the boat and the boat's motion
along the canal, to the point D (having in reality described
the diagonal AD), which point he would have reached in
double the time if he had walked across a bridge from A to
B in the same time that it took him to walk across the boat,
and had then in an equal time walked from B to D along
the opposite side.
THE POLYGON OF MOTION.
102. Let any number of impulses be communicated simulr    E   taneously to a body at 0, one of which
would cause it to move from A to 0 in a
given time, another from B to O in the
^ol^  ~  same time, a third from C to 0 in that time,
~~i /   and a fourth from D to O. Complete the
parallelogram of which AO and BO are adjacent sides; then the impulses AO and BO would simultaneously cause the body to move from E to O through the
diagonal EO in the time spoken of. Complete the parallelogram EOCF, and draw its diagonal OF, then would the impulses EO and CO, acting simultaneously, cause the body to
move through FO in the given time: but the impulse EO
produces the same effect on the body as the impulses AO
and BO; therefore the impulses AO, BO, and C0, will
together cause the body to move through FO in the given
time. In the same manner it may be shown that the impulses AO, BO, CO, and DO, will together cause the body to
move through GO in a time equal to that occupied by the
body's motion through any one of these lines.
It will be observed that GD is the side which completes
the polygon OAEFG, whose other sides OA, AE, EF, FG,
are respectively equal and parallel to the directions OA, OB,
OC, and OD, of the simultaneous impulses.
Instead of the impulses AO, &c. taking place simultaneously, if they had been received successively, the body
moving first from O to A in a given time; then through
AE, which is equal and parallel to OB, in an equal time;
then through EF, which is equal and parallel to OC, in that
time; and lastly through FG, which is equal and parallel to
OD, in that time, it would have arrived at the same point G,
to which these impulses have brought it simultaneously, but
after a period as many times greater as there are motions, so




THE PRINCIPLE OF DALEMBERT.          89
that the principle of the independence of simultaneous
motions obtains, however great may be the number of such
motions.
THE PRINCIPLE OF D'ALEMBERT.
103. Let W1, W2, W3, &c. represent the weights of any
number of bodies in motion, and Pi, P,, P, &c. the moving
forces (Art. 92.) upon these bodies at any given instant of
the motion, i. e. the unbalanced pressures, or the pressures
which are wholly employed in producing their motion, and
pressures equal to which, applied in opposite directions,
would bring them to rest, or to a state of uniform motion.
W           W          W
Then (Art. 95.), P,=    fi P,P= —  f  — f, P=  f,&c.
where., f, f,,  &c. represent the additions of velocity which
the bodies would receive in each second of time, if the
moving force upon each were to become, at the instant at
which it is measured, an uniform moving force. Suppose
these bodies, whose weights are W1, W2, W,3 &c. to form a
system of bodies united together by any conceivable mechanical connection, on which system are impressed, in any
way, certain forces, whence result the unbalanced pressures
PIP, PF3, &c. on the moving points of the system. Now
conceive that to these moving points of the system there are
applied pressures respectively equal to P,, P2, P,, &c. but
each in a direction opposite to that in which the motion of
the corresponding point is accelerated or retarded. Then
will the motion of each particular point evidently pass into
a state of uniform motion, or of rest (Art. 92.). The whole
system of bodies being thus then in a state of uniform
motion, or of rest, the forces applied to its different elements
must be forces in equilibrium.
Whatever, therefore, were the forces originally impressed
upon the system, and causing its motion, they must, together
with the pressures P1, P,, P&, &c. thus applied, produce a
state of equilibrium in the system; so that these forces (originally impressed upon the system, and known in Dynamics
as the IMPRESSED FORCES) have to the forces P1, P2, P,, &c.,
when applied in directions opposite to the motions of their
several points of application, the relation of forces in equilibrium. The forces P,, P, P2, &c. are known in Dynamics
as the EFFECTIVE FORCES.  Thus in any system of bodies
mechanically connected in any way, so that their motions




90            THE PRINCIPLE OF D'ALEMBERT.
may mutually influence one another, if forces equal to the
efective forces were applied in directions opposite to their
actual directions, these would be in equilibrium with the
impressed forces, which is the principle of D'Alembert.
104. The work accumulated in a moving body through any
space is equal to the work which must be done upon it, in
an opposite direction, to overcome the efective force 2upon
it through that space.
This is evident from Arts. 68. and 69., since the effective
force is the unbalanced pressure upon the body.
If the work of the effective force be said to be done upon
the body,* then the work of the effective force upon it is
equal to the work or power accumulated in it, and this work
of the effective force may be all said to be actually accumulated in the body as in a reservoir.
MOTION OF TRANSLATION.
DEFINITION.-When a body moves forward in space, without at the same time revolving, so that all its parts move
with the same velocity and in parallel directions, it is said to
move with a motion of translation only.
105. In order that a body may move with a motion of translation only, the resultant of the forces impressed upon it
must have its direction through the centre of gravity of
the body.
For let w,, w,2 w,) &c. represent the weights of the parts
or elements of the bkdy, and let f represent the additional
velocity per second, which any element receives or would
receive if its motion were at any instant to become uniformly
accelerated. Since the motion is one of translation only,
the value of f is evidently the same in respect to every
other element. The effective forces P1, P2, P,, &c. on the
different elements of the body are therefore represented by
-ff   f, &c. &c.
* This cannot perhaps be correctly said, since work supposes resistance.




MOTION OF ROTATION.              91
Now the forces P1, P2, P, &c. are evidently parallel pressures. Let X be the distance of the centre (see Art. 17.) of
these parallel pressures from any given plane; and let x1, x2,
x3, &c. be the perpendicular distances of the elements w,v w,,
w3, &c. that is, of the points of application of P, P,, P,, &c.
from the same plane. Therefore (by equation 18),
P,+P,+P,+....     } X=Pl+P22 +P3+         * *....;
qe1$ _ + W  + s3s+ * * * _
f f++2+W3+...,
But this is the expression (Art. 19.) for the distance of the
But this is the expression (Art. 19.) for the distance of the
centre of gravity from the given plane; and this being true
of any plane, it follows that the centre of the parallel pressures Pi, P,, P,, &c. which are the effective forces of the
system, coincides with the centre of gravity of the system,
and therefore that the resultant of the effective forces passes
through the centre of gravity. Now the resultant of the
effective pressures must coincide in direction with the resultant of the impressed pressures, since the effective pressures
when applied in an opposite direction are in equilibrium
with the impressed pressures (by D'Alembert's principle).
The resultant of the impressed pressures must therefore have
its direction through the centre of gravity. Therefore, &c.
MOTION OF ROTATION ABOUT A FIXED AXIS.
106. Let a rigid body or system be capable of motion
about the axis A. Let m,, mn, m,, &c. represent the volumes
of elements of this body, and P. the weight of each unit
of volume. Also let f,ff,,  &c. represent the increments
of velocity per second, communicated to these elements
respectively by the action of the forces imrpressed upon the
system. Let P,, P,, P,, &c. represent these impressed forces,
and p,p,, &c. the perpendicular distances from the axis at
which they are respectively applied.
Now since ml, mn, um,, &c. are the weights of the elements, andf, f,, &c. the increments of velocity they receive




92                MOTION OF ROTATION.
per second, it follows that m~ fi, Im/,2,  mSf,, &c. are
g a a
the effective forces upon them (Art. 103.). Let pl, p2, ps, &c.
represent the distances of these elements respectively from
the axis of revolution, then since their effective forces are
in directions perpendicular to these distances, the moments
of these effective forces about the axis are mii fP, fi 2  P,  
g       9
3-fgps, &c. Also Pp1, P2p2, Psp, &c. are the moments of
g
the impressed forces of the system about the axis. Now the
impressed forces P,, P, P,, &c., together with the resistance
of the axis, which is indeed one of the impressed forces, are
in equilibrium with the effective forces by D'Alembert's
principle. Taking then the axis as the point from which the
moments are measured, the sum of the moments of P,, P2
&c. must equal the sum of the moments of the effective
forces, or
PITfip, +     2P2+ P.... =P  I+P2+....
a         g
Now letf represent that value of f,f, &c. which corresponds to a distance unity from the axis. Since the system
is rigid, and f, f, f,, &c. represent arcs described about
it in the same time at the different distances 1, pi, p,, &c. it
follows that these arcs are as their distances, and therefore
thatf= — fp,,,=fp,,J/s=fp,, &c. Substituting these values
in the preceding equation, we have.m,./fp,+mfp,...       =P,+fP,        ~....=P +P +..;.fmp {   +mp +... * =Pp, +P2P2+....
or f           2...
g
ff.....(78),
I
where I represents the moment of inertia of the mass about
its axis of revolution.*
* If a represent the angular velocity, or the velocity of an element at disdte  a  _      ZPpa
tance unity, then by equation (72), f — +,..ad   - ~;Ppa;




MOTION OF ROTATION.               93
107. If the impressed forces P be the weights of the parts
of the body and 6 be, in any position of.cexde]~i     the body, the inclination to the vertical
Ay of the line AG, drawn from A to the:;,K        centre of gravity G, then since the sum of
J —- \     ^the moments of the weights of the parts is
equal to the moment of the weight of the
~wI iwhole mass collected in its centre of
gravity (Art. 17.), we have, representing
AG by G,
zPp=MI. GG,=Mpa. G. sin. 6
MG 
therefore tequation 78),f= g —  sin....... (79).
108. To find the resultant of the ejfective forces on a body
which revolves about a fixed axis.
The resultant of the effective forces upon a body which
revolves about a fixed axis, is evidently equal to that single
force which would just be in equilibrium with these if there
were no resistance of the axis. Let R be that single force,
then the moment of R about any point must equal the sum
of the moments of the effective forces about that point.
QF,               Take a point in the axis for the point' —.............  about which the moments are measured,
and let L be the perpendicular distance. Ad\  ~   from A of the resultant R. Now, as in.  <j    Art. 106. it appears that the sum of the
a,!   3        moments of the effective forces about A is
represented by f-zmp',
l.at  - la22+ =  PJfPpadt.
0
Now pa is the velocity of a point at distance p, therefore Ppa is the work
(Art. 50.) of the force P per second; thereforef padt is the work of P
0
(equation 40) in the time t, which is represented by U, therefore a-2 -a,2
g~ I which corresponds with the result already obtained. See equation
(61).




94                MOTION OF ROTATION.. L=fmp2.....   (80).
To determine the value of R let it be observed that the
effective force -fn~pI on any particle m, acting in a direction nrlm, perpendicular to the distance Am, from the axis
A, may be resolved into two others, parallel to the two
rectangular axes Ay and Ax, each of which is equal to the
product of this effective force, whose direction is n1m,, and
the cosine of the inclination of n~m, to the corresponding
axis. Now the inclination of mln to Ax is the same as the
inclination of Am1 to Ay, since these two last lines are perpendicular to the two former. The cosine of this inclination
equals thereforeAN- or l, if AN1 —Y1. Similarly the cosine
Am,,
of the inclination of n1ml to Ay equals Al or x, if AM  = x,.
Am     P1i
The resolved parts in the directions of Ay and Ax of the
effective force E fmp1 are therefore f f/lpl,, and - fmp,
g                  g       P1     g
o, or _ fmy, and - fmla.
P1    g          g
Similarly the resolved parts in the directions of Ax and
Ay of the "effective force upon mB are fm2y, and f- fmx,,
9          g
and so of the rest.
The sums X and Y of the resolved forces in the directions
of Ax and Ay respectively (Art. 11.) are therefore
fmy1~ + - fmy + ~fmy,3. +. =x,
and  f fmlx +  fm2x2 +  fmSw +..  = Y;
9        g        g
or-f{m~y1 +~y +m~y2+.....}-X,
and f   mx, + mx 2+m,3+m +.....    =Y.
Now let G1 and G2 represent the distances G2G and G1G
of the centre of gravity of the body from Ay and Ax respectively, and let the whole volume of the body be represented
by M,




MOTION OF ROTATION.                95. (equation 18), MG2=my +m,  y +my+ S....,
MG1 =m1   + m22, +33+....;..X=fMG2, Y=-fMG,.....          (81).
6   te'.c. Now (Art. 11.), R =      /X +Y2, therefore
ftj.~ -...\      f
~is"\,    R   PfM VG12+G 22.?          N) ~Now if G be the distance AG of the
|i    ~   centre of gravity from A, G= -/G12+G2,
aR=   fMGr..... (82).
Substituting in equation (82) the value off from equation
(78,) we have
R   M     p..... (83).
And substituting in equation (80) for R its value from
equation (82),
IJ       I"
f MGL=f- I,.L= MG....(84),
where L is the distance of the point of application of the
resultant of the effective forces from the axis.
Now let 8 be the inclination of the resultant ZR to the
axis A?,. (Art. 11.), R cos. =X, R sin. 0=Y,
~.tan. 0=X; but by equations (81),
Y G, AG,
YXGIGAG'tan. AGG,,
tan. 6=tan. AGG,,.- =AGG,.
The inclination of the resultant R to Am is therefore
equal to the angle AGG,, but the perpendicular to AG is
evidently inclined to Ax at this same angle. Therefore the
direction of the resultant R is perpendicular to the line AG,
drawn from the axis to the centre of gravity. Moreover




96            THE CENTRE OF OSCILLATION.
its magnitude and the distance of its point of application
from A have been before determined by equations (83)
and (84).
THE CENTRE OF PERCUSSION.
109. It is evident, that if at a point of the body through
which the resultant of the effective forces upon it passes,
there be opposed an obstacle to its motion, then there will
be produced upon that obstacle the same effect as though
the whole of the effective forces were collected in that
point, and made to act there upon the obstacle, so that the
whole of these forces will take effect upon the obstacle, and
there will be no effect of these forces produced elsewhere, and therefore no repercussion upon the axis.
It is for this reason that the point O in the resultant,
where it cuts the line AG drawn from the axis to the
centre of gravity, is called the CENTRE OF PERCUSSION.' Its distance L from A is determined by the equation...K.. (85),
which is obtained from equation (84) by writing MK2 for I
(Art. 80.), K being the radius of gyration. If at the centre
of percussion the body receive an impulse when at rest,
then since the resultant of the effective forces thereby produced will have its direction through the point where the
impulse is communicated, it follows that the whole impulse
will take effect in the production of those effective forces,
and no portion be expended on the axis.
THE CENTRE OF OSCILLATION.
110. It has been shown (Art. 98.) that in the simple pendulum, supposed to be a single exceedingly small element
of matter suspended by a thread without weight, the time
of each oscillation is dependent upon the length of this
thread, or the distance of the suspended element from the
axis about which it oscillates. If therefore we imagine a
number of such elements to be thus suspended at different
distances from the same axis, and if we suppose them, after
having been at first united into a continuous body, placed
in an inclined position, all to be released at once from this




THE CENTRE OF OSCILLATION.           9T
union with one another, and allowed to oscillate freely, it is
manifest that their oscillations will all be performed in
different times. Now let all these elements again be conceived united in one oscillating mass. All being then compelled to perform these oscillations in the same time, whilst
all tend to perform them in different times, the motions of
some are manifestly retarded by their connexion with the
rest, and those of others accelerated, the former being those
which lie near to the axis, and the others those more remote;
so that between the two there must be some point in the
body where the elements cease to be retarded and begin to
be accelerated, and where therefore they are neither accelerated nor retarded by their connexion with the rest; an element there performing its oscillations precisely in the same
time as it would do, if it were not connected with the rest,
but suspended freely from the axis by a thread without
weight. This point in the body, at the distance of which
from the axis a single particle, suspended freely, would perform its oscillations precisely in the same time that the body
does, is called the CENTRE OF OSCILLATION.
The centre of  oscillation coincides with the centre of
percussion.
111. For (by equation 79) the increment of angular velocity per secondf of a body revolving about an hori-:   zontal axis, the forces impressed upon it being the
i   weights of its parts only, is represented by the forti  mula g-   sin. 8, where 8 is the inclination to the vertical of the line AG, drawn from the axis to its
centre of gravity. But (by equation 84), L=-7  where L
is the distance AO of the centre of percussion from the
axis,
=g sin. 
L. J.....(86)
fL=g sin. 6
Now it has been shown (Art. 98.), that the impressed
moving force on a particle whose weight is w, suspended
from a thread without weight, inlined to the vertical at an
angle 0, is represented by w sin. 6; moreover if f' represent




98            THE CENTRE OF OSCILATION.
the increment of velocity per second on this particle, then
-f is the effective force upon it. Therefore by D'Alembert's principle,
w sin. =  f,:. f=g sin. 6,:. =fL.
Now fL is the increment of velocity at the centre of
percussion, andf' is that upon a single particle suspended
freely at any distance from the axis. If such a particle
were therefore suspended at a distance from the axis equal
to that of the centre of percussion, since it would receive,
at the same distance from the axis, the same increments of
velocity per second that the centre of percussion does, it
would manifestly move exactly as that point does, and perform its oscillations in the same time that the body does.
Therefore, &c.
112. The centres of suspension and oscillation are reciprocal.
{a    Let O represent the centre of oscillation of a body
A) when suspended from the axis A; also let G be its
centre of gravity.   Let AO=L, AG=G, OG-=G1;
(   also let the radius of gyration about A be represented by K2, and that about G by Ak. Therefore
(equation 59), K2=G- +k-2;
(equation 85), L=  G  =G+G......8:. G+G,=G+(,,:.G,=   ~....  (88).
Now let the body be suspended from O instead of A;
when thus suspended it will have, as before, a centre of
oscillation. Let the distance of this centre of oscillation
from O be L1,
k by equation (87), L G +,
by equation (87), L1=G1+Q,




PROJECTILES.                 99
k2. by equation (88), LI=  +G=L.
Since then the centre of oscillation in this second case is at
the distance L from 0, it is in A; what was before the
centre of suspension has now therefore become the centre
of oscillation. Thus when the centre of oscillation is converted into the centre of suspension, the centre of suspension is thereby converted into the centre of oscillation.
This is what is meant, when it is said that the centres of
oscillation and suspension are reciprocal.
PROJECTILES.
113. To determine the path of a body proected obliquely
in vacuo.
Suppose the whole time, T seconds, of the flight of the
body to any given point P..I of its path, to be divided:..... — into equal exceedingly small.:-; -...... —.-4 intervals, represented  by
-  -rv —:. aT, and conceive the whole
-~. { effect of gravity upon the.~f^E~ ~         projectile during each one
of these intervals to be collected into a single impulse at the termination of that interval, so that there may be communicated to it at once, by
that single impulse, all the additional velocity which is in
reality communicated to it by gravity at the different periods
of the small time AT.
Let AB be the space which the projectile would describe,
with its velocity of projection alone, in the first interval of
time; then will it be projected from B at the commencement of the second interval of time in the direction ABT
with a velocity which would alone carry it through the distance BK=AB in that interval of time; whilst at the same
time it receives from the impulse of gravity a velocity such
as would alone carry it vertically through a space in that interval of time which may be represented by BF. By reason
of these two impulses communicated together, the body will
therefore describe in the second interval of time the diagonal BC of the parallelogram of which BK and BF are adja



100                 PROJECTILES.
cent sides. At the commencement of the third interval it
will therefore have arrived at C, and will be projected from
thence in the direction BCX, with a velocity which would
alone carry it through CX=-BC in the third interval; whilst
at the same time it receives an impulse from gravity communicating to it a velocity which would alone carry it
through a distance represented by CG=BF in that interval
of time. These two impulses together communicate therefore to it a velocity which carries it through CD in the third
interval, and thus it is made to describe all the sides of the
polygon ABCD... P in succession. Draw the vertical PT,
and produce AB, BC, CD, &c. to meet it in T, N, O...,
and produce GC, HD, &c. to meet BT in K, L, &c.
Now, since BC is equal to CX, and CK is parallel to XL,
therefore KL is equal to BK or to AB.
Again, since CD is equal to DZ, and DL is parallel to ZM,
therefore LIM is equal to KL or to AB; and so of the rest.
If therefore there be n intervals of time equal to AT, so
that there are n sides AB, BC, CD, &c. of the polygon, and
n divisions AB, BK, &c. of the line AT, then AT, —=AB and
BT= (n-1)AB,. TN  - (-1)KC= (n-1)BF.
Similarly CN=(n —2)CX, therefore NO=(~ —2)DX=
(1n —2)BF; and so of the remaining parts of TP.
Now these parts of TP are (n-1) in number, therefore
TP=(n-l)BF~+(n-2)BF+(n-3)BF+... (n-1) terms};
or TP=I(n1)+(-la)+( 2)+... }BF.
Therefore, summing the series to (n-1) terms., TP-=(-I1)BF.
2
Now g represents the additional velocity which gravity
would communicate to the projectile in each second, if it
acted upon it alone. gAT is therefore the velocity which it
would communicate to it in each interval of AT seconds.
qAT is therefore the velocity communicated to the body by
each of the impulses which it has been supposed to receive
from gravity.




PROJECTILES.                101
Now BF is the space through which it would be carried
in the time AT by this velocity,
BF= (gAT)aT=g(AT)2.TP=-gn(n-1)(AT)2.
AT.
Also AT=-,
n.TP=lg-      l)(n(1) l=' -T.
Now this is true, however small may be the intervals of
time AT, and therefore if they be infinitely small, that is, if
the impulses of gravity be supposed to follow one another at
infinitely small intervals, or if gravity be supposed to act, as
it really does, continuously.
But if the intervals of time AT be infinitely small, then
the number n of these intervals which make up the whole
finite time T, must be infinitely great. Also when n is infinitely great, -=0.
In the actual case, therefore, of a projectile continually
deflected by gravity, the vertical distance TP between the
tangent to its path at the point of projection, and its position
P after the flight has continued T seconds, is represented by
the formula
TP=gT..... (89).
Moreover AT=nAB, and AB is the space which the body
would describe uniformly with the velocity of projection in
the time AT, so that nAB is the space which it would describe in the time n. AT or T with that velocity. If therefore V equal the velocity of projection, then
AT=V. T.... (90);
so that the position of the body after the time T is the same
as though it had moved through that time with the velocity
of its projection alone, describing AT, and had then fallen
through the same time by the force of gravity alone, describing TP (see Art. 101.)...,' t114. Let AM=x, MP=y, angle of
projection TAM=oa, velocity of projec-...  C.tion=V.
--  "*4"-M —-— /.




102                  PROJECTILES.
x sec. C. sec. a=AT=V. T,         ec. -
a tan. a-y=MT- MP-TP =-gT2..... (91).
Substituting the value of T from the preceding equation,
2 xseC.2 a
tan. a-y =  - --
q sec.2a  2.y=  tan. -- sc. a.
2V'2
Let H be the height through which a body must fall freely
by gravity to acquire the velocity V, or the height due to
that velocity; then V2=2gII (Art. 47.), therefore 4H  2
therefore, by substitution,
y=X tan. c_ —-.... (92).
115. To find the time of the fight of a projectile.
It has been shown (equation 91), that if T represent the
time in seconds of the flight to a point whose co-ordinates
are x and y, then
g9T2=  tan. a-y,  ~ T-=2 {x tan. C-y},.T   _ V  ta.   -y.....(93).
Now, -232 = 1nearly,.. T=i/x tan. a-y nearly.
g 32 16
If the projectile descend again to the horizontal plane from
which it was projected, andT be the whole time of its flight,
and X its whole range upon the plane, then, since at the expiration of the time T, y=O and x=X,
T=/2VX tan. a=-/     tan. a nearly.




PROJECTILES.                 103
116. To find the greatest horizontal distance X, to which a
projectile ranges, having given the elevation a and the
velocity V of its projection.
When the projectile attains its greatest horizontal range,
its height y above the horizontal plane
becomes 0, whilst the abscissa x of the,/*.~    point P, which it has then reached in,' |  its path, becomes X.   Substituting
~~.. ~these values 0 and X, for y and x in
I       i,    equation (92), we have O=X tan. a-...........-.j^ - X2 sec.2 a
4H.X=41H tan. a cos.'2=41 sin. a cos. a..- X=2H1 sin. 2a... (94).
If the body be projected at different angular elevations,
but with the same velocity, the horizontal range will be the
Ir     Or
greatest when sin. 2a is the greatest, or when 2a=, or a=4.
117. To find the greatest height which a projectile will
attain in itslight if projected with a given velocity, and
at a given inclination to the horizon.
Multiplying both sides of equation.1      (92) by 4t cos.2 a we have 4H cos.2 a
i/. y=4H cos.2 a tan. a. x —x=2H (2
cos. a sin. a)  -x=21 sin2. 2a. -.
/,'   o nSubtracting both sides of this equar.. —. —-- s  c tion from Si2 sin.2 2a, we have
H2 sin2 2a-4H cos.2 a. y==HI sin.2 2a-2H sin. 2a. x+x.
But sin.2 2a=4 sin.2 a cos.'a,.*. 4H cos.2 a IH sin.2 a -y} = iH sin. 2a-x  2... (95).
Now the second member of this equation is always a
positive quantity, being a square. The first member is
therefore always positive; therefore 1H sin.2 a-y is always
positive. Whence it follows that y can never exceed H
sin.2, so that it attains its greatest possible value when it
equals H sin.2 a, a value which it manifestly attains when




104                  PROJECTmLES.
the first member of the above equation vanishes, or when
x=1H sin. 2o., that is, when x becomes equal to half the
greatest horizontal range, as is apparent from the last proposition; so that the greatest height BD of the projectile
is represented by H sin.2 a a height which it attains when
AD equals half the horizontal range.
118. The path of a projectile in vacuo is a parabola.
/'~    -Let B be the highest point in the
flight of the projectile, and BD its
greatest height. Draw PM, perpendicular to BD.   Let BM1,=-, MP.......... — ^..... ^.  1
x1=BD-M1D=BD-PM==H sin.2 a-y,
Y1=DM=AM-AD=x —H sin. 2a.
Substituting these values in equation (95),
1=4H cos.2 a. x..... (96),
which is the equation to a porabola whose vertex is in
B, whose axis coincides with BD, and whose parameter is
4H cos.2.
The path of a projectile in vacuo is therefore a parabola,
whose vertex is at the highest point attained by the projectile, and whose axis is vertical.
119. To find the range of a projectile upon an inclined
plane.
Let R represent the range AP of a projectile upon an
inclined plane AB, whose inclination is
i. Then H and a being taken to repre-._... A: sent the same quantities as before, and...-......., y being the co-ordinates of P to the
horizontal azis AC, we have
x=AM=AP cos. PAM=R cos. I,
y=PM=AP sin. PAM=R sin. i.
Substituting these values of x and y in the general equation (92) to the projectile we have




PROJECTILES.                105:2 coS.2 1 sec.  M
R sin. i== cos. i tan. a —  4Cos e. 
Dividing by R, multiplying by cos. a, and transposing
I cos.2 sec. a 
41i s  =cos.   1 sin. ua-sin., cos. c=sin. (a-),
R=4H- sin ( —,) cos.. (97).
COS. J
Now sin. (2a-s)-sin. =-sin. ac+(oa-i) -sin. la-(ao —
()) =2 sin. (a-i) cos. a.
Substituting this value of 2 sin. (a-i) cos. a in the preceding equation, we have
R=2H    sin. (2a-i)-sin. I.. (98).
{      COS.2.2  f 
Now it is evident that if a be made to vary, remaining
the same, R will attain its greatest value when sin. (2a —)
is greatest, that is when it equals unity, or when 2a —_=
In'          q t^  I
2 or when a=-+-. This, then, is the angle of elevation
corresponding to the greatest range, with a given velocity
upon an inclined plane whose inclination is;.
If in the preceding expression for the range we substitute
i 2-(a —) } for a, the value of the expression will be found
to remain the same as it was before; for sin. (2ca-i) will, by
this substitution, become sin. ~r —2(a-i) —} =sin. \t-(2a
-) =-sin. (2a -t). The value of R remains therefore the
same, whether the angle of elevation be c or 2-(a — )
And the projectile will range the same distance on the
plane, whether it be projected at one of these angles of
elevation or the other.
Let BAC be the inclination of the plane on which the
projectile ranges, and AT the direcD 5s  *t       t0 ion of projection. Take DAS equal
to BAT. Then BAT=TAC-BAC,.../  \      =a —,. And SAC=DAC —DAS=
^:f —:.... — BAT= —(a —). The range AP
is therefoe the same, whether TA  or SA  be the angle of
is therefore the same, whether TAC or SAC be the angle of




106              CENTRIFUGAL FORCE.
elevation, and therefore whether AT or AS be the direction
of projection.
Draw AE bisecting the angle BAD, then the angle EAC
=BAO+BAE=BAO+iBAD=,+i(~ ~-)=           +~
The angle EAC is therefore that corresponding to the
greatest range, and AE is the direction in which a body
should be projected to range the greatest distance on the
inclined plane AB.
It is evident that the directions of projection AS and AT,
which correspond to equal ranges, are equally inclined to
the direction AE corresponding to the greatest range.
120. T7e velocity of a projectile at different points of its
path. It has been shown (Art. 56.), that if a body move in
any curve acted upon by gravity, the work accumulated or
lost is the same as would be accumulated or lost, provided
the body, instead of moving in a curve, had moved in the
direction of gravity through a space equal to the vertical
projection of its curvilinear path.
Thus a projectile moving from A to P will accumulate or
lose a quantity of work, which is equal to that which it would
accumulate or lose, had it moved vertically from M to P, or
from P to ], PM being the projection of its path on the
direction of gravity. Now the work thus accumulated or
lost equals one half the difference between the vires vivce at
the commencement and termination of the motion.
Let V equal the velocity at A, and v equal the velocity at
W W
P, therefore the work =-g-V2 —— v2. Moreover, the work
w      w
g      g
W. PM, therefore V2-v'=2gMP. Let PM=y,: v'=2-gy..... (99),
which determines the velocity at any point of the curve.
CENTRIFUGAL FORCE.
121. Let a body of small dimensions move in any curvi



CENTRIFUGAL FORCE.                    107
linear path AB, impelled continually towards
H     o:s;a given point S (called a centre of force) by a
given force, whose amount, when the body
fi a   I   has reached the point P in its path, is reprej   sented by F.*    Let PQ    be an exceedingly
"  -/  small portion of the path of the body, and
conceive the force F to remain constant and
parallel to itself, whilst this portion of its path is being described. Then, if PR be a tangent at P, and QR be drawn
parallel to SP, PR is the space which the body would have
traversed in the time of describing PQ, if it had moved
with its velocity of projection from P alone, and had not been
attracted towards S, and RQ or PT (QT being drawn parallel to RP) is the space through which it would have fallen
by its attraction towards S alone, or if it had not been projected at all from P.t Let v represent the velocity which
it would have acquired on this last supposition, when it
reached the point T. Therefore (Art. 66.), if w represent the
weight of the body,
FxPT=-v2.
Now the velocity v, which the body would have acquired in
falling through the distance PT by the action of the constant
force F, is equal to double that which would cause it to de
scribe the same distance uniformly in the same time.:
Representing therefore by V the actual velocity of the
body in its path at P, we have
1_PT.                 PT; -.v=2V. pP.
=~-PR;'   *-PR
Substituting this value of v in the preceding equation,
* The force here spoken of, and represented by F, is the moving force, or
pressure on the body (see Art. 92.), and is therefore equal to that pressure
which would just sustain its attraction towards S.
f See Art. 113. (equations 89 and 90); what is proved there of a body acted
upon by the force of gravity which is constant, and whose direction is constantly parallel to itself, is evidently true ff any other constant force similarly
retaining a direction parallel to itself. To apply the same demonstration to
any such case, we have only indeed to assume g to represent another number
than 32*.
If f represent the additional velocity per second which F would communicate to the body, and t the time of describing PT, then (Art. 44.)
v=ft; but (Art. 46.)PT=t2= V=  t=-t; so that  is the velocity with
which PT would be described  formly in the time 
which PT would be described uniformly in the time t.




108               CENTRIFUGAL FORCE.
FxPT=2V2()vPT     2   F2WV2 QR.
PR              (PR)
Now let a circle PQV be described having a common tangent with the curve AB in the point P, and passing through
the point Q. Produce PS to intersect the circumference of
this circle in V, and join QV; then are the triangles PQV
and QPR similar, for the angle RQP is equal to the angle
QPVY(QR and VP being parallel), and the angle QPR is
equal to the angle QVP in the alternate segment of the cirQR PQ              or,(rQ)
cle. Therefore QR-  P; therefore QR.   SubstiPQP.....Y''? — PVY
tuting this value of QR in the last equation, we have
y   PV   \PRNow this is true, however much PQ may be diminished.
Let it be infinitely diminished, the supposed constant amount
and parallel direction of F will then coincide with the actual
case of a variable amount and inclination of that force, the
PQ
ratio ~- will become a ratio of equality, and the circle
PQV. will become the circle of curvature at P, and PV that
chord of the circle of curvature, which being drawn from P
passes through S. Let this chord of the circle of curvature
be represented by C,.F=2.       (100).
gC
The force or pressure F thus determined is manifestly
exactly equal to that force by which the body tends in its
motion continually to fly from the centre S, and may therefore be called its centrifugal force. This term is, however,
generally limited in its application to the case of a body revolving in a circle, and to the force with which it tends to
recede from the centre of that circle; or if applied to the
case of motion in any other curve, then it means the force
with which the body tends to recede from the centre of the
circle of curvature to its path at the point through which it
is, at any time, moving. When the body revolves in a circular path, the circle of curvature to the path at any one
point evidently coincides with it throughout, and the chord
of curvature becomes one of its diameters. Let the radius
of the circle which the body thus describes be represented
by R, then C=2R;




CENTRIFUGAL FORCE.             109:. F=w-V.... (101).
g R.
Since in whatever curve a body is moving, it may be conceived: at any point of its path to be revolving in the circle
of curvature to the curve at that point, the force F, with
which it then tends to recede from the centre of the circle
of curvature is represented by the above formula, R being
taken to represent the radius of curvature at the point of its
path through which it is moving.
If a be the angular velocity of the body's revolution about
the centre of its circle of curvature, then V= aR;
F= -WR..... (10).
122. From equation (100) we obtain
v2_(F-  )C  2Q( )(IC).
Now (Art. 94.)   represents the additional velocity per
second f, which would be communicated to a body falling
towards S, if the body fell freely and the force F remained
constant. Moreover, by Art. 47. it appears, that V is the
whole velocity which the body would on this supposition
acquire, whilst it fell through a distance equal to i0, or to
one quarter of the chord of curvature. Thus, then, the velocity of a body revolving in any curve and attracted towards
a centre of force is, at any point of that curve, equal to that
which it would acquire in falling freely from that point towards the centre of force through one quarter of that chord
of curvature which passes through the centre of force, if the
force which acted apon it at hat point in the curve remained constant during its descent. It is in this sense that
the velocity of a body moving in any curve about a centre
of force is said to be THAT DUE TO ONE QUARTER THE CHORD
OF CURVATRE.
123. The centrifugalforce of a mass of finite dimensions.
y:            Let BC represent a thin lamina or slice
of such a mass contained between two planes
q: +.& )    exceedingly near to one another and bdth,'.::.. perpendicular to a given axis A, about
-...i...........-. which the mass is nade to revolve.




110               CENTRIFUGAL FORCE.
Through A draw any two rectangular axes Ax and Ay,
let m, be any element of the lamina whose weight is wo, and
let AM1 and AN,, co-ordinates of m,, be represented by x,
and y,. Then by equation (102), if a represent the angular
velocity of the revolution of the body, the centrifugal force
on the element mn is represented by- wlAmn. Let now this
g
force, whose direction is Amn be resolved into two others,
whose directions are Ax and Ay. The former will be repre2                       a2
sented by  wAm,. cos. xAm,, or by — w,, and the latter
g                        g
~2    ____C2
byv-   Ami cos. yAml, or by _wyl; and the centrifugal
v                         g
forces and all the other elements of the lamina being similarly resolved, we shall have obtained two sets of forces,
those of the one set being parallel to Ax, and represented
2      c2     2
by - wl1,, _w, _x,- m, &c. and those of the other set
g      g      g
02     a2     2
parallel to Ay represented by -wy,- wy,, a   wy-,y, &c.
g      g     g
Now if X and Y represent the resolved parts parallel to
the directions of Ax and Ay, of the resultant of these two
sets of forces, then (Art. 11.)
a.......2       2a           a2 
X=   zwl,x,      +,... -   -     -lx= "-WG,;
2     - 2       2              2      02
Y= -, wyn, G+ rp ys+   o-y +  wY3 +  zwy=  WG,,
if Gi and G2 represent the co-ordinates AG, and AG, of the
centre of gravity G of the lamina, and W its weight.
(Art. 18.).
Now the whole centrifugal force F on the lamina is the
resultant of these two sets of forces, and is represented by
X-2+Y2 (Art. 11.),
a~4   054        2
*- F-=  -W'2G  +  W2 G2=   -W    /G-     2, or
F t   W. G...... (03),
where G is taken to represent the distance AG of the centre
of gravity of the lamina from the axis of revolution.
Moreover, the direction of this resultant centrifugal force




CENTRIFUGAL FORCE.               111
is through A, since the direction of all its components are
through that point
124. From the above formula, it is apparent that if a body
revolving round a fixed axis be conceived to
be divided into laminae by planes perpendicu-.A-g: —B lar to the axis, then the centrifugal force of
each such laminwe is the same as it would
have been if the whole of its weight had
been collected in its centre of gravity; so
that if the centres of gravity of all the laminae
be in the same plane passing through the
axis, then, since the centrifugal force on each laminahas its
direction from the axis through the centre of gravity of that
lamina, it follows that all the centrifugal forces of these
laminae are in the same plane, and that they are PARALLEL
forces, so that their resultant is equal to their suum, those
being taken with a negative sign which correspond to
laminae whose centres of gravity are on the opposite side of
the axis from the rest, and whose centrifugal forces are
therefore in the opposite directions to those of the rest.
Thus if F' represent the whole centrifugal force of such a
mass, then F' - 2WG. Now let W' represent the weight
of the whole mass, and G' the distance of its centre of gravity from the axis, therefore zWG-=W'G';. F'= -W'G'......      (104).
In the case, then, of a revolving body capable of being
divided into laminae perpendicular to the axis of revolution,
the centres of gravity of all of which laminae are in the
same plane passing through the axis, the centrifugal force is
the same as it would have been if the whole weight of the
body had been collected in its centre of gravity, the same
property obtaining in this case in respect to the whole body
as obtains in respect to each of its individual lamine.
Since, moreover, the centrifugal forces upon the laminae are
parallel forces when their centres of gravity are all in the
same plane passing through the axis of gravity, and since
their directions are all in that plane, it follows (Art. 16.),
that if we take any point O in the axis, and measure the
moments of these parallel forces from that point, and call
a the perpendicular distance OA of any lamina BC from




112       THE PRINCIPLE OF VIRTUAL VELOCITIES.
that point, and H the distance of their resultant from that
point, then
H - WG- zWGx,
g         g.vWGx:. E   WC.       (105)...
The equations (103) and (104) determine the amount and
the point of application of the resultant of the centrifugal
forces upon the mass, upon the supposition that it can be
divided into laminae perpendicular to the axis of revolution,
all of which have their centres of gravity in the same plane
passing through the axis.
It is evident that this condition is satisfied, if the body be
symmetrical as to a certain axis, and that axis be in the
same plane with the axis of revolution, and therefore if it
intersect or if it be parallel to the axis of revolution.
If, in the case we have supposed, zWG=0, that is, if the
centre of gravity be in the axis of revolution, then the centrifugal force vanishes. This is evidently the case where a
body revolves round its axis of symmetry.
125. If the centres of gravity of the laminae into which
the body is divided by planes perpendicular to
the axis of revolution be not in the same plane
(as in the figure), then the centrifugal forces of
the different laminae will not lie in the same
J/   plane, but diverge from the axis in different
j   directions round it. The amount and direction.r ]  of their resultant cannot in this case be deteri.o   mined by the equations which have been given
above.
THE PRINCIPLE OF VIRTUAL VELOCITIES.
126. If any pressure P, whose point of application A is
made to move through the straight line AB, be resolved
into three others X, Y, Z, in the directions of the three
rectangular axes, Ox, Oy, Oz; and if AC, AD, and AE,
be the projections of AB upon these axes, then the work of
P through AB is equal to the sum of the works of X, Y, Z,
through AC, AD, and AE respectively, or X. AC+Y.
+Z. AE=P.AM.




THE PRINCIPLE OF VIRTUAL VELOCITIES.             113,         U        P
Jj /  ~    Let the inclinations of the direction
I:: 1 of P to the axes Oz, Oy, Oz respeci — i:-Jo  tively, be represented by a, /, y, and''''       the inclniations of AB     to the same.. —1 —.    axes by c,, /3, 7,,
y... (Art. 12.) X=P cos., Y=P cos. /, Z=P cos. y; also AC
=AB cos. %, AD=AB cos. /1, AE=AB cos. y,t
X. XAC=P. AB cos. c cos. a, Y. AD=P. AB cos. F cos. P,,
Z. AE==P     AB cos.     cos. y,,.X. A    +Y. AD+Z. AE-P. AB {cos. a cos. a1+cos. /
cos. 3    o. y c os. y cs..
But by a well-known theorem of trigonometry, cos. a cos., +cos.8   cos. c +cos. y cos.       cos.. PAB,..X. AC+Y. AD+Z. AE=P. AB cos. PAB;
but AB cos. PAB-=AM;..X. AC+Y. AD+Z. AE-P. AM.
But (Art. 52.) the work of P through AM        is equal to its
work through AB. Therefore, &c.*
127. If any number of forces be in equilibrium         (being in
any way mechanically connected with one another), and {f,
subject to that connection, their different points of application be made to move, each through any exceedingly small
distance, then the aggregate of the work of those forces,
whose points of application are made to move towards the
* This proposition may readily be deduced from Art. 53., for pressures equal
and opposite to X, Y, Z, would just be in equilibrium with P, and these tending to move the point A in one direction along the line AB, P tends to move
it in the opposite direction, therefore in the motion of the point A through AB,
the sum of the works of X, Y, Z, must equal the work of P. But the work of
X, as its point of application moves through AB, is equal (Art. 52.) to the
work of X through the projection of AB upon Ax, that is, through AC; similarly the work of Y, as its point of application moves through AB, is equal to
its work through the projection of AB upon Ay, or through AD; and so of Z.
The sum of the works of X, Y, and Z, as their point of application is made to
move through AB, is therefore equal to what would have been the sum of their
works had their points of application been made to move separately through
AC, AD, AE; this last sum is therefore equal to the work of P through AB,
which is equal to the work of P through AM, AM being the projection of AB;
upon the direction of P.
8




114          PRINCIPLE OF VIRTUAL VELOCITIES.
directions in which the several forces applied to them act,
shall equal the aggregate of the work of those forces, the
motions of whose points of application are opposed to the
directions of the forces applied to thewm.
For let all the forces composing such a system be resolved into three sets of forces parallel to three rectangular
axes, and let these three sets of parallel forces be represented by A, B, and C respectively. Then must the resultant of the parallel forces of each set equal nothing. For if
any of these resultants had a finite value, then (by Art. 12.)
the whole three sets of forces would have a resultant, which
they cannot, since they are in equilibrium.
Now let the motion of the points of application of the
forces be conceived so small that the amounts and directions,of the forces may be made to vary, during the motion, only'
iby an exceedingly small quantity, and so that the resolved
forces upon any point of application may remain sensibly
>unchanged. Also let ul, u2, u, represent the works of these
resolved forces respectively on any point, and 2u1 the sum
lof all the works of the resolved forces of the set A, u,2 the:sun of all the works of the forces of the set B, and 2, of the
set C. Now since the parallel forces of the set A have no
resultant, therefore (Art. 59.) the sum of the works of those
forces of this set, whose points of application are moved;towards the directions of their forces, is equal to the sum of
the works of those whose points of application are moved
froom the directions of their forces, so that u,_-o0, if the
values of u, which compose this sum, be taken with the
positive or negative sign, according to the last mentioned.condition.
Similarly, 2u -0  and 23,=,.. 2(1- +f   + U2- 0.
Now let U represent the actual work of that force P1, the
~works of whose components parallel to the three axes are
represented by ul, u,, u%,; then by the last proposition,
U I + f2 +  3 — U..   =o0.... (106);
in which expression U is to be taken positively or negatively
according to the same condition as u,, u,, u,3; that is, according as the work at each point is done in the direction of the
corresponding force, or in a direction opposite to it. Hence
therefore it follows, from the above equations, that the sum




THE PRINCIPLE OF VIS VIVA.            115
of the works in one of these directions equals their sum in
the opposite direction. Therefore, &c.
The projection of the line described by the point of application of any force upon the direction of that force is called
its VIRTUAL VELOCITY, SO that the product of the force by its
virtual velocity is in fact the work of that force; hence
therefore, representing any force of the system by P, and
its virtual velocity by p, we have Pp=U, and therefore,
zPp=0, which is the principle of virtual velocities.*
128. If there be a system offorces such that their points of
cpplication being moved through certain consecutive positions, those forces are in all such positions in equilibrium,
then in respect to any finite motion of the points of application through that series of positions, the aggregate of the
work of those, forces, which act in the directions in which
their several poits of application are made to move, is equal
to the aggregate of the work in the opposite direction.
This principle has been proved in the preceding,proposition, only when the motions communicated to the several
points of application are exceedingly small, so that the work
done by each force is done only through an exceedingly
small space. It extends, however, to the case in which each
point of application is made to move, and the work of each
force to be done, through any distance, however great, provided only that in all the different positions which the points
of application of the forces of the system are thus made to
take up, these forces be in equilibrium with one another; for
it is evident that if there be a series of such positions
immediately adjacent to one another, then the principle
obtains in respect to each small motion from one of this set
of positions into the adjacent one, and therefore in respect
to the sum of all such small motions as may take place in the
system in its passage from any one position into any other,
that is, in respect to the whole motion of the system through
the intervening series of positions. Therefore, &c.
THE PRINCIPLE OF VIS VIVA.
129. If the forces of any system be not in equilibrium with
one another, then the difference between the aggregate work
* This proof of the principle of virtual velocities is given here for the first
time.




116            1ITHEI PRINCIPLE OF VIS VIVA.
of those whose tendency is in the direction of the motions
of their several points of application, and those whose tendency is in the opposite direction, is equal to one half the
aggregate vis viva of the system.
In each of the consecutive positions which the bodies composing the system are made successively to take up, let there
be applied to each body a force equal to the efective force
(Art. 103.) upon that body, but in an opposite direction;
every position will then become one of equilibrium.
Now, as the bodies which compose the system and the
various points of application of the impressed forces move
through any finite distances from one position into another,
let zu, represent the aggregate work of those impressed
forces whose directions are towards the directions of the
motions of their several points of application, and let 2u,
represent the work of those impressed forces which act in the
opposite directions; also let z2u represent the aggregate
work of forces applied to the system equal and opposite to
the efective forces upon it; the directions of these forces
opposite to the effective forces are manifestly opposite also
to the directions of the motions of their several points of
application, so that on the whole zu, is the aggregate work
of those forces whose directions are towards the motions of
their several points of application, and  2,+ itt, the aggregate work opposed to them. Since therefore, by D'Alembert's principle, an equilibrium obtains in every consecutive
position of the system, it follows by the last proposition,
that
Mu 1 =u2 + mu3,.   u. l- C1ub= zu3..... (107).
Now u, is taken to represent the work of a force equal and
opposite to the effective force upon any body of the system;
but the work of such a force through any space is equal to
the work which the effective force (being unopposed) accumulates in the body through that space (Art. 69.), or it is
equal to one half the difference of the vires vivse of the body
at the commencement and termination of the time during
which that space is described (Art. 67.). Therefore 2;u,
equals one half the aggregate difference of the vires vivw of
the system at the two periods;
1
m..u- u,.   - W(v,-v2).....    ( os).
I      2g




POSITION OF MAXIMUM OR MINIMUM VIS VIVA.    117
Thus then it follows, that the difference between the aggregate work zu, of those forces, the tendency of each of which
is towards the direction of the motion of its point of application, and that zu2 of those the direction of each of which is
opposed to the motion of its point of application (or, in other
words the difference between the aggregate work of the
accelerating forces of the system and that of the retarding
forces), is equal to one half the vis viva accumulated or lost
in the system whilst the work is being done, which is the
PRINCIPLE OF VIS VIVA.
130. One half the vis viva of the system measures its
accumulated work; the principle of vis viva amounts,
therefore, to no more than this, that the entire difference
between the work done by those forces which tend to accelerate the motions of the parts of the system to which they
are applied, and those which tend to retard them, is accumulated in the moving parts of the system, no work
whatever being lost, but all that accumulated which is done
upon it by the forces which tend to accelerate its motion,
above that which is expended upon the retarding forces.
This principle has been proved generally of any mechanical system; it is therefore true of the most complicate i
machine. The entire amount of work done by the movin 
power, whatever it may be, upon that machine, is yielde I
partly at its working points in overcoming the resistances
opposed there to its motion (that is, in doing its useful
work), it is partly expended in overcoming the friction anrl
other prejudicial resistances opposed to the motion of the
machine between its moving and its working points, and all
the rest is accumulated in the moving parts of the machine,
ready to be yielded up under any deficiency of the moving
power, or to carry on the machine for a time, should the
operation of that power be withdrawn.
131. When the forces of any system (not in equilibrium? irn
every position which the parts of that system may be
made to take up) pass through a position of equilibrium,
the vis viva of the system becomes a maximum    or a
minimum.
For, as in Art. 129., let the aggregate work done in the
directions of the motions of the several parts of the system




118    POSITION OF MAXIMUMt OR MINIMUM VIS VIVA.
be represented by uz,, and the aggregate work done in
directions opposed to the motions of the several parts by
u2, then (Art. 129.), one half the acquired vis viva of
system =-u-u - 2.  Now as the system passes from any one
position to any other, each of the quantities xu1 and 2?u is
manifestly increased. If Mu, increases by a greater quantity than Mu2, then the vis viva is increased in this change
of position; if, on the contrary, it is increased by a less
quantity than Xu2, then the vis viva is diminished. Thus if
Az2u  and Azz2 represent the increments of 2u1 and  u,2 in
this change of position, then (2lu + Azw,)- (2 + uA:2), or
(u1 —Xu2) + (^z1-X — a2), representing one half the vis
viva after this change of position, it is manifest that the vis
viva is increased or diminished by the change according as
al2 is greater or less than zu2; and that if Azul be equal
to Azu2 then no change takes place in the amount of the
vis viva of the system as it passes from the one position to
the other.
Now from the principle of virtual velocities (Art. 127.),
it appears, that precisely this case occurs as the system
passes through a position of equilibrium, the aggregate
work of those forces whose tendency is to accelerate the
motions of their points of application then precisely equalling that of the forces whose tendency is opposed to these
motions. For an exceeding small change of position therefore, passing through a position of equilibrium, A2z=axu2,
an equality which does not, on the other hand, obtain,
unless the body do thus pass through a position of equilibriun.
Since then the sum u21,-u2, and therefore the aggregate
vis viva of the system, continually increases or diminishes
up to a position of equilibrium, and then ceases (for a certain finite space at least) to increase or diminish, it follows,
that it is in that position a maximum   or a minimum.
Therefore, &c.
132. When the forces of any system pass through a cosition
of equilibrium, the vis viva becomes a maximum or a
minimum, according as that position is one of stable or
unstable equilibrium.
For it is clear that if the vis viva be a maximum in any
position of the equilibrium of the system, so that after it
has passed out of that position into another at some finite




STABLE AND UNSTABLE EQUILIBRIUM.          119
distance from it, the acquired vis viva may have become
less than it was before, then the aggregate work of the
forces which tend to accelerate the motion between these
two positions must have been less than that of the forces
which tend to retard the motion (Art. 131.). Now suppose
the body to have been placed at rest in this position of
equilibrium, and a small impulse to have been communicated to it, whence has resulted an aggregate amount of
vis viva represented by nmY2. In the transition from the
first to the second position, let this vis viva have become:mv2; also let the aggregate work of the forces which have
tended to accelerate the motion, between the two positions,
be represented by.U1, and that of the forces which have
tended to retard the motion by YU2; then, for the reasons
assigned above, it appears that TU2 is greater than 2U1.
Moreover, by the principle of vis viva,
ma-SmY V2 =zU1 -U2,... zmv'=zmV2 —2(U2 -UU,);
in which equation the quantity 2(zU,2 -  U,) is essentially
positive, in respect to the particular range of positions
through which the disturbance is supposed to take place.*
For every one of these positions there must then be a
certain value of -mVz, that is, a certain original impulse
and disturbance of the system from its position of equilibrium, which will cause the second member of the above
equation, and therefore its first member zmV2, to vanish.
Now every term    of the sum  mqnv2 is essentially positive;
this sum  cannot therefore vanish unless each term  of it
vanish, that is, unless the velocity of each body of the
system vanishes, or the whole be brought to rest.    This
repose of the system can, however, only be instantaneous;
for, by supposition, the position into which it has been displaced is not one of equilibrium. Moreover, the displacement of the system cannot be continued in the direction in
which it has hitherto taken place, for the negative term in
the second member of the above equation would yet farther
be increased so as to exceed the positive term, and the first
a The disturbance is of course to be limited to that particular range of
positions to which the supposed position of equilibrium stands in the relation
of a position of maximum vis viva. If there be other positions of equilibrium of the system, there will be other ranges of adjacent positions, in
respect to each of which there obtains a similar relation of maximum or minimum vis viva.




120        STABLE AND UNSTABLE EQUILIBRIUM.
member.mv2 would thus become negative, which is
impossible.
The motion of the system can then only be continued by
the directions of the motions of certain of the elements
which compose it being changed, so that the corresponding
quantities by which zU1 and zU2 are respectively increased
may change their signs, and the whole quantity zU1 - 2zU
which before increased continually may now continually
diminish. This being the case,.mv2 will increase again
until, when zU, - 2zU2-0, it becomes again equal to:mV2;
that is, until the system acquires again the vis viva with
which its disturbance commenced.
Thus, then, it has been shown, that in respect to every
one of the supposed positions of the system' there is a certain impulse or amount of vis viva, which being commumicated to the system when in equilibrium, will just cause it
to oscillate as far as that position, remain for an instant at
rest in it, then return again towards its position of equilibrium, and re-acquire the vis viva with which its displacement commenced. Now this being true of every position
of the supposed range of positions, it follows that it is true
of every disturbance or impulse which will not carry the
system beyond this supposed range of positions; so that,
having been displaced by any such disturbance or impulse,
the system will constantly return again towards the position
of equilibrium from which it set out, and is STABLE in
respect to that position.
On the other hand, if the supposed position of equilibrium be one in which the vis viva is a minimum, then the
aggregate work of the forces which tend to accelerate the
motion must, after the system has passed through that position, exceed that of the forces which tend to retard the
motion; so that, adopting the same notation as before, zU,
must be greater than.U2, and the second member of the
equation essentially positive. Whatever may have been the
original impulse, and the communicated vis viva zmV2,
2mV2 must therefore continually increase; so that the whole
system can never come to a position of instantaneous repose;t
but on the contrary, the motions of its parts must continuously increase, and it must deviate continually farther from
its position of equilibrium, in which position it can never
* That is, of that range of positions over which the supposed position of
equilibrium holds the relation of a position of maximum vis viva.
t Within that range of positions over which the supposed position of
equilibrium holds the relation of minimum vis viva.




DYNAMICAL STABILITY.                121
rest.  The position is thus one of unstable equilibrium.
Therefore, &c.
DYNAMICAL STABILITY.*
If a body be made, by the action of certain disturbing
forces, to pass from one position of equilibrium into another,
and if in each of the intermediate positions these forces are
in excess of the forces opposed to its motion, it is obvious
that, by reason of this excess, the motion will be continually
accelerated, and that the body will reach its second position
with a certain finite velocity, whose effect (measured under
the form of vis viva) will be to carry it beyond that position.
This however passed, the case will be reversed, the resistances will be in excess of the moving forces, and the body's
velocity being continually diminished and eventually destroyed, it will, after resting for an instant, again return
towards the position of equilibrium  through which it had
passed. It will not however finally rest in this position until
it has completed other oscillations about it. Now the amplitude of the first oscillation of the body beyond the position in which it is finally to rest, being its greatest amplitude of oscillation, involves practically an important condition of its stability; for it may be an amplitude sufficient to
carry the body into its next adjacent position of equilibrium,
which being, of necessity, a position of unstable equilibrium,
the motion will be yet further continued and the body
overturned. Different bodies requiring moreover different
amounts of work to be done upon them to produce in all the
same amplitude of oscillation, that is (relatively to that amplitude) the most stable which requires the greatest amount
of work to be so done upon it. It is this condition of stability, dependent upon dynamical considerations, to which, in
the following paper, the name of dynamical stability is
given.
I cannot find that the question has before been considered
in this point of view, but only in that which determines
whether any given position be one of stable, unstable, or
mixed equilibrium; or which determines what pressure is
necessary to retain the body at any given inclination from
such a position.
* Extracted from a paper " On Dynamical Stability, and on the Oscillations
of Floating Bodies," by the author of this work, published in the Transactions
of the Royal Society, Part. II. for 1850. The remainder of the paper will be
found in the Appendix.




122              DYNAMICAL STABILITY.
1. To the discussion of the conditions of the dynamical
stability of a body the principle of vis viva readily lends
itself.  That principle,* when translated into a language
which the labours of M. PONCELET have made familiar to
the uses of practical science, may be stated as follows:" When, being acted upon by given forces, a body or system of bodies has been moved from a state of rest, the difference between the aggregate work of those forces whose
tendencies are in the directions in which their points of
application have been moved, and that of the forces whose
tendencies are in the opposite direction, is equal to one-half
the vis viva of the system."
Thus, if 2u, be taken to represent the aggregate work of
the forces by which a body has been displaced from a position in which it was at rest, and Y22 the aggregate work
(during this displacement) of the other forces applied to it;
and if the terms which compose:u, and:2; be understood
to be taken positively or negatively, according as the tendencies of the corresponding forces are in the directions in
which their points of application have been made to move
or in the opposite directions; then representing the aggregate vis viva of the body by -~ wv2.
2 + z    2   2 w',..... (1').
2g
Now:u2 representing the aggregate work of those forces
which acted upon the body in the position from which it has
been moved, may be supposed to the known; zzu may therefore be determined in terms of the vis viva, or conversely.
2. In the extreme position into which the body is made to
oscillate and from which it begins to return, it, for an instant,
rests. In this position, therefore, its vis viva disappears, and
we have
ZY + Z   0=......  (2').
This equation, in which zu and z2% are functions of the
impressed forces and of the inclination, determines the extreme position into which the body is made to roll by the
action of given disturbing forces; or, conversely, it determines the forces by which it may be made to roll into a
given extreme position.
* See Art. 129.




DYNAMICAL STABILITY.               123
3. The position in which it will finally rest is determined
by the maximum value of au + u2 in equation (1'); for, by
a well-known property, the vis viva of a system  attains a
maximum value when it passes through a position of stable,
and a minimum, when it passes through a position of unstable
equilibrium. The extreme position into which the body
oscillates is therefore essentially different from that in which
it will finally rest.
4. Different bodies, requiring different amounts of work to
be done upon them to bring them to the same given inclination, that is (relatively to that inclination) the most stable
which requires the greatest amount of work to be so done
upon it, or in respect to which z12 is the greatest. If, instead of all being brought to the same given inclination, each
is brought into a position of unstable equilibrium, the corresponding value of 2z, represents the amount of work which
must be done upon it to overthrow it, and may be considered
to measure its absolute, as the former value measures its
relative dynamical stability.t  The absolute dynamical stability of a body thus measured I propose to represent by the
symbol U, and its relative dynamical stability, as to the
inclination 0, by U(6).
The measure of the absolute dynamical stability of a body
is the maximum value of its relative stability, or U the maximum of U((); for whilst the body is made to incline from
its position of stable equilibrium, it continually tends to
return to it until it passes through a position of unstable
equilibrium, when it tends to recede from it; the aggregate
amount of work necessary to. produce this inclination must
therefore continually increase until it passes through that
position and afterwards diminish.
5. The work opposed by the weight of a body to any
change in its position is measured by the product of the
vertical elevation of its centre of gravity by its weight.;
Representing therefore by W   the weight of the body, and
by AH the vertical displacement of its centre of gravity
when it is made to incline through an angle 0, and observing that the displacement of this point is in a direction opposite to that in which the force applied to it acts, we have
u== —W.AH, and by equation (2'),
* Art. 132.
I It is obvious that the absolute dynamical stability of a body may be
greater than that of another, whilst its stability, relatively to a given inclination, is less; less work being required to incline it than the other at that
angle, but more, entirely to overthrow it.
t Art. 60.




124                    FRICTION.
I ()-W.AH=O0.... (3).
If therefore no other force than its weight be opposed to a
body's being overthrown, its absolute dynamical stability,
when resting on a rigid surface, is measured by the product
of its weight by the height through which its centre of gravity
must be raised to bring it fromC a stable into an unstable
position of equilibrium.
6. The Dynamical Stability of Floating Bodies.-The
action of gusts of wind upon a ship, or of blows of the sea,
being measured in their effects upon it by their work, that
vessel is the most stable under the influence of these, or will
roll and pitch the least (other things being the same), which
requires the greatest amount of work to be done upon it to
bring it to a given inclination; or, in respect to which the
relative dynamical stability U (t) is the greatest for a given
value of 6. In another sense, that ship may be said to be the
most stable which would require the greatest amount of work
to be done upon it to bring it into a position from which it
would not again right itself, or whose absolute dynamical
stability U is the greatest. Subject to the one condition,
the ship will roll the least, and subject to the other, it will
be the least likely to roll over.
Thus the theory of dynamical stability involves a question
of naval construction. It will be found discussed in its application to this question in the Appendix.
FRICTION.
133. It is a matter of constant experience, that a certain
resistance is opposed to the motion of one body on the surface of another under any pressure, however smooth may be
the surfaces of contact, not only at the first commencement,
but at every subsequent period of the motion; so that, not
only is the exertion of a certain force necessary to cause the
one body to pass at first from a state of rest to a state of motion upon the surface of the other, but that a certain force is
further requisite to keep p this state of motion. The resistance thus opposed to the motion of one body on the surface
of another when the two arepressed together, is called fric



FRICTION.                     125
tion; that which opposes itself to the transition fiom a state
of continued rest to a state of motion is called the friction
of quiescence; that which continually accompanies the state
of motion is called the friction of motion.
The principal experiments on friction have been made by
Coulomb*, Vince, G. Renniet, N. WoodT, and recently
(at the expense of the French Government) by Morin.~
They have reference, first, to the relation of the friction
of quiescence to the friction of motion; secondly, to the
variation of the friction of the same surfaces of contact under
different pressures; thirdly, to the relation of the friction to
the extent of the surface of contact; fourthly, to the relation
of the amount of the friction of motion to the velocity of the
motion; fifthly, to the influence of unguents on the laws of
friction, and on its amount under the same circumstances of
pressure and contact. The following are the principal facts
which have resulted from these experiments; they constitute the laws of friction.
Ist. That the friction of motion is subject to the same
laws with the friction of quiescence (about to be stated), but
agrees with them more accurately.    That, under the same
circumstances of pressure and contact, it is nevertheless different in amount.
2ndly. That, when no unguent is interposed, the friction
of any two surfaces (whether of quiescence or of motion) is
directly proportional to the force with which they are pressed
perpendicularly together (up to a certain limit of that pressure per square inch), so that, for any two given surfaces
of contact, there is a constant ratio of the friction to the perpendicular pressure of the one surface upon the other,
Whilst this ratio is thus the same for the same surfaces of
contact, it is different for different surfaces of contact. The
particular value of it in respect to any two given surfaces
of contact is called the CO-EFFICIENT of friction in respect to those surfaces. The co-efficients of fiiction in respect
to those surfaces of contact, which for the most part form the
moving surfaces in machinery, are collected in a table, which
will be found at the termination of Art. 140.
3rdly. That, when no unguent is interposed, the amount
of the friction is, in every case, wholly independent of the
extent of the surfaces of contact, so that the force with which
two surfaces are pressed together being the same, and
* Mem. des Sav. Etrang. 1781.         f Phil. Trans. 1829.
A Practical Treatise on Rail-roads, 3d ed. chap. 76.
Mem. de l'Institut. 1833, 1834, 1838.




126                      FRICTION.
not exceeding a certain limit (per square inch), their friction
is the same whatever may be the extent of their surfaces of
contact.
4thly. That the friction of motion is wholly independent
of the velocity of the motion.*
5thly. That where unguents are interposed, the co-efficient
of friction depends upon the nature of the unguent, and upon
the greater or less abundance of the supply. In respect to
the supply of the ungent, there are two extreme cases, that
in which the surfaces of contact are but slightly rubbed with
the unctuous mattert, and that in which, by reason of the
abudant supply of the unguent, its viscous consistency, and
the extent of the surfaces of contact in relation to the insistent pressure, a continuous stratum of unguent remains continually interposed between the moving surfaces, and the
friction is thereby diminished, as far as it is capable of being
diminished, by the interposition of the particular unguent
used. In this state the amount of friction is found (as might
be expected) to be dependent rather upon the nature of the
unguent than upon that of the surfaces of contact; accordingly M. Morin, from the comparison of a great number of
results, has arrived at the following remarkable conclusion,
easily fixing itself in the memory, and of great practical
value:-" that with unguents. hog's lardc and olive oil, interposed in a continuous stratum between them, surfaces of wood
on metal, wood on wood, metal on wood, and metal on metal
(when in motion), have all of them very nearly the same coefficient of friction, the value of that co-efficient being in all
cases included between -07 and'08.' For the unguent tallow, the co-efficient is the same as for
the other unguents in every case, except in that of metals upon
metals.  This unguent appears, from the experiments of Morin, to be less suited to metallic substances than the others,
and gives for the mean value of its co-efficient under the same
circumstances'10."
134. Whilst there is a remarkable uniformity in the results
thus obtained in respect to the friction of surfaces, between
which a perfect separation is effected throughout their whole
extent by the interposition of a continuous stratum of the
* This result, of so much importance in the theory of machines, is fully established by the experiments of Morin.
f As, for instance, with an oiled or greasy cloth.




FRICTION.                     12T
unguent, there is an infinite variety in respect to those states
of unctuosity which occur between the extremes, of which
we have spoken, of surfaces merely unctuous* and the most
perfect state of lubrication attainable by the interposition
of a given unguent. It is from this variety of states of the
unctuosity of rubbing surfaces, that so great a discrepancy
has been found in the experiments upon friction with unguents, a discrepancy which has not probably resulted so much
from a difference in the quantity of the unguent supplied to
the rubbing surfaces in different experiments, as in a difference of the relation of the insistent pressures to the extent
of rubbing surface. It is evident, that for every description
of unguent there must correspond a certain pressure per
square inch, under which pressure a perfect separation of
two surfaces is made by the interposition of a continuous
stratum of that unguent between them, and which pressure
per square inch being exceeded, that perfect separation cannot be attained, however abundant may be the supply of the
unguent.
The ingenious experiments of Mr. Nicholas Woodt, coInfirmed by those of Mr. G. Rennie:, have fully established
these important conditions of the friction of unctuous surfaces.
It is much to be regretted that we are in possession of no
experiments directed specially to the determination of that
particular pressure per square inch, which corresponds in
respect to each unguent to the state of perfect separation,
and to the determination of the co-efficients of frictions in
those different states of separation which correspend to pressures higher than this.
It is evident, that where the extent of the surface sustaining a given pressure is so great as to make the pressure per
square inch upon that surface less than that which corresponds to the state of perfect separation, this greater extent of
surface tends to increase the friction by reason of that cadhesiveness of the unguent, dependent upon its greater or less
viscosity, whose effect is proportional to the extent of the
surfaces between which it is interposed. The experiments
of Mr. Wood~ exhibit the effects of this adhesiveness in a
remarkable point of view.
* Or slightly rubbed with the unguent.
t Treatise on Rail-roads, 3rd ed. p. 399.
Trans. Royal Soc. 1829.
i It is evident that, whilst by extending the unctuous surface which sustains
any given pressure, we diminish the co-efficient of friction up to a certain
limit, we at the same time increase that adhesion of the surfaces which results




128                         FRICTION.
It is perhaps deserving of enquiry, whether in respect to
those considerable pressures under which the parts of the
larger machines are accustomed to move upon one another,
the adhesion of the unguent to the surfaces of contact, and
the opposition presented to their motion by its viscidity, are
causes whose influence may be altogether neglected as compared with the ordinary friction.        In the case of lighter
machinery, as for instance that of clocks and watches, these
considerations evidently rise into importance.
135. The experiments of M. Morin show the friction of
two surfaces which have been for a considerable time in contact, to be not only different in its amount from     the friction
of surfaces in continuous motion, but also, especially in this,
that the laws of friction (as stated above) are, in respect to
the friction of quiescence, subject to causes of variation and
uncertainty from    which the friction of motion is exempt.
This variation does not appear to depend upon the extent of
the surfaces of contact, in which case it might be referred to
adhesion; for with different pressures the co-efficient of the
friction of quiescence was found, in certain cases, to vary
exceedingly, although the surfaces of contact remained the
same.*    The uncertainty which would have been introduced
into every question of construction by this consideration, is
removed by a second very important fact developed in the
course of the same experiments. It is this, that by the
slightest jar or shock of two bodies in contact, their friction
is made to pass from that state which accompanies quiescence
from the viscosity of the unguent, so that there may be a point where the gain
on the one hand begins to be exceeded by the loss on the other, and where
the surface of minimum resistance under the given pressure is therefore
attained.
Mr. Wood considers the pressure per square inch, which corresponds to the
minimum resistance, to be 90lbs. in the case of axles of wrought iron turning
upon cast iron, with fine neat's foot oil. The experiments of Mr. Wood, whilst
they place the general results stated above in full evidence, can scarcely however be considered satisfactory as to the particular numerical values of the constants sought in this inquiry. In those experiments, and in others of the same
class, the amount of friction is determined from the observed space or time
through which a body projected with a given velocity moves before all its
velocity is destroyed, that is, before its accumulated work is exhausted. This
is an easy method of experiment, but liable to many inaccuracies. It is much
to be regretted that the experiments of Morin were not extended to the friction of unctuous surfaces, reference being had to the pressure per square
inch.
* Thus in the case of oak upon oak with parallel fibres, the co-efficient of
the friction of quiescence varied, under different pressures upon the same surface, from'55 to'76.




FRICTION.                   129
to that which accompanies motion; and as every machine or
structure, of whatever kind, may be considered to be subject
to such shocks or imperceptible motions of its surfaces of
contact, it is evident that the state of friction to be made
the basis on which all questions of statics are to be determined, should be that which accompanies continuous motion.
The laws stated above have been shown, by the experiments
of Morin, to obtain, in respect to that friction which accompanies motion, with a precision and uniformity never before
assigned to them; they have given to all our calculations in
respect to the theory of machines (whose moving surfaces
have attained their proper bearings and been worn to their
natural polish) a new and unlooked-for certainty, and may
probably be ranked amongst the most accurate and valuable
of the constants of practical science.
It is, however, to be observed, that all these experiments
were made under comparatively small insistent pressures as
compared with the extent of the surface pressed (pressures
not exceeding from one to two kilogrammes per square centimeter, or from about 14'3 to 28-6 lbs. per square inch.) In
adopting the results of M. Morin, it is of importance to bear
this fact in mind, because the experiments of Coulomb, and
particularly the excellent experiments of Mr. G. Rennie, carried far beyond these limits of insistent pressure*, have fully
shown the co-efficient of the friction of quiescence to increase
rapidly, from some limit attained long before the surfaces
abrade. In respect to some surfaces, as, for instance, wrought
iron upon wrought iron, the co-efficient nearly tripled itself
as the pressure advanced to the limits of abrasion. It is
greatly to be regretted that no experiments have yet been
directed to a determination of the precise limit about which
this change in the value of the co-efficient begins to take
place. It appears, indeed, in the experiments of Mr. Rennie in respect to some of the soft metals, as, for instance, tin
upon tin, and tin upon cast iron; but in respect to the harder
metals, his experiments passing at once from a pressure of
32 lbs. per square inch to a pressure of 1*66 cwt. per square
inch, and the co-efficient (in the case of wrought iron for instance) from about -148 to'25, the limit which we seek is
lost in the intervening chasm. The experiments of Mr. Rennie have reference, lowever, only to the friction of quiescence. It seems probable that the co-efficient of the fric* Mr. Rennie's experiments were carried, in some cases, to from 5 cwt. to
V cwt. per square inch.
9




130                    FRICTION.
tion of motion remains constant under a wider range of pressure than that of quiescence. It is moreover certain, that
the limits of pressure beyond which the surfaces of contact
begin to destroy one another or to abrade, are sooner reached
when one of them is in motion upon the other, than when
they are at rest: it is also certain that these limits are not independent of the velocity of the moving surface. The discussion of this subject, as it connects itself especially with
the friction of motion, is of great importance; and it is to be
regretted, that, with the means so munificently placed at his
disposal by the French Government, M. Morin did not extend his experiments to higher pressures, and direct them
more particularly to the circumstances of pressure and velocity under which a destruction of the rubbing surfaces first
begins to show itself, and to the amount of the destruction
of surface or wear of the material which corresponds to the
same space traversed under different pressures and different
velocities. Any accurate observer who should direct his
attention to these subjects would greatly promote the interests of practical science.
SUMMARY OF THE LAWS OF FRICTION.
136. From what has here been stated it results, that if P
represent the perpendicular or normal force by which one
body is pressed upon the surface of another, F the friction of
the two surfaces, or the force, which being applied parallel
to their common surface of contact, would cause one of them
to slip upon the surface of the other, andf the co-eficient of
friction, then, in the case in which no unguent is interposed,
f represents a constant quantity, and (Art. 133.)
F=fP....(109);
a relation which obtains accurately in respect to the friction
of motion, and approx'mately in respect to the friction of
quiescence.
137. The same relation obtains, moreover, in respect to
unctuous surfaces when merely rubbed with the unguent, or
where the presence of the unguent has no other influence
than to increase the smoothness of the surfaces of contact
without at all separating them from one another.
In unctuous surfacespartially lubricated, or between which




THE LIMITING ANGLE OF RESISTANCE.       131
a stratum of unguent is partially interposed, the co-efficient
of frictionaf is dependent for its amount upon the relation of
the insistent pressure to the extent of the surface pressed,
or upon the pressure per square inch of surface.  This
amount, corresponding to each pressure per square inch in
respect to the different unguents used in machines, has not
yet been made the subject of satisfactory experiments.
The amount of the resistance F opposed to the sliding of
the surfaces upon one another is, moreover, as well in this
case as in that of surfaces perfectly lubricated, influenced by
the adhesiveness of the unguent, and is therefore dependent
upon the extent of the adhering surface; so that, if S represent the number of square units in this surface, and a the
adherence of each square unit, then aS represents the whole
adherence opposed to the sliding of the surfaces, and
F=fP+aS..... (110);
where fis a function of the pressure per square unit S- and
c is an exceedingly small factor dependent on the viscosity
of the unguent.
THE LIMITING ANGLE OF RESISTANCE.
We shall, for the present, suppose the parts of a solid body
to cohere so firmly, as to be incapable of separation by the
action of any force which may be impressed upon them.
The limits within which this suposition is true will be discussed hereafter.
It is not to this resistance that our present inquiry has
reference, but to that which results from the friction of the
surface of bodies on one another, and especially to the direction of that resistance.
138. Any pressure applied to the surface of an imnmoveable
solid body by the intervention of anot/ter body moveable
upon it, will be sustained by the resistance qf the surfaces
of contact, whatever be its direction, provided only the angle uihich that direction makes with the perpendicular to
the surfaces of contact do not exceed a certain angle called
the LIMITING ANGLE OF RESISTANCE of those SURFACES.




132        TIE LIMITING ANGLE OF RESISTANCE.
This is true, however great the pressure may be. Also, if
the inclination of the pressure to the perpendicular exceed
the limiting angle of resistance, then this pressure will not
be sustained by the resistance of the surfaces of contact;
and this is true, however small the pressure may be.
Let PQ represent the direction in which the surfaces of
A   two bodies are pressed together at Q, and let
A    QA be a perpendicular or normal to the sur-.4 ZBk  faces of contact at that point, then will the pres-'-*..sure PQ be sustained by the resistance of the
surfaces, however great it may be, provided its
direction lie within a certain given angle AQB,
called the limiting angle of resistance; and it will not be sustained, however small it may be, provided its direction lie
without that angle. For let this pressure be represented by
PQ, and let it be resolved into two others AQ and RQ, of
which AQ is that by which it presses the surfaces together
perpendicularly, and RQ that by which it tends to cause
them to slide upon one another, if therefore the friction F
produced by the first of these pressures exceed the second
pressure RQ, then the one body will not be made to slip
upon the other by this pressure PQ, however great it may
be; but if the friction F, produced by the perpendicular
pressure AQ, be less than the pressure RQ, then the one
body will be made to slip upon the other, however small PQ
may be. Let the pressure in the direction PQ be represented by P, and the angle AQP by 6, the perpendicular
pressure in AQ is then represented by P cos. 0, and therefore
the friction of the surfaces of contact byfP cos. 0, f representing the co-efficient of friction (Art. 136.). Moreover, the
resolved pressure in the direction RQ is represented by P
sin. 6. The pressure P will therefore be sustained by the
friction of the surfaces of contact or not, according as
P sin. 6 is less or greater than fP cos. 0;
or, dividing both sides of this inequality by P cos. 8, ac
cording as
tan. B is less or greater than f.
Let, now, the angle AQB equal that angle whose tangent is
f, and let it be represented by /, so that tan. =-f. Substituting this value of f in the last inequality, it appears that
the pressure P will be sustained by the friction of the surfaces of contact or not, according as




THE TWO STATES BORDERING UPON MOTION.    133
tan. 0 is greater or less than tan. p,
that is, according as
0 is less or greater than b,
or according as
AQP is less or greater than AQB.
Therefore, &c.                             [Q. E. D.]
THE CONE OF RESISTANCE.
139. If the angle AQB be conceived to revolve about the
axis AQ, so that BQ may generate the surface of,/\A  a cone BQC, then this cone is called the CONE OF
R /  RESISTANCE: it is evident, that any pressure, howQ\B    ever great, applied to the surfaces of contact at
\  Q will be sustained by the resistance of the surfaces of contact, provided its direction be any
where within the surface of this cone; and that it will not
be sustained, however small it may be, if its direction lie any
where without it.
THE Two STATES BORDERING UPON MOTION.
140. If the direction of the pressure coincide with the surface of the cone, it will be sustained by the friction of the
surfaces of contact, but the body to which it is applied will
be upon the point of slipping upon the other. The state of
the equilibrium of this body is then said to be that BORDERING UPON MOTION. If the pressure P admit of being applied
in any direction about the point Q, there are evidently an
infinity of such states of the equilibrium bordering upon motion, corresponding to all the possible positions of P on the
surface of the cone.
If the pressure P admit of being applied only in the same
plane, there are but two such states, corresponding to those
directions of P, which coincide with the two intersections of
this plane with the surface of the cone; these are called the
superior and inferior states bordering upon motion. In the
case in which the direction of P is limited to the plane AQB,
BQ and CQ represent its directions corresponding to the




131      THE TWO STATES BORDERING UPON MOTION.
two states bordering on motion. Any direction of P within
the angle BQC corresponds to a state of equilibrium; any
direction, without this angle, to a state of motion.
141. Since, when the direction of the pressure P coincides
with the surface of the cone of resistance, the equilibrium is
in the state bordering upon motion; it follows, conversely,
and for the same reasons, that this is the direction of the
pressure sustained by the surfaces of contact of two bodies
whenever the state of their equilibrium is that bordering upon
motion. This being, moreover, the direction of the pressure
of the one body upon the other is manifestly the direction of
the resistance opposed by the second body to the pressure of
the first at their surface of contact, for this single pressure
and this single resistance are forces in equilibrium, and therefore equal and opposite. All that has been said above of the
single pressure and the single resistance sustained by two
surfaces of contact, is manifestly true of the resultant of any
number of such pressures, and of the resultant of any number of such resistances.  Thus then it follows, that when any
znumber of presures applied to a body moveable upon another
which is fixed, are sustained by the resistance of the surfaces
of contact of the two bodies, and are in the state of equilibrium
bordering upon motion, then the direction of the resultant of
these pressures coincides with the surface of the cone of resistance, as does that also of the resultant of the resistances of the
different points of the surfaces of contact*, that is, they are
both inclined to tie perpendicular to the surfaces of contact
(at the point where they intersect it), at an angle equal to the
limiting angle of resistance.
* The properties of the'limiting angle of resistance and the.ona of vesatance, were first given by the author of this work in a paper published xu tihe
Cambridge Philosophical Transactions, vol. v.




FRICTION.                           135
TABLE I.
Friction of Plane Surfaces, when they have been some time in Contact.
Limiting
Surfaces in Contact.  Disposition of  State of the  Co-efficient Angle of
the Fibres.   Surfaces.  of Friction. Resistance.
EXPERIMENTS OF M. MORIN.
parallel   { wi    t    }   062     310 48'
unguent
ditto                       0 44    23 46
perpendicu-  without        05
lar          unguent
Oak uoon oak       -
ditto        with water      o71    35  23
endways of 
the  flat  it   ut        043     23  1
surface of   unguelt        4
the other J
Oak upon elm     -       parallel     ditto           0'38   20  49
ditto        ditto           0-69   34  37
rubbed with
Elm upon oak     -       ditto           dry soap     041    22  18
perpendicu-  without               1 
lar          unguent       0          41
Ash, fir, beech, service-  parallel   ditto           053    2   56
tree, upon oak
the leather
a    ditto          0'61   31  23
flat
Tanned leather upon oak    length-    ditto           043    23  16
ways, but  steeped  in            3 
sideways     water 
Black    upon a plane                      u4                36  30
Bdresksed  surface of   parallel                            36  30
erd       oak                          unguent
eather, upon a round- 
orstrap    ed surface   perpendicu-  ditto                  2511
leather    of oak         lar
parallel      ditto          0560   26  34
Hemp matting upon oak    ditto        steeped in      087    41   2
water
Hemp cords upon oak  -   ditto          unuent         80    38  40
unguent
ditto          0-62    31  48
Iron upon oak    -   -   ditto        steeped  in      65     33 
water 
Cast-iron upon oak   -   ditto        ditto           0*65   33   2
Copper upon oak -        ditto          unguent       0'62   3148
steeped  in     062    31  48
Ox-hide as a piston sheath  flat or side- water
Ox-hide as a piston sheath  fI fat or side-  with  oil,
upon cast-iron           ways         tallow, or    012     6   1
hog's lard




136                           FRICTION.
-~ _ _ _ _ _ _ -  _ _ _ _Limiting
Surfaces in Contact.  Disposition of  State of the  Co-efficient Angle of
the Fibres.   Surfaces,  of Friction. Resistance.
EXPERIMENTS OF M. MORIN.
-continued.
Black dressed leather, or             without         02     1 
strap leather, upon a  fat            unguent 
cast-iron pulley                    steeped        0-38    20  49
Cast-iron upon cast-iron -  ditto           u        016      9   6
ditto unguent  0'16     9   6
Iron upon cast-iron  -   ditto        ditto          0'19    10 46
Oak, elm, yoke elm, iron,             with tallow    0lOf     5 43
cast-iron, and brass   ditto        with oil or
sliding two and two,                  ho'sla       015         32
one upon another                      hog'
Calcareous oolite stone  dit          without    t      4    3 
upon calcareous oolite  d             unguent 
Hard calcareous stone, 
called  muschelkalk,   ditto        ditto           0'75   36  62
upon calcareous oolite 
Brick upon calcareous    ditto        ditto           06'7   33  50
oolite 
Oak upon calcareous      wood end-    ditto           0'63   32  13
oolite                   ways
Iron upon calcareous oolite flat       ditto          0'49   26   7
Hard calcareous stone, or
muschelkalk,   upon    ditto         ditto         0'0     35   0
muschelkalk
Calcareous oolite stone  ditto         ditto          075    36  52
upon muschelkalk 
Brick upon muschelkalk -  ditto        ditto          0'6    33  50
Iron upon muschelkalk -  ditto         ditto          0'42   22 47
Oak upon muschelkalk  -  ditto        ditto           0-64   32  38
with a coat- 
ing of mortar, of three
Calcareous oolite stone  ditto          artoffine     0'e4~  8   80
upon calcareous oolitesand                  and
one part of
slack lime
L._.      hcotcha ______               -     -. —---   __ 7-  _
* The surfaces retaining some unctuousness.
Jt When the contact has not lasted long enough to express the grease.
t When the contact has lasted long enough to express the grease and bring
back the surfaces to an unctuous state.
~ After a contact of from ten to fifteen minutes.




UICTION.                           137
Co-eficient Limiting
Nature of Bodies and Unguents.       of Friction.  Angle.
Soft calcareous stone, well dressed, upon the same  -  04  36~ 30'
Hard calcareous stone, ditto  -    -      -5 -             36  52
Common brick, ditto   -     -      -     -      -   06     33  50
Oak, endways, ditto   -            -     -      -   063    32  13
Wrought iron, ditto         -      -     -      -   049    26   7
Hard calcareous stone, well dressed, upon hard calcareous stone   0- -          -70                            35   0
Soft, ditto    -      -     -      -            -   075    36  52
Common brick, ditto                                 06 7   33  50
Oak, endways, ditto   -     -      -     -      -   064    32  3
Wrought iron, ditto   -                         -   042     22 47
Soft calcareous stone upon soft calcareous stone, with
fresh mortar of fine sand  -     -     -      -   074     36  30
EXPERIMENTS BY DIFFERENT OBSERVERS.
Smooth free-stone upon smooth free-stone, dry (Rennie)  0-71  35  23
Ditto, with fresh mortar (Rennie)  -      -     -   066     33  26
Hard polished calcareous stone upon hard polished calcareous stone -     -                         -     58    30   7
Calcareous stone upon ditto, both surfaces being made
rough with a chisel (Bonchardi)         -        - 078    37  58
Well dressed granite upon rough granite (Rennie)  -  066    33  26
Ditto, with fresh mortar, ditto (Rennie) -   -      049     26   7
Box of wood upon pavement (Regnier) -               058     30   7
Ditto upon beaten earth (Herbert)        -      -   033     18  16
Libage stone upon a bed of dry clay  -   -      -   051     2    2
Ditto, the clay being damp and soft      -      -   034     18  47
Ditto, the clay being equally damp, but covered with
thick sand (Greve)  -     -      -                040    21 48




138                          FRICTION.
TABLE II.
Friction of Plane Surfaces, in Motion one upon the other.
Limiting
Surfaces in Contact.  Disposition of  State of the  Co-efficient Angle of
the Fibres.   Surfaces.  of Friction. Resistance.
EXPERIMENTS OF M. MORIN.
parallel     withounguent   048     25039'
rubbedwith     0-16     9 6
ditto                       0        9 6
perpendicu-  without        0 34    18 47
lar          unguent
Oak upon oak             ditto        steeped  in    05      14  3
wood end-'
ways on    without
|wood       1unguent       019       46
length-      unguen
ways    J
parallel     ditto          0.43     23 17
Elm upon oak     -       perpen   di    tto          0       24 14
lar                    [ do45     2414
parallel     ditto          0-25    14  3
Ash, fir, beech,wild pear-                           03 t    19 48
tree, and service-tree,  ditto      ditto          0
upon oak                                           040    )21 49
ditto          0-62    31 48
with water     0-26    14 35
J rubbed with
Iron upon oak   -    -   ditto          dry so       021     11 52
without
unguent      0-49    26  7
with water     0-22    12 25
Cast-iron upon oak       ditto     - rubbed with s'  019    10 46
Copper upon oak -    -   ditto              t        0-62    31 48
unguent
Iron upon elm    -   -   ditto        ditto          0-25    14  3
Cast-iron upon elm-  -   ditto        ditto          0-20    11 19
Black  dressed leather   ditto        ditto                  15
upon oak               itto         ditto          027     15 7
Tanned leather upon oak   lengtw,     ditto          035     19 18
edgeways
without          -5         1
unguent           6    29 
Tanned   leather  upon   ditto          water        036     19 48
cast-iron and brass               l greased and
water
with oil       0215     8 32..with oil       0-51 o8 82




FRICTION.                           139
Limiting
-Surfaces in Contact.  Disposition of  State of the  Co-efficient Angle of
the Fibres.   Surfaces,  of Friction. Resistance.
EXPERIMENTS OF M. MORIN.
-continued.
Hemp, in threads or in   parallel     without        052      2029'
iron                              (   unguent
cord, upon oak      l perpendicu- ) w    wt        0 3      1
Wild peartree, ditto  -  ditto        ditto          0.44     23 45
Iron upon iron   -    -  ditto        ditto           0-44    23 45*6
Iron upon cast-iron and               ditto          018t     10 13 
braso         ditto          0'18~   10 13
brass
Cast-iron, ditto  -   -  ditto        ditto           0-15     8 32
upon brass -   -  ditto         ditto          0-20    11 19
Brass upon cast-iron  -  ditto        ditto           022     12 25
upon iron -        ditto        ditto           0-16t    9  6
greased in
the usual
Oak, elm, yoke elm, wild                wal with      00to    4   1
pear, cast-iron,wrought                og'slard,        ~ 
iron, steel, and moving  ditto        hog'sof, 
one upon another,o or on                o
themselves                           slightly 
greasy to >   0-15      832
the touch 
Calcareous oolite stone  dito         without         0-64    82 37
upon calcareous oolite  d(             unguent 
Calcareous stone, called
muschelkalk, upon cal-  ditto        ditto          0-67    33 50
careous oolite 
Common brick upon cal-   ditto        ditto           0-65    33  2
careous oolite
Oak upon calcareous   ( wood end-       itto.0 38    20 49
oolite                   ways
Wrought iron, ditto      parallel     ditto           0-69    34 3'7
Calcareous stone, called 
muschelkalk,upon mus-. ditto        ditto          0-38     20 49
chelkalk             )    t
Calcareous oolite stone  ditto         ditto          0       33  2
upon muschelkalk
Common brick, ditto   -  ditto        ditto           0-60    30 58
Oak upon muschelkalk     wood  end-    ditto          038     20 49
ditto           0-24 
Iron upon muschelkalk -  parallel     l saturated     0.30    16 42
with water
* The surfaces wear when there is no grease.
The surfaces still retaining a little unctuousness. c          Ibid.
When the grease is constantly renewed and uniformly distributed, this
proportion can be reduced to 0'05



140                             FRICTION.
TABLE III.
Friction of Gudgeons or Axle-ends, in Motion, upon their Bearings.
WFrom the experiments of Morin.)
Co-efficient of Friction when
the Grease is renewed.  Limiting
Surfaces in Contact. State of the Surfaces. _____               Angle of
In the usual   ontinuously.Resistance
Way.
coated with oil of                               4  0'
lard, tallow, and                              3  6
bearings        coatedumwith   as-      0054          0.19       3  6
greasy                  0-14                     7 58
greasy and wetted       0-14      -         -      58
coated with oil of                               4  0
olives,with hog's      to          0   4       4 35
Castiron axles lard, tallow, and  07              6
Cast-iron axles,    soft gom w
ditto             softr t gom3
greasy                  0-16              -      9  6
greasy and damped       0'16                     9  6
scarcely greasy         0-19                    10 46
without unguent         018       -             10 12
with oil or hog's                   0 090        5  9 
axles in catof lig   vea, tallow, n d [7to..0'054 4
in lignum vit   greasy with ditto       0-10      -       -      5 43
bearings        g reasy,o wth 
mixture of og's        0-14                       58
Wrought-iron  coated with   oil                                4 
axles in cast     of olives, tallow,  00o to 0-08    0-054    -     5
hog's lard, or                                 3  6
iron bearings l   soft gom
olives,hog's lard,  007 to 008    0054         4 35
I   l in       or tallow                                      3  6
Iron  axles ing   coated with hard        0,09                     5  9
brass bearings     gom                                           5 
greasy and wetted       0-19      -       -     10 46
scarcely greasy         0'25          X         14  2
Iron  axles  in   coated  with   oil,     011                      6 
lignum vitas      or hog's lard    )' 
bearings      ) greasy                  019               -     10 46
Brass axles in    coated with oil        0o10                      5 43
brass bearings  with hog's lard         009       -       -      59
Brass axles in)                                                    - 35 i' 1
cast-iron bear-  cor tallow     i        -      0-045 to 0-052   2 35
ings          or tallow                                     2 59
ings                                      to wear.
* The surfaces beginning to wear.




FRICTION.                            141
Co-efficient of Friction when
the Grease is renewed.  Limiting
Surfaces in Contact. State of the Surfaces. __                Angle of
In the usual             Resistance.
Way.      Continuously.
Lignum     vits   coated with hog's      012                     65
lard
axles, ditto    greasy                 01                      8  2
Lignum     vitae
axles in lig-   coated with hog's                              4
num      vitae   lard 
bearings 
TABLE IV.
Co-efficients of Friction under Pressures increased continually up to the
Limits of Abrasion.
(From the experiments of Mr. G. Rennie.*)
Co-efficients of Friction.
Pressure per
Square Inch.  Wrought-iron  Wrought-iron    Steel upon    Brass upon
Wrought-iron.   Cast4iron.
32- 5 lbs.     -140            1'74         -166           1567
1-66 cwt.     -250          -275           -300          -225
2-00           *271           292          -333          -219
2-33           -285          -321          -340          -214
2-66'     297           -329          -344          -211
3-00           -312          -333          -347           -215
3-33           -350'     351          *351          -206
3-66'376          -353          -353          -205
4-00           *376          -365          -354          -208
4-33           -395          -366          -356          -221
4-66          -403           -366          -357          -223
5-00          -409           *36'7          358          -233
5-33                          367          -359          -234
5-66'367                      -367          -235
6-00                         *376          -403          -233
6'33                         -434                         -234
6-66                                                      -235'700                                                      -232
7-33                                                      -273
* Phil. Trans. 1829, table 8. p. 159.




142             THE RIGIDITY OF CORDS.
THE RIGIDITY OF CORDS.
142. It is evident that, by reason of that resistance to,-,  *deflexion which constitutes the ri4L1 ~~, 6%gidity of a cord, a certain force or
pressure must be called into action
zjl JB    whenever it is made to change its., i:t}  ~rectilineal direction, so as to adapt
D~ "'e ^    itself to the form of any curved surP,        -     i v  face over which it is made to pass;
~" )i'    ^>',as, for instance, over the circumference of a pulley or wheel. Suppose such a cord to sustain tensions represented by P1 and
P1, of which P1 is on the point of preponderating, and let
the friction of the axis of the pulley be, for the present,
neglected. It is manifest that, in order to supply the force
necessary to overcome the rigidity of the cord and to produce its deflection at B, the tension P1 must exceed P2;
whereas, if there were no rigidity, P, would equal P,; so
that the effect of the rigidity in increasing the tension P1 is
the same as though it had, by a certain quantity, increased
the tension P2. Now, from a very numerous series of
experiments made by Coulomb upon this subject, it appears
that the quantity by which the tension P2 may thus be considered to be increased by the rigidity, is partly constant
and partly dependent on the amount of P,; so as to be
represented by an algebraical formula of two terms, one
of which is a constant quantity, and the other the product
of a constant quantity by P,. Thus if D represent the
constant part of this formula, and E the constant factor
of P2, then is the effect of the rigidity of the cord the same
as though the tension P2 were increased by the quantity
D+E.P,.
When the cord, instead of being bent, under different
pressures, upon circular arcs of equal radii, was bent upon
circular arcs of different radii, then this quantity D +E. P;
by which the tension P, may be considered to be increased
by the rigidity, was found to vary inversely as the radii
of the arcs; so that, on the whole, it may be represented
by the formula




THE RIGIDITY OF CORDS,                    143
D+E. P,)
where R represents the radius of the circular arc over which
the rope is bent. Thus it appears that the yielding tension
P, may be considered to have been increased by the rigidity
of the rope, when in thestate bordering upon motion, so as
to become
D+E. P,
P,+       R
This formula applies only to the bending of the scme cord
under different tensions upon different circular arcs: for different cords, the constants D and E vary (within certain
limits to be specified) as the squares of the diameters or of the
circumferences of the cords, in respect to new       cords, wet or
dry; in respect to old cords they vary nearly as the power 3
of the diameters or circumferences.
Tables have been furnished by Coulomb of the values of
the constants D    and E.    These tables, reduced to English
measures, are given on the next page.*
The rigidity of the cord exerts its influence to increase resistance only at
that point where the cord winds upon the pulley; at the point where it leaves
the pulley its elasticity favours rather, and does not perceptibly affect, the
conditions of the equilibrium.
In all calculations of machines, in which the moving power is applied by the
intervention of a rope passing over a pulley, one-half the diameter of rope is
to be added to the radius of the pulley, or to the perpendicular on the direction
of the rope from the point whence the moments are measured, the pressure
applied to the rope producing the same effect as though it were all exerted
along the axis of the rope.




144                   THE RIGIDITY OF CORDS.
TABLE V. RIGIDITY OF ROPES.
Table of the values of the constants D and E, according to the experiments of
Coulomb (reduced to English measures). The radius R of the pulley is to be
taken in feet.
No. 1. New dry cords. Rigidity proportional to the square of the
circumference.
Circumference of
t Circumference of  Value of D in lbs. Value of E in lbs.
the Rope in Inches.
1                -131528'033533       Squares of propor2             5b26108.'o023030      ttions of the in2'      526108'023030       termediate cir4               2-104451          *073175        cumferences to
8               8-413702          -368494        those  of the
table.
No. 2. New ropes dipped in water. Rigidity proportional  Prp..S- Squares.
to the square of the circumference.          ____
1-0   1-00oo
1'1   1.21
1'2   1'44
Circumference of                                          1'3  1'69
ttC ircuferene of  Value of D in lbs. Value of E in lbs.  14    19
the Rope in Inches.                                       1'5   2'25
~~________________~_______~____ _____~________1'6 2'56
1'7   2 89
1                -263053'0057576          1-   3-24
1'9   8'61
2               1'052217          0230303         2.0    4001
4               4-208902          0731755
8              16-835606'3684860
No. 3. Dry half-worn ropes.   Rigidity proportional to the square root
of the cube of the circumference.
Circumference of  Value of D in lbs. Value of E in lbs.
the Rope in Inches.
1               -146272'0064033     Squareroots of the
2'413656'0180827        cubes of proportions of the in4              1'169641'0512115       termediate cir8              3'308787'1448238        cumferences to
______________________ _        ____ those of the
table.
No. 4. Wetted half-worn cords. Rigidity proportional
to the square root of the cube of the circumference.  ro   Powers 3
~~~~~~~~~~~~Circumference of               1'1  1'154
Circumference of  Value of D in lbs. Value of E in lbs.  1. 1.315
the Rope in Inches.                                        1s   1-48-2
1'4   1'657
1-5   1 837
1'292541'006401          1.6  S24
2'827328          *018107          158  2'415
4              2-339675'051212          2-0  2.828
8              66616589'144822




TUE RIGIDITY OF CORDS.                     145
No. 5. Tarred rope. Rigidity proportional to the number of strands.
Number of Strands. Value of D in lbs. Value of E in lbs.
6              0-33390          0-009305
15             0-17212          0021713
30      1'25294:                 0044983
To determine the constants D and E for ropes whose circumferences are
intermediate to those of the tables, find the ratio of the given circumference
to that nearest to it in the tables, and seek this ratio or proportion in the first
column of the auxiliary table to the right of the page. The corresponding
number in the second column of this auxiliary table is a factor by which the
values of D and E for the nearest circumference in the principal tables being
multiplied, their values for the given circumference will be determined.*
* Note (s) Ed. App.
10




146            THE THEORY OF MACHINES.
P AR T III.
THE THEORY OF MACHINES.
143. THE parts of a machine are divisible into those which
receive the operation of the moving power immediately, those
which operate immediately upon the work to be performed,
and those which communicate between the two, or which
conduct the power or work from the moving to the working
points of the machine. The first class may be called RECEIVERS, the second OPERATORS, and the third COMMUNICATORS of
work.
THE TRANSMISSION OF WORK BY MACHINES.
144. The moving power divides itself whilst it operates in
a machine, first, Into that which overcomes the prejudicial
resistances of the machine, or those which are opposed by
friction and other causes uselessly absorbing the work in its
transmission.  Secondly, Into that which accelerates the
motion of the various moving parts of the machine; so long
as the work done by the moving power upon it exceeds that
expended upon the various resistances opposed to the motion
of the machine (Art. 129.). Thirdly, Into that which overcomes the useful resistances, or those which are opposed to
the motion of the machine at the working point or points
by the useful work which is to be done by it. Thus, then,
the work done by the moving power upon the moving points
of the machine (as distinguished from the working points)
divides itself in the act of transmission, first, Into the work
expended uselessly upon the friction and other prejudicial
resistances opposed to its transmission. Secondly, Into that
accumulated in the various moving elements of the machine,
and reproducible. Thirdly, Into the useful work, or that
done by the operators, whence results immediately the useful
products of the machine.




THE THEORY OF MACHINES.            147
145. The aggregate number of units of useful works yielded
by any machine at its orking points is less than the number received upon the machine directly from the moving
power, by the number of units expended upon the prejudicial resistances and by the number of units accumulated
in the moving parts of the machine whilst the work is being
done.*
For, by the principle of vis viva (Art. 129.), if TzU represent the number of units of work received upon the machine
immediately from the operation of the moving power, 2u
the whole number of such units absorbed in overcoming the
prejudicial resistances opposed to the working of the machine, zU, the whole useful work of the machine (or that
done by its operators in producing the useful effect), and
-w(v2" —v,) one half the aggregate difference of the vires
vivae of the various moving parts of the machine at the
commencement and termination of the period during which
the work is estimated, then, by the principle of vis VIVA
(equation 108),
U1 =2U2,+    +u -w(v22-v12)..... (112);
2g
in which vl and v2 represent the velocities at the commencement and termination of the period, during which the work
is estimated, of that moving element of the machine whose
weight is w. Now one-half the aggregate difference of the
vires vivse of the moving elements represents the work accumulated in them during the period in repect to which the
work is estimated (Art. 130.). Therefore, &c.
146. If the same velocity of every part of the machine return after any period of time, or if the motion be periodical,
then is the whole work received upon it from the moving power
during that time exactly equal to the sum of the useful work
done, and the work expended upon the prejudicial resistances.
For the velocity being in this case the same at the commencement and expiration of the period during which the
work is estimated, w(vl2 - 2 2)=0, so that
* Note (t) Ed. App.




148           THE MODULUS OF A MACHINE.
zU,-=2U2 + 4......(113).
Therefore, &c.
The converse of this proposition is evidently true.
147. If the prime mover in a machine be throughout the
motion in equilibrium with the useful and the prejudicial
resistances, then the motion of the machine is uniform.
For in this case, by the principle of virtual velocities
(Art. 127.), zU=zU=2T,+z;    therefore  (equation 112)
w(v2 —v22)=-0; whence it follows that (in the case supposed) the velocities v1 and v2 of any moving element of the
machine are the same at the commencement and termination of any period of the motion however small, or that
the motion of every such element is a uniform motion.
Therefore, &c.
The converse of this proposition is evidently true.
THE MODULUS OF A MACHINE MOVING WITH A UNIFORM OR
PERIODICAL MOTION.
148. The modulus of a machine, in the sense in which the
term is used in this work, is the relation between the work
constantly done upon it by the moving power, and that constantly yielded at the working points, when it has attained
a state of uniform motion if it admit of such a state of
motion; or if the nature of its motion be periodical, then
is its modulus the relation between the work done at its
moving qand at its working points in the interval of time
which it occupies in passing from any give velocity to the
same velocity again.
The modulus is thus, in respect to any machine, the particular form applicable to that machine of equation (113), and
being dependent for its amount upon the amount of work:lu
expended upon the friction and other prejudicial resistances
opposed to the motion of the various elements of the machine, it measures in respect to each such machine the loss
of work due to these causes, and therefore constitutes a true
standardfor cornparing the expenditnre of moving power necessary to the production of the same efects by diferent ma



THE MODULUS OF A MACHINE.            149
chines: it is thus a measure of the working qualities of
machines.*
Whilst the particular modulus of every differently constructed machine is thus different, there is nevertheless a
general algebraical type or formula to which the moduli of
machines are (for the most part and with certain modifications) referable. That form is the following,
U,=A.U,+B.S......        (114),
where U, is the work done at the moving point of the machine through the space S, U2 the work yielded at the working points, and A and B constants dependent for their value
upon the construction of the machine: that is to say, upon
the dimensions and the combinations of its parts, their
weights, and the co-efficients of friction at their various rubbing surfaces.
It would not be difficult to establish generally this forn of
the modulus under certain assumed conditions. As the modulus of each particular machine must however, in this work,
be discussed and determined independently, it will be better
to refer the reader to the particular moduli investigated in
the following pages.  He will observe that they are for the
most part comprised under the form above assumed; subject to certain modifications which arise out of the discussion of each individual case, and which are treated at length.,
149. There is, however, one important exception to this
general form of the modulus: it occurs in the case of machines, some of whose parts move immersed in fluids. It is
only when the resistances opposed to the motion of the parts
of the machine upon one another are, like those of friction,
proportional to the pressures, or when they are constant resistances, that this form of the modulus obtains. If there be
resistances which, like those of fluids in which the moving
parts are immersed (the air, for instance), vary with the velocity of the motion, and these resistances be considerable,
then must other terms be added to the modulus. This subject will be further discussed when the resistances of fluids
are treated of.  It may here, however, be observed, that if
the machine move uniformly subject to the resistance of a
fluid during a given time T, and the resistance of the fluid
* The properties of the modulus of a machine are here, for the first time,
discussed.




1.50         THE MODULUS OF A MACHINE.
be supposed to vary as the square of the velocity V,>then
will the work expended on this resistance vary as V2. S, or
as V. T, since S=Y. T. If then U, and U, represent the
work done at the moving and working points during the
time T, then does the modulus (equation 114) assume, in this
case, the form
U,=A. U,+B. V. T+.. V3. T...  (115).
THE MODULUS OF A MACHINE MOVING WITH AN ACCELERATED
OR RETARDED MOTION.
150. In the two last articles the work U,, done upon the
moving point or points of the machine, has been supposed to
be just that necessary to overcome the useful and prejudicial resistances opposed to the motion of the machine, either
continually or periodically; so that all the work may be expended upon these resistances, and none accumulated in the
moving parts of the machine as the work proceeds, or else
that the accumulated work may return to the same amount
from period to period. Let us now suppose this equality to
cease, and the work U, done by the moving power to exceed
that necessary to overcome the useful and prejudicial resistances; and to distinguish the work represented by UT in the
one case from that in the other, let us suppose the former
(that which is in excess of the resistances) to be represented
by U'; also let U, be the useful work of the machine, done
through a given space S,, and which is supposed the same
whatever may be the velocity of the motion of the machine
whilst that space is being described; moreover, let S, be the
space described by the moving point, whilst the space S, is
being described by the working point.
Now since U, is the work which must be done at the
moving point just to overcome the resistances opposed to
the motion of that point, and U' is the work actually done
upon that point by the power, therefore U'-U1 is the excess
of the work done by the power over that expended on the
resistances, and is therefore equal to the work accumulated
in the machine (Art. 130.); that is, to one half of the
increase of the vis viva through the space S, (Art. 129.); so
that, if v, represent the velocity of any element of the
machine (whose weight is w) when the work UT began to be
done, and v, its velocity when that work has been completed, then (Art. 129.),




THE VELOCITY OF A MACHIE.          1
TI-u    I   2(V2V1i).'2g
Now by equation (114) U,=AU, +BS,,
1
U' =A. U,+B. S, +-  (2-    )..... (116).
If instead of the work U' done by the power exceeding that
U, expended on the resistances it had been less than it, then,
instead of work being accumulated continually through the
space S,, it would continually have been lost, and we should
have had the relation (Art. 129.),
1 12g- 2   2
so that in this case, also,
Th1 e- 1 e  (W(, t e2). 
The equation (116) applies therefore to the case of a
retarded motion of the machine as well as to that of an
accelerated motion, and is the general expression for the
modulus of a machine moving with a variable motion.
Whilst the co-efficients A and B of the modulus are dependent wholly upon the friction and other direct resistances to
the motion of the machine, the last term of it is wholly
independent of all these resistances, its amount being determined solely by the velocities of the various moving elements of the machines and their respective weights.
THE VELOCITY OF A MACHINE MOVING WITH A VARIABLE
MOTION.
151. The velocities of the different parts or elements of
every machine are evidently connected with one another by
certain invariable relations, capable of being expressed by
algebraical formulae, so that, although these relations are
different for different machines, they are the same for all
circumstances of the motion of the same machine. In a
great number of machines this relation is expressed by a
constant ratio. Let the constant ratio of the velocity 1v of
any element to that V, of the moving point in such a




152           THE VELOCITY OF A MACHINE.
machine be represented by X, so that v =XV,, and let v2 and
V2 be any other values of v, and V,; then v-=XV,. Substituting these values of v, and v, in equation (116), we
have
U-==A. U,+B. S,+    (V2    2)w.... (117);
in which expression.wX2 represents the sum of the weights
of all the moving elements of the machine, each being multiplied by the square of the ratio X of its velocity to that of
the point where the machine receives the operation of its
moving power. For the same machine this co-efficient:ZwX2
is therefore a constant quantity. For different machines it
is different. It is wholly independent of the useful or prejudicial resistances opposed to the motion of the machine,
and has its value determined solely by the weights and
dimensions of the moving masses, and the manner in which
they are connected with one another in the machine.
Transforming this equation and reducing, we have
V 2 v2= +2g { UB-A  U-B S.     118).
by which equation the velocity V, of the moving point of
the machine is determined, after a given amount of work
U' has been done upon it by the moving power, and a given
amount U, expended on the useful resistances; the velocity
of the moving point, when this work began to be done
being given and represented by V1.
It is evident that the motion of the machine is more
equable as the quantity represented by 2wX2 is greater.
This quantity, which is the same for the same machine and
different for different machines, and which distinguishes
machines from one another in respect to the steadiness of
their motion, independently of all considerations arising out
of the nature of the resistances useful or prejudicial opposed
to it, may with propriety be called the CO-EFFICIENT OF
EQUABLE MOTION.* The actual motion of the machine is
more equable as this co-efficient and as the co-efficients A
and B (supposed positive) are greater.
* The co-efficient of equable motion is here, for the first time, introduced
into the consideration of the theory of machines.




CO-EFFICIENTS OF THE MODULUS.        153
TO DETERMINE THE CO-EFFICIENTS OF THE MODULUS OF A
MACHINE.
152. Let that-relation first be determined between the
moving pressure P1 upon the machine and its working pressure P1, which obtains in the state bordering upon motion by
the preponderance of P1. This relation will, in all cases
where the constant resistances to the motion of the machine
independently of P2 are small as compared with P,, be
found to be represented by formulae of which the following
is the general type or form:P,=P. *P, +..... (119);
where 4, and ~ represent certain functions of the friction
and other prejudicial resistances in the machine, of which
the latter disappears when the resistances vanish and the
former does not; so that if,(O) and  2(~) represent the
values of these functions when the prejudicial resistances
vanish, then  2(0~)O=  and  I(0>)=  a given finite quantity
dependent for its amount on the composition of the machine.
Let P1(O) represent that value of the pressure P1 which would
be in equilibrium with the given pressure P2, if there were
no prejudicial resistances opposed to the motion of the
machine. Then, by the last equation, P1()=P,. (0).
But by the principle of virtual velocities (Art. 127.), if
we suppose the motion of the machine to be uniform, so
that Pi and P. are constantly in equilibrium upon it, and if
we represent by S, any space described by the point of
application of P,, or the projection of that space on the
direction of P, (Art. 52.), and by S, the corresponding
space or projection of the space described by P2, then
pI(c. Si=P2. S..  Therefore, dividing this equation by
the last, we have
S,=..   (120).
Multiplying this equation by equation (119),
P,.S1=P,. S     ) + S2 {   } =P 2S2         +S.;. TUJj 4( t) I   U  + S.     (121)
4)i= ~6') U,+~,,......(




154                      AXES.
which is the modulus of the machine, so that the constant
A in equation (114) is represented by (-a  and the constant
B by.
The above equation has been proved for any value of S,,
provided the values of P1 and P2 be constant, and the
motion of the machine uniform; it evidently obtains, therefore, for an exceedingly small value of Si, when the motion
of the machine is variable.
GENERAL CONDITION OF THE STATE BORDERING UPON MOTION
N A BODY ACTED UPON BY PRESSURES IN THE SAME PLANE,
AND MOVEABLE ABOUT A CYLINDRICAL AXIS.
153. If any number of pressures Pi, P,, PS, &c. applied in
the same plane to a body moveable about a cylindrical
axis, be in the state bordering upon motion, then is the
direction of the resistance of the axis inclined to its radius,
at the point where it intersects the circumference, at an
angle equal to the limiting angle of resistance.
For let R represent the resultant of P P,P, &c. Then,
since these forces are supposed to be upon the!la point of causing the axis of the body to turn
-upon its bearings, their resultant would, if made
to replace them, be also on the point of causing
the axis to turn on its bearings. Hence it foljL lows that the direction of this resultant R cannot
be through the centre C of the axis; for if it
were, then the axis would be pressed by it in the
direction of a radius, that is, perpendicularly
upon its bearings, and could not be made to turn upon them
by that pressure, or to be upon the point of turning upon
them. The direction of iR must then be on one side of C,
so as to press the axis lupon its bearings in a direction RL,
inclined to the normal CL (at the point L, where it intersects the circumference of the axis) at a certain angle RLO.
Moreover, it is evident (Art. 141.), that since this force R
pressing the axis upon its bearings at L is upon the point of
causing it to sli upon them, this inclination RLC of R to
the perpendicular CL is equal to the limiting angle of




THE WHEEL AND AXLE.               155
resistance of the axis and its bearings.*  Now the resistance
of the axis is evidently equal and opposite to the resultant
R of all the forces P1, P2, &c. impressed upon the body.
This resistance acts, therefore in the direction LR, and is
inclined to CL at an angle equal to the limiting angle of
resistance. Therefore, &c.
THE WHEEL AND AXLE.;/   \   154. The pressures P, and P2 applied verA     X\B A    tically by means of parallel cords to a
wheel and axle are in the state bordering
upon motion by the preponderance of P,,
it is required to determine a relation
3        tS;  tbetwoen P, and P2.
The direction LR of the resistance of the axis is on that
side of the centre which is towards P1, and is inclined to the
perpendicular CL at the point L, where it intersects the
axis at an angle CLR equal to the limiting angle of resistance. Let this angle be represented by p, and the radius
CL of the axis by p; also the radius CA of the wheel by a,,
and that OB of the axle by a2; and let W be the weight of
the wheel and axle, whose centre of gravity is supposed to
be C. Now, the pressures P,, P,, the weight W   of the
wheel and axle, and the resistance R of the axis, are pressures in equilibrium. Therefore, by the principle of the
equality of moments (Art. 7.), neglecting the rigidity of the
cord, and observing that the weight W may be supposed to
act through C, we have,
P. CA=P,. CB+R. Cm.
If, instead of P, preponderating, it had been on the point
of yielding, or P2 had been in the act of preponderating,
then R would have fallen on the other side of C, and we
should have obtained the relation P1. CA=P. CBR. CCm; so that, generally, P,. CA=P,. CB~R. Cm;
the sign ~ being taken according as P, is in the superior or
inferior state bordering upon motion.
Now CA-a,, CB=a., Cm=CL sin. CLR=p sin., and
* The side of C on which RL falls is manifestly determined by the direction
towards which the motion is about to take place. In this case it is supposed
about to take place to the right of C. If it had been to the left, the direction of R would have been on the opposite side of C.




156             THE WHEEL AND AXLE.
R=P,~+P2~ W; the sign ~ being taken according as the
weight W of the wheel and axle acts in the same direction
with the pressures P1 and P2, or in the opposite direction;
that is, according as the pressures P1 and P, act vertically
downwards (as shown in the figure) or upwards;.. Pa =PA + (P1 + P2 ~ W) p sin. p,..P(a — p sin. qp)=P,(2 +p sin. p)~ Wp sin. I.
Now the effect (Art. 142.) of the rigidity of the cord BP,
is the same as though it increased the tension upon that cord
from P. to P2+ D- E     2) allowing, therefore, for the
rigidity of the cord, we have finally
PI(a-p sin. p)=(P,+ D+E. P) (a,+p sin. p)~W p sin. p,
or reducing,
P=P(IE) + p sin.     D+(    W)p sin. p
P, —PU +Ea+ p i.+       a+- i.. (122),
2 _a, - p sin.    i al-p sin. p  v
which is the required relation between P, and P, in the
state bordering upon motion.
P           P
P sin. g and - sin. q are in all cases exceedingly small;
a,         a2J
we may therefore omit, without materially affecting the
result, all terms involving powers of these quantities above
the first, we shall thus obtain by reduction
PiP2(j){ 1         +(   P)sin. }      { i +
P sin. p }.. (123).
155. The modulus of uniform motion in the wheel and, axle.
It is evident from equation (122), that, in the case of the
wheel and axle, the relation assumed in equation (114).btaicI I=(1      E\a,4+p sin. p
obtains, if we take  2[ 1 + a a-o sin.;
( 




THE WHEEL AND AXLE.                157
D+D -+W)p sn.ip
and                =...,,-p sin. (p
Now observing that F1(~) represents the value of *, when
the prejudicial resistances vanish (or when p=0 and E=O),
we have (o) =.
al
(  1+  sin.1
({ 1+ E          pa,(a,+ p sin.:  1     1+ E;  a.
~(?:  +3,,k2        n               - a-.  sin. p
\a
Therefore by equation (121),
U,=u( 1 + — { 1      (   sin.   +
+ ~WP            sinsin. 
S,... (124),
a, —p sin. q   }
L
which is the modulus of the wheel and axle.
Omitting terms involving dimensions of -- sin. qp, and
P           E
- sin. p, and - above the first, we have
aI +a+aPs                 a
(a, s             i    ().
156. The modulus of variable motion in the wheel and axle.
If the relation of P1 and P, be not that of either state
bordering upon motion, then the motion will be continually
accelerated or continually retarded, and work will continually accumulate in the moving parts of the machine, or the
work already accumulated there will continually expend




158              THE WHEEL AND AXLE.
itself until the whole is exhausted, and the machine is
brought to rest. The general expression for the modulus in
this state of variable motion is (equation 116)
U' AU, + BS + 1zw(v, -   12).
2g
Now in this case of the wheel and axle, if V1 and V2 represent the velocities of Pi at the
commencement and completion
of the space S,, and a the angular
\;3_  velocity of the revolution of the
(  wheel and axle; if, moreover, the
pressures P1 and P2 be supposed
to be supplied by weights suspended from   the cords; then,
since the velocity of P2 is reprer2    i    Psented by  -, v we have zwvp,B                P,2 ( V   + a"2pIl + a2fI,2 if
I, represent the moment of inertia of the revolving wheel,
and I, that of the revolving axle, (Art. 75.), and if p, represent the weight of a unit of the wheel and P2 of the axle;
since zwv12 represents the sum of the weights of all the moving elements of the machine, each being multiplied by the
square of its velocity, and that (by Art. 75.) a2&fI, represents
this sum in respect to the wheel, and ac22I, in respect to the
axle. Now, V -=aa,: Vi.22=Pl.r a+P+   2  2+I     2+I2
a,                 a,      a       al
W-  =(v,- a2 +Pa 2 +PI +,2I
2 *      1   a2         if+ P111.
Similarly gwv  V 2 { Pla, + P222 + -1I-  + PJI2
* w(v2 -1)=(V2 -V12) { Pai       2 +P2a~ KFIlpI2 
Substituting in the general expression (equation 116), we
have




THE WHEEL AND AXLE.               159
T'J= AU,+BS,+    (V22-V,32)
Pal +P2 a2 + PLI +..  (26),
which is the modulus of the machine in the state of variable
motion, the co-efficients A and B being those already determined (equation 124), whilst the co-efficient
Pia1 +P a2 +   ^lI + 2I+2 is the co-efficient Zwx2 (equation
aI
117) of equable motion. Ifthe wheel and axle be each of them
a solid cylinder, and the thickness of the wheel be b, and the
length of the axle b2, then (Art. 85.) I1=-brbla4 II= — rba4.
Now if W, and W. represent the weights of the wheel and
axle respectively, then W.=eal2bipt1 W2=rab 22b2.2; therefore
p1II=W!a2, 2I2-= Waa22.   Therefore the co-efficient of
equable motion is represented by the equation
Ws =P1a12 + P-a2 +i~-W1a12 + W2a2' or
+_Wl + (P2+ 2W)           2.:=p_.,+6.(12^72......(127.
157. To determine the velocity acquired through a given
space when the relation of the weights P1 and P2, suspended
from a wheel and axle, is not that of the state bordering
upon motion.*
Let Si be the space through which the weight P, moves
whilst its velocity passes from V, to Y,: observing that
Saa
U1-=PS1, and that U2.S=PS_-P2 1, substituting in equation (126), and solving that question in respect to V2, we
have
V2=vY1+2galS, Ppa1+P2a22~,^I-+N:... *(18);
Note (la) Ed. App.+
4 Note (u) Ed. App.




-A1;^0  ~THE PULLEY.
making the same suppositions as in formula 127, and representing the ratio a2 by m, we have
a,
Y   _22=Y +g         P-A. P2m-B 
V2 1 +  {(P1 +oW) + (P2 +rW2)m2 }
THE PULLEY.
158. If the radius of the axle be taken equal to that of the
wheel, the wheel and axle becomes a pul111        ley.  Assuming then in equation 122,
a \  =a2=a, we obtain for the relation of the
moving pressures P1 and P, in the state
p r <  -  bordering upon motion in the pulley, when
the strings are parallel,
D
{ l+!sin. p1  D+(~W) sin.p 
P =P     (1+ E     a     l+   ---.. (129);
E    1-]sin.p sa — sin. q             ) (1
a
and by equation 124 for the value of the modulus,
1r     sin. q(
a        1a l-sin.,
b  D+ip-~W)p sin.p 
+S J... (130);
a-p sin. qp
in which the sign ~ is to be taken according as the pressures
Pi and P2 act downwards, as in the first pulley of the preceding figure; or upwards, as in the second. Omitting
dimension of P- sin, P sin. (p, and - above the first, we have
a      a           a
by equations (123, 125)
p,^       ^tll+  E           {^         ),a a.,a D




SYSTEM OF ONE FIXED ONE MOVEABLE PULLEY.  161
UT. {       +     in. }    i +(2~i) sin. p }(-132).
1,,1                      I                1     ~
Also observing that a1=a,, and I —O, the modulus of variable motion (equation 126) becomes
U= =AU,+ BS(+ -(    -VY2( IP+P,+W..... (133),
and the velocity of variable motion (equations 118, 128) is
determined by the equation
V 2V2+ 2S P —A. P, —B..(134);
in which two last equations the values of A and B are those
of the modulus of equable motion (equation 125).
SYSTEM OF ONE FIXED AND ONE MOVEABLE PULLEY.
159. In the last article (equation 131) it was
shown that the relation between the tensions
P, and P, upon the two parts of a string passing over a pulley and parallel to one another,
was, in the state bordering upon motion by the
T, r  preponderance of PF, represented by an expression of the form P-=aP2+b, where a and b are
constants dependent upon the dimensions of the
r   pulley and its axis, its weight, and the rigidity
of the cord, and determined in terms of these
ok — elements by equation 131; and in which ex~L2 J Jpression b has a different value according as the
tension upon the cord passing over any pulley
acts in the same direction with the weight of that pulley (as
in the first pulley of the system shown in the figure), or in
the opposite direction (as in the second pulley): let these
different values of b be represented by b and bl. Now it is
evident that before the weight P2 can be raised by means of
a system such as that shown in the figure, composed of one
fixed and one moveable pulley, the state of the equilibrium
of both pulleys must be that bordering upon motion, which
is described in the preceding article; since both must be
upon the point of turning upon their axes before the weight
P. can begin to be raised. If then T and t represent the
tensions upon the two parts of the string which pass round
the moveable pulley, we have
11




162          SYSTEM OF ONE FIXED AND ANY
P1-aT+b, and T=at+b,.
Now the tensions T and t together support the weight P,,
and also the weight of the moveable pulley,.. T+t=P2+W.
Adding aT to both- sides of the second of the above equations, and multiplying both sides by a, we have
a(1 + a)T=a2(T + t) + ab =a2(P + W) + ab1.
Also multiplying the first equation by (1 +a),
(1 +a)P=-a(1 +a)T+b(1 +a)= 2(P + W)+ ab, +(1+ a),
a,   p    aW + b(l + a) + ab
f                          (1-35).
Now if there were no friction or rigidity, a would evi2
dently become 1 (see equation 131), and,1~= -a would
become; the co-efficients of the modulus (Art. 148.) are
a J   \          W ~  + (l + a) +
therefore A=2 1 -, and B= —a W+(a
\1+a/               1+a
U1. =2()U T      W+.(l+a+abS... (136),
which is the m6dulus of uniform motion to the single move-:able pulley.*
If this system  of two pulleys had been
arranged thus, with a different string passing
over each, instead of with a single string, as
shown in the preceding figure, then, representing by t the tension upon the secondpart of
the string to which P, is attached, and by T
that upon the first part of the string to which
P is attached, we have
P1at+b, T=aP,+b, P,+t-4-W=T.
* The modulus may be determined directly from equation (135); for it is,evident that if Si and S2 represent the spaces described in' the same time by
P, and P2, then S1=2S2. Multiplying both sides of equation (135) by this
equation, we have,
2S2;
PSo- 2(iia) S-, a2b(li+a)Uab2t
now PS1 =U1,, and P2S2 Ut2, therefore &c.




NUMBEPU OF MOVEABLE PULLEYS.        163
Multiplying the last of these equations by a, and adding it
to the first, we have P,(1 +a)+Wa=Ta+b=a2P2,+(1+a)b;: P     = —\2  +.  1 + -a-.
and for the modulus (equation 121),
U1=2(1+a)U2+(b- +a)S-....        (138).
It is evident that, since the co-efficient of the second term
of the modulus of this systen is less than that of the first
system (equation 136) (the quantities a and b being essentially positive), a given amount of work U2 may be done by
a less expense of power U,, or a gived weight P, may be
raised to a given height with less work, by means of this
system than the other; an advantage which is not due
entirely to the circumstance that the weight of the moveable pulley in this case acts in favour of the power, whereas
in the other it acts against it; and which advantage would
exist, in a less degree, were the pulleys without weight.
A SYSTEM OF ONE FIXED AND ANY NIUMBER OF MOVEABLE
PULLEYS.
G[^^ -^   160. Let there be a system of n moveable
pulleys and one fixed pulley combined as
shown in the figure, a separate string passing
over each moveable pulley; and let the tensions on the two parts of the string which
passes over the first moveable pulley be represented by T, and t,, those upon the two
parts of the string which passes over the
second by T, and t,, &c. Also, to simplify
the calculation, let all the pulleys be supposed of equal dimensions and weights, and
the cords of equal rigidity;
P.~ T1=at,  + b, and T,+W=T+ tl;
eliminating, T (+ a  T2 +  l +''(139).
Let the co-efficients of this equation be represented by a
and /3;




164:         SYSTEM OF ONE FIXED AND ANY. T1= T, + P.
Similarly, T,-aT,+3, T,=oaT,+/, T,=aT,+P, &c.=&c.,
T_,-1= aT +, T.= aP2 +.
Multiplying these equations successively, beginning from
the second, by a, a,3, &c., a-1, adding them together, and
striking out terms common to both sides of the resulting
equation, we have
T,= a~P, +/ +a/3  +  23 +.... +  -n/;
or summing the geometrical progression in the second
member,
T'=aP2+(         ). --..... (140);
Substituting for a and F their values from equation (139),
and reducing
T1=( +a) P2+(Wa+         + 1 -(I 1a) 
Now Pl=aT,+b;
P=*** a( 1 -  P2+ a(Wa+ b)    1- -       +b.. (141).
Whence observing, that, were there no friction, a would
become unity, and — +) -   2. We have (equation 121)
for the modulus of this system,,=a( 2+       "I) w2+  (fWa+b) 1   a   }+b sl,...(142).
161. If each cord, instead of having one of
its extremities attached to a fixed obstacle, had
been connected by one extremity to a moveable bar carrying the weight P2 to be raised
(an arrangement which is shown in the second
figure), then, adopting the same notation as
before, we have
T1=atl+b, at,+b=T,, T,=T1 + t+W.
Adding these equations together, striking out
terms common to both sides, and solving in
respect to T,, we have
t, f  i a \f    1 \W;
0         ^-^,,^T;^a+1   a+1




NUMBER OF MOVEABLE PULLEYS.            165
in which equation it is to be observed, that the symbol b
does not appear; that element of the resistance (which is
constant), affecting the tensions t, and t2 equally, and there
fore eliminating with T, and T,. Let -a+  be represented
by, then
t -t -- W. Similarly, t2=~at —W. 
1  2 a...... (143).
t=at-w, W&c.&c.,& t-     -a     W   J
Eliminating between these equations precisely as between
the similar equations in the preceding case (equation 140),
observing only that here / is represented by — aW, and that
the equations (143) are n-1 in number intead of n, we have
t        ^ -tt  - ( " -1 )...(144)
a —1
Also adding the preceding equations (143) together, we have
caW
t,+t,+ *... +ti- =  (t,+t+... t)-(n-1)w.
Now the pressure P, is sustained by the tensions t,, t,, &c.
of the different strings attached to the bar which carries it.
Including P2, therefore, the weight of the bar, we have
l+t2+..    +t-l+t t=P;. t,+t,+.. +t-_=   Pl —t;
and t,+... +t-=P,-t,;
aw
P. -     (P _tt_)-  (nl) (-1.
t=(1-c_)P2 + ctt + (n -1)_.
Substituting this value of t. in equation (144),
I                           a-<  N^I-D  ^. /  i^W  a'W/a -l-l\
t'=(- )    -~P' + ~t' + (-n  % — (Xf \ W -1 )'
Transposing and reducing,
(1-  )t  l-=(1  a)a(-lp- +-+W na — 1 — A;.t_( -   P     W  B(1an  a-  }
1K ~     3  al         1-a2 a




166        TACKLE OF ANY NUMBER OF SIIEAVES.
qow 11           1      l-aW1-  1 +  a-1
a-1       a                   1         an
(l1        - +a-1) — 1- I  - -'              Ian;
_      a* aP__-          i+' 
(+ arl)       a  (a 1+a-1)-1 
Now P,=atl+b;' Pi=(I (       -)1  (+ --              +' }   (1 5).
Whence observing that when a=1, (1 +a-) —1}- =2"-1,
we obtain for the modulus of uniform   motion (equation
121),
U- (1 + a-)n T-1  
1      a-1 au       + t Si... (146).
A TACKLE OF ANY NUMBER OF SHEAVES.
162. If an number of pulleys (called in this case sheaves)
be made to turn on as many different centres in the same
H   block A, and if in another block B there be simiA        larly placed as many others, the diameter of each
of the last being one half that of a corresponding pulley or sheave in the first; and if the same
cord attached to the first block be made to pass
in succession over all the sheaves in the two
blocks, as shown in the figure, it is evident that
the parts of this cord 1, 2, 3, &c. passing between
the two blocks, and as many in number as there
are sheaves, will be parallel to each other, and
will divide between them the pressure of a weight
X   P2 suspended from the lower block: moreover,
that they would divide this pressure between
them equally were it not for the friction of the
sheaves upon their bearings and the rigidity of
the rope; so that in this case, if there were n sheaves, the
tension upon each would be -P2; and a pressure P, of that




TACKLE OF ANY NUMBERE OF SHEAVES.     167
amount applied to the extremity of the cord would be sufficient to maintain the equilibrium of the state bordering upon
motion. Let T, T2, T2, &c. represent the actual tensions
upon the strings in the state bordering on motion by the preponderance of P,, beginning from that which passes from P2
over the largest sheaf; then
P,=aT1+b, T=-aT, +b,, T=aT+,,T,+b,
&c.==&c., T_= a T + b,;
where a,, a &c., b, b2, &c. represent certain constant coefficients, dependent upon the dimensions of the sheaves and
the rigidity of the rope, and determined by equation (131).
Moreover, since the weight P2.is supported by the parallel
tensions of the different strings, we have
P2=T,1+T,+...   +T,
It will be observed that the above equations are one more
in number than the quantities T1, T, T3, &c.; the latter may
therefore be eliminated among them, and we shall thus obtain a relation between the weight P2 to be raised and that
P1 necessary to raise it, and from thence the
modulus of the system.
To simplify the calculation, and to adapt
it to that form of the tackle which is commonly in use, let us suppose another arrangement of the sheaves. Instead of their
being of different diameters and placed all
in the same plane, as shown in the last
P/g  |   figure, let them be of equal diameter and
placed side by side, as in the accompanying
figure, which represents the common tackle.
The inconvenience of this last mode of arrangement is, that the cord has to pass front
the plane of a sheaf in one block to the plane
of the corresponding sheaf in the other oba  liquely, so that the parts of the cords between the blocks are not truly parallel to
one another, and the sum of their tensions is not truly equal
to the weight P2 to be raised, but somewhat greater than it.
So long, however, as the blocks are not very near to one another, this deflection of the cord is inconsiderable, and the
error resulting from it in the calculation may be neglected.
Supposing the different parts of the cord between the blocks
then to be parallel, and the diameters of all the sheaves and




168        TACKLE OF ANY NUMBER OF SHEAVES.
their axes to be equal, also neglecting the influence of the
weight of each sheaf in increasing the friction of its axis,
since these weights are in this case comparatively small, the
co-efficients a1, a2, a will manifestly all be equal; as also
bFl b2 b3;. P -aT,+b, T,=aT+b, T,=aT,+ b, T,           (14 i
&c.=&c., T,-1=aTn+b.....(1
also         P2=T,+T,+T3+..... +T,.
Multiplying equations (147) successively (beginning from the
second) by a, a', a', and a-l1; then adding them together,
striking out the terms common to both sides, and summing
the geometric series in the second member (as in equation
140), we have
P, =anT.+b a. -
Adding   equations (147), and  observing that T,+T,+.... X+T,=P,, and that P,+T,+T,+....          +T,_=
P +P, —T, we have
P,+P,-T,=aP,2+b.
Eliminating T,m between this equation and the last,
P,=a" P —P,( --   )- - b} +ba-'1;
I      I         - 2- - 1 a_ —)r +nba"         b
P=                    - -   a-1_*.. (148).
To determine the modulus let it be observed, that, neglecting friction and rigidity, a becomes unity; and that for this
value of a, a(a —1 becomes a vanishing fraction, whose
value is determined by a well known method to be -.
Hence (Art. 152.),
* Dividing numerator and denominator of the fraction by (a —) it becomes
- +, —a  +1' which evidently equals - when a=1. The modulus
a'l-'!-... + —' -n
may readily be determined from equation (148). Let Si and S2 represent the
spaces described by P1 and P2 in any the same time; then, since when the
blocks are made to approach one another by the distance S2, each of the n portions of the cord intercepted between the two blocks is shortened by this dis



THE MODULUS OF A COMPOUND MACHINE.         169
U=t   a-        + {   an -   b Is..... (149).
a-        a? - -l  a-2 +  1...
Hitherto no account has been taken of the work expended
in raising the rope which ascends with the ascending weight.
The correction is, however, readily made. By Art. 60. it
appears that the work expended in raising this rope (different parts of which are raised different heights) is precisely
the same as though the whole quantity thus raised had been
raised at one lift through a height equal to that through
which its centre of gravity is actually raised. Now the cord
raised is that which may be conceived to lie between two
positions of P2 distant from one another by the space S2, so
that its whole length is represented by nS,; and if uh represent the weight of each foot of it, its whole weight is represented by PnS,: also its centre of gravity is evidently raised
between the first and second positions of P2 by the distance
IS,; so that the whole work expended in raising it is repre-,P'S 2
sented by  nS22 or by -, since S_-nS,.   Adding this
work expended in raising the rope to that which would be
necessary to raise the weight P2, if the rope were without
weight, we obtain*
TT=n..       nU  {   i-2. Si+. S2..... (150),
1     1 U2+    an-1   a-1}S+ 2n
which is the MODULUS of the tackle.
THE MODULUS OF A COMPOUND MACHINE.
163. Let the work of a machine be transmitted from one
to another of a series of moving elements forming a compound machine, until from the moving it reaches the working
point of that machine. Let P be the pressure under which
the work is done upon the moving point, or upon the first
moving element of the machine; P, that under which it is
tance S2, it is evident that the whole length of cord intercepted between the
two blocks is shortened by nS2; but the whole of this cord must have passed
over the first sheaf, therefore Si=nS2. Multiplying equation (148) by this
equation, and observing that Ul=P1S1 and U2=P2S2, we obtain the modulus
as given above.
* A correction for the weight of the rope may be similarly applied to the
modulus of each of the other systems of pulleys. The effect of the weight of
the rope in increasing the expenditure of work on the friction of the pulleys is
neglected as unimportant to the result.




170        MODULUS OF A COMPOUND MACHINE.
ielded from the first to the second element of the machine;
P2 from the second to the third element, &c.; and P1 the
pressure under which it is yielded by the last element upon
the useful product, or at the working point of the machine.
Then, since each element of the compound machine is a simple machine, the relation between the pressures applied to
that element when in the state bordering on motion will be
found to present itself under the form of equation (119)
(Art. 152), in all cases where the pressure under which the
work upon each element is done is great as compared with
the weight of that element (see Art. 166.).
Representing, therefore, by a, 2, a,... *b, b, b3.., certain constants, which are given in terms of the forms and
dimensions of the several elements and the prejudicial resistances, we have
P=a P,+,,       P, =2P2+         P,=-aP3 + 6b,
&.= &c., P _=anP. + b.
Eliminating the n-1 quantities P 2, P 3, P..., P,, between
these n equations, we obtain an equation, of the form,
P —=aP+b....(151);
where a=ala2a1... a;, and                    152
b=aa..a a,1b+aa2...a _+  a.*..+  + ab+ b, (
If the only prejudicial resistance to which each element is
subjected be conceived to be friction, and the limiting angle
of resistance in respect to each be represented by p; then
considering each of the quantities a, bZ, a, b, as a function
of p, expanding each by Maclaurin's theorem into a series
ascending by powers of that variable, and neglecting terms
which involve powers of it above the first, we. have
aL=a, ) +     ) (     b()=b(~) a+ -  d+  ).p
a=a(0)~ + (da,   b2=b2) + ( d,)    &c.=&c.
where, a,(0, b,(O), a(, b2>, represent the values of a, b,, a2
b, &c., when =0 and (    )       (    )  &c. represent
the similar values of their first differential co-efficients.




AXES.                     171
Let
(da(0l)  A_     _       d\(0 )      ~
Io). %              \=a l a — --  Y
(da2)(O) -a(0)              pb)  &(). c= &c.
Therefore a =a (0) (1 +  b= bl() (1 + P/), a — a, (1)(l+a,
I5- 62()(1+/32), &c.=&c.; where ac, f1, %, 2, &c., each
involving the factor p, are exceedingly small. Substituting
the values of a, c2, &c. in the expression for a, and neglecting terms which involve dimensions of al, a2, &c. above the
first, we have
a= 1(~) a2. *. a( 0) i1 + c1+ c2+  +  *....+.M... (153).
Now the co-efficient of the first term of the modulus is
represented (equation 121) by a —,  representing the coefficient of the first term of equation (119), also substituting
the value of a from  equation (153), and observing that
a
a(o) =(0) a. a.... a(), we have  =  {1+ +.   +   };.. U={ 1 +    4+c3+...- +u. s+j U,+..... (154),
which is the modulus of a compound machine of n elements,
U  representing the work done at the moving point, U,,
that at the workting point, S the space described by the
moving point, and b a constant determined by equation
(152).
164. THE CONDITIONS OF THE EQUILIBRIUM OF ANY TWO PRESSURES P1 AND P2 APPLIED IN THE SAME PLANE TO A BODY
MOVEABLE ABOUT A FIXED AXIS OF GIVEN DIMENSIONS.
In fi. 1. the pressure P1 and P. are shown acting on opposite sides of the axis
\ Figl:m..     r Q3 2 whose centre is C, and?_/ \. \  B'      ging. 2. upon the same.l -—.N....i              side.  Let the direcY  tion of the resultant
J,7'~ of PI and P2 be repre3  X  \   sented, in the first,I ycase, by IR, and in
the second by R1. It




172                        AXES.
is in the directions of these lines that the axis is, in the two
cases, pressed upon its bearings.    Suppose the relation
between Pi and P. to be such that the body is, in both
cases, upon the point of turning in the direction in which
P, acts. This relation obtaining between P, and P,, it is
evident that, if these pressures were replaced by their resultant, that resultant would also be upon the point of causing the body to turn in the direction of P,. The direction
IR of the resultant, thus acting alone upon the body, lies,
therefore, in the first case, upon the same side of the centre
C of the axis as P, does, and in the second case it lies upon
the opposite side; and in both cases, it is inclined to the
radius (K at the point K, where it intersects the axis at an
angle CKR, equal to the limiting angle of resistance (see
Art. 153.). Now, the resistance of the axis acts evidently in
both cases in a direction opposite to the resultant of P, and
P,, and is equal to it; let it be represented by R. Upon
the directions of P,, P,, and R, let fall the perpendiculars
CA1, CA,, and CL, and let them be represented by a,, a,,
and X. Then, by the principle of the equality of moments,
since P,, P,, and R are pressures in equilibrium,..Pa,=Pa, +R.
If P, had been upon the point of yielding, or P, on the
point of preponderating, then R would have had its direction
(in both cases) on the other side of C; so that the last equation would have become
Pa, + XR-Pa,.
According, therefore, as P, is in the superior or inferior
state bordering upon motion,
Pa,-PA,-C=(~X)R.
And if we assume X to be taken with the sign + or -, according as P, is about to preponderate or to yield, then
generally
P~a —Pra=kR.....  (155).
Now, since the resistance of the axis is equal to the resultant
of P, and P,, if we represent the angle P1IP2 by if, we have
(Art. 13.)
* The arrows in the figure represent, not the directions of the resultants
but of the resistances of the axis, which are opposite to the resultants.
C Care must be taken to measure this angle, so that P, and P2 may have




AXES.                      178
R= 4/P12+ 2PP, cos. +P'.
Substituting this value of R in the preceding equation, and
squaring both sides,
(P1a,-Pa,)`= X(P, +2P1P, cos.,+P, );
transposing and dividing by P22,'(   ~    )- 2     ^ (a,a, +  COS.O.  -(^-2);
solving this quadratic in respect to \ P5
P1 (laa+X2 cos.,) ~    V2(a1a2  cos. )-(a2-X2) (a2  2).
2,-                    1a2 2X 2
P__(ala +X2 cos.') ~  X  (a12 - 2aa cos. + a,22) _X2 sin.2.
P2                   2 1-x2
Now let the radius CK of the axis be represented by p,
and the limiting angle of resistance CKR by p; therefore
X=CL=CK sin. CKR=p sin. p. Also draw a straight line
firom A, to A2 in both figures, and let it be represented by L;.a. 2-2,aa2 cos. A1CA2+22=L2. Now, since the angles at
A, and A2 are right angles, therefore the angles AJIA2 and
ACA2 are together equal to two right angles, or ACA2 +
=ir; therefore AICA2=r —,, and cos. A1CA2 —cos. i;
therefore L2=a12 + 2aa cos.  a22: substituting these values
of L2 and X in the preceding equation,
p   (aa2 + p2 cos. sin. 2') + p sin. p (L2- p2 sin. 21 sin. 2p)1
(a — p2 sin. 2)
P2. (156).
The two roots of the above equation are given by positive
and negative values of x, they correspond therefore (equation 155) to the two states bordering upon motion. These
two values of X are, moreover, given by positive and negative values of q; assuming therefore p to be taken positively
or negatively, according as P1 preponderates or yields, we
may replace the ambiguous by the positive sign.    The
their directions both towards or both from the angular point I (as shown in the
figure), and not one of them towards that point and the other from it. Thus,
in the second figure, the inclination t of the pressures Pi and Pa is not the
angle A2IP1, but the angle P2aIP. It is of importance to observe this distinction (see note p. 194.).




174                     AXES.
relation above determined between P1 and P2 evidently
satisfies the conditions of equation (119). We obtain therefore for the modulus (equation 121)
U   lad (a1a + p cos.' sin. )) + p(L-p sin.'I sin. I)+ sin. p
kaW'                (,p sin.2^)
U2... (157).
If terms involving powers of (a-) sin. p above the first be
neglected, that quantity being in all cases exceedingly
small, we have
{ (              in    P..... (158),
U,= {   +. a )sin. p u2... (159).
165. To determine the resultcant R of any number of pressures P  P, P...., in terms of those pressures, and the
cosines of their inclinations to one another.
Let a, a, a  &c. represent the inclinations
IAC, IBC, &c. of the several pressures P,,
P, &c. to any given axis CA in the same
plane; and let,,12,i, 1  &c. represent the
inclinations of these pressures severally to one
another.
Now ZAIB     Z IBOC- ZIAC (Euc. I. 32.);
12,  2 = —  C    os. =   C=os. cs. 2+sin. c sin. a.
Similarly, cos. 13-=cos. al cos. a +sin. al sin. a;,
cos. y23=cos. %2 cos. a%+sin. 2 sin. a%.
Now I2=(P, cos. 1 +P, cos.   + P, cos. +.. )2+ (P
sin. al+P2 sin. a2-+P sin. a3+... )2, (equation 9, Art. 11.).
Squaring the two terms in the second member, adding the
results, and observing that cos. 2a,+ sin. 2a1,
R2=P+2 +P,2 P32... + 2PiP, (cos. a1 cos. a2+sin. a sin. a)
+2PIPA (cos. al cos. a+ sin. a, sin. a)+....;




AXES.                        175
R2*'=Pl2+P+P2+'-... +2PCP, cos.,, +2PP cs. Icos
+2P2P, cos. 2,,+ &c..... (160).
166. THE CONDITIONS OF THE EQUILIBRIUM      OF THREE PRESSURES P   P, P, IN THE SAME PLANE APPLIED TO A BODY
MOVEABLE ABOUT A FIXED AXIS, THE DIRECTION OF ONE OF
THEM, P2, PASSING THROUGH THE CENTRE OF THE AXIS, AND
THE SYSTEM BEING IN THE STATE BORDERING UPON MOTION
BY THE PREPONDERANCE OF P,.
Let,, 3, I%3 represent the inclinations of the directions of.B         -i ^the pressures P1, P2, P3 to one
another, a and a the perpen-.l pA  n      diculars let fall from  the cent/ K         ^i~l.       tre of the axis upon P; and P2,
and x the    perpendicular let
J/  \     wl\       iIfall from the same point upon
the resultant R  of P,, P,, P,.
~p  ~\           \\~ Then, since   R is equal and
e  e     A\ft  opposite to the resistance of
~\ -'      ^*    ~   the axis (Art. 153.), we have,
by the principle of the equality of moments, Pa1 -P2c2=
h;, for P3 passes through the centre of the axis, and its
moment about that point therefore vanishes.
Substituting the value of R from equation (160),
PAa -PA -      PiP +Pl 2 + P3 + 2P1P2 cos. 12 +
2P1P3 cos. 1 + 2P2P3 coS. 12' 
Squaring both sides of this equation, and transposing,
P1 (a-a)-2P, 1P2ala2-      (P2 cos.12+P3 cOS. 113)1 =
-P22a22 + x2 p, + P32 + 2PP, cos. 23.
If this quadratic equation be solved in respect to P,, and
* In which expression it is to be understood that the inclination t12 of the
directions of any two forces is taken on the supposition that both the forces
act from or both act towards the point in which they intersect,
and not one towards and the otherfrom that point; so that in
the case represented in the accompanying figure, the inclination t12 of the two forces P1 and P2 represented by the arrows,
is not the angle PiIP2, but the angle QIP1, since IQ and IP1 are
directions of these two forces, both tending from their point 
of intersection; whilst the directions of P2I and IPi are one
of them towards that point, and the otherfrom it.




176                       AXES.
terms which involve powers of X above the first be omitted,
we shall obtain the equation
P   2il2-=P2ala2+X
VP,2(a1,+2aa2 cos.,2+a22)+Ps2al2P2P3a,(a2 cos.l,,+a, cos.2$);
or representing (as in Art. 164.) the line which joins the
feet of the perpendiculars, a, and a2 by L, and the function
a, (a2 cos.,13+a, cos. I,,) by M, and substituting for X its
value p sin. p,
_=(  CY2\P  [p sin. ~j  + P 2aI2
Pi= ()a P2+ (P a 2   i {P22L2 +Pa + 2P2PM    *.. (161).
Representing (as in Art. 152.) the value of P1 when the
prejudicial resistances vanish, or when -p=O, by 1(o), we
have P1(0)= (a P. Also by the principle of virtual velocities P(0). S-=P2. S,. Eliminating P,(O) between these
equations, we have S-== al' S. Multiplying equation (161) by
this, PS1=P2,S + P i {P2 2S22L + 2P2P3S22+P,,, 2S22a 12}.
Substituting U, for PS,, U, for P,S2, and observing that
a.
s2    SO
U   U,+a sin     U22L2 + j + 2U2PSM (a!) +
P32S12a2... (162.)
which is the MODULUS of the system.
If P, be so small as compared with P2 that in the expansion of the binomial radical (equation 161), terms involving
powers of 3 above the first may be neglected; then,
P2
* It will be shown in the appendix, that this equation is but a particular
case of a more general relation, embracing the conditions of the equilibrium
of any number of pressures applied to a body moveable about a cylindrical
axis of given dimensions.




AXES.                     i77
-P  ^P  JL ^8pm sin. ~t j P  PW
s (a')  +(T;) {^4       (); p      *... (163);
which equation may be placed under the form
al)     6aa 
Whence observing that the direction of P, being alwa?,
through the centre of the axis, the point of applicationi of
that force does not move, so that the force P, does not work
as the body is made to revolve by the preponderance of PI;
observing, moreover, that in this case the conditions of
equation (119) (Art. 152.) are satisfied, we, obtain for the
modulus
p                 } U+ (   ) ()     Ssin      (164)
167. The conditions of the equilibrium of two pressures P,
and P2 applied to a body moveable about a cylindrical axis,
taking into account the Weight of the body and supposing it
to be symmetrical about its axis.
The body being symmetrical about its axis, its centre of
gravity is in the centre of its axis, and its weight produces
the same effect as though it acted continually through the
centre of its axis. In equation (161.) let then P, be taken to
represent the weight W of the body, and 13, 23 the inclinations of the pressures P1 and P2 to the vertical. Then
P1()P2~+ (       2p ) { P  +L22P2WM+WW2a+2... (165.)
Also by the equation (162) we find for the modulus
I=n,+     sin. ( p  V j  u +aaw S
U2 =U 2+           U22L2 + 2U2WS Mj2 I
+W   ja... (166.)
And in the case in which P, is considerable as compared
with W, by equations (163, 164).
12




178              THE DIRECTION OF THE
p(A) {+         sin   } P      )L 2+ ) ()W  sin.**  (167.)
UIT { 1 + p sin. q ) U2+ (a  ) WS1 Sin-  **  (68.)
168. A MACHINE TO WHICH ARE APPLIED ANY TWO PRESSURES
P1 AND P2, AND WHICH IS MOVEABLE ABOUT A CYLINDRICAL
AXIS, IS WORKED WITH TH;E GREATEST ECONOMY OF POWER
WHEN THE DIRECTIONS OF THE PRESSURES ARE PARALLEL.
AND WHEN THEY ARE APPLIED ON THE SAME SIDE OF THE
AXIS, IF THE WEIGHT OF THE MACHINE ITSELF BE SO SMALL
THAT ITS INFLUENCE IN INCREASING THE FRICTION MAY BE
NEGLECTED.
For, representing the weight of such a machine by W, and
neglecting terms involving W sin. p, it appears by equation
(168) that the modulus is
Ui=U2,   1 +a- sin.e;
whence it follows that the work Ul, which must be done at
the moving point to yield a given amount TT at the working
point, is less as L is less.
2-q^ ^Now           L   represents
/'s  \  mi2    the distance A1A, ber/  a  > \ Itween the feet of the
A/.-"-                  -   \-M. S^^  perpendiculars CA, and
Xr      OACA2, which distance is
evidently least when P,
/ \                    \ %,,and P2 act on the same,'7  \  a4       side of the axis, as in
fig. 2, and when CA,
and CA2 are in the same straight line; that is, when P, and
P2 are parallel.
169. A MACHINE TO WHICH ARE APPLIED TWO GIVEN PRESSURES P1 AND P2 AND WHICH IS MOVEABLE ABOUT A CYLINDRICAL AXIS, IS WORKED WITH THE GREATEST ECONOMY OF
POWER, THE INFLUENCE OF THE WEIGHT OF THE MACHINE
BEING TAKEN INTO THE ACCOUNT, WIEN THE TWO PRESSURES




GREATEST ECONOMY OF POWER.         179
ARE APPLIED ON THE SAME SIDE OF THE AXIS, AND WHEN
THE DIRECTION OF THE MOVING PRESSURE P1 IS INCLINED TO
THE VERTICAL AT A CERTAIN ANGLE WHICH MAY BE DETERMINED.
Let P1 be taken to represent the weight of the machine,
and let its centre of gravity coincide with the centre of its
axis, then is its modulus represented (equation 166.) by
U =U2+P sin.   { U2L2 + 2U2P3SM () + P2S12a22;
a a                   a
in which expression the work U1, which must be done at the
moving point to yield a given amount UT of work at the
working point, is shown to be greater than that which must
have been done upon the machine to yield the same amount
of work if there had been no friction by the quantity
p sin.{  U222+2     S      +2S12a2 
The machine is worked then with the greatest economy of
power to yield a given amount of work, U2, when this function is a minimum. Substituting for L2 its value
ac2 + 2ala2 cos.,2 + a23, and for M its value a, Ia2 cos.,1 +
acos. 2,} (see Art. 166.), also for S1(,) its value S,. it becomes
p sm. 4 {J. (a,2 + 2acaoS.cos.i,+ a2) + 2U2P2S2a(a2cos.1 +
alcos.i2)+P2812a }....(169.)
Now let us suppose that the perpendicular distance a, from
the centre of the axis at which the work is done, and the inclination 23 of its direction to the vertical, are both given, as
also the space S2 through which it is done, so that the work
is given in every respect; let also the perpendicular distance
al at which the power is applied, and, therefore, the space S,
though which it is done, be given; and let it be required to
determine that inclination,2 of the power to the work which
will under these circumstances give to the above function its
minimum value, and which is therefore consistent with the
most economical working of the machine.
Collecting all the terms in the function (169.) which con



180              THE DIRECT'ION OF THE
tain (on the above suppositions) only constant quantities, and
representing their sum by C, it becomes
p s. {q 2a,a2U,(U2 cos. 12 + PS, cos. y1) +  ( 
Now C being essentially positive, this quantity is a minimum when 2aa2U2(U2 cos.,2+P3S2 cos. 1,, is a minimum; or,
observing that U2=P S, and dividing by the constant factor
2a1a2U2S2, when
P2 cos. I + P cos. i3 is a minimum.
~ A   From the centre of the axis C let lines Cp,
2 be drawn parallel to the directions of the
pressures P1P, respectively; and whilst C2,
and Cp2 retain their positions, let the angle
CP, or ~3 be conceived to increase until P,
attains a position in which the condition
P, cos. I2+P, cos. s,3=a minimum is satisfied.,     Now   2CPa=p,0 2-P-,CP,, or,1=3 -12- 23
substituting which value of 23 this condition
becomes
P2 cos. t 1 + P cos. (ti — -t) a minimum,
or P2 cos. 1,2 + Ps cos. 12 cos. 3 + P3 sin.,2 sin. i, a minimum
or (P2 + P, cos.'23) cos. 12 + Pg sin. 3,, sin. i,2 a minimum.
P. sin.,
Let now  P         = tan. 7,
P2 +r. cos. *2S. (P +P, cos.,a2) cos.,2 + (P2+P3 cos.,23) tan. y sin.,,1 is a minimum, or dividing by the constant quantity (P2 + P3 cos. 23)
and multiplying by cos. y,
c. os. + cos i. y +   sin. s  =cos. (i2-y) is a minimum.
*  122 —-"',l=,a,- ~ n.~  P. sin. ~2S  }...(1Z0)
P2 +P8 cos....
To satisfy the condition of a minimum, the angle plCp2
must therefore be increased until it exceeds 180~ by that
P8 sin. I
angle y, whose tangent is represented by P+ ci2. 3.  To
P,2   8. c. os. 23
determine the actual direction of Pi producee then pC to q,
make the angle qCr equal to 7; and draw Cm perperendcular




GREATEST ECONOMY OF POWER.           181
to Or, and equal to the given perpendicular distance a, of
the direction of P, from the centre of the axis. If mP1 be
then drawn through the point m parallel to Cr, it will be in
the required direction of Pi; so that being applied in this
direction, the moving pressure P, will work the machine with
a greater economy of power than when applied in any other
direction round the axis.
It is evident that since the value of the angle i1 or p2Cp1,
which signifies the condition of the greatest economy of
power, or of the least resistance, is essentially greater than
two right angles, Pi and P2 must, TO SATISFY THAT CONDITION,
BOTH BE APPLIED ON THE SAME SIDE OF THE AXIS. It is then
a condition necessary to the most economical working of any
machine (whatever may be its weight) whicA is moveable about
a cylindrical axis under two given pressures, that THE MOVING PRESSURE SHOULD BE APPLIED ON THAT SIDE OF THE AXIS
OF THE MACHINE ON WHICH THE RESISTANCE IS OVERCOME, OR
THE WORK DONE.   It is a further condition of the greatest
economy of power in such a machine, that the direction in
which the moving pressure is applied should be inclined to
the vertical at an angle %2,, whose tangent is determined by
eguation (170.).
When 23=0, or when the work is done in a vertical
direction, tan. y=0; therefore i2=<, whence it follows that
the moving power also must in this case be applied in a vertical direction and on the same side of the axis as the work.
When 23 =    or when the work is done horizontally, tan.
P 
2 _l,=<+tan.-1 p
The moving power must, therefore, in this case, be applied
on the same side of the axis as the work, and at an inclination to the horizon whose tangent equals the fraction
obtained by dividing the weight of the machine by the
working pressure.
3ir
Since the angle 12 is greater than v and less than -
cos. 12 is negative; and, for a like reason, cos. i,, is also in
certain cases negative. Whence it is apparent that the
function (169.) admits of a minimum value under certain




182                 THE PULLEY.
conditions, not only in respect to the inclination of the
moving pressure, but in respect to the distance a, of its
direction from the centre of the axis. If we suppose the
space S1 through which the power acts whilst the given
amount of work U. is done to be given, and substitute in
that function for the product S2a, its value Sla2, and then
assume the differential of the function in respect to a, to
vanish, we shall obtain by reduction
U22 + SU PS cos. is +P(2S1'
ea= —a.. U2 cos. + TUPS cos. 2.      (171)
If we proceed in like manner assuming the space S2 instead
of S, to be constant and substituting in the function (169.)
for S a, its value S2a,, we shall obtain by reduction
P2a2
a1- -p2 cos. 12+Ps cos. y13.
It is easily seen that if when the values of t,, and i2, determined by equation (170.) are substituted in these equations,
the resulting values of a1 are positive, they correspond in
the two cases to minimum values of the function (169.), and
determine completely the conditions of the greatest economy
of power in the machine, in respect to the direction of the
moving pressure applied to it.
170. THE PULLEY, WHEN THE TENSIONS UPON THE TWO
EXTREMITIES OF THE CORD HAVE NOT VERTICAL DIRECTIONS.
I,/     In the case in which the two parts of the
/.    string which pass over a pulley are not
Zy     parallel to one another, the relations estabJ/'    lished in Article 158. no longer obtain;
and we must have recourse to equation
(167.) to establish a relation between the
tensions upon them in the state bordering'    upon motion.  Calling W  the weight of
the pulley, a its radius, and observing
that the effect of the rigidity of the cord,
in increasing the tension P1, is the same
as though it caused the tension P, to bei,    I   ce      E1       D
-2 o |  come P, (1+ -) + c(Art. 142.), we have




THE PULLEY.                    183
P  a   1 +   sBin.;   P     (1+  )+a   +MP Wsin.,;
I a,       -        (2\      a   a   &   L1
l     (  +a)  1 + asin'p } P2+
D DL           MW
a+     P sin. ++.    sin. P;.;
or,
P,  1+ a)      1l+ ia {1 +
r(2  LDa) P  l i.  *... (172.)
where L represents the chord AB of the arc embraced by
the string, and M=a2 (cos.,, + cos. 3,,),,3 and 23 representing the inclinations of P1 and P2 to the vertical: which
inclinations are measured by the angles PEP, and P2FP,,
or their supplements, according as the corresponding pressures P, and P2 act downwards, as shown in the figure, or
upwards (see note to Article 165.); so that if both these
pressures act upwards: then the cosines of both the angles
become negative, and the value of M   becomes negative;
whilst if one only acts upwards, then one term only of the
value of M becomes negative.
Substituting this value for M, observing that L=2a cos.,,
where 2, represents the inclination of the two parts of the
cord to one another (so that 2= -3+23 ), and omitting terms
which involve products of two of the exceedingly small
DE         P
quantities -) a, and -sin. 9 we have
Ip (,,  pE D2.     _     Wp(cos.,, + cos.,,)sin. ~p
- 1 +_= + =p - cos. sin. } U3+    a +
Wp(cos.,, +cos.,) sin. (p SI. (173);
2a cos.; i




184                 THE PULLEY.
which last equation is the modulus to the pulley, when the
two parts of the string are inclined to the vertical and to
one another.
171. If both the strings be inclined at equal angles to the
vertical, on opposite sides of it; or if,-i=g23=l, so that cos.
1,+cos. 2,,=2 cos. a, then equations (172.) and (173.) become
P,    1 +l   + Pcos. si    +n. +  sin.   (174),
a CaC               a   a
U      -1+a-+- pcos. sin. P U2,+  +  sin s.   (175.)
172. If both parts of the cord passing over a pulley be in
the same horizontal straight line, so that the
pulley sustains no pressure resulting from the
tensio~n upon the cord, but only bears its
weight, then i —, and the term involving
cos. in each of the above equations vanishes. It is, however, to be observed that the weight bearing upon the axis
of the pulley is in, this case the weight of the pulley
increased by the weight of cord which it is made to support.
So that if the length of cord supported by the pulley be
represented by s, and the weight of each foot of cord by k,
then is the weight sustained by the axis of the pulley represented by W+ks. Substituting this value for W in equation (175.), and assuming cos.,=0, we have
UT=(l +   U+ -    D+(W +t8) p sin. p S..... (176.)
\.....    I
173. Let us now suppose that there are n equal pulleys
sustaining each the same length
i   s of cord, and let Un represent
the work yielded by the rope
(through the space SI) after it
has passed over the nth, or last
pulley of the system, UT being
that done upon it before it
-=i"  -L - -    passes over the first pulley;
then by Art. 163., equations




THE PULLEY.                 185
152. 154. and 176., neglecting terms involving powers of
EDP
--'- sin. p above the first, and observing that a1=a2=
E              E
&c.=1+-,     o1=a,=&c.=a- bl=b5,&c.=- D+(W+s)
p sin. p }, we have
U1+ (+,+   { D+(W~+8)p sin. 9 } S,.
Representing the whole weight of the cord sustained by the
pulleys by w, and observing that Pn-s=w, we have
1= (l + a n"a nD+(nW+w)p sin. p S1,.. (177.)
In the above equations it has been supposed, that although
the direction of the rope on either side of each pulley is so
nearly horizontal that cos., may be considered = 0, yet that
it does so far bend itself over each pulley as to cause the
surface of the rope to adapt itself to the circumference of
the pulley, and thereby to produce the whole of that resistance which is due to the rigidity of the cord. If the tension
were so great as to cause the cord to rest upon the pulley
only as a rigid rod or bar would, then must we assume E=O
and D=0 in the preceding equations.
174. If one part of the cord passing over a pulley have a
horizontal, and the other a vertical direction, as, for instance,
when it passes into the shaft of a mine over the sheaf or
wheel which overhangs its mouth; then one of the
angles,, or i,, (equation 173.) becomes;, and the
other 0 or ~, according as the tension on the vertical cord is downwards or upwards, so that cos.
j, + cos. 2,3= i~1, the sign ~ being taken according
as the tension upon the vertical cord is downwards
or upwards.  Moreover, in this case (Art. 169.)
1
~1 and cos. i=; therefore (equation 173.)
4         1              P/2
pi a+  +  n^P+pD~          s..- (18),
) a  a            a c   4/2 




186                 THE PULLEY.
U,=   1 +E + p 42 sin.  U, +   D   sin.  S     (179).
aO   a            aO
174. The modulus of a system of any number of pulleys, over
one of which the rope passes vertically, and over the rest
horizontally.
Let U, represent the work
- Ad'r     done upon the
rope  through
the space S, before it passes
horizontally
over the first
pulley of the
_ =..... system, and let
it pass horizontally over n such pulleys; and then, after having passed
over another pulley of different dimensions, let it take a
vertical direction, descending, for instance, into a shaft. Let
U2 be the work yielded by it through the space Si immediately that it has assumed this vertical direction: also let u1
represent the work done upon it in the horizontal direction
immediately before it passed over this last pulley of the
system. Then, by equation (179.),
ui-{ 1++     P  sin. p Uq2+q  D+ Tsin. p Si.
a    a            a [    2
Also, by equation (177.) representing the radius of each of
the pulleys which carry the rope horizontally by a, the radius
of its axis by p,, and its weight by W1, and observing that
UT is here the power and u. the work, we have
uI   ( 1+  -)ui+ - n7D+(W,+w)p, sin. g S,1.
Eliminating the value of u, between these equations, and
neglecting powers above the first in, &c., we have
U          \= 1+E (+  \+ p /2sin  U2+ D(    +




rTHE PIVOT.                187
+. (W+w)p      sin.  S.... (180o.)
a,   ]a 1       a,
175. If the strings be parallel, and their common
inclination to the vertical be represented by a, so
that i1, —  =; then, since in this case L=-2a, we
have (equation 172.), neglecting terms of more than
one dimension in E and P
t + (a+D_)__    sin. }.. (181.)
therefore the sign of cos. f is to be taken (as before explained)
positively or negatively, according as the tensions on the
cords act downwards or upwards. If the tensions are vertial,  0 or, according as they at upwards or downwards,
so that cos.~ +d 1. The above equations agree in this case,
E $ 2P D.2
ing cos. in the above equations vanish.
176. THE FRICTION OF A PIVOT.
in which equation, is to be taken greater or less than,and
herefore te sign of cos.   axi s to b rests upotake n as   bearings,
not by its convex circumference, but by
positively or negativey, according ashown the tensions on tying figure, it is called a pivot. Let W
corct downrepresent. If the pressure tensions are verti-such a
cal, or accpivot suppose they act upwards or direction peards,
so ta cos.   1. Te above equations surface, and this case
as they ought with equations (131.) and (132.). If the parallel tensions are horizontal, then, —, and the termns involving cos., in the above equations vanisA.
176. THE FRICTION OF A PIVOT.
When an axis rests upon its bearings,
not by its convex circumference, but by
its extremity, as shown in the accompanying figure, it is called a pivot. Let W
represent the pressure borne by such a
pivot supposed to act in a direction perpendieular to its surface, and to press,




188'HE PIVOT.
equally upon every part of it; also let pi represent the
radius of the pivot; then will rpl2 represent the area of the
pivot, and W  tie pressure sustained by each unit of tlat
area. And iff represent the co-efficient of friction (Art.
13.), Wf will represent the force which must be applied
parallel to the surface of the pivot to overcome
{US,:-  the friction of each such unit. Now let the dotted lines in the accompanying figure represent
an exceedingly narrow ring of the area of the pivot, and let
p and p + Ap represent the extreme radii of this ring; then
will its area be represented by +(p + Ap)2_ —p2, or by r 12p(Ap) +
(Ap)2, or, since Ap is exceedingly small as compared with p,
by 2~pap. Now the friction upon each unit of this area is
represented by Wf; therefore the whole friction upon the'Ir  pPi2P1
ring is represented by -- p. 2pap, or by  pap, and the
moment of that friction about the centre of the pivot by
2WWf. p-2, and the sums of the moments of the frictions of
Pi1
all such rings composing the whole area of the pivot by
pI
2,Wf. pap, or by z2p2^, or by 2Wf      dp or by
P1i            PI             Pi 2
sWf
p2Wf pi3 or by Wfp..... (83.);
whence it appears that the friction of thepivot produces the
same effect to oppose the revolution of the mass which rests
2pon it, Gas though the whole pressure which it sustains were
collected over a point distant by two-thirds of its radius from
its centre.
If 0 represent the angle through which the pivot is made
to revolve, then -p16 will represent the space described by
the point last spoken of; so that the work expended upon
the resistance Wf acting there, would be represented by
{Wp1f8, which therefore represents the work expended upon
the friction of the pivot, whilst it revolves through the angle




AXES.                     189
&;. so that the work expended on each complete revolution
of the pivot is represented by
rpfW........      (184)
177. If the pivot be hollow, or its surface be an annular
instead of a continuous circular area, then
representing its internal radius by p2, and
observing that. its area is represented by
r (pl —p2), and therefore the pressure upon
each unit of it by r(P_22) and the fricWf
tion of each such unit by  (22, we obtain, as before,
for the friction of each elementary annulus the expression
2w.Y1  papz and for the sum of the moments of the frictions
Pi' —P2
w2f   Fp,
of all the elements of the pivot,-t j  dp, or
P2
*WS (_z3).
Let r represent the mean radius of the pivot, i. e. let
r= —(p, + 2); and let I represent one half the breadth of the
ring,?.e. let l=-(p —2); therefore p,=r+t  and p,=r-l.
These values of p1 and p, being substituted in the above formula, it becomes
t^f   (r+1)-  (rl)y } oQr iWf {   4r l;
o-rm r{1+1() }.or,
orW     l+t...... (18;
whence it follows that the friction of an ctnnular pivot produces the same eefect as though the whole pressure were collected over a point in it distant by r l +1(-) } from ito
centre, where r represent its mean radius awd I one half its
breadth. From this it may be shown, as before, that the




190                    AXES.
whole work expended upon each complete revolution of the
annular pivot is represented by the formula,
2fr { i+.. W..    (186.)
178. To DETERMINE THE MODULUS OF A SYSTEM OF TWO PRESSURES APPLIED TO A BODY MOVEABLE ABOUT A FIXED AXIS,
WHEN THE POINT OF APPLICATION OF ONE OF THESE PRESSURES IS MADE TO REVOLVE WITH THE BODY, THE PERPENDICULAR DISTANCE OF ITS DIRECTION FROM THE CENTRE REMAINING CONSTANTLY THE SAME.
Let the pressures PI and P2, instead of retaining constantly
(as we have hitherto supposed them to do)
l - A  the same relative positions, be now conceived
A      continually to alter their relative positions by
1   —..... ---- the revolution of the point of application of
P1 with the body, that pressure nevertheless
retaining constantly the same perpendicular
distance a from the centre of the axis, whilst
the direction of P, and its amount remain
constantly the same.
It is evident that as the point A, thus continually alters its
position, the distance A1A2 or L will continually change, so
that the value of P1 (equation 158.) will continually change.
Now the work done under this variable pressure during one
revolution of Pi is represented (Art. 51.) by the formula
27r
U fPlaidd, if 0 represent the angle A1CA described at
0
any time about C, by the perpendicular ClA1, and therefore
a16, the space S described in the same time by the point of
application A1 of P, (see Art. 62.).
Substituting, therefore, for P1 its value from equation
(158.), we have
l.+   2) sin. pp afdo=
0 
0           1  0




AXES.                     191
v.l,=Uo,+   — "I fP*.Ld.         (1(7.)
1  0
Let now P2 be assumed a constant quantity;
27r                2ir.:1 fiPL=P2a2 x -f-    L.
I O              1a2o
Now L=AA2= la2 + 2aa cos. 0 + a2 I;
27r       27r.'. 1aL -id,  1 f(a12 +2aa2 cos. + a22)d=
0          o
27w
(a1 ~a2 )/  {     2a2 coa'- do
a( l i^1 +-2(    -— ) cos.8  d=
27t
neglecting powers of      above thefirst, since in all
cases its value is less than unity. Integrating this quantity
between the limits 0 and 2r the second term disappears, so
that
27r
a-.2 2 fL a= (     2+ a ) 2A)   nearly;
0
29w
P2a2 j12    Ld- p (2ra) (   a       ~ 2; 2 +2';
e ctin        p    ao      o    a a(  -
o
since 2a2 is the space through which the point of application of the constant pressure Pt is made to move in each retion of the constant pressure P2 ismade to move in each re



192                    AXES.
volution. Therefore by equation (187), in the case in which
P2 is constant,
U1I2 1 + (+       )P sin...... (188).
179. If the pressure P2 be supplied by the tension of a
rope winding upon a drum whose radius is a2 (as in the capstan), then is the effect of the rigidity of the rope (Art. 142.)
the same as though P2 were increased by it so as to become
+D+EP or (1E) P         D
a2          a2t    a2
Now, assuming P, to be constant, and observing that
U2=24Pa2, we have, by equation (187),
2.r
U1 _P2a2 { 2ir~ + Ps.ifLd6 }
ala2
o
Substituting in this equation the above value for P2
2pc
U =a{ (4)      2~       2    ~ P sin.+fLd
n2 A  (a )-  a,}      a 
Performing the actual multiplication of these factors, observing that D is exceedingly small, and omitting the term
a,
involving the product of this quantity and P —n, we have
2a
U   P,=P2( +I-) { 2\  p a, fjd } +25D.
0
Whence performing the integration as before, we obtain
Us ^TjjUs (- -) j 1+ ( t a) sin. } } +2-D.
a       \a^  a 
If this equation be multiplied by n, and if instead of ITU and
JU representing the work done during one complete revolution, they be taken to represent the work done through n
such revolutions, then




AXES.                     193
T,     +    l{ 1 + (  +a  P Sin. } + 2nD....(189),
which is the MODULUS.
180. If the quantity  - +    be not so small that terms
a.2 a1
of the binomial expansion involving powers of that quantity above the first may be neglected, the value of the
-27r
definite integral Ld  may be determined as follows:
0
0                       0
=   +a   f{ l-( 42in.^            Let   = k2=  42
-(a1 ~- a~  1 -(a1 + a.)s'n.     Let   -(a+ 2)2'
0
0             0
(a, + a) (i1 -   sin. ll) d
0
7r
=2(al + a2/ (1-k' sin.'2) d*= 2( + a2)E,(k), where E,(k)
o
represents the complete elliptic function of the second order,
whose modulus is kA.  The value of this function is given
for all values of k in a table which will be found at the end
of this work.
Substituting in equation (187),
U1=U+, P sn. 2(+a2,+  ). E1(k)t P,=U2, +
* See Encyc. Met. art. DEF. INT. theorem 2.
) An approximate value of EI(k) is given when k is small by the formula
w 2 /k
Ei(k)=-(] +K-1), where K= 1-+k  (See Encyc. Met. art. DEF. INT. equation
13




194                 THE CAPSTAN.
(2a2P2)! (+) p sm.;
1 I   1\.U    U{+l+( 1+ +) P sin..... (190).
THE CAPSTAN.
181. The capstan, as used on shipboard, is represented in
the accompanying figure.
It consists of a solid timber
CC, pierced through the
greater part of its length byan  aperture  AD, which
receives the upper portion
of a solid shaft AB of great
strength, whose lower extremity is prolonged, and
strongly fixed into the timber framing of the ship. The piece CC, into the upper portion -of which are fitted the moveable
arms of the capstan, turns upon the shaft
AB, resting its weight upon the crown of
/  the shaft, coiling the cable round its central portion CC, and sustaining the tension of the cable by the lateral resistance
of the shaft. Thus the capstan combines,    i/-,  the resistances of the pivot and the axis,
so that the whole resistance to its motion
is equal to the sum of the resistances due separately to the
axis and the pivot, and the whole work expended in turning
it equal to the whole work which would be expended in
turning it upon its pivot were there no tension of the cable
upon it, added to the whole work necessary to turn it upon
its axis under the tension of the cable were there no friction
of the pivot. Now, if UT represent the work to be done
upon the cable in n complete revolutions, the work which
must be done upon the capstan to yield this work upon the
cable is represented (equation 189.) by
E          1(       sin. 1.  
1 + a2 1 +    1t 2+a 2 psin. U2 + 2nYD,




THE CAPSTAN.                  195
where a, represents the length of the arm, and a2 the radius
of that portion of the capstan on which the cable is winding.
Moreover (Art. 175.), the work due to the friction of the
4
pivot in n complete revolutions is represented by 3npilfW.
On the whole, therefore, it appears that the work U,
expended upon n complete revolutions of the capstan is
represented by the formula
Ul1        ) i(i +(alr+)) p 1+sin. }i +
\1a= +  ~T+       )
2nt D+2p/fW... (191).
which is the MODULUS of the capstan.
A single pressure P, applied to a single arm has been
supposed to give motion to the capstan; in reality, a number of such pressures are applied to its different arms when
it is used to raise the anchor of a ship. These pressures,
however, have in all cases,-except in one particular case
about to be described,-a single resultant. It is that single
resultant which is to be considered as represented by PI,
and the distance of its point of application from the axis
by a,, when more than one pressure is applied to move the
capstan.
The particular case spoken of above, in which the pressures applied to move the capstan have no resultant, or cannot be replaced by any single pressure, is that in which
they may be divided into two sets of pressure, each set having a resultant, and in which these two resultants are equal,
act in opposite directions, on opposite sides of the centre,
perpendicular to the same straight line passing through the
centre, and at equal distances from it.*
Suppose that they may be thus compounded into the
equal pressures R, and RI, and let them be replaced by
these. The capstan will then be acted upon by four pressures,-the tension P2 of the cable, the resistance R of the
shaft or axis, and the pressures R, and R2. Now these pressures are in equilibrium. If moved, therefore, parallel to
their present directions, so as to be applied to a single point,
* Two equal pressures thus placed constitute a STATICAL COUPLE. The properties of such couples have been fully discussed by M. Poinsot, and by Mr.
Pritchard in his Treatise on Statical Couples; some account of them will be
found in the Appendix to this work.




196                   THE CAPSTAN.
they would be in equilibrium  about that point (Art. 8.).
But when so removed, R, and R, will act in the same
straight line and in opposite directions. Moreover, they
are equal to one another; R, and R, will therefore separately be in equilibrium with one another when applied to that
point; and therefore P, and R will separately be in equilibrium; whence it follows, that R is equal to P2 or the whole
pressure upon the axis, equal in this case to the whole tension
P2 upon the cable. So that the friction of the axis is represented in every position of the capstan by P2 tan. g (tan. (,
being equal to the co-efficient of friction (Art. 138.)), and
the work expended on the friction of the axis, whilst the
capstan revolves through the angle 6 by P2,p tan. p, or by
Pa (a) tan. p, or by UI (-) tan. p; so that, on the whole,
introducing the correction for rigidity and for the friction of
the pivot, the modulus (equation 191) becomes in this case
U    U 2(l+  ){1+(    )tan. }+
2nt { D +  p.f W }.... (192).
This is manifestly the least possible value of the modulus,
being very nearly that given. (equation 191) by the value
infinity of a,.*
Thus, then, it appears generally from equation (191), that
the loss by friction is less as a, is greater, or as P, is applied
at a greater distance from the axis; but that it is least of all
when the pressures are so distributed round the capstan as
to be reducible to a COUPLE, that case corresponding to the
value infinity of ai. This case, in which the moving pressures upon the capstan are reducible to a couple, manifestly
occurs when they are arranged round it in any number of
pairs, the two pressures of each pair being equal to one another, acting on opposite sides of the centre, and perpendicular to the same line passing through it. This symmetrical
distribution of the pressures about the axis of the capstan is
therefore the most favourable to the working of it, as well
as to the stability of the shaft which sustains the pressure
upon it.
* q being exceedingly small, tan. o is very nearly equal to sin. 6.




AXES.                   197
182. THE MODULUS OF A SYSTEM OF THREE PRESSURES APPLIED
TO A BODY MOVEABLE ABOUT A CYLINDRICAL AXIS, TWO OF
THESE PRESSURES BEING GIVEN IN DIRECTION AND PARALLEL TO ONE ANOTHER, AND THE DIRECTION OF THE THIRD
CONTINUALLY REVOLVING ABOUT THE AXIS AT THE SAME
PERPENDICULAR DISTANCE FROM IT.
Let P, and IP represent the parallel pressures of the sys^.A    tern, and P1 the revolving pressure.?'-.. a From the centre of the axis C, let fall..   the perpendiculars CA,, CA,, CA, upon
_/:|/         the directions of the pressures, and let
2.......-_..  |{A,  0 represent the inclination of CA, to
j  ~'_      (. CA, at any period of the revolution of
8    fi n*3    Pi. Let Pi be the preponderating
pressure, and let P, act to turn the
system in the same direction as P,, and P, in the opposite
direction; also let R represent the resultant of P, and P,,
and r the perpendicular distance CA of its direction from C.
Suppose the pressures P, and P, to be replaced by R; the
conditions of the equilibrium of P, throughout its revolution, and therefore the work of P, will remain unaltered by
this change, and the system will now be a system of two
pressures P, and R instead of three; of which pressures R
is given in direction. The modulus of this system is therefore represented (equation 187) by the formula
U,=U,-r+. Ld.....(193);
0
where U, represents the work of R, and L represents the distance AA1 between the feet of the perpendiculars r and a,,
so that L2-a'-2_a r cos. 0 +r2 =(a,-r cos. 8) +r' sin. 2;.R. R2L2=(Ra,-Rr cos. o8) + Rr sin.O.
Now, R=P, +P, Rr=Pa. —Pa2;. 2L2= (P, +P,)a -(Pa,-Pa)os.1 2 + (Pa,-Pa,)2sin.20
[Now if the relations of a, to a3 are such that
(P, +P2)al-(Pa —Pa) cos. 6 } > (Pa,-Pa2) sin.'
then the value of R2L' will be represented by the sum of the




198                      AXES.
squares of two quantities the first of which is greater than
the second. ED.]  Therefore, extracting the square root by
Poncelet's theorem, (see Appendix B.)
RL=c   (P,+P2)a —(Pa3 -P2a2) cos. ot +f3(Pa — P2A2) sin. 0
very nearly; or,
RL=-a,(P3+P,) —(P3a~ —P2a2)(a cos. — 8 sin. ).... (194).
0              0          0
o               o           o
Jf Pa- -Pa,)(Px cos. o -P sin. a)d,
o
0                      0
(a cos. - - sin  d..).. (195).
If P2 and P, be constant, the integral in the second member
of this equation becomes (Pa —P2a2) ( sin. +/ cos. 8);,   2 2) (a S. r  T  P -P 3 COS —',
whence observing that Pa, —P a =P- IA- -PaO  U -B
also, that U-,=0Rr —Pa-PaP a=_-U — U2, and substituting
in equation (193), we have
U -=,-J-U2+~p sin. (p {  ( L+Th) -
c = - -   2+P S i  4    a +'2'
a8,  2
('a,') (  sin.  +  cos. 6)..... (196);
for complete revolution making -=22r, we have
U1,=   -U,+p sin. pl. (a-i ~ )-     3 (,-2) 
reducing,
i-I —psin.((tz 2al}   3   {
P sin.            U    (9.7)which is the modulus of the system where a and /3 are to be
determined, as in Note B, (Appendix.)




THE CHINESE CAPSTAN.             199
183. If the pressure P3 be supplied by the tension of a cord
which winds upon a cylinder or drum at the point A,, then
allowance must be made for the rigidity of the cord, and a
correction introduced into the preceding equation for that
purpose.  To make this correction let it be observed
(Art. 142.) that the effect of the rigidity of the cord at As is
the same as though it increased the tension there from
PtoP, l+E +;
or (multiplying both sides of this inequality by a,, and integrating in respect to 6,) as though it increased
27r              27r      27r
/P,a   to (l ~) fPa,+ fDd
0                0         0
or, U, to (l+  )U,+2D.
Thus the effect of the rigidity of the rope to which P, is applied upon the work U, of that force is to increase it to
1+) TUJ+2iD. Substituting this value for U, in equation (197), and neglecting terms which involve products of.    E   p sin. q, p sin. (p
the exceedingly small quantities E, p sin. p sn.  and D,
we have
1~ t + P sin.            )   sh-p sin.(p(+-)       U,42tD...   (198).
l        ( az  27al, 
To determine the modulus for n revolutions we must substitute in this expression wf for m.
THE CHINESE CAPSTAN.
184. This capstan is represented in the accompanying




200              THE CHINESE CAPSTAN.
figure under an exceedc 7,.                            ingly portable and cont3      venient form.* The axle
or drum of the capstan
is composed of two parts
of different diameters.
One extremity of the
cord is coiled upon one
of these, and the other, in an oposite direction, upon the
other; so that when the axle is turned, and the cord is
wound upon one of these two parts of the drum, it is, at the
same time, wound of the other, and the intervening cord is
shortened or lengthened, at each revolution, by as much as
the circumference of the one cylinder exceeds that of the
other.  In thus passing from one part of the drum. to the
other, the cord is made to pass round a moveable pulley
which sustains the pressure to be overcome.
To determine the modulus of this machine, let u2 and u3
represent the work done upon the two parts of the cord
respectively, whilst the work U1 is done at the moving point
of the machine, and U2 yielded at its working point.
Then, since in this case we have a body moveable about a
cylindrical axis, and acted upon by three pressures, two of
which are parallel and constant, viz. the tensions of the two
parts of the cord; and the point of application of the third
is made to revolve about the axis, remaining always at the
same perpendicular distance from it; it follows (by equation
198), that, for n revolutions of the axis,
UI,=Au,-Bu2 +2nID..... (199);
where
A=      1+ +~psin. (i    2na,  \ ),and
B=. 1-p sin. (~+2      f) V
a, and as representing the radii of the two parts of the drum,
a, the constant distance at which the power is applied, and p
the radius of the axis.
* A figure of the capstan with a double axle was seen by Dr. O. Gregory
among some Chinese drawings more than a century old. It appears to have
been invented under the particular form shown in the above figure by Mr. G.
Eckhardt and by Mr. M'Lean of Philadelphia. (See Professor Robinson's Mech.
Phil. vol. ii. p. 255.)




THE CHINESE CAPSTAN.             201.1s, since tihe two parts of the cord pass over a pulley,
and the pulley is made
to revolve under the ten-,P-/  H a^^^^^^^3  ~  sions of the two parts of. = = -t  the cord, t, being the
work of that tension
///  2  which preponderates, we
have (by equation 181),
if S represents the length
of cord which passes
over the pulley,
t= A ta+Bl;
where
Ai= 1+-2P+ sin.m, and
a a     a
B=- D   (+ 2 W.. pco. sin.;
a representing the radius of the pulley, p, the radius of its
axis, W its weight, and the inclination of the direction of
the tensions of the two parts of the cord to the vertical, the
axis of the pulley being supposed horizontal, and the two
parts of the cord parallel. Now t-,=  3 ) t2=-2  -  Sub2- ^wra,  2na,
stituting these values, and multiplying by 2nwa,, we have
u, =Au +2naB.....  (200).
a.
Since the tensions t, and t, of the two parts of the cord,
and the pressure P, overcome by the machine, are pressures
applied to the pulley and in equilibrium, and that the points
of application of t, and P. are made to move in directions
opposite to those in which those pressures act, whilst the
point of application of t, is made to move in the same direction; therefore (Art. 59.),
U,+u%=U2,' U  3-U-,.
Eliminating u. and u, between this equation and equation
(200), we have
AU,-2nraB,      a2U 2X'  aB
A,   2            A, 
A3s               a.




202               THE HORSE CAPSTAN.
Substituting these values in equation (199), and reducing,
A(A- 2B        (A-B)Ba, 2A
U=(A      B)      {(A   B)B   } 2ntr + 2njrD.
Substituting their values for A, A,, B, B1, neglecting terms
sin. p E
involving more than one dimension of sm,, &c. and
reducing, we obtain for the MODULUS of the machine,
E,.      (2cc    1a,\ 31__
- -++Ep+2   sin. -
{1l        a.i  E (+  2  sin.lI q)
13  a   a.  THE  ORSD A2n.P.T (201).
a (-2) ~E+2psin. p
From which expression it is apparent that when the radii as
and ae3 of the double axle are nearly equal, a great sacrfice
of power is made, in the e se of this machine, by reason of
the rigidity of the cord.
TiE HORSE CAPSTAN, OR THE WHIM Gm.
185. The whim is a form of the capstan, used in the first
operations of mining, for raising materials from the shaft and
levels by the power of horses, before the quantity excavated
is so great as to require the application of steam power, or
before the valuable produce of the mine is sufficient to give
a return upon the expenditure of capital necessary to the
erection of a steam engine. The construction of this machine
will be sufficiently understood from the accompanying figure.
It will to5 observed that there are two ropes wound upon the
drum in opposite directions, and which traverse the space




THE HORSE CAPSTAN.             203
between the capstan
and the mouth of the
shaft. One of these
carries at its extremity  the  descending
U~_LT~b~~~kB  _~~  ~    (empty) bucket, and is
continually in the act
of winding off the drum of the capstan as it revolves; whilst
the other, from whose extremity is suspended the ascending
(loaded) bucket, continually winds on the drum. The pressure exerted by the horses is that necessary to overcome the
friction of the different bearings, and the other prejudicial
resistances, and to balance the difference between the weight
of the ascending load, bucket, and rope, and that of the
descending bucket and rope. The rope, in passing from the
capstan to the shaft, traverses (sometimes for a considerable
distance) a series of sheaves or pulleys, such as those shown
in the accompanying figure.
Let now a2 represent the radius of the drum on which the
rope is made to wind, and n the number of revolutions
which it must make to wind up the whole cord; also let f
represent the weight of each foot of cord, and 0 the angle
which the capstan has described between the time when the
ascending bucket has attained any given position in the
shaft and that when it left the bottom; then does a20 represent the length of the ascending rope wound on the drum,
and therefore of the descending rope wound of it. Also,
let W  represent the whole weight of the rope; then does
W —a2a represent the weight of the ascending rope, and'a2i that of the descending rope, both of which hang suspended in the shaft. Let P, represent the load raised at
each lift in the bucket, and w the weight of the bucket;
then is the tension upon the ascending rope at the mouth of
the shaft represented by W —va2 +P2 + w, and that upon
the descending rope by aua2 +w.
Let, moreover, p, and p represent the tensions upon these
ropes after they have passed from the mouth of the shaft,
over the intervening pulleys, to the circumference of the
capstan.
Now, since the tension upon the ascending rope, which is
W —a20 + P, + w at the mouth of the shaft, is increased to
p, at the capstan, and that the tension upon the descending
rope, which is P2 at the capstan, is increased to a2O8+w at
the mouth of the shaft, if we represent by (1 + A) and B the
constants which enter into equation 180 (Art. 174.), we have,




204              THE HORSE CAPSTAN.
by that equation (observing that U,=P,S, and U-=P S,. so
that S1 disappears from both sides of it),
p=(1 +A)(    + P2 + w —ca2)+ B,....(202),
and         a2 +w =(1 +A)p2+B..... (203).
Multiplying the former of the above equations by 1 +A,
adding them, transposing, dividing by (1+ A), and neglecting terms of more than one dimension in A and B,
p3-p- (1 + A)(W    P+ +  P 2Aw + 2B - 2a2.
Now U, in equation (193) represents the work of the
resultant of p3 and p during n revolutions of the capstan, it
therefore equals the difference between the work of p. and
that of p, (see p. 198).
2wnT     27rw        2w7r. Ur= fp3a2d -p2a2d =- af(P,-p,) do;
0        0           0
2n7r
Ur=a     (1 + A)(W+P2) + 2Aw + 2B —2pa2O dO=
0
(1 + A) (W + P,) + 2Aw+- 2B } (2nra2) —,(2n2wa)2;:.- =(1+A) -U,+ (1+A)W+2Aw +2B-pS2 S2,..(204);
observing that 2nra2-=S2, and that PS2 -U2.
Now, let it be observed that the pressures applied to the
capstan are three in number; two of them, p, and p2, being
parallel and acting at equal distances a2 from its axis; and
the third, P,, being made to revolve at the constant distance
a, from the axis (P1 representing the pressure of the horses,
or the resultant of the pressures of the horses, if there be
more than one, and a, the distance at which it is applied);
so that equation 193 (Art. 182.) obtains in respect to the
pressures P1, p1, p2; a, being assumed equal to a.
Substituting p2 and p, for P, and P2 in equation (194),
RL= al(p2~p2) —a2(p2-p- ) (a cos. 0 -/ sin. 0);
2nwr       2nr            27Tw.. RLdO=aa(f.3p +p2) dO - a2f(p3-p2)
0           0              0
(a cos. O —r sin. 0) dO.




THE HORSE CAPSTAN.              205
Now, the terms of equation (180), represented in the above
equations by A and B, are all of one dimension in the exceedingly small quantities D, E, sin. p. If, therefore, the values
of p and p. given by these equations be substituted in the
2nnr
value of P sin. /RLdO (equation 193), then all the terms
a1
0
of that expression which involve the quantities A and B will
be at least of two dimensions in ID E, sin. p, and may be neglected. Neglecting, therefore, the values of A and B in
equations (202, 203), we obtain
p, +p2=W  + P +  P+2w, and, —p2=W  + P2-2a20;
2m7w. 
+(2nra,2)(W+2w)= }() {S2P2,+S2(W+2w)}
(    U2 + S, (w+2w);
representing by S, the
space described bythe
load, and by UT the
useful work done upon
it, during n revolutions of the capstan.
Similarly,
2nr                            27r
2 (.PsA-Pa cos. 0- /3 sin. 0)dO = a2f IW+P,- 2a2,0
0                              0
(a cos. 0 —3 in. 6)d=/a(W+P2)    cos. 0-/ sin. -)dO0.:
27r
2ap2f(a cos. 0 —3 sin. 0)0o0.
0




206              THE HORSE CAPSTAN.
2nfr                       2n7r
Nowf (a cos. 0-/3 sin. 0)d0=13, and (a cos. 0-f3 sin. 0)
o                          0
Od= 2[3nr*;
2nr.a. f    -P C-p O)(a   cos. 0-3 sin. O)do0=3S2(W+P2) - 23a2
o
U                                  U2
(2n9a2)=-3a2 2- +a2(W-2-S,2); observing that P,-S
S2                                  2
27nz
* IRLd0= a           ) U+ S2(W + 2w) -a U2 -a(W - 2pS);
o  27T
ps. i                           o —i 5(+.  (W +2w)
0+   2&   a2 }S2-W
+ 2i a+               (w+ }o2 )W2
Substituting this value, and also that of Ur (equation 204)
in equation (193), and assuming
C= (t A)W+2Aw+2B and C2          - (W+2w) ^   + 2a,,
we have
1=-(1 +A)U + OS2 —PS22+e P sin. _
a,
TU  2 + CA-Walt
0              0 
*For j0 cos. d6.    in          -vers. 0; sin. s.  =s. ver.; ls o sin. d
o              o                      o
0
= —0 cos. 0 +'cos. OdO*=-0 cos. 0+sin. 0. Now, substituting 2nw for 0,
these integrals become respectively 0 and -2nir.
* Church's Diff. and Int. Cal. Art. 140.




THE FRICTION OF CORDS.           207.UI= I+A+ Fp sin. (p  - S2W  2I Sm. ap
al;          1 i  2
(01+ pC~ sinP) s. f3Wp'sin.p
which is the MODULUS of the machine, all the various elements, whence a sacrifice of power may arise in the working
of it, being taken into account.
THE FRICTION OF CORDS.
186. Let the polygonal line ABC... YZ, of an infinite
-,_ -- >.     number of sides, be taken to represent'    \ e|"'"  ~.: the curved portion of a cord embracing'.. \   d.  pl any arc of a cylindrical surface (whe-Y':,,b:". ther circular or not), in a plane perPA —'....'"i\  i u pendicular to the axis of the cylinder;
d  \  ~also let Aa, Bb, Cc, &c., be normals
a\  V 5      or perpendiculars to the curve, inclined
a  ~  to one another at equal angles, each
represented by As. Imagine the surface of the cylinder to
be removed between each two of the points A, B, &c., in
succession, so that the cord may be supported by a small
portion only of the surface remaining at each of those
points, whilst in the intermediate space it assumes the direction of a straight line joining them, and does not touch the
surface of the cylinder. Let P, represent the tension upon
the cord before it has passed over the point A; T, the tension upon it after it has passed over that point, or before it
passes over the point B; T2 the tension upon it after it has
passed over the point B, or before it passes over C; T, that
after it has passed over C; and let P2 represent the tension
upon the cord after it has passed over the nth or last
point Z.
Now, any point B of the cord is held at rest by the tensions TI and T2 upon it at that point, in the directions BC
and BA, and by the resistance R of the surface of the cylinder there; and, if we conceive the cord to be there in the
state bordering upon motion, then (Art. 138.) the direction
of this resistance R is inclined to the perpendicular bB to
the surface of the cylinder at an angle RBb equal to the
limiting angle of resistance p.




208       T      THE FRICTION OF CORDS.
Now T, T2, and R are pressures in equilibrium; therefore (Art. 14.)
T, sin. T2BR
T,~sin. T1BR;
but TBR=ABb-RBb=i(~ —AaB)-RBb)=           ~    -,
TBR= CBb+RBb=-( —BbC)+ RBb)=           - -- -  p;
sin. {2- (       -        cos.(. -a -,
/2    2             /A2 
sin.            q —    OS     + q~
T      os. 2-os. (i + )c s.   + p
2 sin. - sin. (P
2
A^..AO.
cos.  cos. p - sin. - sin. q
or dividing numerator and denominator of the fraction in the
second member by cos. 2 cos.,
2 tan. - tan. p
T,- T           2
T             A2
T2    1 —tan., tan. q
Suppose now the angles Aab, BbC, &c., each of which
equals aO, to be exceedingly small, and therefore the points
A, B, C, &c., to be exceedingly near to one another, and
exceedingly numerous. By this supposition we shall manifestly approach exceedingly near to the actual case of an infinite number of such points and a continuous surface; and




THE FRICTION OF CORDS.            209
if we suppose Ad infinitely small, our supposition will coincide
with that case. Now, on the supposition that Ad is exceedingly small, tan.. tan. p is exceedingly small, and may
be neglected as compared with unity; it may therefore be
neglected in the denominator of the above fraction. Moreover ad being exceedingly small, tan. - -=-.         tan... T  T (1 tan.. p T T. A. A).
Now the number of the points A, B, 0, &c. being represented by n, and the whole angle AdZ between the extreme
normals at A and Z by 0, it follows (Euclid, i. 32.) that
8=n. a4; therefore a=-;
T,=T, (1 + -tan. ).
Similarly,         PI=T, (1 +- tan.!),
T,=T, (1 +tan. p),
&c. =&c. =&c.
T1-=P2 (1 + tan. q).
Multiplying these equations together, and striking out factors common to both sides of their product, we have
P1=P, ( +   tan. p)n;
* If we consider the tension T as a function of 8, of which any consecutive
values are represented by T and Ta, and their difference or the increment of
— AT                 1  AT
T by AT, then — =- tan.. AO; therefore  T.  -  - tan.; therefore,
T    an o   t helT iA 
1 dT
passing to the limit T- = -  tan. 0, and integrating between the limits 0
and 6, observing that at the latter limit T==P, and that at the former it equals
P,,we have log. (2  -0 tan. <; therefore Pe-P1a
14




210             THE FRICTION OF CORDS.
or P=P{ 121+n-tan. p+n —    -stan.'8+
n_2-   ~  -tan. + &c.
11 —
orP,=P, 1 l+0 tan. (p+-    2 tan. q +
(1 -^ 1(  - ).    9+    }
(i^- -       s tan. ta +....
Now this relation of P, and P2 obtains however small al
be taken, or however great n be taken. Let n be taken
infinitely great, so that the points A, B, C, &c. may be
infinitely numerous and infinitely near to each other.  The
supposed case thus passes into the actual case of a con123
tinuous surface, the fractions n, -  f-5 &c. vanish, and the
above equation becomes
P -P    1  08 tan. p 08 tan. 2 +83 tan.  +,
=P 2         1 +. 2 +1. 2          +. 3
But the quantity within the brackets is the well known expansion (by the exponential theorem) of the function e0tan.,.      P, =P^tan. 0..... (205).
Since the length of cord S,, which passes over the point
A, is the same with that S, which passes over the point Z,
it follows that the modulus (Art. 152.) of such a cylindrical
surface considered as a machine, and supposed to be fixed
>and to have a rope pulled and made to slip over it, is
UTl    UE tan...... (206).
It is remarkable that these expressions are wholly independent of the form and dimensions of the surface sustaining the tension of the rope, and that they depend exclusively upon the inclination 6 or AdZ of the normals to the
points A and Z, where the cord leaves the surface, and upon
the co-efficient of friction (tan. p), of the material of which
the rope is composed and the material of which the surface
is composed. It matters not, for instance, so far as thefric



THE FRICTION OF CORDS.            211
tion of the rope or band is concerned, whether it passes
over a large pulley or drum, or a small one, provided the
angle subtended by the arc which it embraces is the same,
and the materials of the pulley and rope the same.
In the case in which a cord is made to pass m times round
such a surface, 6=2mr;:. p: =p g2m rt tan..
And this is true whatever be the form of the surface, so
that the pressure necessary to cause a cord to slip when
wound completely round such a cylindrical surface a given
number of times is the same (and is always represented by
this quantity), whatever may be the form or dimension of
the surface, provided that its material be the same. It
matters not whether it be square, or circular, or elliptical.
18'. If P', P1", Pf"', &c. represent the pressures which
must be applied to one extremity of a rope to cause it to
slip when wound once, twice, three times, &c. round any
such surface, the same tension P2 being in each case supposed to be applied to the other extremity of it, we have
P /= =E2 tan. < p/ "=pe4r tan. 0, P"r=p86r tan, &c.-&c.
So that the pressures P', P1", P1 " &c. are in a geometrical progression, whose common ratio is e2r tan.0, which
ratio is always greater than unity. Thus it appears by the
experiments of M. Morin (p. 135.), that the co-efficient of
friction between hempen rope and oak free from unguent is
~33, when the rope is wetted. In this case tan. (=-33 and
2r tan. p=2 x 3-14159 x'33-=2-0345.  The common ratio
of the progression is therefore in this case e2'0745, or it is the
number whose hyperbolic logarithm is 2-07345. This number is 7'95; so that each additional coil increases the friction nearly eight times.  Had the rope been dry, this
proportion would have been much greater. If an additional
/half coil had been supposed continually to be put upon the
rope instead of a whole coil, the friction would have been
found in the same way to increase in geometrical progression, but the common ratio would in this case have
been e8tan' instead of e2ttan-'. In the above example the
value of this ratio would for each half coil have been
2-82.
The enormous increase of friction which results from




212             THE FRICTION OF CORDS.
each additional turn of the cord upon a capstan or drum,
may from these results be understood.
188. We may, from what has been stated above, readily
explain the reason why a knot connecting the two extremities of a cord effectually resists the action of any force
tending to separate them. If a wetted cord be wound round
Fig. 1.              Fig..  F..  a cylinder of oak as
in fig. 1., and its extremities  be  acted
upon by two forces P
and R, it has been'/ y~rr"       shown that P will not
x!.  J^~ H~ ~overcome R, unless it
be equal to somewhere about eight times that force. Now if the string to
which R is attached be brought underneath the other string
so as to h}o pressed by it against the surface of the cylinder,
as at m, jbg. 2.; then, provided the friction produced by
this pressure be not less than one eighth of P, the string will
not move even although the force R cease to act. And if
both extremities of the string be thus made to pass between
the coil and the cylinder, as in fig. 3., a still less pressure
upon each will be requisite. Now, by diminishing the
radius of the cylinder, this pressure can be increased to any,extent, since, by a known property of funicular curves, it
varies inversely as the radius.* HWe may, therefore, so far
diminish the radius of a cylinder, as that no force, however
great, shall be able to pull away a rope coiled upon it, as
represented in fig. 3., even although one extremity were
loose, and acted upon by no force.
Fig.'4         Let us suppose the rope to be
doubled as in fig. 4., and coiled
- ~' ~   as before. Then it is apparent,
from what has been said, that
the cylinder may be made so
small, that no forces P and P'
applied to the extremities of
either of the double cords will
be sufficient to pull them from
it, in whatever directions these are applied.
* This property will be proved in that portion of the work which treats of
the THEORY OF CONSTRUCTION.




THE FRICTION BREAK.             213
Now let the cylinder be removed. The cord then being
drawn tight, instead of being coiled round the cylinder, will
be coiled round portions of itself, at the points m and n;
and instead of being pressed at those points upon the cylinder, by a force acting on one portion of its circumference, it
will be pressed by a greater force acting all round its circumference. All that has been proved before, with regard
to the impossibility of pulling either of the cords away from
the coil when the cylinder is inserted, will therefore now
obtain in a greater degree; whence it follows that no forces
P and P' acting to pull the extremities of the cords asunder,
may be sufficient to separate the knot.
THE FRICTION BREAK.
189. There are certain machines whose motion tends, at
certain stages, to a destructive acceleration; as, for instance,
a crane, which, having raised a heavy weight in one'position
of its beam, allows it to descend by the action of gravity in
another; or a railway train, which, on a certain portion of
its line of transit, descends a gradient, having an inclination
greater than the limiting angle of resistance. In each of
these cases, the work done by gravity on the descending
weight exceeds the work expended on the ordinary resistance due to the friction of the machine; and if some other
resistance were not, under these circumstances, opposed to
its motion, this excess (of the work done by gravity upon it
over that expended upon the friction of its rubbing surfaces)
would be accumulated in it (Art. 130.) under the form of
vis viva, and be accompanied by a rapid acceleration and a
destructive velocity of its moving parts. The extraordinary
resistance required to take up its excess of work, and to
prevent this accumulation, is sometimes supplied in the
crane by the work of the laborer, who, to let the weight
down gradually, exerts upon the revolving crank a pressure
in a direction opposite to that which he used in raising it.
It is more commonly supplied in the crane, and always in
the railway train, without any work at all of the laborer, by
a simple pressure of his hand or foot on the lever of the friction break, which useful instrument is represented in the
accompanying figure under the form in which it is com



214:             THE FRICTION BREAK.
monly applied to the crane,-a form of it which may serve
to illustrate the principle of its applicai on under every
other. BC represents a wheel
47\$    <fixed commonly upon that
I  //    axis of the machine to which
B32~1'    the crank is attached, and
which axis is carried round
by it with greater velocity
than any other.  The periphery of this wheel, which is
usually of cast iron, is embraced by a strong band* ABCE of wrought iron, fixed
firmly by its extremity A to the frame of the machine, and
by its extremity E to the short arm AE of a bent lever PAE,
which turns upon a fixed axis or fulcrum, at A, and whose
arm PA, being prolonged, carries a counterpoise D just
sufficient to overbalance the weight of the arm AP, and to
relieve the point E of all tension, and loosen the strap from
the periphery of the wheel, when no force P is applied to the
extremity of the arm AP, or when the break is out of
action.
It is evident that a pressure P applied to the extremity of
the lever will produce a pressure upon the point E, and a
tension upon the band in the direction ABCE, and that
being fixed at its extremity A, the band will thus be tightened upon the wheel, producing by its friction a certain
resistance upon the circumference of the wheel.
MIoreover, it is evident that this resistance of friction upon
the circumference of the wheel is precisely equal to the
tension upon the extremity A of the band, being, indeed,
wholly borne by that tension; and that it is the same
whether the wheel move, as in this case it does, under the
band at rest, or whether the band move (under the same
tensions upon its extremities, but in the opposite direction)
over the wheel at rest. Let R and Q represent the tensions
upon the extremities A and E of the band; then if we suppose the wheel to be at rest, and the band to be drawn over
it in the direction ECB by the tension R, and d to represent
the angle subtended at the centre of the wheel by that part
of its circumference which the band embraces, we have
(equation 205)
* Blocks of wood are interposed between the band, the periphery of the
break wheel. This case will be discussed in the Appendix.




THE BAND.                 215
RI= Q60tan. $.
Let a, represent the length of the arm AP, and a2 the
length of the perpendicular let fall from A upon the direction of a tangent to that point in the circumference of the
wheel where the end EC of the band leaves it.
Then, neglecting the friction of the axis A, we have
(Art. 5.)
P. 1=Q. a2;
Pa 0tan. 0
R=^ a..... (207).
If P1 represent any pressure applied to the circumference of
the break wheel, and P2 a pressure applied to the working
point of the machine, whatever it may be, to which the
break is applied, and if P,=aP2+b (Art. 152.) represent the
relation between P1 and P2 in the inferior state bordering
upon motion by the preponderance of P2; then, when P2 is
taken in this expression to represent the pressure W, whose
action upon the working point of the machine the break is
intended to control, Pi will represent that value R of the
friction upon the break which must be produced by the
intervention of the lever to control the action of the pressure
W upon the machine; so that taking R to represent the
same quantity as in equation (207), we have
R=aW+b.
Eliminating R between this equation and equation (207),
and solving in respect to P,
P=(aW +b)-~.tan..... (208).
THE BAmD.
190. When the circular motion of any shaft in a machine,
and the pressure which accompanies that motion, constituting together with it the work of the shaft, are to be communicated to any other distant shaft, this communication is




216                    THE BAND.
usually established by means of a band of
leather, which passes round drums fixed upon
the two shafts, and has its extremities drawn
together with a certain pressure and united,
so as to produce a tension, which should be
just that necessary to prevent the band from
slipping upon the drums, subject to the pressure under which the work is transferred.
The facility with which this communication
of rotatory motion may be established or
broken at any distance and under almost
every variety of circumstance, has brought
the band so extensively into use in machinery,
that it may be considered as a principal channel through which work is made to flow in its distribution
to the successive stages of every process of mechanism,
carried on in the same workshop or manufactory.
191. The sum of the tensions upon the two parts of a band
is the same, whatever be thepressure under which the band
is d'iven, or the resistance overcome, the tension of the
driving part of the band being always increased by just so
much as that of the driven part is diminished.
This principle was first given by M. Poncelet; it has since
been amply confirmed by the experiments of M. Morin.*  It
may be proved as followst:-In the very commencement of
the motion of that drum to which the driving pressure is
applied, no motion is communicated by it to the other drum.
Before any such motion can be communicated to the latter,
a diference must be produced between the tensions of the
two parts of the band sufficient to overcome the resistance,
whatever it may be, which is opposed to the revolution of
the driven drum. Now, an increase of the tension on the
driving side of the band must be followed by an elongation
of that side of the band (since the band is elastic), and by
the revolution of the circumference of the driving drum
through a space precisely equal to this elongation. Supposing, then, the other, or driven side of the band, to
remain extended, as before, in a straight line between its two
points of contact with the drums, this portion of the band
* Nouvelles Experiences sur le Frottement, &c. Metz.
i No demonstration appears to have been given of it by M. Poncelet.




THE BAND.                  217
must evidently have contracted by precisely the length
through which the circumference of the driving drum has
revolved, or the driving side of the band elongated. Thus,
the elongation of the driving side of the band is precisely
equal to the contraction of the driven side. Now, the band
being supposed perfectly elastic, the increase or diminution of its tension is exactly proportional to the increase
or diminution of its length. The increase of tension on the
one side, produced by a given elongation, is therefore precisely equal to the diminution of tension produced by a contraction equal to that elongation on the other side. Thus,
if T represent the tension upon each side of the band before
the driving pressure, whatever it may be, was applied,
and if T, and T2 represent the tensions upon the driving
and the driven sides of the band after that pressure is
applied; then, since T,-T represents the increase of tension
on the one side, and T —T2 the diminution of tension on the
other, T,-T=T —T,;. T,+T=2T..... (209).
It is a great principle of the economy of power in the use
of the band to adjust this initial tension T, so that it may
just be sufficient to prevent the band from slipping upon
the drum under any pressure which it is required to transmit.
The means of making this adjustment will be explained
hereafter.
THE IMODULUS OF THE BAND.
192. For simplifying the consideration of this important
element in machinery, we shall first consider a particular
case of its application. Let the two drums, whose axis are
C, and C2, be supposed equal to one another, so that the two
parts of the band which pass round them may be parallel.
Pig I.    Fig. 2.  Let, moreover, the centres of the
A *              two drums be in the same vertirt- ^  /  \ ~ cal straight line, so that the two
--    parts of the band may be vertical.
K            4      c a.Let P, and P2 be pressures applied, in vertical directions, to
turn the drums, and at perpendicular distances from their cen-.           tres, represented by C(P, and
-r 4f    YCPl?2; of which pressures P, is
the working or driven pressure,




218                   THE BAND.
or that which is upon the point of yielding by the preponderance of the other P1. In fig. 1. P2 is seen applied on
the same side of the centre of the drums as P1, and in fig.
2. on the opposite side. Let T, and T2 represent the tensions
upon the two parts of the band, T1 being that on the driving,
and T, that on the driven side.
a,=CP,, a2-=C P,
r=radius of each drum,
W=weight of each drum,
p=radius of axis of each drum,
R, and tR 2resistances of axes of drums,
p=limiting angle of resistance.
Now, the parallel pressures P,, W, T1, T2, R,, applied to the
lower drum, are in equilibrium; therefore (Art. 16.),
R, ~(T+T -P, -W);
or substituting for T1-+T2 its value 2T (equation 209),
R1=~(2T-P, —W)..... (210).
The sign ~ being taken according as 2T is greater or less
than P,+Wl, or according as the axis of the lower drum
presses upon the upper surface of its bearings, as shown in
fig. 1., or upon the lower surface, as shown in fig. 2. In like
manner, the pressures P,, W, T1, T2 R2, applied to the upper
drum, being in equilibrium,
R, =T, T,: P 2+W,
or (equation 209) IR,=2TTP2+ W.... (211),
where the sign T is to be taken according as P2 is applied
on the same side of the axis as P,, or on the opposite side.
Since, moreover, R1 and R, act, in the state bordering
upon motion, at perpendicular distances from the centre of
the axis, which are each represented by p sin. p (Art. 153.),
we have, by the principle of the equality of moments,
P1a +T2r=T1r + Rp sin. q)       (212
P2a2+T2r+Rp sin. p=Tr          ( 12)
observing that the resultant of all the pressures applied to
each drum (excepting only the resistance of its axis) must be
such as would alone communicate motion to it in the direction in which it actually moves, and therefore that the resistance of the axis, which is opposite to this resultant, must
tend to communicate motion to the drum in a direction opposite to that in which it actually moves.




THE BAND.                  219
Subtracting the above equations, and transposing,
P1a1-P2a2=(R,+ R) p sin. p.
Substituting the values of R1 and R2 from equations (210)
and (211), we obtain, in the case in which the negative sign
of R1 is to be taken, or in which 2T is less than P + W, the
axis C1 resting upon the lower surface of its collar as shown
in fig. 2.,
P1a —Pa=((PIF P +2W) p sin. p;
and in the case in which the positive sign of iR is to be
taken, 2T being greater than P1+ W, and the axis C, pressing against the upper surface of its collar, as shown in ig. 1.,
P1a,-Pa2 =(4T-P,:FP2)p sin. p.
Transposing and reducing, we obtain for the relation between the driving and driven pressures in these two cases
respectively,
-P -P /ap sin. p   2Wpsin.p        (
-aI-p sin. p  a, -p sin. p,p _p  _a,2 +p sin..) 4Tp sin... (21,
a,p p sin. q a+ - p sin.p 
and therefore (by equation 121), for the moduli in the two
cases2
U. = Us —             + % -P- sin.  I  ***(9
(P)       sin. }(
alq  ~
U =Ua1Ps                4S(Tp sin. q.  6  (216).
2 +  )sa,+sin.
In all which equations the sign F is to be taken according
as Pa is applied on the same side of the line C1C, joining the
axis as P1, or on the opposite side.
193. To determine the initial tension T upon the band, so that
it may not slip upon the surface of the drum when subjected to the given resistance opposed to its motion by the
work.




220                  THE BAND.
Suppose the maximum resistance which may, during the
action of the machine, be opposed to the motion.    of the drum to be represented by a pressure P
applied at a given distance a from its centre C,.
Suppose, moreover, that the band has received
S such an initial tension T as shall just cause it to
be on the point of slipping when the motion of
V 4_ N   the drum is subjected to this maximum resistance; and let t, and t be the tensions upon
the two parts of the band when it is thus
just in the act of slipping and of overcoming the resistance
P. Now, the two parts of the band being parallel, it embraces one half of the circumference of each drum; the relation between t, and t2 is therefore expressed (equation 205)
by the equation
T tan. m
t == t28, tan.,'whence we obtain  -    But t, +=t
e +  1
2T (equation 209),
<r tan. <  \:t t,-=2T  e_,l
~~1  2  r tan. 0 
e + 1/
Also, the relation between the resistance P, opposed to the
motion of the upper drum, and the tensions t, and t2 upon
the two parts of the band, when this resistance is on the
point of being overcome, is expressed (equation 212) by the
equation
Pa+ tr + Rp sin. qp= tr;
or substituting the value of R, (equation 211), and transposing,
Pa+(2T:PP+W)p sin. p=)t~ —t2)r;
whence, substituting the value of t —t2, determined above,
and transposing, we have
7r tan.  r   s
V+V 




THE BAND.                  221
i tan. q    -...... (217).
r —p sin.:r tan.    s
\i +    /
194. The modulus of the band under its most general form.
The accompanying figure represents an elastic band passing round drums of unequal radii, the
/d....^ line joining whose centres C. and C2
is inclined at any angle to the vertical,
and which are acted upon by any
given pressures P, and P2, P1 being
supposed to be upon the point of giving motion to the system.
Let T, and T2 represent the tensions
upon the two parts of the band, T, be"-' ~    ~ ing that on the driving side.
ae, a perpendiculars upon the directions of P1 and P2 respectively.
01, 01 the inclinations of the directions of P1 and P2 to the
line CoC1.
r1, r1 the radii of the drums.
W, W2 the weights of the drums., the inclination of the line 0C12 to the vertical, and o' the
inclination of the two parts of the band to one another.
P? p2 the radii of the axes of the drums.
q the limiting angle of resistance between the axis of the
drum and its collar.
R1, R2 the resistances of the collars in which the axes of
the drums turn in the state bordering upon motion, or the
resultants of the pressures upon these axes.  The perpendicular distances at which these resistances act from the centres of the axes are (Art. 153.) p, sin. 0 and p2 sin. 0. Since
the pressures acting upon the lower drum are TO, T2, P., W,
and R1, and that these pressures are in equilibrium, W1 acting through the centre of the axis, and T1 and IR acting to
turn the drum in one direction about the axis, and Pi and T,
to turn it in the opposite direction; we have, by the principle of the equality of moments (Art. 153.),
P a +Trl-=Tr, + Rp, sin. p.
And since T, T2, P2, W2, R. are similarly in equilibrium




222                   THE BAND.
on the upper drum, W, acting through the centre, and P,,
R2, T2 acting to turn it in one direction, whilst T1 acts to
turn it in the opposite direction,.P2a + Tr, -T  R+2p sin. p=Tlr2;:.P'a1-(T,-T)r==R1psin.  p  l
Pa —(T,-T2)r,- -R2p2 sin. P 
Let T1-T2=2t, and T,+T,=2T,
P aP -2tr=RElp, sin. p   }       ).
P. a — 2tr --- Rp sin..    (218)
To determine the values of R1 and R2 let the pressures
applied to each drum be resolved (Art. 11.) in directions
parallel and perpendicular to the line C012; those applied to
the lower drum which, being thus resolved, are parallel to
CCO, are
+T, cos. a +T. cos. a — P cos. 8, — W1 cos. i,
those pressures being taken positively which tend to move
the axis of the drum from C, towards C0 and those negatively whose tendency is in the opposite direction.
In like manner the pressures resolved perpendicular to
CCA are
-T1 sin. a, +T, sin. ac +~P1 sill. 8,, — W\ sin. a,
those pressures being taken negatively whose tendency
when thus resolved perpendicular to CG2 is to bring that
line nearer to a vertical direction, and those positively whose
tendency is in the opposite direction.
Observing that R, is the resultant of all these pressures,
we have (Art. 11.)
R2= (T, +T,) cos. a~-P, cos. — W, cos. I} +!P, sin. 8,-(T,-T2) sin. a,-W, sin. 2.
Proceeding similarly in respect to the pressures applied to
the upper drum, we shall obtain
R22= -(T+T2) cos. -a-P2 cos.,2+W, cos.  +
jP, sin. 6,+(T,-T,) sin. a,-W, sin. i,};
or substituting 2T for T,+T,, and 2t for T,-T,
R,2= 2T cos. a,-P, cos. 8~ —W, cos. 1+ 
{P, sin. 81-2t sin. ao-W, sin.,2 
R2= 12T cos. a,-P2 cos.,+W, cos.'+      (2) 
{P, sin. 02+ 2t sin. a, -W2 sin.  J




THE BAND.                    223
By eliminating R1, R2, and t between the four equations
(218) and (219), a relation is determined between the three
quantities P, P,, T. To simplify this elimination let us suppose that the preceding hypothesis in respect to the directions in which the pressures are to be taken positively and
negatively is so made, that the expressions enclosed within
the brackets in the above equations (219) and squared may,
each of them, represent a positive quantity. Let us, moreover, suppose the first of the two quantities squared in each
equation to be considerably greater than the second, or the
pressure upon the axis of each drum in the direction of the
line C1 Cl joining their centres, greatly to exceed the pressure upon it in a direction perpendicular to that line; an
hypothesis which will in every practical case be realised.
These suppositions being made, we obtain, with a sufficient
degree of approximation, by Poncelet's Theorem*,
RIR1=a2T cos. c —P, cos. 01-W1 cos. i} +
/ {P1, sin. 01,-2t sin. al —W sin. 1},
R2= a2T cos. O -P2 cos. 02 + 2 Cos. 1} +
/ {P, sin. 02+2t sin., —W, sin. I}.
Substituting these values of R1 and R2 in equation (218),
and reducing, we have
Plal-2t(rl-/P 1 sin. a, sin. p)= 
pl {2aT cos. al1-P1/e-WTyl1 sin. q    (
P2a2-2t(r2-1p2, sin. aX sin. (p)=...
-p, 2 T cos. al-P2P2 +W2y2  sin. J
where /1=(a cos. 01-/ sin. 01),
/2 =(a cos. 0,- 2   sin. 0,),
Y=:(a cos. (+/P sin.,),
y2=( ( cos. i —/ sin. i).
Eliminating t between these equations, and neglecting
terms above the first dimension in P1 sin. p and p, sin. p,
I +P1a1(r2-/3p2 sin. l sin. p) )
— P2a2(rl-3Pi sin. U, sin. p) )+ pr2(2aT cos. al-PP-Wl) + W))
( +p2r(2aT cos. al — PJi-Wl) f sn.... (221,, being for the most part exceedingly small, the terms
* See Appendix.




C4                    T HE BA-ND.
ppi sin.,I sin,. and Pp2 sin. -   sin. p may be neglected; we
shall then obtain by transposition and reduction
+Palr2(1 + PI sin. p))
pP
-Par,(1 —   sin. q)
J
t+ 2xT(plr, + p2r) cos. c 1
I  sin. (p... (222).
l - W1P1ylrr2-W,P2Yrl)
If this equation be compared with equation (214), it will
be found to agree with it, mutatis mutandis, except that
the co-efficient'2 is in that equation 2. This difference
manifestly results from the cpprowimate character of the
theorem of Poncelet.
Substituting the latter co-efficient for the former, multiplying both sides of the equation by (1 —P —sin. p), neglecting
al
terms of more than two dimensions in -1, 2 and sin. p, and
reducing,
P   (a)      ( a + a2 ) sin. pP2+
a1r,      a,/ 71  aa2
{' 2T(palr2 + p s cos.  sn.  (223}
(WPlylr 2P..2l
which is the relation between the moving and working
pressures in the state bordering upon motion. From this
relation we obtain for the MODULUS of the band (equation
121)
+2-(Wpyr,W -p,Y,o) sin. q... (224).
If the angle, be conceived to increase until it exceed, P, will pass to the opposite side of QC,, and, will
become negative; whence it is apparent, that equation




THE BAND.                   225
(224) agrees with equation (214) in other respects, and in
the condition of the ambiguous sign. It is moreover apparent, from the form assumed by the modulus in this case
and in that of the preceding article, that the greatest
economy of power is obtained by applying the moving and
the working pressures on the same side of the line C1C2 joining the axes of the drums. This is in fact but a particular
case of the general principle established in Art. 168.
195. The initial tension T of the band may be determined precisely as in the former case (equation 217).
Representing by 0 the angle subtended by the circumference which
71-X-  f  the band embraces on the second
or driven drum, by P the maxi- /  /  mum resistance opposed to its motion at the distance a, by 4 the
the band and the surface of the
(::j  ~  drum, and by t, and t2 the tensions
upon the two parts of the band,
when its maximum resistance being opposed, it is upon the
point of slipping; observing, moreover, that in this case
a tan. 42
1
2(t_-t2) or 2t is represented (Art. 193.) by 2T -t; then
e + 1
substituting in the second of equations (220) this value for
2t, and P and a for P2 and a, and neglecting the exceedingly small term which involves the product sin. a, sin. p,
we have
0 tan. 4 
Pa-2T    6  i2a 1 r2,= —p,{2aT cos. a,-P/,+W,22, sin. q.
E + 
Also, since cl represents the inclination of the two parts of
the band to one another; since, moreover, these touch the
surfaces of the drums, and that 0 represents the inclination
of the radii drawn from the centre of the lesser drum to the
touching points, therefore 0-=r —o. Substituting this value
of 0 in the above equation, and solving it in respect to T, we
have
15




226                   THE BAND.
1 f     P(a-pP, sin. /)+W,p2py, sin. c  1
2     (T r-a,) tan. Pr,-p2 a cos. a, sin. + L  *  *   * ( 2 )
(-Tr-al) tan.               l
is    +   1i                   J
196. The modulus of the band when the two parts of it,
which intervene between the drums, are made to cross one
another.
If the directions of the two parts of the band be made
to cross, as shown in the accompanying
figure, the moving pressure T, upon the
second drum is applied to it en the side
*e opposite to that on which it is applied
T  /y       when the bands do not cross; so that in
Lp^!,'/ /    this case, in order that the greatest ecoT- ~~' /s2    nomy of power may be attained (Art.
168.), the working pressure or resistance
P should be applied to it on the side
opposite to that in which it was applied
in the other case, and therefore on the side of the line CC2
opposite to that on which the moving pressure P1 upon the
first drum is applied. This disposition of the moving and
working pressures being supposed, and this case being investigated by the same steps as the preceding, we shall arrive
at precisely the same expressions (equations 223 and 224)
for the relation of the moving and the working pressures,
and for the modulus.
In estimating the value of the initial tension T (equation
225) it will, however, be found, that the angle 0, subtended
at the centre C0 of the second drum by the arc KML, which
is embraced by the band, is no longer in this case represented by e —al but by ~+ a. This will be evident if we
consider that the four angles of the quadrilateral figure
C2KIL being equal to four right angles, and its angles at K
and L being right angles, the remaining angles KIL and
KC2L are equal to two right angles, so that KC2L=- r —
but the angle subtended by KML equals 2S —KC2L; it
equals therefore ~ +C. If this value be substituted for ~ —o,
in equation (225) it becomes




THE TEETH OF WHEELS.             227
T=i   P(a-p2 p  sin. p)+Wp sin.          (226.)
( +a) tan.  r2-P2 cos. a sin. g
i\   +    1I.
Now the fraction in the denominator of this expression
being essentially greater in value than that in the denominator of the preceding (equation 225), it follows that the
initial tension T, which must be given to the band in order
that it may transmit the work from the one drum to the
other under a given resistance P, is less when the two parts
of the band cross than when they do not, and, therefore, that
the modulus (equation 224) is less; so that the band is
worked with the greatest economy of power (other things
being the same) when the two parts of it which intervene
between the drums are made to cross one another. Indeed it
is evident, that since in this case the arc embraced by the
band on each drum subtends a greater angle than in the
other case, a less tension of the band in this case than in the
other is required (Art. 185.) to prevent it from slipping
under a given resistance, so that the friction upon the axis
of the drums which results from the tension of the band is
less in this case than the other, and therefore the work
expended on that friction less in the same proportion.
THE TEETH OF WHEELS.
197. Let A, B represent two circles in contact at D, and
A     moveable about fixed centres at C1 and C2. It
-  XC)  is evident that if by reason of the friction of
these two circles upon one another at D any
motion of rotation given to A be communicated
to B, the angles PC1D and QC2D described in
the same time by these two circles, will be such
iB    as will make the arcs PD and QD which they
subtend at the circumferences of the circles equal to one
another. Let the angle PC1D* be represented by 8, and the
angle QC2D by 82; also let the radii C1D and C2D of the circles be represented by r, and r2. Now, arc PD=rd1, arc
QD=r2,,; and since PD=QD, therefore r21,8r2=2;' Or rather the arc which this angle subtends to radius unity.




228             THE TEETH OF WHEELS.
2   r1
=    r-.....
The angles described, in the same time, by two circles
which revolve in contact are therefore inversely proportional
to the radii of the circles, so that their angular velocities
(Art. 74.) bear a constant proportion to one another; and if
one revolves uniformly, then the other revolves uniformly;
if the angular revolution of the one varies in any proportion,
then that of the other varies in like proportion.
When the resistance opposed to the rotation of the driven
circle or wheel B is considerable, it is no longer possible to
give motion to that circle by the friction on its circumference of the driving circle. It becomes therefore necessary in the great majority of cases to cause the rotation of
the driven wheel by some other means than the friction of
the circumference of the driving wheel.
One expedient is the band already described, by means of
which the weels may be made to drive one another at any
distances of their centres, and under a far greater resistance
than they could by their mutual contact. When, however,
the pressure is considerable, and the wheels may be brought
into actual contact, the common and the more certain
method is to transfer the motion
from one to the other by means of
projections on the one wheel called
TEETH, which engage in similar projections on the other.
In the construction of these teeth
the problem to be solved is, to give
such shapes to their surfaces of mutual contact, as that the wheels shall
be made to turn by the intervention
of their teeth precisely as they would by the friction of
their circumferences.
198. That it is possible to construct teeth which shall
answer this condition may thus be shown.
a/^        Let mn and m'n' be two curves, the one
(   < )  described on the plane of the circle A, and
the other on the plane of the circle B; and
^ ^ 3 a let them be such that as the circle A rec   ) volves, carrying round with it the circle B,
by their mutual contact at D, these two
curves mu and m'n' may continually touch




THE TEETH OF WHEELS.              229
one another, altering of course, as they will do continually,
their relative positions and their point of contact T.
It is evident that the two circles would be made to
revolve by the contact of teeth whose edges were of the
forms of these two curves mn and m'n' precisely as they
would by their friction upon the circumferences of one
another at the point D; for in the former case a certain
series of points of contact of the circles (infinitely near to
one another) at D, brings about another given series of points
of contact (infinitely near to one another) of the curves mn
and m'n' at T; and in the latter case the same series of
points in the curves mn and m'n' brought into contact necessarily produces the contact of the same series of points in
the two circumferences of the two circles at D.
To construct teeth whose surfaces of contact shall possess
the properties here assigned to the curves mn and m'n' is
the problem to be solved. Of the solution of this problem
the following is the fundamental principle:
199. In order that two circles A and B may be made to
revolve by the contact of the surfaces mn and m'n' of their
teeth, precisely as they would by the friction of their cir^a       ctuwmferences, it is necessary, and it is suf\         fi\~tcient, that a line drawn from the point
{^j^-^-    of contact T of the teeth to the point of'.:    B    CO >   contact D of the circumferences should, in
d   every position of the point T, be perpendi\    cular to the surfaces in contact there, i. e.,
a normal to both the curves mn and m'n'.
To prove this principle, we must first establish the following LEMMA:-If two circles M and N be made to revolve
about the fixed centres E and F by their mutual contact at L, and if the planes of these
circles be conceived to be carried round with
them in this revolution, and a point P on the
\  N /  plane of M to trace out a curve PQ on the
plane of N whilst thus revolving, then is this
curved line PQ precisely the same as would
have been described on the plane of N by the same point P,
if the latter plane, instead of revolving, had remained at
rest, and the centre E of the circle M having been released




230               THE TEETH OF WHEELS.
from its axis, that circle had been made to roll (carrying its
plane with it) on the circumference of N. For conceive O
to represent a third plane on which the centres of E and F
are fixed. It is evident that if, whilst the circles M and N
are revolving by their mutual contact, the plane 0, to which
their centres are both fixed, be in any way moved, no change
will thereby be produced in form of the curve PQ, which the
point P in the plane of M is describing upon the plane of N,
such a motion being common to both the planes M and N.*
Now let the direction in which the circle N is revolving be
that shown by the arrow, and its angular velocity uniform;
and conceive the plane O to be made to revolve about F with
an angular velocity (Art. 74) which is equal to that of N,
but in an opposite direction, communicating
this angular velocity to M  and N, these revolving meantime in respect to one another,
and by their mutual contact, precisely as they.        / did before.t
\x _'/    It is clear that the circle N being carried
round by its own proper motion in one direction, and by the motion common to it and the plane 0 with
the same angular velocity in the opposite direction, will, in
reality rest in space; whilst the centre E of the circle M,
having no motion proper to itself, will revolve with the
angular velocity of the plane 0, and the various other points
in that circle with angular velocities, compounded of their
proper velocities, and those which they receive in common
with the plane 0, these velocities neutralising one another
at the point L of the circle, by which point it is in contact
with the circle N.  So that whilst M revolves round N, the
point L, by which the former circle at any time touches the
other, is at rest; this quiescent point of the circle M nevertheless continually varying its position on the circumferences
of both circles, and the circle M being in fact made to roll
on the circle N at rest.
Thus, then, it appears, that by communicating a certain
common angular velocity to both the circles M and N about
Thus for instance, if the circles M and N continue to revolve, we may
evidently place the whole machine in a ship under sail, in a moving carriage,
or upon a revolving wheel, without in the least altering the form of the curve,
which the point P, revolving with the plane of the circle M, is made to trace
on the plane of N, because the motion we have communicated is common to
both these circles.
M and N may be imagined to be placed upon a horizontal wheel 0, first at
rest, and then made to revolve backwards in respect to the motion of N.




THE TEETH OF WHEELS.               231
the centre F, the former circle is made to roll upon the other
at rest; and, moreover, that this common angular velocity
does not alter the form of the curve PQ, which a point P in
the plane of the one circle is made to trace upon the plane
of the other, or, in other words, that the curve traced under
these circumstances is the same, whether the circles revolve
round fixed centres by their mutual contact, or whether the
centre of one circle be released, and it be made to roll upon
the circumference of the other at rest.
This lemma being established, the truth of the proposition
stated at the head of this article becomes evident; for if M
roll on the circumference of N, it is evident that P will, at
any instant, be describing a circle about their point of contact L.*
Since then P is describing, at every instant, a circle about
L when M rolls upon N, N being fixed, and since the curve
described by P upon this supposition is precisely the same
as would have been traced by it if the centres of both circles had been fixed, and they had turned by their mutual
contact, it follows that in this last case (when the circles
revolve about fixed centres by their mutual contact) the
point P is at any instant of the revolution describing, during
that instant, an exceedingly small circular arc about the
point L; whence it follows that PL is always a perpendicular to the curve PQ at the point P, or a normal to it.
Now let p be a point exceedingly near to T in the curve
-n m'n', which curve is fixed upon the plane
of the circle A. It is evident that, as the
[ Id_  p  point p passes through its contact with the
-: _    curve mn at T (see Art. 195.), it will be
made to describe, on the plane of the circle
~t aB, an exceedingly small portion of that
curve mnn.   But the curve which it is
(under these circumstances) at any instant
describing upon the plane of B has been shown to be
always perpendicular to the line DT; the curve mnn is therefore at the point T perpendicular to the line DT; whence it
follows that the curve m'n' is also perpendicular to that line,
and that DT is a normal to both those curves at T. This is
the characteristic property of the curves mn and m'n', so that
they may satisfy the condition of a continual contact with
* For either circle may be imagined to be a polygon of an infinite number
of sides, on one of the angles of which the rolling circle will, at any instant,
be in the act of turning.




232             THE TEETH OF WHEELS.
one another, whilst the circles revolve by the contact of
their circumferences at D, and therefore conversely, so that
these curves may, by their mutual contact, give to the circles the same motion as they would receive from the contact
of their circumferences.
200. To describe, by means of circular arcs, the form of a
tooth on one wheel which shall work truly with a tooth of
any given form on another wheel.
Let the wheels be required to revolve by the action of
their teeth, as they would by the
_ — ~contact of the circles ABE and
~/-  \A      )ADF, called theirprimitive or pitch
circles. Let AB represent an are
of the pitch circle ABE, included
between any two similar points A:\ |-  /       and B of consecutive teeth, and let
\v>?S "      JAD represent an are of the pitch
circle ADF equal to the arce AB, so
that the points D and B may come
simultaneously to A, when the circles are made to revolve by their
mutual contact. AB and AD are
called the pitches of the teeth of the two wheels. Divide
each of these pitches into the same number of equal parts
in the points a, 6, &c., a', b, &c.; the points a and a', b and
b', &c., will then be brought simultaneously to the point A.
Let mn represent the form of the face of a tooth on the
wheel, whose centre is C, with which tooth a corresponding
tooth on the other wheel is to work truly; that is to say,
the tooth on the other wheel, whose centre is C, is to be cut,
so that, driving the surface min, or being driven by it, the
wheels shall revolve precisely as they would by the contact of their pitch circles ABE and ADF at A. From A
measure the least distance Aa to the curve mnn, and with
radius Aa and centre A describe a circular arc a/3 on the
plane of the circle whose centre is C2. From a measure, in
like manner, the least distance an', to the curve mn, and
with this distance au' and the centre a, describe a circular
arc /y, intersecting the arc a/3 in F/. From the point b
measure similarly the shortest distance by" to mnn, and with




THE TEETH OF WHEELS.                    233
the centre b' and this distance be" describe a circular arc
ya, intersecting /3 in 7, and so with the other points of
division.  A curve touching these circular arcs a/, f/, 76,
&c., will give the true surface or boundary of the tooth.*
In order to prove this let it be observed, that the shortest
distance a'c from  a given point a to a given curve mn is a
normal to the curve at the point a' in which it meets it; and
therefore, that if a circle be struck from this point a with this
least distance as a radius, then this circle must touch the
curve in the point a', and the curve and circle have a common normal in that point.
Now the points a and a' will be brought by the revolution
of the pitch circles simultaneously to the point of contact A,
and the least distance of the curve mn from the point A will
then be ax', so that the arc 3y will then be an arc struck
from the centre A, with this last distance for its radius. This
circular arc f/  will therefore touch the curve mun in the point
a' and the line a!,, which will then be a line drawn fiomn
the point of contact A of the two pitch circles to the point
of contact a' of the two curves mn and m'n', will also be a
normal to both curves at that point. The circles will therefore at that instant drive one another (Art. 196.) by the contact of the surfaces mn and m'I', precisely as they would by
the contact of their circumferences. And as every circular
arc of the curve m'n' similar to fy becomes in its turn the
acting surface of the tooth, it will, in like manner, at one
point work truly with a corresponding point of mn, so that
the circles will thus drive one another truly at as many
points of the surfaces of their teeth, as there have been taken
points of division a, b, &c. and arcs a/, /3. &c.t
* This method of describing, geometrically, the forms of teeth is given, without
demonstration, by M. Poncelet in his Mecanique Industrielle, 3me partie, Art. 60.
t The greater the number of these points of division, the more accurate the
form of the tooth. It appears, however, to be sufficient
in most cases, to take three points of division, or even
two, where no great accuracy is required.  M. Poncelet
(/Mec. Indust. 3me partie, Art. 60.) has given the following,
yet easier, method by which the true form of the tooth
may be approximated to with sufficient accuracy in most
cases. Suppose the given tooth N upon the one wheel to
Y/ ~x/          be placed in the position in which it is first to engage or
- Gil.^       disengage from the required tooth on the other wheel,
W     <;2  S0'^  and let Aa and Ab be equal arcs of the pitch circles of
1-""1 7     the two wheels whose point of contact is A. Draw Aa
the shortest distance between A and the face of the tooth
N; join aa; bisect that line in m, and draw mn perpendicular to aa intersecting the circumference Aa in n. If
from the centre n a circular arc be described passing
_   through the points a and a, it will give the required form
of the tooth nearly.




234                 INVOLUTE TEETH.
INV OLUTE TEETH.
201. The teeth of two wheels will work truly together if they
be bounded by curves of the form traced out by the extremity
of a flexible line, unwinding from the circumference of a
circle, and called the involute of a circle, provided that the
circles of which these are the involutes be concentric with
the pitch circles of the wheels, and have their radii in the
same proportion with the radii of the pitch circles.
Let OE and OF represent the pitch circles of two wheels,
AG   and BH   two circles concentric with
/  &-"      them  and having their radii C1A and C2B
( /...\. in the same proportion with the radii CO
Wi-    and C0O of the pitch circles. Also let mns
-\'-    and m'n' represent the edges of teeth on the
two wheels struck by the extremities of flexible lines unwinding from the circumferences
s1      of the circles AG and BH respectively. Let,\ -I!    these teeth be in contact, in any position
\   -....  of the wheels, in the point T, and from the
_ _A     point T draw TA and TB tangents to the
generating circles GA and BH in the points
A and B. Then does AT evidently represent the position of
the flexible line when its extremity was in the act of generating the point T in the curve mnn; whence it follows, that
AT is a normal to the curve mn at the point T*; and in
like manner that BT is a normal to the curve m'n' at the
same point T. Now the two curves have a
/o G        common tangent at T; therefore their nor{   ^     emals TA and TB at that point are in the same
\,.  ) i  straight line, being both perpendicular to their
tangent there. Since then ATB is a straight
line, and that the vertical angles at the point
/-"o^ where AB and CC.2 intersect are equal, as
l a^  | \  \ also the right angles at A and B, it follows
l —'d  j }that the triangles AoCland BoC are similar,
-\ --  / and that Co: 2o:: 1A: CB. But C0A:
C2B::         020;.C;. CGo: 2o::c 0
C0O; therefore the points O and o coincide,
and the straight line AB, which passes through the point of
* For if the circle be conceived a polygon of an infinite number of sides, it
is evident that the line, when in the act of unwinding from it at A, is turning
upon one of the angles of that polygon, and therefore that its extremity is,
through an infinitely small angle, describing a circular arc about that point.




INVOLUTE TEETH.                        235
contact T of the two teeth, and is perpendicular to the surfaces of both at that point, passes also through the point of
contact O of the pitch circles of the wheels.        Now    this is
true, whatever be the positions of the wheels, and whatever,
therefore, be the points of contact of the teeth.       Thus then
the condition established in Art. 199. as that necessary and
sufficient to the true action of the teeth of wheels, viz. " that
a line drawn from the point of contact to the pitch circles to
the point of contact of the teeth should be a normal to their
surfaces at that point, in all the different positions of the
teeth," obtains in regard to involute teeth.*
The point of contact T of the teeth moves along the straight
line AB, which is drawn touching the generating circles BH
and AG of the involutes; this line is what is called the locus
of the different points of contact. Moreover, this property
obtains, whatever may be the number of teeth in contact at
once, so that all the points of contact of the teeth, if there
be more than one tooth in contact at once, lie always in this
line; which is a characteristic, and a most important property   of teeth of the involute form.       Thus in the above
* The author proposes the following illustration of the action of involute
teeth, which he believes to be new. Conceive AB to represent a band passing
round the circles AG and BH, the wheels would evidently be driven by this
band precisely as they would by the contact of their pitch circles, since the
radii of AG and BH are to one another as the radii of the pitch circles. Conceive, moreover, that the circles BI- and AG carry round with them their
planes as they revolve, and that a tracer is fixed at any point T of the band,
tracing, at the same time, lines mn and m'n', upon both planes, as they revolve
beneath it. It is evident that these curves, being traced by the same point,
must be in contact in all positions of the circles when driven by the band, and
therefore when driven by their mutual contact.  The wheels would therefore
be driven by the contact of teeth of the forms mn and m'n' thus traced by the
point T of the band precisely as they would by the contact of their pitch circles. Now it is easily seen, that the curves mn and m'n', thus described, by the
point T of the band, are involutes of the circles AG and BH.




236      EPICYCLOIDAL AND HYPCCYCLOIDAL TEETH.
figure, which represents part of two wheels with involute
teeth, it will be seen that the points r s of contact of the
teeth are in the same straight line touching the base* of one
of the involutes, and passing through the point of contact A
of the pitch circles, as also the points A and b in that touching the base of the other.
EPICYcLOIDAL AND HYPOCYOLOIDAL TEETH.
202. If one circle be made to roll externally on the cir3^i       cumference of another, and if, whilst this motion is taking place, a point in the circumfe$    r     c rence of the rolling circle be made to trace
\         out a curve upon the plane of the fixed circle,
the curve so generated is called an EPICYCLOID,
the rolling circle being called the generating
circle of the epicycloid, and the circle upon
which it rolls its base.
If the generating circle, instead of rolling
"- ^ /  on the outside or convex circumference of its
base, roll on its inside or concave circumference, the curve generated is called the HYPOOYcLOID.
Let PQ and PR be respectively an epicycloid and a hypocycloid, having the same generating circle APH, and
having for their bases the pitch circles AF and AE of two
wheels. If teeth be cut upon these wheels, whose edges
coincide with the curves PQ and PR, they will work truly
with one another; for let them be in contact at P, and let
their common generating circle APH be placed so as to
touch the pitch circles of both wheels at A, then will its circumference evidently pass through the point of contact P
of the teeth: for if it be made to roll through an exceedingly small angle upon the point A, rolling there upon the
circumference of both circles, its generating point will
traverse exceedingly small portions of both curves; since
then a given point in the circumference of the circle APII
is thus shown to be at one and the same time in the perimeters of both the curves PQ and PR, that point must of
necessity be the point of contact P of the curves; since,
* The circles from which the involutes are described are called their bases.
This cut and that at page 237. are copied from Mr. Hawkins' edition of Camus
on the Teeth of Wheels.




EPICYCLOIDAL AND HYPOOYCLOIDAL TEETH.               237
moreover, when the circle APH        rolls upon the point A, its
generating point traverses a small portion of the perimeter
of each of the curves PQ and PR at P, it follows that the
line AP is a normal to both curves at that point; for whilst
the circle APH      is rolling through    an  exceedingly small
angle upon A, the point P in it, is describing a circle about
that point whose radius is AP.*         Teeth, therefore, whose
edges are of the forms PQ and PR satisfy the condition
that the line AP drawn from the point of contact of the
pitch circles to any point of contact of the teeth is a normal
to the surfaces of both at that point, which condition has beern
shown (Art. 199.) to be that necessary and sufficient to the
correct working of the teeth.t
Thus then it appears, that if an epicycloid be described
/- ~
-' /', 
-..   _.      j,.      ~~^\ 
* The circle APH may be considered a polygon of an infinite number of
sides, on one of the angles of which polygon it may at any instant be conceived to be turning.
f The entire demonstration by which it has been here shown that the
curves generated by a point in the circumference of a given generating circle
APR rolling upon the convex circumference of one of the pitch circles, and
upon the concave circumference of the other are proper to form the edges of
contact of the teeth, is evidently applicable if any other generating curve be
substituted for APH. It may be shown precisely in the same manner, that
the curves PQ and PR generated by the rolling of any such curve (not being
a circle) upon the pitch circles, possess this property, that the line PA drawn
from any point of their contact to the point of contact of their pitch circles
is a normal to both, which property is necessary and sufficient to their correct
action as teeth. This was first demonstrated as a general principle of the construction of the teeth of wheels by Mr. Airy, in the Cambridge Phil. Trans.
vol. ii. He has farther shown, that a tooth of any form whatever being cut
upon a wheel, it is possible to find a curve which, rolling upon the pitch circle
of that wheel, shall by a certain generating point traverse the edge of the
given tooth. The curve thus found being made to roll on the circumference
of the pitch circle of a second wheel, will therefore trace out the form of a
tooth which will work truly with the first. This beautiful property involves




238       EPIOYCLOIDAL AND HYPOCYCLOIDAL TEETH.
on the plane of one of the wheels with any generating
circle, and with the pitch circle of that wheel for its base;
and if a hypocycloid be described on the plane of the other
wheel with the pitch circle of that wheel for its base; and
if the faces or acting surfaces of the teeth on the two
weeels be cut so as to coincide with this epicycloid and this
hypocycloid respectively, then will the wheels be driven
correctly by the intervention of these teeth.  Parts of two
wheels having epicycloidal teeth are represented in the preceding figure.
203. LEMMA.-If the diameter of the generating circle of a
hypocycloid equal the radius of its base, the hypocycloid
becomes a straight line having the direction of a radius of
its base.
Let D and d represent two positions of the centre of such
A          a generating   circle, and  suppose  the
generating point to have been at A in:   /.,,; the first position, and join AC; then
will the generating point be at P in the
second position, i. e. at the point where
CA   intersects the. circle in its second
position; for join   Ca and Pd, then
ZPdea= ZPCd+ Z CPd=2ACa.            Also
2da=CA;. 2daxPda=2CAxACa;.~.daxPda=-CAx
ACa;.'.arc Aca=arc Pa. Since then the arc aP equals
the arc aA, the point P is that which in the first position
coincided with A, i. e. P is the generating point; and this
is true for all positions of the generating circle; the generating point is therefore always in the straight line AC.
The edge, therefore, of a hypocycloidal tooth, the diameter
of whose generating circle equals half the diameter of the
pitch circle of its wheel, is a straight line whose direction
is towards the centre of the wheel.*
the theoretical solution of the problem which Poncelet has solved by the
geometrical construction given to Article 200. If the rolling curve be a
logarithmic spiral, the involute form of tooth will be generated.
* The following very ingenious application has been made of this property
of the hypocycloid to convert a circular into an alternate rectilinear motion.
AB represents a ring of metal, fixed in position, and having teeth cut upon its




TO SET OUT THE TEETH OF WHEELS.                 239
TO SET OUT THE TEETH OF WHEELS.
204. All the teeth of the same wheel are constructed of
the same form and of equal dimensions: it would, indeed,
evidently be impossible to construct two wheels with different numbers of teeth, which should work truly with one
another, if all the teeth on each wheel were not thus alike.
All the teeth of a wheel are therefore set out by the workman from the same pattern or model, and it is in determining
the form and dimensions of this single pattern or model of
one or more teeth in reference to the mechanical effects
which the wheel is to produce, when all its teeth are cut out
upon it and it receives its proper place in the mechanical
combination of which it is to form      a part, that consists the
art of the description of the teeth of wheels.
The mechanical function usually assigned to toothed wheels
is the transmission of work under an increased or diminished
velocity. If CD, DE, &c., represent arcs of the pitch circle
concave circumference. C is the centre of
a wheel, having teeth cut in its circumo^A     B y*        ference to work with those upon the circum9Ax           Ah^ \      ference of the ring, and having the diameter of its pitch circle equal to half that of
the pitch circle of the teeth of the ring.
This being the case, it is evident, that if the
l  Vy Dp | pitch circle of the wheel C were made to
roll upon that of the ring, any point in its
circumference would describe a straight line
passing through the centre D of the ring;
~\^'      i/'        ^but the circle C would roll upon the ring by
the mutual action of their teeth as it would
by the contact of their pitch circles; if the
circle C then be made to roll upon the ring
by the intervention of teeth cut upon both, any point in the circumference of
C will describe a straight line passing through D. Now, conceive C to be thus
made to roll round the ring by means of a double or forked link CD, between
the two branches of which the wheel is received, being perforated at their
extremities by circular apertures, which serve as bearings to the solid axis of
the wheel. At its other extremity D, this forked link is rigidly connected
with an axis passing through the centre of the ring, to which axis is communicated the circular motion to be converted by the instrument into an alternating rectilinear motion. This circular motion will thus be made to carry
the centre C of the wheel round the point D, and at the same time, cause it to
roll upon the circumference of the ring. Now, conceive the axis C of the
wheel, which forms part of the wheel itself, to be prolonged beyond the collar
in which it turns, and to have rigidly fixed upon its extremity a bar CP. It is
evident that a point P in this bar, whose distance from the axis C of the wheel
equals the radius of its pitch circle, will move precisely as a point in the pitch circle
of the wheel moves, and therefore that it will describe continually a straight
line passing through the centre D of the ring.  This point P receives, there
fore, the alternating rectilinear motion which it was required to communicate.




240        TO SET OUT THE TEETH OF WHEELS.
z/ I       i        i
of a wheel intercepted between similar points of consecutive
teeth (the chords of which arcs are called the pitches of the
teeth), it is evident that all these arcs must be equal, since
the teeth are all equal and similarly placed; so that each
tooth of either wheel, as it passes through its contact with a
corresponding tooth of the other, carries its pitch line through
the same space CD, over the point of contact C of the pitch
lines. Since, therefore, the pitch line of the one wheel is
carried over a space equal to CD, and that of the other over
a space equal to cd by the contact of any two of their teeth,
and since the wheels revolve by the contact of their teeth
as they would by the contact of their pitch circles at C, it
follows that the arcs CD and cd are equal. Now let r, and
r, represent the radii of the pitch circles of the two wheels,
then will 2vrr, and 2rrrr represent the circumferences of their
pitch circles; and if n, and n2 represent the numbers of
27rrr       2irr
teeth cut on them respectively, then CD=  1 and cd =
27riw  2-irr.i
therefore, 2- - —,
ln,   n,:. - = -nl..... ( (227);
r2   N
Therefore the radii of the pitch circles of the two wheels
must be to one another as the numbers of teeth to be cut
upon them respectively.
Again, let m, represent the number of revolutions made
by the first wheel, whilst m, revolutions are made by the
second; then will 2rrrm, represent the space described by




A TRAIN OF WHEELS.              241
the circumference of the pitch circle of the first wheel while
these revolutions are made, and 27rr2m, that described by the
circumference of the pitch circle of the second; but the
wheels revolve as though their pitch circles were in contact,
therefore the circumferences of these circles revolve through
equal spaces, therefore 27rrm,-=27rr,2;
r   m
r,   _-......(228).
r,  m,
The radii of the plnch circles of the wheels are therefore
inversely as the numbers of revolutions made in the same
time by them.
Equating the second members of equations (227) and (228),
m bs o   r (229).
ml   n2
The numbers of revolutions made by the wheels in the same
time are therefore to one another inversely as the numbers
of teeth.
205. JI a train of wheels, to determine how many revolutions
the last wheel makes whilst the first is making any given
number of revolutions.
When a wheel, driven by another, carries its axis round.
with it, on which axis a third
_9  8::', wheel is fixed, engaging with and;{ ^1,~  /\    \  giving motion to afourth, which,
iI in like manner, is fixed upon its
I {  ~ r a           3 axis, and carries round with it a.~ \   -9 B l   fifth wheel fixed upon the same
axi, which fifth wheel engages
with a sixth upon another axis,
and so on as shown in the above figure, the combination
forms a train of wheels. Let n, nA, n... np represent the
numbers of teeth in the successive wheels forming such a
train of p pairs of wheels; and whilst the first wheel is
making m revolutions, let the second and third (which revolve
together, being fixed on the same axis) make m, revolutions;
the fourth and fifth (which, in like manner, revolve together)
m n revolutions, the sixth and seventh mn,, and so on; and let
the last or 21711 wheel thus be made to revolve mp times whilst
16




242               A TRAIN OF WHEELS.
the first revolves m times. Then, since the first wheel which
has n. teeth gives motion to the second which has n2 teeth,
and that whilst the former makes m revolutions the latter
makes m, revolutions, therefore (equation 229), m =-;
m    n2
and since, while the third wheel (which revolves with the
second, makes ml revolutions, the fourth makes m2 revolutions; therefore, 3 = -.  Similarly, since while the fifth
ml   n
wheel, which has n. teeth, makes m, revolutions (revolving
with the fourth), the sixth, which has n6 teeth, makes mn revoq 3  n m   5     4     m    nF7
lutions; therefore -   -6. In like manner -  - &c. &c.
m2   n6                m3   n8
MP  2p-1. Multiplying these equations together, and
striking out factors common to the numerator and denominator of the first member of the equation which results from
their multiplication, we obtain
MhP fa, c t  * th  * * nt f      r    n h.(230).
m    n2.4 ~. nf.... vn 2p'The factors in the numerator of this fraction represent the
numbers of teeth in all the driving wheels of this train,
and those in the denominator the numbers of teeth in the
driven wheels, or followers as they are more commonly
called.
If the numbers of teeth in the former be all equal and
represented by n,, and the numbers of teeth in the latter.also equal and represented by n,, then
m      n.......(231).
Having determined what should be the number of teeth
in each of the wheels which enter into any mechanical
combination, with a reference to that particular modification
of the velocity of the revolving parts of the machine which
is to be produced by that wheel,* it remains next to consider,
what must be the dimensions of each tooth of the wheel, so
* The reader is referred for a more complete discussion of this subject (which
belongs more particularly to descriptive mechanics) to Professor Willis's Principles of Mechanism, chap. vii., or to Camus on the Teeth of Wheels, by Hawkins, p. 90.




THE STRENGTH OF TEETH.             2 13
that it may be of sufficient strength to transmit the work
which is destined to pass through it, under that velocity, or
to bear the pressure which accompanies the transmission of
that work at that particular velocity; and it remains further
to determine, what must be the dimensions of the wheel
itself consequent upon these dimensions of each tooth, and
this given number of its teeth.
206. To determine the pitch of the teeth of a wheel, knowiny
the work to be transmitted by the wheel.
Let U represent the number of units of work to be transmitted by the wheel per minute, m the number of revolutions
to be made by it per minute, n the number of the teeth to
be cut in it, T thepitch of each tooth in feet, P the pressure
upon each tooth in pounds.
Therefore nT represents the circumference of the pitch
circle of the wheel, and mnT represents the space in feet
described by it per minute.  Now U represents the work
transmitted by it through this space per minute, therefore- T
represents the mean pressure under which this work is transmitted (Art. 50.);
P........... (232).
The pitch T of the teeth would evidently equal twice the
breadth of each tooth, if the spaces between the teeth were
equal in width to the teeth. In order that the teeth of
wheels which act together may engage with one another and
extricate themselves, with facility, it is however necessary
that the pitch should exceed twice the breadth of the tooth
by a quantity which varies according to the accuracy of the
construction of the wheel from b'oth to -,'th of the breadth.*
Since the pitch T of the tooth is dependant upon its
breadth, and that the breadth of the tooth is dependant, by
the theory of the strength of materials, upon the pressure 
which it sustains, it is evident that the quantity P in the
above equation is a function of T. This functiont may be
assumed of the form
* For a full discussion of this subject see Professor Willis's Principles of
Mechanism, Arts. 107-112.
f See Appendix, on the dimensions of wheels.




244            THE STRENGTH OF TEETH.
T=c /P........ (233);
where c is a constant dependant for its amount upon the
nature of the material out of which the tooth is formed.
Eliminating P between this equation and the last, and solving
in respect to T,
T-    2U
T=V/.
mn
The number of units of work transmitted by any machine
per minute is usually represented in horses' power, one
horse's power being estimated at 33,000 units, so that the
number of horses' power transmitted by the machine means
the number of times 33,000 units of work are transmitted by
it every minute, or the number of times 33,000 must be
taken to equal the number of units of work transmitted by
it every minute. If therefore H represent the number of
horses' power transmitted by the wheel, then JU=33,000H.
Substituting this value in the preceding equation, and representing the constant 33,000c2 by C3, we have
T=C 0/ —....... (234).
mn
The values of the constant C for teeth of different materials are given in the Appendix.
207. To determine the radius of the pitch circle of a wheel
which shall contain n teeth of a given pitch.
7-^  -   Let AB represent the pitch T of a tooth,
(.-A-.  and let it be supposed to coincide with its
j\ /-... chord AMB. Let R represent the radius AC
I /  of the pitch circle, and n the number of teeth
to be cut upon the wheel.
Now there are as many pitches in the cira'il   cumference as teeth, therefore the angle ACB
2V
subtended by each pitch is represented by-.
Also T=2AM=2AC sin. iACB=2R sin.-;..R=jT cosec...... (235).




TO DESCRIBE EPICYCLOIDAL TEETH.       245
208. To make the pattern of an epicycloidal tooth.
-a._U  ___Having determined,.as above,
the pitch of the teeth, and the
radius of the pitch circle, strike
an arc of the pitch circle on a
thin piece of oak board or metal plate, and, with a fine saw,
^"  Cl    cut the board through along
^aC^//  ~    the circumference of this cirri,^     d~cle, so as to divide it into two
parts, one having a convex and
a~, /_    ____   I\      the other a corresponding conR    \"x. cave circular edge.  Let EF
represent one of these portions
of the board, and GiH another.
Describe an arc of the pitch circle upon a second board or
plate from which the pattern is to be cut.  Let MIN represent this arc. Fix the piece GH upon this board, so that its
circular edge may accurately coincide with the circumference
of the arc MN. Take, then, a circular plate D of wood or
metal, of the dimensions which it is proposed to give to the
generating circle of the epicycloid; and let a small point of
steel P be fixed in it, so that this point may project slightly
from its inferior surface, and accurately coincide with its circumference. Having set off the width AB of the tooth, so
that twice this width increased by from f —th to J-th of that
width (according to the accuracy of workmanship to be
attained) may equal the pitch, cause the circle D to roll upon
the convex edge GK of the board GH, pressing it, at the
same time, slightly upon the surface of the board on which
the arc MN is described, and from which the pattern is to be
cut, having caused the steel point in its circumference first
of all to coincide with the point A; an epicycloidal arc AP
will thus be described by the point P upon the surface MN.
Describe, similarly, an epicycloidal arc BE through the point
B, and let them meet in E.
Let the board GH now be removed, and let EF be applied
and fixed, so that its concave edge may accurately coincide
with the circular arc MN. With the same circular plate D
pressed upon the concave edge of EF, and made to roll upon
it, cause the point in its circumference to describe in like
manner, upon the surface of the board from which the pattern is to be cut, a hypococloidal arc BH passing through the




24 G        TO DESCRIBE EPIOYCLOIDAL TEETH.
point B, and another AI passing through the point A. HEI
will then represent the form of a tooth, which will work correctly (Art. 202.) with the teeth similarly cut upon any other
wheel; provided that the pitch of the teeth so cut upon the
other wheel be equal to the pitch of the teeth upon this, and
provided that the same generating circle D be used to strike
the curves upon the two wheels.
209. To determine the proper lengths of epicycloidal teeth.
The general forms of the teeth of wheels being determined
by the method explained in the preceding article, it remains
to cut them off of such lengths as may cause them successively to take up the work from one another, and transmit it
under the circumstances most favourable to the economy of
its transmission, and to the durability of the teeth.
In respect to the economy of the power in its transmission,
it is customary, for reasons to be assigned hereafter, to provide that no tooth of the one wheel should come into action
with a tooth of the other until both are in the act of passing
through the line of centres. This condition may be satisfied
in all cases where the numbers of teeth on neither of the
wheels is exceedingly small, by properly adjusting the
lengths of the teeth. Let two of the teeth of the wheels be
in contact at the point A in the line CD, joining the centres
of the two wheels; and let the wheel whose centre is C be
the driving wheel. Let AH   be a portion of the circumference of the generating circle of the teeth, then will the
points A and B, where this circle intersects the edges of the
teeth O and K of the driving wheel, be points of contact




TO DESCRIBE EPICYCLOIDAL TEETH.          24:7
with the edges of the teeth IMI and L of the driven wheel
(Art. 202.). Now, since each tooth is to come into action
only when it comes into the line of centres, it is clear that
the tooth L must have been driven by K from the time when
their contact was in the line of centres, until they have come
into the position shown in the figure, when the point of contact of the anterior face of the next tooth O of the driving
wheel with the flank* of the next tooth M of the driven
wheel has just passed into the line of centres; and since the
tooth 0 is now to take up the task of impelling the driven
wheel, and the tooth K to yield it, all that portion of the
last-mentioned tooth which lies beyond the point B may evidently be removed; and if it be thus removed, then the tooth
iK, passing out of contact, will manifestly, at that period of
the motion, yield all the driving strain to the tooth 0, as it
is required to do. In order to cut the pattern tooth of the
E     ___  _      proper length, so as to satisfy
the proposed condition, we have
only then to take Aa (see the
E=r     accompanying figure) equal to,^\  ~    the pitch of the tooth, and to
3 \^  ~    bring the convex circumference
\e,}   i A~of the generating circle, so as
X")   /           to touch the convex circumfe-,,? T!            rence of the arc MN    in that
-/ -#-  a..-. /\.  point a; the point of interseca / —  ___     \''.  tlion e of this circle with the
~;  face AE of the tooth will be
the last acting point of the tooth; and if a circle be struck
from the centre of the pitch circle passing through that
point, all that portion of the tooth which lies beyond this circle may be cut off.+
The length of the tooth on the wheel intended to act with
this, may be determined in like manner.
210. In the preceding article we have supposed the same
generating circle to be used in striking the entire surfaces
of the teeth on both wheels. It is not however necessary to
* That portion of the edge of the tooth which is without the pitch circle is
called its face, that within it its flank.
f The point e thus determined will, in some cases, fall beyond the extremity
E of the tooth. In such cases it is therefore impossible to cut the tooth of
such a length as to satisfy the required conditions, viz. that it shall drive only
after it has passed the line of centres. A full discussion of these impossible
cases will be found in Professor Willis's work (Arts. 102-104.).




248          TO DESCRIBE EPICYCLOIDAL TEETH.
the correct working of the teeth, that the same circle should
thus be used in striking the entire surfaces of ttwo teeth
which act together, but only that the generating circle of
every two portions of the two teeth which come into actual
contact should be the same. Thus the flank of the driving
tooth and the face of the driven tooth being in contact at
P in the accompanying figure,* this face of the one tooth
and flank of the other must be respectively an epicycloid
and a hypocycloid struck with the same generating circle.
Again, the face of a driving tooth and theflan/k of a driven
tooth being in contact at Q, these, too, must be struck by
the same generating circle. But it is evidently unnecessary
that the generating circle used in the second case should be
the same as that used in the first. Any generating circle
will satisfy the conditions in either case (Art. 202.), provided
it be the same for the epicycloid as for the hypocycloid
which is to act with it.
According to a general (almost a universal) custom among
mechanics, two diferent generating circles are thus used for
striking the teeth on two wheels which are to act together,
the diameter of the generating circle for striking the faces
of the teeth on the one wheel being equal to the radius of
the pitch circle of the other wheel. Thus if we call the
wheels A and B, then the epicycloidal faces of the teeth on
A, and the corresponding hypocycloidal flanks on B, are
generated by a circle whose diameter is equal to the radius
of the pitch circle of B. The hypocycloidal flanks of the
teeth on B thus become straight lines (Art. 203.), whose
directions are those of radii of that wheel. In like manner,
* The upper wheel is here supposed to drive the lower.




TO DESCRIBE EPICYCLOIDAL TEETH.       249
the epicycloidal faces of the teeth on B, and the corresponding hypocycloidal flanks of the teeth on A, are struck by a
circle whose diameter is equal to the radius of the pitch circle of A; so that the hypocycloidal flanks of the teeth of A
become in like manner straight lines, whose directions are
those of radii of the wheel A. By this expedient of using
two different generating circles, the flanks of the teeth on
both wheels become straight lines, and the faces only are
curved. The teeth shown in the above figure are of this
form. The motive for giving this particular value to the
generating circle appears to be no other than that saving of
trouble which is offered by the substitution of a straight for
a curved flank of the tooth. A more careful consideration
of the subject, however, shows that there is no real economy
of labour in this. In the first place, it renders necessary
the use of two different generating circles or templets for
striking the teeth of any given wheel or pinion, the curved
portions of the teeth of the wheel being struck with a circle
whose diameter equals half the diameter of the pinion, and
the curved portions of the teeth of the pinion with a circle
whose diameter equals half that of the wheel. Now, one
generating circle would have done for both, had the workman been contented to make the flanks of his teeth of the
hypocycloidal forms corresponding to it. But there is yet a
greater practical inconvenience in this method. A wheel
and pinion thus constructed will only work with one another;
neither will work truly any third wheel or pinion of a different number of teeth, although it have the same pitch. Thus
the wheels A and B having each a given number of teeth,
and being made to work with one another, will neither of
them work truly with C of a different number of teeth of
the same pitch. For that A may work truly with C, the
face of its teeth must be struck with a generating circle,
whose diameter is half that of C: but they are struck with
a circle whose diameter is half that of B; the condition of
uniform action is not therefore satisfied. Now let us suppose that the epicycloidal faces, and the hypocycloidal flanks
of all the teeth A, B, and C had been struck with the same
generating circle, and that all three had been of the same
pitch, it is clear that any one of them would then have
worked truly with any other, and that this would have been
equally true of any number of teeth of the same pitch.
Thus, then, the machinist may, by the use of the same generating circle, for all his pattern wheels of the same pitch, so
construct them, as that any one wheel of that pitch shall




250           TO DESCRIBE EPICYCLOIDAL TEETH.
work with any other. This offers, under many circumstances,
great advantages, especially in the very great reduction of
the number of patterns which he will be required to keep.
There are, moreover, many cases in which some arrangement similar to this is indispensable to the true working of
the wheels, as when one wheel is required (which is often
the case) to work with two or three others, of different numbers of teeth, A for instance to turn B and C; by the ordinary method of construction this combination would be
impracticable, so that the wheels should work truly. Any
generating circle common to a whole set of the same pitch,
satisfying the above condition, it may be asked whether
there is any other consideration determining the best dimensions of this circle. There is such a consideration arising
out of a limitation of the dimensions of the generating circle
of the hypocycloidal portion of the tooth to a diameter not
greater than half that of its base. As long as it remains
within these limits, the hypocycloidal generated by it is of
that concave form by which the flank of the tooth is made
to spread itself, and the base of the tooth to widen; when
it exceeds these limits, the flank of the tooth takes the convex form, the base of the tooth is thus contracted, and its
strength diminished.    Since then, the generating circle
should not have a diameter greater than half that of any of
the wheels of the set for which it is used, it will manifestly
be the greatest which will satisfy this condition when its
diameter is equal to half that of the least wheel of the set.
Now no pinion should have less than twelve or fourteen
teeth. Half the diameter of a wheel of the proposed pitch,
which has twelve or fourteen teeth, is then the true diameter or the generating circle of the set. The above suggestions are due to Professor Willis.*
* Professor Willis has suggested a new and very ingenious method of
striking the teeth of wheels hy means of circular arcs. A detailed description
of this method has been given by him in the Transactions of the Institution
of Civil Engineers, vol. ii., accompanied by tables, &c., which render its practical application exceedingly simple and easy.




TO DESCRIBE INVOLUTE TEETH.            251
211. To DESCRIBE INVOLUTE TEETH.
Let AD and AG represent the pitch circles of'   two wheels intended to work together. Draw a
(    straight line FE through the point of contact A. i'/'  of the pitch circles and inclined to the line of
centres CAB of these wheels at a certain angle?".' FAG, the influence of the dimensions of which
on the action of the teeth will hereafter be explained, but which appears usually to be taken
not less than 80~.*   Describe two circles eEK
and fFL from    the centres B and C, each touching the
straight line EF. These circles are to be taken as the bases
from which the involute faces of the teeth are to be struck.
It is evident (by the similar triangles ACF and AEB) that
their radii CF and BE will be to one another as the radii
CA and BA of the pitch circles, so that the condition necessary (Art. 201.) to the correct action of the teeth of the
wheels will be satisfied, provided their faces be involutes to
E
these two circles. Let AG and AH      in the above figure
represent arcs of the pitch circles of the wheels on an
enlarged scale, and eE, fL, corresponding portions of the
circles eEK and fEL of the preceding figure. Also let Aa
represent the pitch of one of the teeth of either wheel.
Through the points A and c describe involutes ef and mn.t
* See Camus on the Teeth of Wheels, by Hawkins, p. 168.
I Mr. Hawkins recommends the following as a convenient method of striking
involute teeth, in his edition of ii Camus on the Teeth of Wheels," p. 166. Take
a thin board, or a plate of metal, and reduce its edge MN so as aIccurately to




252             TO DESCRIBE INVOLUTE TEETH.
Let b be the point where the line EF intersects the involute
inn; then if the teeth on the two wheels are to be nearly of
the same thickness at their bases, bisect the line Ab in c; or
if they are to be of different thicknesses, divide the line Ab
in c in the same proportion*, and strike through the point c
an involute curve Ay, similar to ef, but inclined in the opposite direction.  If the extremityfg of the tooth be then cut
off so that it may just clear the circumference of the circle
fL, the true form     of the pattern involute tooth will be
obtained.'
There are two remarkable properties of involute teeth, by
the combination of which they are distinguished from teeth
of all other forms, and coeteris paribus rendered greatly preferable to all others.   The first of these is, that any two
wheels having teeth of the involute form, and of the same
pitch,t will work correctly together, since the forms of the
teeth on any one such wheel are entirely independent of
those on the wheel which is destined to work with it (Art.
201.)  Any two wheels with involute teeth so made to work
together will revolve precisely as they would by the actual
contact of two circles, whose radii may be found by dividing the line joining their centres in the proportion of the
radii of the generating circles of the involutes.    This property involute teeth possess, however, in common with the
epicycloidal teeth of different wheels, all of which are struck
with the same generating circle (Art. 210.)     The second no
less important property of involute teeth-a property which
distinguishes them   from  teeth of all other forms-is this,
that they work equally well, howeverfaor the centres of the
coincide with the circular arc
Is  ee M/      -//,,w' eE, and let a piece of thin
^^^^^^^,         X7      watch-spring OR, having two
projecting points upon it as
shown at P, and which is of a
width equal to the thickness of the plate, be fixed upon its edge by means of
a screw O. Let the edge of the plate be then made to coincide with the arc
eE in such a position that, when the spring is stretched, the point P in it may
coincide with the point from which the tooth is to be struck; and the spring
being kept continually stretched, and wound or unwound from the circle, the
involute arc is thus to be described by the point P upon the face of the board
from which the pattern is to be cut.
This rule is given by Mr. Hawkins (p. 170.); it can only be an approximation, but may be sufficiently near to the truth for practical purposes. It is to
be observed that the teeth may have their bases in any other circles than
those, fL and eE, from which the involutes are struck.
t The teeth being also of equal thicknesses at their bases, the method of
ensuring which condition has been explained above.




THE TEETH OF A RACK AND PINION.       253
wheels are removed asunder from one another; so that the
action of the teeth of two wheels is not impaired when
their axes are displaced by that wearing of their brasses or
collars, which soon results from a continned and a considerable strain. The
(   1 X  existence of this property will readily be
admitted, if we conceive AG and BH to
represent the generating circles or bases
of the teeth, and these to be placed with
/  -*     their centres C1 and C2 any distance
-a,/'     asunder, a band AB (p. 235., note) passing
I  2 j j    round both, and a point T in this band
\ \../    generating a curve mn, m' n' on the plane
of each of the circles as they are made to
revolve under it. It has been shown that
these curves mn and m' n' will represent the faces of two
teeth which will work truly with one another; moreover,
that these curves are respectively involutes of the two
circles AG and BH, and are therefore wholly independent
in respect to their forms of the distances of the centres of
the circles from one another, depending only on the dimensions of the circles. Since then the circles would drive at
any distance correctly by means of the band; since, moreover, at every such distance they would be driven by the
curves mn and m'n' precisely as by the band; and since
these curves would in every such position be the same
curves, viz. involutes of the two circles, it follows that the
same involute curves mn, and m'n' would drive the circles
correctly at whatever distances their centres were placed;
and, therefore, that involute teeth would drive these wheels
correctly at whatever distances the axes of those wheels
were placed.
THE TEETH OF A RACK AND PINION.
212. To determine the pitch circle of the pinion. Let H
represent the distance through which the rack is to be
moved by each tooth of the pinion, and let these teeth be
N in number; then will the rack be moved through the
space N. H during one complete revolution of the wheel.
Now the rack and pinion are to be driven by the action of
their teeth, as they would by the contact of the circum



254         THE TEETH OF A RACK AND PINION.
ference of the pitch circle of the
__  I  ~   pinion with the plane face of the
rack, so that the space moved through
by the rack   during one complete
revolution of the pinion must pre_BB   ^scisely equal the circumference of the
pitch circle of the pinion. If, therefore we call R   the radius of the
pitch circle of the pinion, then
B __        2&RR=N. H;.'. R-N. IH.
213. To describe the teeth of the
pinion, those  of the rack   being
straight. The properties which have
been shown to belong to involute
teeth (Art. 201.) manifestly obtain,
however great may be the dimensions of the pitch circle
of their wheels, or whatever disproportion
may exist between them. Of two wheels
~OF and OE with involute teeth which
\,>; 1     work together, let then the radius of the
pitch circle of one OF become ininite, its.-;f'i     circumference will then become a straight
/.     line represented by the face of a rack.
",'.   Whilst the radius C)O of the pitch circle
\....  OF thus becomes infinite, that C2B of the
circle from     which its involute teeth are'    struck (bearing a constant ratio to the first)
will also become infinite, so that the involute m'' will become a straight line* perpendicular to the
line AB given in position. The involute teeth on the
wheel OF will thus become straight teeth (see fi. 1.), having their faces perpendicular to the line AB determined by
drawing through the point O a tangent to the circle AC,
from which the involute teeth of the pinion are struck. If
the circle AC from which the involute teeth of the pinion
are struck coincide with its pitch circle, the line AB becomes
* For it is evident that the extremity of a line of infinite length unwinding
itself from the circumference of a circle of infinite diameter will describe,
through a finite space, a straight line perpendicular to the circumference of
the circle. The idea of giving an oblique position to the straight faces of the
teeth of a rack appears first to have occurred to Professor Willis.




THE TEETH OF A RACK AND PINION.         255
parallel to the face of the rack, and the edges of the teeth
of the rack perpendicular to its face (fig. 2.).
Now, the involute teeth of the one wheel have remained
unaltered, and the truth of their action with teeth of the'
other wheel has not been influenced by that change in the
dimensions of the pitch circle of the last, which has converted it into a rack, and its curved into straight teeth.
Thus, then, it follows, that straight teeth upon a rack, work
truly with involute teeth upon a pinion. Indeed it is evi(1.)                    (2.)
\dent, that if from the point of co
dent, that if from the point of contact P (fig. oo of such an
involute tooth of the pinion with the straight tooth of a
rack we draw a straight line PQ parallel to the face ab of
the rack, that straight line will be perpendicular to the
surfaces of both the teeth at their point of contact P, and
that being perpendicular to the face of the involute tooth,
it will also touch the circle of which this tooth is the involute in the point A, at which the face ab of the rack would
touch that circle if they revolved by mutual contact. Thus,
then, the condition shown in Art. 199. to be necessary and
sufficient to the correct action of the teeth, namely, that a
line drawn fronm their point of contact, at any time, to the
point of contact of their pitch circles, is satisfed in respect
to these teeth.  Divide, then, the circumference of the
pitch circle, determined as above (Art. 212.), into NT equal




256         THE TEETH OF A RACK AND PINION.
parts, and describe (Art. 211.) a pattern involute tooth from
the circumference of the pitch circle, limiting the length of
the face of the tooth to a little more than the length 3P of
the involute curve generated by unwinding a length AP of
the flexible line equal to the distance H through which the
rack is to be moved by each tooth of the pinion. The
straight teeth of the rack are to be cut of the same length,
and the circumference of the pitch circle and the face ad of
the rack placed apart from one another by a little more
than this length.
It is an objection to this last application of the involute
form of tooth for a pinion working with a rack, that the
point P of the straight tooth of the rack upon which it acts
is always the same, being determined by its intersection with
a line AP touching the pitch circle, and parallel to the face
of the rack. The objection does not apply to the preceding,
the case (fig. 1.) in which the straight faces of each tooth of
the rack are inclined to one another. By the continual
action upon a single point of the tooth of the rack, it is
liable to an excessive wearing away of its surface.
214. To describe the teeth of the pinion, the teeth of the rack
being curved.
This may be done by giving to the face of the tooth of
the rack a cycloidal form, and making the face of the tooth
of the pinion an epicycloid, as will be apparent if we con-:fa   ceive the diameter of the circle whose!., Ccentre is C (see fig. p. 236.) to become.p I infinnite, the other two circles remain-'(  ~B C;~gI  ing unaltered. Any finite portion of
the circumferen'ce of this infinite circle
S "~' will then become a straight line. Let
-']: ~ AE in the accompanying figure repre 




THE TEETH OF A WHEEL WITH A LANTERN.     257
sent such a portion, and let PQ and PR represent, as
before, curves generated by a point P in the circle whose
centre is D, when all three circles revolve by their mutual
contact at A. Then are PR and PQ the true forms of the
teeth which would drive the circles as they are driven by
their mutual contact at A (Art. 202). Moreover, the curve
PQ is the same (Art. 199.) as would be generated by the
point P in the circumference of APH; if that circle rolled
upon the circumference AQF, it is therefore an epicycloid;
and the curve PR is the same as would be generated by the
point P, if the circle APH rolled upon the circumference
or straight line AE, it is therefore a cycloid. Thus then it
appears, that after the teeth have passed the line of centres,
when the face of the tooth of the pinion is driving the flank
of the tooth of the rack, the former must have an epicycloidal, and the latter a cycloidal form. In like manner, by
transferring the circle APH to the opposite side of AE, it
may be shown, that before the teeth have passed the line of
centres when the flank of the tooth of the pinion is driving
the face of the tooth of the wheel, the former must have a
hypocycloidal, and the latter a cycloidal form, the cycloid
having its curvature in opposite directions on the flank and
the face of the tooth. The generating circle will be of the
most convenient dimensions for the description of the teeth
when its diameter equals the radius of the pitch circle of
the pinion. The hypocycloidal flank of the tooth of the
pinion will then pass into a straight flank. The radius of
the pitch circle of the pinion is determined as in Art. 212.,
and the method of describing its teeth is explained in
Art. 208,
215. THE TEETH OF A WHEEL WORKING WITH A LANTERN OR
TRUNDLE.
In some descriptions of mill work the ordinary form of
the toothed wheel is replaced by a contrivance called a lantern or trundle, formed by two circular discs, which are connected with one another by cylindrical columns called
staves, engaging, like the teeth of a pinion, with the teeth
of a wheel which the lantern is intended to drive. This
combination is shown in the following figure.
It is evident that the teeth on the wheel which works with
the lantern have their shape determined by the cylindrical
17




25S     THE TEETH OF A WHEEL WITH A LANTERN.
shape of the staves. Their forms may readily be found by
the method explained in Art. 200.
Having determined upon the dimensions of the staves in
reference to the strain they are to be subjected to, and upon
the diameters of the pitch circles of the lantern and wheel,
and also upon the pitch of the teeth; strike arcs AB and
AC of these circles, and set off upon them
/ the pitches Aa and Ab from the point of
contact A of the pitch circles (if the teeth
^    are first to come into contact in the line'1'":::   j7    of centres, if not, set them off from the
2"'... points behind the line of centres where
the teeth are first to come into contact).
Describe a circle ae, having its centre in
AB, passing through a, and having its
diameter equal to that of the stave, and divide each of the
pitches Aa and Ab into the same number of equal parts
(say three).  From  the points of division A, o,, P in the
pitch Act, measure the shortest distances to the circle ae, and
with these shortest distances, respectively, describe from the
points of division y, 6 of the pitch Ab, circular arcs intersecting one another; a curve ab touching all these circular
arcs will give the true face of the tooth (Art. 200.). The
opposite face of the tooth must be struck from similar centres, and the base of the tooth must be cut so far within the
pitch circle as to admit one half of the stave ae when that
stave passes the line of centres.




PRESSURES UPON WHEELS.           259
216. THE RELATION BETWEEN TWO PRESSURES P1 AND P2
APPLIED TO TWO TOOTHED WHEELS IN THE STATE BORDERING UPON MOTION BY THE PREPONDERANCE OF P1.
Let the influence of -the weights of the wheels be in the
first place neglected. Let B and C represent the centres of
the pitch circles of the wheels, A their point of contact, P
the point of contact of the driving and driven teeth at any
period of the motion, RP the direction of the whole
resultant pressure upon the teeth at their point of contact,
which resultant pressure is equal and opposite to the resistance R of the follower to the driver, BM and CN perpendiculars from the centres of the axes of the wheels upon RP;
and BD and CE upon the directions of P1 and P2.
BD-,, CE=a,, BM=m,, CN=m,.
BA=rl, CA=.r2
p1, p2=radii of axes of wheels.
p1, p2=limiting angles of resistance between the axes of
the wheels and their bearings.
Then, since P, and R applied to the wheel whose centre is
1w1'
i                   \/
1         ~ f~~~~\',
i:         /
XI'




260      RELATION OF THE DRIVING AND WORKING
B are in the state bordering upon motion by the preponderance of P1, and since a, and m, are the perpendiculars on
the directions of these pressures respectively, we have (equation 158)
Pi=     + (- l) sin.       m lR=   + (Lsin)  CR (... (236),
where L1 represents the length of the line DM joining the
feet of the perpendiculars BM and BD.
Again, since R and P., applied to the wheel whose centre
is C, are in the state bordering upon motion by the yielding
of P, (Art. 163.),
P  M2_ (P2L22 I1           Ip2L
*2.=     -   a.2    =   ^    -( a2)   R.(237)
where L2 represents the distance NE between the feet of the
perpendiculars CE and CN. Eliminating R between these
equations, we have
L+(!P "b 1sin. p,
P. ()       -             flu.P..2.. (238).
M2 a' P-(L )sin. 9p
Now let it be observed, that the line AP, drawn from the
point of contact A of the pitch circles to the point of contact
P of the teeth is perpendicular to their surfaces at that point
P, whatever may be the forms of the teeth, provided that
they act truly with one another (Art. 199.); moreover, that
when the point of contact P has passed the line of centres,
as shown in the figure, that point is in the act of moving on
the driven surface Ppfrom the centre C, or from P towards
p, so that the friction of that surface is exerted in the opposite
direction, or from p towards P; whence it follows that the
resultant of this friction, and the perpendicular resistance aP
of the driven tooth upon the driver, has its direction rP
within the angle aPp and that it is inclined (Art. 141.) to the
perpendicular aP at an angle aPr equal to the limiting angle
of resistance. Now this resistance is evidently equal and
opposite to the resultant pressure upon the surfaces of the
teeth in the state bordering upon motion; whence it follows
that the angle RPA is equal to the limiting angle of resistance between the surfaces of contact of the teeth. Let this
angle be represented by p, and let AP=X. Also let the




PRESSURES UPON WHEELS.            261
inclination PAC of AP to the line of centres BC be represented by 0.. Through A draw An perpendicular to RP, and
sAt parallel to it. Then,
m,=BM=Bt+tM=Bt+An=BA sin. BAt+AP sin. APR.
Also BAt=BOR=PAC+APR= + p;
* n1=r sin. ( -+p)+ X sin. p..... (239);
- =CN=CSs-sN=Cs — An=CA sin. CAs —AP sin. APR.
But As is parallel to PR, therefore CAs=BOR =0+p;
-. m2=rsin. (0+p)-  sin. p.... (240.).
Substituting these values of m, and m2 in the preceding
equation,
r rsin. ( + )+X sin. + (pL)- sin..4-1 —( —)               1 -P —..( —-)I-_. —-  - -. (241).
~ ii   r, sin. (O +(p)-X sin. p~- (!'P2J2 )sin. (p,
217. In the preceding investigation the point of contact P,.
_.. Y%- \,12




262        RELATION OF THE DRIVING AND WORKING
of the teeth of the driving and driven wheels is supposed to
have passed the line of centres, or to be behind that line;
let us now suppose it not to have passed the line of centres,
or to be before that line.
It is evident that in this case the point of contact P is in
the act of moving upon the surface pPq of the driven tooth
towards the centre C, or from P towards q, as in the other
case it is from the centre, or from P towards p.  In this case,
therefore, the friction of the driven surface is exerted in the
direction qP; whence it follows, that in this state bordering
upon motion the direction of the resistance R of the driven
upon the driving tooth must lie on the other side of the
normal APQ, being inclined to it at an angle APN equal to
the limiting anglei of resistance.  Thus the inclination of R
to the normal APQ is in both cases the same, but its position
in respect to that line is in the one case the reverse of its
position in the other case.*
The same construction being made as before,
ml, =BM =Bt+ tM = Bt t- An = BA. sin. BAt+ AP. sin. APO.
Also BAt=BOR=BAP-APO=-               -p;t:. m1,=r sin. (8 —p)+   sin. p,
m,=ON=Cs- sN=Cs -A           =CA. sin. CAs-AP. sin. APO.
But As is parallel to PN,. CAs=iBOR=BAP-APO=- p;'. n2 =r, sin. (8-p )-X sin. p.
Substituting these values of ml and m2 in equation (238),
r, sin. (-p)+X sin. +P        ) sin. P
p~,i~pL,~)sinir,            P,.(...).
=    r,2 sin. (- p)-X sin.. pp-  2p-) sine 
This expression differs from the preceding (equation 241)
only in the substitution of ( — q) for (8 +  ) in the first terms
of the numerator and denominator.: Hence it follows, that when the point of contact is in the act of crossing
the line of centres, the direction of the resultant pressure R is passing from
one side to the other of the perpendicular APQ; and therefore that when the
point of contact is in the line of centres, the resultant pressure is perpendicular to that line, and the angle BOR a right angle; a condition which cannot
however be assumed to obtain approximately in respect to positions of any
point of contact exceedingly near to the line of centres.
t The angle 0 being here taken as before to represent the inclination BAP
of the line AP, joining the point of contact of the pitch circles with the point
of contact of the teeth, to the line of centres.




PRESSURES UPON WHEELS.           263
Dividing numerator and denominator of the fraction in
the second member of that equation by sin. ( + (), and
throwing out the factors r, and r,, we have
-    1+     r ( -  s.( )      p
=I)        Xs in.cp + (Pl) sin. 
r sin. (d + p)
Now it is evident, that if in this fractional expression 8 —q
be substituted for 8 +  the numerator will be increased and
the denominator diminished, so that the value of P, corresponding to any given value of P, will be iroreased. Whence
it follows, that the resistance to the motion of the wheels by
the friction of the common surfaces of contact of their teeth
and of the bearings of their axes is greater when the contact
of their teeth takes place before than when it takes place,
at an equal angular distance, behind the line of centres-a
principle confirmed by the experience of all practical mechanists.
218. To DETERMINE THE RELATION OF THE, STATE BORDERING
TUPON MOTION BETWEEN THE PRESSURE P1 APPLIED TO THE
DRIVING WHEEL AND'THE RESISTANCE P2 OPPOSED TO THE
MOTION OF THE DRIVEN WHEEL, THE WEIGHTS OF THE
WHEELS BEING TAKEN INTO THE ACCOUNT.
Now let the influence of the weights W, and W2 of the
two wheels be taken into the account. The pressures applied
to each wheel being now three in number instead of two, the
relations between P1 and R, and P2 and R are determined
by equation (163) instead of equation (158). Substituting
W1 and W. for Ps in the two cases, we obtain, instead of
equations (136) and (237), the following,.P   \, Ml + (Lp, siRn. +,  +, Plsin. g l
i- whi m- -ati2    asn.,2 R-p2 a 2p sin. 2p,
Pan whi     \ aW.  a       LA (p
in which equations M, and M, represent certain functions




264      RELATION OF THE DRIVING AND WORKING
~'!
determined (Art. 166.) by the inclinations of the pressures
P1 and P2 to be vertical.
Eliminating R between the above equations, neglecting
terms above the first dimensions in sin. p1 and sin. p2, and
multiplying by aa2,
Lp..
Pa   qn,2-   sin. p2  -P2a2 { n+LP sin. 1 =
MW,.    MW
dLetmn pi sin. 1 +  2y P2 sin i2 * * *..  * (244).
Substituting the values of ml and m, from equations (239)
and (240), and neglecting the products of sin. p, sin. (p and
sin. 92(p we obtain
Pla  r2 sin. ( + p). sin. p-  2 sin. p2 -
P  +X     ai.2
P2a2 {r sin. (6 + q)-+X sin. ~-+-sin.  p,




PRESSURES UPON WHEELS.              265
-_ La    sin.,+   -2p   sin. i, sin. ( + ).... (245.)
Now (Art. 166.)   M        [=n cos. 1,,+a1 cos. 3, where,, reprea,
sents the inclination WiFP1 of Pi to the vertical, and 3, the
inclination RrF of R to the vertical.*
Let the inclination W1BD of the perpendicular upon P1 to
the vertical be represented by a,, that angle being so measured that the pressure Pi may tend to increase it; let a, represent, in like manner, the inclination EGG of CE to the
vertical; and let [A represent the inclination ABr of the
line of centres to the vertical,.,      P=WiFP-=WBD-BDF=a,-,
I2=RrF=BOR-OBr=O + -I;
*     -l= m, sin. a  +a cos. (0+p-~-/).
M,
Similarly  -2im   cos. P2GIH+a, cos. RqW,.ft      Now
P,GH=ECG+GEC=a,+;  and RqW,=~ —RrF, and
RrF was before shown to be equal to (0 +p —;
*M- -m, sin. a, —~ cos. (8+a —o)
Substituting the values of mn and n,, from equations (239)
and (240),
M                                        1
a-M, sin. (0 +p) sin. a, +  sin. a, sin. q +
al cos. (O+  -13)      i.... (246).
a2= -_  sin. ( +P) sin. a2+  sin. a sin. -
a2 cos. (8+p-1)
* See note, p. 172.
f It is to be observed that the direction of the arrow in the figure represents that of the resistance opposed by the driven wheel to the motion of the
driving wheel, so that the direction of the pressure of the driving upon the
driven wheel is opposite to that of the arrow.




266       RELATION OF THE DRIVING AND WORKING
Let it be supposed that the distances DM and EN, represented by L, and L2, are of finite dimensions, the directions
of neither of the pressures P, and P. approaching to coincidence with the direction of R,-a supposition which has been
virtually made in deducing equation (163) from equation
(161), on the former of which equations, equations (243) depend. And let it be observed that the terms involving sin. p
in the above expressions (equations 246) will be of two dimensions in (p, p, and p, when substituted in equation (245),
and may therefore be neglected.  Moreover, that in all cases
the direction of RP is so nearly perpendicular to the line
of centres BC, that in those terms of equation (245), which
are multiplied by sin. p, and sin. 9p, the angle 0 +p, or BOR,
may be asssumed=; any error which that supposition involves, exceedingly small in itself, being rendered exceedingly less by that multiplication. Equations (246) will then
become
=r sin         sin. sin. 2a+a2 sin. 3,  -
a1                       a2
Substituting these values in the first factor of the second
member of equation (245), and representing that factor by
Nr,r2, we have
Nr2r2r= r22 (r, sin.   + a sin. 1) in.  --
-- rp2(r. sin. 2 -+  sn. /3) sin. q,;
L2
and dividing by rer,,
N* W=   Pi (sin. a,+ a' sin. /)sin. (1LI          ri
Wp(s in. a2 -a sin. 3) sin. q,.... (247).
L2           2
* If the direction of P1 be that of a tangent at the point of contact A of
the wheels, a case of frequent occurrence, the value of LI vanishing, that of N
would appear to become infinite in this expression. The difficulty will however
be removed, if we consider that when al becomes, as in this case, equal to rl,
and the point M is supposed to coincide with A, L1. becomes a chord of the pitch




PRESSURES UPON WHEELS.               267
Substituting lNr, for the factor, which it represents in
equation (245), we have
L22
Pia { sin-. (+@)-  sin. g --- sin. 2} — Pa,, {r sin. (0 +p)+
^a.
X sin. p + L-Pl sin. 91 =Nr1r, sin. ( +).....  (248).
a
Solving this equation in respect to P,
Xsin. q + -  sin. qp,
Palr,+ r sin. (O+p)
ar2         sin.+L2P2sin.
r, 2sin. (O+)  J
Nr,
P2 +            a.
Xsin. p + L-2P sin. p,
a1
r, sin. (0 +p)
Whence, performing actual division by the denominators of
the fractions in the second member of the equation, and
omitting terms of two dimensions in sin. p1,, sin. 9p,, sin. p
(observing that N is already of one dimension in those variables), we have
p    al +'      -H  + Bsin. 9 +  sin. q + L sin.p (P
a1r2    t\r' r             aIr        a2r,
circle, and is therefore represented by 2r1 sin. jDBA, or 2ri sin. (as+-fi)-; so
that       ri    _ sin. ali+sin. / _2sin. (al+-/) cos. (ai+-)_
L1      2ri sin. i (a1+ir-  2rl sin. (ai+fl) 
-cos. I (al+0).
If, therefore, we take the angle al —/f, so as to give to P1 the direction of
1        1
a tangent at A, this expression will assume the value, -cos. O, or-; so that
in this case
N,=w1ip1.,W2p (,.  +-2 n. s). a2
N=_=_  sin. 0i — 2 (sin. a+-sin. p) sin.:.
ri        L2    a.r2




268           THE MODULUS OF A SYSTEM
cosec. (0=) P2p+.... (249).
In this expression it is assumed that the contact of the teeth
is behind the line of centres.
219. THE MODULUS OF A SYSTEM OF TWO TOOTHED WHEELS.
Let n1 and n2 represent the numbers of teeth in the
driving and driven wheels respectively, and let it be observed that these number are one to another as the radii of
the pitch circles of the wheels; then, multiplying both sides
of equation (249) by a 2, we shall obtain
1>,'/'I p~( 13-~(1 rz  Lipi  LP
Pal    PA l 1 + X     ( + —,  sin. +a  sin. s, +l  sin. qoa
P =  rP\+        + r)      air,       a2r2
cosec. (0 +p) +Nr.
Now let A-4 represent an exceedingly small increment of
the angle +, through which the driven wheel is supposed to
have revolved, after the point of contact P has passed the
line of centres; and let it be observed that the first member
r Ar4         r
of the above equation is equal to Pal2- -, and that -A 
represents the angle described by the driving wheel (Art.
204.), whilst the driven wheel describes the angle A,;
whence it follows (Art. 50.) that Pla(, A)2 represents the
work aUT, done by the driving pressure P1, whilst this angle
a{ is described by the driven wheel,
A-P        L+                            LA
1 =P~a2. 1         +  +   sin. + a  sin. p%+ L  sin. *qa
cosec. (0 + ) t +Nr.
Let now 4a be conceived infinitely small, so that the first
member of the above equation may become the differential
co-efficient of U,, in respect to 4. Let the equation, then,
be integrated between the limits 0 and 4; P,, L,, and L2
and therefore N (equation 247) being conceived to remain




OF TWO TOOTHED WHEELS.            269
constant, whilst the angle 4 is described; we shall then
obtain the equation
==Pa fl~+ xL+4      sin.+. sin.+!.sin. p.'J<rri.r,   1     r2
cosec. ( + )   d+N..... (250),
where S is taken to represent the arc r,4t described by the
pitch circle of the driven wheel, and therefore by that of the
driving wheel also, whilst the former revolves through the
angle 4.
220. THE MODULUS OF A SYSTEM OF TWO TOOTHED WHEELS,
THE NUMBER OF TEETH ON THE DRIVEN WHEEL BEING CONSIDERABLE, AND THE WEIGHTS OF THE WHEELS BEING TAKEN
INTO ACCOUNT.
It is evident that the space traversed by the point of contact of two teeth on the face of either of them is, in this case,
small as compared with the radius of its pitch circle, and
that the direction of the resultant pressure R (see fig. p. 259.)
upon the teeth is very nearly perpendicular to the line of
centres BC, whatever may be the particular forms of the
teeth; provided only that they be of such forms as will
cause them to act truly with one another. In this case,
therefore, the angle BOR represented by + q is very nearly
equal to -, and cosec. ( + q)=1.
Since, moreover, RP is very nearly perpendicular to the
line of centres at A, and that the point of contact P of the
teeth deviates but little from that line, it is evident that the
line AP represented by x differs but little from an arc of
the pitch circle of the driven wheel, and that it differs the
less as the supposition made at the head of this article more
nearly obtains. Let us suppose,4 to represent the angle
subtended by this arc at the centre C of the pitch circle of
the driven wheel, then will the arc itself be represented by
r,., and therefore X=r,{ very nearly.  Substituting this
value of X in equation (250), observing that cosec. ( + -)=1,
and that -2- _  (equation 227), and integrating,
r1 n+
U,=1+       +j n(1,+ Sin.~ psin.  L2p2 sin. p,
ar             ar,~




270                INVOLUTE TEETH.
P2a2+ N+.+.... (251).
But the driven or working pressure P2 being supposed to
remain constant, whilst any two given teeth are in action,
Pa2; represents the work UI yielded by that pressure whilst
those teeth are in contact: also r,2 represents the space S,
described by the circumference of the pitch circle of either
wheel whilst this angle is described. Now let + be conceived to represent the angle subtended by the pitch of one
of the teeth of the driven wheel, these teeth being supposed
to act only behind the line of centres, then 4=-, n2 representing the number of teeth on the driven wheel, and 24
(1 + -  -1+               + );
\ nl  n2    nli    \n   n2.U. 1U=  1+q (-+-1 sin. p+-  -sin.+     sin. 
\l n2 all)'Ia
T2+ N.S..... (252),
which relation between the work done at the moving and
working points, whilst any two given teeth are in contact, is
evidently also the relation between the work similarly done,
whilst any given number of teeth are in contact. It is therefore the MODULUS of any system of two toothed wheels, the
numbers of whose teeth are considerable.
221. THE MODULUS OF A SYSTEM OF TWO WHEELS WITH INVOLUTE TEETH OF ANY NUMBERS AND DIMENSIONS.
The locus of the points of contact of the teeth has been
shown (Art. 201.) to be in this case,/?'" "           a straight line DE, which passes,1 gOI,'     through the point of contact A of.'.\ ^,....... the pitch circles, and touches the
^.      -.........'-.  circles (EF and DG) from which the
~///     /'    involutes are struck. Let P repre/i'  (  Y     v' sent any position of this point of,   v' \/; i contact, then is AP measured along
\\,', the given line DE the distance re"" /"      presented by x in Art. 216., and the
" —"    angle CAD, which is in this case
constant, is that represented by 0. Since, moreover, the
point of contact of the teeth moves precisely as a point P
upon a flexible cord DE, unwinding from the circle EF and
winding upon DG, would (see note, p. 238.), it is evident




INVOLUTE TEETH.                 271
that the distance AP, being that which such a point would
traverse whilst the pitch circle AH revolved through a certain angle 4, measured from the line of centres is precisely
equal to the length of string which would wind upon DG
whilst this angle is described by it; or to the arc of that
circle which subtends the angle 4. If, therefore, we represent the angle ACD by i, so that CD=CA cos. ACD-r,
cos. I, then X=r-4 cos. A. Substituting this value for X in
equation (249), and observing that 8 + p — =  +q  ---
('i-p), and that 2  2 we have
r,  n,
a2r(       In                 Lip        L p
P^^=''d-I +^ {4Al~-1+n) cos. X sin. -+' sin. p,+' sin. p2}
sec. (-    )P +P)P,-r..... (253);
a)  i
from  which equation we obtain by the same steps as in
Art. 219, observing that X is constant,
U =   1+    (-+ 4  )cos. sin..+ LIN sin. -+ (   sin. q +
1  ( \nnf n               ar,        a^ 2
sec. ( -p)TIJ,2+NS..... (254),
which is the modulus of a system of two wheels having any
given numbers of involute teeth.
222. THE INVOLUTE TOOTH OF LEAST RESISTANCE.
It is evident that the value of U, in equation (254), or of
the work which must be done,-'-7 —.             upon the driving wheel to cause',''\\,         a given amount U, to be yielded,i, /o.,,'by the driven wheel is dependent
for its amount upon the value of,...z. -..   the   co-efficient of U, in the
3'/' A   ","\\  second member of that equation;
\,' \ ^\^";  and that this co-efficient, again, is,q\ \k /  J    dependent for its value (other
\. /,/     things being the same) upon the' —'    value of i representing the angle
ACD, or its equal the angle DAI,




$272    THE INVOLUTE TOOTH OF LEAST RESISTANCE.
which the tangent DE to the circles from which the involutes are struck makes with a perpendicular AI to the line
of centres. Moreover, that the co-efficient N not involving
this factor X (equation 247), the variation of the value of
U,, so far as this angle is concerned, is wholly involved in
the corresponding variation of the co-efficient of U, and
becomes a minimum with it; so that the value of m which
gives to the function of X represented by this co-efficient, its
minimum value, is the value of it which satisfies the condition of the greatest economy ofpower, and determines that
inclination DAI of the tangent DE to the perpendicular to
the line of centres, and those values, therefore, of the radii
CD and BE of the circles whence the involutes are struck,
which correspond to the tooth of least resistance.
To determine the value of' which corresponds to a minimum value of this co-efficient, let the latter be represented
by u; then, for the required value of i,
du        d2u
a=0, and,>0.
~/1 ~1\L  ~p,    L2p,
Let qr j +  =A),     sin. p,+  sin. 2,=B;
11 2      ar, aLr2.' =1 + (A cos. i sin. + B) sec. (m-(p);..=  + B sec. (^ —)+A sin. p cos. m sec. (m — );
=B sec. (n- p) tan. ( —)-)-A sin. p9 sin., sec. ( -I)cos. n tan. (n-p) sec. (i-q)t;.*.- B sec. "(i- ) sin. (7-)A sin. p sec. 2(i-p) sin. 4 cos. (i-qp)-cos. X sin. (mi-)};
dd.: -=sec. 2(X-p)lB sin. (X- p)-A sin.'~}..... (255).
In order, therefore, that lnay vanish for any value of
a, one of the factors which compose the second member of
the above equation must vanish for that value- of i; but
this can never be the case in respect to the first factor, for
the least value of the square of the secant of an arc is the
square of the radius. If, therefore, the function u admit of




THE INVOLUTE TOOTH OF LEAST RESISTANCE.   273
a minimum value, the second factor of the above equation
vanishes when it attains that value; and the corresponding
value of X is determined by the equation,
B sin. ( —qp)-A sin. 2p=O..... (256).
AA                            2
or by sin. (       — B)=  sin. p or by =  + sin.  Bsin. 2);
or substituting the values of A and B,
t.==Im - (+-I-)sin.p2,,
1        2,72 this value of X gives to that second differential co-efficient of u in respect to X apositive value.
Differentiating equation (255), we have
-2 =2 sec    -p) t. (-. ){B sin.'(5 — )A    si n. p +B sec.  -( ) os. (-)
But the functiproposed valuedmits of (equation 256) has beenthis
value of n provided that which, being substituted in the factor 
sin. (m — ) —A sin. 2t, will cause it to vanish, and therefore,
with it, the whole of the first term of the value of fs' it
corresponds, this valuerefore, to af it gives to that seond differential co-ef
second term B sec. 2( —) cos. (I — I) a positive value; or,
since sec. ~( —a) is essentially positive, and B does not
involve  if it givespect to  cos. ( a  positi ve value, o if
Differentiating equation (255), we have
-p <    or if sec.'( ) tan. ( -p)B,  sin. 2(p<; o
~A         s in  +. 2  2<+B; or if)c
But the proposed value of i (equation 256) has beeni
shown to be that which, being substituted in the factor JB
sin. (I)-A  sin.  will cause it to vanish and therefore
iths condit the whole of the satisfirst terd, the value of, determined
corresponds, therefore, to a mini   if it gives to the
second term B sec. 2(  ) Cos. (I- p) a positive value; or,
since sec. 2()_ —) is essentially positive, and B doeS not
involve', if it gives to cos. (i-q) a positive value, or if
-<- or ifsin.      I- sin.   <-, or  A
A sin. 2qp<B; or if
iI    sin -   LA        LAP2
+ sn. <'sin. p1,  I sin. p,....(258).
This condition being satisfied, the value of i, determined
is




274    THE BEST DIVISION OF THE ANGLE OF CONTACT.
by equation (257), corresponds to a minimum, and determines the INVOLUTE TOOTH OF LEAST RESISTANCE.*
223. To DETERMINE IN WHAT PROPORTION THE ANGLE OF
CONTACT OF EACH TOOTH SHOULD BE DIVIDED BY THE LINE
OF CENTRES; OR THROUGH HOW MUCH OF ITS PITCH EACH
TOOTH SHOULD DRIVE BEFORE AND BEHIND THE LINE OF
CENTRES, THAT THE WORK EXPENDED UPON FRICTION MAY
BE THE LEAST POSSIBLE.
Let the proportion in which the angle of contact of each
tooth is divided by the line of centres be represented by x,
so that - may represent the angular distance from the line
of centres of a line drawn from the centre of the driven
wheel to the point of contact of the teeth when they first
come into action before the line of centres, and (1-x)the corresponding angular distance behind the line of centres
when they pass out of contact; and let it be observed that,
on this supposition, if UT represent as before the work
yielded by the driven wheel during the contact of any two
teeth, wUT will represent the portion of that work done
before, and (1 -x)U2 that done behind, the line of centres.
Then proceeding in respect to equation (253) by the same
method as was used in deducing from that equation the
modulus (Equation 254), but integrating first between the
limits 0 and x-, in order to determine the work ul done by
the driving pressure before the point of contact passes the
line of centres, and then between the limits 0 and (1 —x)
to determine the work u, done after the point of contact has
passed the line of centres; observing moreover, that in the
former case -p is to be substituted in sec. (n —p) for p (Art.
217.), we have
* It may easily be shown by eliminating q between equations (254) and
(256) that the modulus corresponding to this condition of the greatest economy,of power, where involute teeth are used, is represented by the formula
Ul= i 1+U sin. 20+(B'-A' sin. 40) lU+NS.




THE BEST DIVISION OF THE ANGLE OF CONTACT.  275
1= {1+       (x  + -) cos. X sin. p +L —sin., 1+
p sin. 9} sec. (+p) xTU2 +N 1;
a r2
Or assuming
(lix\                    L~p1       L2p12
( +_ -os. m sin. q =a, and   sin., + -  sin.,,=b
1a 1        a r.
1= l + (ax+b) sec. ( +(p) IxUT2+Ns1.', representing the space described by the pitch circle of
either wheel before the line of centres is passed; similarly,
u2= { 1+ a(1-x)+ b} sec. (m-p) } (l-)U2 + Ns2.
Adding these equations together, and representing by U, the
whole work us +fu2 done by the driving pressure during the
contact of the teeth, and by S the whole space described by
the circumference of either pitch circle, we have
U,= { 1+(a + bx)sec.(+ ) 4
la(-x )2+b(1-x)} sec.( -p) } U+NS... (259)
by which equation is determined the modulus of two wheels
driven by involute teeth, when the contact takes place partly
before and partly behind the line of centres.
Let the portion of the work U1, which is expended upon
the friction of the teeth be represented by u. Then
u=   (ax +  x) sec. ( + p) +
a(l —x)'+(1 — x  sec. (n —) } UT+NS.
Now the value of x, which gives to this function its minimum, and which therefore determines that division of the
driving arc which corresponds to the greatest economy of
power, is evidently the value which satisfies the condition
dx=                    2>0
But differentiating and reducing
But differentiating and reducing




276    TIHE BEST DIVISION OF THE ANGLE OF CONTACT.
-dd= 2a2asec. (1+p)+sec.(-~) )t+
— 2= 2{sec. ( + )+sec. (~-()} uT'
Whence it appears that the second condition is always satisfied, and that the first condition is satisfied by that value of
x, which is determined by the equation
2a{xsec. (4 +p)+sec. (- )}t +b{sec. ( +@)-sec. (m-p)} -
2a sec. (X-0 )=0;
Whence we obtain by transposition and reduction
x==   1 —(1 +)tan. h   tan. t.
So that the condition of the greatest economy of power is
satisfied in respect to involute teeth, when the teeth first
come into contact before the line of centres at a point whose
angular distance from it is less than one half the angle subtended by the pitch by that fractional part of the last-mentioned angle, which is represented by the formula i ( +- )
tan. ir tan. q, or substituting for b and a their values by the
formula
l   sin. 0 +  sin.  1
f        1 ar,  aOp,'1 1 +.. -/- 2 —\tatan. tan, p.. (260)
I   ( — + —co&s. sin~. 4
That division of the angle of contact of any two teeth by
the line of centres, which is consistent with the greatest
economy of power, is always, therefore, an unequal division,
the less portion being that which lies before the line of centres; and its fractional defect from one half the angle of contact, as also the fractional excess of the greater portion above
one half that angle, is in every case represented by the above
formula, and is therefore dependent upon the dimensions of
the wheels, the forms and numbers of the teeth, and the circumstances under which the driving and working pressures
are applied to them.*
* The division of the are of contact which corresponds to the greatest economy of power in epicycloidal teeth, may be determined by precisely the same
steps.




THE MODULUS OF A SYSTEM OF TWO WHEELS.    277
224. THE MODULUS OF A SYSTEM OF TWO WHEELS DRIVEN BY
EPICYCLOIDAL TEETH.
The locus of the point of contact P of any two such teeth
is evidently the generating circle APH of
the epicycloidal face of one of the teeth, and
(  B  8    the hypocycloidal flank of the other (Art.
\ AQ     202.); for it has been shown (Art. 199.),
that if the pitch circles of the wheel and the
generating circle APH of the teeth be conceived to revolve about fixed centres B, C,
D by their mutual contact at A, then will a
point P in the circumference of the last-mentioned circle move at the same time upon
the surfaces of both the teeth which are in
contact, and therefore always coincide with their point of
contact, so that the distance AP of the point of contact P of
the teeth from A, which distance is represented in equation
(250) by x, is in this case the chord of the arc AP, which
the generating circle, if it revolved by its contact with
the pitch circles, would have described, whilst each of the
pitch circles revolved through a certain angle measured
from the line of centres.  Let the angle which the driven
wheel (whose centre is C) describes between the period
when the point of contact P of the teeth passes the line of
centres, and that when it reaches the position shown in the
figure be represented as before by -, the arc of the pitch
circle of that wheel which passes over the point A during that
period will then be represented by r,2. Now the generating
circle APH having revolved in contact with this pitch circle,
an equal arc of that circle will have passed over the point A;
whence it follows that the arc AP=r-; and that if the radius
of the generating circle be represented by r, then the angle
ADP subtended by the arc AP is represented by, or
r
by 24, if 2e be taken to represent the ratio - of the radius
r
of the pitch circle of the driven wheel to the radius of the
generating circle.  Now the chord AP=2AD sin. i ADP;
r
therefore X=2r sin. e=-2 sin. e4.  Substituting this value
of X in equation (250); observing, moreover, that the angle




278     THE MODULUS OF A SYSTEM OF TWO WHEELS
PAD represented by 0 in that equation is equal to- - 
ADP, or to 2-e+, and that the whole angle +t through
which the driven wheel is made to revolve by the contact of
each of its teeth is represented by-, we have
n2rL
~22
U,=Pa    f{ l++      +     sin.p sin. e~.+-  sin. qp+
L2p2 sin.p2}sec. (4-9) } d +NS N
or, assuming Li and L2 to remain constant during the contact of any two teeth representing the constant 1 + Lsin.q +
a r1
2Psin. q) by A, and observing thatr2 _r 2,
a^2       v                       ri ni
2ir                           27r
n2                            w2
U-=Pa2 4 A    sec. (e4-()d~ /+  (1 +n2) sin. / sin. e~ sec.
0                              0
(e4- 9)d     N +NS.
Now the general integral/sec. (e+ —p)d, or
-   sec. (e4 —)d(e —1) being represented* by the function
- fsec. (e —-b)d(e — >)=e  cos d(e-~I —) - J  -(e —- 
ea                 ej cos.(e*-)  e   cos.2(e*-0)
1 Fcos. (ep —f)d(e'tP-f)  1 _os.(e_-_)_d(e_ -)_ _
eQJ  1-s —.     - g =-2e. J- +sin. (ep —0)
~cos.i-   ()-o    lo+g.. sin.      —.-l o
21J -(11i-sin. ~(e —0b) -2 ~g'e - -sin. (e -  2e -g
*i 1-sin (eW-~) r    log.I - -1+sin-. ( —-? t 2_




HAVING EPICYCLOIDAL TEETH.         279
log. tan.  4+,(+-~), its definite integral between the
limits 0 and - has for its expression,
tan.   +   e- e-   c
log.            2 
_e   Eo       tan. ({)
2irn
27r                 _I
Alsof sec.^(e~-) sin.;4d=f sec.(e —) sin. I(e4 —) +)  1 dC
o                   o
2r
=/sec. (e + —q~) { sin. (e4 —q) cos. qp+cos. (e — p) sin.(p d
0
27r
=J   o. p tan. (      -p)+smin. P}l
0
27r
2                          nw
ecos. <tan. (e4-p) d (e —)~ -- sin. (.
0
Now the general integralj tan. (e —p) d (e —p) has for
its expression-log.: cos. (ed —qp).* Taking its definite integral between the limits 0 and 2-  we have, therefore,
"sec.                 1        COS., I-ere fo
2s.                               I 2eC
sec. (e —p ) sin. e4dJ  — cos.Plog.              p.
1    4             4
Ion.   n. I           0s.
e 6    i 2 sin- T-ateW-0) tcose  { 7r+(eV-p) t 4
* Jtan. (e4'-9) d. (e4+ _~)=J'sifl (eV-4) d (e2-P-)
r-dce os. (e-)(e2 sin. l(eI _   1 —0) I-  cos. (e{-r ) 
llog. tan. +i-(-0) +.
ftan. ( —. (,-='e-,
1-dcos. (e, —) --
C —Oos. (            ep —)).




280     THE MODULUS OF A SYSTEM OF TWO WHEELS
Substituting these expressions in the modulus, representing
-- by p', and observing that if U, represent the work
42
yielded by the driven wheel during the action of each tooth,
then P2a2.2=U,, so that p 22=2U, we have
n,                    21
ncos. A   tan.p  e.                  cos.p
Now      A      log.     j 1s+tan.   an1 +     log.
/.2e  r   \. r.,COS. — 7-,  2.     ( U, +   NS...(1).
cos. p            ti
Cs' 2e                    2e           2en
Now log. no    2      log.   1 +tan. —-tan.qp cos.-'cos.                     h 
2e,                2et                  2 
log. cos. n  +log.   1 + tan. — tan.p - =log. cos.    +
tan. 2e tan. p — tan.'2. tan.29 + &c.  Substituting this
n.        2      n
expression in the preceding equation, and neglecting terms
above the first dimension in tan. p and sin. p,
tan. (  q'
U, ===- {Alog.     taI'2       1n. 2' log.
Y 2e-,; i  0  tan. <p     2e   ~   /
cos.-U2+NS...(262).
225. If the radius r of the generating circle be equal to
one half the radius 2 of the pitch circle of the driven wheel,
according to the method generally adopted by mechanics
(Art. 203.), then e=- =2 = =1.
r     r
In this case, therefore-that is, where the flanks of the
driven wheel are straight (Art. 210.)-the modulus becomes
tan.    - + ( -'
U= =   {AlQog. — an     +  _-  1 + - sin. 2p log. cos.tan.,        n(263)
u, +  s..... (263).




HAVING EPICYCLOIDAL TEETH.           281
226. Substituting (in equation 262.) for q' its value -,
log.  tan p    = log.          2 
e tan. 9              I tan.(  )
2+rt ("   L   1-tan. 
I+tan. -_            1 -tan.
n,   2              2         el,
2 tan. +- -tan. ( —   ) +  tan.S- +&c.
If, therefore, we assume the teeth in the driven wheel to
be so numerous, or n2 to be so great a number, that the third
power and all higher powers of tan. (     -  may be neglected as compared with its first power, and if we neglect
powers of tan. 2 above the second,
2
log. ( t'  = 2  tan.  -    — ) + tan. };
log.,   tan.      -          2-  
2'n'
which expression becomes- if we suppose the two arcs
which enter into it to be so small as to equal their respective tangents.
2er,    i[2e,~
Again, log. 6 cos. -  --  -2 —) very nearly.*
* For assume log,e cos. x=alx2+ a2x4+~as36+-....; then differentiating,
-tan. x=2alx+4a2ax3+6asx3+...;
2
but (Miller, Diff. Cal. p. 95.)-tan. x= —x —~ ——....; equating,
therefore, the co-efficients of these identical series, we have
1           1             2
al-,      a2   3     - 3   3. 5. 6'    -
X2  x'   2x6.'.log.e cos. X- - -2.
2  3.45 3. —.6




282     THE MODULUS OF THE RACK AND PINION.
Substituting these values in equation (262), and performing actual multiplication by the factor —, we have
1J=I   A+  (n  + -)sin. 2(p } U2+S;
or substituting for A its value; and assuming i sin. 2p=
sin. p, since p is exceedingly small,
Ui= {(1+ L, sin.,+ p sin..) +
(l+n    s )sin. T 2,+NS....   (264)2
which is the modulus of a wheel and pinion having epicycloidal teeth, the number of teeth n2 in the driven wheel
being considerable (see equation 252).
It is evident that the value of U, in the modulus (equation 261), admits of a minimum in respect to the value of g;
there is, therefore, a given relation of the radius of the
generating circle of the driving, to that of the driven wheel,
which relation being observed in striking the epicycloidal
faces and the hypocycloidal flanks of the teeth of two wheels
destined to work with one another, those wheels will work
with a greater economy of power than they would under any
other epicycloidal forms of their teeth. This value of e may
be determined by assuming the differential co-efficient of the
co-efficient of U. in equation (261) equal to zero, and solving
the resulting transcendental equation by the method of
approximation.
227. THE MODULUS OF THE RACK AND PINION.
If the radius r2 of the pitch circle of the driven wheel be
supposed infinite (Art. 213.), that wheel becomes a rack, and
the radius r1 of the driving wheel remaining of finite dimensions, the two constitute a rack and pinion. To determine
the modulus of the rack and pinion in the case of teeth of
any form, the number upon the pinion being great, or in
the case of involute teeth and epicycloidal teeth of any
number and dimensions, we have only to give to r2 an
infinite value in the moduli already determined in respect




THE MODULUS OF THE RACK AND PINION.     283
to these several conditions. But it is to be observed in
respect to epicycloidal teeth, that n2 becomes infinite with
r,, whilst the ratio-2 remains finite, and retains its equality
to the ratio -L (equation 227), so that — i   -i =  1;
ni                     n2     r     1nr ni
if we represent the ratio - by 2e,. Making n2 and r2 infinite
in each of the equations (252), (254), and (261), and substituting -l for - in equation (262); we have
1. For the modulus of the rack and pinion when the teeth
are very small, whatever may be their forms, provided that
they work truly.
Ul= -   1 +  r Lsin. <p,+-sin. (-} U,+NS.....(265).
2. For the modulus of a rack and pinion, with involute
teeth of any dimensions (seefig. 1. p. 255),
U,   { 1+ (- cos. n   sin.    0 +  s ec. ( —  ) } U,+
NS.. (266).
3. For the modulus of the rack and pinion, with cycloidal
and epicycloidal teeth respectively (equation 261),
u 2e, +L   si. p log.6 
— 2ei~    ai                tan.
sin. 29* log.  c  x   U2+NS.
2e1   ~    cos. p
In each of which cases the value of N is determined by
making r2 infinite in equation (247).
e       e  n2
n2  nl'  el  nil.. (equation 262)+ 1( + ^)- = (  )+W =(~ +- =-, because
e is infinite. The friction of the rack upon its guides is not taken into account
in the above equations.




284                   CONICAL WHEELS.
CONICAL OR BEVIL WHEELS.
228. These wheels are used to communicate a motion of
rotation to any given axis from another, inclined to the first
at any angle.
Let AF be an axis to which a motion of rotation is to be
communicated from      another axis AE
inclined to the first at any angle EAF,
a'<   by means of bevil wheels.
/    Divide the angle EAF by the straight
line AD, so that DO and DN, perpen-.<'-' ——, —-.   diculars from  any point D in AD  upon
AE   and AF    respectively, may be to
one another as the numbers of teeth
which it is required to place upon the
two wheels.*
Suppose a cone to be generated by the revolution of the
line AD   about AE, and another by the revolution of the
line AD about AF. Then if these cones were made to
revolve in contact about the fixed axes AE and AF, their
surfaces would roll upon one another along their whole line
of contact DA, so that no part of the surface of one would
slide upon that of the other, and thus the whole surface
of the one cone, which passes in a given time over the line
of contact AD, be equal to the whole surface of the other,
which passes over that line in the same time.       For it is
evident that if nI times the circumference of the circle DP
be equal to n. times that of the circle DI and these circles
be conceived to revolve in contact carrying the cones with
them, whilst the cone DAP makes nl revolutions, the cone
* This division of the angle EAF may be made as follows:-Draw ST and
UW from any points S and U in the straight lines AE and AF at right angles
to those lines respectively, and having their
lengths in the ratio of the numbers of teeth
which it is required to place upon the two wheels;
U           and through the extremities T and W of these
\  /  /^   lines draw TD and WD parallel to AE and AF
o -                 respectively, and meeting in D. A straight line
drawn from A through D will then make the
required division of the angle; for if DO and
----— IT \     DN be drawn perpendicular to AE and AF, they
will evidently be equal to UW and ST, and therefore in the required proportion of the numbers
of the teeth; moreover, any other two lines
drawn perpendicular to AE and AF from any
other point in AD will manifestly be in the same proportion as DO and DN.




'CONICAL WHEELS.               285
DAI will make n, revolutions; so that whilst any other
circle GH of the one cone makes n, revolutions, the corresponding circle iK of the other cone will make n2 revolutions: but n, times the circumference of the circle GH
is equal to n2 times that of the circle HIK, for the diameters
of these circles, and therefore their circumferences, are to
one another (by similar triangles) in the same proportion as
the diameters and the circumferences of the circles DP and
DI. Since, then, whilst the cones make n, and n2 revolutions
respectively, the circles HG and IlK are carried through n,
and n2 revolutions respectively, and that n, times the circumference of HG is equal to n2 times that of HK, therefore
the circles HG and HK roll in contact through the whole of
that space, nowhere sliding upon one another. And the
same is true of any other corresponding circles on the cones;
whence it follows that their whole surfaces are made to roll
upon one another by their mutual contact, no two parts
being made to slide upon one another by the rolling of the
rest.
The rotation of the one axis might therefore be communicated to the other by the rolling of two such cones in contact, the surface of the one cone carrying with it the surface
of the other, along the line of contact AD, by reason of the
mutual friction of their surfaces, supposing that they could
be so pressed upon one another as to produce a friction equal
to the pressure under which the motion is communicated, or
the work transferred. In such a case, the angular velocities
of the two axes would evidently be to one another (equation
227) inversely, as the circumferences of any two corresponding circles DP and DI upon the cones, or inversely as their
radii ND and OD, that is (by construction) inversely as the
numbers and teeth which it is supposed to cut upon the
wheels.
When, however, any considerable pressure accompanies
the motion to be communicated, the friction of two such
cones becomes insufficient, and it becomes necessary to
transfer it by the intervention of bevil teeth. It is the characteristic property of these teeth that they cause the motion
to be transferred by their successive contact, precisely as it
would by the continued contact of the surfaces of the
cones.




286                 CONIC L WHEELS.
229. To describe the teeth of bevil wheels.*
From D let FDE be wrawn at right angles to AD, intersecting the axes AE and AF of the two cones in E and F;
suppose conical surfaces to be generated by the revolution
of the lines DE and DF about AE and AF respectively;
\   ^im —---- __,'and let these conical surfaces be truncated by planes LM
and XY respectively perpendicular to their axes AE and
AF, leaving the distances DL and DY about equal to the
depths which it is proposed to assign to the teeth. Let now
the conical surface LDPM  be conceived to be developed
upon a plane perpendicular to AD, and passing through the
point D, and let the conical surface XIDY    be in like
manner developed, and upon the same plane. When thus
developed, these conical surfaces will have be-,t,,,. come the plane surfaces of two segmental annuli
~/J2     I MIPpm and IXxit, whose centres are in the
points E and F of the axes AE and AF, and
\     r~-,-~,  which touch one another in the point D of the
line of contact AD of the cones.
Let now plane or spur teeth be struck upon
Y.,   the circles Pp and Ii, such as would cause them
* The method here given appears first to have been published by Mr. Tredgold in his edition of Buchanan's Essay on Millwork, 1823, p. 103.
t The lines MP and pm in the development, coincided upon the cone, as also
the lines IX and ix; the other letters upon the development in the above




CONICAL WHEELS.                      287
to drive one another as they would be driven by their
mutual contact; that is, let these circles Pp and Ii be taken
as the pitch circles of such teeth, and let the teeth be
described, by any of the methods before explained, so that
they may drive one another correctly. Let, moreover, their
pitches be such, that there may be placed as many such
teeth on the circumference Pp as there are to be teeth
upon the bevil wheel HRP, and as many on Ii as upon the
wheel HI.
Having struck upon a flexible surface as many of the first
teeth as are necessary to constitute a pattern, apply it to
the conical surface DLMP, and trace off the teeth from it
upon that surface, and proceed in the same manner with the
surface DIXY.
Take DH equal to the proposed lengths of the teeth, draw
ef through H perpendicular to AD, and terminate the wheels
at their lesser extremities by concave surfaces HGlml and
HKxZy described in the same way as the convex surfaces
which form    their greater extremities.    Proceed, moreover,
in the construction of pattern teeth precisely in the same
way in respect to those surfaces as the other; and trace out
the teeth from these patterns on the lesser extremities as on
the greater, taking care that any two similar points in the
teeth traced upon the greater and lesser extremities shall lie
in the same straight line passing through A. The pattern
teeth thus traced upon the two extremities of the wheels are
the extreme boundaries or edges of the teeth to be placed
upon them, and are a sufficient guide to the workman in
cutting them.
230. To prove that teeth thus constructed will work truly
with one another.
It is evident that if two exceedingly thin wheels had been
taken in a plane perpendicular to AD (fig. p. 286.) passing
figure represent points which are identical with those shown by the same letters in the preceding figure. In that figure the conical surfaces are shown
developed, not in a plane perpendicular to AD, but in the plane which contains
that line and the lines AE and AF, and which is perpendicular to the last-mentioned plane. It is evidently unnecessary, in the construction of the pattern
teeth, actually to develope the conical extremities of the wheels as above
described; we have only to determine the lengths of the radii DE and DF by
construction, and with them to describe two arcs, Pp, Ii, for the pitch circles
of the teeth, and to set off the pitches upon them of the same lengths as the
pitches upon the circles DP and DI, which last are determined by the numbers
of teeth required to be cut upon the wheels respectively.




288            THE MODULUS OF A SYSTEM
through the point D, and having their centres in E and F;
and if teeth had been cut upon these wheels according to
the pattern above described, then would these wheels have
worked truly with one another, and the ratio of their angular velocities have been inversely that of ED to FD, or (by
similar triangles) inversely that of ND to OD; which is the
ratio required to be given to the angular velocities of the
bevil wheels.
Now it is evident that that portion of each of the conical
surfaces DPML and DIXY which is at any instant passing
through the line LY is at that instant revolving in the plane
perpendicular to AD which passes through the point D, the
one surface revolving in that plane about the centre E, and
the other about the centre F; those portions of the teeth of
the bevil wheels which lie in these two conical surfaces will
therefore drive one another truly, at the instant when they
are passing through the line LY, if they be cut of the forms
which they must have had to drive one another truly (and
with the required ratio of their angular velocities) had they
acted entirely in the above-mentioned plane perpendicular
to AD and round the centres E and F. Now this is precisely the form in which they have been cut. Those portions of the bevil teeth which lie in the conical surfaces
DPML and DIXY will therefore drive one another truly at
the instant when they pass through the line LY; and therefore they will drive one another truly through an exceedingly
small distance on either side of that line. Now it is only
through an exceedingly small distance on either side of that
line that any two given teeth remain in contact with one
another. Thus, then, it follows that those portions of the
teeth which lie in the conical surfaces DM and DX work
truly with one another.
Now conceive the faces of the teeth to be intersected by an
infinity of conical surfaces parallel and similar to DM and
DX; precisely in the same way it may be shown that those
portions of the teeth which lie in each of this infinite number ofconical surfaces work truly with one another; whence
it follows that the whole surfaces of the teeth, constructed as
above, work truly together.
231. THE MODULUS OF A SYSTEM OF TWO CONICAL OR
BEVIL WHEELS.
Let the pressure P, and P2 be applied to the conical




OF TWO CONICAL WHEELS.           289
wheels represented in the accompanying figure at perpendicular distances a, and 2 from their axes CB and COG; let
the length AF of their teeth be represented by b; let the
distance of any point in this line from F be represented by
x, and conceive it to be divided into an exceedingly great
number of equal parts, each represented by Ax. Through
each of these points of division imagine planes to be drawn
/   /I,
/.' /,-;.-. -'-o.....  ~
perpendicular to the axes CB and OG of the wheels, dividing
the whole of each wheel into elements or laminae of equal
thickness; and let the pressures P1 and P2 be conceived to be
equally distributed to these laminae. The pressure thus distribnted to each will then be represented by' asx on the
one wheel, and  al2' on the other. Let p, and p represent
the two pressures thus applied to the extreme laminae AH
and AK of the wheels, and let them be in equilibrium when
thus applied to those sections separately and independently
of the rest; then if R represent the pressure sustained along
that narrow portion of the surface of contact of the teeth of
the wheels which is included within these laminae, and if R,
and R2 represent the resolved parts of the pressure R in the
directions of the planes AH and AK of these laminae, the
pressuresp, and R1 applied to the circle All are pressures
in equilibrium, as also the pressures p2 and R2 applied to the
circle AK. If, therefore, we represent as before (Art. 216.)
by m, and mn, the perpendiculars from B and G upon the
19




290            THE MODULUS OF A SYSTEM
directions of R, and R2, and by L, and L2, the distances between the feet of the perpendiculars a,, m  and a2, mr2, we
have (equation 236, 237), neglecting the weights of the
wheels,
=ai M+      sLin.p,..  ( 6).
p, and p2 representing the radii of the axes of the two wheels,
p? and p, representing the radii of the axes of the two wheels,
and p1 and p2 the corresponding limiting angles of resistance.
Let 7, and 72 represent the inclinations of the direction of R
to the planes of AH and AK respectively; then
R1= R cos. 71, R2= R cos. 7Y2
Now it has been shown in the preceding article, that the:action of that part of the surface of contact of the teeth which
is included in each of the laminae AH, AK, is identical with
the action of teeth of the same form and pitch upon two
cylindrical wheels AD and AL of the same small thickness,
situated in a plane EAD perpendicular to AC, and having
their centres in the intersections, b and q, with that plane of
the axes CB and CG produced. The reciprocal pressure R
of the teeth of the element has therefore its direction in the
plane EAD; and if its direction coincided with the line of
centres DL of the two circles EA and AD, then would its
inclinations to the planes of AH and AK be represented by
DAH and LAK, or by ACB and ACG.
The direction of R is however, in every case, inclined to
the line of centres at a certain angle, which has been shown
(Art. 216.) to be represented in every position of the teeth,
after the point of contact has passed the line of centres by
( + p); where 0 represents the inclination to AL of the line
X, which is drawn fiom the point of contact A of the pitch
circles to the point of contact of the teeth, and where qp represents the limiting angle of resistance between the surfaces of
the teeth. To determine the inclination y/ of RA to the
plane of the circle AH, its inclination RAD to the line of
centres being thus represented by (0 + p), and the inclination
of the plane AD, in which it acts, to the plane AH being
DAH, which is equal to ACB, let this last angle be repre



OF TWO CONICAL WHEELS.            291
sented by I,; and let Aa in the accom-.. —". panying figure represent the intersection'-'-'j::! of the planes AD and AH; Aard repres/ enting a portion of the former plane and.^ A x    -    AAach of the latter. Let moreover Ar
represent the direction of the pressure R.in the former plane and let Ad and Ah be portions of the
lines AD and AH of the preceding figure. Draw re perpendicular to the plane Aach, and rd and ch parallel to Aa,
and join dh; then rAc represents the inclination y, of the
direction of R to the plane AD, dAr represents the inclination (p + ) of AR to AD, and dAh represents the inclination
I1 of the planes AD and AH to one another. Also, since Aa
is perpendicular to the plane Ahd, therefore dr is perpendicular to that plane,
re = Ar sin. 7 = Ad sec. (0 + p) sin. y,
Also hd = Ad sin. i,, but re = Ad,.. Ad sec. (O + () sin. y,=Ad sin.,,;. sin. 7y = cos. ( +p) sin. r,.
In like manner it may be shown that sin. 72 = cos. ( + P)
sin. 2, 1being taken to represent the inclination KAL of the
planes AE and AK, which angle is also equal to the angle
ACG.
From the above equations it follows that
R,=R lcos. 7y=R $/1 - cos. 2( +p) sin. 2t     (268)
R=-R cos. 7,=IR 4/1 - cos. 2(6 + ) sin., 1
From the centre b of the circle AD draw bm perpendicular
to RA, then is BM  (the perpendicular let fall from  the
centre of the circle AI{ upon the direction of R1) the projection of bm upon the plane of the circle AH. To determine
the inclination of bm to the plane AH, draw An parallel to
bm; the sine of the inclination of An to the plane AH is
then determined to be cos. DAn. sin. y, precisely as the sine
of the inclination of Am to the same plane was before determined to be cos. DAm. sin.,.
Now DAn =-Abr, =- -DAR =         — ( + q); therefore the
sine of the inclination of An, and therefore of bm, to the plane




292            THE MODULUS OF A SYSTEM
//X"'.y.'
CF
AH is represented by the formula sin. (+ p) sin.,, and the
cosine of its inclination by  /1 —sin. 2( +p) sin.; 2;. m,-=BM=bm     l-sin. 2(0+q) sin.',.
Now it has been shown (Art. 216.) that the perpendicular
bm let fall from the centre of a spur wheel upon the direction of the pressure upon its teeth is, in any position of their
point of contact, represented (equation 239) by the formula,
r, sin. (0 + )+  sin. p,
where 0, p, X represent the same quantities which they have
been taken to represent in this article; but r, represents the
radius hA of the circle AD, instead of the radius BA of the
circle AH; now bA=BA sec. DAH=r, sec. qi; substituting
this value for r, in the preceding formula, we have
bm=r, sin. ( + () sec. i +x sin. q;
~,       sin. (= 0r -) sec. (, +X sin. p} 
l-sin. + ) sin..
Similarly it may be shown that....  (269).
m2-= 2J sin. ( + ) sec. i, — sin. }p
4/1-sin. 2( + ) sin. 2,.
Substituting the values of m, and m. above determined,
and also the values of R, and R, (equations 268) in equations




OF TWO CONICAL WHEELS.           293
(267), and eliminating R between those equations, a relation
will be determined between pl and p2 which is applicable to
any distance of the point of contact of the teeth from the
line of centres.
Let it now be assumed that the number of the teeth of
the driven wheel is considerable, so that the angle - tra-'2
versed by the point of contact of each tooth may be small,
and the greatest value of the line x, the chord of an exceedingly small arc of the pitch circle of the driven wheel. In
this case + p will very nearly equal (Art. 220.); so that
cos. 2(0 + ) will be an exceedingly small quantity and may
be neglected, and sin. (8 + p) very nearly equal unity. Substituting these values in equations (268) and (269) we have
R=R,- R, R,
m-n=,+ Xsin. c cos. i, m2=-r-X sin. p cos. 2.
Substituting these values in equations (267) and dividing
those equations by one another so as to eliminate R,
p1  a2 f+~sin.p cos. i,+ (j I sin.,.
1 —X sin. qp cos. I- ( -2) sin. qp,
1 + - sin. lp cos. ql+ (- sin. 
p    2 a1r2   X             p2L2 \J
p, a_ - - sin. qp cos.',2- ) sin.,2
Whence performing actual division by the denominator of
the fraction, and neglecting terms involving dimensions
above the first in sin. p, sin. %p, sin. 2,,
(+ A) sin.  pI
-P, sn           -i
Now if 4I represent the angle described by the driven
wheel or circle ELA, whilst any two teeth are in contact,
since x is very nearly a chord of that circle subtending this
small angle, (Art. 220.);:. X = 2. Let q,  represent the




294:           THE MODULUS OF A SYSTEM.
angle described by the conical wheel FE, whilst the circle
ELA describes the angle +4; then, since the pitch circle of
the thin wheel AK and the circle ELA revolve in contact at
A, they describe equal arcs whilst they thus revolve, respectively, through the unequal angles.~ and {. Moreover, the
radius Ag of the circle AL=AG sec. GAg=r, sec. I, therefore r,2 sec. i- =r2;.'. 4=  cos.,..... (270).
Substituting the above valves of. and X, and observing
rz  n22
that  =,
^   nH
~~~~(               O'1 O1
P2   a^n L  1+  (cos.;, COS., \  o  *in.'+
-PI-=  - \ 1     +       m  - _+..- ~ cos. sin. (p +,2   a,11      2  nI     n2         2
p,L, sin. (p +  P(a  sin. 2..  (271).
\r aairj     \,r
Multiplying both sides of this equation by p2 <',, and observing that pla, 2 =-a, 2 —A  and that a.k is the exceedingly small angle described by the driving wheel AN, whilst
the driven wheel describes the angle AT, so that if au, represent the work done by the pressure p, upon the lamina AH,
whilst the angle ad is described by the driven wheel, then
pa, -2Ai=azI, we have
att6            RCOS. \, COS. 2\
A    Pl  $1 Jr,+  n  + Cos; 2  cos.:2 sin. q +
(pL,) sin. P, +  P2L2 Sin. 921;
or assuming At infinitely small, and integrating between the
limits 0 and 21 (Art. 220.),
n2
=2  a2 tI +~ (coS+. iO 1+co  2) cos., sin. +
p(    sin. (p,+ (P2 sin. 9.
a(           \^ a 




OF TWO CONICAL WHEELS.            295
Now the above relation between the work u, done by the
pressure p, upon the extreme element AH of the driving
wheel whilst any two teeth are in contact, and the pressure
p, opposed to the motion of the corresponding element of
the driven wheel, is evidently applicable to any other two
corresponding elements; the values of p,, r, r, L, and L,
proper to those elements being substituted in the formula.
If, therefore, we represent by AU, that increment of the
whole work U, done upon the driving wheel, which is due
to any one of the elements into which we have imagined
that wheel to be divided, and if we substitute for p. its
value -2A, assign to L,, L,,  r,, their values proper to that
element, and represent those values by L, L', r, r',
Pcos.  cos.s\
AU- 1            +      J O. -  COS. + cos., sin. p+
1   n    (       n,    nI
Lp,.       /L'p,
( r) s —in. p, +   sin. (p, A^x
or assuming ar infinitely small, and integrating between the
limits 0 and b, and observing that Pa2-  represents the
n2
whole work U, done upon the driven wheel under the constant pressure P, during the contact of any two teeth,
U,=-   l+U2  COS,   +-, cos. I2 sin. +
1      2
b
pi sin. p, fLd   sin.p  Ld
a 9       - r a   bad..     2-dx. (.... (2t2).
Now a+x being taken to represent the distance of the
point of contact of any two such elements from C, and a to
represent the distance CF, the radii r and r' of these elements are evidently (by similar triangles) represented by
za+-x    I   x        a+x          x\.
----  or 1 +   r,, and — 2, or 1 + - r,, and r representing the radii of the extreme elements NF and OF, or of
the pitch circles of the lesser extremities of the wheels.
Also assuming, as we have done, the pressures R, and R,




296            THE MODULUS OF A SYSTEM
to be perpendicular to the lines BA,
/4^'^~^ a GA joining the centre of each eleo  )  /  ment with their point of contact A,
g:/ ~  so that the points M and N (see fig.
p. 292.) coincide with the point A
R ^  )k (see accompanying figure)*; and representing the angles ABD and ACE
made by the perpendiculars DB and
CE with the line of centres by 60 and
o0 respectively; observing also that AD2=BA2- 2BA. BD
AD(BD
cos. ABD+BD2, so that (A) =1-2           ) cos. ABD(+
(BD\), we have, substituting, in the second number of this
equation, for BA or r its value r, (1 +
(   -L 1  2(/l) (1 + f   cos. +       + 
\r                 a    \/ a\7/
or expanding the binomials in this expression, observing
that x is an exceedingly small quantity, neglecting terms
a
involving powers of that quantity above the first, and
reducing,
2( )' Cos. 41 ( s... (273).
Now L, representing the value of L when a-=O, and 0 re
maining constant,
alu\                  al     L  2.( )2    2( s. 0, —  1- +  -
* The circles in this figure represent two of the corresponding lamina into
which wheels have been imagined to be divided; they are not, therefore, in the
same plane. Their planes intersect in AH.




OF TWO CONICAL WHEELS.             297
Let now the angle ADB, made in respect to the first element of the driving wheel between the perpendicular BD or
a, and the chord AD or L, be represented by y1, and let a,
represent the corresponding angle in the driven wheel, then
2
L - 2La, cos., + a2 = r2,.. ()
-- cos. + -17;
4..2  c 2 - l  (L 2) =1;
— 2L  con   = 1- a   )2   (L)    2 () (cos.O.-.
ri'l       \'rJ    rI       r\         rl
Substituting these values of (f') and 2 (r) ( cos. —  )
in equation (273);
(a)    ()-2(L- (La) (a) cos. 11=     )
{ 1 -2 (.L)() cos., }
Extracting the square root of the binomial, and neglecting
terms involving powers of - above the first,
L    L*      p sin., rL  pasin. p   L,    b
(Equation 272) pl   - d-   x  Sf  idi mco.Pl1
ba',  0 r      ri a,       a
b
Similarly       I /  dx      -        -     Cos. B d.
ba2v ^ *0 ra. 2                a o.  ( 
Substituting these values in the modulus (equation 272),
-U=     1io    +      -    cos., sin. p +
p, sin. p, L,  b         p, sin. p, L, 
-r,    a.    a           -r.    a.       0^ a




298            THE MODULUS OF A SYSTEM
Now let the angle BCG, or the inclination of the axes,
from one to the other of which motion is transferred by the
wheels, be represented by 2,; therefore  +, =2s. Also a
sin.,1=r, and a sin. 2,=r,,
sin. 1'   n~
*  sin. ~'2-  n2'
sin.',   sin. 1      co      1    cos. 2'
n2       na2     -2      2     2     n 2 
i 1 cos.\  COS. 2  ICOS. I_ COS. I, cos., Cos. Is
ni2 N22   ni2 ~n        n \           nl    n2 ^ } \  /"-'
(COS. 2 cos. 2    COs. c, _ )1\ 
**2  --  +   } —-' +2      J72 cos.
n       n2 1\ cos. 12  n
(. ICOS.  COS 2 
+ -n  + ^  ) cos. = rcos S      -
+     COS. 1    c2
n1 COS. 12 1%
Now cos.,   cos.'+i('1 -'1)}  1-tan. ( —2,) tan.,
Noco.       cos. l' —4('1-'1,) -  1+tan. ('-2,) tan.,;
al  sin.   sin. +('1-'2,) _ tan. +4- tan.- (,- 2,)
also  sin.,~ sin. 4-(-12-')  tan. - tan. — tn  i(-2). tan. i     (,- I,)= - n —   tan. s;
nl+n2.-1     _tan.,
cos., I     nl - n       (n, + -) — (+  - ~n) tan. -1
cos       + n      tan.   (n=  + n,) + (n(+ — )+,) tan.',
C05.2 -  - tan. 2
1 cos.,         1    COS.     1
n cos. 1 2 n    n     2 COS. 12  I
-1 (n,2 - n22) + /2 -n2)tan.2'
n, ( + 2)+ (n1-n2) tan. -




OF TWO CONICAL WHEELS.            299
I-(ll2  -  a) sec. 2            1 2    )2
_\        n_ —4__,                             sin.^ __)_n
2 4- +              -2C
(LL z  (-* I +P  2)in OS. (2  (  +  COS. 21
1  /          1.     \    2 in. l a2l
Substituting in thle preceding relation, between U, and U,,
U,-  { I 1+ Jr 4  -2 sin.) 2     p, sin. p,
U1 = \'          -sn.p+ as           r    IL,
T al-1                 s   in.     1 *   * 
which is the modulus of the conical or bevil wheel, neglecting
the influence of the weight of the wheel.
If for cos.  th and cos. 2 we substitute their values (see
p 29), we shall obtain by reduction
(  ) ( /I   1     2 sin.. 2 L 1
4aL         U2.... (275),
from which equation it is manifest that the most favourable
directions of the driving or working pressures are those
determined by the equations
L,=    (a ^   ), b  I  ^  (' a2 )2
II  4a-&           \ 4a-b 
232. It is evident, that if the plane of the revolution
of such a wheel be vertical, the influence of its weight must
be very nearly the same as that of a cylindrical or spur




300  THE MODULUS OF A SYSTEM OF TWO CONICAL WHEELS.
wheel of the same weight, having a radius equal to the mean
radius of the conical wheel, and revolving also in a vertical
plane. If the axis of tie wheel be not horizontal, its weight
must be resolved into two pressures, one acting in the plane
of the wheel, and the other at right angles to it; the latter is
effective only on the extremity of the axis, where it is borne
as by a pivot, so that the work expended by reason of it may
be determined by Art. 175, and will be found to present
itself under the form of N. S, where N2 is a constant and S
the space described by the pitch circle of the wheel, whilst
the work IU1 is done. The resolved weight in the plane
of the wheel must be substituted for the weight of the wheel
in equation (247), which determines the value of N. Assuming the value of N, this substitution being made, to be represented by N, the whole of the second term of the modulus
will thus present itself under the form (N1 + -N)S.
a U/I       1I   2 sin.'          sin,, q
(b     -~ cos.  + P2 -^2(2 -     cos.2)   U2+
(N, + N2) S..... (276).
233. Comparing the modulus of a system of two conical
wheels with that of a system of two cylindrical wheels
(equation 252), it will be seen that the fractional excess
of the work U2 lost by the friction of the latter over that
lost by the friction of the former is represented by the
formula
2   sin.'i sin, (p  b  p,        P2 p 
+  s.        + ~  -  cos., sin. 1 +      si...
rl              r2 
The first term of this expression is due to the friction of
the teeth of the wheels alone, as distinguished from the friction of their axes; the latter is due exclusively to the friction
of the axes. Both terms are essentially positive, since y,
and /2 are in every case less than 
Thus, then, it appears that the loss of power due to the
friction of bevil wheels is (other things being the same)
essentially less than that due to the friction of spur wheels,
so that there is an economy of power in the substitution of




THE MODULUS O0 A TRAIN OF WHEELS.     301
a bevil for a spur wheel wherever such substitution is practicable. This result is entirely consistent with the experience
of engineers, to whom it is welt known that bevil wheels run'ighter than spur wheels.
234. THE MODULUS OF A TRAIN OF WHEELS.
In a train of wheels such as that shown in the accompany-.9                   ing figure, let the radii of their
PI e  s-   1  pitch circles be represented in
IEmd  {'^  4  \  order by r,, r,, 3.. r4, begint} 57 6(~^ ~  f3  < Sning from the driving wheel;.It S elf X ni | o 1-and let a, represent the perpen-'-j \i  ldicular distance of the driving
Hi    J  pressure from the centre of that
wheel, and a2 that of the driven
pressure or resistance from the centre of the last wheel of the
train; UT the work done upon the first wheel, u, the work
yielded by the second wheel to the third, u, that yielded by
the fourth to the fifth, &c., and U2 the work yielded by the
last or nth wheel upon the resistance, then is the relation between U1 and u2 determined by the modulus (equation 252),
it being observed that the point of application of the resistance on the second wheel is its point of contact b with the
third wheel, so that in this case a=r,3.
These substitutions being made, and L2 being taken to
represent the distance between the point b and the projection
of the point a upon the third wheel, we have
UJ= {        + l+(   + S   sin. q + sin1+
Ls, in.     +N,. S,.
To determine, in like manner, the relation between u2 and
u, or the modulus of the third and fourth wheels, let it be
observed that the work uz which drives the third wheel has
been considered to be done upon it at its point of contact b
with the fourth; so that in this case the distance between the
point of contact of the driving and driven wheels and the
foot of the perpendicular let fall upon the driving pressure
from the centre of the driving wheel vanishes, and the term
* See note p. 266.




302      THE MODULUS OF A TRAIN OF WHEELS.
which involves the value of L representing that line disappears from the modulus, whilst the perpendicular upon the
driving pressure from the centre of the driving wheel becomes r. Let it also be observed, that the work of the
fourth wheel is done at the point of contact c of the fifth and
sixth wheels, so that the perpendicular upon the direction of
that work from the axis of the driven wheel is r. We shall
thus obtain for the modulus of the third and fourth wheels,
u2= 1(i+4~'l-,+siln.i+ L1P
- 2   +(n +)sin.+ r4r i   g     03  3+NS2.
In which expression L, represents the distance between the
point c and the projection of the point b upon the fifth
wheel.
In like manner it may be shown, that the modulus of the
fifth and sixth wheels, or the relation between u3 and u4 is
U3,  { l~(-+     sin s)6in. + 4 94  4+N9.S S3;
r \   n6
and that of the seventh and eighth wheels, or the relation
between u4 and u%,
U,-=    1 + ~ ( +  s   in. +.l sin.  p u+, 4. S4;
4    \nV,  I       rr,    6     N   S 4
and that, if the whole number of wheels be represented by
2p, or the number of pairs of wheels in the train by p, then
is the modulus of the last pair,
Ulz  i+~  1  1 )
Up   { 1+  (-     i) +  sin. p4+n2\p-l qp
L3+lP+lsin.  +l } U2+N r. Sp;
In which expressions the symbols N1, N, N3...Np, are
taken to represent, in respect to the successive pairs of wheels
of the train, the values of that function (equation 247),
which determines the friction due to the weights of those
wheels; and each of the symbols L,, L,, L4.... Lp, the distance between the point of contact of a corresponding pair
of wheels and the projection upon its plane of the point of
contact of the next preceding pair in the train; whilst the
symbols nL n2%,.... N2P, represent the numbers of teeth in
the wheels; l, r2, r,... rp, the radii of their pitch circles;
and S1, SO, S.. Sp, the spaces described by their points of




THE MODULUS OF A TRAIN OF WHEELS.      303
contact a, b, c, &c. whilst the work U1 is done upon the first
wheel of the train.
Let us suppose the co-efficients of u,  u,  U,, in these
moduli to be represented by (1 +  ), (1 +  ), (1+If).* *
(1 + p); they will then become
U,= ( + i1) u2+N,. S,,
Zu=(1+p2) zu+N,. S,,
= (14+ )^ U4+N 3. S3,
&c.=-&c.
p =(l+ P) UT +-Np. Sp.
Eliminating u2, u, u4.. u, between these equations, we
shall obtain an equation of the form
U,=(1+(i1) (1 +~2) (1 +f)... (1+P )U2+N. S..  (277),
where
NS =N S + (1 + 1)N2S + (1 + 1)(1 + 2)NS, +....
+ (1 + a)(1 + 2).... ( _ +  _-)N p.... (278).
Now let it be observed, that the space described by the first
wheel, at distance unity from its centre, whilst the space S1
is described by its circumference, is represented by Si, and
S
that this same space is represented by - if S represent the
space described in the same time by the foot of the perpendicular,, or the space through which the moving
pressure may be conceived to work during that time; so
that'_.  Also let it be observed that the space derl a1
scribed by the third wheel, at distance unity from its centre,
is the same with that described at the same distance from
its centre by the second wheel, so that S=S-; in like
r3 r2
manner that the spaces described at distances unity from
their centres by the fourth and fifth wheels are the same, so
S   SS                     S,
that -  -; and similarly, that  —,  &c.=&c.; and
r    4                     r, r,
finally, S- _S_.
rp-1 r2p-2




804       THE MODUILUS OF A TRAIN OF WHEELS.
Multiplying the two first of these equations together, then
the three first, the four first, &c., and transposing, we have
S11=S, 2 —           S.= S(
al        a. r         \' n2 
1 r2 * 4      a1 \2 *. n4
r1,. r5. r5      (   n.n. n     s,
S   <             _...../..... njSs
&c.-=&c.
g~ l * r' * b  *2p-1 I-1 f n3 n*.  *. n2P-\ S.
a1. r2. r4... r2-2  a\a  n2.n... * n2p-21
Substituting these values of S1, S,, &c. in equation (278),
and dividing by S, we have
N        {i   + (1 + (1) ( N2 + (1 +F-)
(1 +ffiU2). n4)  +'    };
or if we observe that the quantities t%, 2,,,3 are composed
of terms all of which are of one dimension in sin. p, sin. p,1
sin. p,, &c. and that the quantities N1, N2, N,, &c. (equation
247) are all likewise of one dimension in those exceedingly
small quantities; and if we neglect terms above the first
dimension in those quantities, then
N=(S) { N +("     )N +()3 6     +
(!-'+.....       (279).
If in like manner we neglect in equation (277) terms of
more than one dimension in p), P,) P, &c. we have
U1,=-{1+  +   +   + **++... +  J} + U2+N. S.
Now P,= r (    t~ +  isin. p  +  l sin. + - -sin. p,
A\^ n    aii r,      r,2r, =       + ~ (- n _ -) sin. p +  -LsP sin. pr,,
noa    N          r~y,




THE MODULUS OF A TRAIN OF WHEELS.      305
f=h   (a- q+ )sin. + -p sin. p,,
\5 e,6 rf'r
&c.=&c.
I    (  I(  + -) sin. +     sin. P.
PIP - _   p ns pas
Substituting these values of p1, P, &c. in the preceding
equation,
n1  n2    P    n2p        ar
Lp U,= jl +~$-     3-        a )sin. q~q   sin.s q, +
L  si P. + s  in.pn +.... n.    S.... (280),
2r3r.       4 r5          r,,a,
which is a general expression for the modulus of a train of
any number of wheels.
235. The work U1 which must be done upon the first
wheel of a train to yield a given amount U, at the last wheel,
exceeds the work UT, or, in other words, the work done upon
the driving point exceeds that yielded at the working point,
by a quantity which is represented by the expression
(1+ +.. +    s)in. * u2+   sin. + s    in. q,
+.. + Lp     sin. ) U+NS... (281).
In which expression the first term represents the expenditure
of work due to the friction of the teeth, and varies directly as
the work U2, which is done by the machine.  The second
term represents the expenditure of work due to the friction
of the axes of the wheels, and varies in like manner directly
as the work done. Whilst the third term represents the
expenditure of work due to the weights of the wheels of the
train, and is wholly independent of the work done, but only
upon the space S, through which that work is done at the
point where the driving pressure is applied to the train.
20




306        FRIOTION OF THE AXES OF A TRAIN.
236. The expenditure of work due to the friction of the teeth.
The work expended upon the friction of the teeth is represented by the formula
~'            ++, - ) +. + — ) sin. q.....(282),
\nPIn,, nWl'          \
whose value is evidently less as the factor sin. p is less, or as
the coefficient of friction between the common surfaces of the
teeth is less; and as the numbers of the teeth in the different
wheels which compose the train are greater. The number
of teeth in any one wheel of the train may, in fact, be taken
so small, as to give this formula a considerable value as compared with TU, or to cause the expenditure of work upon the
friction of the teeth to amount to a considerable fraction of
the work yielded by the train: and the numbers of teeth
of two or more wheels of such a train might even be taken
so small as to cause the work expended upon their friction to
equal or to surpass by any number of times the work yielded
by the train at its working point.  This will become the
more apparent if we consider that the surfaces of contact of
the teeth of wheels are for the most part free from unguent
after they have remained any considerable time in action, so
that the limiting angle of resistance assumes in most cases
a much greater value at the surfaces of the teeth of the
wheels than at their axes.  From this consideration the
importance of assigning the greatest possible number of
teeth to the wheels of a train individually and collectively
is apparent.
237. The expenditure of work due to the friction of the axes.
This expenditure is represented by the formula
( sin. p, +  sin. +...+  -   sin.  )... (283),
alC r,     f         "     rr r'
forming the second term of formula 280. Now, evidently, the
value of this formula is less as the quantities sin. p, sin. p,,
&c. are less, or as the limiting angles of resistance between
the surfaces of the axes and their bearings are less, or the




FRICTION OF THE AXES OF A TRAIN.       307
lubrication of the axes more perfect; and it is less as the
fractions L,, L2P2 LP3, &c. are less.
fra r rsr r)r4a
Now, L2 being the distance between the point of contact b
of the third and fourth wheels
\ 8   and the projection of the point of
contact a of the first and second
Ir    s H o         upon the plane of those wheels,,|   oI  ~|2n I't it follows that generally, L2 is
4T V\                  least when the projection of a
\Ji "~ ffalls on the same side of the axis
as the point b;- and that it
is least of all when this line falls on that side and in the line
joining the axis with the point b; whilst it is greatest of all
when it falls in this line produced to the opposite side of the
axis. In the former case its value is represented by r,-r,,
and in the latter by r +r2; so that, generally, the maximum
and minimum values of L2 are represented by the expression
r:r2, and the maximum and minimum values of L2P_ by
(j - -) P2. And similarly it appears that the maximum and
minimum values of L2P are represented by(:  ) pa; and
r4r5                  4 7'5
so of the rest. So that the maximum and minimum values of
the work lost by the friction of the axes are represented
by the expression
i -;  * —-F P, sin. q, + ( )       - sin  +
~-; PS sin. q +.}.. U2;
from which expression it is manifest, that in every case the
expenditure of work due to the friction of the axes is less as
the radii of the axes are less when compared with the radii
of the wheels; being wholly independent of actual dimensions
of these radii, but only upon the ratio or proportion of the
radius of each axis to that of its corresponding wheel: more* This important condition is but a particular case of the general principle
established in Art. 168.; from which principle it follows, that the driving
pressure on each wheel should be applied on the same side of the axis as the
driven pressure.




30.8          THE WEIGHTS OF THE WHEELS.
over, that this expenditure of work is the least when the
wheels of the train are so arranged, that the projection of the
point of contact of any pair upon the plane of the next
following pair shall lie in the line of centres of this last pair,
between their point of contact and the axis of the driving
wheel of the pair; whilst the expenditure is greatest when
this projection falls in that line but on the other side of the
axis. The difference of the expenditures of work on the
friction of the axes under these two different arrangements
of the train is represented by the formula
2  e sin. p, +    Ps         sin. p  + P i. 3 +   in. +.. } U,;
r i        3        r5        r7
which, in a train of a great number of wheels, may amount
to a considerable fraction of U,; that fraction of U2 representing the amount of power which may be sacrificed by a
false arrangement of the points of contact of the wheels.
238. The expenditure of work due to the weiqhts of the several
wheels of the train.
The third and last term N. S of the expression (280) represents the expenditure of work due to tile weights of the
several wheels of the train; of this term the factor N is
represented by an expression (equation 279), each of the
terms of which involves as a factor one of the quantities N,,
N,, N,, &c., whose general type or form is that given in
equation (247), it being observed that the direction of the
driving pressure on any pair of the wheels being supposed
that of a tangent to their point of contact; the case is that
discussed in the note to page 266. The other factor of each
term of the expression (equation 279) for N, is a fraction
having the product n, n,.... of the numbers of teeth in all
the preceding drivers of the train, except the first, for its
numerator, and the product n. n. n. n.., of the numbers of
teeth in the preceding followers of the train for its denominator; so that if the train be one by which the motion is to
be accelerated, the numbers of teeth in the followers being
small as compared with those in the drivers, or if the multilying power of the train be great, and if the quantities
1, N2, 3,, &c., be all positive; then is the expenditure of
work by reason of the weights of the wheels considerable, as




MODULUS OF A TRAIN IN WIICH THE DRIVERS ARE EQUAL. 309
compared with the whole expenditure. Since, moreover, the
coefficients of N., N2 N, &c., in the expression for N (equation 279) increase rapidly in value, this expenditure of work
is the greatest in respect to those wheels of the train which
are farthest removed from its first driving wheel: for which
reason, especially, it is advisable to diminish the weights
of the wheels as they recede from the driving point of the
train, which may readily be done, since the strain upon each
successive wheel is less, as the work is transferred to it under
a more rapid motion.
239. 2The modulus of a train in which all the drivers are
equal to one another and all the followers, and in which
the points of contact of the drivers and followers are all
similarly situated.
The numbers of teeth in the drivers of the train being in
this case supposed equal, and also the radii of these wheels,
n=n3,=-n6-=n=&c., rl=r3 -=r,-r=-&c. The numbers of
teeth in the followers being also equal, and also the radii of
the followers n2 =-4=n6= &c., r,=r4=r6 —&c.
If, moreover, to simplify the investigation, the driving
work U1 be supposed to be done upon the first wheel of the
9                    train at a point situated in res<^ *5     %   spect to the point of contact a of'a||=|  f     \ ~ that wheel with its pinion pret,'  7: e a |a      cisely as that point of contact is
in respect to the point of contact
b of the next pair of wheels of
J     the train; and if a similar supposition be made in respect to
the point at which the driven work U. is done upon the last
pinion of the train, then, evidently, L, —L-L-..  Lp,
and (see equation 247) N,=N2-.. -=N.
The modulus (equation 280) will become, these substitutions being made in it, the axes being, moreover, supposed
all to be of the same dimensions and material, and equally
lubricated, and it being observed that the drivers and the
followers are each p in number,
U   iT = 1 +P  +   in. +lP sin. 1 U, +NS..   (284),
which is the modulus required.
which is the modulus required.




310         THE TRAIN OF LEAST RESISTANCE.
Moreover, the value of N (equation 277) will become by
the like substitutions,
N=Nr)               +         +)+    *
or       NN        ))        -j(.285).
THE TRAIN OF LEAST RESISTANCE.
240. A train of equal driving wheels and equal followers
being required to yield at the last wheel of the train a
given amount of work U2, under a velocity m times greater
or less than that wuder which the work UT which drives the
train is done by the moving power upon thefirst wheel, it
is required to determine what should be the number p of
pairs of wheels in the train, so that the work UT expended
through a given space S, in driving it, may be a minimum.
Since the number of revolutions made by the last wheel
of the train is required to be a given multiple or part of the
number of revolutions made by the first wheel, which multiple or part is represented by m, therefore (equation 231),
ni P         23 
n^2     n.,p   ri
1
/ nP..  i  n1
m=I-I..          m,
*and -      --;1                 1
I1   1    m2 +1        1    m;
\n +n= ----, and.
\n -  -- -'  qrr  r2
Substituting these values in the modulus (equation 284);
substituting, moreover, for N its value from equation (285),
we have
1                    si. 1'in='l+^p    +l)8in. p4 (-+) ^misin.  q
n^  ~       ~~~~ 2 PM BI.




THE TRAIN OF LEAST RESISTANCE.          311
UT+N,(m-1) (     X    )S.... (286).
mp-1
It is evident that the question is solved by that value of p
which renders this function a minimum, or which satisfies
dU,         d2U1
the conditions -  = 0 and d   > 0.   The first condition
dp          dp
gives by the differentiation of equation (286),,(_l --     ).+sin. q)  + -in.  q  + sin. 4
IB
rn
U +    (-     log.   NS=0.. (       287).
ffE~(m - 1)'.p2(MP)2
This equation may be solved in respect to p, for any given
values of the other quantities which enter into it, by approximation. If, being differentiated a second time, the above
expression represents a positive quantity when the value of
p (before determined) is substituted in it, then does that
value satisfy both the conditions of a minimum, and supplies, therefore, its solution to the problem.
If we suppose g,-=0 and N1=0, or, in other words, if we
neglect the influence of the friction of the axes and of the
weights of the wheels of the train upon the conditions of the
question, we shall obtain
(-   lop    sn in. +        sin. p=0;
whence by reduction,
plog..         (288).
1+m p
* This formula was given by the late Mr. Davis Gilbert, in his paper on the
"Progressive improvements made in the efficiency of steam engines in Cornwall," published in the Transactions of the Royal Society for 1830. Towards
the conclusion of that paper, Mr. Gilbert has treated of the methods best
adapted for imparting great angular velocities, and, in connection with that
subject, of the friction of toothed wheels; having reference to the friction of
the surfaces of their teeth alone, and neglecting all consideration of the influ



312                    THE INCLINED PLANE.
THE INCLINED PLANE.
241. Let AB represent the surface of an inclined plane on
which is supported'a body whose centre of gravity is C, and
its weight W, by means of a pressure acting in any direction,
and which may be supposed to be supplied by the tension of
a cord passing over a pulley and carrying at its extremity
a weight.
Let OR represent the direction of the resultant of P and
W. If the direction of this line be inclined to the perpendicular ST to the surface of the plane, at an angle OST
equal to the limiting angle of resistance, on that side of ST
which is farthest from      the summit B      of the plane (as in
fiy. 1), the body will be upon the point of slipping upwards;
and if it be inclined to the perpendicular at an angle OST,
ence due to the weights of the wheels and to the friction of their axes. The
author has in vain endeavoured to follow out the condensed reasoning by which
Mr. Gilbert has arrived at this remarkable result; it supplies another example
of that rare sagacity which he was accustomed to bring to the discussion of
questions of practical science. Mr. Gilbert has given the following examples
of the solution of the formula by the method of approximation:-If m=120,
or if the velocity is to be increased by the train 120 times, then the value ofp
glven by the above formula, or the number of pairs of wheels which should
c Impose the train, so that it may work with a minimum resistance, reference
1 eing had only to the friction of the surfaces of the teeth, is 3'745; and the value
nl
( tf the factor p(mP+l-) (equation 286), which being multiplied by - sin. f Ua
f presents the work expended on the friction of the surfaces of the teeth, is in
this case 17'9; whereas its value would, according to Mr. Gilbert, be 121 if the
velocity were got up by a single pair of wheels. So that the work lost by the
friction of the teeth in the one case would only be one seventh part of that in
the other. In like manner Mr. Gilbert found, that if m=100, then p should
equal 3'6; in which case the loss by friction of the teeth would amount to the
sixth part only of the loss that would result from that cause if p=l, or if the
required velocity were got up by one pair of wheels.
If m-40, then p=2-88, with a gain of one third over a single pair.
If m=-359, then p=l.
If m=12-85, then p=2.
If m=46'3, then p=3.
If m=166-4, then p=4.
It is evident that when p, in any of the above examples, appears under the
form of a fraction, the nearest whole number to it, must be taken in practice.
The influence of the weights of the wheels of the train, and that of the friction
of the axes, so greatly however modify these results, that although they are
fully sufficient to show the existence in every case of a certain number of
wheels, which being assigned to a train destined to produce a given acceleration of motion shall cause that train to produce the required effect with the
least expenditure of power, yet they do not in any case determine correctly
what that number of wheels should be.




THE INCLINED PLANE.              313
(1.)              (2.)
equal to the limiting angle of resistance, but on the side of
ST nearest to the summit B (as in fig. 2.), then the body will
be upon the point of slipping downwards (Art. 138.); the
former condition corresponds to the superior and the latter
to the inferior state bordering upon motion (Art. 140.).
Now the resistance of the plane is equal and opposite to
the resultant of P and W; let it be represented by R.
There are then three pressures P, W, and R in equilibrium.
P     sin. WOE.~(Art. 14.)  - sin. W OR
VW- sin. POR.
Let ZBAC=, ZOST-=limg. Z of resistance-, let o
represent the inclination PQB of the direction of P to the
surface of the plane, and draw OV perpendicular to AB;
then,
infig. 1, WOR=WOV +SOV=BAC+OST= +,
and POR=PQB + OSQ=PQB +          OST    +    —,;
2        2
in fig. 2., WOR=WOV-SOV=BAC-OST=I —p,
and POR=PQB+OSQ=PQB+ + OST=             +0+q;
2        2:.WOR=,~q;        and POR=++(0~,t   );
the upper or lower sign being taken according as the body
is upon the point of sliding up the plane, as in fig. 1, or
down the plane, as in fig. 2. Or if we suppose the angle p
to be taken positively or negatively according as the body is
on the point of slipping upwards or downwards; then generally WOR=i +p     POR=-+(4 —);




314              THE INCLINED PLANE.
P     sin. (, + )  sin. ( + p)
sin. (+      p)cos ('
PW-      sin. (~ )):P(W)...s.. (289).
cos.(-qp)''
If the direction of P be parallel to the plane, Z PQB or
8=0; and the above relation becomes
p^^8m..in. (I +,)
P=W. sin.(+).          (290).
cos. p
(3.)                        (4.)
=     te p    e b     s hriz   l (. 3, and t 
If,-=0 the plane becomes horizontal (fig. 3)., and the relation between P and W assumes the form
P=W.  si (.. (291).
cos.(0-)         (291)
If 0=:0, P=W. tan. p, as it ought (see Art. 138.).
If the angle PQB or 0 (fig. 1.) be increased so as to become ~ —, PQ will assume the direction shown in fig. 4,
and the relation (equation 289), between P and W will become
p=_W     sin. (, +.(.) (292).
cos. (0+ 9).......
The negative sign showing that the direction of P must,
in order that the body may slip up the plane, be opposite to
that assumed in fig. 1.; or that it must be a pushing pressure in the direction PO instead of a pulling pressure in the
direction OP.
If, however, the body be upon the point of slipping down
the plane, so that gp must be taken negatively; and if, moreover, qp be greater than,, then sin. (j+ p), will become sin.
(s —p)= —sin. (p —), so that P will in this case assume the
positive value
P=W.  -sin, (-.93),
cos. (6-p).     (9




THE MOVEABLE INCLINED PLANE.         315
which determines the force just necessary under these circumstances to pull the body down the plane.
If i=-, P=O, the body will therefore, in this case, be upon
the point of slipping down the plane without the application
of any pressure whatever to cause it to do so, other than its
own weight. The plane is under these circumstances, said
to be inclined at the angle of repose, which angle is therefore equal to the limiting angle of resistance.
242. The dAirecton of least traction.
Of the infinite number of different directions in which the
pressure P may be applied, each requiring a different amount
to be given to that pressure, so as to cause the body to slide
up the plane, that direction will require the least value to be
assigned to P for this purpose, or will be the direction of
least traction, which gives to the denominator of the fraction
"p             in equation (289) its greatest value, or
which makes -(pO=0 or 8=p. The din    \ /\.T  rection of P is therefore that of least
\(S't, <?\ Y traction when the angle PQB is equal to
t    the limiting angle, a relation which ob*'  Az.1  tains in respect to each of the cases dis^-~"  -    _t.A. cussed in the preceding article.
243. THE MOVEABLE INCLINED PLANE.
Let ABC represent an inclined
plane, to the back AC of which.~   a//      is applied a given pressure P,,
~, \,/          and which is moveable between
the two resisting surfaces GH and
BG =iP"-?    p KL, of which either remains fixed,
(        /,,"  _  I   and the other is upon the point
of yielding to the pressure of the
I2 ss       plane.
If we suppose the resultants of the resistances upon the
different points of the two surfaces AB and BC of the plane
to be represented by R1 and R2 respectively, it is evident
that the directions of these resistances and of the pressure P,




316         THE MOVEABLE INCLINED PLANE.
will meet, when produced, in the same point O*; and that,
since the plane is upon the point of slipping upon each of
the surfaces, the direction of each of these resistances is
inclined to the perpendicular to the surface of the plane, at
the point where it intersects it, at an angle equal to the corresponding limiting angle of resistance.
So that if ET and FS represent perpendiculars to the
surfaces AB and BC of the plane as the points E and F and
p,, p2, the limiting angles of resistance between these surfaces
of the plane and the resisting surfaces GH and KL respectively, then R1ET=-p, R2FS=,.
Now the pressures P1, R,, R, being in equilibrium (Art.
14),
P, sin. EOF       P, sin. EOF
R,-sin. DOE'      R- ~sin. DOE'
But the four angles of the quadrilateral figure BEOF
being equal to four right angles (Euc. 1'32), EOF =2zEBF-OEB-OFB; but EBF=i, OEB=-+p, OFB=
2+P.2.. EOF={ —,I ——,.
Similarly, DOE=27r-ADO-AEO-DAE; but ADO=
2 AEO=- —, BAC= —I:.*.DOE=-++,1.
2'       2            2             2
Since, moreover, DO is parallel to BC, both being perpendicular to AC,.'DOF=7r-OFC; but OFC= —~,2:.DOF=-  +2.
P*  sin. Ar-( +1+P,)1  sin. (+~p1+2P).
R,      sin.(I+2)    -    cos. 2. p=R sin. (    +.. (294.)
COS. p,
P,   sin. s:-(+,++) _sin. (; + p,+ 2)
R2    sin. ++i)        - o  oos-t (f te or 
* Since either is equal and opposite to the resultant of the other two.




A SYSTEMi OF TWO AIOVE'ABLE INCLINED PLANES.  317
sin. (I +    +%)           (295.)
c1 os. (, +.p,)
In the case in which the surface GH yields to the pressure
of the plane, KL remaining fixed, we obtain (equation 121.)
for the mnodtulus (see Art. 148.) observing that P,(~)-R, sin. 
(equation 294),
sin. (0+  -+p,)    ^ c   ^
Usin. (      1    ).(296).
sin.  *  cos. p,
In the case in which the surface KL yields, CH remaining
fixed, observing that P,(~)=RE tan. (equation 295), we have
sin. (     ), +  +97).)
cos. (; + -,)tan. 
Equations (296) and (297) may be placed respectively under the forms
U,=Usi    (1 +2) cot. (, + 2) + cot. }
Cos. P2
and      U =U cos. () +  { tan.'+tan. (p,+p2) 
sin.     cot. q,-tan. i) tan.'
The value of U, corresponding to a given value of U, is in
the former equation a minimum when'=-, and in the latter
2
when
tan. =  j       s-1            tain. (p+p).... (298).
From the former of these equations it follows, that the work
lost by friction (when the driving surface of the plane is its
hypothenuse) is less as the inclination of the plane is greater,
or as its mechanical advantage is less.
244. A system of two moveable inclined planes.
Let A and B represent two inclined planes, of which A




318    A SYSTEM OF TWO MOVEABLE INCLINED PLANES.
rests upon a horizontal surface, and
V.i2       receives a horizontal motion from
i..- -  --. the action of the pressure P1; cornv-     - -- ------...  municating to B a motion which is
Ia^'        restricted to a vertical direction by
-.-. —.,       the resistance of the obstacle D,
=j\i.... which vertical motion of the plane
is opposed by the pressure P2 applied to its superior surface. It is
required to determine a relation between the pressures P,
and P2, in their state bordering upon motion; and the modulus of the machine.
Let R~ represent the pressure of the plane A upon the
plane B, or the resistance of the latter plane upon the former,
and R, the resistance of the obstacle D upon the back of the
plane B; then is the relation between R1 and P1 determined
by equation (294). And since R., R,, P2 are pressures in
equilibrium, the relation between R, and P2 is expressed
(Art. 14.) by the relationR  n P2QR3.  Now   RQ is
rat   sin. RiQR'
inclined to a perpendicular to the back of the plane B, at an
angle equal to the limiting angle of resistance between the
surface of that plane and the obstacle D on which it is upon
the point of sliding. Let this angle be represented by pg,
then is the inclination of R. to the back of the plane or P2Q
represented by — 3; so that P2QR23= --.
And if ZRQ be produced so as to meet the surface of the
plane A in V, and VS be drawn horizontally, R,QR,=
QVR1+VR,Q =REVS +SVA+ VRQ= p3 +, +++
where i represents the inclination of the superior surface of
the plane A or the inferior surface of the plane B to the
horizon. Substituting these values of PQR, and R,QR, we
obtain
R,      sin. -- )        cos.+ 
sm. 2+' +(3+(p
Multiplying this equation by equation (294), and solving in
respect to P,,




A SYSTEM OF THREE INCLINED PLANES.     319
pp 8pp sin. (t + + p1 ) cos.     (299)
COS. (I + 91 + 91) Cos.     29)'co s. (t + p, + %) c os.  s. (..(Art. 152.) UU,   sin. (t+ +p)cos..     (300).
cos. (t + p + p3) tan. t cos.p2
A system of three inclined planes, two of which are moveable, and the third fixed.
245. The inclined plane A, in the accompanying figure, is
u          fixed in position, the plane B is
\2          moveable upon A, having its upper
D.a3 surface inclined to the horizon at a
I,,-   less angle than the lower; and C is
C'~' T  an inclined plane resting upon B,
which is prevented from moving
{b  - ---  horizontally by the obstacle D, but
5  {^.~".     /      may be made to slide along this
"aa:"  -  obstacle vertically. It is required
FA%      to determine a relation between
P, and P2, applied, as shown in the figure, when the system
is in the state bordering upon motion.
Let R1, R2, R3 represent the resistances of the surfaces on
which motion takes place, 91 p2 93 their limiting angles of
resistance respectively, and t1, Lt the inclinations of the two
surfaces of contact of B to the horizon. Since Pi, R1, R2 are
pressures in equilibrium, as also P2, R2, R,
P,  sin. R2OR, R2   sin. P2QR3
R,- sin. POR1' P-    sin. R,QRE'
Multiplying these equations together,
P,   sin. ROR,. sin. PQR,
P2   sin. P1OR1. sin. R2QR3
Draw OS and OT parallel to the faces of the plane B; then
R2OR1=R1OS + QOT-TOS; but ROS-= - -p,, since OS is
parallel to the inferior face of the plane B, also QOT= — r
since OT is parallel to the superior face of the plane B; and




320      A SYSTEM OF TIHREE INCLINED PLANES.
TOS =the inclination of the faces of the plane B to one
another -=i- -2*E2~1OE  (2-  )+ (2- )-(-02,-)-(, +)-(-' ).
Also P2QR3= - — RQM= 2 - %p.
Let PO be produced toV; therefore P,OR,==r —ROV=
-r-(,ROS-SOV)=- - j- (   -    - )  -, = +,1. Lastly
RQR = OQM+MQR,. Now, MQR,=p,; also, OQM=.     ~-'~+~,+~,-  -(,~ —0r.  
r-QOV=vr-(QOT+TOV)=ir-       I (-9) +124 = U2 +P2
sin. -(+)sin. +', +pi). sin. - -(s2-2 -~p,). p=P_ sin. {(q +p2)+( (1-)} cos.. 3  (01 )
cos. (', + (,) cos. {,,-(2,+ )' 
Whence we obtain for the modulus (Art. 152.), observing
that     sin. (,1 —2)
COS. 1, Cos.'2
sin. (, ++,-t — ) cos. Cos. 2. o. (302).
1    (cos.(&-2- p,) cos.(t, +p,) sin. (,-L)s *** 




THE WEDGE DRIVEN BY PRESSURE.        321
THE WEDGE DRIVEN BY PRESSURE.
246. Let ACB represent an isosceles wedge, whose angle
ACB is represented by 2t, and which is
driven between the two resisting surfaces
j 1B  DE and DF, by the pressure P1. Let R1
and R2 represent the resistances of these
~f*s> ^       surfaces upon the acting surfaces CA and
C G —  OB of the wedge when it is upon the
\   point of moving forwards. Then are the
1\'J i   a    directions of R, and R2 inclined respectively to the perpendicular Gs and Rt
to the faces CA and CB of the wedge, at
angles each equal to the limiting angle of
resistance p. The pressures R, and R2 are
therefore equally inclined to the axis
of the wedge, and to the direction of P1, whence it follows
that R,=R,, and therefore (Art. 13.) that P =2R, cos. GOR.
Now, since CGOR is a quadrilateral figure, its four angles
are equal to two right angles; therefore GOR-=2 —GCROGC-ORO. ButGCR=2t; OGC-=ORC =            +2:.*.GOR=r-(2t+2q).iG=OR=      -(t+p).:.P,= 2R sin. (t+q).....(303).
Whence it follows (equation 121) that the modulus of the
wedge is
UU, I sin. (t+).(304).
sin. t
This equation may be placed under the form
U, = U {cot. + cot. t} sin. qp.
The work lost by reason of the friction of the wedge is
greater, therefore, as the angle of the wedge is less; and
infinite for a finite value of p, and an infinitely small value
of t.
The angle of the wedge.
247. Let the pressure P,, instead of being that just suffi21




322           THE WEDGE DRIVEN BY PRESSURE.
cient to drive the wedge, be now supposed
to be that which is only just sufficient to
*,i    _     keepit it in its place when driven. The two
surfaces of the wedge being, under these
0~ (A_,._ -_i circumstances, upon the point of sliding
s~ —— t-'0-^  -backwards upon those between which the
wedge is driven, at their points of contact
G and R, it is evident that the directions
of the resistances iG and i2R upon those
X       points, must be inclined to the normals
-   sG and tR at angles, each equal to the
limiting angle of resistance, but measured
on the sides of those normals opposite to
those on which the resistances RG and R2RI are applied.*
In order to adapt equation (303) to this case, we have
only then to give to p a negative value in that equation. It
will then become
PI -2R, sin. ( — ).....(305).
So long as t is greater than p, or the angle C of the wedge
greater than twice the limiting angle of resistance, P, is
positive; whence it follows that a certain pressure acting in
the direction in which the wedge is driven, and represented
in amount by the above formula, is, in this case, necessary
to keep the wedge from receding from    any position into
which it has been driven. So that if, in this case, the pressure P, be wholly removed, or if its value become less than
that represented by the above formula, then the wedge will
recede from any position into which it has been driven, or
it will be started.  If t be less than p, or the angle C of the
wedge less than twice the limiting angle of resistance, P1
will become negative; so that, in this case, a pressure, oppo —
site in direction to that by which the wedge has been driven,
will have become necessary to cause it to recede from the
position into which it has been driven; whence it follows,
that if the pressure P1 be now wholly removed, the wedge
will remain fixed in that position; and, moreover, that it
will still remain fixed, although a certain pressure be applied
to cause it to recede, provided that pressure do not exceed
the negative value of P, determined by the formula.
* This will at once be apparent, if we consider that the direction of the
resultant pressure upon the wedge at G must, in the one case, be such, that if
it acted alone, it would cause the surface of the wedge to slip downwards on
the surface of the mass at that point, and in the other case upwards; and that
the resistance of the mass is in each case opposite to this resultant pressure.




THE WEDGE DRIVEN BY IMPACT.                    323
It is this property of remaining fixed in any position into
which it is driven when the force which drives it is removed,
that characterises the wedge, and renders it superior to
every other implement driven by impact.
It is evidently, therefore, a principle in the formation of a
wedge to be thus used, that its angle should be less than
twice the limiting angle of resistance between the material
which forms its surface, and that of the mass into which it
is to be driven.
THE WEDGE DRIVEN BY IMPACT.
248. The wedge is usually driven by the impinging of a
heavy body with a greater or less velocity upon its back, in
the direction of its axis.    Let W    represent the weight of
such a body, and V its velocity, every element of it being
conceived   to move with      the same velocity.       The work
accumulated in this body, when it strikes the wedge, will
1W
then be represented (Art. 66.) by - -V2. Now the whole of
2g
this work is done by it upon the wedge, and by the wedge
upon the resistances of the surfaces opposed to its motion;
if the bodies are supposed to come to rest after the impact,
and if the influence of the elasticity and mutual compression
of the surfaces of the striking body and of the wedge are
neglected, and if no permanent compression of their surfaces
1 WV2
follows the impact.*. U.   ---
The influence of these elements on the result may be deduced from the
principles about to be laid down in the chapter upon impact. It results from
these, that if the surfaces of the impinging body and the back of the wedge,
by which the impact is given and received, be exceedingly hard, as compared
with the surfaces between which the wedge is driven, then the mutual pressure'of the impinging surfaces will be exceedingly great as compared with the
resistance opposed to the motion of the wedge. Now, this latter being
neglected, as compared with the former, the work received or gained by the
wedge from the impact of the hammer will be shown in the chapter upon
impact to be represented by ( +e)W12-, where W1 represents the
2g(W,+W,)2
weight of the hammer, W, the weight of the wedge, and e that measure of
the elasticity whose value is unity when the elasticity is perfect. Equating
this expression with the value of U, (equation 304), and neglecting the effect
of the elasticity and compression of the surfaces G and R, between which the
wedge. is driven, we shall obtain the approximation
(l+ —e)TWl2WV'   sin. L
2(Wl+2W)2 sin. (t+-)'




324          THE WEDGE DRIVEN BY IMPACT.
Substituting this value of U1 in equation 303, and solving in
respect to UT, we have
1 WV'   sin.          (
2U,      sin.(-).... (306);
2 g sin. (t +q~)
by which equation the work U2 yielded upon the resistances
opposed to the motion of the wedge by the impact of a given
weight W with a given velocity V is determined; or the
weight W necessary to yield a given amount of work when
moving with a given velocity; or, lastly, the velocity V with
which a body of given weight must impinge to yield a given
amount of work.
If the wedge, instead of being isosceles, be of the form of
a right angled triangle, as shown
I/    in the accompanying figure, the
is  "/       relation between the work U1 done
ES ^ l //^  upon its back, and that yielded
upon the resistances opposed to
B   nw~ —-— "" c g  its motion at either of its faces, is
II'A   f1   represented by equations (296);R  c -     and (297). Supposing therefore
r~2 ~     this wedge, like the former, to be
driven by impact, substituting as before for U1 its value
1W'
-  vT2 and solving in respect to U2, we have in the case in
2g
which the face AB of the wedge is its driving surface
1 WV' sin. t cos.,   (
"-2   g   sin.(t++) q7)
when the base BC of the wedge is its driving surface,
U   WV2 tan. tcos.(t+p,)        (30
2      g' sin. (t +  (p, + q,
From this expression it follows, that the useful work is the greatest, other
things being the same, when the weight of the wedge is equal to the weight
of the hammer, and when the striking surfaces are hard metals, so that the
value of e may approach the nearest possible to unity.




THE MEAN PRESSURE OF IMPACT.         325
So^       249. If the power of the wedge
*;-X      be applied by the intervention of
B   o- }-^ + -  aln inclined plane moveable in a
V —-. —:...s 3direction at right angles to the di" fee.          rection of the impact*, as shown in
+ e —— ^  X  the accompanying figure, then sub}  \",~. (   stituting for UT in equation (300)
half the vis viva of the impinging
-' ~   body, and solving, as before, in
respect to U,, we have
1Y WV2 cos. ( +1 + ) tan. cos..          (309).
2 g        sin. (t+,1+o2s) cos' 
P.
1,          If instead of the base of the
c  Qo-::' —  - A  plane being parallel to the direction of impact, it be inclined to
a —^-^;      it, as shown in the accompanying
l ^3../.r     figure, then, substituting as above
e|`~, ^    in equation 302, we have
1 WVY cos. (t2,- q-,) cos. (t1 + 1) sin. (t —t2)  (310)
2'sin. (qP,1+9P+1-t) cos. t, cos. t cos.s''
THE MEAN PRESSURE OF IMPACT.
250. It is evident from equations 306, 307, 308, that, since,
whatever may be the weight of the impinging body or the
velocity of the impact, a certain finite amount of work U2 is
yielded upon the resistances opposed to the motion of the
wedge; there is in every such case a certain mean resistance
R overcome through a certain space S, in the direction in
which that resistance acts, which resistance and space are
such, that
U,
RS=U,, and therefore R== —.
If therefore the space S be exceedingly small as compared
* This is the form under which the power of the wedge is applied for the
expressing of oil.




326                   THE SCREW.
with U,, there will be an exceedingly great resistance R
overcome by the impact through that small space, however
slight the impact. From this fact the enormous amount of
the resistances which the wedge, when struck by the hammer, is made to overcome, is accounted for.  The power of
thus subduing enormous resistances by impact is not however peculiar to the wedge, it is common to all implements
of impact, and belongs to its nature; its effects are rendered
permanent in the wedge by the property possessed by that
implement of retaining permanently any position into which
it is driven between two resisting surfaces, and thereby opposing itself effectually to the tendency of those surfaces, by
reason of their elasticity, to recover their original form and
position. It is equally true of any the slightest direct impact
of the hammer as of its impact applied through the wedge,
that it is sufficient to cause any finite resistance opposed to
it to yield through a certain finite space, however great that
resistance may be. The difference lies in this, that the surface yielding through this exceedingly small but finite space
under the blow of the hammer, immediately recovers itself
after the blow if the limits of elasticity be not passed;
whereas the space which the wedge is, by such an impact,
made to traverse, in the direction of its length, becomes a*
permanent separation.
THE SCREW.
251. Let the system of two moveable inclined planes re-.-_..-...__  presented in fig. p. 318. be formed of ex—. —-.. —.1 ceedingly thin and pliable laminue, and con-'...:  ceive one of them, A for instance, to be
I -...1 wound upon a convex cylindrical surface, as. 1         shown in the accompanying figure, and the
C,,!Q other, B, upon a concave cylindrical surface
"   having an equal diameter, and the same axis
with the other; then will the surfaces
EF and GH of these planes represent truly
the threads or helices of two screws, one of them of the form
called the male screw, and the other the female screw. Let
the helix EF be continued, so as to form more than one spire
or convolution of the thread; if, then, the cylinder which
carries this helix be made to revolve upon its axis by the
action of a pressure PI applied to its circumference, and the
cylinder which carries the helix Gti be prevented from re



THE SCREW.                   327
volving upon its axis by the opposition of an obstacle D,
which leaves that cylinder nevertheless free to move in a
direction parallel to its axis, it is evident that the helix EF
will be made to slide beneath GH, and the cylinder which
carries the latter helix to traverse longitudinally; moreover,
that the conditions of this mutual action of the helical surfaces EF and GH will be precisely analogous to those of the
surfaces of contact of the two moveable inclined planes discussed in Art. 244. So that the conditions of the equilibrium of the pressures P1 and P, in the state bordering upon
motion, and the modulus of the system, will be the same in
the one case as in the other; with this single exception, that
the resistance R2 of the mass on which the plane A rests (see
fig. p. 318.) is not, in the case of the screw, applied only to
the thin edge of the base of the lamina A, but to the whole
extremity of the solid cylinder on which it is fixed, or to a
circular projection from that extremity serving it as a pivot.
Now if, in equation 299, we assume (p2O, we shall obtain
that relation of the pressures P1 and P2 in their state bordering upon motion, which would obtain if there were no friction of the extremity of the cylinder on the mass on which it
rests; and observing that the pressure P2 is precisely that
by which the pivot at the extremity of the cylinder is pressed
upon this mass, and therefore the moment (see Art. 175,
equation 183) of the resistance to the rotation of the cylinder
produced by the friction of this pivot by 3P2p tan. 2,, where
p represents the radius of the pivot; observing, moreover,
that. the pressure which must be applied at the circumference of the cylinder to overcome this resistance, above that
which would be required to give motion to the screw if there
were no such friction, is represented by2 P P tan. p2, r being
taken to represent the radius of the cylinder, we obtain for
the entire value of the pressure P, in the state bordering
upon motion
sin. (t + q) cos.p, 2  p
P =P s ( ) +            - p    tan.,.
cos. (t +- cp91 ~) +32rt
The pressure P, has here been supposed to be applied to
turn the screw at its circumference; it is customary, however,
to apply it at some distance from its circumference by the
intervention of an arm. If a represent the length of such an




328                  THE SCREW.
arm, measuring from the axis of the cylinder, it is evident
that the pressure P, applied to the extremity of that arm,
would produce at the circumference of the cylinder a pressure
represented by PI, which expression being substituted for
P, in the preceding equation, and that equation solved in
respect to P1, we obtain finally for the relation between P,
and P2 in their state bordering upon motion,
/rY sin. (t p) Cos. 2p 2 p
a (  cos. (t +       2 + ()   3  tan.  (311)
If in like manner we assume in the modulus (equation 300)
92=0, and thus determine a relation between the work done
at the driving point and that yielded at the working point,
on the supposition that no work is expended on the friction
of the pivot; and if to the value of Ui thus obtained we add
the work expended upon the resistance of the pivot which is
shown (equation 184) to be represented at each revolution
by 4-pP2 tan. p1, and therefore during n revolutions by
4
-.rnpP2, we shall obtain the following general expression for
the modulus; the whole expenditure of work due to the
prejudicial resistances being taken into account.
U, -=-r U  sin. (t +,) cos. p  4  
U=U,              1 U1      — rnpP tan. (p,.
2 cos. (t-+ i1 +3) tan. + 3
Representing by X the common distance between the threads
of the screw, i. e. the space which the nut B is made to
traverse at each revolution of the screw; and observ4            4: U
ing that nXP-=U_, so that 3rnpP2 tan. p,=-r2 p tan. 92=
2n                                     3Xth  P  1r = 2 1  3  t  3 2
22w  p                               2wrr
-2-;..U  tan. p2, in which expression -= cot. t, we
obtain finally for the modulus of the screw.  sin. (t+1) cos P 32  tan. 2 cot t..... (312).
cos. (t+2+~P3) +3 r
It is evidently immaterial to the result at what distance
from the axis the obstacle D is opposed to the revolution of




APPLICATIONS OF THE SCREW.          329
that cylinder which carries the lamina B; since the amount
of that resistance does not enter into the result as expressed
in the above formula, but only its direction determined by
the angle (,, which angle depends upon the nature of the
resisting surfaces, and not upon the position of the resisting
point.
APPLICATIONS OF THE SCREW.
252. The accompanying figure represents an application
of the screw under the circumstances described in the preceding article, to the well known machine called the VICE.
AB is a solid cylinder carrying on its surface the thread of a
male screw, and within the piece CD is a hollow cylindrical
surface, carrying the corresponding thread of a female
screw; this female screw is prevented from revolving with
the male screw by a groove in the piece CD, which carries
it, and which is received into a corresponding projection EF
of the solid frame of the machine, serving it as a guide;
which guide nevertheless allows a longitudinal motion to
the piece CD. A projection from the frame of the instrument at B, met by a pivot at the extremity of the male
screw, opposes itself to the tendency of that screw to traverse in the direction of its length. The pressure P to be
overcome is applied between the jaws II and K of the vice,
and the driving pressure P, to an arm which carries round
with it the screw AB.
It is evident that, in the state bordering upon motion, the
resistance R upon the pivot at the extremity B of the screw
AB, resolved in a direction parallel to the length of that
screw, must be equal to the pressure P, (see Art. 16.); so
that if we imagine the piece CD to become fixed, and the




330           APPLICATIONS OF THE SCREW.
piece BM to become moveable, being prevented from revolving, as CD was, by the intervention of a groove and guide,
then might the instrument be applied to overcome any given
resistance R opposed to the motion of this piece CD by the
constant pressure of its pivot upon that piece.
The screw is applied under these circumstances in the
common screw press. The piece
c          AA, fixed to the solid frame of the
machine, contains a female screw
whose thread corresponds to that
of the male screw; this screw,
when made to turn by means of a.. B      handle fixed across it, presses by
S|jJ  ~   the intervention of a pivot B, at its
extremity, upon the surface of a
solid piece EF moveable vertically, but prevented from turning
/' —s  X  with the screw by grooves receiv\1. -        -I ing two vertical pieces, which
serve it as guides, and form parts
of the frame of the machine.
\i  The formulae determined in
-----       1       ^Art. 251. for the preceding case
of the application of the screw, obtain also in this case, if
we assume p3=-O. The loss of power due to the friction of
the piece EF upon its guides will, however, in this calculation, be neglected; that expenditure is in all cases exceedingly small, the pressure upon the guides, whence their
friction results, being itself but the result of the friction of
the pivot B upon its bearings; and the former friction being
therefore, in all cases, a quantity of two dimensions in
respect to the coefficient of friction.
If, instead of the lamina A (p. 326.) being fixed upon the
convex surface of a solid cylinder, and B upon the concave
surface of a hollow cylinder, the order be reversed, A being
fixed upon the hollow and B on the solid cylinder, it is evident that the conditions of the equilibrium will remain the
same, the male instead of the female screw being in this case
made to progress in the direction of its length. If; however,
the longitudinal motion of the male screw B (p. 326.) be,
under these circumstances, arrested, and that screw thus
become fixed, whilst the obstacle opposed to the longitudinal
motion of the female screw A is removed, and that screw
thus becomes free to revolve upon the male screw, and also
to traverse it longitudinally, except in as far as the latter




THE DIFFERENTIAL SCREW.          331
motion is opposed by a certain resistance
R, which the screw is intended, under
-- l —  ~ these circumstances, to overcome; then
will the combination assume the well
known form of the screw and nut.
-       =~:__ ______   To adapt the formulae of Art. 251. to' *-==     this case, %, must be, made =0, and
instead of assuming the friction upon the extremity of the
screw (equation 311) to be that of a solid pivot, we must
consider it as that of a hollow pivot, applying to it (by
exactly the same process as in Art. (251.), the formulae of
Art. (177.) instead of Art. (176.).
THE DIFFERENTIAL SCREW.
253. In the combination of three inclined planes discussed
in Art. 245., let the plane B be conceived of much greater
width than is given to it in the figure (p. 319.), and let it
then be conceived to be wrapped upon a convex cylindrical
surface. Its two edges ab and ed will thus become the
helices of two screws, having their threads of different inclinations wound round different portions of the same cylinder,
F
as represented in the accompanying figure, where the thread
of one screw is seen winding upon the surface of a solid
cylinder from A to C, and the thread of another, having a
different inclination, from D to B.
Let, moreover, the planes A and C (p. 319.) be imagined
to be wrapped round two hollow cylindrical surfaces, of
equal diameters with the above-mentioned solid cylinder,
and contained within the solid pieces E and F, through
which hollow cylinders AB passes. Two female screws will
thus be generated within the pieces E and F, the helix of




332                   HUNTER'S SCREW.
the one adapting itself to that of the male screw extending
from A to C, and the helix of the other to that upon the
male screw extending from D to B. If, then, the piece E
be conceived to be fixed, and the piece F moveable in the
direction of the length of the screw, but prevented from
turning with it by the intervention of a guide, and if a pressure Pi be applied at A to turn the screw AB, the action of
this combination will be precisely analogous to that of the
system of inclined planes discussed in Art. 245., and the
conditions of the equilibrium precisely the same; so that the
relation between the pressure P1 applied to turn the screw
(when estimated at the circumference of the thread) and that
P2, which it may be made to overcome, are determined by
equation (301), and its modulus by equation (302).
The invention of the differential screw has been claimed
by M. Prony, and by Mr. White of Manchester.        A comparatively small pressure may be made by means of it to
yield a pressure enormously greater in magnitude.*        It
admits of numerous applications, and, among the rest, of
that suggested in the preceding engraving.
HUNTER'S SCREW.
254. If we conceive the plane B (p. 319.) to be divided
by a horizontal line, and ti'e upper part
to be wrapped upon the inner or concave
surface of a hollow  cylinder, whilst the
lower part is wrapped upon the outer or
convex circumference of the same cylinder, thus generating the thread of a female screw within the cylinder, and a
male screw without it; and if the plane
C be then wrapped upon the convex surface of a solid cylinder just fitting the inside or concave surface of the above-mentioned hollow cylin* It will be seen by reference to equation (301), that the working pressure
Pa depends for its amount, not upon the actual inclinations t t2 of the threads,
but on the difference of their inclinations; so that its amount may be enormously increased by making the threads nearly of the same inclination. Thus,
neglecting friction, we have, by equation (301), P2P —P si. ( c-;  ich
expression becomes exceedingly great when  nerly equals  2
expression becomes exceedingly great when t, nearly equals t2.




VARIABLE INCLINATION OF THE THREAD.    333
der, and the plane A upon a concave cylindrical surface just
capable of receiving and adapting itself to the outside or
convex surface of that cylinder, the male screw thus generated adapting itself to the thread of the screw within the hollow cylinder, and the female screw to the thread of that
without it; if, moreover, the female screw last mentioned
be fixed, and the solid male screw be free to traverse in the
direction of its length, but be prevented turning upon its
axis by the intervention of a guide; if, lastly, a moving pressure or power be applied to turn the hollow screw, and a resistance be opposed to the longitudinal motion of the solid
screw which is received into it; then the combination will
be obtained, which is represented in the preceding engraving,
and which is well known as Mr. Hunter's screw, having been
first described by that gentleman in the seventeenth volume
of the Philosophical Transactions.
The theory of this screw is identical with that of the preceding, the relation of its driving and working pressures is
determined by equation (301), and its modulus by equation
(302).
THE THEORY OF THE SCREW WITH A SQUARE THREAD IN REFERENCE TO THE VARIABLE INCLINATION OF THE THREAD AT
DIFFERENT DISTANCES FROM THE AxIS.
255. In the preceding investigation, the inclined plane
which, being wound upon the cylinder, generates the thread
of the screw, has been imagined to be an exceedingly thin
sheet, on which hypothesis every point in the thread may be
conceived to be situated at the same distance from the axis
of the screw; and it is on this supposition that the relation
between the driving and working pressure in the screw and
its modulus have been determined.
Let us now consider the actual case in which the hread
Let us now consider the actual case in which the thread




334       VARIABLE INCLINATION OF TIHE THREAD.
of the screw is of finite thickness, and different elements of
it situated at different distance from its axis.
Let mb represent a portion of the square thread of a screw,
in which form of thread a line be, drawn from any point b on
the outer edge of the thread perpendicular to the axis ef,
touches the thread throughout its whole depth bd. Let AC
represent a plane perpendicular to its axis, and cf the projection of be upon this plane. Take p any point in bd, and
let q be the projection of p.  Let ep=r, mean radius of
thread -R, inclination of that helix of the thread whose
radius is R*=-I, inclination of the helix passing through p=-t
whole depth of thread =2D, distance between threads (or
pitch) of screw =L.  Now, since the helix passing through
p may be considered to be generated by the enwrapping of
an inclined plane whose inclination is t upon a cylinder
whose radius is r, the base of which inclined plane will then
become the arc tq, we havepq=-tq. tan. t. But, if the angle
Afa be increased to 2X:, pq will become equal to the common distance L between the threads of the screw, and tq
will become a complete circle, whose radius is r; therefore
L=27rr tan. t, and this being true for all values of r, therefore L-=2,R tan. I. Equating the second members of these
equations, and solving in respect to tan. t,
R tan. I
tan. = —...... (313).
r
From which expression it appears, that the inclination of the
thread of a square screw increases rapidly as we recede from
its edge and approach its axis, and would become a right
angle if the thread penetrated as far as the axis.  Considering, therefore, the thread of the screw as made up of an infinite number of helices, the modulus of each one of which
is determined by equation (312), in terms of its corresponding inclination l, it becomes a question of much practical importance to determine, if the screw act upon the resistance
at one point only of its thread, at what distance from its axis
that point should be situated, and if its pressure be applied
at allthe different points of the depth of its thread, as is
commonly the case, to determine how far the conditions of
its action are influenced by the different inclinations of the
thread at these different depths.
* This may be called the mean helix of the thread. The term helix is here
taken to represent any spiral line drawn upon the surface of the thread; the
distance of every point in which, from the axis: of the screw, is the same.




VARIABLE INCLINATION OF THE THREAD.    335
We shall omit the discussion of the former case, and proceed to the latter.
Let P2 represent the pressure parallel to its axis which is
to be overcome by the action of the screw. Now it is evident that the pressure thus produced upon the thread of the
screw is the same as though the whole central portion of it
within the thread were removed, or as though the whole
pressure P2 were applied to a ring whose thickness is As or
2D. Now the area of this ring is represented by  (R+D)2
-(R-D)2}, or by 4iRD. So that the pressure of P2, upon
P
every square unit of it, is represented by 4RD.  Let 
represent the exceedingly small thickness of such a ring
whose radius is r, and which may therefore be conceived to
represent the termination of the exceedingly thin cylindrical
surface passing through the point p; the area of this ring is
then represented by 2wrar, and therefore the pressure upon
i P2. 2Ar      P rAr
it by 24RD, or by 2PD.  Now this is evidently the
pressure sustained by that elementary portion of the thread
which passes through p, whose thickness is Ar, and which
may be conceived to be generated by the enwrapping of a
thin plane, whose inclination is l, upon a cylinder whose radius is r; whence it follows (by equation 311) that the elementary pressure AP,, which must be applied to the arm of
the screw to overcome this portion of the resistance P2, thus
applied parallel to the axis upon an element of the thread.
is represented by
P P    ir  \ ( sin. (, + q)) cos. qp,   -p,. 
l (2RD/ \a)                     r cos.('+,+cp+);
whence, passing to the limit and integrating, we have
R+D
_- _2       sin. ( + (,) cos.p q9,2~r tan.n 
2RDa       - cos.O ( +p)    +    tan.   dr.
R-D
Now
sin. (i + p) cos. p,      tan. i +tan. qt
cos. (,+-p+~) -1-tan. -, tan. (p —tan. i (tan. (p+tan. p,)......tan. 4 +tata n. +  -ta  n. (  )ta.-( - tan.,1 tan.  1)1- tan. stan. (             <.




836      THE SCREW WITH A TRIANGULLAR THREAD.
+ tan. (91 + 9) tan. 2t. Neglecting dimensions of tan. p, and
tan. p, above the first*,
R+D.*. P  RDa      (tan. p,+ tan. i+tan. (91+, ) tan. +)r+
R-D
fprtan.   dr...    (314).
Substituting in this expression for tan.  its value (equation
313), it becomes
R+D
P — 2Da        {f2 tan. 9, + Rr tan. I + R2 tan.2 I tan. (q, + q3) +
R-D
Mpr tan. 9(p dr.
Integrating and reducing,
p, P a     n.   I     - 1ta.  -       tan.I+l),+
tan. 2Itan. (qp+pq).*. * *.(315);
whence we obtain by (equation 121) for the modulus,
UT,= U    1+ l ( 1=~ )tan. ql, +i() tan. p2+
tan.'itan. (1+P) } cot.I....(316).
256. Whence it follows that the best inclination of the
thread, in respect to the economy of power in the use of
the square screw, is that which satisfies the equation
tan. I=.~ (I + i) tan.,l +-I() tan. i92i
tan. ((p, + p) 
The inclination of thread of a square screw rarely exceeds
7~, so that the term tan. 2I tan. ((pI+,) rarely exceeds'015
tan. (p, +q3), and may therefore be neglected, as compared
X The integration is readily effected without this omission; and it might be
desirable so to effect it where the theory of wooden screws is under discussion,
the limiting angle of resistance being, in respect to such screws, considerable.
The length and complication of the resulting expression has caused the omission of it in the text.




THE BEAM OF THE STEAM ENGINE.             337
with the other terms of the expression; as also may the
term ~ r      tan. 9,, since the depth 2D of a square screw
being usually made equal to about -th of the diameter, this
term does not commonly exceed T-3 tan. qp.
Omitting these terms, observing that L=2&R tan. I, and
eliminating tan. I,
P1=P,.. -    +R tan. 4+ iptan. %.... (317).
U1-=TJ   1 + L       n (R taan, +   ta. )....(318).
THE BEAM OF THE STEAM ENGINE.
257. Let P1, P,, P,, P4 represent the pressures applied by
the piston rod, the crank rod, the air pump rod, and the cold
U     P
1                            a
water pump rod, to the beam of a steam engine; and suppose the directions of all these pressures to be vertical.*
Let the rods, by which the pressures P,, P,, P,, P4 are
applied to the beam, be moveable upon solid axes or gudgeons, whose centres are a, d, b, e, situated in the same
straight line passing through the centre C of the solid axis
of the beam.
Let p,, p, P?, p4 represent the radii of these gudgeons, p the
radius of the axis of the beam, and p,,, p,, 9p4,  the limiting angles of resistance of these axes respectively. Then, if
the beam be supposed in the state bordering upon motion
A supposition which in no case deviates greatly from the truth, and any
error in which may be neglected, inasmuch as it can only influence the results
about to be obtained in as far as they have reference to the friction of the
beam; so that any error in the result must be of two dimensions, at least, in
respect to the coefficient of friction and the small angle by which any pressure
deviates from a vertical direction.
22




338          THE BEAM OF THE STEAM ENGINE.
by the preponderance of P1, each gudgeon or axis being
upon the point of turning on its bearings, the directions of
the pressures P,, P,, PS, P) R, will not be through the centres of their corresponding axes, but separated from them by
perpendicular distances severally represented by p, sin. (,, p,
sin. p,, p3 sin. pg, p4 sin. p4, and p sin. p, which distances, being
perpendicular to the directions of the pressures, are all
measured horizontally.
Moreover, it is evident that the direction of the pressure
P, is on that side of the centre a of its axis which is nearest
to the centre of the beam, since the influence of the friction
of the axis a is to diminish the effect of that pressure to turn
the beam. And for a like reason it is evident that the
directions of the pressures P2, P3, P4 are farther from the
centre of the beam than the centres of their several axes,
since the effect of the friction is, in respect to each of these
pressures, to increase the resistance which it opposes to the
rotation of the beam; moreover, that the resistance R upon
the axis of the beam has its direction upon the same side of
the centre C as P1, since it is equal and opposite to the
resultant pressure upon the beam, and that resultant would,
by itself, turn the beam in the same direction as P1 turns it.
Let now a-= Ca, a- -C, a3=Cb, a4=-Ce. Draw the horizontal line offCg, and let the angle aCf=8. Let, moreover,
W be taken to represent the weight of the beam, supposed
to act through the centre of its axis. Then since P1, P2, P1,
PO, W, R are pressures in equilibrium, we have, by the
principle of the equality of moments, taking o as the point
from which the moments are measured, P. of=P2. og+
P. oh+P,.ok+W. oC.
Now of=Cf-Co=ca, cos. 8-p1 sin. p —p sin. p, og=Cg+
Co=a cos. 0+p2 sin. q1~+p sin. p, a=O-C-Co=a,3 cos. 0-+
p, sin. p —p sin. p, ok-Ck+Co=a cos. 8+p4 sin. 4+p sin. ip... PI {a cos. -(p, sin. q1 + p sin. p)} =
P, ta2 cos. o + (p, sin. p, + p sin. p) +
P,{Iac cos. +(p, sin. p3-p sin. p) +.. (319).
P4 a4 cos. 6 + (p4 sin. p4 + p sin. p)1 + Wp sin. qp
Multiplying this equation by 0, observing that a1, represents the space described by the point of application of P1,
so that Pa16 represents the work U1 of Pi; and similarly
that P1a20 represents the work U, of P,, P~a.O, that UT of P,,:and P4av4, that U4 of P4, also that a1c represents the space S,




THE BEAM OF THE STEAM ENGINE.          339
described by the extremity of the piston rod very nearly;
we have
U2 cos.+(p    sin., +  -S- p sin  qA) +
p, sin. q — p sin. P t
u icos. aP3l            P —  +.. (320),
--' I?+w~( )sin.q
lU1 cos. + (p4 sin q4+psin 9) q  +Ws (tp) f.
which is the modulus of the beam.
Its form will be greatly simplified if we assume cos. 8=1,
since 0 is small * suppose the coefficient of friction at each
axis to be the same, so that  1==  2==,,=p4 and divide by
the coefficient of U,, omitting terms above the first dimension in P- sin. p, &c.; whence we obtain by reduction
+         + Ppt  2)sin.  +
l,              I   (   -+ P  P P)sinr. -       (321).
l  4 1  (    +   4   Sin.@ q  WS1 P)Sin. qJ
258. The best position of the axis of the beam.
Let a be taken to represent the length of the beam, and x
the distance aC of the centre of its axis from the extremity
to which the driving pressure is applied.,. P 
8  3A  A
~        n     V    pr      ihgW  4  2
* In practice the angle 6 never exceeds 200, so that cos. 6 never differs from
unity by more than'060307. The error, resulting from which difference, in
the friction, estimated as above, must in all cases be inconsiderable.




340         THE BEAM OF THE STEAM ENGINE.
Let the influence of the position of the axis on the
economy of the work necessary to open the valves, to work
the air-pump, and to overcome the friction produced by the
weight of the axis, be neglected; and let it be assumed to
be that, by which a given amount of work U2 may be
yielded per stroke upon the crank rod, by the least possible
amount U1 of work done upon the piston rod. If, then, in
equation (321), we assume the three last terms of the second
member to be represented by A, and observe that a, in that
equation is represented by a, and a, by a - x, we shall
obtain
U - { 1+ (PPi+P+     ) sin. } U2+A.
The best position of the axis is determined by that value
of x which renders this function a minimum; which value
of x is represented by the equation
= —(....            322.)
1 P+P.
If p,2>P1, then(P-P+P) >1 and x<ia; in this case, therefore, the axis is to be placed nearer to the driving than to
the working end of the beam. If p2<p1, the axis is to be
fixed nearer to the working than to the driving end of the
beam.
259. It has already been shown (Art. 168.), that a
machine working, like the beam of a steam engine, under
two given pressures about a fixed axis, is worked with the
greatest economy of power when both these pressures are
applied on the same side of the axis. This principle is
manifestly violated in the beam engine; it is observed in
the engine worked by Crowther's parallel motion,* and in
the marine engines recently introduced by Messrs. Seaward,
and known as the Gorgon engines. It is difficult indeed to
defend the use of the beam on any other legitimate ground
than this, that in some degree it aids the fly-wheel to
equalise the revolution of the crank arm,t an explanation
* As used in the mining districts of the north of England.
t The reader is referred to an admirable discussion of the equalising power
of the beam, by M. Coriolis, contained in the thirteenth volume of the Journal
de l'Ecole Polytechnique.




THE CRANK.                  341
which does not extend to its use in pumping engines,
where, nevertheless, it retains its place; adding to the
expense of construction, and, by its weight, greatly increasing the prejudicial resistances opposed to the motion of the
engine.
THE CRANK.
260. The modulus of the crank, the direction of the resistance being parallel to that of the driving pressures.
Let CD represent the arm of the crank, and AD the connecting rod. And to simplify the
A-'f'I            investigation, let the connecting
rod be supposed always to retain
p                   its vertical position.*  Suppose the
/ ]_=|              weight of the crank arm CD, acting through its centre of gravity,
w/ (?     to be resolved into two    other,                weights (Art. 16), one of which WV
I  ^""'-  is applied at the centre C of its axis
D:!,'  and the other at the centre c of:@  I  \  the axis which unites it with the
connecting   rod. Let this latter
Y      Q -^? — -t —^ i weight, when added to the weight
v\ i ]s           / of the connecting rod, be repre-,, / ^/, sented by W1. Let P2 represent a
-./ fpressure opposed to the revolution'-,s_.   //'f  of the crank, which would at any
instant be just sufficient to balance
the driving pressure P, transmitted through the connecting
rod; and to simplify the investigation, let us suppose the
direction of the pressure P2 to be vertical and downwards.
Let Cc=a, CA-,       CA2=2, cCW —=, radii of axes C
and c=p?, p,, lim. s of resistance =%,, 2, W=whole weight
of crank arm and connecting rod==W  +WW.
Since the crank arm is in the state bordering upon
motion, the perpendicular distance of the direction of the
resistance upon its axis C from the centre of that axis, is
* Any error resulting from this hypothesis affecting the conditions of the
question only in as far as the friction is concerned, and being of two dimensions at least in terms of the coefficient of friction and the small angular deviation of the connecting rod from the vertical.




3 42                 THE CRANK.
represented by p, sin. p1 (Art. 153.). The resistance is also
equal to P1 i (P, + W); P, being supposed greater than P2 + W,
and the sign ~ being taken
according as the direction of
P\ is downwards or upwards,
-Xg. \     or according as the crank arm,<.1.t..-.   is describing its descending or,, T   ascending  arc.  Whence it
/,! \ 1    follows, that the moment of
^       ] \ \\ the resistance of the axis about
32  _ --— I —  \. irits centre is represented by
i.-. —  d.P   1 ^P ~ (P,+ W )t   P1   sin.  pp1.
\ J.           i- [> Now the pressures P,, P,, and
the resistance of the axis, are
//p    pressures  in   equilibrium.'s...............  Therefore, by the principle of
the equality of moments, observing that the driving pressure is represented by P,~W,,
according as the arm is descending or ascending,
(P~Wl) a,=P~a2+ jP,~(P2+W)} p, sin. 91.
Since moreover the axis c, which unites the connecting
rod and the crank arm, is upon the point of turning upon
its bearings, the direction of the pressure Pi is not through
the centre of that axis, but distant from it by a quantity
represented by P2 sin. 2,, which distance is to be measured
on that side of the centre c which is nearest to C, since the
friction diminishes the effect of P1 to turn the crank arm.:. a a sin.. -p sin. 2..... (323).
Substituting this value of a, in the preceding equation,
(P, ~W) (a sin. -p,2 sin. 2)=P2a2+ Pi~(P,+
W)} p, sin. q9,..... (324).
Transposing and reducing
P1 a sin. O-p, sin.,- p, sin. q} =P2, a2~:p sin.,1} ~
Wp1 sin. pFTWa   sin. sin. -p, sin. %p);
which is the relation between P1 and P. in their state bordering upon motion. Now if AO represent an exceedingly
small angle described by the crank arm, c2AO will represent
the space through which the resistance P2 is overcome
whilst that angle is described, and P2acA6 will represent the




THE CRANK.                  343
increment AU2 of the work yielded by the crank whilst that
small angle is described. Multiplying the above equation
by a2A6, we have
Pa2 a sin. 6-p, sin. qp-p? sin. p1A-p=  a2~p, sin. sn   AU2
Wa21, sin. p1la:FWa2 (a sin. 8-p, sin. 2,)A8.... (325).
whence passing to the limit, integrating from -=0 to 8=
r —e, and dividing by a2
P, J2acos.0-(~-20)(psin.pq+plsin'.p)}- { l esiun. i 2
W (<r-20) p, sin. 9q1:FW  12a cos. o-p, (,r —20) sin. ip,.. (326).
Now, let it be observed that 2a cos. 0 represents the projection of the path of the point c upon the vertical direction
of P1, whilst the arm revolves between the positions 0 and
r —0; so that P12a cos. 0 represents (Art. 52.) the work
U1 done by P, upon the crank whilst the arm passes from
one of these positions to the other, or whilst the work U2 is
U,
yielded by the crank. Whence it follows that P,= 2- sec. 0.
Substituting this value of P1, dividing by a, and reducing,
we obtain
U, 1- ( —O)sec. 0 ( sin. p2+ 1 sin. 91 
i l —?sin.'   U,,~W  (~-20) p, sin. pFW, 122 cos. op2 (7r-2Q) sin. 91\..... (327).
By which equation is determined the modulus of the crank
in respect to the ascending or descending stroke, according
as we take the upper or lower signs of the ambiguous terms.
Adding these two values of the modulus together, and
representing by U, the whole work of P,, and by U2 the
whole work of P2, whilst the crank arm makes a complete
revolution, also by u, the work of P2 in the down stroke,
and 2 in the up stroke, we obtain
Us \ 1- f  ^ sec. ( E- sin. 92+- sin. )\ \ =U,+
i(.  \2        \a        a       fT
(u —Ut2) P sin. qp....  (328),
hich is the modulus of the crank in respect to a 
which is the modulus of the crank in respect to a vertical




344                   THE CRANK.
direction of the driving pressure and of the resistance, the
arm being supposed in each half revolution, first, to receive
the action of the driving pressure when at an inclination of
O to the vertical, and to yield it when it has again attained
the same inclination, so as to revolve under the action of
the driving pressure through the angle r-29.
In the double-acting engine, u1 —u2=0; in the single-acting engine u —O.  The work expended by reason of the
friction of the crank is therefore less in the latter engine than
in the former, when the resistance P2 is applied, as shown
in the figure, on the side of the ascending arc.
If the arm sustain the action of the driving pressure constantly, =0,- and the modulus becomes, for the double-acting engine,
or, dividing by the co-efficient of U, and neglecting dimensions above the first in sin. (,, sin. p,,
U- {1 + (sin. + p     s    in.2) } U2.... (329).
The modulus not involving the symbol W which represents the weight of the crank, it is evident that so long as P,
and P2 are vertical and P, greater than P + WV, the economy
of power in the use of the crank is not at all influenced by
its weight and that of the connecting rod, the friction being
upon the whole as much diminished by reason of that weight
in the ascending stroke as it is increased by it in the descending stroke.
It is evident, moreover, that if the friction produced by
the weight of the crank be neglected, the modulus above deduced, for the case in which the directions of the pressures
P, and P2 are vertical, applies to every case in which the
directions of those pressures are parallel.
The condition PI> P,+W evidently obtains in every other
position of the crank arm, if it obtain in the horizontal position.
Now, in this position, P,=-P,, if we neglect friction. The
reqired condition obtains, therefore, if.    To
required condition obtains, therefore, if P,> —P, +W.  To
a2
satisfy this condition, a, must be greater than a, or the
resistance be applied at a perpendicular distance from the




THE DEAD POINT IN THE CRANK.         345
axis greater than the length of crank arm, and so much
greater, that P  (1 —  may exceed V. These conditions
commonly obtain in the practical application of the crank.
261. Should it, however, be required to determine the modulus in the case in which P1 is not, in every position of the
arm, greater than P2+W, let it be observed, that this condition does not affect the determination of the modulus (equation 327) in respect to the descending, but only the ascending stroke; there being a certain position of the arm as it
ascends in which the resultant pressure upon the axis represented by the formula P1 —(P2 +W)}, passing through zero,
is afterwards represented by {(P2+W)-P1j; and when the
arm has still further ascended so as to be again inclined to the
vertical at the same angle, passes again through zero, and is
again represented by the same formula as before. The value
of this angle may be determined by substituting P, for
P~+W   in equation (324), and solving that equation in respect to 8. Let it be represented by 0; let equation (325)
be integrated in respect to the ascending stroke from 8=0
to 8=81, the work of P2 through this angle being represented
by u,; let the signs of all the terms involving p, sin. p1 then
be changed, which is equivalent to changing the formula representing the pressure upon the axis from  {P1-(P2+W)}
to {(P +W)-P1}; and let the equation then be integrated
from =1 to =-, the work of P2 through this angle being re2'
presented by u2; 2(u1 +u2) will then represent the whole
work 12 done by P, in the ascending arc. To determine
this sum, divide the first integral by the co-efficient of u,,
and the second by that of,, add the resulting equations,
and multiply their sum by 2; the modulus in respect to the
ascending arc will then be determined; and if it be added
to the modulus in respect to the descending arc, the modulus in respect to an entire revolution will be known.
THE DEAD POINTS IN THE CRANK.
362. If equation (324) be solved in respect to P1 it becomes




346                THE DOUBLE CRANK.
p _p i         a, l P1 sin. (p 
1   2 { asin. O-p, sin. 2, —p sin. p1 
WpI sin.g —W(a sin.8-p2 sin. (p)
a sin. O-p sin. sin -p sin. q1
In that position of the arm, therefore, in which
sin.= P2 sin. -2+p1 sin.    (330)
^     ^  si^.... (330),
the driving pressure P1 necessary to overcome any given resistance P2 opposed to the revolution of the crank, assumes
an infinite value. This position from which no finite pressure acting in the direction of the length of the connecting
rod is sufficient to move the arm, when it is at rest in that
position, is called its dead point.
Since there are four values of 6, which satisfy equation
(330) two in the descending and two in the ascending semirevolution of the arm, there are, on the whole, four dead
points of the crank." The value of P1 being, however, in all
cases exceedingly great between the two highest and the two
lowest of these positions, every position between the two
former and the two latter, and for some distance on either
side of these limits, is practically a dead point.
THE DOUBLE CRANK.
263. To this crank, when applied to the steam engine, are
affixed upon the same solid shaft, two arms at right angles
to one another, each of which sustains the pressure of the
steam in a separate cylinder of the engine, which pressure is
transmitted to it, from the piston rod, by the intervention of
a beam and connecting rod as in the marine engine, or a
guide and connecting rod as in the locomotive engine.
* It has been customary to reckon theoretically only two dead points of the
crank, one in its highest and the other in its lowest position. Every practical
man is acquainted with the fallacy of this conclusion.




THE DOUBLE CRANK.               347
Ail.,a ^In either case, the connecting rods
may be supposed to remain conFig. 1.  stantly parallel to themselves, and
the pressures applied to them  in
I 7. different planes to act in the same
i,  X 1|    plane,* without materially affecting
the results about to be deduced.t
II p      Let the two arms of the crank be.... _ --.. —  supposed to be of the same length a;
A\'v let the same driving pressure P1 be
i. \\  "   supposed to be applied to each; and:,1/ _ _ l  let the same notation be adopted in
L —-7-'-; \i )   other respects as was used in the. ye y, case of the crank with a single arm;
and, first, let us consider the case'r         j''/';  represented in fig. 1, in which both
"...       arms of the crank are upon the same
*&-...... side of the centre C.
Let the angle W CB=; therefore WCE-+6: whence
it follows by precisely the same reasoning as in Art. 260.,
that the perpendicular upon the direction of the driving
pressure applied by the connecting rod AB is represented
(see equation 323) by a sin. — p2 sin. p,, and the perpendicular upon the pressure applied by the rod CD by
a sin..( +  -p, sin.,, or a cos. O-p2 sin. 9.  Let now a,
be taken to represent the perpendicular distance from the
axis C, at which a single pressure, equal to 2P1, must be applied, so as to produce the same effect to turn the crank as
is produced by the two pressures actually applied to it by
the two connecting rods; then, by the principle of the equality of moments,
2P,a =Pl(a sin. -p sin.  +P,(a cos. 8-p2 sin. q,);
a. a=-a (sin  + c. )     n. + co.- sin;
* This principle will be more fully discussed by a reference to the theory of
statical couples. (See Pritchard on Statical Couples.)
t The relative dimensions of the crank arm and connecting rod are here supposed to be those usually given to these parts of the engine; the supposition
does not obtain in the case of a short connecting rod.




348               THE DOUBLE CRANK.
a al=   (sin. cos. 4 + cos. 8  sin. 4) -p sin. qaa<.   (    q\
a sin.    -J — P? sin. 7P;
/jsin. 0+     -,s.p
which expression becomes identical with the value of a,, determined by equation (323), if in the latter equation a be
replaced by a, and  by 6 +Z. Whence it follows that the
4/2           4
conditions of the equilibrium of the double crank in the
state bordering upon motion, and therefore the form of the
modulus, are, whilst both arms are on the same side of the
centre, precisely the same as those of the single crank, the
direction of whose arm bisects the right angle BCE, and
the length of whose arm equals the length of either arm of
the double crank divided by 4/2.
Now, if 01 be taken to represent the inclination W,CF of
this imaginary arm to W,C, both arms will be found on the
same side of the centre, from that position in which O,to that in which it equals (r -  ). If, therefore, we substitute - for e, in equations (326), and for a, a-, and add. these
equations together, the symbol 2 U2 in the resulting equation will represent the whole work yielded by the working
pressure, whilst both arms remain on the same side of the
centre, in the ascending and the descending arcs. We thus
obtain, representing the sum of the driving pressures upon
the two arms by P,,
2P, a -  (P2 sin.,+ p, sin.p) t=2U2.... (331).*
Throughout the remaining two quadrants of the revolution
of the crank, the directions of the two equal and parallel
pressures. applied to it through the connecting rods being
opposite, the resultant pressure upon the axis is represented
by (P2 + W), instead of I P, ~ (P, + W); whilst, in other
respects, the conditions of the equilibrium of the state bor
* Whewell's Mechanics, p. 26.




THE DOUBLE CoANK.             849
dering upon motion remain the same as before; that is, the
Fig. 2.     same as though the pressure P, were
applied to an imaginary arm, whose, -w               a
|1|  1I |)II  length is -I, and whose position coincides with CF. Now, referring to
--—.    equation (324), it is apparent that
- E  I,  this condition will be satisfied if, in
that equation, the ambiguous sign of
P'~ (P +W) be suppressed, and the
value of P, in the second member,
0:....-..... I which is multiplied by P1 sin. (p1, be.<    _ assumed =0; by which assumption
the term -p, sin. p, will be made to
B'~,~,  t  X disappear from the left-hand member
of equation (325), and the ambiguous
signs which affect the first and second
terms of the right-hand member will become positive. Now,
these substitutions being made, and the equation being then
integrated, first, between the limits 0 and _, and then be4
tween the limits 3 and or, the symbol U2 in it will evidently
4
represent the work done during each of those portions of a
semi-revolution of the imaginary arm in which the two real
arms of the crank are not on the same side of the centre.
Moreover, the integral of that equation between the limits 0
and 4 is evidently the same with its integral between the
3Ir
limits 4 and r. Taking, therefore, twice the former integral, we have
2Pa2         CO (1-cos.4) -- p2 sin. p  =  a2+ p1 sin. p,
2U +-2 Wc2p, sin.  2 T Wa2    - 1-( cos.: — 4 p sin. i.,,.,i ~t2           g        4)     2 sin. 92
Dividing  this equation  by (a +p,1 sin. %p), or by  a
1 +   sin. pi ), and neglecting terms above the first dimension in sin. q9 and sin. p,,




350               THE DOUBLE CRANK.
P A     (1-a-cos. 4 ) (i-  -- sin. 1 ) 4 P2 sin.P t ==2U, +?Wpsin. p T2W,   -  (1-cos.) (1 —  sin. )
^.     )
P2 sin. 92;
in which equation 2U, represents the work done in the
descending or ascending arcs of the imaginary arm, according as the ambiguous sign is taken positively or negatively.
Taking, therefore, the sum of the two values of the equation
given by the ambiguous sign, and representing by 4U, the
whole work done in the descending and ascending arcs, during those portions of each complete revolution when both of
the arms are not on the same side of the centre, we have
(  a             pi       err
4P    ~a (1-cos.) (     1 S-1       2 in. -  psin -P  =
4 T,2+W'p, sin.p 9;
* r   1
or, observing that cos. 4= -- 2
2P, a 4/2-1)-a( 4/2-1) - sin. q, —  p sin. 92  -
a,       2
4U 2+W7Wp, sin. p,.
Adding this equation to equation (331), and representing by
JU the entire work yielded during a complete revolution of
the imaginary arm,
2PI a 4/2 - a( 4/2 — 1)a asin. p, -2 (2p,2sin.p+ ) psin.,) 
=U2 + +Wp, sin. (,.
But if U, represent the whole work done by the driving
pressures at each revolution of the imaginary arm, then
4 /Pt-UI. Since 2 -    is the projection of the space
4/2 




THE CRANK GUIDE.               351
described by the extremity of the arm during the ascending
and descending strokes respectively, therefore 2P- =- -2a
Substituting this value for 2P1,
1 (,   4/2-1 P,           2P2      )
U 1 1 — 4/ --  ~ alsin. q< — -. —  sin.  )  -
i 4/2  2      2 42 \        /
U, + Wrp, sin. p.... (332),
which is the modulus of the double crank, the directions of
the driving pressure and the resistance being both supposed
vertical; or if the friction resulting from the weight of the
crank be neglected, and W be therefore assumed -0, then
does the above equation represent the modulus of the
double crank, whatever may be the direction of the driving
pressure, provided that the direction of the resistance be
parallel to it.  Dividing by the coefficient of TU, and
neglecting terms of more than one dimension in sin., and
sin. p,,
U, =   us   l+ ~1/ p      + sin.   - 1  2p2 sin. q~,  +
Pisin. p   + 2Wtp, sin. ).... (333).
THE CRANK GUIDE.
264. In some of the most important applications of the
steam engine, the crank is made to receive its continuous
rotatory motion, from the alternating rectilinear motion of
the piston rod, directly through the connecting rod of the
crank, without the intervention of the beam or parallel
motion; the connecting rod being in this case jointed at one
extremity, to the extremity of the piston rod, and the oblique
pressure upon it which results from this connexion being
sustained by the intervention of a cross piece fixed upon it,
and moving between lateral guides.*
* This contrivance is that well known as applied to the locomotive carriage.




352               THE CRANKE GUIDE.
I  -------------  _ 
Let the length CD of the connecting rod be represented
by b, and that BD of the crank arm by a, and let P, and P2
in the above figure be taken respectively, to represent the
pressure upon the piston rod of the engine and the connecting rod of the crank, and RS to represent the direction of
the resistance of the guide in the state bordering upon
motion by the excess of the driving pressure P1. Then is
RS inclined to a perpendicular to the direction of the guides,
or of the motion of the piston rod, at an angle equal to the
limiting angle of resistance (Art. 141) of the surfaces of contact of the guides.
Since, moreover, P., P2, R are pressures in equilibrium,
P,   sin.P,CS
P, - sin. P2CS
Let LBCID=; limiting angle of resistance of guide =qp;
therefore, PCS= —, PCS=- +q-;
2           2. ~in..P,     1 7    /~i   cos. (O -q(
et     n-a C-   b, and  B       and assume P2 to
Let BD =a, CD= b, and DBO==6, and assume P. to
remain constant, Pi being made to vary according to the
conditions of the state bordering upon motion;.~. aUl=P,. AAC= -P,. ABC= -P1. A (a cos., + b cos. O)=
P, sec. p cos. (O —q) (a sin. 0BA01 +b sin. OAO);
all, = -P, (AAC) cos. 8 =P, (a sin. 0BaB, + b sin. 6A0) cos. 0;
7rTT~~         0.*,=Pasec. pa  f sin. cos.(o-p)d  +  sin. cos.(-p)d.
O                   0




THE FLY-WHEEL.                 353
7r                0
U2==P2 a Ssin. 6 cos. ed, + bSsin. cos. da0.
O                 o
The second integral in each of these equations vanishes
between the prescribed limits; also sin. 6 =-  sin. 08; therefore cos. = (1- - sin. 2);
7T r7r
J,=P2as.. 0, cos.  (-)Pa,=(1 - sin. 2) sino. 101=
o                   0
/r
Pa tan.' sin. 0 sin. 0,cd,=U2+ P~ ~a tan. q0ffin. 201 d&U2 +P,   tan. g;
whence eliminating P, and reducing, we obtain
-     2-(2-1) ~og'E (b-a) }..
a          -1). bwhich is the modulus of the crank guide.
THE FLY-WHEEL.
265. The angular elocity o the y-wheel.
Let P, be taken to represent a constant pressure applied
through the connecting rod to the arm of the crank of a
whChurch's Diff. and Int. al. Art. 199.
23




9.54                THE FLY-WHEEL.
G          i
steam engine; suppose the direction of this pressure to
remain always parallel to itself, and let P2 represent a constant resistance opposed to the revolution of the axis which
carries the fly-wheel, by the useful work done and the prejudicial resistances interposed between the axis of the
fly-wheel and the working points of the machine.
Let the angle ACB=-, CB=a, CP=a,.
Now the projection, upon the direction of P1, of the path
of its point of application B to the crank arm, whilst that
arm describes the angle ACB, is AM, therefore (Art. 52.),
the work done by P, upon the crank, whilst this angle is
described, is represented by P1. AM, or by P, a vers. 6.
And whilst the crank arm revolves through the angle 0, the
resistance P2 is overcome through the arc of a circle subtended by the same angle 8, but whose radius is a, or
through a space represented by a2c. So that, neglecting the
friction of the crank itself, the work expended upon the
resistances opposed to its motion is represented by P2aO, and
the excess of the work done upon it through the angle ACB
by the moving power, over that expended during the same
period upon the resistances, is represented by
Pa vers. — P 2a2....  (336).
Now 2aP1 represents the work done by the moving pressure
P2 during each effective stroke of the piston, and 2ra2,P2 the
work expended upon the resistance during each revolution
of the fly-wheel; so that if m represent the number of
strokes made by the piston whilst the fly-wheel makes one




THE FLY-WHEEL..355
revolution, and if the engine be conceived to have attained
its state of uniform or steady action (Art. 146.), then
2maP =2ra2P,,
~2P2=    P.......(337).
Eliminating from equation (336) the value of aP2 determined by this equation, we obtain for the excess of the work
done by the power (whilst the angle 0 is described by the
crank arm), over that expended upon the resistance, the
expression
Pa i vers. e-.. (338).
But this excess is equal to the whole work which has been
accumulating in the different moving parts of the machine,
whilst the angle 0 is described by the arm of the crank (Art.
145). Now, let the whole of this work be conceived to have
been accumulated in the fly-wheel, that wheel being proposed to be constructed of such dimensions as sufficiently to
equalise the motion, even if no work accumulated at the
same time in other portions of the machinery (see Art. 150.),
or if the weights of the other moving elements, or their
velocities, were comparatively so small as to cause the work
accumulated in them to be exceedingly small as compared
with the work accumulated during the same period in the
fly-wheel. Now, if I represent the moment of inertia of the
fly-wheel, p, the weight of a cubic foot of its material, oa its
angular velocity when the crank arm was in the position
CA, and a its angular velocity when the crank arm has
passed into the position CB; then will I- (a2a — 2) represent
the work accumulated in it (Art. 75.) between these two
positions of the crank arm, so that
I (a-,2  )=P-  { vers. 0- m
2a,       I -f { vers....  (339).
266. The positions of greatest and least angular velocity of
the fly-wheel.
If we conceive the engine to have acquired its state of
steady or uniform motion, the aggregate work done by the




356                THE FLY-WHEEL.
/ 
power being equal to that expended upon the resistances,
then will the angular velocity of the fly-wheel return to the
same value whenever the wheel returns to the same position;
so that the value of al in equation (339) is a constant, and
the value of a a function of 8; a assumes, therefore, its minimum and maximum values with this function of 0, or it is a
dz2        2a 2
minimum when - = 0, and d >> 0, and a maximum when
da 2      dg2a        dad        m      d2 2
-=0, and -o<0. But — sin. —, and — = cos. 8,
therefore d- =0, when
sin... (340.)
Now this equation is evidently satisfied by two values of
0, one of which is the supplement of the other, so that if r
represent the one, then will (~r-i) represent the other;
which two values of 0 give opposite signs to the value cos.
O of the second differential co-efficient of a2, the one being
positive or >0, and the latter negative or <0. The one
value corresponds, therefore, to a minimum and the other
to a maximum value of a. If, then, we take the angle ACB
in the preceding figure, such that its sine may equal m
(equation 340), then will the position CB of the crank arm
be that which corresponds to the minimum angular velocity




THE FLY-WHEEL.                357
of the fly-wheel; and if we make the angle ACE equal to
the supplement of ACB, then is CE the position of the
crank arm, which corresponds to the maximum angular
velocity of the fly-wheel.
267. The greatest variation of the angular velocity of the
fly-wheel.
Let 2 be taken to represent the least angular velocity of
the fly-wheel, corresponding to the position CB of the crank
arm, and cc its greatest angular velocity, corresponding to
pI'
the position CE; then does - (a2 —22) represent the work
accumulated in the fly-wheel between these positions, which
accumulated work is equal to the excess of that done by the
power over that expended upon the resistances whilst the
crank arm revolves from the one position into the other,
and is therefore represented by the difference of the values
given to the formula (338) when the two values r —' and
a, determined by equation (340), are substituted in it for 0.
Now this difference is represented by the formula
P,a { vers. (r —) —vers. X —-- ( - d-  > 
Ir
or by P1a{2 cos. m-m(l -   )i;
('(2-p2)Pa { 2 cos. -)m(1 - 2
2    2 2 —   Ia 2 cos.  - (1-)..... (341);
in which equation i is taken (equation 340) to represent
that angle whose sine is-.
268. The dimensions of the fly-wheel, such that its angular
velocity nay at no period of a revolution deviate beyond
prescribed limitsfrom the mean.
Let IN be taken to represent the mean number of revolutions made by the fly-wheel per minute; then will Ji6




358                THE FLY-WHEEL.
represent the mean number of revolutions or parts of a
N        kN
revolution made by it per second, and 1-02., or -60, the
mean space described per second by a point in the fly-wheel
whose distance from the centre is unity, or the mean angular
velocity of the fly-wheel. Now, let the dimensions of the
fly-wheel be supposed to be such as are sufficient to cause
its angular velocity to deviate at no period of its revolution
by more than -th from its mean value; or such that the maximum value a, of its angular velocity may equal, (1 +  )
and that its minimum value a may equal X-  1 -  ); then
2 N-  2nN  IrN 2
(a3-   )=6(^+~)0'6.                sow-  3 ro
Substituting in equation (341)5
q72WN  2P.ga                2\'-_02n  P-J  2 cos. n  I-m  1 — -a 
Let H be taken to represent the horses' power of the
engine, estimated at its driving point or piston; then will
33000H represent the number of units of work done per
minute, upon the piston. But this number of units of work
is also represented by -NNm. 2Pa; since TNmt is the number
of strokes made by the piston per minute, and 2P1a is the
work done on the piston per stroke,. 2P-,o=6600o
Substituting this value for 2PEa in the above equation, we
obtain, by reduction,
i 66000.302g)         l —      2\ ) Hin
PI -    -2 a- 2    2 cos. I-m1l-    )'N... (342).
Let k7 be taken to represent the radius of gyration of the
wheel, and M its volume; then (Art. 80.) [2] -I, therefore
PM. k2 —=f. But FM represents the weight of the wheel
in lbs.; let W  represent its weight in tons; therefore,
tM=-2240W.    Substituting this value, and solving in
respect to W,




THE FLY-WHEEL.                   359
W_ t 66000.30~.g             --       2_ -\)  H n
Substituting their values for r and g, and determining the
numerical value of the co-efficient,
W=86491 { - cos. - (          ) }.-.  (343).
If the influence of the work accumulated in the arms of
the wheel be given in, for an increase of the equalising
power beyond the prescribed limits, that accumulated in the
heavy rim or ring which forms its periphery being alone
taken into the account;* then (Art. 86.) Mk2=I=2qrcbeR
(R2 +c2), where b represents the thickness, c the depth, and
R  the mean radius of the rim. But by Guldinus's first
property (Art. 38.), 2rbeR=M; therefore k2=(R2 +Ic).
ubstituting in equation (343)
W=86491     -- cos. -- (1 -...3 i)        344).
If the depth G of the rim be (as it usually is) small as
compared with the mean radius of the wheel, kc2 may be
neglected as compared with R2, the above equation then
becomes
W=86491        cos.           ) } N         (345);
by which equation the weight W   in tons of a fly-wheel of a
given mean radius R is determined, so that being applied to
an engine of a given horse power H, making a given number of revolutions per minute IN, it shall cause the angular
velocity of that wheel not to vary by more than -th from its
mean value. It is to be observed that the weight of the
wheel varies inversely as the cube of the number of strokes
made by the engine per minute, so that an engine making
twice as many strokes as another of equal horse power,
* If the section of each arm be supposed uniform and represented by ec, and
the arms be p in number, it is easily shown from Arts. 79., 81., that the
momentum of inertia of each arm about its extremity is very nearly represented by j-(R- c)3, where c represents the depth of the rim; so that the
whole momentum of inertia of the arms is represented by Pi(R- -c)3,which
expression must be added to the momentum of the rim to determine the whole
momentum I of the wheel. It appears, however, expedient to give the inertia
of the arms to the equalising power of the wheel




360                THE FLY-WHEEL.
would receive an equal steadiness of motion from a flywheel of one eighth the weight; the mean radii of the
wheels being the same.
If, in equation (342), we substitute for I its value 2<rbcR3,
or 2&KR' (representing by K the section be of the rim), and
if we suppose the wheel to be formed of cast iron of mean
quality, the weight of each cubic foot of which may be
assumed to be 450 lb., we shall obtain by reduction
ER3=68521{ — cos. — 1- -  ) N....(346);
by which equation is determined the mean radius R of a flywheel of cast iron of a given section K, which being applied
to an engine of given horse power H, making a given number of revolutions ~N per minute, shall cause its angular
velocity not to deviate more than -th from the mean; or
conversely, the mean radius being given, the section K may
be determined according to these conditions.
269. In the above equations, m is taken to represent the
number of effective strokes made by the piston of the engine
whilst the fly-wheel makes one revolution, and i to represent
that angle whose sine is 
Let now the axis of the fly-wheel be supposed to be a
continuation of the axis of the crank, so that both turn with
the same angular velocity, as is usually the case; and let its
application to the single-acting engine, the double-acting
engine, and to the double crank engine, be considered separately.
1. In the single-acting engine, but one effective stroke of
the piston is made whilst the fly-wheel makes each revolution.
In this case, therefore, m=l, and sin. i= — =0-3183098=
a   180 33'
r
sin. 18~ 33'; therefore, cos. i =*9480460, also  -  180==
~103055; therefore, 1 - = -793888.
— 2 cos. m-  1 -- I  1-102203.
m                Iff




THE FLY-WHEEL.                361
Substituting in equations (345) and (346),
Hn
W=95330-64 NW
42 27 Vln~  Kn=Th52  fin
R.N           K=755... (347);
by which equations are determined, according to the proposed'conditions, the weight W in tons of a fly-wheel for a
single-acting engine, its mean radius in feet R being given,
and its material being any whatever; and also its mean
radius R in feet, its section (in square feet) K being given,
and its material being cast iron of mean quality; and lastly,
the section K of its rim in square feet, its mean radius X
being given, and its material being, as before, cast iron.
2. In the double-acting engine, two effective strokes are
made per minute by the piston, whilst the fly-wheel makes
2
one revolution. In this case, therefore, m=-2 and sin. n =
0-636619= sin. 39~ 32'; therefore, cos.  =-'7712549 -
39-0 -21963; therefore (1 - -    — 56074;
S2     X        2 ) \ _
[-cos. 4u    -(oi)   =21051.
Substituting in equations (345) and (346),
fIn
W= 18207N H
-   24-3593 V/Hn           Hnt
24=  9 /H      K=14424 N3  -...(348);
by which equations the weight of the fly-wheel in tons, the
mean radius in feet, and the section of the rim in square
feet are determined for the double-acting engine.
3. In the engine working with two cylinders and a double
crank, it has been shown (Art. 263.) that the conditions of
the working of the two arms of a double crank are precisely
the same as though the aggregate pressure 2P1 upon their
extremities, were applied to the axis of the crank by the
intervention of a single arm and a single connecting rod;




362         THE FRICTION OF TIHE FLY-WHEEL.
the length of this arm being represented by - instead of a,
and its direction equally dividing the inclination of the arms
of the double crank to one another.
Now, equations (345) and (346) show the proper dimensions of the fly-wheel to be wholly independent of the
length of the crank arm; whence it follows that the dimension of a fly-wheel applicable to the double as well as a
single crank, are determined by those equations as applied
to the case of a double-acting engine, the pressure upon
whose piston rod is represented by 2P1. But in assuming
~Nm. 2Pa=33000H, we have assumed the pressure upon
the piston rod to be represented by PI; to correct this error,
and to adapt equations (345) and (346) to the case of a
double crank engine, we must therefore substitute ~H for H
in those equations. We shall thus obtain
un
W=9103 -5.
19.3339Vllm       K       lln
N             K' K=7212.....(349);
by which equations the dimensions of a fly-wheel necessary
to give the required steadiness of motion to a double crank
engine are determined under the proposed conditions.
THE FRICTION OF THE FLY-WHEEL.
270. W representing the weight of the wheel and (p the
limiting angle of resistance between the surface of its axis
and that of its bearings, tan. p will represent its coefficient
of friction (Art. 138.), and W tan. p, the resistance opposed
to its revolution by friction at the surface of its axis. Now,
whilst the wheel makes one revolution, this resistance is
overcome through a space equal to the circumference of the
axis, and represented by 2rp, if p be taken to represent the
radius of the axis. The work expended upon the friction of
the axis, during each complete revolution of the wheel, is
therefore represented by 2'pW tan. p; and if N represent
the number of strokes made by the engine per minute, and
thereforeN the number of revolutions made by the fly-wheel
2




MODULUS OF THE CRANK AND FLY-WHEEL.     863
per minute, then will the number of units of work expended
per minute, upon the friction of the axis be represented by
NrpW tan. p; and the number of horses' power, or the fractional part of a horse's power thus expended, by
NW~rp tan.       (350).
33000
If in this equation we substitute for W the weight in lbs.
of the fly-wheel necessary to establish a given degree of
steadiness in the engine, as determined by equations (347),
(348), and (349), we shall obtain for the horse power lost by
friction of the fly-wheel, in the single-acting engine, the
double-acting engine, and the double crank engine, respectively, the formule
lIHnp tan. qp
20329  N21R
8821Hnp tan.p 19412Hnp tan. q 
N38825  1941R25...    (351)
THE MODULUS OF THE CRANK AND FLY-WHEEL.
271. If S1 represent the space traversed by the piston of
the engine in any given time, and a the radius of the crank,
W the weight of the fly-wheel in lbs., and p the radius of its
axis, then will 2a represent the length of each stroke, - the
2Ia
number of strokes made in that time, and 2~rpW tan..,SI
2a'
or W S, P tan. q the work expended upon the friction of the
a
fly-wheel during that time, which expression being added to
the equation (329) representing the work necessary to cause
the crank to yield a given amount of work U2 to the machine driven by it (independently of the work expended on
the friction of the fly-wheel), will give the whole amount of
work which must be done upon the combination of the crank
and fly-wheel, to cause this given amount of work to be
yielded by it, on the machine which the crank drives. Let
this amount of work be represented by U1, then in the case
in which the directions of the driving pressure and the reAistance upon the crank are parallel (equation (329), and the




364                THE GOVERNOR,
friction of the crane guide is neglected, we obtain for the
modulus of the crank and fly-wheel in the double-acting
engine
U= i1 i+   (PI sin. q + P2 sin. 2 } U, + WS, P tan. q (352).
2\a       a                  a
THE GOVERNOR.
272. This instrument is represented in the figure, under
that form in which it is most commonly applied to the steam
c   ~   engine. BD and CE are rods jointed
s1      at A upon the vertical spindle AF,
i,' ~and at D and E upon the rods DP
and EP, which last are again jointed
at their extremities to a collar fitted
Ed -D  ~accurately to the surface of the spin/:,"aJ    dle and moveable upon it. At their
-- C^Sqi      extremities B and C, the rods DB
D  ^    and EC carry two heavy balls, and
being swept round by the spindle —
- _....... which receives a rapid rotation always proportional to the speed of the
machine, whose motion the governor
E —.:  is intended to regulate-these arms
]i  ~   by their own centrifugal force, and
that of the balls, are made to separate, and thereby to cause
the collar at P to descend upon the spindle, carrying with it,
by the intervention of the shoulder, the extremity of a lever,
whose motion controls the access of the moving power to
the driving point of the machine, closing the throttle valve
and shutting off the steam from the steam engine, or closing
the sluice and thus diminishing the supply of water to the
water-wheel. Let P be taken to represent the pressure of
the extremity of the lever upon the collar, Q the strain
thereby produced upon each of the rods DP and EP in the
direction of its length, W the weight of each of the balls, w
the weight of each of the rods BD and CE, AB==a, AD=b,
DP=c, FAB=-, APD=-,. Now upon either of these rods
as BD, the following pressures are applied: the weight of
the ball and the weight of the rod acting vertically, the
centrifugal force of the ball and the centrifugal force of the
rod acting horizontally, the strain Q of the rod DP, and
the resistance of the axis A. If a represent the angular




TIHE GOVERNOR.               365
velocity of the spindle, -     2. FB, or-'a sin. I, will reprey           g
sent the centrifugal force upon the ball (equation 102),
and  -W ca sin. 0 cos. 0 its moment about the point A; also
g
the centrifugal force of the rod BD produces the same effect
as though its weight were collected in its centre of gravity
(Art. 124.), whose distance from A is represented by 1(a-b),
so that the centrifugal force of the rod is represented by
i-2(a —b) sin. 8, and its moment about the point A by:ia2(a-b) sin. 8 cos. 6. On the whole, therefore, the sum of
the moments of the centrifugal forces of the rod and ball are
represented by a {Wa2+t-w(a-b)'} sin. 0 cos. 0. Now if t
represent the weight of each unit in the length of the rod,
w = p(a + b); therefore Wa' + 1w(a - b)2 = Wa'2 +it(a' -  )
(a-b). Let this quantity be represented by Wla',..WI=W+ith( -      2) (a-b).... (353);
t2
then will - W a sin. 6 cos. 0 represent the sum of the moments
g
of the centrifugal forces of the rod and ball about A. Moreover, the sum of the moments of the weights of the rod and
ball, about the same point, is evidently represented by Wa
sin. 6+w{(a — b) sin. 0, or by {Wa+v(a- b —)} sin. d; let this
quantity be represented by W. a sin. 0,
-W,=W+ ia(l -      )... (354).
Also the moment of Q about A=Q. AII=Qb sin. (8+8,).
Therefore, by the principle of the equality of moments, observing that the centrifugal force of the rod and ball tend to
communicate motion in an opposite direction from their
weights and the pressure Q,
-    Wla2 sin. 0 cos. 0=Qb sin. (+~,)+W,a sin. 0.




a3T                THE GOVERNOR.
Now P is the resultant of the pressures Q acting in the
directions of the rods PD and PE, and inclined to one
another at the angle 280; therefore (equation 13),
P=2Q cos.;.Q sin. ( +,) -P sin. ( + 0)P {sin. O+i cos. 0 tan.,l.
COS. 01
But since the sides b and c of the triangle APD are opposite to the angles 8, and 6, therefore sin. =, b; therefore
sin.   c
cos. 8,=  (1- - sin. da );:-. Q sin. (d+ ) -  P I sin(  { +i n sin. 0 cos. 0 (1  -sin.2) t.
Substituting this value in the preceding equation, dividing
by sin. 0, and writing (1-cos. 20) for sin. 28, we obtain
a- W12 cos. 8=         - l+ bcos. 1 — 
2             W  a..2. ( 35
2 cos.^   1+Wa... (355);
which equation, of four dimensions-in terms of cos. 8, being
solved in respect to that variable, determines the inclination
of the arms under a given angular velocity of the spindle.
It is, however, more commonly the case that the inclination
of the arms is given, and that the lengths of the arms,
or the weights of the balls, are required to be determined,
so that this inclination may, under the proposed conditions,
be attained. In this case the values of W, and W2 must be
substituted in the above equation from equations (353) and
(354), and that equation solved in respect to a or W.
The values of b and c are determined by the position on
the spindle, to which it is proposed to make the collar
descend, at the given inclination of the arms or value of 0.
If the distance AP, of this position of the collar from A, be
represented by h, we have h=b cos. 0 + c cos. 8,,.: =b cos. o+c 1 -   sin.28.... (356);
6 2    




THE GOVERNOR.                367
brom which equation and the preceding, the value of one
of the quantities b or c may be determined, according to the
proposed conditions, the value of the other being assumed to
be any whatever.
If N represent the number of revolutions, or parts of a
revolution, made per second by the fly-wheel, and yN the
number of revolutions made in the same time by the spindle
of the governor, then will 27yN represent the space a described per second by a point, situated at distance unity from
the axis of the spindle.  Substituting this value for c in
equation (355), and assuming b=c, we obtain
272N2aW dcos. 0=Pb +W.... (357):
y
also by equation (356),
A =2b cos.......(358).
Eliminating cos. 0 between these equations, and solving in
respect to A,
bg(Pb + W^a)  1
A-  2yr2a2W.... (359);
Let P (1+~) and P (1 —) represent the values of P
corresponding to the two states bordering upon motion
(Art. 140) and let N (1 + ) and N (1- -) be the corresponding values of N; so that the variation either way of -th from
the mean number N of revolutions, may be upon the point
of causing the valve to move. If these values be respectively
substituted for P and N in the above formula, it is evident
that the corresponding values of h will be equal. Equating
those values of h and reducing, we obtain
Wa     1  (1..
Pb    2rn    no 
By which equation there is established that relation between
the quantities W1, a, P, m which must obtain, in order that a
variation of the number of revolutions, ever so little greater




368              THE CARRIAGE-WHEEL.
than the'th part, may cause the valve to move. Neglect1
ing - as small when compared with n.
n=2m (l+    p );
which expression, representing that fractional variation in the
number of revolutions which is sufficient to give motion to
the valve, is the true measure of the SENSIBILITY of the
governor.
273. The joints E and D are sometimes
fixed upon the arms AB and AC as in the
accompanying figure, instead of upon the
prolongations of those arms as in the preE  ^w    ceding figure. All the formulae of the
last Article evidently adapt themselves
\\   y A    ~to this case, if b be assumed =0 (in equations 353, 354). The centrifugal force of
the rods EP and DP is neglected in this
computation.
THE CARRIAGE-WHEEL.
274. Whatever be the nature of the resistance opposed to
the motion of a carriage-wheel, it is evidently equivalent to
that of an obstacle, real or imaginary, which the wheel may
be supposed, at every instant, to be in the act of surmounting. Indeed it is certain, that, however yielding may be the
material of the road, yet by reason of its compression before
the wheel, such an immoveable obstacle, of exceedingly small
height, is continually in the act of being presented to it.
275. The two-wheeled carriage.
Let AB represent one of the wheels of a two-wheeled
carriage, EF an inclined plane, which it is in the act of ascending, O a solid elevation of the surface of the plane, or an
obstacle which it is at any instant in the act of passing over,




THE CARRIAGE-WHEEL.              369
P the corresponding tracA.q^     ~     ~tion, W  the weight of the
wheel and of the load which
J,. f  it supports.
Now the surface of the box
a../ ~ of the wheel being in the
state bordering upon motion' /   t /f' B   on the surface of the axle,
the direction of the resist-'-x \ A.:^    \   ance of the one upon the
other is inclined at the limit^ — o  | d/,/  ing angle of resistance, to a
radius of the axle at their
All  ~   point of contact (Art. 141.).
i \     This resistance has, more5r  \s    over, its direction through
"?   the point of contact 0 of
the tire of the wheel with the obstacle on which it is in the
act of turning. If, therefore, OR be drawn intersecting the
circumference of the axis in a point c, such that the angle
CcR may equal the limiting angle of resistance 9, then will
its direction be that of the resistance of the obstacle upon
the wheel.
Draw the vertical GH representing the weight W, and
through H draw HK parallel to OR, then will this line
represent (to the same scale) the resistance R, and GK the
traction P (Art. 14.);
P   GK   sin. GHK       sin. GHK
WT-GH-sin. GKHI-sin. (PGH-GHK -
sin. WsO
sin. (PLW-WsO)'
Let R=radius of wheel, p=radius of axle, ACO =, ACW;=- inclination of the road to the horizon, O=inclination of
direction of the traction to the road.  Now W80s =WO +
sin. Cos Cco
COs, but WCO=t+7e, and                  Let COs be resin. CcR- CO
presented by a, then W80s= + +a, and
sin. a=  sin. 9... (360).
AlsoPLW=-    ++0; therefore PLW-WsO-=         (1 +a — );
242
24




370                 THE CARRIAGE-WHEEL.
P        sin (~ +  +  ).        (361);
cos..o (,+U-o)
when the direction of traction is parallel to the road, 0=0,..P =W   sin. t+ cos. ttan. (ie+a)}.... (362).
If the road and the direction of traction be both horizontal
-=t=-=0, and
P=W     tan. ( +a)..... (363).
In all cases of traction with wheels of the common dimensions upon ordinary roads, ACO or X is an exceedingly small
angle; a is also, in all cases, an exceedingly small angle
(equation  360); therefore tan. (yO+)=:-+a very nearly.
Now if A   be taken to represent the arc AO, whose length
is determined by the height of the obstacle and the radius;of the wheel, then
=-...   (364).
Substituting the value of a from equation (360),
p   W. (A+p sin.. )
~~~R  ~       (365).
276. It remains to determine the value of the arc A intercepted between the lowest point to which the wheel sinks in
the road, and the summit 0 of the obstacle, which it is at
every instant surmounting.     Now, the experiments of Coulomb, and the more recent experiments of M. Morin,* appear to have fully established the fact, that, on horizontal
roads of uniform quality and material, the traction P, when
its direction is horizontal, varies directly as the load W, and
inversely as the radius R of the wheel; whence it follows
(equation 365), that the arc A is constant, or that it is the
same for the same quality of road, whatever may be the
weight of the load, or the dimensions of the wheel.t       The
Experiences sur le Tirage des Voitures, faites en 1837 et 1838. (See APPENDIX.)
f In explanation of this fact let it be observed, that although the wheel
sinks deeper beneath the surface of the load as the material is softer, yet the
obstacle yields, for the same reason, more under the pressure of the wheel, the
arc A being by the one cause increased, and by the other diminished. Also,
that although by increasing the diameter of the wheel the arc A would be rendered greater if the wheel sank to the same depth as before, yet that it does
not sink to the same depth by reason of the corresponding increase of the surface which sustains the pressure.




THE CARRIAGE-WHEEL.              71
constant A may therefore be taken as a measure of the resisting quality of the road, and may be called the modulus
of its resistance.
The mean value of this modulus being determined in respect to a road, whose surface is of any given quality, the
value of r will be known from equation (364), and the relation between the traction and the load upon that road, under
all circumstances; it being observed, that, since the arc A
is the same on a horizontal road, whatever be the load, if the
traction be parallel, it is also the same under the same circumstances upon a sloping road; the effect of the slope being equivalent to a variation of the load.  The same substitution may therefore be made for tan. ( + a) in equation
(362), as was made in equation (363),:.P=W   { sin. t+ (A+sin. ) cos... (366).
277. The best direction of traction in the two-wheeled
carriage.
This best direction of traction is evidently that which gives
to the denominator of equation (361) its greatest value; it
is therefore determined by the equation,. X +,-A+p sin. 9      (36
-=-= +o=-'c —..... (367).
278. The four-wheeled carriage.
Let W   W, W2 represent the loads borne by the fore and
hind wheels, together with their own weights, R1, R2 their
radii, p, P2 the radii of their axles, and 9p, p2 the limiting angles of resistance. Suppose the direction of the traction P
parallel to the road, then, since this traction equals the sums
of the tractions upon the fore and hind wheels respectively,
we have by equation (366)
P=WI {       s si n+ (-+ in cos. t +
W2 4sin.,+ >+ prsm. cos.       +
W     sin. + (A+p sin. 9) COS.t
2         R~~~~~~~




372               TIRE CRRIAGE-WHEEL.
or,
P=(W,+W,) sin. s+A(i+~W3) cos. I+
RI   R2
i W    (P) sin. p,+W  2 ) sin. p, cos....  (368).
279. The work accumulated in the carriage-wheel.*
Let I represent the moment of inertia of the wheel about
its axis and M its volume; then will MR +I represent its
moment of inertia (Art. 79.) about the point in its circumferences about which it is, at every instant of its motion, in
the act of turning. If, therefore, a represent its angular
velocity about this point at any instant, U the work at that
instant accumulated in it, and (t the weight of each cubical
unit of its mass, then (Art. 75.), U=c a2 (MR  + I)  M
(aeR)2 +a-2I. Now if v represent the velocity of the axis
of the wheel, aR-V;.. U —_ gMV~+~~fgI;
g           u
whence it follows, that the whole work accumulated in the
rolling wheel is equal to the sum obtained by adding the
work which would have been accumulated in it if it had
moved with its motion of translation only, to that which
would have been accumulated in it if it had moved with its
motion of rotation only. If we represent the radius of gyration (Art. 80.) by K, I=MK2; whence substituting and
reducing,
U=-M      (1 +  )2...     (369).
The accumulated work is therefore the same as though the
wheel had moved with a motion of translation only, but with
a greater velocity, represented by the expression (1+  ) RV.
* For a further discussion of the conditions of the rolling of a wheel, see a
paper in the Appendix on the Rolling Motion of a Cylinder.
t The angular velocity of the wheel would evidently be a, if its centre were
fixed, and its circumference made to revolve with the same velocity as now.




ACCELERATED OR RETARDED MOTION.       373
280. ON THE STATE OF THE ACCELERATED OR THE RETARDED
MOTION OF A MACHINE.
Let the work U, done upon the driving point of a machine
be conceived to be in excess of that U, yielded upon the
working points of the machine and that expended upon its
prejudicial resistances. Then we have by equation (117)
U,-AU, + BS, +   2(V>  - V2).WX2;
2g
where V represents the velocity of the driving point of the
machine after the work U1 has been done upon it, V, that
when it began to be done, and 2wX' the coefficient of equable
motion. Now let S, represent the space through which U,
is done, and S, that through which U, is done; and let the
above equation be differentiated in respect to Si,
dU1,  dU,  dS,      1 dV 
dS7    dS   dS d~B  q   dSw
dU
but dS;= P (Art. 51.) if P, represent the driving pressure.
dUS
Alsod- P,, if P, represent the working pressure; also
dV     dY   dt      dV   1    dV
VdS =     t   dS =V *dt      -    dt  f (equation 72).
If, therefore, we represent by A the relation eS, between the
spaces described in the same exceedingly small time by the
driving and working points, we have
P,=AAP2 + B + -    2.....  (370);.f=g.  P — AAP-2-B..       (371);
wheref (Art. 95.) represents the additional velocity actually
acquired per second by the driving point of the machine, if
P, and P, be constant quantities, or, if not, the additional
velocity which would be acquired in any given second, if
these pressures retained, throughout that second, the values
which they had at its commencement.




374:       THE ACCELEIRATION OR RETARDATION
281. To determine the coefficient of equable motion.
2wX2 represents the sum of the weights of all the moving
elements of the machine, each being multiplied by the ratio
of its velocity to that of the driving point, which sum has
been called (Art. 151.) the coefficient of equable motion. If
the motion of each element of the machine takes place about
a fixed axis, and a a, a,, &c., represent the perpendiculars
from their several axes upon the directions in which they
receive the driving pressures of the elements which precede
them in the series, and b,, b,, b,, &c., the similar perpendiculars upon the tangents to their common surfaces at the
points where they drive those that follow them; then,
while the first driving point describes the small space aS,
the point of contact of the pth and p + th elements of the
series will be made (Art. 234.) to describe a space represented by
bb,... b
ba^  S;
a1a,.. *. a 
so that the angular velocity of the pth element will be
represented by
ZA' * * Zag S
a,....bap  S
and the space described by a particle situated at distance p
from the axis of that element by
a,.. a. ptS
and the ratio X of this space to that described by the driving
point of the machine will be represented by
The sum zwX' will therefore be represented in respect to
this one element by
a. a2,..  * ap
Or if Ip represent the moment of inertia of the element, and
tp the weight of each cubic unit of its mass, that portion of
the value of YwX2 which depends upon this element will be
represented by




OF THE MOTION OF A MACHINE.         375
(bb...  -1
a1a2... apI
And the same being true of every other element of the
machine, we have
WV=2 (Ai     i    -  ).. (372),
a a^b b,a, * * ap? (IE
which is a general expression for the coefficient of equable
motion in the case supposed. The value of A in equation
(371) is evidently represented by.....(873)
a;ba;3 a.. a.  (3
282. To determine the pressure upon the point of contact of
any two elements of a machine moving with an accelerated
or retarded motion.
Letp, be taken to represent the resistance upon the point
of contact of the first element with the second, P2 that upon
the point of contact of the second element of the machine
with the third, and so on. Then by equation (370), observing that, P, and p, representing pressures applied to the
same element, zYwX2 is to be taken in this case only in
respect to that element, so that it is represented by 1lIi,
whilst A is in this case represented by al, we have, neglecting friction,
Substituting the value of f from equation (371), and solving
in respect top,
= alPl- bPPl (- AP,) ZW1..... (74),
where the value of A is determined by equation (373), and
that of zwX2 by equation (372).  Proceeding similarly in
respect to the second element, and observing that the
impressed pressures upon that element are p' and 2, we
have
62   f,
P=P +y,,
a.= 2




376        ACCELERATED OR RETARDED MOTION.
f, representing the additional velocity per second of the
point of application of p,, which evidently equals -Jf
Substituting, therefore, the value of f from equation (371)
as before,
b2.    Pi-AP,
P   af a'      wX    I.
Substituting the value of p, from equation (374), and solving in respect to p,, we have
a_ aaa,              2      P —AP
P  bbP,-b     fI + (I )      (I )(,2       (375)And proceeding similarly in respect to the other points of
contact, the pressure upon each may be determined. It is
evident, that by assuming values of A and B in equations
(370) and (371) to represent the coefficients of the moduli in
respect to the several elements of the machine, and to the,whole machine, the influence of friction might, by similar
steps, have been included in the result.




PART IV.
THEORY     OF THE STABILITY       OF STRUCTURES.
GENERAL CONDITIONS OF THE STABILITY OF A STRUCTURE OF
UNCEMETED STONES.*
A STRUCTURE may yield, under the pressures to which it is
subjected, either by the slipping of certain of its surfaces of
contact upon one another, or by their turning over upon the
edges of one another; and these two conditions involve the
whole question of its stability.
THE LINE OF RESISTANCE.
283. Let a structure MNLK, composed of a single row of
uncemented stones of any forms,
and placed under any given circuma<z  Q    f stances of pressure, be conceived to
Ax  bX 2~ /i be intersected by any geometrical
s/ __/       A  surface 1 2, and let the resultant a A
of all the pressures which act upon
d   o/ -f  one of the parts MN21, into which
9/ X, 8  this intersecting surface divides, the
structure, be imagined to be taken.
~:y,  /  ~   Conceive, then, this intersecting
l/.f yr  o       surface to change its form and posixyA \ s    f y     tion so as to coincide in succession
I7::   -  L       with all the common surfaces of
YA/  rl    contact 8 4, 5 6, 7 8, 9 10, of the;~/,  ~.-stones which compose the structure:
and let bB, cC, dD, eE be the re* Extracted from a memoir on the Theory of the Arch by the author of this
work in the first volume of the " Theoretical and Practical Treatise on Bridges,"
by Professor Hosking and Mr. Hann of King's College, published by Mr. Weale.
These general conditions of the equilibrium of a system of bodies in contact
were first discussed by the author in the fifth and sixth volumes of the "Cambridge Philosophical Transactions."
877




378              THE LINE OF RESISTANCE.
sultants, similarly taken with aA, which correspond to these
several planes of intersection.
In each such position of the intersecting surface, the resultant spoken of having its direction produced, will intersect
that surface either within the mass of the structure, or, when
that surface is imagined to be produced, without it. If it
intersect it without the mass of the structure, then the whole
pressure upon one of the parts, acting in the direction of
this resultant, will cause that part to turn over upon the
edge of its common surface of contact with the other part;
if it intersect it within the mass of the structure, it will not.
Thus, for instance, if the direction of the resultant of the
forces acting upon the part NM 1 2 had been a'A', not intersecting the surface of contact 1 2 within the mass of the
structure, but when imagined to be produced beyond it to a';
then the whole pressure upon this part acting in a'A! would
have caused it to turn upon the edge 2 of the surface of contact 1 2; and similarly if the resultant had been in a" A",
then it would have caused the mass to revolve upon the
edge 1. The resultant having the direction aA, the mass
will not be made to revolve on either edge of the surface of
contact 1 2.
Thus the condition that no two parts of the mass should be
made, by the insistent pressures, to turn over upon the edge
of their common surface of contact, is involved in this other,
that the direction of the resultant, taken in respect to every
position of the intersecting surface, shall intersect that surface actually within the mass of the structure.
If the intersecting surface be imagined to take up an ininite number of different positions, 1 2, 3 4, 5 6, &c., and the
intersections with it, a, b, c, d, &c., of the directions of all
the corresponding resultants be found, then the curved line
abcdef, joining these points of intersection, may with propriety be called the LINE OF RESISTANCE, the resisting points
of the resultant pressures upon the contiguous surfaces lying
all in that line.
This line can be completely determined by the methods of
analysis, in respect to a structure of any given geometrical
form, having its parts in contact by surfaces also of given
geometrical forms.  And, conversely, the form of this line
being assumed, and the direction which it shall have through
any proposed structure, the geometrical form of that structure may be determined, subject to these conditions; or
lastly, certain conditions being assumed, both as it regards
the form of the structure and its line of resistance, all that is




THE LINE OF PRESSURE.            379
necessary to the existence of these assumed conditions may
be found. Let the structure ABCD have for its line of rec       sistance the line PQ. Now
i-. —-*. it is clear that if this line;       cut the surface MN of any
a/< ~ R  ~ section of the mass in a point
7/-,'s        f n without the surface of the
mass, en tn the resultant of
the pressures upon the mass._,CMN will act through n,
1Du4 B ~~~and cause this portion of the
1Dai~/  ~mass to revolve about the
- _______  ~      nearest point N of the inBE                    tersection of the surface of
section MN with the surface of the structure.
Thus, then, it is a condition of the equilibrium that the
line of resistance shall intersect the common' surface of contact of each two contiguous portions of the structure actually
within the mass of the structure; or, in other words, that it
shall actually go through each joint of the structure, avoiding none: this condition being necessary, that no two portions of the structure may revolve on the edges of their
common surface of contact.
THE LINE OF PRESSURE.
284. But besides the condition that no two parts of the
structure should turn upon the edges of their common surfaces of contact, which condition is involved in the determination of the LINE OF RESISTANCE, there is a second condition
necessary to the stability of the structure. Its surfaces of
contact must no where slip upon one another. That this
condition may obtain, the resultant corresponding to each
surface of contact must have its direction within certain
limits. These limits are defined by the surface of a right
cone (Art. 139.), having the normal to the common surface
of contact,at the above-mentioned point of intersection of
the resultant) for its axis, and having for its vertical angle
twice that whose tangent is the co-efficient of friction of the
surfaces. If the direction of the resultant be within this
cone, the surfaces of contact will not slip upon one another;
if it be without it, they will.
Thus, then, the directions of the consecutive resultants in




380          THE STABILITY OF A SOLID BODY.
respect to the normal to the point, where each intersects its
corresponding surface of contact, are to be considered as important elements of the theory.
Let then a line ABCDE be taken, which is the locus of
ARt M     the consecutive intersections of the
a'-;/. \     resultants aA, bB, cC, dD, &c. The
Al/i-'~2 I direction of the resultant pressure
upon every section is a tangent to
6    /-'i 4 i' this line; it may therefore with pro71 ^^1\         priety be called the LINE OF PRESSURE.
/Its geometrical form may be deterf///  I   mined under the same circumstances
//''-~.~t  /     as that of the line of resistance. A. /'  1         straight line cC drawn from the point
El? \             C, where the LINE OF RESISTANCE abed
hatf~\~^-  L    I  intersects any joint 5 6 of the struc^*..'  le "ture, so as to touch the LINE OF PRESSURE ABCD, will determine the
direction of the resultant pressure
upon that joint: if it lie within the cone spoken of, the
structure will not slip upon that joint; if it lie without it,
it will.
Thus the whole theory of the equilibrium of any structure
is involved in the determination with respect to that structure of these two lines-the line of resistance, and the line
of pressure: owe of these lines, the line of resistance, determining the point of application of the resultant of the
pressures upon each of the surfaces of contact of the system;
and the other, the line of pressure, the direction of that
resultant.
The determination of both, under their most general forms,
lies within the resources of analysis; and general equations
for their determination in that case, in which all the surfaces
of contact, or joints, are planes-the only case which offers
itself as a practical case-have been given by the author of
this work in the sixth volume of the " Cambridge Philosophical Transactions."
THE STABILITY OF A SOLID BODY.
285. The stability of a solid body may be considered to be
greater or less, as a greater or less amount of work must be
done upon it to overthrow it; or according as the amount




THE STABILITY OF A STRUCTURE.         381
of work which must be done upon it to bring it into
that position in which it will fall over of its own accord is
greater or less. Thus the stability of the solid represented
n fig. 1. resting on a horizontal  Fig.         Fig.2.
plane is greater or less, according
as the work which must be done
upon it, to bring it into the position
represented in Jig. 2., where its centre of gravity,is in the vertical 
passing through its point of sup- 
port, is greater or less. Now this
work is equal (Art. 60.) to that
which would be necessary to raise its whole weight, vertically, through that height by which its centre of gravity
is raised, in passing from the one position into the other.
Whence it follows that the stability of a solid body resting
upon a plane is greater or less, as the product of its weight
by the vertical height through which its centre of gravity is
raised, when the body is brought into a position in which it
will fall over of its own accord, is greater or less.
If the base of the body be a plane, and if the vertical
height of its centre of gravity when it rests upon a horizontal
plane be represented by h, and the distance of the point or
the edge, upon which it is to be overthrown, from the point
where its base is intersected by the vertical through its
centre of gravity, by k; then is the height through which its
centre of gravity is raised, when the body is brought into a
position in which it will fall over, evidently represented by
Th2 +2) —h; so that if W  represent its weight, and U the
work necessary to overthrow it, then
U=W    j(h~+P)+ -Ah....   (376).
U is a true measure of the stability of the body.
THE STABILITY OF A STRUCTURE.
286. It is evident that the degree of the stability of a
structure, composed of any number of separate but contiguous solid bodies, depends upon the less or greater degree of
approach which the line of resistance makes to the extrados
or external face of the structure; for the structure cannot be
thrown over until the line of resistance is so defl ected as to




382                  THE WALL OR PIER.
intersect the extrados: the more remote is its direction from
that surface, when free from any extraordinary pressure, the
less is therefore the probability that any such pressure will
overthrow it. The nearest distance to which the line of resistance approaches the extrados will, in the following pages,
be represented by m, and will be called the MODULUS OF
STABILITY of the structure.
This shortest distance presents itself in the wall and buttress commonly at the lowest section of the structure. It is
evidently beneath that point where the line of resistance intersects the lowest section of the structure that the greatest
resistance of the foundation should be opposed. If that point
be firmly supported, no settlement of the structure car take
place under the influence of thp pressures to which it is ordinarily subjected.*
THE WALL OR PIER.
287. The stability of a wall.
If the pressure upon a wall be uniformly distributed along
its length,t and if we conceive it to be intersected by vertical planes, equidistant from one another and perpendicular
to its face, dividing it into separate portions, then are the
conditions of its stability, in respect to the pressures applied
to its entire length, manifestly the same with the conditions
the stability of each of the individual portions into which it
is thus divided, in respect to the pressures sustained by that
portion of the wall; so that if every such columnar portion
or pier into which the wall is thus divided be constructed so
as to stand under its insistent pressures with any degree of
firmness or stability, then will the whole structure stand with
the like degree of firmness or stability; and conversely.
In the following discussion these equal divisions of the
length of a wall or pier will be conceived to be made one
foot apart; so that in every case the question investigated
will be that of the stability of a column of uniform or varia* A practical rule of Vauban, generally adopted in fortifications, brings the
point where the line of resistance intersects the base of the wall, to a distance
from the vertical to its centre of gravity, of A-ths the distance from the latter
to the external edge of the base. (See Poncelet, Memoire sur la Stabilitg des
Revetemens, note, p. 8.)
e In the wall of a building the pressure of the rafters of the roof is thus
uniformly distributed by the intervention of the wall plates.




THE LINE OF RESISTANCE IN A PIER.      383
ble thickness, whose width measured in the direction of the
length of the wall is one foot.
288. When a wall is supported by buttresses placed at
equal distances apart, the conditions of the stability will be
made to resolve themselves into those of a continuous wall,
if we conceive each buttress to be extended laterally until it meets the adjacent buttress, its material at the same
time so diminishing its specific gravity
that its weight when thus spread along
the face of the wall may remain the
same as before. There will thus be obtained a compound wall whose external
and internal portions are of different
specific gravities; the conditions of
whose equilibrium  remain manifestly
unchanged by the hypothesis which has
been made in respect to it.
THE LINE OF RESISTANCE IN A PIER.
289. Let ABEF be taken to represent a column of uniform dimensions.
/ JLet PS be the direction of any presj |I,' /  sure P sustained by it, intersecting its
1  C   A G,/     axis in 0. Draw any horizontal seci T-,   --      tion IK, and take ON   to represent
I li,          the weight of the portion AKIB of!,,'!//     the column, and OS on the same scale
I~,ffi    to represent the pressure P, and comI/,,,           plete the parallelogram ONRS; then
|. -$  ~  will OR evidently represent, in mag— j -._._    nitude and direction, the resultant of,,v 1M      the pressures upon the portion AKIB
]! i~      ~ of the mass (Art. 3.), and its point of
-I              intersection Q with IK will represent
E  DXn v)F    a point in the line of resistance.
Let PS intersect BA (produced if necessary) in G, and let
GO=k, AB=a, AK=x, MQ=y, POC=a, p=weight of
each cubic foot of the material of the mass. Draw RL perpendicular to CD; then, by similar triangles,




884        TIE LINE OF RESISTANCE IN A PIE.
QM RL
OM- OL
But QM=y, OM=CM-CO= —k cot. o, RL=RN
sin. RNL=P sin. a, OL=ON+NL= ON+RN cos. RNL
=ILaX+ P cos. a;
y         P sin. a
x- k cot. a ~pax+P cos. a'. x sin. a-k cos. a...* y==P. --- -o   ----.      (377);.Lhax+P cos. a''
which is the general equation to the line of resistance of a
pier or wall.
290. The conditions necessary that the stones of the pier may
not sliip on one another.
Since in the construction of the parallelogram ONRS,
whose diagonal OR determines the direction of the resultant
pressure upon any section IK, the side OS, representing the
pressure P in magnitude and direction, remains always the
same, whatever may be the position of IK; whilst the side
ON, representing the weight of AKIB, increases as IK descends: the angle ROM  continually diminishes as IK descends. Now, this angle is evidently equal to that made by
OR with the perpendicular to IK at Q; if, therefore, this
angle be less than the limiting angle of resistance in the
highest position of IK, then will it be less in every subjacent
position. But in the highest position of IK, ON-=0, so that
in this position ROM=a. Now, so long as the inclination
of OR to the perpendicular to IK is less than the limiting
angle of resistance, the two portions of the pier separated by
that section cannot slip upon one another (Art. 141.). It is
therefore necessary, and sufficient to the condition that no
two parts of the structure should slip upon their common
surface of contact, that the inclination a of P to the vertical
should be less than the limiting angle of resistance of the
common surfaces of the stones. All the resultant pressures
passing through the point 0, it is evident that the line of
pressure (Art. 284.) resolves itself into that point.




THE LINE OF RESISTANCE IN A PIER.      385
291. The greatest height of thepier.
At the point where the line of resistance intersects the
external face or extrados of the pier, y=1a; if, therefore, H
represents the corresponding value of w, it will manifestly
represent the greatest height to which the pier can be built,
so as to stand under the given insistent pressure P. Substituting these values for X and y in equation (377), and solving
in respect to H,
H   P(-ia+ k) cos. a..
P sin. a —jCa2.(
If P sin. a=-ia2, H=inmimty; whence it follows that in
this case the pier will stand under the given pressure P however great may be the height to which it is raised.
292. The line of resistance is an equilateral hyperbola.
Multiplying both sides of equation (377) by the denominator of the fraction in the second member,
y(pax+P cos. a)=-Px sin. a —Pk cos. a;
dividing by pa, transposing, and changing the signs of all the
terms,
P sin. a   /    P cos. \  P Cos. 
-_ -y x+ = -                    k;
a               pla       pa
adding Psin. cos, to both sides,
P'sin. a/  Pcos'a\. Pcos. c   Pcos. a k  Psin..a
— (x+ —vi~y   +4 —--— ) —-- 
\ a         \                   \ J    a   I
P sin. a    P cos. a
Let CH be taken equal to --     IIT= —--; and let
VQ=y,, TV=x1,
88j 




386        THE LINE OF REESSTIFANCE IN A PIER.
P sin.. y,= VQ=VM-MQ=CH —MQ=             sinay;
Pcos. a
xa=TV=HV+TH=x+ Pcos;
Pcos. a /   Psin. a: ~,y= = a-   k +   --      a constant quantity.
This is the equation to a rectangular hyperbola, whose
asymtote is TX.* The line of resistr     /''   * ance  continually  approaches TX
7/           therefore, but never meets it; whence
/      ~   it follows, that if TX lie (as shown
in the figure) within the surface of
3          --     the  mass, or if    CH<CB      or
P sin. a
y |         — sin.<ia -  or 2P sin. a <a2, then
[  /1 ~,,     the line of resistance will no where,|/I_     cut the extrados, and the structure
I, gio          will retain its stability under the in-_.-I:g         sistent pressure P, however high it
may be built; which agrees with
the result obtained in the preceding
article.
E    -  1)  F'
293. The thickness of the pier, so that when raised to a given
height it may have a given stability.
Let m be taken to represent the nearest distance to which
the line of resistance is intended to approach the extrados of;the pier, which distance determines the degree of its stability,:and has been called the modulus of stability (Art. 286.). It
is evident from the last article that this least distance will
present itself in the lowest section of the pier.  At this
lowest section, therefore, y=-a-m. Substituting this value
for y in equation (377), and also the height h of the pier for
x, and solving the resulting quadratic equation in respect to
a, we shall thus obtain
= P cos. a  )
/P os. a      2  2P   sin.    (k-M-) os.c.. 37
* Church's Analyt. Geom., Art. 161.




A WALL SUPPORTED BY SHORES.         387
294. To vary the point of application of the pressure P, so
that any required stability may be given to the pier.
It is evident, that if in equation (377) we substitute -a-m
for y, and h for x, the modulus of stability m
may be made to assume any given value for
A.       a given thickness a of the pier, by assigning
a corresponding value to k; that is, by moving the point of application G to a certain
distance from the axis of the pier, deter/ mined by the value of k in that equation.
il'     This may be done by various expedients,
r:  D      and among others by that shown in the
figure. Solving equation (377) in respect to
i?-L         k, we have
k =A tan. a- (-2-m) 1      -)     (380).
2   2  htan. or (1-A +PL Cos. al
It is necessary to the equilibrium of the pier, under these
circumstances, that the line of resistance should no where
intersect its intrados below the point D.
THE STABILITY OF A WALL SUPPORTED BY SHORES.
295. Let the weight of the portion of the wall supported
by each shore or prop, and the
pressure insistent upon it, be imagined to be collected in a single
] i^ ~foot of the length of the wall; the
o''f   conditions of the stability of the
- /.   wall evidently remain unchanged
/,      by this hypothesis.  Let ABOD
/  1' I   represent one of the columns or
piers into which the wall will thus
b e divided, EF the corresponding
shore, P the pressure sustained upon
the summit of the wall, Q the
X ai \thrust upon the shore EF, 2w its
weight, x the point where the line
of resistance intersects the base of
the wall, Cx-m, CF=b FEC=/;
and let the same notation be taken in other respects as in




388           A WALL SUPPORTED BY SHORES.
the preceding articles. Then, since x is a point in the direc
tion of the resultant of the resistances by which the base of
the column is sustained, the sum of the moments about that
point of the pressure P and half the weight of the shore,
supposed to be placed at E*, is equal to the sum of the
moments of the thrust Q, and the weight tah of the column;
or drawing xM  and xN perpendiculars upon the directions
of P and Q,
P. xM+w. xC=Q. xN+ —ah. oK.
Now xM =xs sin. xsM=(HK-Ht) sin. a = ^h —(Hp+st)
cot. aI sin. a=hA sin. oa-(k+a a-rnj cos. a, xN=(b +nm) cos. /,.~. P h sin. — (k+~a-m) cos. a} +
wm= Q(b +m) cos. 3+t ah(ja-m)
Solving this equation in respect to Q, and reducing, we
<obtain.
P {h sin.a - (k + ja)cos. a - ja2h+m(P os.a +  ah + w)
(b +m) cos. t... (381).
This expression may be placed under the form
Q =(P cos. a + a ah+ w) sec. 3P{b cos. a —A sin. +(k+~la) cos. ac + tat h(aa+b)++wb
(b+m) cos. 3 
If the numerator of the fraction in the second member of
this equation be a positive quantity (as in all practical cases
it will probably be found to be) the value of Q manifestly
diminishes with that of m. Now the least value of m, consistent with the stability of the wall, is zero, since the line
of resistance no where intersects the extrados; the least
value of Q (the shore being supposed necessary to the support of the wall) corresponds, therefore, to the value zero of
m; moreover this least value of the thrust upon the shore
consistent with the stability of the wall is manifestly that
which it sustains when the wall simply rests upon it, the
* The weight 2w of the shore may be conceived to be divided into two equal
parts and collected at its extremities.
f The expression (b-m) cos. 3 may be placed under the form b cot, / sin.,3+m cos. -=-c sin. f3-m cos. 8, where c. represents the height CE of the point
against which the prop rests.




A WALL SUPPORTED BY SHORES.          389
shore not being driven so as to increase the thrust sustained
by it beyond that just necessary to support the wall.*
This least thrust is represented by the formula
QP Ahsin. a-(k+i-a) cos. a} - i2h
b cos. /
The thrust which must be given to the prop in order that
there may be given to the wall any required stability, determined by the arbitrary constant m, is determined by equation (381). The stability will diminish as the value of m is
increased beyond ~a, and the wall will be overthrown
inwards when it exceeds a.
296. The stability of a wall sustained by more than one
shore in the same plane.
Let EF, ef be shores in the same plane, sustaining the
wall ABCD, and both necessary to
/  its stability; so that if EF were reop A;      moved, the wall would turn over upon
f, and if ef were removed, upon some
point between F and C.
If the thrust of the shore EF be
only that just necessary to sustain
the tendency of the wall to overturn
— /Tl     upon f, it is evident that the line of
resistance must pass through that
point; but if the thrust exceed that
just necessary to the equilibrium, or
if the shore be driven then the line
E1//  m    ~of resistance will intersect fg in some
- g  m   point x. Letfx=m; then representing the thrust upon EF by Q, the distances fD and fi by h and b, and the angle EFC by /, the
value of Q is evidently determined by equation (381).
If z be taken in like manner to represent the point where
the line of resistance intersects the base of the wall, and
Cz=m,, CE=b,; Ce=b2, Cfe-=/, CD=h,, the thrust upon
the prop ef by Q, and its weight by 2w,; then the sum of
the moments about the point z of Q and Q1, and the weight
* This case presents an application of the principle of least resistance. (See
Theory of the Arch.)




390      TIlE STABILITY OF A GOTHIC STRUCTURE.
a.cth of the wall, equals the sum of the moments of P, w,
and w1; or
Q,(b,+m,) cos. ji3+Q (b,(+l) cos. 3+tfiah (1-a-m0)=
P {A sin. a —(k+ia-im,) cos. c} + (w+o) m...... (382.)
Substituting the value of Q in this equation, from equation
(381), and solving in respect to Q1, the thrust upon the prop
f' will be determined, so that the stability of the wall, upon
its section fg and upon its base CB, may be m and in,
respectively.
If m=s=m, the portions of the wall above and below fg
are equally stable.
If m-=m —0, the thrust upon each shore is only that
which is just necessary to support the wall, or which is produced by its actual tendency to overturn. In this case we
have
(P sin. C —a) (        )+ (hb-hb)+P (b-b) (k + a) cos. 
A1"~- &bb2 cos. C     s0
the value of Q being determined by equation (380).
297. The stability of a structure having parallel walls, one
of which is supported by means of struts resting on the
summit of the other.
Let AB and CD be taken to represent the walls, and EF
one of the struts; the thrust Q upon the
strut may be determined precisely as in
C A  Art 295. So that the line of resistance
i  may intersect the base of the wall AB at
/F      a given distance m from the extrados
(see note, p. 388.)
Let m, represent the distance Dx from
the extrados at which the line of resist-; #r       ance intersects the base of the wall CD;
then taking the moments of the pressures
applied to the wall CD about the point
x, as in Art. 295, and observing that
besides the pressure Q the weight w of
_T-  f _   one half the strut is applied at E, we
u  -  B  ~have
Q {h, sin. +(k, +1 al —n,) cos. } =PlaC1/ (2a, —ml)+
(k1 + Ia,- m,) w;




THE STABILITY OF A GOTHIC STRUOITURE.   391
in which equation h, and a, are taken to represent the
height and thickness of the wall CD, k, the distance of the
point E on which the strut rests from the axis of the wall, P
the inclination of the strut to the vertical, and l, the weight
of a cubic foot of the material of the wall.
Substituting for Q its value from equation (381), and
reducing,
Jh sin. a- (k+Ia) cos. a -  a2h+ m (P cos. a +,ah+ow)
c sin. l3+m cos. /
^1a1 (la, -1) + (k1 +i-1-m) w
A, sin. 3 + (k1 + 1 - -m) cos.  * * *.
By this equation is determined that relation between the
dimensions of the two walls and the amount of the insistent
pressure P, by which any required stability may be assigned
to each wall of the structure. If m=-0 the pressure upon
the strut will be that only which is produced by the tendency of AB to overturn; and the value of mv determined
from the above equation will give the stability of the external wall on this supposition.
If m==O and mv=0, both walls will be upon the point of
overturning, and the above equation will express that relation between the dimensions of the wall and the amount of
the insistent pressure, which corresponds to the state of the
instability of the structure.
The conditions of the stability, when the wall AB is supported by two struts resting upon the summit of the wall
CD, may be determined by a method similar to the above
(see Art. 296).
The general conditions of the stability of the structure
discussed in this article evidently include those of a GOTHIC
BUILDING having a central nave, whose walls are supported,
under the thrust of its roof, by the rafters of the roof of its
side aisles. By a reference to the principles of the preceding
article, the discussion may readily be made to include the
case in which a further support is given to the walls of the
nave by flying buttresses, which spring from the summits of
the walls of the aisles.  The influence of the buttresses
which support the walls of the aisles upon the conditions of
the stability of the structure forms the subject of a subsequent article.




392            THE WALL OF A DWELLING.
298. The stability of a wall sustaining the loors of a
dwelling.
The joists of the floors of a dwelling-house rest at their
extremities upon, and are sometimes
notched into, pieces of timber called
/   wall-plates, which are imbedded in
vD          the masonry of the wall.    They'~c  ~serve thus to bind the opposite sides
e         of the house together; and it is upon
1/<.     the support which the thin walls of
modern houses receive from these:E/.joists, that their stability is some/  i,       times made to depend.*!It E,'..Representing by w the weight of:/  [] ~that portion of the flooring which
/   i          rests upon the portion ABCD of the
i/ t f       wall, and the distance BE  by c,,.~/  E       taking x, as before, to represent the._  ~ _point where the line of resistance....!     intersects the base of the wall, and
measuring the moments from   this
point, we have
N. Q +xK. tzah + xB. w=xM.P;
whence, taking the same notation as in the preceding articles, and substituting,
CQ + (ta-m)) +ah (a- m)w= {- sin. a-(k+a n —m) cos. } P;. Qe= -h sin. a-(k+ -a) cos. aC P —aA-2h-wa+
nl (P cos. oL+JLah+w)..... (384);
from which expression it appears that Q is less as m is less.
When, therefore, the strain upon the joints is that only
which is just necessary to preserve the stability of the wall,
or which it produces by its tendency to overturn, then
m= 0. In this'case, therefore,
* A house thus constructed evidently becomes unsafe when its wall-plates or
the extremities of its joists begin to decay.




A WALL SUPPORTED BY BUTTRESSES.       893
h sin. ac-(k+ia) cos. a} P -&ya'h-wa
Qy=~= —---— ^ —-..... (385).
If 18 be assumed a right angle, and if (a-m)w be substituted for mw, the case discussed in Art. 295. will
/^  evidently pass into that which is the subject of
Aj    the present article, and the preceding equation
may thus be deduced from equation (381) (see
note, p. 388.).
r.: l_.>  In like manner, if the wall sustain the pres_     -- sure of two floors, and h be taken to represent
the distance from its summit to the lower floor,
and h, its whole height; then, representing by m
and mn the distances from the extrados at which
A.:     the line of resistance intersects the sections EG',    3 ~and eg, and substituting (w + w) (a- mn) for
(w +w)m,, the value of the strain Q on the
joists of the lower floor may be determined by
equation (382), it being observed that for the
coefficient of Q, in that equation must be substio'      tuted (as was shown above) the height (h,-h) of
the lower floor from the bottom of the wall. If the strain
be only that produced by the tendency of the wall to overturn at g and C, then
Q,1=(h-c) (Ia-P sin. o)+
P(k+i a)cos. +oa-h - h...(386).
The value of Q is determined by equation (385), c being
taken to represent the distance Ee between the floors. If
the joists be not notched into the wall-plates, the friction of
their extremities upon them, produced per foot of the length
by the weight which they support, must at least equal Q and
Q, respectively.
299. The stability of a wall supported by piers or buttresses
of uniform thicwness.
Let the piers be imagined to extend along the whole




394         WALL SUPPORTED BY BUTTRESSES.
length of the wall, as explained in Art. 288.;
and let ABCD represent a section of the compound wall thus produced. Let the weight of./.  each cubic foot of the material of the portion
ABFE be represented by p1, and that of each
cubic foot of GFCD by,_ EA=,, GD-a,,
BC=a, AB==,, CD-=,, distance from CD,
produced, of the point where P     intersects
AE=-, x the intersection of the line of resistance with CB, Cx=m. By the principle of the
equality of moments, the moment of P about
the point x is equal to the sum of the moments;iof the weights of GC and AF about that point.
But (Art. 295.) moment of P=P hA, sin. a-      (I-mn) cos. a}; also moment of weight of AF=
c: [_r   (a2-m) + ia,)h1acv1; moment of weight of GC=
(a, -m)h2a2Pt2.:. P {h, sin. a —( —m) cos. a}- =(a,-m+~a:)hlaltpl +
(ea2-m)haA.......(387).
If the material of the pier be the same with that of the
wall; then, taking b to represent the breadth of each pier,
and c the common distance of the piers from centre to
centre (Art. 288.), ca2,=b2 =a2t1, therefore c,% =ba-,. Representing 7 by n, eliminating the value of P2 between this
equation and equation (387), writing p, for,, and reducing,
P(h1 sin.-l cos. ca)=-' (aah, + 2alahl +a2h) -
miPcos.+   L(aJ+   a2h2) +..... (388);
by which equation a relation is determined between the
dimensions of a wall supported by piers, having a given
stability m, and its insistent pressure P.  Solving it in
respect to a2, the thickness of the pier necessary to give any
required stability to the wall will be determined. (See
APPENDIX.)
If a2 be assumed to represent that width of the pier by
which the wall would just be made to sustain the given
pressure P without being overthrown; then taking m=O,
and solving in respect to a2,




WALL SUPPORTED BY BUTTRESSES.        395
hA
a, = -na l+
(,in..    cos. a) +   l          * * * (389).
300. The stability of a pier or buttress surm/ounted by a pinnacle.
Let W    represent the weight of the pinnacle,
and e the distance of a vertical through its centre of gravity from the edge C of the pier: then
assuming x to be the point where the line of
resistance intersects the base of the pier, and taking the same notation as before, equation (387)
will evidently become
C
P  lh, sin. a-(l —m) cos. a} = a2 —m~+a, } ha, +
{ja2-m\ h2aa2a  + (e-m)W.
Substituting for p2 its value-^orl, writing P   for p, and
c    n
reducing,
P(hA sin. a-I cos. a)=  (la2h +2aaJ +I~a2uh) +
We-m     P cos. a+W+     (ah,+ a22... (390).
If a2 represent the thickness of that pier by which the wall
will just be sustained under the pressure, taking m=O, and
solving in respect to a2, as -na-l +
A2
JP(h, sin. a-l cos. a)-We +1 a,           (391).




396      WALL SUPPORTED BY GOTHIC BUTTRESSES.
THE GOTHIC BUTTRESS.
301. In Gothic buildings the thickness of
a buttress is not unfrequently made to vary
at two or three different heights above its
base. Such buttress is represented in the
accompanying figure.
The conditions by which any required stability may be assigned to that portion of it
whose base is be may evidently be determined
by equation (390). To determine the condi-,i      tions of the stability of the whole buttress
upon CD, let the heights of the points Q, a,
S     and b above CD be represented by h,, h and.~ +      i h,; let DE=a,, DF=a,, FC=a, Cx= ml;
/  I \ *   then adopting, in other respects, the same
notations as in Arts. 299 and 300. Since the
distances from x of the verticals through the
centres of gravity of those portions of the
l       buttress whose bases are DE, DF, and FC
c I I        respectively, are (a,+a 2+ -a, —O), (ca,+ a,
X I.   -inm) and (1a3-m1) we have, by the equality
of moments,
P!h, sin. a-(l-ml) cos. ac =-(a,+,+ia,1-m) hal +
(a +a2-mL,)h, f + (a -m1) h3a+,' W -(e- m,)..... (392).
n             n
This equation establishes a relation between the dimensions of the buttress and its stability, by which any one of
those dimensions which enter into it may be so determined
as to give to m, any required value, and to the structure any
required degree of stability. (See APPENDIX.)
It is evident that, with a view to the greatest economy of
the material consistent with the given stability of the buttress, the stability of the portion which rests upon the base
be should equal that of the whole buttress upon CE; the
value of ml in the preceding equation should therefore equal
that of m in equation (390) If m be eliminated between
these two equations, it being observed that h, and h2 in equa.
tion (390) are represented by h,-h2 and h2-h8 in equation
(392), a relation will be established between a,, a, a3 h,, h,
h,, which relation is necessary to the greatest economy of




THE STABILITY OF WALLS SUSTAINING ROOFS.  397
material; and therefore to the greatest stability of the structure with a given quantity of material.
THE STABILITY OF WALLS SUSTAININ  ROOFS.
302. Thrust upon the feet of the rafters of a roof, the tiebearm not being suspended from the ridge.
If p be taken to represent the weight of each square foot
of the roofing, 2L the span, t the,  H a  As     i~inclination BAC of the rafters to
^ ///-"   \ ~ the horizon, q the distance between
~f/          -'"'. — each two principal rafters, and a
-~^          Ti the inclination to the vertical of
the resultant pressure P on the
foot of each rafter; then will L sec. t represent the length of
each rafter, and piL  sec. c the weight of roofing borne by
each rafter. Let the weights thus borne by each of the
rafters AB and BC be imagined to be collected in two equal
weights at its extremities; the conditions of the equilibrium
will remain unchanged, and there will be collected at B the
weight supported by one rafter and represented by pLgq
sec.t, and at A and A   weights, each of which is represented
by ~J1Lq sec. t. Now, if Q be taken to represent the thrust
produced in the direction of the length of either of the
rafters AB and BC, then (Art. 13.) C~Lq sec. t=2Q cos.
jABC: but ABC= r — 2; therefore cos. JABC=sin. t;
therefore 2Q sin. t=U BLq sec. t;
sec. l     Ep1Lqg    iL' Q=1Lq_-. sin.  2 sin. t cos. l  sin. 2t
The pressures applied to the foot; A of the rafter are the
thrust Q and the weight jpLq sec. t; and the requiredc pressure P is the: resultant of these two pressures. Resolving Q
vertically and horizontally, we obtain Q sin. t and Q cos. t,
or iLq sec. t and iu1Lq cosec. t. The whole pressure applied
vertically at A is therefore represented by pLq sec. t, and
the whole horizontal pressure by -1iLq cosec. t; whence it
follows (Art. 11.) that
P=,/,2L2q sec. + i2LgSe   C cosec.
PLq sec.  /1 +1 cot. t..... (393).




398                RAFTERS OF A ROOF.
IL cosec. t
tan. Y_ — 2    c   = — cot. t  (394).
If the inclination t of the roof be made to vary, the span
remaining the same, P will attain a minimum value when
1
tan. t —, or when
e=35~ 16'..... (395).
It is therefore at this inclination of the roof of a given
span, whose trusses are of the simple form shown in the
figure, that the least pressure will be produced upon the feet
of the rafters. If qp represent the limiting angle of resistance
between the feet of the rafters and the surface of the tie, the
feet of the rafters would not slip even if there were no mortice or notch, provided that c were not greater than qp (Art.
141.), or 2 cot. t not greater than tan. p, or
t not less than cot.-1 (2 tan. p)*..... (396).
303. The thrust upon the feet of the rafters of a roof in
which the tie-beam is suspended from  the ridge by a
king-post.
It will be shown in a subsequent portion of this work,p  p^~ ^     (see equation 558) that, in this case,
is/  fk_   \    the strain upon the king-post BD is
equal to -ths of the weight of the;^/    in }\t ~\c tie-beam with its load. RepresentrVH    -JP   ---   2ing, therefore, the weight of each
l  l, [~ ~      foot in the length of the tie-beam
by,, and proceeding exactly as in
the last article, we shall obtain for the pressure P upon the
feet of the rafters, and its inclination to the vertical, the
expressions
P-LPL{ (2pLq sec. +  )2 + (q sec.- + 5 P)2 cot. 1} _...(397).
tan. a=cot. (2, q sec. t + 2  @..... (398).
\2q sec. t + ~ ).      (9)
* If the surfaces of contact be oak, and thin slips of oak plank be fixed
under the feet of the rafters, so that the surfaces of contact may present parallel fibres of the wood to one another (by which arrangement the friction will
be greatly increased), tan. -='48 (see p. 133.); whence it follows that the
rafters will not slip, provided that their inclination exceed cot.-''96, or
46~ 10'.




WALL SUSTAINING THE THRUST OF A ROOF.   399
304. The stability of a wall sustaii&ng the thrust of a roof,
having no tie-beam.
Let it be observed, that in the equation to the line of
resistance of a wall (equation
A        ^?37'7), the terms P sin..a and P
cos. c represent the horizontal
V  V^  ^  and vertical pressures on each
foot of the length of the summit
of the wall; and that the former
X^   At\  ~of these pressures is represented
----—. —   in the case of a roof (Art. 302.)
by -&1L cosec. t, and the latter
by AL sec. t; whence, substituting these values in  equation'i^         == Of(377), we obtain for the equation
to the line of resistance, in a wall
sustaining the pressure of a roof,. —------—. —  without a tie-beam
_:T x cot. L —K<
y-L   2....c.  (399);
ax cos.t +L
in which expression a represents the thickness of the wall,
k the distance of the feet of the rafters from the centre of
the summit of the wall, L the span of the roof, t the weight
of a cubic foot of the wall, and P, the weight of each square
foot of the roofing. The thickness a of the wall, so that,
being of a given height A, it may sustain the thrust of a
roof of given dimensions with any given degree of stability,
may be determined precisely, as in Art. 293, by substituting
h for x in the above equation, and ~a-m for y, and solving
the resulting quadratic equation in respect to a.
If, on the other hand, it be required to determine what
must be the inclination t of the rafters of the roof, so that
being of a given span L it may be supported with a given
degree of stability by walls of a given height h and thickness a; then the same substitutions being made as before,
the resulting equation must be solved in respect to l instead
of a.
The value, of a admits of a minimum in respect to the
variable t. The value of t, which determines such a minimum value of a, is that inclination of the rafters which is




400              STABILITY OF A WALL.
consistent with the greatest economy in the material of the
wall, its stability being given.
305. The stability of a wall supported by buttresses, and
sustaining the pressure of a roof without a tie-beam.
The conditions of the stability of such a wall, when supported by buttresses of uniform thickness, will evidently be
determined, if in equation (388) we substitute for P cos. a
and P sin. a their values ~P1L sec. i and OL consec. t; we
shall thus obtain
iLL (/h, cosec. t-l sec. t) =-  (al2h-1+ 2ala2h,  a22)- m
n
XL sec. t+l  (alh, + - a2h2 }... (400).
From  which equation the thickness a2 of the buttresses
necessary to give any required stability m to the wall may
be determined.
If the thickness of the buttresses be different at different
heights, and they be surmounted by pinnacles, the conditions of the stability are similarly determined by substituting for P sin. a and P cos. a the same values in equations
(390) and (392).
To determine the conditions of the stability of a Gothic
building, whose nave, having a roof without a tie-beam, is
supported by the rafters of its two aisles, or by flying buttresses, which rest upon the summits of the walls of its
aisles, a similar substitution must be made in equation (383).
If the walls of the aisles be supported by buttresses,
equation (383) must be replaced by a similar relation
obtained by the methods laid down in Arts. 299 and 301;
the same substitution for P sin. a and P cos. a must then be
made.
306. The conditions of the stability of a wall supporting a
shed roof.
Let AB represent one of the rafters of such a roof, one ex



STABILITY OF A WALL.            401
tremity A resting against the face of
t          the wall of a building contiguous to
the shed, and the other B upon the....i^ summit of the wall of the shed.
It is evident that when the wall
-..........   w BH is upon the point of being overthrown, the extremity A will be upon
the point of slipping on the face of
the wall DC; so that in this state of
the stability of the wall BH, the direction of the resistance R of the wall
E       ~C~  ODC on the extremity A of the rafter
will be inclined to the perpendicular AE to its surface at an
angle equal to the limiting angle -of resistance.  Moreover,
this direction of the resistance R which corresponds to the
state bordering upon motion is common to every other state;
for by the principle of least resistance (see Theory of the
Arch) of all the pressures which might be supplied by the
resistance of the wall so as to support the extremity of the
rafter, its actual resistance is the least.  Now this least resistance is evidently that whose direction is most nearly vertical; for the pressure upon the rafter is wholly a vertical
pressure. But the surface of the wall supplies no resistance
whose direction is inclined farther from the horizontal line
AE than AR; AR is therefore the direction of the resistance.
Resolving R vertically and horizontally, it becomes R sin.
p and R cos. p.  Representing the span BF by L> the inclination ABF by t, the distance of the rafters by q, and the
weight of each square foot of roofing by ii (Art. 10.), R sin.
+ P cos. a=th Lg sec. t and R cos. — P sin. a=0; also the
perpendiculars let fall from A on P and upon the vertical
through the centre of AB, are represented by
L cos. (a +t) sec. t and ~L; therefore (Art. 7).
PL cos. (c+ t)sec. = —L. Lp1 q sec. t, and hence
P cos. (ca+t)-=iLlq. Eliminating between these equations, we obtain
cot. Sa=tan. (p2 tan. t..... (401);
R_- rLpq     p —Lq, q1+( tan. p +2 tan. )2 +    
sin. y, + t'  cos. t (tan. p + tan. )2
26




402               THE PLATE BANDE.
If the rafter, instead of resting at A
~T.   against the face of the wall, be received
E   into an aperture, as shown in the figure,
so that the resistance of the wall may be
applied upon its inferior suface instead of
at its extremity: then drawing AE perI.     pendicular to the surface of the rafter,
the direction AR of the resistance is evidently inclined to that line at the given
limiting angle p. Its inclination to the horizon is therefore represented by 2- +.
Substituting this angle for p in equations (401) and (402),
cot. o=cot. ( -)+ 2 tan...... (403).
~- -       i ~sec. t   P
cos. (t-)p)+sin. (t- p)tan. t -
i1 + [cot. (t —) + 2 tan. t]2}
cos. l Icot. (l- ) +tan.L t.
~Substituting in equations (377) and (379) for P sin. a, P cos. a,
their values determined above, all the conditions of the stability of a wall supporting such a roof will be determined.
307. THE PLATE BANDE OR STRAIGHT ARCH.,W         -    - Let MN represent any joint of
"I-~D  fN11:', = I  the plate bande ABCD, whose
B,      111points of support are A and B;
Q S^  5W ~  PA the direction of the resistance
p]  at A, WQ a vertical through the:E.'';,'' centre of gravity of AMND, TR
the direction of the resultant pressure upon MN; the directions of
TR, WQ, and PA intersect, therefore, in the same point O.
Let OAD=a, AM=x, MR=y, AD=H, AB=2L, weight
of cubic foot of material of arch=v1. Draw Rm a perpendicular upon PA produced; then by the principle of the
equality of moments,
Rm. P=MQ. (weight of DM).




THE PLATE BANDE.              403
But Rm = x cos. a - y sin. a, MQ = JA, weight of DM =
HIIx; also resolving P vertically,
P cos. a=L....(405).
Whence we obtain, by substitution in the preceding equation, and reduction,
L(a -ytan.a)=_3x....(406),
which is the equation to the line of resistance, showing it to
be a parabola. If, in this equation, L be substituted for x,
and the corresponding value of y be represented by Y, there
will be obtained the equation Y tan. a = L, whence it
appears that c is less as Y is greater; but by equation (405),
P is less as a is less. P, therefore, is less as Y is greater;
but Y can never exceed H, since the line of resistance cannot intersect the extrados. The least value of P, consistent
with the stability of the plate bande, is therefore that by
which Y is made equal to H, and the line of resistance
made to touch the upper surface of the plate bande in F.
Now this least value of P is, by the principle of least
resistance (see Theory of the Arch), the actual value of the
resistance at A,
L..tan.a=.... (407).
Eliminating a between equations (405) and (407),
P=LHp,/1 +I... (408).
Multiplying equations (405) and (407) together,
P sin. a=L2^.... (409).
Now P sin. a represents the horizontal thrust on the point
of support A. From this equation it appears, therefore, that
the horizontal thrust upon the abutments of a straight arch
is wholly independent of the depth H of the arch, and that
it varies as the square of the length L of the arch; so that
the stability of the abutments of such an arch is not at all
diminished, but, on the contrary, increased, by increasing
the depth of the arch. This increase of the stability of the
abutment being the necessary result of an increase of the
vertical pressure on the points of support, accompanied by
no increase of the horizontal thrust upon them.




404               TIE PLATE BANDE.
308. The loaded plate bande.
It is evident that the effect of a loading, distributed
m-,-fi.,    uniformly over the extrados of the
l:__1_,_-' \: W  plate bande, upon its stability, is in
X tn/ 1 I i'fa,  every respect the same as would be
produced if the load were removed,,\        /.?/i  and the weight of the material of
e..,/ ~-, the bande increased so as to leave
f? \ ".:m  the entire weight of the structure
unchanged. Let p3 represent the
X\./        weight of each cubic foot when thus
increased, t2 the weight of each
cubic foot of the load, and H, the height of the load; then
^3HL =t^,HL + t,H,L,
Hi
-. p3=, +    * * * * *. (410).
The conditions of the stability of the loaded plate bande
are determined by the substitution of this value of,3 for p,
in the preceding article.
309. Conditions necessary that the voussoirs of aplate bande
may not slip upon one another.
It is evident that the inclination of every other resultant
pressure to the perpendicular to the surface of its corresponding joint, is less than the inclination of the resultant
^w          _n 8pressure or resistance P, to the
V,   r..,,C~. perpendicular to the joint AD.
I":fI:''El111l1 m  If, therefore, the inclination be
Aiffs          not greater than this limiting anPr~                - l  gle of resistance, then will every
pi i?-     other corresponding inclination
be less than it, and no voussoir
e5          Ei —.:~ will therefore slip upon the surface of its adjacent voussoir. Now the tangent of the inclination P to the perpendicular to AD is represented by cot. a
211
or by -  (equation 40T); the required condition is therefore
determined by the inequality,
2H
< tan...... (411).
JL




THE SLOPING BUTTRESS.           405
It is evident that the liability of the arch to failure by the
slipping of its voussoirs, is less as its depth is less as compared to its length. In order the more effectually to protect the arch against it, the voussoirs are sometimes cut of
the forms shown by the dotted lines in the preceding figure,
their joints converging to a point. The pressures upon the
points A and B are dependent upon the form of that portion
of the arch which lies between those points, and independent of the forms of the voussoirs which compose it; these
pressures, and the conditions of the equilibrium of the piers
which support the arch, remain therefore unchanged by this
change in the forms of the voussoirs.
310. To determine the conditions of the equilibrium of
the upright piers or columns of masonry which form the
abutments of a straight arch, supposing them to be terminated, as shown in the figure, on a different level from the
extrados CD of the arch, let b be taken to represent the
elevation of the top of the pier above the point A; then will
Lb
b tan. a, or 2  (equation 407), represent the distance AG
(p. 383), or the value of k —a). Substituting for k in equation (377) and also the values of P sin. a, P cos. a, from
equations (409) and (405), we have
l- E \
y=   -L2... A(412);
(   ax,+HL
which is the equation to the line of resistance of the pier, a
representing its thickness, b the height of its summit above
the springing A of the arch, L the length of the arch, P the
weight of a cubic foot of the material of the arch or abutment (supposed the same).
The conditions of the stability may be determined from
this equation as in the preceding articles. If the arch be
uniformly loaded, the value of P3 given by equation (410)
must be substituted for p'.
311. THE OENTRE OF GRAVITY OF A BUTTRESS WHOSE FACES
ARE INCLINED AT ANY ANGLE TO THE VERTICAL.
Let the width AB of the buttress at its summit be repre



406              THE SLOPING BUTTRESS.
sented by a, its width CD at the base by b,
I   its vertical height AF by c, the inclination
//  /1  of its outer face or extrados BC to the
/! i|  vertical by a,, that of its intrados AD by
/I l^ 7[T..Let I represent the centre of gravity of
/,i i. j the parallelogram ADEB, and K that of
/ /   /|  | I' the triangle BCE, and G that of the butC     f- B" d, tress; draw HM, GL, KN, perpendiculars
upon AF.   Then representing GL by,
and observing that the area ADEB is represented by ac,
the area EBC by ~(bh- )c, and the area ADCB by ~(a+b)c,
ac. H  + (b-a)cKN    2a. HM + (b-a)KN
f(ca+b)c              a+b
Now HM-=Hh +M=-a        -+ c tan. ca= —(a+e tan. a2),
KN-SKl+lA+kS2= (b-a)+a+23 tan. a
l(b-+2a+2c tan. 2);
Substituting these values and reducing,
(-    + ab +  2) + (ca- 2b) c tan. a,
3(a + b)
b= CD=CF-DF=c tan. a, + a-c tan. a,; also (a2+ab+62)
=(b-a)2 + 3ab: =2(tan.al -tan.a2)2 + 3ac(tan.at- tan.2) + 3a2,
(a+2b) c tan. 0a= t2c (tan. ac,-tan. a)+3 3  c tan. a2
= 2ce (tan. ac —tan. a,) tan. ao2 + 3ae tan. ag;. (c +ab +b)+(a+ 2b) c tan. a,=-2 (tan. a —tan. a2)
+ 3ac tan. a + 3a2..-.   2 c(tan. 2a-tan. 22)+ ac tan. a+2  (414) 
c(tan. c1- tan. a2)+ 2a        (4
312. THE LINE OF RESISTANCE IN A BUTTRESS.
Let LM represent any horizontal section of the buttress,
TK a vertical line through the centre of
C.B XA  J  gravity of that portion AMLB of the but"    - W.[''''  tress which rests upon this section. Produce LM to meet the vertical AE in V,
i/i/    and let KIVY=  and AV=x; then is the
/. ----- 1.  value of X determined by substituting x
for c in equation (414).  Let PO be the
direction in which a single pressure P is
applied to overturn the buttress. Take
* This equation is, of course, to be adapted to the case in which the inclination of AD is on the other side of the vertical, as shown by the dotted line
Ad by muaking a,, and therefere tan. ad negative.




THE SLOPING BUTTRESS.            407
OS to represent P in magnitude and direction, and ON to
represent the weight of the portion AMLB of the buttress;
complete the parallelogram SN, and produce its diagonal
OR to Q; then will OR evidently be the direction of the
resultant pressure upon AMLB, and Q a point in the line of
resistance.
Let VQ=y, AG-=k, ZGOT=t, _=weight of each cubic
foot of material; and let the same notation be adopted in
other respects as in the last article. By similar triangles,
QK RI
OK- OI
QK=QV -KV=y-X,
OK=TK-TO=TK-TG cot. GOT=x-(X+k) cot. t,
RI=RN sin. RNI=P sin. t,
OI= ON + NI=IAV(AB + LM)+ RN cos. RNI=
x {2a + x (tan. a — tan. 2) t + P cos. t;
— X                    P sin. t
**x —(X+k) cot. ~i-x {2a+x(tan. a,-tan. an  ) +P cos. 
Transposing and reducing,
~XPx 12a+ x (tan. a- tan. a2)} +P (x sin. -- k cos. t
"       j~x a12a-x (tan. a —tan. a,)} +P cos. t
but substituting x for e in equation (414), and multiplying
both sides of that equation by the denominator of the fraction in the second member, and by the factor ~x, we have
xXZx {2a + x (tan. a- tan. acc) } +X3 (tan. a — tan. 2a) +
BExa tan. aC,+Ixa2.y'.Y
iuYx (tan.2a,-tan.2a2)+-,xa tan. al+-juxa22P(x sin. t-k cos. t)
/x 12a+-x(tan.al-tan.a2) } +2P cos.   41);
which is the equation to the line of resistance in a buttress.
If the intrados AD be vertical, tan. a2 is to be assumed =0.
If AD be inclined on the opposite side of the vertical to that
shown in the figure, tan. ac is to be taken negatively. The
line of resistance being of three dimensions in x, it follows
that, for certain values of y, there are three possible values
of x; the curve has therefore a point of contrary flexure.
The conditions of the equilibrium of the buttress are deter



408              WALL SUSTAINING THE
mined from its line of resistance precisely as those of the
wall.
Thus the thickness a of the buttress at its summit being
given, and its height c, and it being observed that the distance CE is represented by a+ c tan. o,, the inclination a, of
its extrados to the vertical may be determined, so that its
line of resistance may intersect its foundation at a given distance m from its extrados, by solving equation (415) in respect to tan.,, having first substituted c for x and a+ c tan.,-m for y; and any other of the elements determining the
conditions of the stability of the buttress may in like manner
be determined by solving the equation (the same substitutions being made in it) in respect to that element.
313. A WALL OF UNIFORM THICKNESS SUSTAINING THE PRESSURE OF A FLUID.
If E be taken to represent the surface of the fluid, IK any
l__X            section of the wall, and EP two thirds
E__~   ~ the depth EK; then will P be the centre of pressure* of EK, the tendency
of the fluid to overturn the portion, ~  -    IAKIB of the wall being the same as
I —: — =~ —   would be produced by a single pressure
y — B         applied perpendicular to its surface at.- A  t-=~ 1P, and being equal in amount to the
______      weight of a mass of water whose base,' i — lo..is equal to EK, and its height to the
depth of the centre of gravity of EK, or
to yEK. Let AK=x, AE=e, weight of each cubic foot of
thlie fluid=1,;.P. P = (-e).(x-e)  i (x-e)21
Let the direction of P intersect the axis of the wall in O;
let it be represented in magnitude by OS; take ON to
represent the weight of the portion AKIB of the wall; complete the parallelogram SN, and produce its diagonal to
meet IK in Q; then will Q be a point in the line of resistance. Let QM=y, AB=a, weight of each cubic foot of
QM RN        o
material of wall= P. By similar triangles, QMO N   Now
* Treatise on "Hydrostaties and Hydrodynamics," by the author of this
work, Art. 88. p. 26.




PRESSURE OF A FLUID.            409
QM=y, MO=PKE=EK==_(x-e), RN=OS=P=
i,(x —e), NO=weight of ABIK -=ax;
Y __i1( -e)2..           - (-e)3
" (x-e)-    ax; "y 6 { 
Dividing numerator and denominator of this equation by
s,, and observing that the fraction - represents the ratio a
of the specific gravities of the material of the wall and the
fluid, we have
y= (e.....(416);
which is the equation to the line of resistance in a wall of
uniform thickness, sustaining the pressure of a fluid.
314. To determine the thickness, a, of the wall, so that its
height, h, being given, the line of resistance may intersect
its foundation at a given distance, m, within the extrados.
Substituting, in equation (416), h for a, and ~a-m for y,
and solving the resulting equation in respect to a, we obtain
a=m+     m'2+ I    -... (417.)
Equation (416) may be put under the form y=
-x -2 (1 -;;   whence it is apparent that y increases continually with x; so that the nearest approach is made by
the line of resistance, to the extrados of the pier, at its
lowest section. m therefore represents, in the above expression, the modulus of stability (Art. 286).
315. The conditions necessary that the wall should not be
overthrown by the slipping of the courses of stones on one
another.
The angle SRO represents the inclination of the resultant
pressure upon the section IK to the perpendicular; the proposed condition is therefore satisfied, so long as SRO is less
than the limiting angle of resistance p.




410               WALL SUSTAINING THE
Now, tan. SR     OS   Rl N ~t  (x —e)2
Now, tan. SRO-SR      RN —O  -  —; the proposed condition is therefore satisfied, so long as  a-  <tan. qp; or.
reducing this inequality, so long as
x<e+aatan., { 1     + (~) }t) I}.   (418.)
316. TmE STABILITY OF A WALL OF VARIABLE THICKNESS
SUSTAINING THE PRESSURE OF A FLUID.
Let us first suppose the internal face AB of the wall to be
vertical; let XY be any section of it,
P the centre of pressure of EX, and
SM a vertical through the centre of
---    gravity of the portion AXYD of the
wall. Produce the horizontal direc/.' —-  tion of the pressure P of the fluid,
supposed to be collected in its centre
Y __i/  I       of pressure, to meet MS in S, and let
9/ N"            SK be taken to represent it in ma-g____/       - nitude, and ST to represent the weight
c^d'' l         of the portion AXYD of the wall,
and complete the parallelogram   STRK; then will its
diagonal SR represent the direction and amount of the
resultant pressure upon the mass AXYD, and if it be produced to intersect XY in Q, Q will be a point in the line of
resistance.
Let AX-x, XQ=y, MX=X, AE=e, AD=a, inclination
of DC to vertical- a, p=weight of cubical foot of wall,
y,=weight of cubical foot of fluid. By similar triangles,
QM RT
SM-ST' Now
QM=QX-MX=y-X, SM=PX=3EX*=                 -(x —e);
RT=pressure of fluid on EX=    IEX.IEX=  1,(x — e)t+:
ST= weight of mass AY =~ 22a+ x tan. a }xtF.
* The centre of pressure of a rectangular plane surface sustaining the
pressure of a fluid is situated at two thirds the depth of its immersion.Hydrostatics, p. 26.
The pressure of a heavy fluid on any plane surface is equal to the weight
of a prism of the fluid whose base is equal in area to the surface pressed, and
its height to the depth of the centre of gravity of the surface pressed.Hydrostatics, Art. 31.




PRESSURE OF A FLUID.            411
Y- X       jwl(x-e)-.. _ ^
* (x-e)  12ax=x-2tan. a}l'h
Let -=a; then, if the fluid be water, a represents the
specific gravity of the material of the wall; and if not, it
represents the ratio of the specific gravities of the fluid and
wall.
y —=          (x- )s
*  ~y a\37 2a +? x tan. a'
Now making a2a=O in equation (414), and substituting 
for al, and x for c,
2 tal. tan.   a tan a +'3 tan. + ax2 tan. a + a'x
x atan. a + 2a         2ax+ X2 tan. a
Adding this equation to the preceding,
I-(x-e) + jx3 tan.2a + ax tan. a +a x
y=          I2+xtan. a              * * (419);
Y'        2ax+0 tan. a
which is the equation to the line of resistance to the wall,
the conditions of whose stability may be determined from it
as before (see Arts. 291. 293.).
317. The conditions necessary ihat no course of stones composing the wall mcy slip) upon the subjacent course.
This condition is satisfied when the inclination of SQ to
the perpendicular to the surface of contact at Q is less than
the limiting angle of resistance qp; that is, when QSM<<p,
or when
RT           ~(x —e)a
tan. p>tan. QSM, or > -T, or >(    +tn );
(~^\  (^y-e)
or tan.      2    xp > -..-.... (420).
No course of stones will be made by the pressure of the
fluid to slip upon the subjacent course so long as this condition is satisfied.
It is easily shown that the expression forming the second
member of the above inequality increases continually with




412          THE NATURAL SLOPE OF EARTH.
x, so that the obliquity of the resultant pressure upon each
course, and the probability of its being made to slip upon
the next subjacent course, is greater in respect to the lower
than the upper courses, increasing with the depth of each
course beneath the surface of the fluid.
EARTH WORKS.
318. The natural slope of earth.
It has been explained (Art. 241.) that a mass, placed upon
an inclined plane and acted upon by no other forces than its
weight and the resistance of the plane, will just be supported
when the inclination of the plane to the horizon equals the
limiting angle of resistance between the surface of the plane
and that of the mass which it supports; so that the limiting
angle of resistance between the surfaces of the component
parts of any mass of earth might be determined by varying
continually the slope of its surface until a slope or inclination
was attained, at which particular slope small masses of the
same earth would only just be supported on its surface, or
would just be upon the point of slipping down it. Now this
process of experiment is very exactly imitated in the case of
embankments, cuttings, and other earth-works, by natural
causes. If a slope of earth be artificially constructed at an
inclination greater than the particular inclination here
spoken of; although, at first, the cohesion of the material
may so bind its parts together as to prevent them from sliding upon one another, and its surface fiom assuming its
natural slope, yet by the operation of moisture, penetrating
its mass and afterwards drying, or under the influence of
frost, congealing, and in the act of congelation expanding
itself, this cohesion of the particles of the mass is continually
in the process of being destroyed; and thus the particles, so
long as the slope exceeds the limiting angle of resistance,
are continually in the act of sliding down, until, when that
angle is at length reached, this descent ceases (except in so
far as the particles continue to be washed down by the rain),
and the surface retains permanently its natural slope.
The limiting angle of resistance p is thus determined by
observing what is the natural slope of each description of
earth.




THE PRESSURE OF EARTH.                   413
The following table contains the results of some such
observations *:
NATURAL SLOPES OF DIFFERENT KINDS OF EARTH.
Nature of Earth.       Natural Slope.   Authority.
Fine dry sand (a single experiment)-  21~        Gadroy.
Ditto -   -   -   -   -    -    34~ 29'       Rondelet.
Ditto -  -    -   -   -    -    39~           Barlow.
Common earth pulverised and dry -  46~ 60'       Rondelet.
Common earth slightly damp -  -    54~           Rondelet.
Earth the most dense and compact-  55~           Barlow.
Loose shingle perfectly dry  -     39~           Pasley.
SPECIFIC GRAVITIES OF DIFFERENT KINDS OF EARTH.
Nature of Earth.                Specific Gravity.
Vegetable earth  -   -   --                         1-4
Sandy earth --                                      1'6
Marl               -                                19
Earthy sand  -   -   -        -                     1'7
Rubble masonry of calcareous earth or siliceous stones  1-7 to 2'3
Rubble masonry of granite  -      -                 2'3
Rubble masonry of basaltic stones -  -  -  -        25
319. THE PRESSURE OF EARTH.
Let BD    represent the surface of a wall sustaining the
pressure of a mass of earth whose surface AE is horizontal.
Let P represent the resultant of the pressures sustained
by any portion AX of the wall; and let the cohesion of the
particles of the earth to one another be neglected, as also
their friction on the surface of the wall.   It is evident that
* It is taken from the treatise of M. Navier, entitled Resume dun Cours de
Construction, p. 160.




414             THE PRESSURE OF EARTH.
any results deduced in respect to
VI                the dimensions of the wall, these
lw ^' —r,=i-  elements of the calculation being
i::T. %,;;: ^ neglected, will be in excess, and
-.,-;.  err on the safe side... ti.-J~;E'  Now the mass of earth which.i;:c,;&,  presses upon AX may yield in the
i,:^. %  direction of any oblique section
-- a:i^;S,.,  XY, made from X to the surface.  AE of the mass. Suppose YX to
[H' 1.;  be the particular direction in which,,- ~.:.,,  it actually tends to yield; so that
":        lt. if AX  were removed, rupture
would first take place along this
section, and AXY be the portion of the mass which would
first fall. Then is the weight of the mass AYX supported
by the resistances of the different elements of the surface AX
of the wall, whose resultant is P, and by the resistance of
the surface XY on which it tends to slide. Suppose, now,
that the mass is upon the point of sliding down the plane
XY, the pressure P being that only which is just sufficient
to support it; the resultant SR of the resistances of the
different points of XY is therefore inclined (Art. 241.) to the
normal ST, at an angle RST equal to the limiting angle of
resistance p between any two contiguous surfaces of the
earth.
Now the pressure P, the weight W of the mass AXY, and
the resistance R, being pressures in equilibrium, any two of
them are to one another inversely as the sines of their inclinations to the third (Art. 14.).
P   sin. WSR           sin. WSR
\         iWsin. PSR;'  sin. PSR 
But WSR=WST-RST=AYX-RST=i —t-p,
if AXY=t; PSR=PST+RST=AXY+RST=t+qp.:.P=Wcot. (t+p).... (421).
Also W=iP1AX. AY=~ —\P2 tan. t; if p,=weight of each
cubic foot of earth, and AX=az;:. P= =: 2 tan. t cot. ( +).... (422).
Now it is evident that this expression, which represents




THE PRESSURE OF EARTH.                 4.15
the resistance of the wall necessary to sustain the pressure of
the wedge-shaped mass of earth AXY, being dependent for
its amount upon the value of t (so that different sections,
such as XY, being taken, each different mass cut off by such
section will require a different resistance of the wall to support it), may admit of a maximum value in respect to that
variable.*  And if the wall be made strong enough to supply
a resistance sufficient to support that wedge-shaped mass of
earth whose inclination t corresponds to the maximum value
of P, and which thus requires the greatest resistance to support it; then will the earth evidently be prevented by it from
slipping at any inclination whatever, for it will evidently not
slip at that angle, the resistance necessary to support it at
that angle being supplied; and it will not slip at any other
angle, because more than the resistance necessary to prevent
it slipping at any other angle is supplied.
If, then, the wall supplies a resistance equal to the maximum value of P in respect to the variable t, it will not be
overthrown by the pressure of the earth on AX. Moreover,
if it supply any less resistance, it will be overthrown; there
not being a sufficient resistance supplied by it to prevent the
earth from  slipping at that inclination t which corresponds
to the maximum value of P.
To determine the actual pressure of the earth on AX, we
have then only to determine the maximum value of P in respect to t.
This maximum value is'that which satisfies the conditions
dP          d 2P
-=0, and -<0.
di          dt
But differentiating equation (422) in respect to t, we obtain
by reduction
dP =,      sin. 2(t+p))-sin. 2L
di -4       cos. 2t sin. 2(t+)  -         )
Let the numerator and denominator of the fraction in the
* The existence of this maximum will subsequently be shown: it is, however, sufficiently evident, that, as the angle t is greater, the wedge-shaped mass
to be supported is heavier; for which cause, if it operated alone, P would become greater as t increased. But as L increases, the plane XY becomes less
inclined; for which cause, if it operated alone, P would become less as t in
creased. These two causes thus operating to counteract one another, deter
mine a certain inclination in respect to which their neutralising influence is the
least, and P therefore the greatest.
-t Church's Diff. andInt. Cal., Art. 41.




416                 REVETEMENTS.
second member of this equation be represented respectively
dC?        1 idl,?  d
by p and q; therefore  -=i2 * - (3      P   b  when
<2         q2     -d) but\ we
dP                            C12P      id ~   1
d-  0,, 0; in this case, therefore, -P=tl 2l-d-. Whence
it follows, by substitution, that for every value of t by which
the first condition of a maximum is satisfied, the second differential co-efficient becomes
d2P,cos. 2(t p) —cos. 2t
dt2.=~P ~x  2. (424).
-d2 P 1   cos. 2t sin. a(t+p)..  (42
Now it is evident from equation (423) that the condition
d=I0 is satisfied by that value of t which makes 2(+p)= -
di
-r-2L, or
l   -2_..... (425).
And if this value be substituted for t in equation (424), it
becomes
d2              -sin. p..
=Plx -_      -.... (426);
cos. 2-) sin. 4+ )
which expression is essentially negative, so that the second
condition is also satisfied by this value of t. It is that, therefore, which corresponds to the maximum value of P; and
substituting in equation (422), and reducing, we obtain for
this maximum value of P the expression
PhiPh     tan.2a )... (427);
which expression represents the actual pressure of the earth
on a surface AX of the wall, whose width is one foot and its
depth x.
REVETEMENT WALLS.
320. If, instead of a revetement wall sustaining the pres



REVETEMENTS.                 417
sure of a mass of earth, the weight
A.   of each cubic foot of which is ret:..:-.:::-,;i, presented by p,, it had sustained;     i.,,,:.,....;   the pressure of a fuid, the weight:'  i-:..: i  of each cubic foot of which was re^ ^j^^a^^                  \4  2/5
M'..-  X'': presented by,1 tan.2 — 2) then
/"       -...":::::: would the pressure of that fluid.:,,.. -l' 5  upon the surface AX have been.....,....  represented* by lt2tan.2(-)'-:.i::-     so that the pressure of a mass of
earth upon a revetement wall (equation 427), when its surface is horizontal (and when its horizontal surface extends,
as shown in the figure, to the very surface of the wall), is
identical with that of an imaginary fluid whose specific gravity is such as to cause each cubic foot of it to have a weight
t2, represented in pounds by the formula, — ptan.27 —).... (428);
Substituting this value for, in equations (416) and (419),
we determine therefore, at once, the lines of resistance in
revetement walls of uniform and variable thickness, under
the conditions supposed, to be respectively
y-= — =                  - 4 an.. (430);
2ax+x2 tan. a
where a represents the ratio of the specific gravity of the
material of the wall to that of the earth. The conditions of
the equilibrium of the revetement wall may be determined
from the equation to its line of resistance, as explained in
the case of the ordinary wall.
* Hydrostatics, Art. 31.
27




418                  REVETEMENTS.
321. The conditions necessary that a reuetement wall may
not be overthrown by the slipping of the stones of any
course upon those of the subjacent course.
These are evidently determined from the inequality (420)
by substituting a2 (equation 428) for a, in that inequality;
we thus obtain, representing the limiting angle of resistance
of the stones composing the wall by p, to distinguish it from
that p of the earth,
1       en' q    (x-e)2
tan. p,>- tan.   -       - 2ax+x tan.a  (431);
al"G   \4  2 2ax+x' tan.a...
where a represents the ratio of the specific gravity of the
material of the wall to that of the earth.
As before, it may be shown from this expression that the
tendency of the courses to slip upon one another is greater
in the lower courses than the higher.
322. The pressure of earth whose surface is inclined to the
horizon.
Let AB represent the surface of such a mass of earth, YX
the plane along which the
5i-1- f               ~~rupture of the mass in
contact with the surface
- -------- -— ______ E  AX of a revetement wall::I.-n~'.-     tends to take place, AX=
i;::.      x,  AXY =,    XAB =/.
5;~;-.-:_          Then if W   be taken to...-_:::.:... Trepresent the weight of
m -:s:::,:. "; <.     the mass AXY, it may be.?.;..l.'::.         shown, as in Art. 319,
equation (421), that P=
W cot. (l +P).
Now W= I,,AX.AY.sin. P, AY=          - s —'. theresin. (t +)'
2for  sin. t sin. /3   $
fore W — =~,  (sin. + /) -=  cot. t + cot. /;.p=,. cot. (t+) +3).
cot. t cot. 3     (432)
Now the value of t in this function is that which renders
it a maximum (Art. 319). Expanding cot. (t+q), and dif



REVETEMENTS.                 419
ferentiating in respect to tan. l, this value of t is readily
determined to be that which satisfies the equation
cot. t=tan. PA+sec. (p 4/1+cot. /3 cot. qp.... (433).
Substituting in equation (432), and reducing,
Clos.A  o
P =aft    1 + sin. 9 4/ + cot. 9 cot..e  (4c34
From which equation it is apparent, that the pressure of the
earth is, in this case, identical with that of a fluid, of such a
density that the weight 3,, of each cubic foot of it, is represented by the formula
(    cos. qp      )
P IZQ- }l...... (435).
-13   1 + sin. p v4/1 cot. g cot.   (43
The conditions of the equilibrium of a revetement wall
sustaining the pressure of such a mass of earth are therefore
determined by the same conditions as those of the river wall
(Arts. 313 and 316).
323. THE RESISTANCE OF EARTH.
Let the wall BDEF be supported by the resistance of a
mass of earth upon its sury-''-.,:~  pface AD, a pressure P, ap-:T-T      ~     plied to its opposite face,.S.:i-i;^.,S; tending to overthrow it. Let:.:-.    |-.,:.,:.- the surface All of the earth.':.-l:'.be horizontal; and let. Q'L ^^i.m-l|l:|i' represent the pressure which,.^.;.^: I   g.:E-'; being applied to AX, would.;~~.::e::,.k.  just be sufficient to cause the
%z iTI!i= ^^l:mass of' earth in contact
-~-T I^i^; with that portion of the wall:,..:.:,:, to yield; the prism AXY
xS a>              slipping backwards upon the
surface XY. Adopting the same notation as in Art. 319,
and proceeding in the same manner, but observing that RS
is to be measured here on the opposite side of TS (Art. 241),
since the mass of earth is supposed to be upon the point of
slipping upwards instead of downwards, we shall obtain
Q=1pOa tan. I cot.(  ).....- (436).




420            WALLS BACKED BY EARTH.
Now it is evident that XY is that plane along which rup.
ture may be made to take place by the least value of Q; t
in the above expression is therefore that angle which gives
to that expression its minimum value. Hence, observing
that equation (436) differs from equation (422) only in the
sign of p, and that the second differential (equation 426) is
rendered essentially positive by changing the sign of p, it is
apparent (equation 427) that the value of Q necessary to
overcome the pressure of the earth upon AX is represented
by
Q1    ~tan-2 (~+P)... (437).
324. It is evident that a fluid would oppose the same
resistance to the overthrow of the wall as the resistance of
the earth does, provided that the weight P4 of each cubic
foot of the fluid were such that
4-  tan.   + ).... (438);
so that the point in AX at which the pressure Q may be
conceived to be applied, is situated at ids the distance AX.
325. The stability of a wall of uniform thickness which a
given pressure P tends to overthrow, and which is sustained by the resistance of earth.
Let y be the point in which any section XZ of the wall
would be intersected by the
-E?      B               resultant of the pressures
*r-" r  ~     upon the wall above that sec1.A L.yT    tion, if the whole resistance:,.::.     Q, which the earth in con-,,.;Li^^...- tact with AX is capable of
i E Esupplying, were called into'-_:'.::action.       et BX=x, Xy=y,:::.. BA    e, BE-a, B      =k,;.,'ia;.ll-   weight of cubic feet of ma-',,,...:i     terial of wall=-, inclination.,.:Gi,.: of P to vertical=O. Taking
-':.. --  the moments about the point
y of the pressures applied to BXZE, we have, by the principle of the equality of moments, observing that XQ=i




WALLS BACKED BY EARTH.            421
(x-e), and that the perpendicular from y, upon P is represented by x sin. — (k-y) cos. 6,
P x sin. -(k-y) cos. 8t =~(x-e)Q+(a-y(tax;
or substituting for Q its value (equation 437), and solving in
respect to y,.V4((x-e)'+~-Va2x-P(x sin. 6-k cos. 8)
y-    --  P cos. O + a....(439).
Now it is evident that the wall will not be overthrown
upon any section XZ, so long as the greatest resistance Q,
which the superincumbent earth is capable of supplying, is
sufficient to cause the resultant pressure upon EX to intersect that section, or so long as y in the above equation has
a positive value; moreover, that the stability of the wall is
determined by the minimum value of y in respect to x in
that equation, and the greatest height to which the wall can
be built, so as to stand, by that value of w which makes y=O.
326. The stability of a wall which a given pressure tends to
overthrow, and which is supported by a mass of earth
whose surface is not horizontal.
Let 3 represent the inclination of the surface AB of earth
to the horizon.  By reasoning:~^^%~~ ~         similar to that of Art. 322., it is
E~~.~~   ~    apparent that the resistance Q
of the earth in contact with any
given portion AX of the wall to
displacement, is determined by
i,'%     assigning to p a negative value
1 s x  ~ in equation (434). Whence it
I;.S^  follows, that this resistance is
^E —;-r.:: —  equivalent to that which would
-:": Hi\\?     be produced by the pressure of
-;^;^:"^   a fluid upon the wall, the weight
m —E.i:,t:;-  6 of each cubic foot of which
was represented by the formula
cos.          )2..
cos.=|ji^ ------  _..... f - (440).
1 —sin. p 4/1 -cot. p cot. i/
The conditions of the stability of an upright wall subjected to any given pressure P tending to overthrow it, and




42-2                  REVETEMENTS.
sustained by the pressure of such a mass of earth, are therefore precisely the same as those discussed in the last article;
the symbol p, (equation 439) being replaced by J5 (equation
440).
327. The stability of a revetement wall whose interior face
is inclined to the vertical at any angle; taking into account
the friction of the earth upon theface of the wall.
Let ac represent the inclination of the face BD of such a
wall to the vertical, p2 the limiting angle of resistance
between the mass of earth and the surface of the wall; and
let the same notation be adopted as in the last article in
respect to the other elements of the
question, and the same construction.. ja:. made. Draw PQ perpendicular to BD;.I:. g.  then is the direction PS of the resist-: I:-;I.SI;  ance of the wall upon the mass of earth,
i,:_  -~'  evidently inclined to QP at an angle
i -  QPS equal to the limiting angle of
resistance A,, in the state bordering.,:'.  upon motion by the overthrow of the
_,:.      wall* (Art. 241.).
5__,N*^:   Draw Pn horizontally and Xa verti-;  cally, produce TS and RS to meet it in
D"......   m and n, and let aXY=t
P    sin. WSR   sin. (WST-TSR)
W= sin. PSR = sin. (RmP + SPm).
But WST=AYX         -aXY=     —, TSR=p,
2         2
RmP=TnP +mSn= oXY+RST=t +,
SP     SPm = SPQ + QPn=  2 + 2;
p    sin.(- t    )       Cos. (
*'W          s sn. t  sin. (+t+9+M2)   sin (t +  p + 92)'
Also W=-T1aX. AYv-=ix2 (tan. +tan. a,); if aX=x,
* It is not only in the state of the wall bordering upon motion that this
direction of the resistance obtains, but in every state in which the stability of
the wall is maintained. (See the Principle of Least Pesi,;tance.)




REVETEMENTS.                 423
pj. 2p!,.cos. (t + P) tan. t+ tan. a)..  (441).
sin. (C+2P+(2)j     *  *')
Assuming a2 + + 2,=/3, then differentiating in respect to t,
dP
and assuming -- =0, we obtain by reduction
-(tan. t + tan. a,) cos. ( —(p)+
cos. (t+p) sin. (t+ /) sec. 2t=O; or,
-(tan. t + tan. a,) (1 + tan. t tan. p) +
(1-tan. t tan. qp) (tan. t + tan. =)= 0;. tan.2 t + 2 tan. t tan. / - tan. 3 cot. q +
(cot. p + tan. /) tan. aO =0.
Solving this quadratic in respect to tan. t, neglecting the
negative root, since tan. L is essentially positive, and reducing,
tan. t=(tan. /3-tan. ac2) (tan./3 +cot. )1 —tan. /... (442.)
Now the value of t determined by this equation, when
substituted in the second differential coefficient of P in
respect to t, gives to that coefficient a negative value; it
therefore corresponds to a maximum value of P, which
maximum determines (Art. 319.) the thrust of the earth
upon the portion AX of the wall. To obtain this maximum
value of P by substitution in equation (441), let it be
observed that
cos. (t+p) 1-tan. l tan. q (cos. p \
sin. (+ 3)- (tan. t+tan. /)  cos. /'
— tan..       t11 =  +1 + t. /3 tan. p-tan. p (tan. /tan. ao2)(tan. / +cot. p)1,
=tan. q (tan. 3+cot. q) {(tan. / +cot. ) — (tan. /-tan. aC)+
tan. +tan. /=(tan. i/ +cot. 0)(tan. / —tan. a2)i;
cos. (t+q) sin. p i tan. /+cot. qp
" sin. (t+/3)-cos. l  tan. /3-tan. a2 1 
Also tan. + tan. a2=(tan./3 +cot. p)+(tan. /3tan. a,) —(tan. -tan. a,)
-(tan. /-tan. ao2) I(tan. /+cot. ) — (tan. g/-tan. a3)q,
si   tn.                         0
~p=In. - -   (tan. / +cot. p) —(tan. /3-tan. a,)'2;
cos. /3




424                  REVETEMENTS.
which expression may be placed under the following form,
better adapted to logarithmic calculation,
sinp  p i /(cos. (l 3p)\s   (sin. (I3-,) ) 
= Pcos.U         s.COS. Ca
or substituting for f its value a2 + + +,
pp  i   sin.  | Mcoseo (a,+,)
2cos.'2 (,  +2)   sin. P 
cosi, (q)  } q..)(443).
Co(s a2
By a comparison of this equation with equation (427) it
is apparent, that the pressure of a mass of earth upon a
revetement wall, under the supposed conditions, is identical
with that which it would produce if it were perfectly fluid,
provided that the weight of each cubic foot of that fluid had
a value represented by the coefficient of ~-2 in the above
equation; so that the conditions of the stability of such a
revetement wall are identical (this value being supposed)
with the conditions of the stability of a wall sustaining the
pressure of a fluid, except that the pressure of the earth is
not exerted upon the wall in a direction perpendicular to its
surface, as that of a fluid is, but in a direction inclined to
the perpendicular at a given angle, namely, the limiting
angle of resistance.
328. THE PRESSURE OF EARTH WHICH SURMOUNTS A REVETEMENT WALL AND SLOPES TO ITS SUMMIT.
Hitherto we have supposed the surface of the earth whose.a....x pressure is sustained by a revetei,:.: ment wall to be horizontal; let us.^5:&'i-~: now suppose its surface to be ele-.B.t^dl: S.: rvated above the summit of the wall,
/'3  and to descend to it by the natural
/,I:f.gI <.;  slope; the wall is then said to be
p/la —S  "  surccharged, or to carry a parapet.
-/ i', N.   Let EF represent the natural slope
l f~  1of the earth, FY its horizontal sur—'-;r: j   face, BX any portion of the internal./,   J._....:;  face or intrados of the wall, P the
-.,'-i-?  horizontal pressure just necessary




REVETEMENTS.                 425
to support the mass of earth HXYF, whose weight is W,
upon the inclined plane XY. Produce XB and YF to meet
in A, and let AX=x, AH=c, AXY=t, vL,=weight of each
cubic foot of the earth, ( the natural slope of its surface
FE. Now it may be shown, precisely by the same reasoning as before, that the actual pressure of the earth upon the
portion BX of the vwall is represented by that value of P
which is a maximum in respect to the variable t; moreover,
that the relation of P and i is expressed by the function P
='W cot. (t+); where W=P (area HlXYF)==y(AXYAHF)=p,,(9  tan. t-b  cot. <p);.P.P==-1(x2 tan    ot. t) cot. +)cot. )..... (444).
Expanding cot. (t +q),
(2 tan. t-c' cot. p) (1 —tan. I tan. p)
— 2-1      tan. t+tan. p
To facilitate the differentiation of this function, let
tan. + tan. p be represented by z, and let it be observed
that whatever conditions determine the maximum value of P
in respect to z determine also its maximum value in respect
to.* Then tan. t=z-tan. cp; therefore 1-tan. t tan. q=
1-z tan. p +tan. 29= — tan. + sec.'2. Also, x' tan. -
0' cot. qp=x2z- (x' tan. p+' cot. p).
Substituting these values in the preceding expression for
P, and reducing,
p-=  -i     a2     (x tan. p-+c2 cot. (p) sec.'
P=I       z&-z   tan. j-                    +
x (sec.' + tan.') + 2.... (445).
dcP     2i    t      (e' tan. p+c 2 cot. p) sec. 2@:d  =    1  -  tan.             2     - + 
RdP dPdz    d2P d2P/dz \ dP d2z   dz
d     z For d' and — 2=- \ —   -- d-; now - =sec. 2t, thered dz     d   L'dz' \dL/  dz dtO 
dP dP    2
fore -=, sec.'t; and for all values of e less than, sec.'t has a finite
2'
dP        dP
value, so that- 0 when — 0.
dt       dz 
d P  d2P d'P c(dz\   )      d.
When, moreover, dP-~} dPs- dP(;  so that, when   is negative,
di'P
-,- is also negative.
ILZ




426                   REVETEMENTS.
d~P      (za tan. p +-c' cot. p) sec. l
dz2 -- 1 —l        Z
The first condition of a maximum is therefore satisfied by
the equation
( t   ( tan. g + - cot. A) sec. 2  0  
— m'2tan. +    -=0....(446);
or, solving this equation in respect to z, and reducing, it is
satisfied by the equation
=z-  (sec. p + $2 cosec. ).
Now the second condition of a maximum is evidently
satisfied by any positive value of z, and therefore by the
positive root of this equation. Taking, therefore, the positive sign, substituting for z its value, and transposing,
/        <32
tan. t=  sec. 29 + - cosec. 29  -tan. p.....(447);
which equation determines the tangent of the inclination
AXY to the vertical, of the base XY of that wedge-like
mass of earth HXYF, whose pressure is borne by the surface BX of the wall.  To determine the actual pressure
upon the wall, this value of tan. t must be substituted in the
expression for P (equation 445). Now the two first terms
of the expression within the brackets in the second member
of that equation may be placed under the form
-i ta-     (0 tan. (p+c2 cot. p) sec. p }
-z a? tan. p+ — 2                   -v
But it appears by equation (446) that the two terms which
compose this expression are equal, so that the expression is
equivalent to — 2z2 tan. p; or, substituting for the value of
_2
z, to -2   tan. p (sec. 29q-  cosec. 2(p), or to -2x sec. qP
(a2 tan. 29 + 02) T. Substituting in equation (445),
P=V-, t -2x sec. p(x2 tan. 2qp + c) 1 + (x tan. 2p - 2) +  sec. 2}'. P= —, x sec. q —(X tan. 9 p+ c2) 2.. (448);
by which expression is determined the actual pressure upon
a portion of the wall, the distance of whose lowest point
trom A is represented by x.




REVETEMENTS.                  427
329. The conditions necessary that a revetement wall carrying a parapet may not be overthrown by the slipping of
any course of stones on the subjacent course.
Let q, represent the limiting angle of the resistance of the
stones of the wall upon one another; and let OQ represent
&   r    Y the direction of the resultant pressure, -;:P. on the course XZ. The proposed
-.i,;  conditions are then involved (Art.
l.::,:, JI    141.) in the inequality 9p>QOM, or
P,- tan., > tan. QOM, or tan.        1 >
2&':_ ~ RS       P
A;      OS>weight of BZ-    or substituting
]/ 4...: for P its value (equation 448), and
/.,.s ~'5:;i ".4tt;i(2ax + 2 tan. c) for the weight of
-,       ^ - — x,,:  BZ, it appears that the proposed
i/,. conditions are determined by tlhe
].'::-:'a:..::.-^ inequality
ta.,> (il Ixsec9 -(tan. 2+cY2)l 2,)   (449)
ta  p>  q2ax+-   tan. a 
330. The line of resistance in a revetement wall carrying a
parapet.
Let OT be taken to represent the pressure P, and OS the
weight of BZ. Complete the parallelogram ST, and produce its diagonal OR to Q; then will Q be a point in the
line of resistance. Let AX=x, QX=y, AB=b, AP=X,
XM   X, W = weight of BZt. 3y similar triangles, QM
RS
OS; but QM=(y-X), OM=x-X,           RS=P, OS-=W;
y^X    P        Wx+ Px-PX
O-X =W'.Ye= —... (450).
Now the value of x is determined from equation (414), by
* The influence, upon the equilibrium of the wall, of the small portion of
earth BHE is neglected in this and the subsequent computation.
1 The influence of the weight of the small mass of earth BEH which rests
on the summit of the wall is here again neglected.




428                   REVETEMENTS.
substituting in that equation (x-b) for c: whence we obtain,
observing that tan. a2=0, and substituting a for a,,
i=(x-b)2 tan. 2ac+ a(x-) tan. + a
(x-b) tan. + 2 2
Also   W=-it( —b) (- b) tan. a+ 2a..... (451);... WX=e-^( — b) I(x -b)2 tan.'a + a(x-b) tan. a +  }.
It remains, therefore, only to determine the value of the
term P. X. Now it is evident (Art. 16.) that the product
P. X is equal to the sum of the moments of the pressures
upon the elementary surfaces which compose the whole surface BX. But the pressure upon any such elementary surface, whose distance from A is x, is evidently represented
dP                                             dP
by dPAx*; its moment is therefore represented by d-PxaA
and the sum of the moments of all such elementary pressures
dP
by 2 dPxx, or when ax is infinitely small, by
jf e',xdx; therefore P. X= fdPxdc.
b                           b
But differentiating equation (448),
dP                                         xtan.'   )
dP =      se. x-( tan. +c)1      s. - ( tan.2 (p + c2) 
Performing the actual multiplication of the factors in the
x tan.'
second member of this equation, observing that  tan. 2 + c2
(*L'    0      ~(x' tan. +c -)9
(I2tan. +6c)-c       (2 tan. 2 )-   t     + 
(=x' ~. ~~~    =(= tan. p+c, -  (mt~.-+e), and re(X'tan. 2p + Ca)i               ('tan.  + c')'
ducing we obtain
* P being a function of x, let it be represented by f(x); then will f(x) represent the pressure upon a portion of the surface BX terminated at the distance
x from A, and f(x-Ax) that upon a portion terminated at the distance x-+Ax;
therefore f(x+-Ax)-fx will represent the pressure upon the small element Ax
of the surface included between these two distances. But by Taylor's theorem,
dP     d2P (Ax)'
f(x ~- A) -f -fx  Ax ~ +    -4d, &c.; therefore, neglecting terms in.
dx     dxp 1j 2:dP
volving powers of Ax above the first, pressure on element = TAX.
dX




THE ARCH.                          429
-dP='     (sec. p +tan.'a)-2 sec. p (a tan. 2P +')I +
( 2 sec. p  I
(x tan. +c) T
Multiplying this equation by x, and integrating between the
limits b and x,
(isec.p2 q +tan.' p)(x-bS)-  sec. 9 cot.'2p (X tan.2 
P.X= +'        c4)3-   -(b2 tan.2 p+C2(2 +c2 sec. p cot.2 g 
i{(2 tan.2 c2)- -(62 tan.2 q + c2)It.... (452). 
This value of P. X being substituted in equation (450),
and the values of W, W,   P, from   equations (448) and
(451), the line of resistance to the revetement wall will be
determined, and thence all the conditions of its stability
may be found as before.*
THE ARCH.
331. Each of the structures, the conditions of whose stability (considered as a system of bodies in contact), have
hitherto been discussed, whatever may have been the pressures supposed to be insistent upon it, has been supposed to
rest ultimately upon a single resisting surface, the resultant
of the resistances on the different elements of which was at
once determined in magnitude and direction by the resultant
of the given insistent pressurest being equal and opposite
to that resultant.
The arch is a system of bodies in contact which reposes
ultimately upon two resisting surfaces called its abutments.
The resistances of these surfaces are in equilibrium with the
* The limits which the author has in this work imposed upon himself do not
leave him space to enter further upon the discussion of this case of the
revetement wall, the application of which to the theory of fortification is so
direct and obvious. The reader desirous of further information is referred to
the treatise of M. Poncelet, entitled " Memoire sur la Stabilite des Revetements et de leurs Fondations." He will there find the subject developed in all
its practical relations, and treated with the accustomed originality and power
of that illustrious author. The above method of investigation has nothing in
common with the method adopted by M. Poncelet except Coulomb's principle
of the wedge of maximum pressure.
t The weight of the structure itself is supposed to be included among these
pressures.




430         THE PRINCIPLE OF LEAST RESISTANCE.
given pressures insistent upon the arch (inclusive of its
weight), but the direction and amount of the resultant pressure upon each surface is dependent upon the unknown
resistance of the opposite surface; and thus the general
method applicable to the determination of the line of
resistance, and thence of the conditions of stability, in that
large class of structures which repose on a single resisting
surface, fails in the case of the arch.
332. THE PRINCIPLE OF LEAST RESISTANCE.
If there be a system of pressures in equilibrium among which
are a given number of resistances, then is each of these a
minimumn, subject to the conditions imposed by the equilibrium of the whole.*
Let the pressures of the system, which are not resistances,
be represented by A, and the resistances by B; also let any
other system of pressures which may be made to replace the
pressures B and sustain A, be represented by C.
Suppose the system B to be replaced by C; then it is
apparent that each pressure of the system C is equal to the
pressure propagated to its point of application from    the
pressures of the system A; or it is equal to that pressure,
together with the pressure so propagated to it from the
other pressures of the system C. Ini the former case it is
identical with one of the resistances of the system B; in the
latter case it is greater than it. Hence, therefore, it appears
that each pressure of the system  B is a mtnimum, subject
to the conditions imposed by the equilibrium of the whole.
If the resultant of the pressures applied to a body, other
than the resistances, be taken, it is evident from the above
that these resistances are the least possible so as to sustain
that resultant; and therefore that if each resisting point be
capable of supplying its resistance in any direction, then are
all the resistances parallel to one another and to the resultant of the other pressures applied to the body.
* The principle of least resistance was first published by the author of this
work in the Philosophical Magazine for October, 1833.




THE ARCH.                    431
333. Of all the pressures which can be applied to the highest
voussoir of a semi-arch, different in their amounts and
points of application, but all consistent with the equilibrium of the semi-arch, that which it would sustain from
the pressure of an opposite and equal semi-arch is the least.
Let EB represent the surface by which an arch rests upon
D
either of its abutments; then are the resistances upon the
different points of that surface (Art. 331.) the least pressures,
which, being applied to those points, are consistent with the
equilibrium of the arch. They are, moreover, parallel to one
another: their resultant is therefore the least single pressure,
which, being applied to the surface EB, would be sufficient
to maintain the equilibrium of the arch, if the abutment were
removed.
Now, if this resultant be resolved vertically and horizontally, its component in a vertical direction will evidently be
equal to the weight of the semi-arch: it is therefore given in
amount. In order that the resultant may be a minimum, its
vertical component being thus given, it is therefore necessary
that its horizontal component should be a minimum; but
this horizontal component of the resistance upon the abutment is evidently equal to the pressure P of the opposite
semi-arch upon its key-stone: that pressure is therefore a
minimum; or, if the semi-arches be equal in every respect,
it is the least pressure which, being applied to the side of the
key-stone, would be sufficient to support either semi-arch;
which was to be proved.
The following proof of this property may be more intelligible to some readers than the preceding. It is independent
of the more general demonstration of the principle of least
resistance.*
* See Memoir by the author of this work in Mr. Hann's " Treatise on the
Theory of Bridges," p. 10.




432                   THE ARCH.
The pressure which an opposite semi-arch would produce
upon the side AD of the key-stone, is equal to the tendency
of that semi-arch to revolve forwards upon the inferior edges
of one or more of its voussoirs. Now this tendency to motion
is evidently equal to the least force which would support the
opposite semi-arch. If the arches be equal and equally
loaded, it is therefore equal to the least force which would
support the semi-arch ABED.
334. GENERAL CONDITIONS OF THE STABILITY OF AN ARCH.*
Suppose the mass ABDO to be acted upon by any number
of pressures, among which
own\  ^  is the pressure Q, beingthe
resultant of certain resist/,?               ances, supplied by different'       /S           points in a surface BD;
common to the mass and to. /'7                 an immoveable    obstacle
/,ih\,BE.
_jL~ ii_ y -~Now it is clear that un-'D _Q                      der these circumstances we
may vary the pressure P,
] Rae__       both as to its amount, direction, and point of application in AC, without disturbing the equilibrium, provided
only the form and direction of the line of resistance continue
to satisfy the conditions imposed by the equilibrium of the
system.
These have been shown (Art. 283) to be the following:
that it no where cut the surface of the mass, except at P,
and within the space BD; and that the resultant pressure
upon no section MN of the mass, or the common surface BD
of the mass and obstacle, be inclined to the perpendicular to
that surface, at an angle greater than the limiting angle of
resistance.
Thus, varying the pressure P, we may destroy the equilibrium, eithe:, first, by causing the resultant pressure to
take a direction without the limits prescribed by the resistance of any section MN through which it passes, that is,
without the cone of resistance at the point where it inter* Theoretical and Practical Treatise on Bridges, vol. i.; Memoir by the author of this work, p. 11.




THE AERH.                    433
sects that surface; or, secondly, by causing the point Q to
fall without the surface BD, in which case no resistance can
be opposed to the resultant force acting in that point; or,
thirdly, the point Q lying within the surface BD, we may
destroy the equilibrium by causing the line of resistance to
cut the surface of the mass somewhere between that point
and P.
Let us suppose the limits of the variation of P, within
which the first two conditions are satisfied, to be known; and
varying it, within those limits, let us consider what may be
its least and greatest values so as to satisfy the third condition.
Let P act at a given point in AC, and in a given direction.  It is evident that by diminishing it under these
circumstances the line of resistance will be made continually
to assume more nearly that direction which it would have
if P were entirely removed.
Provided, then, that if P were thus removed, the line of
resistance would cut the surface,-that is, provided the
force P be necessary to the equilibrium,-it follows that by
diminishing it we may vary the direction and curvature of
the line of resistance, until we at length make it touch some
point or other in the surface of the mass.
And this is the limit; for if the diminution be carried
further, it will cut the surface, and the equilibrium will be
destroyed. It appears, then, that under the circumstances
supposed, when P, acting at a given point and in a given
direction, is the least possible, the line of resistance touches
the interior surface or intrados of the mass.
In the same manner it may be shown that when it is the
greatest possible, the line of resistance touches the exterior
surface or extrados of the mass.
The direction and point of application of P in AC have
here been supposed to be given; but by varying this direction and point of application, the contact of the line of
resistance with the intrados of the arch may be made to
take place in an infinite variety of different points, and each
such variation supplies a new value of P. Among these,
therefore, it remains to seek the absolute maximum and
minimum values of that pressure.
In respect to the direction of the pressure P, or its inclination to AC, it is at once apparent that the least value of
that pressure is obtained, whatever be its point of application, when it is horizontal.
There remain, then, two conditions to which P is to be
subjected, and which involve its condition of a minimum.
28




434                    THE ARCH.
The first is, that its amount shall be such as will give to the
line of resistance a point of contact with the intrados / the
second, that its point of application in the iey-stone AC
shall be such as to give it the least value which it can receive,
subject to the first condition.
335. PRACTICAL CONDITIONS OF THE STABILITY OF AN ARCH
OF UNCEMENTED STONES.
The condition, however, that the resultant pressure upon
the key-stone is subject, in respect to the position of its
point of application on the key-stone, to the condition of a
minimum, is dependent upon hypothetical qualities of the
masonry. It supposes an unyielding material for the archstones, and a mathematical adjustment of their surfaces.
These have no existence in the uncemented arch. On the
striking of the centres the arch invariably sinks at the
crown, its voussoirs there slightly opening at their lower
edges, and pressing upon one another exclusively by their
upper edges. Practically, the line of resistance then, in an
arch of uncemented stones, touches the extrados at the crown;
so that only the first of the two conditions of the minimum
stated above actually obtains: that, namely, which gives to
the line of resistance a contact with the intrados of the
arch. This condition being assumed, all consideration of
the yielding quality of the material of the arch and its
abutments is elirminated.
The form of the solid has hitherto been assumed to be
given, together with the positions of the different sections
made through it; and the forms of its lines of resistance and
pressure, and their directions through its mass have thence
been determined.
It is manifest that the converse of this operation is possible.
Having given the form and position of the line of resistance or of pressure, and the positions of the different sections
to be made through the mass, it may, for instance, be
inquired what form these conditions impose upon the surface
which bounds it.
Or the direction of the line of resistance or pressure and
the form of the bounding surface may be subjected to certain
conditions not absolutely determining either.
If, for instance, the form of the intrados of an arch be
goiven, and the direction of the intersecting plane be always




THE ARCH.                     485
perpendicular to it, and if the line of pressure be supposed
to intersect this plane always at the same given angle with
the perpendicular to it, so that the tendency of the pressure
to thrust each from its place may be the same, we may
determine what, under these circumstances, must be the
extrados of the arch.
If this angle ezcal constantly the limiting angle of resistance, the arch is in a state bordering upon motion, each
voussoir being upon the point of slipping downwards, or upwards, according as the constant angle is measured above or
below the perpendicular to the surface of the voussoir.
The systems of voussoirs which satisfy these two conditions are the greatest and least possible.
If the constant angle be zero, the line of pressure being
every where perpendicular to the joints of the voussoirs, the
arch would stand even if there were no friction of their surfaces. It is then technically said to be equilibriated; and
the equilibrium of the arch, according to this single condition, constituted the theory of the arch so long in vogue,
and so well known from the works of Emerson, Hutton, and
Whewell. It is impossible to conceive any arrangement of
the parts of an arch by which its stability can be more
effectually secured, so far as the tendency of its voussoirs to
slide upon one another is concerned: there is, however,
probably, no practical case in which this tendency really
affects the equilibrium.  So great is the limiting angle of
resistance in respect to all the kinds of stone used in the
construction of arches, that it would perhaps be difficult to
construct an arch, the resultant pressure upon any of the
joints of which above the springing should lie without this
angle, or which should yield by the slipping of any of its
voussoirs.
Traced to the abutment of the arch, the line of resistance
ascertains the point where the direction of the resultant
pressure intersects it, and the line of pressure determines the
inclination to the vertical of that resultant;* these elements
determine all the conditions of the equilibrium of the abutments, and therefore of the whole structure; they associate
themselves directly with the conditions of the loading of the
arch, and enable us so to distribute it as to throw the points
of rupture into any given position on the intrados, and give
to the line of resistance any direction which shall best con* The inclination of the resultant pressure at the springing to the vertical
may be determined independently of the line of pressure, as will hereafter be
shown




436       THE LINE OF RESISTANCE IN THE ARCH.
duce to the stability of the structure; from known dimensions, and a known loading of the arch, they determine the
dimensions of piers which will support it; or conversely,
from known dimensions of the piers they ascertain the
dimensions and loading of the arch, which may safely be
made to span the space between them.
336. To DETERMINE THE LINE OF RESISTANCE IN AN ARCH
WHOSE INTRADOS IS A CIRCLE, AND WHOSE LOAD IS COLLECTED OVER TWO POINTS OF ITS EXTRADOS SYMMETRICALLY
PLACED IN RESPECT TO THE CROWN OF THE ARCH.
Let ADBF represent any portion of such an arch, P a
W^h ~  ~    pressure applied at its extreme
D           voussoir, and X and Y the hoi j........" —," rizontal and vertical compo-:''r^^^! nents of any pressure borne
~/(~",      I' ~ upon the portion DT of its ex~/\ ~sv:    I ~,trades, or of the resultant of'.e >. ^          many number of such pressures;
let, moreover, the co-ordinates,
l[ I   "    - a, from the centre C, of the point'L,  of application of this pressure,
"  or of this resultant pressure, be
x and y.
c    Let the horizontal force P
be applied in AD at a vertical distance p from C; also let
CT represent any plane which, passing through C, intersects
the arch in a direction parallel to the joints of its voussoirs.
Let this plane be intersected by the resultant of the pressures applied to the mass ASTD in R. These pressures are
the weight of the mass ASTD, the load X and Y, and the
pressure P. Now if pressures equal and parallel to these,
but in opposite directions, were applied at R, they would of
themselves support the mass, and the whole of the subjacent
mass TSB might be removed without affecting the equilibrium. (Art. 8.) Imagine this to be done; call M the weight
of the mass ASTD, and h the horizontal distance of its centre of gravity from C, and let CR be represented by p, and
the angle ECS by 0, then the perpendicular distances from
C of the pressures M1 +Y and P-X, imagined to be applied
to R, are p sin. 6 and p cos.:; therefore by the condition of
the equality of moments,




THE ANGLE OF RUPTURE IN THE ARCH.        437
(M+Y) p sin. O+(P-X) p cos.8=MhA+Yx-Xy+Pp;
MAh+Yx-Xy+Pp
(453),
P'(M+y) sin. + (P-X) cos.         4'' 
which is the equation to the line of resistance.
M and h are given functions of 0; as also are X and Y, if
the pressure of the load extend continuously over the surface
of the extrados from D to T.
It remains from this equation
to determine the pressure P, beD.-     ing that supplied by the opposite'"<    -- < _ semi-arch. As the simplest case,
X.^-^ —'i      let all the voussoirs of the arch
1<jr -.be Of the same depth, and let the
inclination ECP of the first joint
of the semi-arch to the vertical be
represented by o, and the radii
of the —extrados and intrados by
R  and,.. Then, by the known
principles of statics.*
R   0
Mh=     fr2 sin. Oddr= -*(1R-)e(cos. -cos. 0);
r  0
also, M=(R3-r-)(-e);. p I1(R2 — )(0 -) sin. O+Y sin. 8-X cos. 8+P cos. 6 =
~(R8-r~)(cos. o-cos. ) +Yx-Xy+-P.....  (454),
which is the general equation to the line of resistance.
THE ANGLE OF RUPTTUE.
337. At the points of rupture the line of resistance meets
the intrados, so that there p-=r: if then T be the corresponding value of 6,
r {g(Re2-)(T-_) sin. F +Y sin. T -X cos. T +P cos. IF =
(R3-_3)(cos. e-cos.tF) +Yx-Xy+Pip...       (455).
* See Note 1 at end of PART IV.-ED.




438               THE ANGLE OF RUPTURE
Also at the points of rupture the line of resistance touches
the intrados, so that there -dp -   0-; assuming then, to
simplify the results, that the pressure of the load is wholly
in a vertical direction, so that X=0, and that it is collected
dY
over a single point of the extrados, so that  =0,- and dife
d'
ferentiating equation (454), and assuming d=0 when 6=-Y
and p=r, we obtain
-(R2- r) (-    ) cos.' +- (2'-r ) sin.' +
Y cos. — P sin.'} -=(Ri3-r3) sin. a;
hence, assuming R=r (1+ a),
i cP                      6-316+ )o}+
+2 +a (2a+ 3)    tan.Y=  { r-3a(a+2)        +
3 (a + 2)..... (456).
Eliminating (v —o) between equations (455) and (456), we
have
{      r+a' (~c+~) } sec.2 - {      P+(l~ + a+l+1)cos. }
sec. Y=-   (a + )..... (457).
Eliminating P between equations (455) and (456), and
reducing.
Y*   pcos.+sin -        1    (2     )(-os..      -e)
- (a2 +a') sin. y- (a+a +a) cos. O     -(42 + +a)cos.  sin. V..... (458).
C This equation might have been obtained by differentiating equation (454)
dP
in respect to P and 0, and assuming d = 0 when r and TI are substituted for
p and 0; for if that equation be represented by u=0, u being a function of
du dP  du       du dp  du                 du
P, p and 0, -+   =, oand    -+ 0=-. The same result +-=0 is
dP dO  TO       TP TO  TO                 dO
dP      dp
therefore obtained, whether we assume d = 0, or -d — 0 which last supposition is that made in equation (45), whence equation (458) has resulted. The
tion is that made in equation (456), whence equation (458) has resulted. The




IN THE ARCH.                  439
~A:""-k~   i    Let AP=X; therefore       -
\''         (l (1+) cos. 0. Substituting this
"    value of  5
_.,
2 { - sin. Yf +(1 +X)cs) S. O  -1  = (a + a)
3 1-(1+X) cos.c 0 Cos.  ( —~)+(cos.'-cos. 0) sin.Y t +
X(i-a + 18') sin. T cos...... (459),
by which equation the angle of rupture' is determined.
If the arch be a continuous segment the joint AD is vertically above the centre, and CD coinciding with CE, =-0;
if it be a broken segment, as in the Gothic arch, 0 has a
given value determined by the character of the arch. In the
pure or equilateral Gothic arch,  = 30~. Assuming  =-0,
and reducing,
Y     x' /             \ -c
--- -.                  (tan  x       tan. —x co (t.''y-vers. T } +X(2 +ia).....    (460.)
It may easily be shown that as'Y increases in this equation, Y increases, and conversely; so that as the load is
increased, the points of rupture descend. When Y=O, or
there is no load upon the extrados,
tan. —Xcot.T 2Y-vers.+T+-         ci -0...  (461).
dP
hypotheses eO, p r, determine the minimum of the pressures P, which
dO
being applied to a given point of the key-stone will prevent the semi-arch from
turning on any of the successive joints of its voussoirs.




440              THE LINE OF RESISTANCE.
When x=0, or the load is placed on the crown of the
arch,
Y    (2   a) vers. - (2 +    )      +           (462).3)
tan. — 2;cot. Y
When   -- (tan.   - X cot. I )= 0,   becomes infinite;
an infinite load is therefore required to give that value to the
angle of rupture which is determined by this equation.
Solved in respect to tan. -  it gives,
p () ~V()2+(+x)
tan. - -         w.... (463).
2            2+x
No loading placed upon the arch can cause the angle of rupture to exceed that determined by this equation.
THE   LINE  OF RESISTANCE IN   A CIRCULAR ARCH    WHOSE
VOUSSOIRS ARE EQUAL, AND WHOSE LOAD IS DISTRIBUTED
OVER DIFFERENT POINTS OF ITS EXTRADOS.
338. Let it be supposed that the pressure of the load is
wholly vertical, and such that any; —.-^. portion FT of the extrados sustains..."   1 the weight of a mass GFTV imme-, " 1,^ diately superincumbent to it, and.VI 0,''I-T,   bounded by the straight line GV
jF,^'::', "B.inclined to the horizon at the ant \       J',;,   I gle t; let, moreover, the weight' Ably  "a    of each cubical unit of the load be
--',   I   equal to that of the same unit of
l^ y A ^  ar  ithe material of the arch, multiplied
by the constant factor i; then, re-" —-c presenting AD by R3, ACF by o,
-.4~~^          hACT by 0, and DZ by z, we have,
area GFTY =     TV. dz:
0




SEGMENTAL ARCH.                 441
but TV= MZ-(MT+VZ), and MZ = CD = R~.l/l, MT=
R cos. 0, VZ - DZ tan. t = R sin. 0 tan. t. Therefore MT+
VZ = R cos. 0+R sin. 0 tan. t = R {cos. 0 cos. t+sin. 0 sin. t}
sec. t =R cos. (0 —) sec. t;
TV= R {1 + —-cos.i (0 —) sec. t};
also, z=DZ=R sin. 0;
~ dO             J
0               0:area GFTV-.   do  RSI    1+P -
d-o
cos. (0-i) sec. t} cos. OdO;
0:. Y=weight of mass GFTV=~E p1 +^ + —sec.t cos.(0 —t)
cos. 0d0 = pR2  (1 +) (sin. 0-sin. ) — sec. & sin. (2-i) -
sin. (2 o-t)}-  (-(  ) }... (464).*
0
Yx = moment of GFTV=-R3       {(1 + 3)-sec. cos. (0- 
0
sin. 0 cos. Od0=ER3 {1(1 +/ ) (cos. 2 -cos. 20) — (CO.'8 -
cos. 30) — tan. t (sin. 0 —sill. 9)... 465).*
A SEGMENTAL ARCH WHOSE EXTRADOS IS HORIZONTAL.
7v_   _         339. As the simplest case, let us first.. _        suppose DV horizontal, the material of
2-, T,,, 1-1  the loading similar to that of the arch,
_,ZS n - i,   and the crown of the arch at A, so
_~<x >   ~  that t=O=,  =1, and e=0. Substituting the values of Y and Ya (equaY ".,, titions 464, 465) which result from these
-I',.  suppositions, in equation (455, solving
".\ j  that equation in respect to-, and reducing, we have, — =
* See Note 2, at end of PART IV.-ED.




442               THE GOTHIC ARCH.
(1-a) (1f+a)2(1+4-) sin.6.2i (l+a) (l-2) cos.-+( a2+ias-i) cos.I —y ksin.++
1+ —os.'..... (466Q
dP
Assuming — =0 (see note, page 438.), and X =a, and
reducing,
(1 - 2a) cos. 8   {(1- )(1+ 1P) + (1+a)(1- 2a) }   COs. 2 +
(1+ )2 +2(1 — a2) ( 1    cos. * + (1+ a)2   (+a)
2      a + la3
Co8 T  in -(1-) (1 q-i)   3(1+   - 1+i   - 0..(467).
os. } sin. T           3(1 + a)2
In the case in which the line of resistance passes through
the bottom of the key-stone, so that X=0, equation (466)
becomes
-  =i(1 + Y)2(1 + P)(1 -) (1 + cos. T)-.(I -+ a)'(1- 2a)
(1 +cos.'Y) cos.'-Y- Tcot. iY +=0.... (468);
whence assuming     =0, we have.(1 + ()2(1 2x) cos. 2  (1 + a)2 l(-a) 3 +(4-5a)} Cos. T +
- -{(1 + a)2 (1 -a)(1 + )+ ++)- a(1  )   0...(469.)
sin.                                            (469)
A  GOTHIC ARCH, THE EXTRADOS OF EACH SEMI-ARCH BEING
A STRAIGHT LINE INCLINED AT ANY GIVEN ANGLE TO THE
HORIZON, AND THE MATERIAL OF THE LOADING DIFFERENT
FROM THAT OF THE ARCH.
340. Proceeding in respect to this general case of the
stability of the circular arch, by precisely the same steps as
in the preceding simpler case, we obtain from equation
(455),
Yxa Y
(ia+a'l+a ) (cos.0-co.i,)-('a2  -a)(i -  )sin. "^-(-g- -Yin.-^
e"icoS.'k-(1+) Cos.e^*




THE GOTHIC ARCH.               443
in which equation the values of Y and Yx are those determined by substituting I for 0 in equations (464) and (465).
dP
Differentiating it in respect to,5 assuming,_=0 (note,
p. 438.), and X=a, we obtain
(a+la2-lc3-ia4) cos. 0 sin. S-(iab+o) sin.' cos. v(2 + a){ i-  l+ a) Cos. e Cos. } (T -v )
Y Yx
(- l_(l4-t+a) cosin. ccos. -    in(1+a) cos. E I1 d(Yx) sin. T' dY
r( dcs  d}     d'\ =0...(471).
Y     Yx
Substituting in this equation the values of  and -d, determined by equations (464) and (465) the following equa
tion will be obtained after a laborious reduction: it deter
mines the value of,Y:
A+B cos. Ty-   cos.2 3'-D cos.s3;+E sin. 3yF sin. I' cos. Y —G sin.3 I -H cot.' +
(v-e)     L
1(1 —K cos. )      + s -i n sn. 0... (472)
where
A=a(1+a)2 { (1+a) tan t sin8 — (1+~p )(12-1 +a) cos2 }
-( + a) COS. e t} + (2a + a  a-a4) Cos. o
B=(l+ )22i(1-a2) ( 1+t) cos. 0-(1-P)} +1.
C=a(1 +a)2 (l-a) ( +/3)+( +a) (l-2a) cos. }.
D =l.+(1 +)(l -2a).
E-=(l+a )2(1-2a) tan. -=3D tan. t.
F=v(1+a)S(1-2a) tan. t cos. ~=E(l+a) cos. o.
G=b-(l+Ui)\(l1-22a) tan. L=D tan. t.
H=i,(1 +a)j 2(1+P3)-sec. t cos. (0l-t) sin. 20.
I=i-(i-,,) (l+ x)2.
K=(l-a) cos. 0.
L=es(1 + o)2 2(1+ )-sec. t cos. (e-t)} sin. r.
Tables might readily be constructed from this or any of




444   AN ARCH SUSTAINING TIHE PRESSURE OF WATER.
the preceding equations by assuming a series of values of A,
and calculating the corresponding values of / for each given
value of a, L, P, 0. The tabulated results of such a series of
calculations would show the values of Y corresponding to
given values of a/,, A,  ~, 0. These values of Y being substituted in equation (470), the corresponding values of the
horizontal thrust would be determined, and thence the polar
equation to the line of resistance (equation 454).
A CIRCULAR ARCH HAVING EQUAL VOUSSOIRS AND SUSTAINING
THE PRESSURE OF WATER.
341. Let us next take a case of oblique pressure on the
V —_ D extrados, and let us suppose it to be
T       the pressure of water, whose surface
T I__ __B stands at a height /3R above the sumlmit of the key-stone. The pressure of
//  \ \    this water being perpendicular to the
\  i extrados will everywhere have its di-'\ i rection through the centre C, so that its
\   motion about that point will vanish,
and Yx-Xy=0; moreover, by the'  principles of hydrostatics,* the vertical
component Y of the pressure of the water, superincumbent
to the portion AT of the extrados, will equal the weight of
that mass of water, and will be represented by the formula
(464), if we assume t=0. The horizontal component Xt of
the pressure of this mass of water is represented by the
formula
0
X=L R2f{ +S -cos. ( sin. Od-=a(1 +)rl {(1 +/3)(cos. 0 —
cos. I)-X (cos.'0-cos. 8)}..... (473).
Assuming then 0=0, we have (equation 464), in respect
to that portion of the extrados which lies between the crown
and the points of rupture,
Y
=u-(1 +<)2 1 +/3) sin. i'- sin. 2'-il}i,
and (equation 473) 7==(1l+oc)2 1(1+/) vers. T-  sin. St,
* See Hydrostatics and Hydrodynamics, p. 30, 31.
* See Note 3, end of PART IV.-ED.




AN ARMOI SUSTALNING TIlE PRiESSURE OF WATER.  445
Y         X
*  sin. ~T —2 cos. = (1+a)2 { ( +2  ) vers. ~-_1  sin. ~}.... (474).
Substituting this value in equation (455), making Yx-Xy=0,
P
solving that equation in respect to - and making -=1+x,
we have
P  \i+a-~-z-(1+a)2} sin. — aa2+ia3+-u(l+a)' (1+f)} vers. (5)
r                      a+t-vers. t
If, instead of supposing the pressure of the water to be
borne by the extrados, we suppose it to
take effect upon the intrados, tending to
<~ x-_ hblow up the arch, and if P represent the
height of the water above the crown of
the intrados, we shall obtain precisely
the same expressions for X and Y as
before, except that r must be substituted
for (1 + )r, and X and Y must be taken
Y         X
negactively; in this case, therefore, A- sin. ~ --  os. T—,{+(1+ /) vers. T — -' sin.'}; whence, by substitution in
equation (455), and reduction,
P   (12~2+ )+ _i)'sin.-  a+2+       ~M-~3+(l+/1 vers. (
3-                   h X+ vers. T
ld P
Now by note, page 438, dy   =0; differentiating equations (475) and (476), therefore, and reducing, we have,
T   tan. - -Xcot. T  -vers. + AX=O..... (477);
which equation applies to both the cases of the pressure of a
fluid upon an arch with equal voussoirs; that in which its
pressure is borne by the extrados, and, that in which it is
borne by the intrados; the constant A representing in the
Ia+, 0.+ — a(-2+/ ) (1, a 
first case the quantity 2 3      r17) (1,   and in the
As~~ ~  c~ v ~7-.    cc)2. -.,+  t^(.+/3).
second case   -. — +i. If the line of resistance
2 r-  + 2 f  -.
pass through thle surimit of tlie key-stone, x must be taken =h.




446            EQUILIBRIUM OF AN ARCH.
If it pass along the inferior edge of the key-stone,
>-=0. In this second case, tan. -{' —sin. A} =0, therefore,
y-=0; so that the point of rupture is at the crown of the
arch. For this value of' equations (475) and (476) become
vanishing fractions, whose values are determined by known
methods of the differential calculus to be, when the pressure
is on the extrados,
r2=  -^    3 +  I3(1+^ a....(478);
when the pressure is on the intrados,
P
=a 03 —4.... (479).
It is evident that the line of resistance thus passes through
the inferior edge of the key-stone, in that state of its equilibrium which precedes its rupture, by the ascent of its crown.
The corresponding equation to the line of resistance is deterP
mined by substituting the above values of; in equation
(454). In the case in which the pressure of the water is
sustained by the intrados, we thus obtain, observing that
Y.     Y
-,sin. — r cos. = —p {(1+ I) vers. 6 —6 sin. 8);
2-  + 2-~ —(Ai +  2~ +  +a) cos.     4
p' (WM2 +M+1) sinew+(a-ia3 + )cos. -p(l+    * * (  )
If for any value of 0 in this equation, less than the angle
of the semi-arch, the corresponding value of p exceed
(1 + )r, the line of resistance will intersect the extrados, tld
the arch will blow up.
THE EQUILIBRIUM OF AN  RCH, THE CONTACT OF WHOSE
VOUSSOIRS IS GEOMETRICALLY ACCURATE.
842. The equations (459) and (456) completely determine




EQUILIBRIUM OF AN ARCH.           447
the value of P, subject to the first
of the two conditions stated in
D.....,, Art. 333., viz. that the line of resistance passing through a given.X.. d-  point in the key-stone, determined
I\,, |  by a given value of x, shall have a
r W     point of geometrical contact with
-y'"., \   the intrados. It remains now to
B t^^S \'S  determine it subject to the second.\4Q "i      condition, viz. that its point of apc   plication P on the key-stone shall
be such as to give it the least vat  ~I - -T~ lue which it can receive subject to
-—'~- ~     the first condition.  It is evident
that, subject to this first condition, every different value of
X will give a different value of; and that of these values of
I' that which gives the least value of P, and which corresponds to a positie value of X not greater than a, will be the
true angle of rupture, on the hypothesis of a mathematical
adjustment of the surfaces of the voussoirs to one another.
To determine this minimum value of P, in respect to the variation of IF dependent on the variation of X or ofp, let it be
observed that X does not enter into equation (456); let that
equation, therefore, be differentiated in respect to P and T,
dP
and let -  be assumed=O, and Y constant, we shall thence
dT
obtain the equation
sec. 2-         2....(481).
r2 +a2(2a+3)
whence, observing that
tan. T    - +2 (2a + 3)
j sin. 2= — f'    J --    2   )    tan.',
sec.7Y     3a(a + 2)
we obtain by elimination in equation (456)
4Y
sin. 2 —2yr  ( +     -2..... (482),
a(a + 2)r2
from which equation s may be determined.  Also by equation (481)




448     APPLICATIONS OF THE THEORY OF THE ARCH.
-P 13a(a~ + 2) cos. aa(2a + 3).... (483);
and by eliminating sec. fbetween equations (457) and (481),
and reducing,
P=(1 +X)cos. o=p -     a(a+2){    +a      ) 
a(2 + a+l) cos. @ —     }... (484).
The value of X given by this equation determines the actual
direction of the line of resistance through the key-stone, on
the hypothesis made, only in the case in which it is a positive
quantity, and not greater than a; if it be negative, the line
of resistance passes through the bottom of the key-stone, or
if it be greater than a, it passes through the top.
Such a mathematical adjustment of the surfaces of contact
of the voussoirs as is supposed in this article is, in fact, supplied by the cement of an arch. It may therefore be considered to involve the theory of the cemented arch, the influence on the conditions of its stability of the adhesion of its
voussoirs to one another being neglected. In this settlement,
an arch is liable to disruption in some of those directions in
which this adhesion might be necessary to its stability. That
old principle, then, which assigns to it such proportions as
would cause it to stand firmly did no such adhesion exist,
will always retain its authority with the judicious engineer.
APPLICATIONS OF THE THEORY OF THE ARCH.
343. It will be observed that equation (459) or (472)
determines the angle I' of rupture in terms of the load Y,
and the horizontal distance x of its centre of gravity from
the centre C of the arch, its radius r, and the depth owr of its
voussoirs; moreover, that this determination is wholly independent of the angle of the arch, and is the same whether
its arc be the half or the third of a circle; also, that if the
angle of the semi-arch be less than that given by the above
equation as the value of', there are no points of rupture,
such as they have been defined, the line of resistance passing
through the springing of the arch and cutting the intrados
there.




THEORY OF THE ARCH.               449
The value of'F being known from this equation, P is
determined from equation (456), and this value of P being
substituted in equation (454), the line of resistance is completely determined; and assigning to 0 the value ACB
(p. 437.), the corresponding value of p gives us the position
of the point Q, where the line of resistance intersects the
lowest voussoir of the arch, or the summit of the pier.
Moreover, P is evidently equal to the horizontal thrust on
the top of the pier, and the vertical pressure upon it is the
weight of the arch and load: thus all the elements are
known, which determine the conditions of the stability of a
pier or buttress (Arts. 293. and 312.) of given dimensions
sustaining the proposed arch and its oading.
Every element of the theory of the arch and its abutments
is involved, ultimately, in the solution in respect to,' of
equation (459) or equation (472)  Unfortunately this solution presents great analytical difficulties. In the failure of
any direct means of solution, there are, however, various
methods by which the numerical relation of T and Y may
be arrived at indirectly. Among them, one of the simplest
is this:Let it be observed that that equation is readily soluble in
respect to Y; instead, then, of determining the value of'F
for an assumed value of Y, determine conversely the value
of Y for a series of assumed values of T. Knowing the distribution of the load Y, the values of z will be known in
respect to these values of', and thus the values of Y may
be numerically determined, and may be tabulated. From
such tables may be found, by inspection, values of T corresponding to given values of Y.
The values of T, P, and r are completely determined by
equations (482, 483, 484), and all the circumstances of the
equilibrium of the circular arch are thence known, on the
hypothesis, there made, of a true mathematical adjustment
of the surfaces of the voussoirs to one another; and although
this adjustment can have no existence in practice when
the voussoirs are put together without cement, yet may it
obtain in the cemented arch. The cement, by reason of
its yielding qualities when fresh, is made to enter into so
intimate a contact with the surfaces of the stones between
which it is interposed that it takes, when dry, in respect
to each joint (abstraction being made of its adhesive properties), the character of an exceedingly thin voussoir, having
its surfaces mathematically adjusted to those of the adjacent
voussoirs; so that if we imagine, not the adhesive properties
29




450               APPLICATIONS OF THE
of the cement of an arch, but only those which tend to the
more uniform diffusion of the pressures through its mass, to
enter into the conditions of its equilibrium, these equations
embrace the entire theory of the cemented arch. The hypothesis here made probably includes all that can be relied
upon in the properties. of cement as applied to large structures.
An arch may FALL either by the sinking or the rising of
its crown. In the former case, the line of resistance passing
through the top of the key-stone is made to cut the extrados
beneath the points of rupture; in the latter, passing through
the bottom of the key-stone, it is made to cut the extrados
between the points of rupture and the crown.
In the first case the values of X, Y, and P, being determined as before and substituted in equation (454), and p
being assumed =-(1+  )r, the value of 0, which corresponds
to p=-(1 +o)r, will indicate the point at which the line of
resistance cuts the extrados. If this value of 0 be less than
the angle of the semi-arch, the intersection of the line of
resistance with the extrados will take place above the
springing, and the arch will fall.
In the second case, in which the crown ascends, let the
maximum value of p be determined from equation (454), p
being assumed =r; if this value of p be greater than R, and
the corresponding value of 6 less than the angle of rupture,
the line of resistance will cut the extrados, the arch will
open at the intrados, and it will fall by the descent of the
crown.
If the load be collected over a single point of the arch,
the intersection of the line of resistance with the extrados
will take place between this point and the crown; it is that
portion only of the line of resistance which lies between these
points which enters therefore into the discussion. Now if
we refer to Art. 336., it will be apparent that in respect to
this portion of the line, the values of X and Y in equations
(453) and (454) are to be neglected; the only influence of;these quantities being found in the value of P.




THEORY OF THE ARCH.             451
Excamle 1.-Let a circular arch of equal voussoirs have
the depth of each voussoir equal to
Tlth the diameter of its intrados, so
"V  ~  that oc=-2, and let the load rest upon
\______   it by three points A, B3, D of its
A exrados, of which A is at the crown
and B D are each distant from it 45~;:BX/jU       a and let it be so distributed that {ths
of it may rest upon each of the points
B and D, and the remaining - upon,. ~..,1  ~   A; or let it be so distributed within
-.....!-.    60~ on either side of the crown as to
~""               produce the same effect as though it....-;. -rested upon these points.....T:;       Then assigning one half of the load
upon the crown to each semi-arch,
and calling x the horizontal distance
of the centre of gravity of the load upon either semi-arch
from C, it may easily be calculated that - =  sin. 45=
~5303301. Hence it appears from equation (463) that no
loading can cause the angle of rupture to exceed 65~.
Assume it to equal 60~; the amount of the load necessary to
produce this angle of rupture, when distributed as above,
will then be determined by assuming in equation (460),
T=60~, and substituting a for x, -2 for a, and'5303301 for -.
Y                              Y
We thus obtain -='0138. Substituting this value of r, and
also the given values of a and T in equation' (457), and
observing that in this equation  is to be taken =1 +a and
P                                  P
o=0, we find  = — 11832. Substituting this value of  in
the equation (454), we have for the final equation to the line
of resistance beneath the point B'2426 vers. 8 +'1493
p   -r0138 sin. 6 + 1183 cos. 6 + -22 6 sin. 0'




452              APPLICATIONS OF TIHE
Y           If the arc of the arch be a complete semicircle, the value of p in this
equation corresponding to 0=- will
determine the point Q, where the
line of resistance intersects the abutment; this value is p=1-09r.
If the arc of the arch be the third
o of a circle, the value of p at the
C'~-:a abutment is that corresponding to
0-    this will be found to be r, as
3,
A*:if              it manifestly ought to be, since the
points of rupture are in this case at
the springing.
In the first case the volume of the semi-arch and load is
represented by the formula
r2     2 i2+ a 2+ rs,  g39
{(     ~x+ Y}- -3594r,
and in the second case by
(        + (+    } +  =) + 2442
Thus, supposing the pier to be of the same material as the
arch, the volume of its material, which would have a weight
equal to the vertical pressure upon its summit, would in the
first case be *3594r2, and in the second case *2442r2, whilst
the horizonital pressures P would in both cases be the same,
viz. *11832r2; substituting these values of the vertical and
horizontal pressures on the summit of the pier, in equation
(377), and for k writing  a —(p —), we have in the first
case'3594(a — 09r)r2.
*11832r — a 2
and in the second case,
- 2442 ar
~11832r? — ja;




THEORY OF THE ARCH.              453
where H is the greatest height to which a pier, whose width
is a, can be built so as to support the arch.
If a'2 —11832r'=0, or a=-4864r, then in either case the
pier may be built to any height whatever, without being
overthrown. In this case the breadth of tile pier will be
nearly equal to Ith of the span.
The height of the pier being given (as is commonly the
case), its breadth, so that the arch may just stand firmly
upon it, may readily be determined. As an example, let us
suppose the height of the pier to equal the radius of the
arch. Solving the above equations in respect to a, we shall
then obtain in the first case a -=2978r, and in the second
a= — 3r.
If the span of each arch be the same, and r, and r, represent their radii respectively, then r,=r sin. 60'; supposing
then the height of the pier in the second arch to be the same
as that in the first, viz. r,, then in the second equation we
must write for HI, r sin. 60~. We shall thus obtain for a the
value'28r,.
The piers shown by the dark lines in the preceding
figures are of such dimensions as just to be sufficient to
sustain the arches which rest upon them, and their loads,
both being of a height equal to the radius of the semicircular
arch.  It will be observed, that in both cases the load
Y= 0138r2, being that which corresponds to the supposed
angle of rupture 60~, is exceedingly small.
Example 2.-Let us next take the example of a Gothic
arch, and let us suppose, as in the last examples, that the
angle of rupture is 60~, and that a='2; but let the load in
this case be imagined to be collected wholly over the
crown of the arch, so that $ = sin. 30~. Substituting in equar
tion (459), 30~ for o, and 60~ for P, and -2 for a, and sin. 30~
M                                   Y
for -, we shall obtain the value'21015 for -; whence by
equation (457) P-=2405, and this value being substituted,
equation (457) P = -2405, and this value being substituted,




454                 APPLICATIONS OF THE
1....-.-....     equation (454) gives 1'145r for
|           the value of p when 0 =. We
have thus all the data for determining the width of a pier of
given height which will just
support the    arch.   Let the
height of the pier be supposed,
1^       ^:      as before, to equal the radius
of the intrados; then, since the
-/J-I'      X'\  weight of the semi-arch and its
load is'5556r2, and the horizontal thrust'2405r2, the width a
of the pier is found by equation
(379) to be'4195r.
The preceding figure represents this arch; the square,
formed by dotted lines over the crown, shows the dimensions
of the load of the same materials as the arch which will cause
the angle of the rupture to become 60~; the piers are of the
required width'4195r, such that when their height is equal
to AB, as shown in the figure, and the arch bears this insistent pressure, they may be on the point of overturning.
TABLES OF THE THRUST OF ARCHES.
344. It is not possible, within the limits necessarily
assigned to a work like this, to enter further upon the discussion of those questions whose solution is involved in the
equations which have been given; these can, after all, become accessible to the general reader, only when tables shall
be formed from them.
Such tables have been calculated with great accuracy by
M. Garidel in respect to that case of a segmental arch* whose
loading is of the same material as the voussoirs, and the extrados of each semi-arch a straight line inclined at any given
angle to the horizon. These tables are printed in the Appendix (Tables 2, 3).
* The term segmental arch is used, here and elsewhere, to distinguish that
form of the circular arch in which the intrados is a contiguous segment from
that in which it is composed of two segments struck from different centres, as
in the Gothic arch.




THEORY OF THE ARCH.               455
Adopting the theory of Coulomb*, M. Garidel has arrived
at an equationt which becomes identical with equation (472)
in respect to that particular case of the more general conditions embraced by that equation, in which -=1 and 0= 0.
By an ingenious method of approximation, for the. details
of which the reader is referred to his work, M. Garidel has
determined the values of the angle of rupture A, and the
quantity -,, in respect to a series of different values of a and
/3. The results are contained in the tables which will be
found at the end of this volume.
The value of P- being known from   the tables, and the
values of Y and Yx from equations (464), (465), the line of
resistance is determined by the substitution of these values
in equation (454). The line of resistance determines the
point of intersection of the resultant pressure with the summit of pier; the vertical and horizontal components of this
resultant pressure are moreover known, the former being the
weight of the semi-arch, and the other the horizontal thrust
on the key. All the elements necessary to the determination of the stability of the piers (Arts. 289 and 312) are
therefore known.
It will be observed that the amount of the horizontal
thrust for each foot of the width of the soffit is determined
by multiplying the value of -P, shown by the tables, by the
square of the radius of the intrados in feet, and by the
weight of a cubic foot of the material
* See Mr. Hann's Theory of Bridges, Art. 16.; also p. 24. of the Memoir on
the Arch by the author of this work, contained in the same volume.
f Tables des Poussdes des Voutes, p. 44. Paris, 1837. Bachelier.




456                      NOTES TO PART IV.
NOTE 1. PART IV.
The length of an elementary arc ds of the intrados AS subtending the angle
dO is expressed by rdO; an elementary volume of the arch will therefore be
expressed by rdodr; the perpendicular distance of the centre of gravity of
this volume from the vertical line CE is r sin. 0; the moment of this volume,
with regard to CE, is therefore rdOdrXr sin.0=r2dr sin. OdO; then from (Art.
31.) equation (20) there obtains
R     6
Mh=      r2dr sin. OdO.
r     0
NOTE 2. PART IV.-General integrals of equations 464, 465.
The general integral, (equation 464)
J{1+3-cos. (0-2) sec.       cos. dO=J-1+) coo. = 1  dfsec. t (cos. 0 cos. t+sin. 0 sin. ) cos. NOd=J 1+~ ) cos. d0Jsec.   cos. t cos.26d6 — sec. t sin. t sin. 0 cos. OdO.
But /(1+/f) cos. 60d=(1+P) sin.;    cjsec. t cos. t cos.'0dO=
sec. t Cos. fJ(. -os + 2O8)d6=sec. cos. t (,1 6-+   sin. 26 )Jsec. L sin I
P1I                          1
sin. 0 cos. 0d0=sec. sin. J -sin. 2 d(2) = -sec. l sin. I - cos. 2;
)4                         4,    1,(  l+  -cos. (0-t) sec. Lt Cos. 6d=(1+~ ) sin. 08-   sec. ~
I1~~~~~~~e.  1               1
(sin. 20 cos. d-sin.L cos. 20) —  = (1+' ) sin  — s.. sin.(20- ) —1
2                 4                 2
The general integral, f{(1+f)-sec. ~ cos. (0-t )} sin. 0 cos. 6dO, (equation
465), =f+(1+~) sin. 0 cos. OdO -Jsec. I cos.6 cos. t+sin.6 sin. 1  sin.6 os. 6d.
But (1+3) sin. o cos. do= (1+) sin. 20d20) -      (+) cos. 20=
2cos(1+)   26(2)= (    CS+ ) cos 2+ 1
4                         4




NOTES TO PART IV.                        457
Jsec. t cos. 0 cos. t+sin. 0 sin.  sin. 0 cos. Od.=: 0  cos.w sin. HdO+
Jtan. sin. 2'cos. OdO. = cos.20dcos. 0 -Jtan. c sin.'Od sin. 0=
1      0
-  COS.  -3 — sin. 30.'~ (1+B-)-sec. t cos.(0-t ) sin. a cos. Ld. -d0= — (1+3)cos. 0+! (1+-)+ -- os. 10 —- tan. L sin. 80.
4   3    3
NOTE 8. PART IV.
In equation (427), (Art. 319), by making 0=0, we obtain P=- A, xa; since
tan. -=1, and this answers to the case of the horizontal pressure of a perfect
fluid like water. From this expression there obtains dP=-u,xdx, to express
the elementary pressure at any depth x below the surface. This depth in
(Art. 341), equation (473), is TV=AD-AB=AD+AC-BC=fPR+R-R cos.0,.d. dP=uR(1l+-cos. Od) R(1.,/-cos. 0)=zlR2 {1+P-cos. 6} sin. Odo
0.     X. P=X=tR/f 1+~-cos. 0 } sin. odo.
Q




PAR T V.
THE STRENGTH       OF MATERIALS.
ELASTICITY.
345. From numerous experiments which have been made
upon the elongation, flexure, and torsion of solid bodies
under the action of given pressures, it appears that the
displacement of their particles is subject to the following
laws.
1st. That when this displacement does not extend beyond
a certain distance, each particle tends to return to the place
which it before occupied in the mass, with a force exactly
proportional to the distance through which it has been
displaced.
2dly. That if this displacement be carried beyond a
certain distance, the particle remains passively in the new
position which it has been made to take up, or passes finally
into some other position different from that from which it
was originally moved.
The effect of the first of these laws, when exhibited in
the joint tendency of the particles which compose any finite
mass to return to any position in respect to the rest of the
mass, or in respect to one another, from which they have
been displaced, is called elasticity. There is every reason to
believe that it exists in all bodies within the limits, more or
less extensive, which are imposed by the second law stated
above.
The force with which each separate particle of a body
tends to return to the position from which it has been
displaced varying as the displacement, it follows that the
force with which any aggregation of such particles, constituting a finite portion of the body, when extended or
compressed within the limits of elasticity, tends to recover
its form, that is the force necessary to keep it extended or
468




ELONaATION.                           459
compressed, is proportional to the amount of the extension
or compression; so that each equal increment of the extending or compressing force produces an equal increment of its
extension    or compression.       This law, which       constitutes
perfect elasticity, and which obtains in respect to fluid and
gaseous bodies as well as solids, appears first to have been
established by the direct experiments of S. Gravesande on
the elongation of thin wires.*
It is, however, by its influence on the conditions of
deflexion and torsion that it is most easily recognized as
characterizing the elasticity of matter, under all its solid
forms,t within     certain  limits   of the   displacement of its
particles or elements, called its elastic limits.
ELONGATION.
346. To determine the elongation or compression of a bar of
a given section under a given strain.
Let K   be taken to represent the section of the bar in
square inches, L its length in feet, I its elongation or compression in feet under a strain of P pounds, and E the strain
or thrust in pounds which would         be required to     extend a
bar of the same material to double its length, or to compress
* For a description of the apparatus of S. Gravesande, see Illustrations of
Mechanics, by the Author of this work, 2d edition, p. 30. In one of his
experiments, Mr. Barlow subjected a bar of wrought iron, one square inch in
section, to strains increasing successively from four to nine tons, and found the
elongations corresponding to the successive additional strains, each of one ton,
to be, in millionths of the whole length of the bar, 120, 110, 120, 120, 120.
In a second experiment, made with a bar two square inches in section, under
strains increasing from 10 tons to 30 tons, he found the additional elongations,
produced by successive additional strains, each of two tons, to be, in millionths
of the whole length, 110, 110, 110, 110, 100, 100, 100, 100, 95, 90. From an
extensive series of similar results, obtained from iron of different qualities, he
deduced the conclusion that a bar of iron of mean quality might be assumed
to elongate by 100 millionth parts, or the 10,000th part, of its whole length,
under every additional ton strain per square inch of its section. (Report to
Directors of London and Birmingham Railway. Fellowes, 1835.)
The French engineers of the Pont des Invalides assigned 82 millionth parts
to this elongation, their experiments having probably been made upon iron of
inferior quality. M. Vicat has assigned 91 millionth parts to the elongation
of cables of iron wire (No. 18.) under the same circumstances, MM. Minard
and Desormes, 1,176 millionth parts to the elongation of bars of oak.
(lllust. Mech., p. 393.)
t The experiments of Prof. Robison on torsion show the existence of this
property in substances where it might little be expected; in pipe-clay, for
instance.




460        THE WORK E'PENI)ED ON ELONGATION.
it to one half its length, if the elastic limit of the material
were such as to allow it to be so far elongated or compressed,
the law of elasticity remaining the same.*
Now, suppose the bar, whose section is K square inches,
to be made up of others of the same length L, each one inch
in section; these will evidently be K in number, and the
P
strain or the thrust upon each will be represented by R.
Moreover, each bar will be elongated or compressed, by this
strain or thrust, by I feet; so that each foot of the length of
it (being elongated or compressed by the same quantity as
each other foot of its length) will be elongated or compressed
by a quantity represented, in feet, by L. But to elongate
or compress a foot of the length of one of these bars, by one
foot, requires (by supposition) E pounds strain or thrust; to
elongate or compress it by  - feet requires, therefore, E- 
pounds. But the strain or thrust which actually produces
P                     P      1
this elongation is K pounds. Therefore,K-    EL.
PL.-=EK.....(485).
347. To find the number of units of work expended upon the
elongation by a given quantity (1) of a bar whose section is
K and its length L.
If x represent any elongation of the bar (x being a part
of 1), then is the strain P corresponding to that elongation
XE
represented (equation 485) by -"x; therefore the work
done in elongating the bar through the small additional
KE
space Ax, is represented by  L -xx (considering the strain
to remain the same through the small space ax); and the
* The value of E in respect to any material is called the modulus of its elasticity. The value of the moduli of elasticity of the principal materials of construction have been determined by experiment, and will be found in a table at
the end of the volume.




THE WORK EXPENDED ON ELONGATION.                461
whole work U done is, on this supposition, represented by
KE
L 2.ax,    or  (supposing   ax   to  be  infinitely  small) by:   I
KE                   E KE 
L     xdx or by i —l1.
0
~_ KE12.u=      L.....(486).
KE
348. By equation     (485) P =- L, therefore   U =-PZ;
whence it follows that the work of elongating the bar is one
half that which would have been required to elongate it by
the same quantity, if the resistance opposed to its elongation
had been, throughout, the same as its extreme elongation 1.
If, therefore, the whole strain P       corresponding    to  the
elongation I had been put on at once, then, when the elongation I had been attained, twice as much work would have
been done upon the bar as had been expended upon its
elasticity.  This work would therefore have been accumulated in the bar, and in the body producing the strain under
which it yields; and if both had been free to move on (as,
for instance, when the strain of the bar is produced by a
weight suspended freely from       its extremity), then would
this accumulated work have been just sufficient yet further
to elongate the bar by the same distance l,* which whole
elongation of 21 could not have remained; because the
strain upon the bar is only that necessary to keep it
elongated by 1. The extremity of the bar would therefore,
under these circumstances, have oscillated on either side of
that point which corresponds to the elongation 1.
* The mechanical principle involved in this result has numerous applications; one of these is to the effect of a sudden variation of the pressure on a
mercurial column. The pressure of such a column varying directly with its
elevation or depression, follows the same law as the elasticity of a bar;
whence it follows that if any pressure be thrown at once or instantaneously
upon the surface of the mercury, the variation of the height of the column
will be twice that which it would receive from an equal pressure gradually
accumulated. Some singular errors appear to have resulted from a neglect of
this principle in the discussion of experiments upon the pressure of steam,
made with the mercurial column. No such pressure can of course be made to
operate, in the mathematical sense of the term, instantaneously; and the term
gradually has a relative meaning. All that is meant is, that a certain relation
must obtain between the rate of the increase of the pressure and the amplitude
of the motion, so that when the pressure no longer increases the motion may
cease.




462               RESILIENCE AND FRAGLITY.
349. Eliminating I between equations (485) and (486), we
obtain
p2T
=     KE..    (487);
whence it appears that the work expended upon the elongation of a bar under any strain varies directly as the square
of the strain and the length of the bar, and inversely as the
area of its section.*
THE MODULI OF RESILIENCE AND FRAGILITY.
350. Since TT=E (   -4) KL  (equation 486), it is evident
that the different amounts of work which must be done upon
different bars of the same material to elongate them by equal
fractional parts ( ) are to one another as the product KL.
Let now two such bars be supposed to have sustained that
fractional elongation which corresponds to their elastic limit;
let U, represent the work which must have been done upon
the one to bring it to this elongation, and M, that upon the
other: and let the section of the latter bar be one square
inch and its length one foot; then evidently
Ue-= MeKL.....       (488).
Me is in this case called the modulus of longitudinal resilience.t
It is evidently a measure of that resistance which the
material of the bar opposes to a strain in the nature of an
impact, tending to elongate it beyond its elastic limits.
If   f be taken to represent the work which must be similarly done upon a bar one foot long and one square inch in
section to produce fracture, it will be a measure of that
resistance which the bar opposes to fracture under the like
circumstances, and which resistance is opposed to its fra* From this formula may be determined the amount of work expended prejudicially upon the elasticity of rods used for transmitting work in machinery,
under a reciprocating motion-pump rods, for instance. A sudden effort of the
pressure transmitted in the nature of an impact may make the loss of work
double that represented by the formula; the one limit being the minimum, and
the other the maximum, of the possible loss.
T The term "modulus of resilience" appears first to have been used by
Mr. Tredgold in his work on "the Strength of Cast Iron," Art. 304.




A BAR SUSPENDED VERTICALLY.            463
gility; it may therefore be distinguished from the last mentioned as the modulus offragility. If Uf represent the work
which must be done upon a bar whose section is K square
inches and its length L feet to produce fracture; then, as
before,
Ui=MrKL..... (489).
If Pe and Pf represent respectively the strains which
would elongate a bar, whose length is L feet and section K
inches, to its elastic limits and to rupture; then, equation
(487),
U=M0.KL =-;
~?2                 P?.       -Me=-.  Similarly M..... (490).
These equations serve to determine the values of the
moduli M, and M by experiment.*
351. The elongation of a bar suspended vertically, and szstaining a given strain in the direction of its length, the
influence of its own weight being taken into the account.
Let x represent any length of the bar before its elongation, ax an element of that length, L the whole length of the
bar before elongation, w  the weight of each foot of its
length, and K its section. Also let the length x have become
ax when the bar is elongated, under the strain P and its own
weight. The length of the bar, below the point whose distance from the point of suspension was x before the elongation, having then been L-x, and the weight of that portion
of the bar remaining unchanged by its elongation, it is still
represented by (L-x) w. Now this weight, increased by P,
constitutes the strain upon the element Ax; its elongation
under this strain is therefore represented (equation 485) by
P+(E     -)Ax, and the length ax1 of the element when thus
* The experiments required to this determination, in respect to the principal materials of construction, have been made, and are to be found in the
published papers of Mr. Hodgkinson and Mr. Barlow. A table of the moduli
of resilience and fragility, collected from these valuable data, is a desideratum
in practical science.




464          THE VERTICAL OSCILLATIONS OF
elongated, by Ax+ P + (L-x), whence dividing by Ax,
and passing to the limit, we obtain
dx, 1   P+(L-x)w          491
d       H1+  E+ —BE      (491).
Integrating between the limits 0 and L, and representing
by L, the length of the elongated rod,
L=(1+ P     L+   L.L2...(492).
If the strain be converted into a thrust, P must be made
to assume the negative sign; and if this thrust equal one
half the weight of the bar, there will be no elongation at all.
352. THE VERTICAL OSCILLATIONS OF AN ELASTIC ROD OR
CORD SUSTAINING A GIVEN WEIGHT SUSPENDED FROM ITS
EXTREMITY.
Let A represent the point of suspension of the rod (fig. 1.
on the next page), L its length AB before its elongation, and
lt the elongation produced in it by a given weight W suspended from its extremity, and C the corresponding position
of the extremity of the rod.
Let the rod be conceived to be elongated through an
additional distance CD== by the application of any other
given strain, and then allowed to oscillate freely, carrying
with it the weight W; and let P be any position of its
extremity during any one of the oscillations which it will
thus be made to perform. If, then, CP be represented by x,
the corresponding elongation BP of the rod will be represented by i+ x, and the strain which would retain it permanently at this elongation (equation 485) by L- (~i+x); the
unbalanced pressure or moving force (Art. 92.) upon the
weight W, at the period of this elongation, will therefore be
XE'K               XE
represented by -L (1+x) —W, or by - -x; since W, being
the strain which would retain the rod at the elongation il, is
KE
represented by -L-1 (equation 485).
* Whewell's Analytical Statics, p. 113.




A LOADED BAR.                 465
The unbalanced pressure, or moving force, upon the mass
W varies, therefore, as the distance x of the point P from the
given point C; whence it follows by the general principle
established in Art 97., that the oscillations of the point P
extend to equal distances on either side of the point C, as a
centre, and are performed isochronously, the time T of each
oscillation being represented by the formula
T=  (f  )t...(493).
\ gKE/"
The distance from A of the centre C, about which the
oscillations of the point P take place, is represented by
L+jl; so that, representing this distance by L1, and substituting for -l its value, we have
WL       (494).
L.=L+..+.(494).
A        A
353. Let us now suppose that when in
making its first oscillation about C
(fig. 2.) the weight W  has attained its
-   highest position d, and is therefore, for
an instant, at rest in that position, s
second weight w is added to it; a second
~c       c   series of oscillations will then be coimp        c   menced about a new centre C1, whose
distance L2 from A is evidently repre-;. D       sented by the formula
1..-D
L =L+ (W+w)L              (495).
L2 = L    K+  - E........wL
So that the distance CC, of the two centres is; and the
greatest distance CDi, beneath the centre C,, attained in the
second oscillation, equal to the distance, Cd, at which the
oscillation commenced above that point.  Now  CDO =
wL
Cd.=Cd, + CC=CD+(C        c+ C,=c; the amplitude d,D, of
the second oscillation is therefore 2(c +  )
30




466        THE OSCILLATIONS OF A LOADED BAR.
Let the weight w be conceived to be removed when the
lowest point D, of the second oscillation is attained, a third
series of oscillations will then be commenced, the position of
whose centre being determined by equation (494), is identical
with that of the centre C, about which the first oscillation
was performed. In its third oscillation the extremity of the
rod will therefore ascend to a point d2 as far above the point
C as D, is below it; so that the amplitude of this third oscillation is represented by 2CD,, or by 2C,D,+CC1, or by
2 (4+ 2wL   When the highest point d1 of this third oscillation is attained, let the weight w be again added; a fourth
oscillation will then be commenced, the position of whose
centre will be determined by equation (495,) and will therefore be identical with the centre C1, about which the second
oscillation was performed; so that the greatest distance C,D2
beneath that point attained in this fourth oscillation will be
equal to Cld, or to CC, +CD,; and its amplitude will be
represented by 2 (c+ 3wL)  And if the weight w be thus
conceived to be added continually, when the highest point
of each oscillation is attained, and taken off at the lowest
point, it is evident that the amplitudes of these oscillations
will thus continually increase in an arithmetical series; so
that the amplitude An of the nth oscillation will be represented by the formula
A {.C+(a -1) 1...(496).
The ascending oscillations of the series being made about
the centre C, and the descending oscillations about C,
if n be an even number, the centre of the nth oscillation is
C,; the elongation c of the rod corresponding to the lowest
point of this oscillation is therefore equal to BC, + A,; or
substituting for BC, its value given by equation (495), and
for An its value from equation (496),
(W+nw)L            (9).
Thus it is apparent that by the long contined and
Thus it is apparent that by the long continued and




DEFLEXION.                       467
periodical addition and subtraction of a weight w, so small
as to produce but a slight elongation or contraction of the
rod when first added or removed from it, an elongation c,
may eventually be produced, so great as to pass limits of its
elasticity, or even to break it.' Numerous observations have
verified this fact: the chains of suspension bridges have
been broken by the measured tread of soldiers;* and M.
Savart has shown, that by fixing an elastic rod at its centre,
and drawing the wetted finger along it at measured intervals, it may, by the strain resulting from  the slight friction
received thus periodically upon its surface, be made with
great ease to receive an oscillatory movement of sufficient
amplitude to be measured.t     M. Poncelet has compared the
measurement of M. Savart with theoretical deductions
analogous to those of the preceding article, and has shown
their accordance with it.
DEFLEXION.
354. The neutral surface of a deflected beam.
One surface of a beam becoming, when deflected, convex,
and the other concave, it is evident that the material forming that side of the beam which is bounded by the one
surface is, in the act of flexure, extended, and that of the
other compressed. The surface which separates these two
portions of the material being that where its extension terminates and its compression begins, and which sustains,
therefore, neither extension nor compression, is called the
NEUTRAL SURFACE.
355. THE POSITION OF THE NEUTRAL SURFACE OF A BEAM.
Let ABOD be taken to represent any thin lamina of the
* Such was the fate of the suspension bridge at Broughton near Manchester,
the circumstances of which have been ably detailed by Mr. E. Hodgkinson in
the fourth volume of the Manchester Philosophical Transactions. M. Navier
has shown, in his treatise on the theory of suspension bridges (Sur les Ponts
Suspendus, Paris, 1823), that the duration of the oscillations of the chains of
a suspension bridge may in certain cases extend to nearly six seconds; there
-might easily, in such cases, arise that isochronism at each interval, or after
any number of intervals, between the marching step of the troops and the
oscillations of the bridge, whence would result a continually increasing elongation of the suspending chains.
t  eficanique Industrielle, p. 437, Art. 331.-ED.




4@8              THE NEUTRAL SURFACE
io              p   beam contained by planes
parallel to the plane of its
\  \\i      ^D/   deflexion, and P., P2 P3 the
e\ o\  \0 ~  a ^resultants of all the presc\c  \  Add ^ Usures applied to it aclb that
b e'^ I  /   portion of the neutral sur~B.~. /          face of the beam which is
\  7w ~  contained within this laP^   \ f  mina, and may be called its
neutral line; PT and QV
4/         planes exceedingly near to
one another, and perpendicular to the neutral line at the points where they intersect
it; and O the intersection of PT and QV when produced.
Now let it be observed that the portion APTD of the
beam is held in equilibrium by the resultant pressure P1,
and by the elastic f6rces called into operation upon the surface PT; of which elastic forces those acting in PR (where
the material of the beam is extended) tend to bring the
points to which they are severally applied nearer to the
plane SQ, and those acting in RT (where the material is
compressed), to carry their several points of application
farther from the plane ST.
Let aR-x, SR=Ax, and imagine the lamina PQVT to be
made up of fibres parallel to SR; then will Ax represent
the length of each of these fibres before the deflexion of the
beam, since the length of the neutral fibre SR has remained
unaltered by the deflexion. Let dx represent the quantity
by which the fibre pq has been elongated by the deflexion
of the beam, then is the actual length of that fibre represented by Aa- -6dx. Whence it follows (equation 485), that
the pressure which must have operated to produce this
elongation is represented by -  4k, ak being taken to repre
sent the section of the fibre, or an exceedingly small element
of the section PT of the lamina. Now PT and QV being
normals to SR, the point 0 in which they meet, when
produced, is the centre of curvature to the neutral line in
R. Let the radius of curvature OR be represented by R,
and the distance Rp by p. By similar triangles, 02-.OR
pr     R+ p   aAX+d        P       o, P
SR5 or -R- == —, or   E — 1   +.^; therefore,  =
SR' ~R          Ax'




OF A BEAM.                  469
JX                           X
-.   Substituting this value of  in the expression for the
pressure which must have operated to produce the elongation of the fibre p, and representing that pressure by AP,
we have
Ep
AP   W kA.... (498).
If, therefore, RP be represented by k, and RT by k,, then
the sum of the elastic forces developed by the extension of
E.kl
the fibres in RPQS is represented by -20opAk; and, similarly,
the sum of those developed by the compression of the fibres
E k2
in RTVS is represented by zopak. Now let it be observed
that (since the pressures applied to APTD, and in equilibrium, are the forces of extension and compression acting
in RP and RT respectively, and the pressure P,), if the
pressure P1 be resolved in a direction perpendicular to the
plane PT, or parallel to the tangent to the neutral line in R,
this resolved pressure will be equal (Art. 16.) to the difference of the sums of the forces of extension and compression
applied (in directions perpendicular to that plane, but opposite to one another) to the portions RP and RT of it respectively. Representing, therefore, by 6 the inclination ReP,
of the direction of P1 to the normal to the neutral line in R,
we have'wE = k  E kpP. sin. ==-OpAk-   opAk.
But if k be taken to represent the whole section PT, and h
the distance of the point R from its centre of gravity, then
(Art. 18.)
khA=  pak — pAt;.. P sin. = R;
0     0
RP. A= Ek sin......   (499);
which expression represents the distance of the neutral line
from the centre of gravity of any section PT of the lamina,
thit distance being measured towards the extended or the
compressed side of the lamina according as 0 is positive or




470               RADIUS OF CURVATURE.
negative; so that the neutral line passes from one side to
the other of the line joining the centres of gravity of the
cross sections of the lamina, at the point where -=0, or at
the point where the normal to the neutral line is parallel to
the direction of P,.
356. Case of a rectangular beam.
If the form of the beam be such that it may be divided
into laminae parallel to ABCD of similar forms and equal
dimensions, and if the pressure Pi applied to each lamina
may be conceived to be the same; or if its section be a rectangle, and the pressures applied to it be applied (as they
usually are) uniformly across its width, then will the distance
h of the neutral line of each lamina from the centre of gravity of any cross section of that lamina, such as PT, be the
same, in respect to corresponding points of all the laminae,
whatever may be the deflection of the beam; so that in this
case the neutral surface is always a cylindrical surface.
357. Case in which the deflecting pressure P1 is nearly perpendicular to the length of the beam.
In this case 6, and therefore sin. 0, is exceeding small, so
long as the deflexion is small at every point R of the neutral
line; so that h is exceedingly small, and the neutral line
of the lamina passes very nearly, or accurately, through the
centre of gravity of its section PT.
358. THE RADIUS OF CURVATURE OF THE NEUTRAL SURFACE
OF A BEAM.
Since the pressures applied to the portion APT) of the
ID
lamina ABCD are in equilibrium, the principle of the equality
of moments must obtain in respect to them; taking, there



RADIUS OF CURVATURE.             471
fore, the point R, where the neutral axis of the lamina intersects PT, as the point from which the moments are measured,
and observing that the elastic pressures developed by the
extension of the material in RP and its compression in RT
both tend to turn the mass APTD in the same direction
about the point R, and that each such pressure upon an
element Ak of the section PT is represented (equation 498) by
pAk, and therefore the moment of that pressure about the
E
point R by E-p 2k, it follows that the sum of the moments
about the point R of all these elastic pressures upon PT is
E            El
represented by -p2 ak, or by, if I be taken to represent
ZR           R'
the moment of inertia of PT about R. Observing, moreover,
that if p represent the length of the perpendicular let fall
from R upon the direction of any pressure P applied to the
portion APTD of the beam, Pp will represent its moment,
and zPp will represent the sum of the moments of all the
similar pressures applied to that portion of the beam;
we have by the principle of the equality of moments,
EI
zR
1   P3....(500):
R-R EI 
359. The neutral surface of the beam is a cylindrical surface, whatever may be its deflection or the direction of its
deflecting pressure, provided that its section is a rectangle
(Art. 353.); or whatever may be its section, provided that its
deflection be small, the direction of the deflecting pressure
nearly perpendicular to its length, and its form before deflexion symmetrical in respect to a plane perpendicular to the
plane of deflexion. In every such case, therefore, the neutral
lines of all the laminee similar to ABOD, into which the
beam may be divided, will have equal radii of curvature at
points similar to R lying in the same right line perpendicular
to the plane of deflection; taking, therefore, equations similar to the above in respect to all the laminae, multiplying
both sides of each by I, adding them together, and observing that R and E are the same in all, we have RI  Pp
n t    E
In this case, therefore, I may be taken in equation (p00) to:




4i72              MOMbNENT OF INERTIA.
represent the moment of inertia of the whole section of the
beam, and P the pressure applied across its whole width.
360. The radius of aurvature of a beam whose deflexion is
small, and the direction of the deflecting pressures nearly
perpendicular to the length of the beam.
In this case the neutral line is very nearly a straight line,
perpendicular to the directions of the deflecting pressures;
so that, representing its length by x, we have, in this case,
p=x; and equation (500) becomes
1   zPx
R    EI.... (501);
which relation obtains, whatever may be the form of the
transverse section of the beam, I representing its moment of
inertia in respect to an axis passing through its centre of
gravity and perpendicular to the plane of deflexion.
361. The moment of inertia I of the transverse section of a
beam about the centre of gravity of the section.
In treating of the moments of inertia of bodies of different
geometrical forms in a preceding part of this work (Art. 82,
&c.), we have considered them as solids; whereas the moment of inertia I of the section of a beam which enters into
equation (500) and determines the curvature of the beam
when deflected, is that of the geometrical area of the section.
Knowing, however, the moment of inertia of a solid about
any axis, whose section perpendicular to that axis is of a
given geometrical form, we can evidently determine the
moment of the area of that section about the same axis, by
supposing the solid in the first place to become an exceedingly thin lanrina (i. e. by making that dimension of the
solid which is parallel to the axis exceedingly small in the
expression for the moment of inertia), and then dividing
the resulting expression by the exceedingly small thickness
of this lamina. We shall thus obtain the following values
of I:




MOMENT OF INERTIA.              473
362. For a beam with a rectangular section,)
whose breadth is represented by b and its depth > I=-b6'.
by c (equation 61),
363. For a beam with a triangular )
section, whose base is b and its height ce Il- bo(ib+ c).
(equation 63), 
364. For a beam or column with a circular  I = - r
section, whose radius is c (equation 66), 
365. To determine the moment of inertia I in respect to a
A^_ Bbeam whose transverse section is of the
c,-  form represented in the accompanying
figure, about an axis ab passing through
C- -   d    its centre of gravity; let the breadth of
~am-     _  bthe rectangle AB be represented by b, and
its depth by d,, and let b, and d, be simiK        -i larly taken in respect to the rectangle EF,
and b, and d, in respect to CD; also let I,
represent the moment of inertia of the section about the axis
cd passing through the centre of CD, A,, A, A, the areas
of the rectangles respectively, and A the area of the whole
section.
Now the moments of inertia of the several rectangles,
about axes passing through their centres of gravity, are
represented by -Tbaiq Tbd, AbcSd,, and the distances of
these axes from  the axis cd are respectively i(di+d,),
j(d,+d3), 0. Therefore (equation 58),
I= -bld3 + +i(d, + ds)'A,-+ lib,2d2,+i(d2+ d,)A, ++ Abci;
but A,=b6,i, A,=bdc,, A,=bd,;
-:. I,=-(A,d2+Ad,'+ Ad,)+i(d, + d,) A, +  +(d + d+,)2 A2.
Also if h represent the distance between the axes ab and cd,
then (Art. 18) hA=-i(d,+ d)A, —(d, + d)A,, and (equation
58) I=I,-h'A.
J. I  =A   2 + A&dc + Ac) +-4i (d, + I)'2A, + (c + ci)'2A, -
(cd, + ds(A - (d, + d3,)A}  (5.
If  and   be exceedingly small as compared with (5
if d~ and d, be exceedingly small as compared withd,,




474              DEFLEXION OF A BEAM.
neglecting their values in the two last terms of the equation
and reducing, we obtain
I=AT(Ald ~+A2d2 +A8d) + i (A     A~ AA       ),..... (503).
If the areas AB and EF be equal in every respect,
I=- {d        + 3(d1 + d)2 A + lAd2..... (504).
366. THE WORK EXPENDED UPON THE DEFLEXION OF A BEAM
TO WHICH GIVEN PRESSURES ARE APPLIED.
If AP represent the pressure which must have operated
-0o~  ~to produce the elongation or?~\0.      iny  compression which the ele*':\.     /'  mentary fibre pq receives,' it\  \.  ^     by reason of the deflexion
\ C\  \'' S Aoof the beam, Ax the length
^beC^-^      / ~of that fibre before the de-'B'       3v`^ ^ -.  /  nflexion of the beam, and Ak
its section; then the work
gp  X;   /        which must have been done: |\  /upon it, thus to elongate
or compress it, is represented, equation (487) by
(EP)2. A    But (equation 498) AP=   -Ak. The work expended upon the extension or compression of pq is therefore represented by
E.Ax
R2 (WAk).
And the same being true of the work expended on the
compression or extension of every other fibre composing the
elementary solid VTPQ, it follows that the whole work
expended upon the deflexion of that element of the beam
is represented by I —  p2k, or by iA; for p2   represents the moment of inertia I of the section PT, about an
axis perpendicular to the plane of ABCD, and passing through
the point R. If, therefore, a, be taken to represent the
length of that portion of the beam which lies between D




DEFLEXION OF A BEA.              475
and M before its deflexion, and therefore the length of the
portion ac of its neutral line after deflexion, then the whole
work expended upon the deflexion of the part AM of the
al I                     I
bea/m is represented by i Ez -R.  But (equation 500) RjE'I; whence, by substitution, the above expression
p 2ap 2
becomes B — I- at. Passing to the limit, and representing the work expended upon the deflexion of the part AM
of the beam by u,,
P 2 Pa1l 2
-A 2E    Idx.... " (505).
0
367. The work expended upon the deflexion of a beam of
uniform dimensions, when the deflecting pressures are
nearly perpendicular to the surface of the bearm
In this case I is constant, andp,=x; whence we obtain
by integrating (equation 505) beP1, tween the limits 0 and a,
-D /aa         P,'a,'
A        -  EI.... (506),
hC M' Q             where uq represents the work ex-,~ -~  /*     pended upon the deflexion of the
I'pp  "t/     portion AM  of the beam.  Simi~~8 llarly, if bc=a,, the work expended
upon the deflexion of the portion BM of the beam is represented by
P 2a s
e6 EI 
so that the whole work TU expended upon the deflexion of
the beam is represented by
P 2a' I+ P,2a,'
6EIBut by the principle of the equality of moments, if a
represent the whole length of the beam,




476              DEFLEXION OF A BEAM.
Pla=Psas, P2a=Pa,.
Eliminating P, and P, between these equations and the preceding, we obtain by reduction
a= E.... (507).
If the pressure Pg be applied in the centre of the beam,
a, =cc,=iacta.,..... (508).
1396E. * * *
368. THE LINEAR DEFLEXION OF A BEAM WHEN THE DIRECTION
OF THE DEFLECTING PRESSURE IS PERPENDICULAR TO ITS
SURFACE.
Let the section MN remain fixed, the deflexion taking
place on either side of that section;
at then ul representing the work expended upon the deflexion of the
j, portion AM of the beam, and Di.          A..... the deflexion of the point to which
Pc    \   a//<    | PI is applied, measured in a direc~c_ —--—'i^^    ^tion perpendicular to the surface, we
bo p       have (equation 40), u1 =fP  DD;
du    du, t  dPl
therefore P, =  Di =  P   d
1dD7 dPjI dD1
But by equation (506), d-p  i Ei; therefore Pl =  P1EI 
dP,, t         dD,   a 1a1
DP 1;d therefore  gI-i EI; whence we obtain by integration
D1=   E     * * * (509).
If the whole work of deflecting the beam be done by the
pressure P., the points of application of P and P2 having no
motions in the directions of these pressures (Art. 52.), then
proceeding in respect to equation (507) precisely as before in
respect to equation (506), and representing the deflexion
* Church's Diff. Cal. Art. 17.




DEFLEXION OF A BEAM.             477
perpendicular to the surface of the beam at the point of
application of P, by D,, we shall obtain
D3=   3E I' * a  -(510).
If the pressure P, be applied at the centre of the beam
aTP..D_ 48EI..... (511).
Eliminating P, between equations (506) and (509), and P.
between equations (507) and (510), we obtain
3EID,2      3aIEID2
u E, D,'a  U  _  a,)2......(512);
by which equations the work expended upon the deflexion
of a beam is determined in terms of the deflexion itself, as
by equations (506) and (507) it was determined in terms of
the deflectingpressures.
369. UONDITIONS OF THE DEFLEXION OF A BEAM TO WHICH ARE
APPLIED THREE PRESSURES, WHOSE DIRECTIONS ARE NEARLY
PERPENDICULAR TO ITS SURFACE.
Let AB represent any lamina of the beam parallel to its
plane of deflexion, and acb the neutral line of that lamina
intersected by the direction of P, in the point c.
Draw xx, parallel to the length of the beam before its
deflexion, and take this line as the axis of the abscissae, and
the point c as the origin; then, e.presenting by x and y the
* This result is identical with that obtained by a different method of investigation by M. Navier (Resume de Lepons dc COisttuecton, Art. 359.).




478         EQUATION TO THE NEUTRAL LINE.
co-ordinates of any point in ac, and by R the radius of curvature of that point, we have*
1   dy (    d 2  -
R   x2 (1   dx2)
Now the deflexion of the beam being supposed exceedingly small, the inclination to ex of the tangent to the
neutral line is, at all points, exceedingly small, so that (d)
1 d2y
may be neglected as compared with unity; therefore R- dx
Substituting this value in equation (501), and observing that
in this casep is represented by (a — x) instead of x,
d2y Pl(al-x)         1
d:2 —  E....  (513).
the direction of the pressure P, being supposed nearly per
pendicular to the surface of the beam, and I constant. Let
the above equation be integrated between the limits 0 and
x, 3 being taken to represent the inclination of the tangent
at c to cx, so that the value of  at c may be represented by
tan. /,
dytan. 3=-     tax-jx2... (514).
dx an. /3
Integrating a second time between the limits 0 and x, and
observing that when x=0, y=O,
y^=E ^i^alx  x+xtan. 15..., (515).
Y=EIl 
Proceeding similarly in respect to the portion be of the neutral line, but observing that in respect to this curve the value
of -d at the point c is represented by tan. $, we have
dy   P2(a — x)
dxO    El
dZ+tan./ 3=EIa2-.. (516).
* Church's Diff. Cal. Art. 105.




EQUATION TO THE NEUTRAL LINE.       479
Pyi|2    6Z't-tan.... (517.)
If DI and D2 be taken to represent the deflexions at the
points a and b, and ca and ob be assumed respectively equal
to cd and ce,
by equation (515), D,= P3I +a, tan. 3,
by equation (517), D= P3E1-a2 tan..
If the pressures P1 and P, be supplied by the resistances
of fixed surfaces, then D,-=D. Subtracting the above equation we obtain, on this supposition,
P1 a -P a 3
0=  1 —    23E   +(a1+a,) tan. F.
Now P  — P3, Pka'ala-P sala32)
Now P'A2 -PA3=                 =Pa,aa2(a. — a,); oba
serving that Pa=Pa,, P2a=P3a1, and a, + a2=a,: tan. -    3EIa... (518).
If PI, 2, represent the inclinations of the neutral line to
xx, at the points a and b, then by equations (514) and (516)
tan.l, -tan. /P    tan.,3+.tan.3 P='2
-EI'               2EI'
Substituting for tan. / its value from equation (518), eliminating and reducing,
Pa, a (al  + 2) ta.  _Paa,(a,+2a1)
tanl.               tan.  -     6EIa....(519).
To determine the point m where the tangent to the neutral
line is parallel to cxx1, or to the undeflected position of the
beam, we must assume -=0 in equation (516)*; if we
then substitute for tan. 3 its value from equation (518),
substitute for PI its value in terms of P,, and solve the
* Church's Diff. Cal. Art. 78.




480          LENGTH OF THE NEUTRAL LINE
resulting equation in respect to x, we shall obtain for the
distance of the point m from c the expression
+    *,(+a, + 2a)... (520).
370. THE LENG:TH OF THE NEUTRAL LINE, THE BEAM BEING
LOADED IN THE CENTRE.
Let the directions of the resistances upon the extremities
of the beam be supposed nearly perpendicular to its surface;
then if x and y be the co-ordinates of the neutral line from
the point a, we have (equation 501), representing the horizontal distance AB by 2a, and observing that in this case
-  and that the resistance at A or B =  P,
-EI d= PE.
Integrating between the limits x and a, and observing that
at the latter limit,  0,
EIEIdY =jP(a&-x2).
Now if s represent the length of the curve ac,
a               a
*=    ( 1 + -) df(=(i       (y      dx nearly; since
Ohuttch's mt. CaI. Art. 19l.




THE DEFLEXION OF A BEAM.                    481
the deflexion being small, -, is exceedingly small at every
point of the neutral line.
a:.'= s     1+ 3+E'Ip (a2 —2a%2 +?) do;
0
Pta6. s=a+    60E....(521).
Eliminating P between this equation and equation (511), and
representing the deflexion by D,
D2*
8=a+i a.
371. A   BEAM, ONE PORTION OF WHICH IS FIRMLY INSERTED IN
MASONRY, AND WHICH SUSTAINS A LOAD UNIFORMLY DISTRIBUTED OVER ITS REMAINING PORTION.
Let the co-ordinates of the neutral line be measured from
* The following experiments were made by Mr. Hatcher, superintendant of
the work-shop at King's College, to verify this result, which is identical with
that obtained by M. Navier (Resume des Leqons, Art. 86.).  Wrought iron
rollers'7 inch in diameter were placed loosely on wrought iron bars, the surfaces of contact being smoothed with the file and well oiled. The bar to be
tested had a square section, whose side was *7 inch, and was supported on the
two rollers, which were adjusted to 10 feet apart (centre to centre) when the
deflecting weight had been put on the bar. On removing the weights carefully, the distance to which the rollers receded as the bar recovered its horizontal position was noted.
Deflecting Weight            Distance through which  Distance through which
in lbs   Deflection in Inches. each Roller receded  each Roller would have
in inches.     receded by Formula.
i56           3'7               1                *13
84            5-45.2.29
31




482           TIE DEFLEXION OF A BEAM
the point B where the beam
is inserted in the masonry,
and let the length of the,......   portion AD which sustains
the load be represented by
a, and the load upon each
unit of its length by A,;
then, representing by x and
y the co-ordinates of any
point P of the neutral line,
_~-'A, * ______ __ _-= _  and observing that the pressures applied to AP, and in
equilibrium, are the load t(a-x) and the elastic forces
developed upon the transverse section at P, we have by the
principle of the equality of moments, taking P as the point
from which the moments are measured, and observing that
since the load p,(a-x) is uniformly distributed over AP it
produces the same effect as though it were collected over the
centre of that line, or at distance -(a-x) from P; observing,
moreover, that the sum of the moments of the elastic forces
upon the section at P, about that point, is represented (Art.
358.) by R, or by El $ (Art. 369.);
EI    = i(a-  )2.. (522).
Integrating twice between the limits 0 and a, and observing
that when x=0, -h=-0 and y=O, since the portion BC of the
beam is rigid, we obtain
EllY-     ^(- )93+-    at... (523),
EIy = I-,,(a-)4+-'-0 -214....'* (524),
which is the equation to the neutral line.
Let, now, a be substituted for x in the above equation;
and let it be observed that the corresponding value of y
represents the deflexion D at the extremity A of the beam;
we shall thus obtain by reduction
D=-...  (525).




LOADED UNIFORMLY.              483
Representing by / the inclination to the horizon of the tangent to the neutral line at A, substituting a for x in equation
(523), and observing that when x=a, d= tan. /, we obtain
tan. i  EI..   (526).
372. A BEAM SUPPORTED AT ITS EXTREMITIES AND SUSTAINING
A LOAD UNIFORMLY DISTRIBUTED OVER ITS LENGTH.
Let the length of the beam be represented by 2a, the load
-,,e i',.,,  upon each' unit of length by t; take,',','',',1'',,',' x and y as the co-ordinate of any,   point P of the neutral line, from the,^~B  ~  A   origin A; and let it be observed
that the forces applied to AP, and in
equilibrium, are the load px upon that
_          I~ ~ portion of the beam, which may be
supposed collected over its middle
point, the resistance upon the point A, which is represented
by ta, and the elastic forces developed upon the section
at P; then by Art. 360.,
d2 
EI,2.=9 a2 —^aa.... (527).
Integrating this equation between the limits x and a, and
observing that at the latter limit  0, since y evidently
dx
attains its maximum value at the middle C of the beam,
EI dy - =('-al)-  pa(- a 2).... (528).
Integrating a second time between the limits 0 and a, and
observing that when x=0, y=0,
which is th e eqation to thde nu ) t i. Sub(529),
which is the equation to the neutral line. Substituting a for




484           THE DEFLEXION OF A BEAM
x in this equation, and observing that the corresponding
value of y represents the deflexion D in the centre of the
beam, we have by reduction
D=25EI.      (530).
Representing by f the inclination to the horizon of the tangent to the neutral line at A or B, and observing that when?=0 in equation (528), d-  tan.,
dx
tan. -.....' (531).
oEI
Let it be observed that the length of the beam, which in
equation (511) is represented by a, is here represented by
2'I, and that- equation (530) may be placed under the form
D=  (. a4) (); whence it is apparent that the deflexion
of a beam, when uniformly loaded throughout, is the same
as though -ths of that load (2aa) were suspended from its
middle point.
373. A BEAM IS SUPPORTED BY TWO STRUTS PLACED SYMMETRICALLY, AND IT IS LOADED UNIFORMLY THROUGHOUT
ITS WHOLE LENGTH; TO DETERMINE ITS DEFLEXION.
Let CD-=2a, CA= a load upon each foot of the length
of the beam=af; then load on
each point of support=lta. Take
C as the origin of the co-ordinates;
"~ RIB   E  c Cthen, observing that the forces
impressed upon any portion CP
of the beam, terminating between
C and A, are the elastic forces
upon the transverse section of the
beam at P, and the weight of the
load upon CP; and observing that the weight ^CP of the
load upon CP, produces the same effect as though it were
collected over the centre of that portion of the beam, so that
its moment about the point P is represented by p.. CP. JCP,




LOADED UNIFORMLY.              485
or by i'CP'; we obtain for the equation to the neutral line
in respect to the part CA of the beam (Art. 360)
d~y
EI (g5=-.....(532).
Since, moreover, the forces impressed upon any portion CQ
of the beam, terminating between A and E, are the elastic
forces developed upon the transverse section at Q, the
resistance Pa of the support at A, and the load upon CQ,
whose moment about Q is represented by JiCQ2, we have
(equation 501), representing CQ by x,
-EI  =A- a( —a)   a)~..... (533).
Representing the inclination to the horizon of the tangent to
the neutral line at A by i/, dividing equation (532) by p.,
integrating it between the limits x and a,, and observing
tha at te latter limitdy
that at the latter limit d=tan. /, we have, in respect to the
portion CA of the beam,
EI(- y_ tan./3) =a9 — -..... (534).
Integrating equation (533) between the limits x and a, and
dy
observing that at the latter limit j= 0, since the neutral
line at E is parallel to the horizon,
EI dy
1X-^ a(     - a)2-as+ia(a - al)2..... (535)
which equation having reference to the portion AE of the
beam, it is evident that when x=al, dx=tan. 3.
EI
~p.   — tan.-= a(a- al)' —(a' — a,')=-( a)(2 — 4aa,-.a).... (536).
Substituting, therefore, for tan. 3 in equation (534), and
reducing, that equation becomes
EI dcy8+ia(a-a1)2-..... (537).




486           THE DEFLEXION OF A BEAM
Integrating equation (535) between the limits a, and x, and
equation (537) between the limits 0 and x, and representing
the deflexion at C, and therefore the value of y at A, by D,,
EI
-(y-DI)          -                    a'-$l (- a,)-{- -a8  (a-) se —',14  +
a'al- Iaa(a-a,)'
EI      4
by=  _3+ {2a(a- a1)2-_ asx..   (538);
the former of which equations determines the neutral line
of the portion AE, and the latter that of the portion CA of
the beam. Substituting a, for x in the latter, and observing
that y then becomes D,; then substituting this value of D,
in the former equation, and reducing,
D= 24EI 12a( —a)2    -- a') * * * (539);
EIT
Ey= L'- -a(x- a,)+a 3(a-a2a,)' —   x.... (540).
The latter equation being that to the neutral line of the portion AE of the beam, if we substitute a in it for x, and
represent the ordinate of the neutral line at E by y,, we
shall obtain by reduction
tfha
1= 24EI 4(a, + 2a)(a-a,)2-3a'l.... (541).
If a1-0, or if the loading commence at the point A of the
beam, the value of y, will be found to be that already determined for the deflexion in this case (equation 530).
Now, representing the deflexion at E by D2, we have evidently D,=D,-y,.
D,= (a-EIa  -5)  +laa,+a,'2.....(542).
374. THE CONDITIONS OF THE DEFLEXION OF A BEAM LOADED
UNIFORMLY THROTJGHOIT ITS LENGTH, AND SUPPORTED AT
ITS EXTREMITIES AAND D, AND AT TWO POINTS B AND 
SITUATED AT EQUAL DISTANCES FROM THEM, AND IN THE
SAME HORIZONTAL STRAIGHT LINE.
Let AB=a, AD=2a.
Let A be taken as the origin of the co-ordinates; let the




LOADED UNIFORMLY.             487
_: ~.:'"."';.':','"':'-1  pressure upon that point be'  t,  represented by P1, and the
_t    g pressure upon B by P,; also
m3  V    B   Am  ^the load upon each unit of
the length of the beam by p.
If P be any point in the
neutral line to the portion AB.....  -    of the beam, whose co-ordinates are x and y, the pressures applied to AP, and in equilibrium, are the pressure
Pi at A, the load ax supported by AP, and producing the
same effect as though it were collected over the centre of
that portion of the beam, and the elastic forces developed
upon the transverse section of the beam at P; whence it
follows (Art. 360.) by the principle of the equality of
moments, taking P as the point from which the moments
are measured, that
d2y
EI-dx f-P,-P1x.... (543).
Integrating this equation between the limits a,, and x, and
representing the inclination to the horizon of the tangent to
the neutral line at B by /,,
El(d- tan.  ) =(x-a,)-P,1(x-a,2).... (544).
Integrating again between the limits 0 and x,
EI(y -xtan. 9) — l(;x —a,-) ) —~ P(& -a/,x).. (545),
Whence observing that when x=a1, y-0,
EI tan. /3,= —Pia13- la..... (546).
Similarly observing, that if x and y be taken to represent
the co-ordinates of a point Q in the beam between B and C,
the pressures applied to AQ are the elastic forces upon the
section at Q, the pressures P, and P2 and the load Ax, we
have
d2y
EI d-X2i2 - I   -P2(x- a).... (547).
Integrating this equation between the limits a. and x, and
dy
observing that at the former limit the value of - is represented by tan. 3,, we have




488            THE DEFLEXION OF A BEAM
EI (d-tan. (2) -     -. p(S a)- PI(X2- a2)-iP1(x-ca)'..... (548).
Now it is evident that, since the props B and C are placed
symmetrically, the lowest point of the beam, and therefore
of the neutral line, is in the middle, between B and C; so
that   = _- 0, when x=a. Making this substitution in equation (548),
-EI tan. 2=- (ai -a -) -P1, (a2 -a2)- iP (a -  )2. (549).
Since, moreover, the resistances at C and D are equal to
those at A and B, and that the whole load upon the beam is
sustained by these four resistances, we have
P1+P2,=-a.... (550).
Assuming a,-na, and eliminating P1, P2, tan. P2, between
the equations (546), (549), and (550), we obtain
p a I (n + 12n2-24n+8. (   551);
=8n t       2n-3
a       4& -n2 -8   _  a(n-2) _   -n2+2n+4 )
28n        2n-3         a(8-  ) -2n2+-3+       (5);
-8s-n  ( n58-21 32+24n-8 )
tan./~        -— i          i.
4EI'224 —3E 2n-3
g   ^(n-l)  5Sn2-16n+8 }
24EI          2n-3   }.... (553).
Making x=0 in equation (544); and observing that the corresponding value of dy is represented by tan. /,, we have
EI (tan. 3P,-tan. /2) --  a,'+iP1a, 
Substituting for tan.. and P, their values from equations
(553) and (551), and reducing,
an \ -32n3+18W     24-     8+8 
tan.'/,-48EI        2n —               (4)
Representing the greatest deflexions of the portions AB and




LOADED UNIFORMLY.                489
BC of the beam, respectively, by D, and D2, and by a, the
distance from A at which the deflexion D, is attained, we
have, by equations (544) and (545),
-EI tan. i= — IP (x -3- a)-P1(-12- a) -
EI(D -x ^tan./-)=   (' —a 12 — 8 P,(- (3- $,a ) t*(555)'
The value of D, is determined by eliminating x, between
these equations, and substituting the values of P1 and tan. 3,
from equations (551) and (553).
Integrating equation (548) between the limits a, and a,
and observing that at the latter limit y=D2, we have
EID,=EI(a-a-  ) tan. 2i +* 6 {-j(4 -a4)-a( a-a,) -
iP, {1(a3 -a,)-a2(- a)1 -P,2(a-a,)".
Substituting in this equation for the values of tan. f2, P,, P,,
and reducing, we obtain
D2 4            I((2n) s —2n —8n+6}..... (526).
L48EI(3 -2n)
Representing BC by 2a2, and observing that a= -AE —
AB=-a-na=(1 -n)a,
tia4  n& —2n —8n+6
2 -48EI     (3-2n)(1-)....      (557).
375. A BEAM, HAVING A UNIFORM LOAD, SUPPORTED AT EACH
EXTREMITY, AND BY A SINGLE STRUT IN TIE MIDDLE.
If, in the preceding article, a, be assumed equal to a, or._=1, the two props B and
I I. I. X I, C will coincide in the centre;
~,,-:, B-, -A,!:~ L  and the pressure P, upon;    ~:'_q_ ~the single prop, resulting
from their coincidence, will
be represented by twice the
corresponding value of P, in,.....____~F equation (552); we thus ob-. _.. tain
P2,=FPa, P1=Sta;     )
tan. -1, Itan.=O..... (558)
tn'48EP'




490           THE DEFLEXION OF A BEAM
The distance x, of the point of greatest deflexion of either
portion of the beam from its extremities A or D, and the
amount D, of that greatest deflexion, are determined from
equations (555). Making tan. f/=0 in those equations,
substituting for P, its value, solving the former in respect to
a,, and the latter in respect to D1, we obtain
1+ 4/33,I=   1    a6 -=421535a.....(559).
D    X=,(a-x,)(2x, +a)'259997Ia 
- 48EI-       48EI....(560).
376. A BEAM WHICH SUSTAINS A UNIFORM LOAD THROUGHOUT
ITS WHOLE LENGTH, AND WHOSE EXTREMITIES ARE SO FIRMLY
IMBEDDED IN A SOLID MASS OF MASONRY AS TO BECOME
RIGID.
Let the ratio of the lengths of the two portions AB and
AE of a beam, supported by two props (p. 487), be assumed
to be such as will satisfy the condition 5n2 —16n+8=0; or,
solving this equation, let
n='(4~ 4/6)..... (561).
The value of tan. 23 (equation 553) will then become
r-, r-,,  d..,. -,-r  zero; so that when this relation obtains, the neutral
E.~ illine will, at the point B, be:.-   parallel to the axis of the
I   abscissae; or, in other words,
the tangent to the neutral
line at the point B will retain,
after the deflexion of the beam, the position which it had
before; i. e., its position will be that which it would have
retained if the beam had been, at that point, rigid. Now 
this condition of rigidity is precisely that which results from
the insertion of the beam at its extremities in a mass of
masonry, as shown in the accompanying figure; whence it
follows that the deflexion in the middle of the beam is the
same in the two cases. Taking, therefore, the negative sign
in equation (561), and substituting for n its value 2(4- 4/6)
or -6202041 in equation (557), and observing that, in that




SUPPORTED AT ANY NUMBER OF POINTS.         491
equation, 2a, represents the distance BC in the accompanying figure, we obtain
pJa4
DI24EI... (562).
By a comparison of this equation with equation (530), it
appears that the deflexion of a beam sustaining a pressure
uniformly distributed over its whole length, and having its
extremities prolonged and firmly imbedded, is only one-fifth
of that which it would exhibit if its extremities were free. 
If the masonry which rests upon each inch of the portion
AB of the beam be of the same weight as that which rests
upon each inch of BC, the depth AB of the insertion of each
end should equal'62 of AE, or about three telinhs of the
whole length of the beam.
377. Conditions of the equilibrium of a beam supported at
any?umber of points and deflected by given pressures.
To simplify the investigation, let the points of support
p          P     ABC be supposed to be three
g  3  }  1   in number, and let the directions of the pressures bisect
_:_ICE      Al-          the distances between them;
V1-     r   |     t     l l the same analysis which deI/~!  |    S      -p termines the conditions of the..l........     equilibrium in this case will
be found applicable in the more general case. Let P1, P~,
P5, be taken to represent the resistances of the several points
of support, a, and a2 the distances between them, P, P4 the
deflecting pressures, and x y the co-ordinates of any point in
the neutral line from the origin B. Substituting in equation
(500) for  its value dG, and observing that in respect to the
portion BD of the beam    zPp=P2(1 - a-x)-P(a —x), and
that in respect to the portion DA    of the beam, zPp=
— P1(a, —x), we have for the differential equation to the
neutral line between B and D
* The following experiment was made by Mr. Hatcher to verify this result.
A strip of deal -3 in. byj-F: in. was supported with its extremities resting
loosely on rollers six feet apart, and was observed to deflect 1'2 inch in the
middle by its own weight. The extremities were then made rigid by confining
them between straight edges, and, the distance between the points of support
remaining the same, the defletion was observed to be'22 inch. The theory
would have given it'24.




492    BEAM SUPPORTED AT ANY N UMBER OF POINTS.
E~lY=P(a-)-P,(a,-a)..         (563),
dx    2
between D and A
ElY    -P(a,-).. (564).
Representing by / the inclination of the tangent at B to the
axis of the abscissae, and integrating the former of these
equations twice between the limits 0 and x,
EIdy- P2(a —2)- P      -(a, — x2)+EItan./^.... (565);
EIy=iP2(a1x —jx)-iP,(ap(-x2-) +EIx tan..... (566).
Substituting'a, for x in these equations, and representing
by D1 the value of y, and by y the inclination to the horizon
of the tangent at the point D, we obtain
EI tan. y-'iP2a12 —P1a, 2+EI tan.3... (567),
EID1,=P2a13 — AP5 PaL-3+JEIa, tan.... (568).
Integrating equation (564) between the limits - and x
2
EI=  — P,(, —    2)+EI tan. y+fPal2.
Eliminating tan. y between this equation and equation (567)
and reducing,
EI=y-=-P (a1x-x2) +EI tan. / +P2a12..  (569).
Integrating again between the limits  and  and elimintegrating the value of  om equation (568), d eliminating the value of D, from equation (568),
Ely= -P(ax2 — ) + (EI tan. +~iP2a,2)x- IP2aa3. (570).
Now it is evident that the equation to the neutral line in
respect to the portion CE of the beam, will be determined
by writing in the above equation P5 and P4 for P, and P,
respectively.
Making this substitution in equation (570), and writing
-tan. 3 for +tan. 3 in the resulting equation; then assum



BEAM SUPPORTED AT ANY NUMBER OF POINTS.   493
ing x=a, in equation (570), and x==a in the equation thus
derived from it, and observing that y then becomes zero in
both, we obtain
0= —Plal3 + 5 P2a1 +EIa, tan. 3,
0=P —*Pa2 + —5P4a21-EIa. tan. /3.
Also, by the general conditions of the equilibrium of parallel
pressures (Art. 15.),
Pa +iP 4, +P=P a2+P,.
Pl+P3+P5=P2+P4.
Eliminating between these equations and the preceding, assuming a,+a-=a, and reducing, we obtain
_P2a,(8a2 + 5a1)-3P4a2     (:.16aa..''
P4a2(8a + 5a2)- 3Pa.   (72).
5    16aa, 
P2(    8,    Pi13a  (  3a 4  *.La
By equation (568),
D,=-  8   l Pa(16 +7a)-9Pa2....  (574).
Similarly,
D2=T6       Pa(1t6al,+7a) - 9P a 2 }.    (575);
tan. /=48Ia   P,(8aI2-5a)3P42.... (56).
By equation (567),
tan. y=128I    Pa{ +Pa2.....     (577).
If a, be substituted for a in equation (569), and for P1 and
tan. f3 their values from equations (571) and (576); and if
the inclination of the tangent at A to the axis of x be represented by P,, we shall obtain by reduction




494      BEAM   SUPPORTED AT ANY NUMBER OF POINTS.
tan. /=132EIa      Pa —P2a,(2a+ a        )....(578).
Similarly, if /2 represent the inclination of the tangent at
C to the axis of x,
tan.   = 3a2EI   { P    2a -P4a (2a +    )..... (579).
378. If the pressures P, and P,, and also the distances a,
and a,, be equal,
P:   p =PPT,      P= LPI, tan. /6=0, tan.,2Etan.         A —E
379. If the distances a, and a, be equal, and P,=3P,,
P1=  P, P3-41 2, P5 —P2, tan. /          — 16Ei' tan. P,=0.
380. If a,=a2 and 3P,=13P2, P,0, P, P-            P,, P,=5   P,.
* The following experiments were made by Mr. Hatcher to verify this result.
The bar ACB, on which the experiment was to be tried, was supported on
knife edges of wrought iron at A, C, and B, whose distances AC and CB were
each five feet. The angles of the knife edges were 90~, and the edges were
oiled previous to the experiments. The weights were suspended at points D
C! 
and E intermediate between the points of support. In measuring the angles
of deflexion the instrument (which was a common weighted index-hand turning on a centre in front of a graduated arc) was placed so that the angle c
of the parallelogram of wood carrying the arc was just over the knife-edge B,
the side cd of the parallelogram resting on the deflected bar. This position
gave the angle at the point of support.
1st Experiment.-A bar of wrought iron half an inch square, being loaded
at E with a weight of 18 lb. 13 oz., and at D with 52 lb. 3 oz., assumed a perfectly horizontal position at B, as shown by the needle. The proportion of
these weights is 2'77: 1.
2d Experiment.-A bar'7 inches square, being loaded at E with a weight of
3'-3 lb., and at D with a weight of 112 lb., assumed a perfectly horizontal
position at B. The weights were in this experiment accurately in the proportion 3: 1.
3d Experiment.-A round bar,'75 inch in diameter, being loaded at E with
37-3 lb., and at D with 112 lb., showed a deviation from the horizontal position
at B amounting to not more than 20'. The weights were in the proportion of
3:1.
The influence of the weight of the bar is not taken into account.




A BEAM DEFLECTED BY PRESSURES.        495
381. CURVATURE OF A RECTANGULAR BEAM, THE DIRECTION OF
THE DEFLECTING PRESSURE AND THE AMOUNT OF THE DEFLEXION BEING ANY WHATEVER.
The moment of inertia I (Art. 358.) is to be taken, about
an axis perpendicular to the plane of deflexion, and passing
through the neutral line, the distance h of which neutral
line from the centre of gravity of the section is determined
by equation (499).
Now Tlbc3 representing (Art. 362.) the moment of inertia
of the rectangular section of the beam about an axis passing through its centre of gravity, it follows (Art. 79.) that
the moment I about an axis parallel to this passing through
a point at distance h from it is represented by
I= hbe + ~ —lbe.
Substituting, therefore, the value of h from equation
(499),
I  E 2b sin. +T1bc'.... (580).
Substituting this value in equation (500), and reducing,
1       12P,Ebep,
p a   +....(581).
~- 12RP, 2sin.' + E"'bc4....
q                          Draw ax parallel to the
"\\>.~ position of the beam be"::(...... fore deflexion; take this
\\\ ^~     ~    line as the axis of the
-  - Fasn   abscissae and a as the
\d  A c,C     origin; thenpl =-Rm=Rn
-    +unm=MR    cos. MRm+.C...  M'':. v     aM sin. Mam==y cos. Mam
B Y^l^^ll^^        -+x sin. Manm.
X/B ~~ / \ t         Let, now, the inclination
4:;  DrDaP, of the direction of
P1 to the normal at a be
represented by 8,, and the inclination Mat of the tangent to
the neutral line at a to ax, by /3; then Manm==2-(O8 +,).
~ P,=y sin. (, + S,) +  cos. ( + i,).
Substituting this value of p, in the preceding equation,




496         A BEAM DEFLECTED BY PRESSURES.
1  12PEbc y sin. (1 +,) +x cos. (01 +) ~,)'- R    12R2P,2 sin. 2 + E2b2c4....(
where 0 represents (Art. 355.) the inclination Rqa of the
normal at the point R to the direction of P,.
382. Case in which the deflexion of the beam is small.
If the deflexion be small, and the inclination 8, of the
direction of Pi to the normal at its point of application, be
not greater than 4; then y sin. (0 + 11) is exceedingly small,
and may be neglected as compared with x cos. (,1 +il); in
this case, moreover, 6 is, for all positions of R, very nearly
equal to 01. Neglecting, therefore, fi as exceedingly small,
we have
1      12PEbcx cos. 01
R~12REP2 sin. 20, +E2b2 *. ().
Solving this equation, of two dimensions, in respect to R, and
taking the greater root,
1   6P,
R=Eb6G  { Cos. 0 + 4/? cos.'0 —1'C sin. 0.... (584).
383. THE WORK EXPENDED UPQN THE DEFLEXION OF A UNIFORM RECTANGULAR BEAM, WHEN THE DEFLECTING PRESSURES ARE INCLINED AT ANY ANGLE GREATER THAN HALF A
RIGHT ANGLE TO THE SURFACE OF THE BEAM.
If u1 represent work expended on the deflexion of tle
portion AM of the beam, then (equation 505)
a1
Pi2 rPi 
2/dx;
0
P.e  E   pi
but by equation (500)  1 E. I
IP''R




. BEAM DEFLECTED BY PRESSURES.       497
al..    p u f1Pd7.... (585).
0
1Bt,  6P,
Ru   Ebce {x cos. 1 + 4/2 cos. 2I1-'i sin.,l x cos. 81;
by equation (584), observing that the deflexion being small,
p,=x cos. 81 very nearly. Now the value of  (equation
584) becomes impossible at the point where w cos.,1 becomes
less than —  sin.,1; the curvature of the neutral line commences therefore at that point, according to the hypotheses
on which that equation is founded. Assuming, then, the
corresponding value -c tan. 0, of x to be represented by a,,
the integral (equation 585) must be taken between the limits
s, and a,, instead of 0 and a,;
I 3P,'cos.0,/',,.. s    axli.*. = P 2    cos. 01+w 4/a2 cos. 20 i-2 sin. 2 dx;
a51
3v'~: __ =P~ COS. 2a ---   c3tan., +  (a' —ctan.'O,)~ *(586).
And a similar expression being evidently obtained for the
work expended in the deflexion of the portion BM of the
beam, it follows, neglecting the term involving c3 as exceedingly small when compared with a,1, that the whole work U,
expended upon the deflexion is represented by the equation
U8=E8 i2 COS. 28.   a," + (a1 -~c2 tan. 2d,)2 +
P2 cos. 20 la,'+ (a^2-j2 tan. 0,2)} }
But if 0, be taken to represent the inclination of P, to the
normal to the surface of the beam, as 01 and 0, represent the
similar inclinations of P, and P,, then, the deflexion being
small,
* Church's Int. Cal. Art. 149.
32




498         A BEAM DEFLECTED BY PRESSURES.
P1a cos. 81=Pa2 cos. 0, Pa cos. 3  8=Pa1 cos. 0.
Eliminating P1 and P. between these equations and the
preceding,
US=      S' a26 22 ja a,+(a2 —ei2 tan. 2,)2} +
a,2 a^  + (a22- 2tan. 2)2} }... (587).
If the pressure P, be applied perpendicularly in the centre
of the beam, and the pressures P, and P2 be applied at its
extremities in directions equally inclined to its surface; then.-a2 =a-=, =-,=8 =  and 03=0.  Substituting these values.in the preceding equations, and reducing,
P 2 JaS + (a2 _4tan. 20)31
Us-         16Ebc     -.... (588).'384. THE LINEAR DEFLEXION OF A REOTATNGULAR BEAM.
D, being taken as before (Art. 368.) to represent the deflexion of the extremity A measured in a direction perpendicular to the surface of the beam, we have (Art. 52.),= SP, cos. 1,dD,
P    0 cos.  du,  du,  dPl.. P cos. 8-dD,-dP    dI
But by equation (586), neglecting the term involving c',
dP-Eb 2P cos. 2 a22+(a'-2-c2 tan. 281)2
*P1cos. dP=, 2P13 cos^.2aa1,+(a1-2 a-tan.221)2a
dP    2P,3...? cos. 61        COS. I~ oa,' ~+ (a, —o. ta  )
Dividing both sides by PI, reducing, and integrating,
3
9P                    t   2
D, E  cos.  as    - (a, —   tan. 8).... (589)
Proceeding similarly in respect to the deflection D" perpen



INCLINED AT ANY ANGLE TO ITS SURFACE.   499
dienlar to the surface of the beam at the point of application
of PS, we obtain from equation (587)
I=2P     3 Cs 6  +   -^' 
D 3=1 COJ3 a2'     a'+ (a,2 —c2 tan. x81)2 +
al2 la2+ (a2 —  tan. 2) }..  (590)
In the case in which P1 and Pi are equally inclined to the
extremities of the beam and the direction of P, bisects it,
this equation becomes
3
D   P. a+(n — c tan.           (591).
D2     S      48Ebc2           (591).
385. The work expended upon the deflexion of a beam subjected to the action of pressures applied to its extremities,
and to a single intervening point, and also to the action
of a system of parallel pressures uniformly distributed
over its length,
Let u represent the aggregate amount of the parallel
D
pressures distributed over each unit of the length of the
beam, and a their common inclination to the perpendicular
to the surface; then will Ux represent the aggregate of those
distributed uniformly over the surface DT, and these will
manifestly produce the same effect as though they were
collected in the centre of DT. Their moment about the
point R is therefore represented by Ix 2x cos. a, or by -ax2
cos. a; and the sum of the moments of the pressures applied
to AT is represented by (Px cos. O,-1-t2 cos. ). Substituting this value of the sum of the moments for Pp, in
equation (505), we obtain
al
1 f(PIX cos. O,-IjPx cos. a)2d.
0




500     DEFLEXION OF A BEAM BY PRESSURES.
386. If the pressures be all perpendicular to the surface of
the beam,,=O, ac=O, and I is constant (equation 499);
whence we obtain, by integration and reduction,
1-2EI Vi    4 1   2. 2 )
u=j {*P2 —P1a +.ea.. (592).
If the pressure P, be applied in the centre of the beam,
P1=jP,+4pa, and a,=-a, also the whole work UT of
deflecting the beam is equal to 2u,; whence, substituting
and reducing,
U,-48EI {PlP +P Pa+ifPa  1P 2*.  (593).
387. A RECTANGULAR BEAM IS SUPPORTED AT ITS EXTREMITIES
BY TWO FIXED SURFACES, AND LOADED IN THE MIDDLE: IT
IS REQUIRED TO DETERMINE THE DEFLEXION, THE FRICTION
OF THE SURFACES ON WHICH THE EXTREMITIES REST BEING
TAKEN INTO ACCOUNT.
It is evident that the work which produces the deflexion
of the beam is done upon it partly by the deflecting pressure
P, and partly by the friction of the surface of the beam
upon the fixed points A and B, over which it moves whilst
in the act of deflecting. Representing by p the limiting
angle of resistance between the surface of the beam and
either of the surfaces upon which its extremity rests, the
friction Q, or Q2 upon either extremity will be represented
by JP tan. qp; and representing by s the length of the
curve ca or eb, and by 2a the horizontal distance between
the points of support; the space through which the surface
of the beam would have moved over each of its points of
support, if the point of support had been in the neutral line,
is represented by 8-a, and therefore the whole work done
upon the beam by the friction of each point of support by
i tan. qfPdc. Moreover, D representing the deflexion of




THE SOLID OF THE STRONGEST FORM.      501
the beam under any pressure P, the whole work done by P
is represented by  PdD.  Substituting, therefore, for the
work expended upon the elastic forces opposed to the
deflexion of the beam its value from equation (588), and observing that the directions of the resistances at A and B are
inclined to the normals at those points at angles equal to
the limiting angle of resistance, we have
jfPdD + tan.     Pg= p      (a2-   tan.)}
frd          + fGEb
r f   dD    I    _     ds
But   PdD =       fP dP; and  Pds= jPf-pdP
30E2j2 dPp by equation (521).
Substituting these values in the above equation, and differentiating in respect to P, we have
dD   P {a3 + (a2 — 2 tan. 2q  PWa
dP-          8Ebe     - — 30E2tan 
Dividing by P, and integrating in respect to P,
PD )'a+(a2 —c 4tan. 2q))}  PVa
=    -a'+ 8Ebc  -   -6OE2-2tan.... (594).
388. THE SOLID OF THE STRONGEST FORM WITH A GIVEN
QUANTITY OF MATERIAL.
The strongest form which can be given to a solid body in
the formation of which a given quantity of material is to be
used, and to which the strain is to be applied under given
circumstances, is that form which renders it equally liable to
rtpture at every point.. So that when, by increasing the
strain to its utmost limit, the solid is brought into the state
bordering upon rupture at one point, it may be in the state
bordering upon rupture at every other point. For let it be
supposed to be constructed of any other form, so that its
rupture may be about to take place at one point when it is
not about to take place at another point, then may a portion
of the material evidently be removed from the first point
without placing the solid there in the state bordering upon




502             THE RUPTURE OF A BAR.
rupture, and added at the second point, so as to take it out
of the state bordering upon rupture at that point; and thus
the solid being no longer in the state bordering upon
rupture at any point, may be made to, bear a strain greater
than that which was before upon the point of breaking it,
and will have been rendered stronger than it was before.
The first form was not therefore the strongest form of which
it could have been constructed with the given quantity of
material; nor is any form the strongest which does not
satisfy the condition of an equal liability to rupture at every
point.
The solid, constructed of the strongest form, with a given
quantity of a given material, so as to be of a given strength
under a given strain, is evidently that which can be constructed, of the same strength, with the least material; so
that the strongest form is also the form of the greatest
economy of material.
RUPTURE.
389. The rupture of a bar of wood or metal may take
place either by a strain or tension in the direction of its
length, to which is opposed its TENACITY; or by a thrust or
compressing force in the direction of its length, to which is
opposed its resistance to COMPRESSION; or each of these
forces of resistance may oppose themselves to its rupture
transversely, the one being called into operation on one side
of it, and the other on the other side, as in the case of
a TRANSVERSE STRAIN.
TENACITY.
390. The tenacities of different materials as they have
been determined by the best authorities, and by the mean
results of numerous experiments, will be found stated in a
table at the end of this volume. The unit of tenacity is that
opposed to the tearing asunder of a bar one square inch in
section, and is estimated in pounds. It is evident that the
tenacity of a fascile of n such bars placed side by side, or
of a single bar n square inches in section, would be equal
to n such units, or to n times the tenacity of one bar.
To find, therefore, the tenacity of a bar of any material
in pounds, multiply the number of square inches in its sec



RUPTURE OF A BAR SUSPENDED VERTICALLY.   503
tion by its tenacity per square inch, as shown by the
table.
391. A BAR, CORD, OR CHAIN IS SUSPENDED VERTICALLY, CARRYING A WEIGHT AT ITS EXTREMITY: TO DETERMINE THE
CONDITIONS OF ITS RUPTURE.
First. Let the bar be conceived to have a uniform section
represented in square inches by K; let its length in inches
be L, the weight of each cubic inch a, the weight suspended
from its extremity W, the tenacity of its material per square
inch T; and let it be supposed capable of bearing m times
the strain to which it is subjected. The weight of the bar
will then be represented by ELK, and the strain upon its
highest section by pLK +W. Now the strain on this section
is evidently greater than that on any other; it is therefore at
this section that the rupture will take place. But the resistance opposed to its rupture is represented by KT; whence it
follows (since this resistance is m times the strain) that
K-r=m(pLK + W),
mW
_..... (595).
By which equation is determined the uniform section K of a
bar, cord, or chain, so that being of a given length it may be
capable of bearing a strain m times greater than that to
which it is actually subjected when suspended vertically.
The weight Wi of the bar is represented by the formula
KL~,.*MpWL.W, —-m     L * *   *.(596).
392. Secondly. Let the section of the rod be variable; and
let this variation of the section be such that its strength, at
every point, may be that which would cause it to bear,
without breaking, m times as great a strain as that which it
actually bears there. Let K represent this section at a point
whose distance from the extremity which carries the weight
W is x; then will the weight of the rod beneath that point
be represented by f_ Kdx; or, supposing the specific gravity




504     RUPTURE OF A BAR SUSPENDED VERTICALLY.
of the material to be every where the same, by f Kdx: also
the resistance of this section to rupture is KT.
m (W +PfKdx)=KT
Differentiating this expression in respect to x, observing that
K is a function of x, and dividing by Kr, we obtain
1 dK  _ m
K dx -- 
Integrating this expression between the limits 0 and w, and
representing by Ko the area of the lowest section of the rod,
log.       x; K   K: =Koe r
EKO- T
But the strain sustained by the section Ko is W, therefore
Ko07=mW;
K-     e T X....... (597).
The whole weight W, of the rod, cord, or chain, is represented by the formula
LWfMPf,                 (m L       \
W    ^=tffKdx  = — e r 0 d=W(         _         (598).
0           ~
A rope or chain, constructed according to these conditions,
is evidently as strong as the rope or chain of uniform section
whose weight W, is determined by equation (596), the value
of m being taken the same in both cases. The saving of material effected by giving to the cord or chain a section varying according to the law determined by equation (598) is
represented by W1-W,, or by the formula
L-          (T-1...... (599).
Church's t. al. Art. 19.
* Church's Int. Cal. Art. 159.




THE SUSPENSION BRIDGE.          505
THE SUSPENSION BRIDGE.
393. General conditions of the equilibrium of a loaded
chain.
Let AEH represent a chain or cord hanging freely from
two fixed points A and H,
and having certain weights
A^-ER.^- <w,, w, w3,) &c., suspended by, rods or cords from  given
points B, C, D, &c., in its
s </ S    length. Through the lowest
^..    point E of the chain draw
^ ^..;   the vertical Ea, containing
i^  "'I-, o  as many equal parts as there
E>^..^    f       J. are units in the weight of
the chain between E and any
E J123  4 6 6  8 Lpoint of suspension B, together with the suspending
rods attached to it, and the weights which they severally
carry; draw aP parallel to the direction of a tangent to the
curve at B, and produce the tangent at E to meet aP in P;
then will aP and EP contain as many equal parts as there
are units in the tensions at B and E respectively; and if Eb
and Ec be taken to represent the whole weights sustained by
EC and ED, and Pb and Pc be joined, these lines will in
like manner represent the tensions upon the points C and D.
For the pressures applied to EB, and in equilibrium, being
the weight of the chain, the weights of the suspending rods,
the weights attached to the rods, and the tensions upon B
and E, the principle of the polygon of pressures (Art. 9.)
obtains in respect to these pressures. Now the lines drawn
to complete this polygon, parallel to the weights, form
together the vertical line Ea, and the polygon (resolving
itself into a triangle) is completed by the lines aP and EP
drawn parallel to the tensions upon B and E. Each line
contains, therefore, as many equal parts (A.t. 9.) as there
are units in the corresponding tension. Also, the pressures
applied to the portion EC of the curve, being the weights
whose aggregate is represented by Eb, and the tensions upon
E and C, of which the former is represented in direction
and amount by EP, it follows (Art. 9.) that the latter is
represented also in direction and amount by the line Pb,




506                 THE CATENARY.
which completes the triangle aPb; so that bP is parallel to
the tangent at C.
In like manner it is evident that the tension upon D is
represented in magnitude and direction by cP; so that cP is
parallel to the tangent to the curve at D.
THE CATENARY.
394. If a chain of uniform  section be suspended freely
between two fixed points A and B, being acted upon by no
other pressures than the weights of its parts, then it will
assume the geometrical fornm of a curve called the
catenary.
Let PT be a tangent to any point P of the curve intersecting the vertical CD passing through its lowest point D
in T; draw the horizontal line DM intersecting PT in Q;
take this line as the axis of the abscissae; and let DM =x,
MP=y, DP=s, weight of each unit in the length of the
chain ==P, tension at D=c. Now DT being taken to represent the weight Ps of DP, it has been shown (Art. 393.)
that DQ will represent the tension c at D, and TQ that
at P.
dy                        DT   Pus
Also, -= tan. PQM = tan. DQT =D-     s
*rdx-1                  -'? DQ-'"
dy P's
c.c =.... (600).
gain, s -  +l                     Integrating  bedsin ~-'-~




THE CATENARY.                 507
tween the limits 0 and s, and observing that when s=0,
x=O,
=-lo.g,  { c      + (1  c2..... (601).
t:., +c  +  )   }.....  
Ps  ( I&5 2     aX.     C   + 1+/  E C.s  / 1+ -— s     - i P-'c  \      I      c
By addition and reduction,
s=1,  -                  (602).
Substituting this value for s in equation (600), and integrating between the limits 0 and x,
/                    /           2      (603)2
Y   -   G                 - 2c    2c..... (603);
which is the equation to the catenary.
395. The tension (c) on the lowest point of the catenary.
Let 2S represent the whole length of the chain, and 2a
the horizontal distance between the points of attachment.
Now when x=a, s=S; therefore (equation 602),
/  aI -    \
S=   -   c    c-.     (604);
for which expression the value of c may be determined by
approximation.
396. The tension at any point of the chain.
The tension T at P is represented by TQ= 4/D-Q'+]P;
* Church's Int. Cal. Art. 144.




508                 THE CATENARY.. T =(c' +s')..... (605).
Now the value of c has been determined in the preceding
article; the tension upon any point of the chain whose distance from its lowest point is s is therefore known.
397. The inclination of the curve to the vertical at any
point.
dy
Let l represent this inclination, then cot. = —;.(equation 603) cot. t=  ~c - c -.....(606).
The inclination may be determined without having first
determined the value of c, by substituting cot. t for  in
c
equation (601); we thus obtain, writing also a and S for x
and s,
=-tan. l log. (cot. t+cosec. t)=tan. t log. 6 cot. $t;. tan. t log. tan...... (607).
This equation may readily be solved by approximation; and
the value of c may then be determined by the equation
c=p-S tan. t.
398. A chain of given length being suspended between two
given points in the same horizontal line: to determine the
depth of the lowest point beneath the points of attachment;
and, conversely, to determine the length of the chain whose
lowest point shall hang at a given depth below its points
of attachment.
The same notation being taken as before,




THE CATENARY.                  509
Integrating between the limits 0 and 8, and observing that
y=O when s=0,
1
y^= p;(6+F282c- Icl...  (608).
Solving this equation in respect to s,
S=      (+ 2c           (609).
If H represent the depth of the lowest point, or the ersed
sine of the curve, then y=H when s=S.
H=-(c2+-2S2)-.....(610).
S  V/H (H+     ). *. -  (611).
399. The centre of gravity of the catenary.
If G represent the height of the centre of gravity above
the lowest point, we have (Art. 32.)
ds
S. G=jyds      fy  dx.
Substituting, therefore, for y and  their values from equations (602) and (603), we have
a
S. G =-$,c f* i" fx  -,ax  yix  -fix
S.G=        c     c (  c       c )d
0   a( +a    -2''a   +    a         )
a/,
C     2iYX  - /x       ( fx  _/axv i
o ] s        +s  +  +2s
C ( C / 2ya  -2a    \  2C / ma  -ma t
i{ * 2l        +2a       V     -   n




510             TIE SUSPENSION BRIDGE
0  cifa  -jua  j a  -I \La
t F a )(+ S    -4
But by equation (602) S=-(           and by equation
603),:.SG..  {s(H-.) +.
G=    H-( 1-      ). (613).
400. THE SUSPENSION BRIDGE OF GREATEST STRENGTH, THE
WEIGHT OF THE SUSPENDING RODS BEING NEGLECTED.
Let ADB represent the chain, EF the road-way; and let
the weight of a bar of the material of the chain, one square
the weight of a bar of the material of the chain, one square
inch in section and one foot long, be represented by P,j the
weight of each foot in the length of the road-way by 2, the
aggregate section of the chains at any point P (in square
inches) by K, the co-ordinates DM and MP of P by x and y,
and the length of the portion DP of the chain by s. Then
will the weight of DP be represented by p,/KKds, and the
weight of the portion CM of the roadway by px; so that
the whole load (u) borne by the portion DP of the chain
will be represented (neglecting the weight of the suspending
rods) by
I rKds+p2x,. AU=   fKds+f2....      (614).




OF GREATEST STRENGTH.              511
Let this load (u), supported by the portion DP of the
chain, be represented by the line Da, and draw Dp in the
direction of a tangent at D, representing on the same scale
the tension c at that point; then will ap be parallel to a
tangent to the chain at P (Art. 393).
ady u.  - o..... (615).
dx/    - c
Now let it be assumed that the aggregate section of the
chains is made so to vary its dimensions, that their strength
may at every point be equal to m times the strain which
they have there to sustain. But this strain is represented in
magnitude by the line ap (Art. 393.), or by (c2+3~)t; if,
therefore, r be taken to represent the tenacity of the material of the chain, per square inch of the section, then
KTm( + U)....... (616).
Therefore K==mc ( +       m  + ( + dy ) (equation 615)
ds           ds Kr          r           dsr^8
=mc-n; therefore- =-. Also      Kd=      K-d
dSx          d~x  mc       J me
fK2dx=    cf (c + u)dx (equation 616);.(equation 614)u=-'(c2 +u)dx + Px.
du.
Differentiating in respect to:, and observing that   =
du dy   u du,
— dy dx    (equation 615), we have
du  u du   mPl(     2      f     (      T,
dj —  dy- -- (0< +,)+) + "      2 + 62 +  -P I;
~v                    6
du                      udu
x_   r                      r Y=ud
* 331L~k m    AESIM   mMj,,U 4   2''*^ lu+c'fm ~ ^~^ —




512              TIHE SUSPENSION BRIDGE
Integrating these expressions," we obtain
TO
-=     1c2 +       tan.'c2 +       ~..... (61).
Y=  - log.        c m1 
c"Substituting in this equation the value of u given by the
preceding equation, and reducing,
y- (2log. sec. f      ( J + Cm,).. (8)
which is the equation to the suspension chain of uniform
strength, and therefore OF THE GREATEST STRENGTH WITH A
GIVEN QUANTITY OF MATERIAL.
401. To determine the variation of the section K of the
chain of the suspension bridge of the greatest strength.
Let the value of u determined by equation (617) be substituted in equation (616); we shall thus obtain by reduction
EK=    \+ 1+ (+  +  " itan..     +',)   }..(619).t
T (.     x \'mc&, /      \    cmTl    5,'
It is evident from this expression that the area of the section of the chains, of the suspension bridge of uniform
strength, and therefore of the greatest economy of material,
increases from the lowest point towards the points of suspen-.
sion, where it is greatest.
* Church's Int. Cal. Art. 133, Case IV.
ds IrK       r p
- -'. s= -j /Kdx. Now the function K (equation 619) may be
dx mc'      me =
integrated in respect to x by known rules of the integral calculus; the value
of s may therefore be determined in terms of x, and thence the length in
terms of the span. The formula is omitted by reason of its length.
Church's Int. Cal. Art. 129, Case II.




OF GREATEST STRENGTH.               513
402. To determine the weight W  of the chain of the suspension bridge of the greatest strength.
Let it be observed that W=,f Sids=u-,2x       (equation
614); substituting the value of u from equation (617), we
have
W=c(l+       — i2) tan.    1 ( +-  )     -Px... (620).
l mcll         T.mp
403. To determine the tension c upon the lowest point D of
the chain of uniform strength.
Let H be taken to represent the depth of the lowest point
D, beneath the points of suspension, and 2a the horizontal
distance of those points: and let it be observed that H and
a are corresponding values of y and x (equation 618);.-H — log., sec. - 11+- 2         a
mpi e1       (' cmJ 
Solving this equation in respect to c,
C=   ~i m(- -sec          F. C T — 1...(621).
L L( 1          -   oli  2   ( 1
404. THE SUSPENSION BRIDGE OF GREATEST STRENGTH, THE
WEIGHT OF THE SUSPENDING RODS BEING TAKEN INTO ACCOUNT.
Conceive the suspending rods to be replaced by a con-" —-      -'"'..   -.B.....
33




514              TH-E SUSPENSION BRIDGE
tinuous flexible lamina or plate connecting the roadway with
the chain, and of such a uniform thickness that the material
contained in it may be precisely equal in weight to the material of the suspending rods.  It is evident that the conditions of the equilibrium will, on this hypothesis, be very
nearly the same as in the actual case. Let Vp, represent the
weight of each square foot of this plate, then will 3,fyd
rrersent the weight of that portion of it which is suspended
from the portion DP of the chain, and the whole load u upon
that portion of the chain will be represented by
U=~ ds f    [-2++fydx.... (622).
It may be shown, as before (Art. 400.), that,   = r M   +.....(623).
ds         2 lnear e uandx.  Substituting in eqration w  b (622),
T        -2ay                -.,e-...~y,+ i2e    +.      d d  u d
differentiat in respect to  and observing that when yb, 
du  u du  mu(rch's Int ).       (624).
dx c dy- -TO
Transposing, reducing, and assuming,
M-Ioc.        (625);
dy
A linear equation in u, the integration of which by a well
known method gives
-2ay   -*-2ay
U2    =2     3y  + ac'+5)f   dy+C.*
Assuming the length of the shortest connecting rod DO to
be represented by b, integrating between the limits b and y,
and observing that when y=b, u=0,
* Church's Int. Cal. Art. 176.




OF GREATEST STRENGTH.               515,-2 $-   ^()                ( +-2ab  -2ays; c+9  \ -2ab  -. —aY).}
Tee ~2 C ~bs 
gt) =     -8~    jfY  4 * +' 2~        -f2   )
) (                                         62a(y-A)'/  v2a(y-b)'.    I \          -)+ (a+^+F        2a(y-)( b  1).(626).
Substituting this value of u' in equation (623), and
reducing,
K   (=4     (  r / + c+ U}    -     3a(y —).   * (627);
by which expression the variation of the section of the chain
of uniform strength is determined.
Differentiating the equation  y —=  in respect to a, and
du
substituting for d  its value from equation (624).
d'y
cd=3    (o2 +X2)+2 +  y.
Substituting for u2 its value from equation (626),
Cd2o -    +=+ (2 +?+  ) ~ b+c 
d 2r                    b 2a'
dy
Mltiplying both sides of this equation by d-, and integrating between the limits b and y, observing that when y=b,
dy
— 0
(dy 2  /    b             a(y-b)
\xi) -      tb+ ac + p2) (s    -1)-    (y-).
Now let it be observed, that the value of I, being in all
practical cases exceedingly great as compared with the
values of p and m, the value of a (equation 625) is exceedingly small; so that we may, without sensible error, assume
those terms of the series 2a(y-b) which involve powers of
2a(y-b) above the first, to vanish as compared with unity.
* Church's Int. Cal. Art. 140.




516             THE SUSPENSION IBRIDGE.
This supposition being made, we have e2a(-b) —l-2X,(-b),
whence, by substitution and reduction,
c_)Y =2(=  b+awo+  ) (y-b).
Extracting the square root of both sides, transposing, and
integrating,
i= [ +c 2  ) (yl —b)... (628);
the equation to a parabola whose vertex is in D, and its
axis vertical.*
The values a and H of x and y at the points of suspension
being substituted in this equation, and it being solved in
respect to c, we obtain
C   2H1-2b a+c2) a2 *...  (629);
by which expression the tension c upon the lowest point of
the curve is determined, and thence the length y of the suspending rod at any given distance x from the centre of the
span, by equation (628), and the section K of the chain at
that point by equation (627), which last equation gives by a
reduction similar to the above
K=  2(y-b) (         a + 1.....(630).
405. The section of the chains being of uniform dimensions,
as in the common suspension bridge, it is required to
determine the conditions of the equilibrium.t
The weight of the suspending rods being neglected, and
the same notation being adopted as in the preceding articles, except that p, is taken to represent the weight of one
foot in the length of the chains instead of a bar one square
inch in section, we have by equation (614), since K is here
constant,
u=s + F2.x.....  (631).' Church's Analyt. Geom. Art. 191.
f This problem appears first to have been investigated by Mr. Hodgkinson
in the fifth volume of the Manchester Transactions; his investigation extends
to the case in which the influence of the weights of the suspending rods is
included.




THE COMMON SUSPENSION BRIDGE.        517
Differentiating this equation in respect to a, and observing
th late
that -=(l + i-)- (1 +         (equation 615), and that
du  du dy  dbu u    /    e\
-,-==^  — = -- - = (1 1 + - 2,'
dx dy de-dy c-       (+   ) +.
-'    ou                 udu
j(02 +2)'+ U2Ce 0J (0       + u2)+ d
The former of these equations may be rationalised by
assuming (c2+2- -2)   c + u, and the latter by assuming
(0+fu2) —; there will thus be obtained by reduction
r- y  (i- l^K+(12)dz            -rzdz
The latter equation may be placed under the form
which expression being integrated and its value substituted
for z, we obtain
y=    (c2+ I 2)+ —     log.P 1(02 + qe), +  }...(632).
The method of rational fractions (Church's Integ. Calc.
Art. 135) being applied to the function under the integral
sign in the former equation, it becomes
- 2C     { 1-    (___              dz.e
The integral in the first term in this expression is represented by i log. (\1- ) and that of the second term by
(a', a  tan.,       or!'+I, + [2 o




518            RUPTURE BY COMPRESSION.
og. (t 2+ thi). +(P'-t)
2(pfi2  g)  s (a +)     ^   ZX
according as j, is greater or less than,, or according as the
weight of each foot in the length of the chains is greater or
less than the weight of each foot in the length of the roadway.
Substituting for z its value, we obtain, therefore, in the
two cases,
co    (U-c)+(u2++2)   2s2
c (   (u+-c) -( u+c)-   2          ( 
l-g-el (U+ )-(jU2+-2) (l 2 —2+2) 
(^,+~0zG+(~-/Z0~i {(~+c2):-*})
If the given values, a and H, of x and y at the points of
suspension, be substituted in equations (633) and (632),
equations will be obtained, whence the value of the constant
c and of u at the points of suspension may be determined by
approximation. A series of values of u, diminishing from
the value thus found to zero, being substituted in equations
(633) and (632), as. many corresponding values of x and y
will then become known. The curve of the chains may thus
be laid down with any required degree of accuracy.
This common method of construction, which assigns a
uniform section to the chains, is evidently false in principle;
the; strength of a bridge, the section of whose chains varied
according to the law established in Art. 401. (equation 619),
would be far greater, the same quantity of iron being
employed in its construction.
RUPTURE BY COMPRESSION.
406. It results from the experiments of Mr. Eaton Hodgkinsbon* on the compression of short columns of different
heights but of equal sections, first, that after a certain height
is passed the crushing pressure remains the same, as the
* Seventh Report of the British Association of Science.




RUPTURE BY COMPRESSION.              519
heights are increased, until another height is attained, when
they begin to break; not as they have done before, by the
sliding of one portion upon a subjacent portion, but by
bending. Secondly, that the plane of rupture is always
inclined at the same constant angle to the base of the
column, when its height is between these limits. These two
facts explain one another; for if K represent the transverse
section of the column in square inches, and c the constant
inclination of the plane of rupture to the base, then will
K see. a represent the- area of the plane of rupture. So that
if y represent the resistance opposed, by the coherence of
the material, to the sliding, of one square inch upon the surface of another,* then will yK sec. a represent the resistance
which is overcome in the rupture of the column, so long as
its height lies between the supposed limits; which resistance being constant, the pressure applied upon the summit
of the column to overcome it must evidently be constant.
Let this pressure be represented by P, and let CD
#   be the plane of rupture. Now it is evident that
the inclination of the direction of P to the perpen- Xdicular QR to the surface of the plane, or its
R! c equal, the inclination a of CD to the base of the'. column, must be greater than the limiting angle
of resistance of the surfaces; if it were not, then
would no pressure applied in the direction of P
be sufficient to cause the one surface to slide upon the other,
even if a separation of the surfaces were produced along
that plane.
Let P be resolved into two other pressures, whose directions are perpendicular and parallel to the plane of rupture;
the former will be represented by P cos. a, and the friction
resulting from it by P cos. a tan. p; and the latter, represented by P sin. a, will, when rupture is about to take place,
be precisely equal to the, coherence: Ky sec. a of the;plane of
rupture increased by its friction P cos. a tan. Ap, or P  sin.
a=Ky sec. a+P cos. a tan. p, whence by reduction
p  K   cos. p         2Ky cos.             (634).
sin. (a -) cos. a  siAn (;:2 —, — si:n. 
It is evident from this expression that if the coherence of
the: material were the same in all directions, or if the unit of
* The force necessary to overcome a. resistance, suchas- that. here spoken
of' has been. appropriately called, by Mr. Hodgkinson the force necessary tV6
s/ewr it-across.




520              THE PLANE OF RUPTURE.
coherence y opposed to the sliding of one portion of the
mass upon another were accurately the same in every direction in which the plane CD may be imagined to intersect
the mass, then would the plane of actual rupture be inclined
to the base at an angle represented by the formula
a=+......(635);
since the value of P would in this case be (equation 634)
a minimum when sin. (2a —) is a maximum, or when
2a —q=-, or = —+-; whence it follows that a plane inclined to the base at that angle is that plane along which the
rupture will first take place, as P is gradually increased beyond the limits of resistance.
The actual inclination of the plane of rupture was found
in the experiments of Mr. Hodgkinson to vary with the material of the column. In cast iron, for instance, it varied
according to the quality of the iron from 48~ to 58~*, and
was different in different species. By this dependence of
the angle of rupture upon the nature of the material, it is
proved that the value of the modulus of sliding coherence
y is not the same for every direction of the plane of rupture, or that the value of g varies greatly in different qualities of cast iron.
Solving equation (634) in respect to y we obtain
7= — sin. (a —) cos. a sec...... (636);
from which expression the value of the modulus y may be
determined in respect to any material whose limiting angle
of resistance qp is known, the force P producing rupture,
under the circumstances supposed, being observed, and also
the angle of rupture.t
THE SECTION OF RUPTURE IN A BEAM.
407. When a beam is deflected under a transverse strain,
* Seventh Report of British Association, p. 349.
t A detailed statement of the results obtained in the experiments of Mr.
Hodgkinson on this subject is contained in the Appendix to the " Illustrations
of Mechanics " by the author of this work.




GENERAL CONDITIONS OF RUPTURE.       521
the material on that side of it on which it sustains the strain
is compressed, and the material on the opposite side
extended.  That imaginary surface which separates the
compressed from the extended portion of the material is
called its neutral surface (Art. 354.), and its position has
been determined under all the ordinary circumstances of
flexure. That which constitutes the strength of a beam is
the resistance of its material to compression on the one side
of its neutral surface, and to extension on the other; so that
if either of these yield the beam will be broken.
The section of rupture is that transverse section of the
beam about which, in its state bordering upon rupture, it is
the most extended, if it be about to yield by the extension
of its material, or the most compressed if about to yield by
the compression of its material.
In a prismatic beam, or a beam of uniform dimensions, it
is evidently that section which passes through the point of
greatest curvature of the neutral line, or the point in
respect to which the radius of curvature of the neutral line
is the least, or its reciprocal the greatest.
GENERAL CONDITIONS OF THE RUPTURE OF A BEAM.
408. Let PQ be the section of rupture in a beam sustaining any given pressures, whose;i  resultants are represented, if
they be more in number than
m  -.D  three, by the three pressures P,
~p2-  \  a /-   P2 Pa. Let the beam be upon,.........  the point of breaking by the
1'I ~ yielding of its material to extenB'~ \      31  sion at the point of greatest ex-':' S3 ]tension P; and let R represent,
in the state of the beam bordering upon rupture, the intersection of the neutral surface
with the section of rupture; which intersection being in
the case of rectangular beams a straight line, and being in
fact the neutral axis, in that particular position which is
assumed by it when the beam is brought into its state bordering upon rupture, may be called the axis of rupture;
AK the area in square inches of any element of the section
of rupture, whose perpendicular distance from the axis of
rupture R is represented by p; S the resistance in pounds




522           GENERAL CONDITIONS OF RUPTURE
opposed to the rupture of each square inch of the section at
P; c, and c, the distances PR and QR in inches.
The forces opposed per square inch to the extension and
compression of the material at different points of the section of rupture are to one another as their several perpendicular distances from the axis of rupture, if the elasticity
of the material be supposed to remain perfect throughout
the section of rupture, up to the period of rupture.
Now at the distance c, the force thus opposed to the
extension of the material is represented per square inch by
S; at the distance p the elastic force opposed to the extension or compression of the material (according as that
distance is measured on the extended or compressed side), is
S
therefore represented per square inch by -p, and the elastic
force thus developed upon the element AK of the section of
rupture by -pS K, so that the moment of this elastic force
Cl
about R is represented by -p2AK, and the sum     of the moments of all the elastic forces upon the section of rupture
about the axis of rupture by S p2aK;t or representing the
moment of inertia of the section of rupture about the axis
of rupture by I, the sum, of the moments of the elastic
forces upon the section of rupture about its axis of rupture
SI
is represented, at the instant of rupture, by  f.t  Now the
elastic forces developed upon PQ are in equilibrium with
the pressures applied to either of the portions APQDI or
BPQC, into which the beam is divided by that section; the
sum of their moments about the point P is therefore equal
to the moment of P, about that point. Representing,
therefore, by p, the perpendicular let fall from the point R
upon the direction of P,, we have
X It will be observed, as in Art. 358., that the elastic forces of extension
and those of compression tend to turn the surface of rupture in the same
direction about the axis of rupture.
f This expression is called by the French writers, the moment of rupture;
the beam is of greater or less strength under- given circumstances according
as it has a greater or less value.




BY TRANSVERSE STRAIN.             523
SI
Pp-...... (637).
409. If the deflexion be small in the state bordering upon
rupture, and the directions of all the deflecting pressures be
perpendicular to the surface of the beam, the axis of rupture
passes throulgh the centre of gravity of the section, and the
value of cl is known. Where these conditions do not obtain,
the value of c1 might be determined by the principles laid
down in Arts. 355. and 381. This determination would,
however, leave'   the theory of the-rupture of beams still incomplete in one important particular. The elasticity of the
matemrial has been supposed to remain perfect, at every point
of the section of rupture, up to the instant when rupture is
about to take place. Now it is to be observed, that by reason of its greater extension about the point P than at any
other point of the section of rupture, the elastic limits are
there passed before rupture takes places and before they are
attained at points nearer to the axis of rupture; the forces
opposed to the extension of the material cannot therefore be
assumed to vary, at all points of PR, accurately as their distances from the point R, in that state of the equilibrium of
the; beam. which immediately precedes its rupture; and the
sum of their mom ents eannot therefore be assumed to be acSI
curately represented by the expression -. This remark affects, moreover, the determination of the values of h and R
(Arts. 355. and 381.), and therefore the value of c.
To determine the influence upon the conditions of rupture
by transverse strain of that unknown direction of the insistent
pressures, and that variation from the law of perfect elasticity which belongs to the state bordering upon rupture, we
must fall back upon experiment. From this it has resulted,
izn  spect to rectangular beams,, that the error produced by
theses different causes in equation (637) will be corrected if
a value be assigned to cl bearing, for each given material, a
constant ratio to the distance of the point P from the centre
of gravity of the section of rupture; so that c representing
the depth of a rectangular beam, the error will be corrected;
in respect to a beam of any material, by assigning to c, the
value m-ic, where qm is a certain constant dependent upon
the nature of the material. It is evident that this correction is equivalent to assuming c-=ic, and assigning
to S the: value  S instead of that which it has hitherto




524            GENERAL CONDITIONS OF REUPTURF
been supposed to represent, viz. the tenacity per square inch
of the material of the beam.
It is customary to make this assumption. The values of S
corresponding to it have been determined, by experiment,
in respect to the materials chiefly used in construction, and
will be found in a table at the end of this work. It is to
these tables that the values represented by S in all subsequent formulae are to be referred.
410. From the remarks contained in the preceding article,
it is not difficult to conceive the existence of some direct relation between the conditions of rupture by transverse and by
longitudinal strain. Such a relation of the simplest kind appears recently to have been discovered by the experiments
of Mr. E. Hodgkinson*, extending to tie conditions of rupture by compression, and common to all the different varieties of material included under each of the following great
divisions-timber, cast iron, stone, glass.
The following tables contain the summary given by Mr.
Hodgkinson of his results:Mean Transverse
Assumed Crushing  Mean Tensiler
Description of Material.  Strength per Square Strength per Squaretregh of a Bar
Inch.   Inch.   ~1 Inch Square and
I1 Foot Long.
Timber -   -   -    -     1000          1900           851
Cast-iron  -   -    -     1000           158          19.8
Stone, including marble-  1000           100            9-8
Glass (plate and crown)-  1000           123          10'
The following table shows the uniformity of this ratio in
respect to different varieties of the same material:Mean Transverse
i                      Assumed Crushing  M   Tensile  Mean Transverse 
Description of Material.  Strength per Square Strength per Square S  Inch uof a Band
Description  Inch.  Inch.  1 Inch Square and
Inch.  Inch.  Foot Long.
Black marble -  -   -      1000          143           10'1
Italian marble  -   -     1000            84           10'6
Rochdale flagstone-  -    1000           104           9'9
High Moorstone  -   -      1000          100
Yorkshire flag  -   -      1000          -              95.Stone from Little Hulton,  1000           0            88
near Bolton            1000            (0           88
* This discovery was communicated to the British Association of Science at
their meeting in 1842; it opens to us a new field of theoretical research.




THE STRONGEST FORM OF SECTION.           525
411. THE STRONGEST FORM OF SECTION AT ANY GIVEN POINT
IN THE LENGTH OF THE BEAM.
Since the extension and the compression of the material
are the greatest at those points which are most distant from
the neutral axes of the section, it is evident that the material cannot be in the state bordering upon rupture at every
point of the section at the same instant (Art. 388.), unless all
the material of the compressed side be collected at the same
distance from the neutral axis, and likewise all the material
of the extended side, or unless the material of the extended
side and the material of the compressed side be respectively
collected into two geometrical lines parallel to the neutral
axis: a distribution manifestly impossible, since it would
produce an entire separation of the two sides of the beam.
The nearest practicable approach to this form of section is
that represented in the accompanying figure, where the
material is shown collected in two thin butwide flanges,
united by a narrow rib.
That which constitutes the strength of the
beam being the resistance of its material to compression on the one side of its neutral axis, and
its resistance to extension on the other side, it is
evidently (Art. 388.) a second condition of the
strongest form  of any given section that when
the beam is about to break across that section by
extension on the one side, it may be about to break by compression on the other. So long, therefore, as the distribution
of the material is not such as that the compressed and
extended sides would yield together, the strongest form of
section is not attained.  Hence it is apparent that the
strongest form of the section collects the greater quantity
of the material on the compressed or the extended side of
the beam, according as the resistance of the material to
compression or to extension is the less. Where the material
of the beam is cast iron*, whose resistance to extension is
greatly less than its resistance to compression, it is evident
that the greater portion of the material must be collected on
the extended side.
Thus, then, it follows, from the preceding condition and
* It is only in cast iron beams that it is customary to seek an economy of
the material in the strength of the section of the beam; the same principle of
economy is surely, however, applicable to beams of wood.




526          THE STRONGEST FORM OF SECTION.
this, that the strongest form of section in a cast iron beam is
that by which the material is collected into two unequal
flanges joined by a rib, the greater flange being on the
extended side; and the proportion of this inequality of the
flanges being just such as to make up for the inequality of
the resistances of the material to rupture by extension and
compression respectively.
Mr. Hodgkinson, to whom     this suggestion is due, has
directed a series of experiments to the determination of that
proportion of the flanges by which the strongest form of
section is obtained.2
The details of these experiments are found in the following
table:
Number of Ratio of the Sections  Area:of whole
Number of Ratio of the Slanges.  Section in Square Strength per Square
Experiment.  of the Flanges.    e   Inch of Section in Ibs.
Inches.  Inch of Section in lbs.
1        1 to 1         2'82         2368
2        1 to 2-        2-87         2567
3        1 to 4-       3802          27 7
4        1 to 4'5       33'7          3183
5        1 to 5-5       5'0           3346
6        1 to 6-1       6-4           4075
In the first five experiments each beam broke by the tearing asunder of the lower flange. The distribution by which
both were about to yield together-that is, the strongest
distribution-was not therefore up to that period reached.
At length, however, in the last experiment, the beam yielded
by the compression of the upper flange. In this experiment,
therefore, the upper flange was the weakest; in the one before it, the lower flange was the weakest. For a form
between the two, therefore, the flanges were of equal strength
to resist extension and compression respectively; and this
was the strongest form of section (Art. 388.).
In this strongest form the lower flange had six times the
material of the upper. It is represented in the accompanying figure.
In the best form of cast iron beam or
girder used    before these experiments,
there was never attained a strength of
-e    -— d    more than 2885 lbs. per square inch of
a.   6 &  section. There was, therefore, by this
form, a gain of 1190 lbs. per square inch
of the section, or of 2ths the strength of
the beam.
* Memoirs of Manchester Philosophical Society, vol. iv. p. 453. Illustrations of Mechanics, Art. 68.




THE BEAM2 OF G-RATJEST STRENGTH.      527
412. THE SECTION OF RUJPTURE.
The conditions of rupture being determined in respect to
any section of the beam by equation (637), it is evident that
the particular section across which rupture will actually take
place is that in respect to which equation (637) is first satisfied, as P, is continually increased; or that section in respect
to which the formula
-  e.... e(638)
P10'
is the least.
If the beam be loaded along its whole length, and x represent the distance of any section from the extremity at which
the load commences, and IJ the load on each foot of the
length, then (Art. 371.) Pp, is represented by }x2. The
section of rupture in this case is therefore that section in
respect to which tP is first made to satisfy the equation
fi-l= --; or in respect to which the formula.... a e(639)
is the least.
If the section of the beam be uniform, - is constant; the
section of rupture is therefore evidently that which is most
distant from the free extremity of the beam.
413. THE BEAM OF GREATEST STRENGTH.
The beam of greatest strength being that (Art. 388.) which
presents an equal liability to rupture across every section, or
in respect to which every section is brought into the state
bordering upon rupture by the same deflecting pressure, is
evidently that by which a given value of P is made to satisfy
equation (637) for all the possible values of I, p,, and c,, or
in respect to which the formula
PI..... e e  (640)
is constanto




04528         TTHE STRENGTH OF BEAMS.
If the beam be uniformly loaded throughout (Art. 371.),
this condition becomes.....(641),
or constant, for all points in the length of the beam.
414. ONE EXTREMITY OF A BEAM IS FIRMLY IMBEDDED IN
MASONRY, AND A PRESSURE IS APPLIED TO THE OTHER
EXTREMITY IN A DIRECTION PERPENDICULAR TO ITS LENGTH:
TO DETERMINE THE CONDITIONS OF THE RUPTURE.
If x represent the distance of any section of the beam
from the extremity A to which the load P
-— e- S-^-  is applied, and a its whole length, and if the
section of the beam  be everywhere the
same, then the formula (638) is greatest
at the point B, where x is greatest: at
this point, therefore, the rupture of the
beam  will take place.  Representing by
P the pressure necessary to break the
beam, and observing that in this case the
B  perpendicular upon the direction of P
from the section of rupture is represented
by a, we have (equation 637)
SI..... (642).
If the section of the beam be a rectangle, whose breadth is b and its depth G,
then I=-2bc3, co,=c.
bC2
P=-.IS.....(643).
If the section be a solid cylinder, whose radius is c, then
(Art. 364.) I=jc4, c-=c.. P=VS -.....(644).
If the section be a hollow cylinder, whose radii are rf and
r,, I=-r(r14-r24); which expression may be put under the
form  rcr(r2 + c) (see Art. 86.), r representing the mean




THE STRENGTH OF BEAMS.             529
radius of the hollow cylinder, and c its thickness. Also
(r + 1c)a: p=~S(r +i~c...(645).
415. The strongest form of beam under the conditions supposed in the last article.
1-st. Let the section of the beam be a
rectangle, and let y be the depth of
this rectangle at a point whose distance
from its extremity A is represented by
a, and let its breadth b be the same
throughout.  In   this  case  I= —i'byS
c, — y; therefore (equation 637) P=
SI        y2
-e-=Sb. If, therefore, P be taken
C x-    X
C ~    to represent the pressure which the
beam is destined just to support, then
the form of its section ABC is determined (Art. 413.) by the equation,6P
y  P   x....(646);
it is therefore a parabola, whose vertex
is at A.*
If the portion DO of the beam    do not rest against
eA    masonry at every point, but only at its
extremity D, its form should evidently be
the same with that of ABC.
2d. Let the section be a circle, and
let y represent its radius at distance x
from  its extremity  A, then I=-ry4,
lic,=y;  therefore P-=rS   so that the
geometrical  form  of its longitudinal
section  is determined  by  the  equation
D
* The portion of the beam imbedded in the masonry should have the form
described in Art. 417.
34




530             THE STRENGTH OF BEAMS.
4P
y     PS......(647),
P representing the greatest pressure. to which it is destined
to be subjected..
416. THE CONDITIONS OF THE RUPTURE OF A BEAM SUPPORTED
AT ONE EXTREMITY, AND LOADED THROUGHOUT ITS WHOLE
LENGTH.
Representing the weight resting upon each inch of its
_,___ _,_ -,, ^length a by a, and observI      r,  I I  i, I, II I. II  ing that the moment of the
~_ __'[_x~'^-r',_ —r-,.weight upon a length x of
_____     A  the beam from A, about the.  T - 1B  ~ corresponding neutral axis,
~ E1-  l {rg -is represented (Art. 371.)
I I,, I I I             by ix2', it is apparent (Art.
412.) that, if the beam  be.    __L~.r _ _,.of uniform dimensions, its."~-p:-L% l!'section of rupture is BD.
_  i' i  rL I* i n    Its strength is determined
by substituting jac2 for Pp,
in equation (637), and solving in respect to a; we thus obtain
2SI..*..c(648);
=......w;
"by which equation is determined the uniform load to which
fhe beam may be subjected, on each inch of its length.
For a rectangular beam, whose width is b and its depth'c, this expression becomes
Sbc'
= 3..... (649).
417. To determine the form of greatest strength (Art. 413.)
in the case of a beam having a rectangular section of uniform breadth, jx2 must be substituted for Pp2 in equation
(637), and -by' for I, and ~y for cl; whence we obtain by
reduction
Y=(..... (650.




THE STRENGTH OF BEAMS.             531' -~ l I  _I  I  i i  I  I ~ 
-  1  3  A._1  i  I  l.I
~  ~F    I  I_  I. _ _  I t 
I      I  I i
The form of greatest strength is therefore, in this case, the
straight line joining the points A and B; the distance DB
being determined by substituting the distance AD for x in
the above equation.
That portion BED of the beam which is embedded in the
masonry should evidently be of the same form with DBA.*
418. If, in addition to the uniform load upon the beam, a
given weight W be suspended from A, ip2+Wx must be
i  I  I - 1   I    [ I - I  IT-. L I
substituted for Pop, in equation (637); we shall thus obtain
for the equation to the form of greatest strength
I        I    I     I I 2W 
y-sb tM     +a? **-*v(651),
which is the equation tho an hyperbola having its vertex
at A.t
* It is obvious that in all cases the strength of a beam at each point of its
length is dependent upon the dimensions of its cross section at that point, and
stat its general form may in anyequation (63 weithout impairing its strength,
provided those dimensions of the section be everywhere preserved.
t Church's Anal. Geom. Art. 124.




532            THE STRENGTH OF BEAMS.
419. THE BEAM OF GREATEST STRENGTH IN REFERENCE TOTHE
FORM OF ITS SECTION AND TO THE VARIATION OF THE
DIMENSIONS OF ITS SECTION, WHEN SUPPORTED AT ONE
EXTREMITY IN A HORIZONTAL POSITION, AND LOADED UNIFORMLY THROUGHOUT ITS LENGTH.
The general form of the section must evidently be that
described in Art. 411. Let.I... -,         I:  the same notation be taken
as in Art. 365., except that
the depth MQ of the plate
iL-,,j-t r', I         or rib   joining the  two
flanges is to be represented
by y, and its thickness by c,
so that d,-y, and A-=cy;
therefore by equation (503),
T=1-Ad12. + A d2 +oy)+i {,4A1A2 + (A, + A2)cy,
1=:A- (A+a+A.+<d     +  A++A^              \yj 
Also representing by c, the distance of the centre of gravity
of the whole section from the upper surface of the beam,
we have c,(A,+A + y)= (jy+docy+(y+d2 +        i)A + 
A,. Substituting for I and cl in equation (637), and for P1p its
value i-x2, x being taken to represent the distance AM, and
P, the load on each inch of that length, we have (Art.
413.)
3p,
-a-Xa2 --
(Ald2+ A' +  A cy) (Al + ( A + cy) + {12AA2,+ 3(A + A,)cy} y'      +';  (y+ 2d;)c + 2(y + d + d,)A, + Ad,2.....(652).
Let the area cy of the section -of the rib now be neglected,
as exceedingly small when compared with the areas of the
sections of the -flanges, an hypothesis which assigns to the
beam somewhat less than its actual strength; let also the
area of the section of the upper flange be assumed equal to
n times that of the lower, or A.=nA,,
h 3e  =(7 + 1) (d2 +d22) + l2ny2  (
flanes be e   edingly tn,d       d ae 2)  excee
If the flanges be exceedingly thin, d an t   w aretexceedingly small and may be neglected.'The equation will then




THE STRENGTH OF BEAMS.            533
become that to a parabola whose vertex is at A and its axis
vertical. This may therefore be assumed as a near approximation to the true form of the curve AQO.
Where the material is cast iron, it appears by Mr. Hodgkinson's experiments (Art. 411.) that n is to be taken=-6.
420. A  BEAM OF UNIFORM SECTION IS SUPPORTED AT ITS
EXTREMITIES AND LOADED AT ANY POINT BETWEEN THEM:
IT IS REQUIRED TO DETERMINE THE CONDITIONS OF RUPTURE.
The point of rupture in the case of a uniform: section
is evidently (Art. 412.) the point
C, from which the load is suspended; representing AB, AC,
wge  s      BoC; by a, a,, and a; -'and observing that the - pressure P,
Wa
upon the point B of the beam -=, so that the moment
a
of P,, in respect to the section of rupture C =  a', we
a
Waiao   SI
have, by equation (637),- a=
SIa
=....... (654).
If the beam be rectangular I1= bc, c,=-c,
S bea.W=....(655);
6- a'as )
where W represents the breaking weight, S the modulus of
rupture, a the length, b the breadth, c the depth, and a,, a,
the distances of the point c from the two extremities, all
these dimensions being in inches.
If the load be suspended in the middle, a,=a,=ja,
2S be'
W=~. —..-a.    (656).
If the beam be a solid cylinder, whose radius =c, then I=
Frc, c,=c; therefore, equation (654),
WrS ac'
W=-    ~...... (657).
4   a  Ia.,




534            THE STRENGTH OF BEAMS.
If the beam be a hollow cylinder, whose mean radius is r,
and its thickness c, I='cr(r2+ic2), c,=r++c; therefore,
equation (654),
acr(r2 + ~6)
W=S     ( +..... (658).
If the section of the beam be that represented in Art. 411.,
being everywhere of the same dimensions, then, observing
that Ac=-IdA, + dA1, nearly, we have, (equations 503 and
654)
Sa A(Ad2+A2d22+A3d32)+3(4AA+AlA3+A2As)d832 
W   6             (2Al+As)ala2d3..   9)
where Al, A2 represent the areas of the sections of the upper
and lower flanges, and A, that of the connecting rib or plate,
and d,, d,, d their respective depths.
421. A BEAM IS SUPPORTED AT ITS EXTREMITIES, AND LOADED
AT ANY GIVEN POINT BETWEEN THEM; ITS SECTION IS OF A
GIVEN GEOMETRICAL FORM, BUT OF VARIABLE DIMENSIONS:
IT IS REQUIRED TO DETERMINE THE LAW OF THIS VARIATION, SO THAT THE STRENGTH OF THE BEAM MAY BE A
MAXIMUM.
W representing the breaking load upon the beam, and
a,, ab the distances of its point
B. —~~-       — ~A    ~of suspension C, from A and
_ —  X~^    B, the pressure P, upon A is
Wa
X w         ^^s=  represented by -2 If,therefore (Art. 388.), x represent
the horizontal distance of any section MQ from the point of
support A, and I its moment of inertia, and c, the distance
from its centre of gravity to the point where rupture is about
to take place (in this case its lowest point); then by equation (637)
Wa2=_..... (660).
a     01
1st. Let the section be rectangular; let its breadth b be
constant; and let its depth at the distance w from A be




THE STRENGTH OF BEAMS.            535
represented by y; therefore I= —-by3, cl=-y.  Substituting
in the above equation and reducing,
ySa...  (661).
The curve AC is therefore a parabola, whose vertex is at
A, and its axis horizontal. In like manner the curve BC is
a parabola, whose equation is identical with the above, except that a, is to be substituted in it for a2.
2d. Let the section of the beam be a circle. Representing the radius of a section at distance x from A by y, we
have I=-ry4, 1c=y; therefore by equation (660)
s.(4Wa".   (662).
3d. Let the section of the beam be circular; but let it be
hollow, the thickness of its material being every where the
same, and represented by c. If y= mean radius of cylinder
at distance x from A, then I-=oy(y2+ic2), c1=(y+jc);
Sracy 4y2+G\ +.    (6.
~ ~=o,~-Yy-     )... (663).
422. THE BEAM OF GREATEST ABSOLUTE STRENGTH WHEN
LOADED AT A GIVEN POINT AND SUPPORTED AT THE EXTREMITIES.
Let the section of the beam be that of greatest strength
(Art. 411.). Substituting in equation (660) the value of 
as before in equation (652), and reducing,
6Wa2x _(Aldi2+A2d22a+cy)(A1+.A2+cy)+12AA2y2+3(Al+A2)cy3
Sa -         (y2d2)cy+2(y+d2+di )Ai+A-2d2      (664).
If the section cy of the rib be every where exceedingly
small as compared with the sections of the flanges, and if
A, =nA,
12Wa2    ( + 1) (d2 + d2)+12ny2
SAa 2= 2y+d +(n+),             *'(665).
There is a value of x in this equation for which y becomes




536            THE STRENGTH OF BEAMS.
impossible. For values less than this, the condition of uniform strength cannot therefore obtain. It is only in respect
to those parts of the beam which lie between the values of
w (measured from the two points of support) for which y
thus becomes impossible, that the condition of greatest
strength (Art. 388.) is possible.  If its proper value be
assigned to n (Art. 411.), this may be assumed as an approximation to the true form of beam of THE GREATEST ABSOLUTE
STRENGTH. When the material is cast iron, it appears by the
experiments of Mr. Hodgkinson (Art. 411.) that n_=6. A2
represents in all the above cases the section of the extended
flange; in this case, therefore, it represents the section of
the lower flange.
The depth CD at the point of suspension may be determined by substituting a, for x in equation (665); its value is
thus found to be represented by the formula
Wa a,
CD=-... (666).
SAna....
423. If instead of the depth of the beam being made to
vary so as to adapt itself to the condition (Art. 388.) of uniform strength, its breadth b be made thus to vary, the depth
o remaining the same; then, assuming the breadth of the
upper flange at the distance x from the point of support A
to be represented by y, Ind the section of the lower flange
to be n times greater than that of the upper; observing,
moreover, that in equation (503) A =yd1, A-=nA,=nyd,;
neglecting also A, as exceedingly small when compared with
Al and AS, and writing c for d,, we have by reduction,
I= ly(d,+ nd,')d + y+Cd1
Also c, being the distance of the lower surface of the beam
from the common centre of gravity of the sections of the
two flanges, we have c,(n + 1)=_. Eliminating, therefore,
the values of I and c, from equation (660),
Xaw       -ln+ l) (d+ nd2)l +nd.....  (667),
the equation to a straight line. Each flange is therefore in
this case a quadrilateral figure, whose dimensions are determined from the greatest breadth; this last being known, for




THE STRENGTH OF BEAMS.           537
the upper flange, by substituting a, for x in the above equation, and solving in respect of y, and for the lower flange
from the equation nbd,-bdc, in which b,, b, represent the
greatest breadths of the two flanges, and d, d2 their depths.
424. A BEAM IS LOADED UNIFORMLY THROUGHOUT ITS WHOLE
LENGTH, AND SUPPORTED AT ITS EXTREMITIES: IT IS REQUIRED
TO DETERMINE, 1. THE CONDITIONS OF ITS RUPTURE WHEN ITS
CROSS SECTION IS UNIFORM THROUGHOUT; 2. THE STRONGEST
FORM OF BEAM HAVING EVERY WHERE A RECTANGULAR CROSS
SECTION; 3. THE BEAM OF GREATEST STRENGTH IN REFERENCE BOTH TO THE -FORM AND THE VARIATION OF ITS CROSS
SECTION.
1. If the section of the beam be uniform, its point of rup— r-,              t 1',,.-, t  ture is determined by formula (639)'r, — I   I I  to be its middle point. Representing,....           It therefore, in this case, the length of
the beam by 2a, the weight on each
inch of its length by p, and its breadth
_:J:~.       Tby b; and observing that in this case
PI,p-='- - a    -— jp- a, we have by
equation (637)
2SI
-i......(668),
where P; represents the load per inch of the length of the
beam necessary to produce rupture. In the case of a rectangular beam, this equation becomes
Sbc\
- 3a2....(669).
2. To determine the form of the beam of greatest strength
having a rectangular section of
I "-. l^,"l- given breadth b, let y be taken to
I~- I represent its depth PQ at a point P,
~T::~ ~and x its horizontal distance from
---.    = Ithe point A.   Then   I - by",
c,=-y; also P,pl (equation 637)
representing the moment of the resultant of the pressures
upon AP about the centre of gravity of PQ=-Pax — a2;
therefore by equation (637) pax-j-ix'=-Sby2;




538             THE STRENGTH OF BEAMS..y =-S (2ax —9),
the equation to an ellipse, whose vertex is in A, and its
centre at C.
3. To determine the beam of absolute maximum strength,
let it be assumed, as in Art. 422, that the area of the section
of the rib is exceedingly small as compared with the areas
of the sections of the flanges; and let the area of the section
of the lower or extended flange be n times that of the upper;
I       ( ~ A +  + nd22) +12 ny
then, as in Art. 422,  6-{ _+ 1 (,y      n+)d12 
also Plpl,=ax —* anx; whence, by equation (637),
SA1 (n + 1)(4d1 + ndo)+ 12ny2
- axA    =6        2y+dl+(n+2 )*             (70).
4. If it be proposed to make the rib or plate uniting the
two flanges everywhere of the same depth,* and so to vary
the breadths of the flanges as to give to the beam a uniform
strength at all points under these circumstances; representing by y the breadth of the upper flange at a horizontal
distance x from the point of support, we shall obtain, as in
Art. 423,
I=y(?c a, + nd2)dl + yncd,.
Moreover, Plp =pax-    x%-BJx2(2a-x); whence we obtain
by substitution in equation (637), and reduction,
x( a-x)= (6) (n++1) (dl+nd)+12n}y..... (673);
the equation to a parobola,t whose axis is in the horizontal
line bisecting the flange at right angles, its parameter repre* As in Mr. Hodgkinson's construction.
f Church's Anal. Geom. Art. 171.




THE STRENGTH OF BEAMS.           539
sented by the coefficient of y in the preceding equation, and
half the breadth of the flange in the middle determined by
the formula
~    _____ 6c a2
i)  s d, (6.4).
{(n+1) (dl2+nd,22)+ 12n2c Sd,  * * (64).
The equation to the lower flange is determined by substiyd,
tuting for y, in equation (673), -i; whence it follows that
the breadth of the lower flange in the middle is equal to
that of the upper multiplied by the fraction ~-.
425. A RECTANGULAR BEAM OF UNIFORM SECTION, AND UNIFORMLY LOADED THROUGHOUT ITS LENGTH, IS SUPPORTED BY
TWO PROPS PLACED AT EQUAL DISTANCES FROM ITS EXTREMITIES: TO DETERMINE THE CONDITIONS OF RUPTURE.
It is evident from formula (639) that the section of rupture of the portion CA of the
beam is at A, and therefore that
the conditions of its rupture are
determined (Art. 416.) by the
equation
Sbc2
l-3(1a,..... (65);
where K represents, as before, the
load upon each inch of the length of the beam, b its
breadth, c its depth, and a, the length of the portion AC.
Again, it is evident that the point of rupture of the portion AB of the beam is at E. Now the value of Pp,
(equation 637) is, in respect to the portion AE of the beam,
2a(a-a,) —P2a2; 2a representing the whole length of the
beam 2 the load upon each inch of the length of the beam
which would produce rupture at E, and therefore ha the
resistance of each prop in the state bordering upon rupture;
also - =-bc2.  Whence, by equation (637), pqa(a-a,)C1
ima2=2a(ia-a,)-=bc2S;
Sbc2
=3a(a —2a..... (676).




540             THE STRENGTH OF BEAMS,
426. THE BEST POSITIONS OF THE PROPS.
If the load P be imagined to be continually increased, it
is evident that rupture will eventually take place at A or at
E according as the limit represented by equation (675), or
that represented by equation (676), is first attained, or
according as,,l or I, is the less.
Let s, be conceived to be the less, and let the prop A be
moved nearer to the extremity C; a, being thus diminished,
p, will be increased, and p, diminished. Now if, after this
change in the position of the prop, 1, still remains less than
EP,2 it is evident that the beam will bear a greater load than
it would before, and that when by continually increasing
the load it is brought into the state bordering upon rupture
at A it will not be in the state bordering upon rupture at E.
The beam may therefore be strengthened yet further by
moving the prop A towards C; and thus continually, so
that the beam evidently becomes the strongest when the
prop is moved into such a position that A, may just equal
p,. This position is readily determined from equations (675)
and (676) to be that in which
a,=a( 4/2-1)=-44225a..... (677).
427. A RECTANGULAR BEAM OF UNIFORM SECTION AND UNIFORMLY LOADED IS SUPPORTED AT ITS EXTREMITIES, AND BY
TWO PROPS SITUATED AT EQUAL DISTANCES FROM THEM: TO
DETERMINE THE CONDITIONS OF RUPTURE.
Adopting the same notation as in Art. 374., it appears by
equation (543) that the dis-,     I' —','.. r \'L^ tance a, of the point of greatI-',','   il t   --   est curvature of the neutral
_d  io    ~   -  B    -line, and therefore of the sec-, I |  |    - r tion of rupture in AB from
F  A  (Art. 407.) being that
-_ —— E  ~  |,-     where     is the least, is determined by the equation
* The curvature of the neutral line being everywhere exceedingly small,
ds
may be assumed =1. The expression for the radius of curvature in terms
of the rectangular co-ordinates resolves itself therefore, in this case, into the
Second differential coefficient.




THE STRENGTH OF BEAMS..            541,xi=-P, it being observed that, at the section of rupture, the
neutral line is concwae to the axis of x, and therefore the
second differential coefficient (equation 543) negative. The
value of P is that determined by equation (551); so that
n3 + 12n2-24n +( 8
x1=a-.    - n(2,n-3     *** 
where a represents the distance AE, and na the distance AB.
Let P represent the intersection of the neutral line with
the plane of rupture, and PI the load per inch of the whole
length of the beam which would produce a rupture at P.
Now the sum of the moments of the forces impressed on
AP (other than the elastic forces on the section of rupture)
is represented in the state bordering upon rupture, by
Pi_-1xi,1x2; or, since P=',1x,, it is represented by  P2;
whence it follows by equation (637) that the conditions of
the rupture of the beam between A and B are determined
by the equation  p2. - -Sbc2, or,
P,2=,Sbc2.....   (679).
Eliminating the value of Pi between equations (551] and
(679), we obtain
-36~    1 4+12W2 -24n +8 f..  (680).
Substituting this value of, in equation (679), and
reducing
S    sona i3,(2,3)
p   Sa     n+ 1    -2n  +8.         (681).
-— 3a- Wq 21W     24A + 8,,, -,,',Ifh the points A and C
_5:~  "    A    I:~  coincide, or the beam be
supported by a single prop
-_ - _ _                  fore, by equations (680) and
S  (681),
64Sbc':i"     -- 27'...... (682);




542            THE STRENGTH OF BEAMS.
8Sbc2
P,= 9a... (683).
Similarly, it appears by equation (54:) that the point of
greatest curvature between B and C is E; if the rupture of
the beam take place first between these points, it will therefore take place in the middle. Let ~p, represent the load,
per inch of the length, which would produce a rupture at E.
Now, the sum of the moments about E of the forces impressed upon AE is P1a+P2(a-na)-J-i2a= (P1+P2)aP~na —i 2a2 =pta -(2 a- P,)na —i~2a2 (since P, + P2 =2a)=
\(1-2n)p2a2+Plna. Therefore by equation (637)
J(1-2n)pr,2a +P1na= 1Sbc2..... (684).
Substituting for P1 its value from equation (551), and
solving in respect to p2,,
Sbc, — n4(1-3    }(685)~
I23- a2   ns- 4(1-n)2.(685)
If the load be continually increased, the beam will break
between A and B, or between B and C, according as a,
(equation 680) or Ad2 (equation 685) is the less.
428. THE BEST POSITIONS OF THE PROPS.
It may be shown, as in Art. 426., that the positions in
which the props must be placed so as to cause the beam to
bear the greatest possible load distributed uniformly over its
whole length, are those by which the values of P1 (equation
680) and t2 (equation 685) are made equal; the former of
these quantities representing the load per inch of the length,
which being uniformly distributed over the whole beam
would just produce rupture between A and B, if it did not
before take place between B and C; and the latter that
which would, under the same circumstances, produce rupture between B and C if it had not before taken place
between A and B.
Let, then, qna represent the distance at which the prop B
must be placed from A to produce this equality; and let the
value of il, given by equation (679) be substituted for pa in
equation (684); we shall thus obtain by reduction
Sbcn2P   (SbcT)2
Pa2+2Pia. qu3(1a-2n)-2 9(1 -2n)
Solving this quadratic in respect to Pa,




THE STRENGTH OF BEAMS.             543
Pa=. Sbc2   ~(    )-n  }.
FI b 8S  1 —2n-'
The negative sign must be taken in this expression, since
the positive would give P=-apa by equation (679), and corresponds therefore to the case n=0. Assumingthe negative
sign, and reducing, we have 3(2n —1)Pla=Sb2.  Substituting in this expression for P. its value from equation (681),
and reducing,
8Sn(2n-1) (2n —3)1:n +12?n-_24 + 8
The three roots of this equation are 1-57087, *61078, and
*26994. The first and last are inadmissible; the one carrying the point B beyond E, and the other assigning to P, a
negative value.  The best position of the prop is therefore
that which is determined by the value
n=  61078..... (686).
429. THE CONDITIONS OF THE RUPTURE OF A RECTANGULAR
BEAM  LOADED UNIFORMLY THROUGHOUT ITS LENGTH, AND
HAVING ITS EXTREMITIES PROLONGED AND FIRMLY IMBEDDED
IN MASONRY.
It has been shown (Art. 376.) that the conditions of the
deflexion of the beam are, in this case, the same as though
its extremities, having been prolonged to a point A (see jig.
p. 540.), such that AB might equal'6202AE, had been supported by a prop at B, and by the resistance of any fixed
surface at A. The load which would produce the rupture
of the beam is therefore, in this case, the same as that which
would produce the rupture of a beam supported by props
(Art. 427.) between the props, and is determined by that
value of 2 (equation 685) which is given by the value -6202
of n. It is, however, to be observed that the symbol a
" We may, nevertheless, suppose the extremity A, instead of being sup
ported from beneath, to be pinned down by a resistance or a pressure acting
from above. This case may occur in practice, and the best position of the
props corresponding to it is that which is determined by the least root of the
equation, viz. *26994.




544            THE STRENGTH OF COLUMNS.
represents in that equation
I=,..'',__ - Ithe distance AE (fig. Art.
7L[   A..   -:      427.); and that if we take
i,- -: it to represent the distance
BE in that or the accompa-  T1:'  1 1  _.:,D nying figure, we must substitute 1-  for a in equa1-n
tion (685), since a=BE=AE-AB=(1-n)AE; so that
AE_== i. This substitution being made, equation (685),
becomes
Sbc2 (2n- 3)(1-gn)y.
t   4 3 C  3-4(-n)2 
and substituting the value -6202 for n, we obtain by reduction......             (687),
by which formula the load per inch of the length of the
beam necessary to produce rupture is determined.
If the beam had not been prolonged beyond the points of
support B and D and imbedded in the masonry, then the
load per inch of the length necessary to produce rupture
would have been represented by equation (669): eliminating between that equation and equation (687), we obtain
s,=3pt; so that the load per inch of the length necessary to
produce rupture is 3 times as great, when the extremities
of the beam are prolonged and firmly imbedded in the masonry, as when they are free; i. e. the strength of the beam
is 3 times as great en the one case as in the other.
430. THE STRENGH OF COLUMNS.
For all the knowledge of this subject on which any reliance can be placed by the engineer he is endebted to experiment.*
* The hypothesis upon which it has been customary to found the theoretical
discussion of it, is so obviously insufficient, and the results have been shown
by Mr. Hodgkinson to be so little in accordance with those of practice, that
the high sanction it has received from labours such as those of Euler, Lagrange,
Poisson, and Navier, can no longer establish for it a claim to be admitted
among the conclusions of science. (See Appendix G.)




THE STRENGTH OF COLUMNS.                 545
The following are the principal results obtained in the
valuable series of experimental inquiries recently instituted
by Mr. Eaton Hodgkinson.*
FORMULY REPRESENTING THE ABSOLUTE STRENGTH OF A CYLINDRICAL COLUMN TO SUSTAIN A PRESSURE IN THE DIRECTION
OF ITS LENGTH.
D=external diameter or side of the square of the column
in inches.
DI=internal diameter of hollow cylinder in inches.
L=length in feet.
W=breaking weight in tons.
Both Ends being
rounded, the Length of Both Ends being flat, the
Nature of                                th of the Column exceeding
fifteen times its  exceeding thirty times its
Diameter.        Diameter.
Solid cylindrical column of  -   LD-76    W446Ds
cast iron....    -               -     L'7
Hollow cylindrical column of   D" l3 -Ds"e        p    p'.57
cast iron  -... W=l-7            W=44-34- L-7
Solid cylindrical column of}     Ds'76             D3.55
wrought iron   - -       W=42'8         W=133-75 L'
Solid square pillar of Dantzic      _              4
oak (dry) -  -   -                          l- W O-9
Solid square pillar of red deal }                D 4
(dry)-..              W=781 
In all cases the strength of a column, one of whose ends
was rounded and the other flat, was found to be an arithmetic mean between the strengths of two other columns of
the same dimensions, one having both ends rounded and the
other having both ends flat.
The above results only apply to the case in which the
length of the column is so great that its fracture is produced
wholly by the bending of its material; this limit is fixed by
Mr. Hodgkinson in respect to columns of cast iron at about
fifteen times the diameter when the extremities are rounded,
* From a paper by Mr. Hodgkinson, published in the second part of the
Transactions of the Royal Society for 1840, to which the royal medal of the
Society was awarded. The experiments were made at the expense of
Mr. Fairbairn of Manchester, by whose liberal encouragement the researches
of practical science have been in other respects so greatly advanced.
35




546                    TORSION.
and thirty times the diameter when they are fiat.  In
shorter columns fracture takes place partly by the crushing
and partly by the bending of the material. To these shorter
columns the following rule was found to apply with sufficient accuracy:-" If W, represent the weight in tons
which would break the column by bending alone (or if it
did not crush) as given by the preceding formula, and W
the weight in tons which would break the column by crushing alone (or if it did not bend) as determined from the
preceding table, then the actual breaking weight W of the
column is represented in tons by the formula
WW
W   w+w,.. (688).
Columns enlarged in the middle. It was found that the
strengths of columns of cast iron, whose diameters were from
one and a half times to twice as great in the middle as at
the extremities, were stronger by one seventh than solid
columns, containing the same quantity of iron and of the
same length, when their extremities were rounded; and
stronger by one eighth or one ninth when their extremities
were flat and rendered immoveable by discs.
431. RELATIVE STRENGTH OF LONG COLUMNS OF CAST IRON,
WROUGHT IRON, STEEL, AND TIMBER OF THE SAME DIMENSIONS.
-Calling the strength of the cast iron column 1000, the
strength of the wrought iron column will, according to these
experiments, be 1745, that of the cast steel column 2518, of
the column of Dantzic oak 108'8, and of the column of red
deal 78-5.
Effect of drying on the strength of columns of timber.It results from these experiments, that the strength of short
columns of wet timber to resist crushing is not one half that,of columns of the same dimensions of dry timber.
TORSION.
432. The elasticity of torsion.
Let ABOD represent a solid cylinder, one of whose trans



TORSION,                   547
verse sections AEB is immoveably fixed,.T-     and every other displaced in its own plane,.-D     about its centre, by the action of a presr   sure P applied, at a given distance a from
the axis, to the section CD of the cylinder
in the plane of that section and round its
centre; the -cylinder is said, under these
circumstances, to be subjected to torsion,
and the forces opposed to the alteration of
its form, and to its rupture, constitute its
resistance to torsion.
Let arb/ be any section of the cylinder
whose distance fromr the section AEB is
/  represented by x, and let a/3 represent that
/  //    diameter of the section aabf3 which was
parallel to the diameter AB: before the torsion commenced;
let ab be the projection of the diameter AB upon the section aab/3, and let the angle aca be represented by 0.
Now the elastic forces called into action upon the section
cabP are in equilibrium with the pressure P. But these
elastic forces result fiom the displacement of the section
aab,8 upon its immediately subjacent section. Moreover,
the actual displacement of any small element AK of the
section aab3, upon the subjacent section, evidently depends
partly upon the angular displacement of the one section
upon the other, and partly upon the distance p of the
element in question from the axis of the cylinder. Now the
angle ac'a or 6 is evidently the sum of the angular displacements of all the sections between acbP and AEB upon their
subjacent sections; and the angular displacement of each
upon its subjacent section is the same, the circumstances
affecting the displacement of each being obviously the same:
also the number of these sections varies as a, and the sum
of their angular displacements is represented by 0; therefore the angular displacement of each section upon its sub0
jacent section varies as, and the actual displacement of
the small element AK of the section acb3 varieasa -p. Now
the material being elastic, the pressure which must be
applied to this element in order to keep it in this state of
displacement varies as the amount of the displacement
(Art. 345.), or as -p. Let its actual amount, when referred




548                    TORSION.
to a unit of surface, be represented by Gp, where G is a
certain constant dependant for its amount on the elastic
qualities of the material, and called the modulus of torsion;
then will the force of torsion required to keep the element
aK in its state of displacement be represented by G-pAK, and
its moment about the axis of the cylinder by G-p2'aK. So
that the sum of the moments of all such forces of torsion in
respect to the whole section aacb3 will be represented by
G - zp2K, or by G-I, if I represent the moment of inertia
of the section about the axis of the cylinder. Now these
forces are in equilibrium with P; therefore, by the principle
of the equality of moments,
Pa=GI-....(689).
X
If r represent the radius of the cylinder, I=r'4 (Art.
85.). Substituting this value, representing by L the whole
length of the cylinder, and by o the angle through which
its extreme section CD is displaced or through which OP is
made to revolve, called the angle of torsion, and solving in
respect to o,
l2a\   PL
— (-) -/'r.. 7. (690)o
Thus, then, it appears that when the dimensions of the
T t     cylinder are given, the angle of torsion @ varies
directly as the pressure P by which the torsion
is produced; whence, also, it follows (Art. 97.)
that if the cylinder, after having been deflected
through any distance, be set free, it will oscillate isochronously about is position of repose,
the time T of each oscillation being represented
in seconds (equation 76) by the formula
T  9       r......(69),
ince by equation (690) P= (2aL   );( in which expression
~24eL ()




TORSION.                    549
(oa) represents the length of the path described by the
point P from its position of repose, so that the moving force
upon the point P, when the pressure prducing torsion is
removed, varies as the path described by it from its position
of repose.
The above is manifestly the theory of Coulomb's Torsion
Balance.*  W represents in the formula the weight of the
mass supposed to be carried round by the point P, and the
inertia of the cylinder itself is neglected as exceedingly
small when compared with the inertia of this weight.
The torsion of rectangular prisms has been made the subject of the profound investigations of MM. Cauchyt, Lame,
et Clapeyron:, and Poisson.~  It results from these investigations I| that if b and c be taken to represent the sides of the
rectangular section of the prism, and the same notation be
adopted in other respects as before, then
3PLa(b'+c)            (69).
GbVQc-      * * * (692).
M. Cauchy has shown the values of the constant G 1o
be related to those of the modulus of elasticity E by the
formula
G= —E... (693).
In using the values of G deduced by this formula from
the table of moduli of elasticity, all the dimensions must be
taken in inches, and the weights in pounds.
433. ELASTICITY OF TORSION IN A SOLID HAVING A CIRCULAF
SECTION OF VARIABLE DIMENSIONS.
Let ab represent an element of the solid contained by
* Illustrations of Mechanics, Art. 37.
Exercices de Mathematiques, 4e annee.
Crelle's Journal.       ~ Memoires de rAcadmie, tome viii.
Navier, ResumE des Leqons, &c., Art. 159.




550                   TORSION.
planes, perpendicular to the axis, whose dis5..RP  tance from one another is represented by
~.I.   the exceedingly small increment Ax of the
distance x of the section ab from the fixed
section AB, and let its radius be represented by y; and suppose the whole of the
solid except this single element to become..s rigid, a supposition by which the conditions
of the equilibrium of this particular element
lM,~,    will remain, unchanged, the pressure P remaining the same, and being that which
produces the torsion of this single element.
Whence, representing by ad the angle of
/, /'   torsion of this element, and considering it
a cylinder whose length is Ax, we have by equation (689),
substituting for I its value fy'4,
A6
Pa=-Gt/4-yPassing to the limit, and integrating between the limits 0
and L, observing that at the former limit 8=0, and at the
latter 0=o.
L
o_2Pa f dx
e^/2PaJ f....(694.)
0
If the sides AC and BD of the solid be straight lines, its
form being that of a truncated cone, and if r, and r, represent its diameters AB and CD respectively; then
dx       L
dy     rl-r2
Also,
y-f4 dx   f               Ly-4dy=   y
0         9l1. r1
L,(  =_ r)_
- =~-r2 
* 1   r2
PLa ( r, _r\-)  (695).




TORSION.                     551
434. THE RUPTURE OF A CYLINDER BY TORSION.
It is evident that rupture will first take place in respect
to those elements of the cylinder which are nearest to its
surface, the displacement of each section upon its subjacent
section being greatest about those points which are nearest
to its circumference. If, therefore, we represent by T the
pressure per square inch which will cause rupture by the
sliding of any section of the mass upon its contiguous section,* then will T represent the resistance of torsion per
square inch of the section, at the distance r from the axis, at
the instant when rupture is upon the point of taking place,
the radius of the cylinder being represented by r. Whence
it follows that the displacement, and therefore the resistance
to torsion per square inch of the section, at any other distance p from the axis, will be represented at that distance by
TP the resistance upon any element AK, by - pAK, and the
r                                         r
sum of the moments about the axis, of the resistances of all
such elements, by T- p'aK, or by   I, or substituting for I
r              r
its value (equation 64) by Tnr.  But these resistances are
in equilibrium with the pressure P, which produces torsion,
acting at the distance a from the axis;.Pa=P  Tr'... (696).
It results from the researches of M. Cauchy, before referred
to, that in the case of a rectangular section whose sides are
represented by b and c, the conditions of rupture are determined by the equation
Pat-=T (2 + b... (697).
The length of a prism subjected to torsion does not -affect
the actual amount of the pressure required to produce rupture, but only the angle of torsion (equation 690) which
precedes rupture, and therefore the space through which
* Or the pressure per square inch necessary to shear it across (Art. 406.).
t Navier, Resume d'un Cours, &c. Art. 167.




552                  TORSION.
the pressure must be made to act, and the amount of worx
which must be done to produce rupture.
According to M. Cauchy, the modulus of rupture by torsion T is connected with that S of rupture by transverse
strain by the equation
T=4S.... (698).




PART VI.
IMPACT.*
435. THE IMPACT OF TWO BODIES WHOSE CENTRES OF GRAVITY
MOVE IN THE SAME RIGHT LINE, AND WHOSE POINT OF CONTACT IS IN THAT LINE.
From the period when a body first receives the impact of
another, until that period of the impact when both move for
an instant with the same velocity, it is evident that the su'faces must have been in a state of continually increasing
compression: the instant when they acquire a common velocity is, therefore, that of their greatest compression. When
this common velocity is attained, their mutual pressures will
have ceased if they be inelastic bodies, and they will move
with a common motion; if they be elastic, their surfaces
will, in the act of recovering their forms, be mutually
repelled, and the velocities will, after the impact, be different from one another.
436. A BODY WHOSE WEIGHT IS W1, AND WHICH IS MOVING
IN A HORIZONTAL DIRECTION WITH A UNIFORM VELOCITY
REPRESENTED BY VY, IS IMPINGED UPON BY A SECOND BODY
WHOSE WEIGHT IS Ws, AND WHICH IS MOVING IN THE SAME
STRAIGHT LINE WITH THE VELOCITY V2: IT IS REQUIRED TO
DETERMINE THEIR COMMON VELOCITY V AT THE INSTANT OF
GREATEST COMPRESSION.
Letf, represent the decrement per second of the velocity
of W, at any instant of the impact (Art. 94.), or rather the
decrement per second which its velocity would experience
if the retarding pressure were to remain constant; then will
* Note (u), Ed. App..s.




554                    IMPACT.
-ijl represent (Art. 95.) the efective force upon W1; and if
f2 be taken to represent, under the same circumstances, the
increment of velocity received by W2, then will -2/' represent the effective force upon W,. Whence it follows, by the
principle of D'Alembert (Art. 103.), that if these effective
forces be conceived to be applied to the bodies in directions
opposite to those in which the corresponding retardation
and acceleration take place, they will be in equilibrium with
the other forces applied to the bodies. But, by supposition,
no other forces than these are applied to the bodies: these
are therefore in equilibrium with one another,
W, W
-=-^a ~..... (699).
Let now an exceedingly small increment of the time from
the commencement of the impact be represented by At, and
let av, and AV2 represent the decrement and increment of
the velocities of the bodies respectively during that time,:. (Art. 95.)fAt=Av1,f^tt=AV2;. (equation 699) WI. AV1z=Wg. AV2;
and this equality obtaining throughout that period of the
impact which precedes the period of greatest compression, it
follows that when the bodies are moving in the same direction
W,(V,-Y)=W2(VY-2).....       (700);
since VY —   represents the whole velocity lost by W, during
that period, and V-V, the whole velocity gained by W,.
If the bodies be moving in opposite directions, and their
common motion at the instant of greatest compression be in
the direction of the motion of W,, then is the velocity lost
by W, represented as before by (V,-V); but the sum of
the decrements and increments of velocity communicated to
W,, in order that its velocity V2 may in the first place be
destroyed, and then the velocity V communicated to it in an
opposite direction, is represented by (V +V)...W(V —V)=W2,(V +    ).
Solving these equations in respect to V, we obtain




IMPACT.                   555
VW1 +      2...   (701);
W- W+W,...u1
the sign ~ being taken according as the motions of the
bodies before impact are both in the same direction or in
opposite directions.
If the second body was at rest before impact, V,=0, and
WYV
V=w +,..... 702)If the bodies be equal in weight,
V=(YVI~YV  )
The demonstration of this proposition is wholly independent of any hypothesis as to the nature of the impinging
bodies or their elastic properties; the proposition is therefore true of all bodies, whatever may be their degrees of
hardness or their elasticity, provided only that at the
instant of greatest compression every part of each body
partakes in the common velocities of the bodies, there being
no relative or vibratory motion of the parts of either body
among themselves.
437. To DETERMINE THE WORK EXPENDED UPON PRODUCING
THE STATE OF THE GREATEST COMPRESSION OF THE SURFACES OF THE BODIES.
The same notation being taken as before, the whole work
accumulated in the bodies, before impact, is represented
W       W
by V-l3+ 4 -2YV22; and the work accumulated in them
at the period of greatest compression, when they move with
the common velocity V, is represented by  — 1V.
Now the difference between the amounts of work accumulated in the bodies in these two states of their motion has
been expended in producing their compression; if, therefore, the amount of work thus expended be represented by
u, we have
=W,+W          V    3-9 W v
g V I      9  2            V




556                    IMPACT.
or substituting for V its value from equation (701), and
reducing,
=- 2          (V1TV)2.... (0-3).
This expression represents the amount of work permanently
lost in the impact of two inelastic bodies, their common
velocity after impact being represented by equation (701).
If W, be exceedingly great as compared with W.,
W,
1=- (V1,Vy... (704).
438. Two ELASTIC BODIES IMPINGE UPON ONE ANOTHER: IT IS
REQUIRED TO DETERMINE THE VELOCITY AFTER IMPACT.
If the impinging bodies be perfectly elastic, it is evident
that after the period of their greatest compression is passed,
they will, in the act of expanding their surfaces, exert
mutual pressures upon one another, which are, in corresponding positions of the surfaces, precisely the same with
those which they sustained whilst in the act of compression;
whence it follows that the decrements of velocity experienced by that body whose motion is retarded by this
expansion of the surfaces, and the increments acquired by
that whose velocity is accelerated, will be equal to those
before received in passing through corresponding positions,
and therefore the whole decrements and increments thus
received during the whole expansion equal to those received
during the whole compression.
Now the velocity lost by W, during the compression is
represented by (V1-V); that lost by it during the expansion, or from the period of greatest compression to that
when the bodies separate from one another, is therefore
represented by the same quantity. But at the instant of
greatest compression both bodies had the velocity V; the
velocity %I of W1 at the instant of separation is therefore
V —(V — ~), or 2V-V,. In like manner, the velocity
gained by W2 during compression, and therefore during
expansion, being represented by (V:FV,), and its velocity
at the instant of greatest compression by V, its velocity v,
at the instant of separation is represented by V +(V:T V),
or by 2V:FV2, the sign F being taken according as the




IMPACT.                    557
motion of the bodies before impact was in the same or
opposite directions.
Substituting for V its value in these expressions (equation
701), and reducing, we obtain
e  (Wx-w   2)V a 2W 2Vw,.  (705)
v2= T (W,- W)V, + 2WxV... (706).
W +W,''
If the bodies be perfectly elastic and equal in weight,, =V2,, v2,=V; they therefore, in this case, interchange their
velocities by impact; and if either was at rest before-impact,
the other will be at rest after impact.
If W2 be exceedingly great as compared with W1, v-=
-V1~2V2, 2-= IV2. In this case %v is negative, or the
motion of the lesser body alters its direction after impact,
when their motions before impact were in opposite directions; or when they were in the same direction, provided
that 2V: be not greater than V1.
439. If the elasticities of the balls be imperfect, the force
with which they tend to separate at any given point of the
expansion is different from that at the corresponding point
of the compression; the decrements and increments of the
velocities, produced during given corresponding periods of
the compression and expansion, are therefore different;
whence it follows that the whole amounts of velocity, lost
by the one and gained by the other during the two periods,
are different: let them bear to one another the ratio of 1 to e.
Now the velocity lost during compression by W1 is under
all circumstances represented by (VI-V); that lost during
expansion is therefore represented, in this case, by e (V, -V);
therefore, v -=V-e(V —V)-(1 +e)V -eV.   In like manner, the velocity gained by W2 during compression is in all
cases represented by (V:F V,); that gained during expansion
is therefore represented by e(VYV2); therefore, v2-=V+
e(V T  ) — (1 + e)V FeV2. Substituting for V, and reducing,
v     (W -eW,)V, ~ (1+ e)W2V,2... (707);
--      W,(+W,08
v=_ (w, - eW,)V, + (1 + e)WV,      (708).
W. +- Wz




558                   IMPACT.
440. IN THE IMPACT OF TWO ELASTIC BODIES, TO DETERMINE
THE ACCUMULATED WORK, OR ONE HALF THE VIS VIVA, LOST
BY THE ONE AND GAINED BY THE OTHER.
The vis viva lost by W, during the impact is evidently
W       WW     W W
represented by  V- -V,'vI2   (V1'-2) -       V2{(1 + e)YV-eV l2  =' {(1 - e2)VYs + 2e(l + e)VV,- (1 + e)'
V2=      (1 +e) (V,-V) {,(l-e) +V(1 + e).
Substituting in this expression its value for V (equation
701) reducing and representing by u, one half the vis viva
lost by W1 in its impact, or the amount by which its accumulated work is diminished by the impact (Art. 67.),
(1 + e)WW. (V, ) V2)'2WVI + (1 e)WV i
2g(W,+- W)2
(1+ e)W,V,}... (709).
Similarly, if u2 be taken to represent one half the vis viva
gained by W,, or the amount by which its accum ulated work
is increased by the impact, then' =~ (1 + e)WW(ViF2) 2W2, + (1 -e)WV
2g(W, +W,)2
(1 + e)W,V.....(710).
441. Let u be taken to represent the whole amount of the
work accumulated in the two bodies before their impact,
which is lost during their impact. This amount of work is
evidently equal to the difference between that gained by the
one body and lost by the other; so that u=u, —z,. Substituting the values of u, and u, from the preceding equations,
and reducing, we obtain
O(1-e')W,W,(V,V,)2
U   --  2g(W,).... (711).
This expression is equal to one half the vis viva lost during
the impact of the bodies. If the bodies be perfectly elastic,
e=1, and u=0. In this case there is no real loss of vis viva




IMPACT.                          559
in the impact, all that which the one body yields, during the
impact, being taken up by the other.*
442. In the preceding propositions it has been supposed
that the motions of the impinging body, and the body impinged upon, are opposed by no resistance whatever during
the period of the impact. There is no practical case in
which this condition obtains accurately.        If, nevertheless,
the resistance opposed to the motion of each body be small,
as compared with the pressure exerted by each upon the
other, at any period of the impact, then it is evident that all
the circumstances of the impact as it proceeds, and the motion of each body at the instant when it ceases, will be very
nearly the same as though no resistance were opposed to the
motion of either.t
443. As an illustration of the principle established in the
last article, let it be required to determine the space through
* It has been customary, nevertheless, to speak of a loss of vis viva in the
impact of perfectly elastic bodies. This loss is in all such cases to be understood only as a loss experienced by one of the bodies, and not as an absolute
loss. When the impinging bodies are perfectly elastic, it is evident that the
one flies away with all the vis viva which is lost in the impact by the other.
f Let Pi and P2 represent resistances opposed to the motions of two imWI        W
pinging bodies whose weights are W1 and W2; also let -fi, and -if2 represent the effective forces upon. the two bodies at any period of the impact,;
then, by D'Alembert's principle,
Wi,:     W2-W f,-i —Pi-f2-2=o;
1          g
or representing by t the time occupied in the impact, up to the period of
greatest compression, by V their common velocity at that period, and by v,
and v2 their velocities at any period of the impact, and substituting for fi and
f2 their values (equation 72),
W1 dvl       W2 dv2
_ —     -P            P2 — O.
g  dt        g  dt
Transposing and integrating between the limits 0 and t,
t
WI (VY-V)=    2 (V-V,)+    Pl+P2)at.
0
Now if Pi and P2 be not exceedingly great, the integral in the second member
of the equation is exceedingly small as compared with its other terms, and may
be neglected; the above equation will then become identical with equation
(0oo).




560                    IMPACT.
which a nail will be driven by the blow of a hammer; and
let it be supposed that the resistance opposed to the driving
of the nail is partly a constant resistance overcome at its
point, and partly a resistance opposed by the friction of the
mass into which it is driven upon its sides, varying in amount
directly with the length of it x, at any time imbedded in the
wood. Let this resistance be represented by a+3x; then
will the work which must be expended in driving it to a
depth D be represented (Art. 51.) by
D
+ Z)d x, or by (aD+ i3D).
0
Let W2 represent the weight of the nail, and V the velocity
with which a hammer whose weight is W, must impinge
upon it to drive it to this depth, and let the surfaces of the
nail and hammer both be supposed inelastic; then will the
work accumulated in the hammer before impact be repreW
sented by 1'V2, and the amount of this work lost during
the impact by the compression of the surfaces of contact will
be represented (equation 711) by  ( WW   V2 The work
remaining, and effective to drive the nail, is therefore represented by the difference of these quantities; and this work
being that actually expended in driving the nail, we have
2-g  2g W, + W,       =
1      12W a__
w   w-2D + D'.... (712);
by the solution of which quadratic equation, D may be determined.
444. Two SOLID PRISMS HAVE A COMMON AXIS; THE EXTREMITY OF ONE OF THEM RESTS AGAINST A FIXED SURFACE, AND
ITS OPPOSITE EXTREMITY RECEIVES THE IMPACT, IN A HORIZONTAL DIRECTION, OF THE OTHER PRISM: IT IS REQUIRED
TO DETERMINE THE COMPRESSION OF EACH PRISM,. THE LIMITS
OF PERFECT ELASTICITY NOT BEING PASSED IN THE IMPACT.
Let W represent the weight of the impinging prism, and




IMPACT.                    561
V its velocity before impact; L, and L, the lengths of the
prisms before compression; E, and E2 their moduli of elasticity; K1 and K, their sections; 1, and,1 the greatest compressions produced in them respectively by the impact;
then will the amounts of work which must have been done
upon the prisms to produce these compressions be represented (equation (486) by the formulae
2K L    and 1K2H
and the whole work thus expended by
/K E    KEAI
i(KL1   +  LK2E)
W
But this work has been done by the work  - V2, accumulated (Art. 66) before impact in the impinging body, and
that work has been exhausted in doing it;
/K1,E2112KE 122 ^w
i (      L, +..
Moreover, the mutual pressures upon the surfaces of contact are at every period of the impact equal, and at the
instant of greatest compression they are represented respectively (equation 485) by  L  and --- 
1        2
- L'L2           (713).
Li -   L,
Eliminating 12 between this equation and the preceding, and
reducing,
I       (K1,    L2 )i 4vW ^.       (714);
P=V { (    1+      ) } 4/W.*      (5);
in which expressions l represents the greatest compression
36




562                    IMPACT.
of the prism whose section is K1, and P the driving pressure
at the instant of greatest compression.
445. The mutual pressures P of the surfaces of contact at
any period of the impact.
Let I represent the space described by that extremity of
the impinging prism, by which it does not impinge: it is
evident that this space is made up of the two corresponding
compressions of the surfaces of impact of the prisms; so that
if these be represented by 1, and 2,, then 1=1, +l. But
(equation 713) 1    E, I =     =E; therefore I=P (KE +'I~aE4La )E).**P=^(716).
AP=l E    + KE (.....
446. A measure of the compressibility of the prisms.
If X be taken to represent the space through which that
extremity of the impinging prism by which it does not
impinge will have moved when the mutual pressure of the
surfaces of contact is 1 lb.; or, in other words, if X represent the aggregate space through which the prisms would
be compressed by a pressure of 1 lb.; then, by the preceding equation,
L,   L2
KE,      KE,.
X may be taken as a measure of the aggregate compressibility of the prisms, being the space through which their
opposite extremities would be made to approach one another
by a pressure of 1 lb. applied in the direction of their
length.
If \X and 2 represent the spaces through which the
prisms would severally be compressed by pressures of 1 lb.
L,      L,
applied to each, then X =K-' x2=; therefore X=x +
\, or the aggregate compressibility of the two prisms is
*equal to the sum of their separate compressibilities.




IMPACT.                    563
447. The,work u expended upon the compression of the
prisms at any period of the impact.
The work expended upon the compression 11 is repreXET
sented by 2   l112; or substituting its value for 11 (equation
1 LL
713), it is represented by i2K  P. And, similarly, the work
expended on the compression 12 is represented by K 2P;
J2E2
therefore u=    -- KIE   j- P; or substituting for P its
value from equation (716),
2u (iE +E)-= i * ***.8).
448. The velocity of the impinging body at any period of the
impact, the impact being supposed to take place vertically.
It is evident that at any period of the impact, when the
velocity of the impinging body is represented by v, there
will have been expended, upon the compression of the two
bodies, an amount of work which is represented by the
work accumulated in the impinging body before impact,
increased by the work done upon it by gravity during the
impact, and diminished by that which still remains accumulated in it, or by i- V + Wl — g v.
g           g
Representing, therefore, by u the work expended upon
W
the compression of the bodies, we have   - -V+WWISubstituting, therefore, for u its value from  equation
(718),
i-Y  +Wl-j' — v 2        Ej12
9          9         1 1   2 g




5(64                   IMPACT.
V2 ~v~2+ 2l  12 (KLi E  )   *   (719).
K-];YE2I + ^   l....'
Or substituting for I its value in terms of P (equation 716),
v2=Y     (jE,+KI   ) j( 2P -w)... (720).
THE PILE DRIVER.
449. It is evident that the pile will not begin to be
driven until a period of the impact is at-.j..   tained, when the pressure of the ram upon
its head, together with the weight of the
pile, exceeds the resistance opposed to its
motion by the coherence and the friction of
the mass into which it is driven. Let this
resistance be represented by P; let V represent the velocity of the ram at the instant of
impact, and v its velocity at the instant when
the pile begins to move, and W, W, the,l 1~ A    weights of the ram and pile; then, since the
pile will have been at rest during the whole
of the intervening period of the impact, since
/ W^   moreover, the mutual pressures Q of the sur_i]i ifaces of contact are at the instant of motion'ilI    represented by P-W,, we have by equation
(720)
2=Y2     (I4          ((P-W
ff (@,Ez -+- 2)      X (v — 2(P-We) }... (721).
If the value of v determined by this equation be not a
possible quantity, no motion can be communicated to the
pile by the impact of the ram; the following inequality is
therefore a condition necessary to the driving of the pile,
> gV2, +       { i-w  2   2(P-W )... (722).
After the pile has moved through any given distance, one
portion of the work accumulated in the ram before its
impact will have been expended in overcoming, through
that distance, the resistance opposed to the motion of the




IMPACT.                    565
pile; another portion will have been expended upon the
compression of the surfaces of the ram and pile; and the
remainder will be accumulated in the moving masses of the
ram and pile. The motion of the pile canhot cease until
after the period of the greatest compression of the ram and
pile is attained; since the reaction of the surface of the pile
upon the ram, and therefore the driving pressure upon the
pile, increases continually with the compression. If the
surfaces be inelastic, having no tendency to recover the
forms they may have received at the instant of greatest
compression, they will move on afterwards with a common
velocity, and come to rest together; so that the whole work
expended prejudicially during the impact will be that
expended upon the compression of the inelastic surfaces of
the ram and pile. If, however, both surfaces be elastic,
that of the ram will return from its position of greatest
compression, and the ram will thus acquire a velocity relatively to the pile, in a direction opposite to the motion of
the pile. Until it has thus reached the position in respect to
the pile in which it first began to drive it, their mutual
reaction Q will exceed the resistance P, and the pile will
continue to be driven. After the ram has, in its return,
passed this point, the pile will still continue to be driven
through a certain space, by the work which has been accumulating in it during the period in which Q has been in
excess of P. When the motion of the pile ceases, the rain
on its return will thus have passed the point at which it
first began to drive the pile: if it has not also then passed
the point at which its weight is just balanced by the elas
ticity of the surfaces, it will have been continually acquiring
velocity relatively to the pile from the period of greatest
compression; it will thus have a certain velocity, and a
certain amount of work will be accumulated in it when the
motion of the pile ceases: this amount of work, together
with that which must have been done to produce that compression which the surfaces of contact retain at that instant,
will in no respect have contributed to the driving of the
pile, and will have been expended uselessly. If the rain in
its return has, at the instant when the motion of the pile
ceases, passed the point at which its weight would just be
balanced by the elasticity of the surfaces of contact, its
velocity relatively to the pile will be in the act of diminishing; or it may, for an instant, cease at the instant when the
pile ceases to move. In this last case, the pile and ram, for
an instant, coming to rest together, the whole work accumu



566                    IMPACT.
lated in the impinging ram will have been usefully expended
in driving the pile, excepting only that by which the remaining compression of the surfaces has been produced; which
compression is less than that due to the weight of the rain.
This, therefore, may be considered the case in which a maximum useful effect is produced by the ram. The following
article contains an analytical discussion of these conditions
under their most general form.
450. A prism rimpinged upon by another is moveable in the
direction of its axis, and its motion is opposed by a constant pressure P: it is required to determine the conditions of the motion during the period of impact, the
circumstances of the impact being in other respects the
same as in Article 448.
Let f, and f, represent the additional velocities which
would be lost and acquired per second (see Art.
95) by the impinging prism     and the prism
impinged upon, if the pressures, at any instant
Q    operating upon them, were to remain from that
W    W,
instant constant; then will -f, -f  represent
the effective forces upon the two bodies (Art. 103)
or the pressures which would, by the principle of
D'Alembert, be in equilibrium with the unbalaI inced pressures upon them, if applied in opposite
1  directions..I  I  aNow the unbalanced pressure upon the system
BP composed of the two prisms is represented by
(W-W2- P),
f      -   =W,+ W, —   P..... (723);
also the unbalanced pressure upon the prism PQ=W2,+
Q-P, where Q represents the mutual pressure of the prisms
atQ;. -f=W2+ Q-P..... (24).
Let A have been the position of the extremity B of the
impinging prism at the instant of impact; and let x, represent the space through which the aggregate length BP of
the two prisms has been diminished since that period of the




IMPACT.                    567
impact, and w, the space through which the point P has
moved; then (equation 716)
Q=X( L       L   )             75).
Also AB=xa1+x2; therefore velocity of point B — d( +x
(Art. 96); therefore f*-    +- 2  dFt+f2
dt2       +     A -d=   f,.
Substituting these values of f and Q in equations (723) and
(724), and eliminatingf, between the resulting equations,
d2x,    g1     1      Pg
t2     xW    W+, +W.  * * * *.(726).
Integrating this equation by the known rules,t we obtain
Pg
x,=A sin. 7t+B cos. zt+ 2..... (727);
in which expression the value of y is determined by the
equation
g (11 ~i \)           W-+W-1.
x   +LW      l L,(KLE,)-'+L2(K2E2)-I... (728);
and A and B are certain constants to be determined by the
conditions of the question. Substituting in equation (724)
the value of Q from equation (725), and solving in respect
tof2,
-f2=Wx     (      y.....  (729).
Substituting for xl its value from equation (727), and for fs
its value -d, and reducing,
d2cx  Ag          Bg                P (
dit  Wx sin. yWt+w   cos. yt+  1W      _
Integrating between the limits 0 and t, and observing that
when t=0, d-=0; the time being supposed to commence
with the motion of the prism PQ,
* Art. 96. Equations (72) and (74).
f Church's Int. Cal. Art. 183.




568                     IMPACT.
d       (I — os. Yt) +   sin. yt +  1 +       gt.
dt -W.2Y              W                W —     /Integrating a second time between the same limits,
= w   y2 (yt  sin. t) + W y (1- cos.yt) +
Now when the motion of the second prism ceases - O 20;
dt
whence, if the corresponding value of t be represented by T,
A(1-cos. yT)+B sin. 7 T +   ( 1  -  p  W,2X7T=0.(731).
To determine the constants A and B, let it be observed
that the motion of the prism QP cannot commence until the
pressure Q of the impinging prism upon it, added to its own
weight W,, is equal to the resistance P opposed to its motion.
So that if c be taken to represent the value of x, (i. e. the
aggregate compression of the two prisms) at that instant,
then, substituting for Q its value from equation (725), - +
W — P;.c=(P-W2)X=-(P-W2) (2E +KE)... (732).
Now since the time t is supposed to commence at the
instant when this compression is attained, and the prism PQ
is upon the point of moving, substituting the above value of
c for x, in equation (727), and observing that when x=c,
Pg s
t=0, we obtain (P-W)X=B +     -; whence by substitution from equation (728); and reduction,
~ (P-W — +        W, —W)g  -1.... (733).
B-(        W   2)  x (w-W-1]^' * * * 
So long as the extremity P, of the prism impinged upon,
is at rest, the whole motion of the point B arises from the
compression of the two prisms, and is represented by d-t
dt,




IMPACT.                        569
The value of   b5 when t=0, is represented therefore by v
dt
(equation 721). Differentiating, therefore, equation (727),
assuming t=O, and substituting v for dt7 we obtain v=7A;
whence it appears that the value of A is determined by
dividing the square root of the second number of equation
(721) by y
Substituting for A and B their values in equation (734),
( )
Y(1-cos. 7 T) +xW,(       +W   -   )1 sin. T+
(1    W, iw)W2X7T=O.
Reducing, and dividing by the common factor of the two
last terms,
x vP(-cos. yT)      +sin. zT-7yT=O.... (734.)
iX —W21P(W1+W)-1-4^             7    r      *   "(
Substituting for A and B their values in equation (730), and
representing by D the value of x2, when t=T,
D=        - y(7T —sin.7T)+g(    +     — )(vers     - T iT2)
w2^W                                             2J.... (735).
The value of T determined by equation (734) being substituted in equation (735), an expression is obtained for the
whole space through which the second prism is driven by
the impact of the first.*
* The method of the above investigation is, from equation (726), nearly the
same with that given by Dr. Whewell, in the last edition of his Mechanics; the
principle of the investigation appears to be due to Mr. Airey. If the value
of y, as determined by equatin1 (734), were not exceedingly great, then, since
the value of T is in all practical cases exceedingly small, the value of yT would
in all cases be exceedingly small, and we might approximate to the value of
T in equation (735), by substituting for cos. yT and sin. yT, the two first terms
of the expansions of those functions, in terms of yT.








EDITORIAL APPENDIX.
NOTE (a).
BESIDES its direction defined (Art. 1), we have also to take
into consideration, in estimating the effects of a force, its
point of application, or the point of the body where it acts,
either directly or through the medium of some other body,
as a rigid bar, or an inextensible cord in its line of direction;
the point on its line of direction towards which the point of
application has a tendency to move; and finally the intensity, or magnitude of the force as expressed in terms of some
settled unit of measure.
NOTE (b).
This result of experiment also admits of the following
_  a'proof: Let A be the point of appli<2 -< —s i-cation of a force P, and let this point
22 APdf       be invariably connected with another
point B, in the line of direction towards which A tends to
move from the action of P; suppose now two other forces
P, and P,, each equal to P, to be applied; the one at A, in
a direction opposite to P, and the other at B, in the same
direction as P; the introduction of these two equal forces,
acting in opposite directions,will evidently in no wise change
the direction or intensity of P; but as P1 is equal and opposite to P its effect will be to balance the action of P at A,
whilst it leaves P2 to exert an action at B precisely the same
as P was exerting at A before the introduction of P, and P,.
NOTE (6).
Suppose two forces P1 and P, applied to the same point A,
571




572              EDITORIAL APPENDIX.
the direction of the one being AB, that of the
other AC; no was these forces make an angle
with each other, it is evident, as the point of
application can move but in one direction, and
B           as it is solicited to move towards B and C at
2 the same time, that it must move in some
P  GBs   direction which is coincident with neither of
these; this direction, it is equally evident,
must be in the same plane as the directions AB and AC, for
there is no argument in favor of a direction assumed exterior
to the plane and on one side of it which would not equally
apply to a symmetrical direction assumed on the other side;
it is also evident that this direction must be some one AF
within the angle formed by AB and AC, for the point, if
solicited by P, alone, would take the direction AB, and as it
cannot take a direction to the left of BD, as there is no force
that solicits it on that side, and, for like reasons, cannot take
one to the right of CE, it must therefore take the one
assigned somewhere within the angle BAC.
Now suppose further that P1 and P2 are equal, it is evident that the direction assigned to their resultant, or that of
the motion of their point of application, must be the one
which bisects the angle BAC, for the argument for any
direction on the left of this line would be equally cogent for
the like position on the other side.
If Pi and P2 are unequal then will the direction of their
resultant divide the angle BAC unequally, the
smaller portion being next to the greater force;
/ \c   for suppose Pi divided into two portions, one
P   IF  2 of which P shall be equal to P,; P and P2 can
be replaced by their resultant iR, the direction
eB /  IBl  of which AF bisects the angle BAC; we shall
Pi        then have two forces R, and the remaining
\     portion of Pi, the resultant of which R must lie
somewhere within the angle BAF, and therefore nearer to Pi than to P,; but R is the resultant of the
two forces P, and P2. Therefore, &c.
Hence it is seen that two forces whose directions form an
angle between them and meet, 1st, have a resultant; 2nd, that
the direction of this resultant lies in the plane of the two
forces; 3d, that it passes through the point where the directions meet, and lies within the angle contained between
them; 4th, that it bisects this angle when the forces are
equal; 5th, that when the forces are unequal it divides this
angle unequally, the smaller portion being next to the greater
force.




EDITORIAL APPENDIX.             573
Now as the two forces P, and P2 can be replaced by their
resultant R, and as the effect of this will be the
same if applied at any point F in its line of
direction as at the point of application of the
a two forces, it is evident, if we transfer P1 and P,
also to F, preserving their new parallel to their
Pa /v  A  original directions, that they, in turn, can be made
to replace R. It thus appears that the point of
application of two forces may be transferred to
any point of the line of direction of their resultant without changing the effects of these forces, provided
their new directions are kept parallel to their original ones.
It is upon the preceding propositions, in themselves selfevident, that the mode of demonstration, known as Duchayla's, of the proposition, termed the parallelogram of forces,
or of pressures, is based.
NOTE (d).
When two parallel forces are applied to two points invariably connected, their resultant can be found by applying
the propositions in (Arts. 1, 2, 3).
Let P1 and P2 be two parallel forces applied at the points
A and B invariably connected, as by a
Q/;~.;.2  rigid bar. Let two equal forces Q1 and
~',.',-  Q2 be so applied, the one at A the other
\  at B, as to act in opposite directions
eQ   uc-r, - Q, along AB.  These two will evidently
- a, /  have no effect to change the action of
Bal, /;      P1 and P2. Now the two forces P1 and
Q applied at A will have a resultant R,,
the intensity and direction of which can
be found by the preceding method. In like manner the
resultant R, of P. and Q can be obtained. Now the forces'being replaced by their resultants, the equilibrium will still
subsist, and the effect will remain the same whether R1 and
R2 act at A and B, or at o their point of meeting. But as
R1 and R2 can each be replaced by their components at any
point of their direction, let these components be transferred
to the point o. In this position Q, and Q2 will destroy each
other, whilst P1 and P, will act in the same direction along
oC and parallel to their original ones, with an intensity equal
to their sum P + P,.
Now from the similar triangles AoC, rom; and BoO, son,
there obtains,




57?              EDITORIAL APPENDIX.
om: mr:: oC: CA, or P1: Q,:: oC: CA.
ns: on:: CB: oC, or Q  P,:: CBP: oG.
Multiplying the two last proportions, there obtains,
P,: P2:: B: CA,
and
P,: P,: P,+P,:: B: CA: CB+ CA orAB.
From this we see that two parallel forces acting in the
same direction, 1st, have a resultant which is equal to their
sum; 2nd, that the direction of this resultant is parallel to
that of the forces; 3d, that it divides the line joining the
points of application of the two forces into parts reciprocally
proportional to the forces; 4th, that either force is to the
resultant as the portion of the line between the resultant and
the other force is to the total distance between the points of
application; 5th, that the foregoing propositions hold true
for any position of the line AB with respect to the two
parallel forces and their resultant.
When the two forces act in opposite directions at the
points A and B, by following
-.._____.__ -—' a like process, we obtain the
C."T/....    two resultants R1 and R,,; I/'....-....  which being prolonged to
&-;:?r!,;^            their point of meeting o we,,,-.-:-: —-%             can again replace them by
their components P1, Q, and
i.".......               P,, Q,; of which P, and P,,
acting parallel to their original positions but in opposite directions, will have for their
resultant P,-P,.
Now prolonging the direction of this resultant until it
meets AB prolonged at C, there obtains as in the preceding
case, from the similar triangles AoC, rom, and BoC, son,
om: mry:: oC: CA, or P1: Q:: oC: CA,
ns: on:: B: oC, or Q,: P,:: CB: oC,
hence,
P,: P,: P1-P,:: B: A: CB-CA or AB.




EDITORIAL APPENDIX.             575
Remark.-Although it may be assumed, as self-evident,
that any resultant can be replaced
—..... —---        by two equivalent components,
ai.<' ""-...:.. without disturbing the equilibrium,.: —~ ~'g, \ -.    and that each of these in turn
may be replaced by two other
equivalent components, and so on
for any number of components; still like compositions and
resolutions of forces are of such frequent occurrence in estimating the pressures, or strains on the various points of any
mechanical contrivance, as a machine, a frame of timber,
&c., arising from a resultant pressure, that the student cannot be too familiar with the processes of effecting such compositions and resolutions.
To show by a simple illustration this truth, let the resultant AR be replaced by its two equivalent components AP,
and AP, in any assumed positions; and let each of these
components be resolved into two others, AQ1, AR, for AP,;
and AQ,, AR, for AP,, taken respectively perpendicular and
parallel to AR. Now it is evident, from the figure, that the
two components AQ,, AQ, of this last resolution are equal
and opposite in direction, and therefore destroy each other;
whilst the two AR,, AR, act in the direction of AR, and
their sum is equal to AR. The same may in like manner be
shown for any number of sets of components by which AR
might be replaced.
NOTE (e).
If the point o from which perpendiculars are drawn to the
directions of two forces P1 and P,, is
taken on the direction of their re-,2, \', sultant, then will nmtP_=ntP.
/A    * ^ o       "  For from o draw the perpendicu\  1 -[.' ~    lars om, on, to P, and P, and join
the points m and n of their interm-'-e;    section. The quadrilateral Amon,
having the angles at m and n right
angles, can be inscribed in a circle, therefore the two angles
at m and A, subtended by the chord on, will be equal. In
the triangles mon and ABC, the angle o is the supplement
of the angle A of the quadrilateral, and B, being the adjacent angle of the parallelogram, is also the supplement of A;




5T6              EDITORIAL APPENDIX.
the two triangles, having two angles equal, are similar,
therefore,
AB: BC     om:: o: on, or P.: P,:: om: on;
hence
P, x om=P2 x on. Therefore, &c.
From this proposition the relations of two parallel forces
to their resultant can be readily deduced from the limiting
case of the angle mon of the triangle; for from the two similar triangles there obtains as before
P.: P: R or AC:: m: on: mn.
Now as this is true for any value of the angle o, when it
becomes 180~, the forces P., P. having the same direction,
and their resultant R become parallel; the perpendiculars
om and on become portions of the line mn; and, as mn —=om
+on, it follows, from the above proportion, that R=P,+P2.
When P, and P2 have opposite directions, we can suppose
the force P1, for example, and its perpendicular turned about
the point o in the plane of the forces until the point m falls
on the prolongation of on on the opposite side from o, in
which position P1 and P2 again become parallel, but act in
opposite directions. During this rotation of P,, the resultant
still passes through o, and there still obtains
P,: P,::   mR:   o  n: mnn;
but, as mn now is equal to om-on, it follows, from the.proportion, that R-P, -P,   Hence the same relations
between P,, P2 and R as already established, NOTE (d).
NOTE (f).
Otherwise, since in any number of forces in equilibrium
either of them is equal and directly opposed to the resultant
of all the rest, the whole may be replaced by these two
without disturbing the equilibrium. If now through the
point of application of these two we draw any two lines at
right angles to each other, and then resolve each of the two
forces into two components parallel to these two lines, it
will be at once seen, from the diagram, that the like com



EDITORIAL APPENDIX.             577
ponents will be equal and opposite to each other, and this
would evidently be the same for the components of the original forces resolved in the same manner, otherwise there
would be a resultant for all the forces, which is contrary to
the supposition of an equilibrium.
Remark.-As this method of resolving a system of forces
into sets of components parallel to any assumed rectangular
axes, in order to determine their algebraical values, is of
frequent use, in simplifying the numerical calculations
necessary in the applications of the principles of statics, the
student should make himself perfectly familiar with the propositions that precede and follow Art. 11.
NoT (g).
Otherwise, join DE which will be parallel to AC, thus
forming with it and the lines AD and CE two equi-angular
triangles, from which there obtains
DE: DG:: AC: AG;
but DE=-AC, therefore DG=-iAG=j-DA.
NOTE (h).
Otherwise, join Gil which, as AG and CH intersect, will
be in the same plane with them and with the line AC. As
AH and CG are respectively i of the lines drawn from A
and C to the middle of BD, it follows that GH is parallel to
AC and forms with it and the lines AG and CH, by their
intersection at K, the two equi-angular triangles CEH and
and AKC, from which there obtains
GH: GK:: AC: AK,
but GH=ABC, therefore GK=-=AK=-     AG.
NOTE (~).
As the methods employed in (Art. 45, &c.) to represent,
by geometrical diagrams, what are termed the laws of
motion, or the relations which exist at any two given
37




578              EDITORIAL APPENDIX.
instants between the velocity, the space, and the time of a
body's motion, although very simple in themselves, are
sometimes found to offer difficulties to the student, particularly as to the representation of spaces by areas, a few additional marks on this point may not be here misplaced.
Taking, in the first place, the case of a body M moving in
D           a rectilinear path from A towards
e!   7[1E           B with a uniform motion.  Acn.t                    cording to the definition, the
~__1  ]   -':         body will move over the suc-'M.I;  I B cessive equal portions Ab, be, cd,
b   c  d  e  &c., of its path in equal successive portions of time, however small or great these portions
may be. Taking now anyportion of time as a unit, as a second,
a minute, &c., and supposing Ab the portion of its path, or
the space through which the body has moved during this
unit, Ab will represent what is termed the velocity, or rate
of motion of the body; and when the path itself is expressed
in terms of any linear unit, as a foot, a yard, a mile, &c., the
number of these units in Ab will measure the velocity; for
*example, if'the unit of path, or space is a foot, and there
were four of these units in Ab, and the unit of time is a
second, then the velocity would be termed a velocity of four
feet per second, &c. Supposing the body to start from A.,
with this velocity, it will successively move over distances,
each of four feet in length, along its path, in successive
seconds of time; consequently any distance, or space, as Ad,
will be equal to Ab taken as many times as the number of
seconds elapsed from the time the body started from A until
it reached d; or, in other words, the number of units in the
space Ad is expressed by the abstract number obtained by
multiplying the number of units in the velocity by the number of units in the time. This, like all other similar products, can be expressed algebraically, or geometrically; but
by whatever symbol expressed, the signification is the same.
For example, on any two lines, as AB and AC, taken at
right angles, set off any number of equal parts as Ab, be,
ed, &c., as unit sof time, and on AC any number also of
equal parts, which may be the same in length, or otherwise,
as those on AB, to represent the units in which the velocity
is expressed. Suppose the latterto be composed of the four
units Am, mn, &c.; and that the number of units of time
considered is three; on the lines Ad, AC construct the
rectangle AD; then is the area of the rectangle said to,express the space corresponding to the velocity and time




EDITORIAL APPENDIX.             579
here assumed; that is, the number of units in area of this
rectangle, expressed in terms of the unit of area on Ab and
Am for example, is equal to the number of units of space.
In like manner the area of the rectangle AE expresses the
space corresponding to the velocity and the time Ae, &c.
In uniformly varied motion, as the velocity increases in
the same proportion as the time
increases, or, in other words, as
the augmentations of the velocity
~..   ^  ~ for equal intervals of time is the
*: t |          same, these relations between the
FAC    \ -o e  e-,... times, velocities and spaces, can,
6       in like manner, be expressed by
a geometrical diagram as follows: On any line, as AB, set
off a numler of equal parts as Ab, be, cd, &c., to represent
equal intervals of time; at the points b, c, d, &c., having
drawn perpendiculars to AB, set off on them distances bm,
cn, do, &c., to represent the corresponding velocities; in
which cn=2bim do=3bm; or Ad: Ac: Ab:: do: cn: bn,
&c. Now, as the same relations obtain between all the dis,
tances set off on AB and their corresponding perpendiculars, it follows that thle line AC, drawn through the points
Mn, n, o, &c., is a right line, and that the triangles Abm, Acn,
&c., are therefore similar.  As the relations between the
times and velocities are true, however great, or however
small the equal portions of time may be assumed, let us suppose these portions, as Ab, bc, cd, to be taken so small that
the velocity of the body during any one of them may be
considered uniform, and as a mean between what it actually
is at the commencement and end of this portion; that is en
and do, for example, representing the actual velocities at the
beginning and end of the interval of time represented by cd,
then -1 (cn+-do) represents the mean, or uniform velocity
during this interval. This being premised, the number of
units of space over which the body will pass whilst moving
with a uniform velocity, expressed by i (cn+ do), during the
interval cd, will be represented, according to the preceding
proposition, by cdx  (cn +do), but this also expresses the
area of the trapezoid cdno; and as the same is true for all
the like trapezoids it will also be true for their sums, or for
the triangles, as Ado and Afq for example, the areas of which
are equal to the sum of the areas of the trapezoids of which
they are composed. Supposing the body to move from a
state of rest with a uniformly accelerated motion, and that
at the intervals of time, represented by Ad and Af, its




580               EDITORIAL APPENDIX.
respective velocities are do andfq, then will the number of
units of space which the body will have moved over in these
two intervals be respectively expressed by the number of
units of area in the triangles Ado and Afq. As the triangles are similar their areas are as the squares of their like
sides; it therefore follows that in uniformly varied motion,
the spaces are as the squares of the times, or as the squares
of the velocities.
As do represents the velocity acquired during the time
Ad, supposing the body to have moved from a state of rest,
and the number of units of area in the triangle Ado represents the corresponding number of units of space, it follows,
that if the body had moved, during the same interval, with
the velocity do which it actually acquired in it, the number
of units of space it would then have passed over would have
been represented by the number of units of area in the rectangle Ao, constructed on Ad and do. But, as the area of
the rectangle is double that of the triangle, the space that
would have been passed over in the supposed case would
have been double that passed over in the actual case.
If we take any portion, as Ae, to represent the unit of
time, then the corresponding perpendicular ep will represent
the velocity, or the quantity f used in (Arts. 46. 47) following.
NOTE (j).
As the propositions under this head, and those under the
heads of Accumulation of Work in a Moving Body (Art. 64)
and Principle of Vis Viva (Art. 129) constitute the basis of
what may be termed Industrial Mechanics, or the applications of the principles of abstract mechanics to the calculation of the effects of motive power transmitted by machines
and employed to produce some useful mechanical end, it is
very important that the student should have a clear and
definite apprehension of their signification in this point of
view. Work, as here defined, supposes two conditions as
essential to its production: a continued resistance, or obstacle
removed by the action of a force, and a motion of the point
of application of the force in a direction opposite to that in
which the resistance acts. Its measure is expressed by the
product arising from multiplying the number of units of the
resistance, or of its equivalent force directly opposed to it,
by the number of units of path which the point of application of this force has described during the interval consi



EDITORIAL APPENDIX.             581
dered, in which the force acts to overcome the resistance.
It follows that the work will be 0 when this product is 0;
that is, when either of the factors, the resistance, or its equivalent force, or the path described, is 0.
In estimating work, that which is external and which alone
generally we have the means of measuring, is alone considered. For example, if with a flexible bar a person attempts
to push before him any obstacle, the first effect observed will
be a certain deflection of the bar, during which the hand, at
one end of the bar, will have moved forward a certain distance in the direction of the point of application at the other,
producing an amount of work which is expressed by the
product of the pressure, or force exerted by the hand, supposing this pressure to remain constant during this period,
and the path described, in the line of direction of this pressure, by the point where it is applied. During this period,
as the obstacle to be moved has remained at rest, no path
has been described by the point where the bar rests against
it, therefore, according to our definition, no work has been
done upon the resistance.  The effect produced by the
pressure has been simply to bend the bar, and the work is
therefore due only to the resistance offered by the molecular
forces of the material composing the bar to the force that
tends to bend it. This portion of the work, although in this
case we have the means of measuring it, being what may be
termed internal, is not taken into the account in estimating
that due to the resistance to be overcome, which would have
been the same had a perfectly rigid bar been used instead of
the flexible one.
In like manner, when an animal carries a burthen on his
back from one point to another on a horizontal plane no
work is produced according to our definition; for no resistance has been overcome in the direction in which the burthen has been carried, and therefore the product that represents the work is 0. The work in this case, as in that of the
flexible bar, is internal; and similar to that arising from a
burthen borne by an animal whilst standing still; and therefore although both of them may be very useful operations
and have a marketable value, still they can neither be measured by the standard by which it is agreed to estimate
work.
Every mechanical operation performed by machinery presents a case of work. Take for example the simple operation of planing, in which the hand moves a plane, which is
but a rigid bar to which is fixed an iron tool like a chisel for




582               EDITORIAL APPENDIX.
removing successive thin portions from the edge, or surface
of a board. In this case the resistance offered, and which is
sensibly in'the same direction as the power applied, is that
arising from the cohesion of the fibres of the material, and
is measured by the pressure applied; the path which the
point of application of the iron tool describes is the same as
that described by the hand; and the work will be expressed
by the product of these two elements, each estimated in
terms of its own unit of measure. The case of the common
grindstone presents an example of a rather more complicated
character.  Here the instrument to be ground is pressed
against the periphery of the stone with sufficient force to
cause a certain resistance to any power however applied to
put the stone in motion. The direction however in which
this resistance acts at the point of application is in the
direction of the tangent to the periphery at this point, and,
in one revolution of the stone, it will describe a path equal
in length to the circle described by the point of application.
The work therefore for each revolution will be the product
of the resistance, estimated in the direction of the tangent,
and the circumference described by the point of application.
It may be as well to remark, in this place, that although
the work done in overcoming the molecular resistances of
the materials by means of which the action of a force or
pressure is transmitted, as in the example above cited of a
fexible bar, is not taken into account in estimating the
external work, there are cases in which this work constitutes
the entire work done, and which again is reproduced in
external work; as for example in the cases of the common
bow used for projecting arrows, and the springs by which
the machinery of some time-pieces is moved. In each of
these the resistance offered by the molecular forces of the
material is overcome by the action of some external force,
whose point of application is made to describe a given path;
by this action a certain amount of work is expended in
bringing the spring to a certain degree of tension which,
when the force is withdrawn, will reproduce the same amount
of external work in an opposite direction to that in which the
force acted.
NOTE (k).
The work of a pressure of constant intensity acting in the
same direction as the path described by its point of application may be represented by a geometrical diagram in the




EDITORIAL APPENDIX.             583
same way as that used for representing the space described
by a body moving with a uniform velocity in any given
time; by constructing a rectangle, one side of which represents the number of units of force, the other the number of
units of path; the number of units of area of the rectangle
will express the number of units of work.
NOTE (1).
The method given (Art. 51) for estimating, by a geometrical diagram, the work of a pressure which varies in intensity at different points of the path described in its line of
direction by its point of application, finds its application and
has to be used whenever there is no geometrical law of continuity by which the pressure can be expressed in terms of
the path; and, even when such a law obtains, it is sometimes found to be a more convenient method of obtaining an
approximate value of the amount of work than the more
rigorous one expressed by the formula
U =fPds;
Si
in which IT can be rigorously found whenever P, which being
a function of S can be expressed algebraically in terms
of it.
As an example of these two methods of estimating the
work of a variable pressure, acting in the
*c  _   XD direction of the rectilinear path described by... "   its point of application, let the familiar case
of the action of steam on the piston of the
steam-engine be taken.
Let ABCD represent the steam-tight cy| 0' | linder in which the piston is driven from the
position at a, at one end, to c at the other, in
-:::::-'  the direction of the axis ac, of the cylinder,
by means of the pressure of the steam on the
Jend of the piston. Let us suppose that the
-A        B steam acts with a constant pressure, represented by P1, whilst the piston is driven through the portion
ba of the path, and, having reached this point, the communication between the cylinder and the boiler being then cut




58i              EDITORIAL APPENDIX.
off, that the steam already admitted acts, through the remainder of the path described by the piston, by what is
termed its expansive force, in which the pressure continually
decreases as the piston approaches the point c. Let us suppose that the law of variation of this pressure on the piston
at different points is such that the pressures at any two
points are inversely proportional to their distances from the
point a. P, then denoting the pressure when the piston is
at b, let P denote the pressure when it has reached another
point o at a distance S from a, and S, and S, denote the
lengths ac and ab, then according to the above law there
obtains
P: P:: S: S,, therefore P=P.
Let the elementary portion of the path be denoted by dS,
then by multiplying the variable force by the elementary
path there obtains
PdS=P, SdS,
which may be termed the elementary work, or in other
words, the work done whilst the variable pressure acts
through the elementary path, during which period the variable pressure may be regarded as constant.
To obtain the total work whilst the variable pressure acts
from b to c, or through the path S - S, there obtains
S,          S,
U -=PdS = PS f         ==P1S, (log.e S-log. SI).
~S,      S,
SI          Si
If instead of the exact work due to the expansive force of
the steam, and which is given by the foregoing formula, an
approximate value only was required, it could be obtained by
a geometrical diagram as follows.
""7~ Having set off to any scale a numI ".,,b her of units representing the path
*'"'.be, calculate the pressures at the'-....  points b, c, and at the middle point. o, for a first approximation. That
/ _____  iJ-o?, at b will be simply P1; that at c,
Pl, and that at o Pa (Si
151 P)I 1(- (SSI)




EDITORIAL APPENDIX.             85
Having drawn perpendiculars to be at b, o, and c, set off on
them the distances bin, on and cp respectively equal to the
corresponding pressures, estimated in terms of the unit of
pressure, and according to anlly given scale. Join mn and
np; the number of units of area, in the figure thus formed,
estimated in units of path and pressure, will be an approximate value of the required number of units of work.
The greater the number of parts into which be is divided and
the corresponding pressures calculated, the nearer will the
enclosed area approach to the true value of the work.
The mean pressure, or that force which, acting with a
constant intensity along the same path as that described by
the point of application of the variable pressure, would give
the same work, is found either by dividing the result of the
integration by S2-S1, or by dividing the area in the last
method by be.
NOTE (m).
As an example of the manner of obtaining the work done,A..    by a constant pressure acting always
-/'.... in parallel directions whilst its inclina-..-.. — c  tion to the path described by its point,  G      ~^ of application is continually varying,
let the well known mechanism of the
"  \   crank arm and connecting rod be taken.
-..     Let O be the centre around which the
3     F'    crank arm is made to revolve, by the
application of a constant pressure P,,
transmitted through a connecting rod
CD, all of whose positions during the motion are parallel to
the diameter AB. The path described by the point of application C will be the circumference of which OC is the
radius, and the inclination of P, to this path will be the
variable angle DON, between its direction and the tangent
to the circle at C, of which the variable angle AOC, that
measures the inclination of the crank arm to the diameter
AB, is the complement. Denote this last angle by a, and
the length of the crank arm OC by b. Now decomposing
PI into components in the direction of the tangent CX and
the radius OC, we obtain for the first P1 sin. a, and for the
second P1 cos. a, of which P, sin. a is alone effective to produce work, since Pi cos. a acts constantly towards the fixed
point O without describing any path in the direction of its




486              EDITORIAL APPENDIX.
iaction.  But the elementary path described by the point
of application is evidently bda, the infinitely small arc Cn of
the circle. The elementary work of the variable component
P1 sin. a will therefore be expressed by
P1 sin. a x bda.
The total work for any portion of the path, as AC, will
therefore be
a
f    P sin. a bda=P1 (1-cos. a)=P1b ver. sin. a.
o
lmd for a=xr, it becomes
P, x 2b, or P, x AB;
a result which might have been foreseen, since AB is the
path described by the point of application of P1 in its line
of direction, whilst the actual path is the semi-circumference ACB.
As C-W=bda, if through n a perpendicular nm is drawn to
CD, the line of direction of P,, the distance Cm is evidently
the projection of the elementary path actually described on
the line of direction of Pi, and is therefore the corresponding
elementary path of P, in its line of direction; but Cm=Cn
Ain. a=bda sin. a. Denoting AB by h, then Cm=dh; and
there obtains
dh=bda sin. tr; and P, dh=P1 bda sin. a;
and
h              ^
I Plh=PA=          Plb sin. a da=P x 2b.
0               0
A result the same as is shown to obtain by the preceding
proposition.
To find the mean, or constant pressure which, acting in
the direction of the circular path, would produce the same
amount of work as the variable force does in acting through
the semi-circumference; call Q this mean force, its path
being Trb, its work will be Q x rb; and as this is to be equal
to the work of P, sin. a, there obtains
Q x 7rb=P, x 2b, hence Q=P, - =0'6366 P, nearly,
-for the value of the force.




EDITORIAL APPENDIX.             587
It may be well to observe here that the mean pressures
have no farther relations to the actual pressures than as
numerical results which are frequently used instead of the
actual pressures to facilitate calculations; and also as a
means of comparing results, or work actually obtained from
a force of variable intensity, at different epochs of its action,
with what would have been yielded at the same epochs by
the equivalent mean force.
To show the manner of making the comparison in this
case, let us take the two expressions for the quantity of work
due the mean force, and also to the variable component, for
a portion of the path corresponding to any angle a. Since
Q=P, -, its work corresponding to a will be'2         2a
P, - x ba = PThe corresponding work of the variable component P1 sin. a
will be
Pb (1- cos. a).
The difference therefore between these two amounts of work
will be
P2- P b(1-cos. a)=Pm    b (2-a  + cos. a
Now this difference will be  fo r the following values of a,
Now this difference will be 0 for the following values of a,
a=0,    a=,   and a =r.
The maximum value of this difference can be found by the
usual method of differentiation and placing the first differential coefficient equal to 0.  Performing this operation,
there obtains
sin. a  - -06366;
the corresponding values of a being respectively
a = 0-21964 7r,and a - =  - 0-21964: r.
Substituting these values of a and the corresponding values
of cos. a in the preceding expression for the difference there
obtains, for the first,




588              EDITORIAL APPENDIX.
Pb ( —2a+-cos. a) =P1 (2x0-21964-1+     /1-      =
+ 0-21039P, b;
and for the second,
Pb (  -  +tcos. a) =PB (2 x 021964-1-1/1- 4) =
- 0-21039 Pb.
From these two expressions it is seen that the greatest excess
of the work of the mean force over that of the other would
be +0-21039Pb= +0-1052 x P12b, or about -I- of the total
work of Pi corresponding to the path 2b; whilst that of the
work of P, over the mean force, represented by -0 21039Pb,
is the same in amount.
If now we suppose the direction of the constant force P,
to be changed, when its point of application reaches the point
B, so as to act parallel to the direction BA until the point of
application arrives at A, it is clear that the work of P1 due
to the path described from B to A will also be expressed by
Pi x 2b, so that the work due to an entire revolution of the
point of application will be P1 x 4b. As the mean force will
evidently be the same for the entire revolution of the point
of application, it follows that the greatest positive, or negative excess, as stated above, will be 0-0526 x P,b or'- of
the work for one entire revolution.
It is thus seen that although the work of the effective
variable component P, sin. a of P, is not, like that of the mean
force, uniform for equal paths, still it at no time falls short
of nor exceeds the work of the mean force by more than
about -- of the entire work for each revolution. Were any
mechanism, as that for pumping water for example, so
arranged that either the constant force P1, or a mean force
equal to 0'6366P1, acting as above described, were applied
to it, the quantity of water delivered by the one would at no
time exceed, in any one revolution, that delivered by the
other by more than -I of the total quantity delivered by
either during the entire revolution.
NOTE (n).
If P,, for example, were the resultant of the other pressures, its component P. cos. a2 would be equal to the alge



EDITORIAL APPENDIX.             589
braic sum of the components P, cos. a,, P, cos. a,, &c., of the
other pressures P, P,, &c.; the work therefore of P,, estimated in the direction of the given path AB, and corresponding to any portion of this path, will be equal to the
algebraic sum of the work of the other pressures P,,,, &c.,
corresponding to the same portion of the given path.
NOTE (o).
Since at the point E, taken as the point of application, the
line of direction of the pressure becomes a tangent to the
arc described with the radius OE, it follows that the infinitely small arc described with the radius OE may be taken
for the infinitely small path described by the point of application in the direction of the tangent. Denoting by da the
infinitely small angle described by the radius OE, then
OE x da will express the infinitely small path, or arc; and
P x OEda will represent the elementary work of the
pressure.
If the pressure remains constant in intensity and direction
during an entire revolution of the body about 0, then will
the work of P for this revolution be represented by
P x circum. OE.
NOTE (p).
The term liing force is more generally used with us by
writers on mechanics instead of its Latin equivalent vis vvwa,
to designate the numerical result arising from multiplying
the quantity denominated the mass of a body by the square
of the velocity with which the body is moving at any
instant. It will be readily seen that this product does not
represent a pressure, or force, but the numerical equivalent
of the product of a certain number of units of pressure and a
certain number of units of path. The one magnitude being
of as totally a distinct order from the other as an area is
different from a line, and therefore having no common unit
of measure.
Besides this expression, which serves no other really use.
ful purposes than as a name to designate a certain numerical
magnitude which is of constant occurrence in the subject of
mechanics, there is another also of frequent use, termed




590               EDITORIAL APPENDIX.
quantity of motion, wnich is the product of the mass and
the velocity, or w -.  This is also termed the dynamical
measure of a force in contradistinction to pressure, as usually
estimated, which is termed the statical measure of a force.
NOTE (q).
In estimating the accumulated work in the pieces of a
machine which have either a continuous or a reciprocating
motion of rotation it is necessary to find expressions for the
moments of inertia of these pieces with respect to their axis
of rotation, and this may, in all cases, be done, within a certain degree of approximation to the true value, by calculating separately the moment of inertia of each of the component parts of each piece and taking their sum for its total
moment of inertia, on the principle that these moments may
be added to or subtracted from each other in a manner
similar to that in which volumes, or areas are found from
their component parts.
In making these approximate calculations, which in many
cases are intricate and tedious, it will be well to keep in view
the two or three leading points following, with the examples
given in illustration of some of the more usual forms of
rotating pieces.
1st. The general form for the moment of inertia of a body
rotating around an axis parallel to the one passing through
its centre of gravity as given in equation 58, (Art. 79) is
I,  a2M+I.
Now if the distances of the extreme elements of the body
from the axis passing through its centre of gravity are small
compared with that of A, the distance between the two axes,
the second term I of the second member of this equation
may be neglected with respect to the first, and h2M be taken
as the approximate value of the required moment. This
consideration will find its application in many of the cases
referred to, as, for example, in that of finding the moment of
inertia of the portion of a solid, like the exterior flanch of the
beam of a steam-engine, the volume of which may be approximately obtained by the method of Guldinus (Art. 39.). In
this case, A representing the area of the cross section of the




EDITORIAL APPENDIX.              591
flanch, and s the path which its centre of gravity would
describe in moving parallel to itself in the direction of the
fianch around the beam, any elementary volume of the
flanch between two parallel planes of section will be expressed by Ads. Now the moment of inertia of this elementary volume from equation 58 is
I, =A/r'ds+I;
in which the first term of the second member, which
expresses the sum of the elementary volumes Ads into the
squares of their respective distances r from the axis of rotation, may be taken as the approximate value required; inasmuch as I, the sum of their moments of inertia with respect
to the parallel axes through their centres of gravity, may be
neglected with respect to the first term. The problem will
therefore reduce to finding the moment of inertia of the line
represented by s, which would be described by the centre of
gravity of A, with respect to the assumed axis of rotation,
and then multiplying the result by A.
2nd. As the line s is generally contained in a plane perpendicular to the axis of rotation, and is given in kind, as
well as in position with respect to this axis, being also generally symmetrically placed with respect to it, its required
moment of inertia may, in most cases, be most readily
obtained' by finding- the moment of inertia of s separately,
with respect to two rectangular axes contained in its plane,
and taken through the point in.whichthe given axis of rotation
pierces this plane, and then adding these two moments.
The moment of inertia of a line taken in this way with
respect to a point in its plane has been called' by some
writers the polar moment of inertia.
This method is also equally applicable to finding the moment of inertia of a plane thin disk revolving around an, axisperpendicular to its plane, and to solids which can be divided
into equal laminae by planes passed perpendicular to the axis
of rotation.
(a1) The moment of inertia of the arc of a parabola with
respect to an axis perpendicular to the plane of the curve at
a given point on the axis of the curve.
Let BAC be the. given arc; A the vertex of the parabola 




592               EDITORIAL APPENDIX.
Rthe point on its axis at which the
_~rI       -   axis is taken.  Through R  draw
the chord PQ.      Represent the
v k —D       chord BC of the given arc by b;
its corresponding abscissa AD by a;
and AR by c. Let y represent the
ordinate p, and x the corresponding abscissa of any element dz of the arc.
1rom the preceding remarks, the moment of inertia of dz
with respect to the axis AD will be expressed by y2dz; and
that of the entire arc BAC by
~~b       ~~lb
2 fdz* =     2 fyb\   + 64ay) dy;
o            o
b
as from the equation of the parabola, y' - 4ax.
By integration t
lb
I1=2,y(b + 64 ay2)id    b (b' 16a'   -    b   z
b=y'ca           =   a'  64         4.64ag;
o
in which Z is the length of the arc BAG.
In like manner the moment of inertia of dz with respect
to the chord PQ is
f(o-w)dz
and for the entire arc BAG,
lb            lb
(,O-,-             8 ac   16ad',
I,= 2 f (o  ydz s= 2f(c     2, y+       (1- +y)(6=+64a y2)dy;
o             o
which integrated as above,
-(1    + 32    + 3 2.64 2'   -
9 b        \(b  + 16a2)
a + 32a      )     64
Church's Int. Cal. Art. 199.    t Ibid. Art. 150.




iEDTORIAL APSi'VE 3X.          J5!3
From the preceding remarks, the moment of inertia of Z,
w1h respect to the axis at the point R perpendicular to the
$ptane of Z, is
I,+ 2cZ     + 32ac  32. 64 ac',L    b2   -2 C  I Y
(i +    1)  \a  32a 
The value of Z in the above -expressions is
Z= i- (b +  6a+ 8a, log.e (  +       + 16a).*
Each of the preceding expressions may be simplified, and
an (approximate value obtained, sufficiently near for practical
applications, when the ratios of b and c to a are given. For
example, when b Zja there obtains
11.+iJ=cZa (          -3,.a: (h 37Sbg  t);
Z   2a + -    log. -y;
the terms omittred, being small fractions with respect to
unity, do not materially affect the: esult.
Having found the moment of inertia of a parabolic curve,
that of a parabolic ring of utniform cross section, taken perpendicular to the direction of the curve at any point, and
having its centre of gravity at its point of intersection with
tile curve, can be obtained by simply- ultiplying 1 +I, by
S /the area of the given cross section.
(b') The moment of nertia of the segment of a parabola -with
respect to an axis perendicula to its plane at a gien
point of the axis of the curve.
Let -BAC be the given segment; A the vertex; AD the
* Chutchs Int. Cal. -rt. 199,
38




594              EDITORIAL APPENDIX.
axis of the curve; and D the point on
-  the axis with respect to which the mo-'i!        ments of inertia are estimated. Denote
a~.       -..-D the chord BC by b; the abscissa AD
by a.
p    *3  3By (Art. 81) the moment of inertia of
an elementary area pq with respect to AD is -(p q)3dx-=
~ (2y)'dx. That of the segment therefore will be
fb           tb
i1      f8y'dx =A-f82   y'dy -= 3 ab'.
0            0
In like manner the moment of inertia of an elementary
area as ps, with respect to the axis BC, is ~ (ps)'dy-= (a —w)
dy. That of the segment therefore will be
I    i,/ (a - xfdy  a   (a - 4 y2)3dy -  I-ab
0                0. I+I =-  ab +~VP a'b.
From this last expression we readily obtain the moment,of inertia of a disk having the segment for its base and its
thickness represented by c, with respect to an axis at D perpendicular to its base by simply multiplying I,+ I, by c; or
(I,+I)e = —   abc c  +  = -bo --  abc (q-b1 + I4 a2);
in which ] ab = V, the volume of the disk.
(c') The moment of inertia of a parabolic disk, or prism,
with respect to an axis parallel to the chords which terminate the upper and lower bases and midway between the
chords.
Let pq be an elementary volume of the disk contained
between two planes parallel to the base
L\   ~BC of the disk.     Adopting the same
notation as in the preceding article, the
7  \  \  -volume pq is expressed by.... -------      2y. c. dx.
The moment of inertia of this elementary volume with respect to an axis
^ -— ~ bthrough its centre of gravity and parallel
to BC is (Art. 83)




EDITORIAL APPENDIX.              595
12.2y..     da {c2+(dx) 
and the moment of inertia of the same volume with respect
to the axis, parallel to the one through its centre of gravity,
taken on the base BC of the disk and midway between the
upper and lower chords is (Art. 79 Eq. 58)
-1 22. y. c. d +x (dx)} +2y. c. dx (a-x)';
the moment of inertia of the entire disk with respect to the
same axis is
lb                       lb
- * = -   2y * c. dx 1a' + (dx)'} +f 2y. e. dx (a — ).
Q. O                     O
Substituting for x and dx in terms of y, omitting the term
containing (dx)9, and integrating as indicated, there obtains,
I =abe (-1  a +   e- c') = V(- a' +/ c');
in which V= iabo.
(d') The moments of inertia of a right prism with a trapezoidal base with respect to axes perpendicular and parallel
to the base at the middle point of the face terminated by the
broader side of the trapezoid.
Let AGHC be the trapezoid forming the base of the prism.
Represent the altitude EF of the trapeC:'.  zoid by a; AG by b; CH by b'; and the
1  \ \   heightGB of the prismbyc. Letpq bean,  \ \    elementary volume of the prism between.......     two planes parallel to the face AB and
at a distance Ee=x from the face CD.
z:  From C drawing Cc parallel to HG there
vA      m:::   _ obtains
Ee         x
pr=EF. c         (b- b);. ps =pr + rs = - (b - b) + b.
a8 =.Pr)+




U596             EDITORIAL APPENDIX.
The elementary volumep"  iS therefore
( ( - b) + b1 cd0
The moment of inertia of pq with respect to an axis through
its centre of gravity and perpendicular to the base of the
prism is (Art. 83).
-12 a (b   )   }       (d) +   (b - b')+ b
and that of the entire prism with respect to an axis at F, the
middle point of AG, and parallel to the preceding axis, is
a      b)+b       c      (b-I)+      +
0
a
i  (b  l~)+b  ~. da(a- x;
a
-omitting the term containing d()'a, and integrating, as indi-:cated, <there obtains
2     24         b+b-''a: b+31'  -+ b'' t.  r     +.4a.
ia which V =;a+ b.
By a like series of operations the:-moment of inertia of the
entire prism, with respect to an axis perpendicular to the
preceding one at its middle point:between the upper and
lower bases of the prism, will be
T   nWl b 5+361-!.> \
~L^ -  "t-      b'    r




EDITORIAL APPENDIX.             597
(e') The moment of inertia.of a right prismoid with rectanr
gular bases with respect to an axis XY through the centre
of gravity of the lower base and parallel to one of its
sides.
Let AB = b, BO = c be the sides of the rectangle of the
lower base; ab = bl, bc = c the sides
ij —-~        Qof. the upper base. Let pqrs be any
a. — \\   section of the prismoid parallel to
the- lower base and at a distance x
^.......     from it; and let a be the altitude of
the prismoid, or the distance between
//...'     i.ts:.ipper and lower bases.
D. —- --.... —    From the relations between the
_..-........    dimensions of the prismoid there obtains (Art. d')
aB
^ P~= - (b-b'^== ~ (bl-  b1) + ab'-              ((  - C') + ao)
r ~= C (e -  3k-).~aG=.-  - -.
and to express the elementary- solid contained between two
pianes parallel to the base of the prismoid and at the height
x above it,
Xa (b - b)) + abl  x(c - c1). + ac
-..      -....-  x  —.dxi,.
a              a
The moment of inertia of this solid, with respect to an
axis xy:through its centre of gravity and parallel to XY, is
(Art. 83)
(b -- b) + ab  x(c -c )+ac 
T  o - -a        a 
The mpiment of inertia of the prismoid (Art. 79 Eq. 58) is
J_*'xb')+a       x(c-1c') +C ac1' 7 X((-CIl)+alCe)  d.'
o




598S             EDITORIAL APPENDIX.
a
xZ(b - b')+abV x(c -- c) +     a  d
0
omitting the term containing (dw)3 and integrating as indicated, there obtains,
I=   (b      ) + +  b'be1) + 2h-a b  ab(cl + 2c1c+ 3coc + 4ce) +
-4 ab4 (c+ 22ccl + 3 c+ 4c').
By integrating the expression for the elementary volume
between the same limits, there obtains to express the volume
of the prismoid
Y=   ab ((2+ ce) + b(2c1 +c)1,
which is the formula usually given in mensuration.
In each of the preceding examples, the quantities I, I, &c.
are expressed only in terms of certain linear dimensions; to
obtain therefore the moments of inertia proper these results
must be multiplied by the quantity -, or the unit of mass
corresponding to the unit of volume, in which V represents
the weight of the unit of volume of the material and
g= 32 -feet.
Each of the above values of I may be placed under more
simple forms for the greater readiness of numerical calculation by throwing out such terms as will visibly affect the
result in only a slight degree. But as such omissions depend
upon the numerical relations of the linear dimensions of the
parts no rule for making them can be laid down which will
be applicable to all cases.
(f') The nmoment of inertia of a trip hammer.
These hammers consist of a head of iron of which A represents a side and A' a
A     A,                      front elevation; of a
--- -........   1.        handle  of wood B,
^. a....^';    which is either of the
shape of a rectangular
parallelopiped, or of




EDITORIAL APPENDIX.             599
two rectangular prismoids, having a common base at the
axis of rotation C where the trunnions, upon which the
hammer revolves, are connected firmly with the handle by
an iron collar. Another iron collar is placed at the end of
the handle, and is acted on by that piece of the mechanism
which causes the hammer to rotate.
To obtain the moment of inertia of the whole, that of each
part with respect to the axis is separately estimated and
the sum then taken.
The head A, A' may be regarded as a parallelopiped of
which the side A', reduced to its equivalent rectangle by
drawing two lines parallel to the vertical line that bisects
the figure, is the end, and the breadth of the side A is the
length. If then from the moment of inertia of this parallelopiped that of the void a, or eye of the hammer, which is
also a parallelopiped, be taken, the difference will be the
moment of inertia of the solid portion of the head, The
moments of inertia of these parallelopipeds may be calculated, with respect to the axis C, by first estimating them
with respect to the axes through their respective centres of
gravities G and g, parallel to C, by (Art. 83) and then with
respect to C by (Art. 79. Eq. 58). Or if the moments of
inertia with respect to G and g are small with respect to the
product of their volumes and the squares of the distances
GC and gC, then the difference of the latter products may
be taken as the approximate value.
The moment of inertia of the handle, if also a parallelopiped, will be found with respect to C by (Arts. 79, 83). If
it is composed of two rectangular prismoids, then the moment of the parts on each side of the axis must be found by
(e') and their sum taken.
The moment of inertia of the trunnions and the iron hoop
to which they are attached may be found by (Arts. 85, 87)
and their sum taken. But as this quantity will be generally
small with respect to the others it may be omitted.
That of the hoop at the end of the handle may be taken
approximately as equal to the product of its volume and the
square of the distance between the axis through its centre
of gravity and that of rotation.
(g') The moment of inertia of a cast iron wheel.
These wheels usually consist of an exterior rim A A' of




600              EDITORIAL APPENDIX.
A                   uniform  cross section connected with the boss, or nave
X  lX\  i. ~      aC, C', which is a hollow
cylinder, by radial pieces, or
i-^_ —r -~ )      I B;     arms- B, B', the cross section
of which is in the form of a
cross. Each arm having the
same breadths at top and.A      bottom in the direction of
the axis of the wheel as those of, the rim and nave which it
connects; the thickness perpendicular to tlhe axis being
uniform. The projection or ribs on: the side of each arm,
and which give the cross form to the section, being of uniform breadth and thickness; or else of uniform thickness
but tapering in breadth from the nave to the rim. These
ribs join another of the same thickness' that projects from the
inner surface of the rim.
Represent by R the mean radius -ofthe rim, estimated fiom
the axis to the centre of gravity of its cross section; b its
breadth, and d its meain thickness; Vits volume, and I its
moment of inertia with respect to the axis; v the weight of
its unit of volume, and g=32- feet; then by (Art. 86)
V=2mrRbd andI=' VR2,
omitting Id2 as but a small fractional part of R2.
Representing by b, the breadth of the arm at the axis,
supposing it prolonged to this line; \b its breadth at the rim,
supposing it prolonged also to the mean circle of the rim, d,
its thickness; Ji its volume; I, its moment of inertia, then
by (d')
V>I ^ b+b   andl,-_"' 17R2-  {       +4
~]          =    Rd, 2  +         bl+3b, + 2 d,'~
1W' 6,~+b, IR'
Representing by a, the breadth at bottom, a, the breadth at
top of the ribs, or projections on the sides of each arm, estimated also at the axis and mean circle of the rim; d2 their
thickness; V, their volume; I2 their moment of inertia;
then by (d')
VCiRd2 a+a2     and 1, -  i  aR    8   +- i2)
e sm           wi be the moment of inertia of the
The sum I + I, + Il will be the moment of inertia of the




EDITORIAL APPENDIX.             601
-etire wheel approximately, sice the moment of inertia of
the portions of the boss. between the arms is omitted, this
bei.ng compensated for by supposing the arms prolonged to
the axis and, to the mean circle. of the rim. As the: quantities JF, VJ, I~ and I2 are taken but for one arm, they must
be multiplied by the number of arms to. have the entire
moment.
(h') The moment of inertia of a cast iron steam engine beam.
These beams usually consist of two equal arms symmetrical with respect to a
D_             =          line a a' through the
6     T>     7,7.~-~-.Zz — -.  b, axis of rotation o.
_       -_ iX'  -'-......:'- rEach arm, a' a' and
a b a', consists of a
parabolic disk of uniform thickness; b and b' being the vertices of the exterior bounding curves, aGa' their common
chord, and ob, ob' their axes. The disk is terminated on the
exterior by a flanch B of uniform breadth and thickness. A
rib C, either of uniform breadth and thickness, or else, of
uniform thickness, and tapering in breadth from the centre o
to the ends b, 6, projects from each face of the plane disk
along the axis b b'. The beam is perforated at the centre,
near the two extremities and at intermediate points, to
receive the short shafts, or centres around which rotation
takes place. Around each of these. perforations,, projections,
or bosses D' D, ID, &c,, are cast, to add strength and give. 4
more secure fastening for the shafts.
The beam being symmetrical with respect to a a', it will
be only necessary to calculate the moments of inertia of the
component parts: of each arm with respect to the axis o and
take double their sum for the total moment of inertia of the
beam. These component parts are —ist, the paralolic flanah;
2nd, the parabolic disk of uniform thickness enclosed by the
flanch; 3d, the rib on each side of the disk, running along
the central line b6'; 4th, the projections, or bosses D' &c.,
around the centres.
The moment of inertia of the flanch will be calculated by
(a') as its thickness is small compared with the other linear
dimensions.  That of the disk will be calculated by (b).
That of the rib by (d'). Those of the projections may be
obtained within a sufficient degree of approximation by




602              EDITORIAL APPENDIX.
taking the product of their volumes and the squares of their
respective distances from the axis o.
The sum of these quantities being taken it must be multiplied by - as in the preceding cases; v being the weight of
the unit of volume of the material.
NOTE (s).
The increase of tension due to rigidity and which ig exT)+E. P 
pressed by  +E   P2 may be placed under the following
R
form,
cm. a+n. b. Pa  C (a+b. P,)
R                 R
by writing cm. a for D, and cm. b for E, in which c represents the circumference of the rope, and m the power to
which c is raised.
The increase of tension of any other rope whose circumference is c, bent over the same pulley and subjected to the
same tension P, is, in like manner, expressed by
Cm (a+bP,)
R
Now representing by T and T1 the two values above for the
respective increase of tension for c and cl there obtains, by
dividing the one by the other,
C~m   //~\m    T             /C~\M
scm     ) -.C) - T  hence T      T;
which expresses the rule given above for using the tables in
calculating the increase of rigidity due to a cord whose circumference is different from those in the tables.
NOTE (t).
As one of the chief ends of every machine designed for
industrial purposes is, under certain restrictions as to the




EDITORIAL APPENDIX.            603
quality, to yield the greatest amount of its products for the
motive power consumed, it becomes a subject of prime
importance to see clearly in what way the work yielded by
the motive power to the receiver, at its applied point, is
diminished by the various prejudicial resistances, in its
transmission through the material elements of the machine
to the operator, or tool by which the products in question
are formed.
The most convenient method for doing this will be to
place (equation 112, Art. 145) which expresses the relation
between the work zU, of the motive power at the applied
point and that zU2 the work of the operator at the working
point, with the portion zU+ -1 zw (v,9-v,1) which represents the work consumed by the prejudicial resistances and
the inertia, under a form such that the work of each prejudicial resistance shall be separately exhibited, for the purpose of deducing, from this new form of the equation, the
influence which each of these has in diminishing the work
yielded at the applied point and transmitted to the operator.
To effect this change of form in (equation 112) designate by
P1 the motive power, and S, the path passed over by its
point of application in its line of direction between any two
intervals of time, during which P1 may be regarded as variable both in intensity and direction; P2 and S2 the resistance
and corresponding path at the working point; R the various
prejudicial resistances which, like friction, the stiffness of
cordage, &c., act with a constant intensity, or are proportional to P,, and S their path; wl the weight of the parts the
centre of gravity of which has changed its level during the
period considered, and H its path; and -1  (v2 -v12)= -i
2g
(v22,-v1,) the half of the difference between the living forces
or the accumulated work of the material elements in motion,
of which m = -  is the mass, during the same period, in
which the velocity has changed from vs to v.
Now for an elementary period dt of time, during which
the forces Pi &c., may be regarded as constant, and their
points of application to have described the elementary paths
dS, &c., in their lines of direction, (equation 112) will take
the form,
(mvdv=_PdS, —RdS —zPdS,~i wdh,.... (A),




604              EDITORIAL APPENDIX.
in which the 1st member of the equatio expresses. the increment of the Uvingg force, or the elementary, ecuinulated
work for the interval dt at any instant when the velocity of
the- mass m is: v; and the 2nd member the corresponding
algebraic, sum of the elementary work of P,, IR &c, This
e qiation being integrated between the limits t. and t: in
which. changes from'v to U, there obtains,
~aim (^- v12)'=  f 1dYI-^s f RdS'- fP2d.s,
z      Adh......  (B).
This equation (B) is the same as (equation 112). The
symbol; designating the aggregate of the work of the
various forces of the, same kind; and that as f PdS &c.
the work of each force as P;, supposing it to be either constant or variable. In either case whenever P1 &c., can be
expressed in terms of S, the value offPAdS1 can be found
by one of the methods in (Notes I and m); and supposing Pi
&ec., to represent their mean values, and S1 &c., the paths
described in their true directions during the interval considered, equation (B) may be written under the following form
for the convenience of discussion,
m12-v=2)=pS       RS-P~2S2WH.... (C).
In this. last equation 2, df =WII (Art. 60) represents
the work of the total weight of the parts whose centre of
gravity has changed its level during the interval considered,
and it takes the double sign +, as the path H may be
described either in the same, or a contrary direction to that
in which W always acts.
Before proceeding to discuss the terms of (equation C),
it may be well to remark that the term -RS does not take
into account the work expended by P1 in overcoming the
molecular forces brought into play- by the deflection, torsion,
extension,. &C., of the parts of the machine; for, owing to
the rigidity of these parts, this forms but a very small fractional part of the total work of the exterior forces whilst the
machine operates continuously for some time; as, during




EDITORIAL APPENDIX.               605
this time, the tension of the parts, or the molecular resist
ees remain sensibly the same, and the molecular displace
ients are for the most part inappreciable, or else -very small
Oempared with the paths described by the points of application of the other forces.
This remark, however, does not -apply to the expenditure
of work by the motive power where the operation of the
machine requires that some of the parts in motion shall be
brought into contact with others Which are either at rest, or
moving with a slower velocity so as to produce a shock.
In this case there may be a very appreciable amount of
living force, or accumulated work destroyed by the shock,
owing to the constitution of the material of which the parts
are composed where the shock takes place; and, if the shocks
are frequent during the interval considered, and in which
the other forces continue to act, the accumulated work
destroyed during this interval may form a large portion of
the work expended, or to be supplied by the motive power.
h'T calculating this amount of accumulated work destroyed,
we admit what is in fact true in such machines, that the
interval in which the shock takes place is infinitely small
compared with the interval in which the other forces act
continuously, and therefore,-in estimating the accumulated
work destroyed in each shock, that we can leave out of
account the work of the other forces during this infinitely
small interval. In this way, considering also that the parts
where the shock takes place are usually formed of materials,which undergo an almost inappreciable change of form from
the shock, and that therefore the mechanical combinations:of'the machine are sensibly the same after the shock as
before it, we readily see that, to obtain the total expenditure
of work by the motive power, for any finite interval, we must
ealiculate that which is consumed by all the other resistances,ainting this interval, and add to this that destroyed by the
shocks during the same interval, the latter being calculated
irirespective of the work of the other forces during the short
duration in which each shock occurs.
We thus see that, except in some cases where the great
velocity of the parts in motion may give rise to an appreciable expenditure of work caused by the resistance of the
medium in which these parts may be moving, as the air, &c.,
-the:forces which act -upon any machine in motion are the
miotie power; the resistances, ~such as friction, sftffness of
cordage, &c., which act either with a eonstant intensity
during the motion, dr are proportional to,the motive power;




606               EDITORIAL APPENDIX.
the weight of the parts whose centres of gravity do not remain
on the same level during this interval; the useful resistance
arising from the mechanical functions the machine is designed
for; and the forces of inertia which either give rise to accumulated work, or the reverse, as the velocity increases, or
decreases during the interval considered.
Resuming equation (C) we obtain, by transposition,
PS =P~S, -RS~WH-i mv2'+i -nv2. 
That is the useful work, or that yielded at the working point
and which it is generally the object of the machine to make
as great as possible consistently with the quality of the
required products, will be the greater as the terms in the
second member of the equation affected with the negative
sign are the smaller.
Taking the term — RS, it is apparent that all that can be
done is to endeavor in the case of each machine to give
such forms, dimensions and velocities to those parts where
these resistances are developed as will make it the least
possible.
With respect to WH it will entirely disappear from the
equation when H=o; in which case the centre of gravity of
the entire system will remain at the same level; or else
only that portion of this term will disappear which belongs
to those parts of the machine whose centres of gravity either
remain at rest, as in the case of wheels exactly centered, endless bands and chains, &c.; or in the case of those pieces
which receive a motion simply in a horizontal direction.
This term will also disappear in whole, or in part, in those
cases where the centre of gravity ascends and descends
exactly the same vertical distance in the interval corresponding to the work PS,; for during the ascent, as the direction
of the path H is opposite to that of the weight W, the work
consumed will be -WH, whereas, in the descent, it will
restore the same amount or +WH, and the sum of the two
will therefore be 0. This takes places in the parts of many
machines, for example in crank arms, and in wheels which
are not accurately centered; in both of which cases the
centre of gravity ascends and descends the same distance
vertically in the interval corresponding to each revolution
of these parts whilst in motion; also in those parts of a machine, like the saw and its frame in the saw mill, which rise
and fall alternately the same distance.
In all of these cases then the useful work P,S, will not be




EDITORIAL APPENDIX.             607
affected by the work due to the weight of the parts in
question.
It may be well to observe that the preceding remarks refer
only to the direct influence of the weight of the parts on the
amount of useful work; but whilst directly it may produce
no effect however great its amount, the weight, indirectly,
may cause a considerable diminution of this work, by
increasing the passive resistances and thus the term RS.
The same holds with regard to the accumulated work, represented by the term Jmv22, from which a considerable diminution may be made in PS, if this accumulated work cannot
be converted into useful work, and thus be made to form a
portion of PS,, when the action of the motive power is either
withdrawn, or ceases, by variations in its intensity, to yield
an amount of work which shall suffice for the work consumed
by the resistances.
These last remarks naturally lead us to the consideration
of the two terms _wmlv2, and — jwmv, or half the living forces,
or accumulated work at the commencement and end of the
interval considered. As the machine necessarily starts from
a state of rest under the action of the motive power P1, it
follows that mv2,', the accumulated work due to this action
tends to increase P2,S, whilst that — mv,22 is so much accumulated in the moving parts by which P2S2 is lessened.
This diminution of P2S, is but inconsiderable in comparison
with the total useful work when the interval in question, and
during which the machine operates without intermission, is
great; also in cases where the velocity attained by the parts
in motion is inconsiderable, as for example in machines employed for raising heavy weights, in which imv22 will in
most cases be but a small fraction of the useful work which
is the product of the weight raised and the vertical height it
passes through. In this last example we also see the inconveniences which would result from allowing bodies raised by
machinery to acquire any considerable amount of velocity;
or to quit the machine with any acquired velocity, as, in
this case, the accumulated work generally would be entirely
lost so far as the required useful effect is concerned.
Except in the case where the accumulated work jmv,'
can be usefully employed in continuing the motion of the
machine and gradually bringing it to a state of rest when the
motive power P, has either ceased to act, or has so far
decreased in intensity as to be incapable of overcoming the
resistances, whatever tends to any augmentation of living
force should be avoided, for the term which represents this




608              [EDITOtIaML AIPPENDIX.
being eomV'osed of two factors the one representlng the mass
of the parts in motion and the other the square of its velo4city, it is evident that ithe prejudicial reisgtances such as
-i;ction on the'one hand -and the resistance of the air on the
otherw will increase as either of these factors is increased, and
thus a very appreciable amount of this accumulated work
may be consumed:in useless work caused by the very increase in question. If, moreover,the machine from the nature
of its operations is one that requires to be brought suddenly
to a state of rest, any considerable amount of accumulated
work might so increase the effects of shocks at the points of
articulation as to -endanger the safety of the parts.
The foregoing remarks apply only to those parts of a machine where the direction of motion remains the same whilst
the machine is in operation. Where -any of the parts have
a reciprocating motion, in which case whilst the part is
moving in one direction the velocity increases from 0 up to
a certain limit and then decreases until it again becomes 0
at the moment when the change in the direction of motion
takes place, and so on for each period of change, it will be
readily seen that where the velocity varies by insensible
degrees, the accumulated work of these parts for each period
of change will be 0 and will therefore have no influence on
the amount P2S, of useful work.
The avoidance of abrupt changes of velocity in any of the
parts of a machine is of great importance. The mechanism
therefore?should, as a general rule, be so contrived that there
-shall be the least play possible at the articulations of the
various parts, -and'that the articulations shall receive such
forms as to procure a continuous motion.  In cases also
-where any of the parts have a reciprocating motion such
mechanical contrivances should be used as will cause the
variations of velocity imn these parts, within the range of
their paths, to take place in a very gradual manner; such.fr examples as what obtains itn the cranks and eccentrics
which are mostly employed to convert the continuous circular motion of one part into reeiprocating motion in another,
or the reverse.
There are some industrial operations however which are
performed -by shocks, as in stamping machines, trip hammers, &c., and in these cases the useful work is due to the
work developed by the motive power in raising the pestle
of the'stamping -machine, or the head of the trip hammer
through a certain vertical distance from which it again falls
uon the tmatter to be acted on, having accuired in ia




EDITORIAL APPENDIX.             609
descent an amount of living force, or accumulated work due
to the height through which it has been raised. In such
cases it is to be noted that, independently of the work due
to the motive power consumed by the resistances whilst the
hammer or pestle is kept in motion by the other parts of the
mechanism, and which is so much uselessly consumed so far
as the useful work is concerned, there will be a portion of
the accumulated work in the pestle, or hammer also uselessly
consumed, arising from the want of perfect rigidity and
elasticity in the material of which these two pieces are
usually composed. Besides this, both the pestle and matter
acted on may and generally do have relative velocities after
the shock between them, which as they are foreign to the
purpose of the operation, will also represent an amount of
accumulated work lost to the useful work. From this we
may infer that, as a general rile, other industrial modes of
operating a change of form in matter will be preferable to
those by shocks, whenever they can be employed; and that
such modes are moreover advantageous, as they avoid those
jars to the entire mechanism which accompany abrupt
changes in the velocity of any of the parts, and which, by
loosening the articulations more and more, increase the evil,
and ultimately render the machine unfit for service.
Having examined the influence of all the various hurtful
resistances brought into action in the motion of machines
upon the work PS, expended by the motive power, and
pointed out generally how the consumption of the work may
be lessened, and the useful work to the same extent increased,
we readily infer that like observations are applicable to the
term P2S2 the work of the resistance at the working point.
As the prime object in all industrial operations performed by
machinery is to produce the greatest result of a certain kind
for the amount of work expended by the motive power, it
will be necessary to this end that the velocity, the form, &c.,
of the operator, or tool by which the result sought is to be
obtained, should be such as will not cause any useless expen-~
diture of work. On this point experiment has shown that
for certain operators there is a certain velocity of motion
by which the result produced will be the most advantageous
both as to the quality and quantity.
With respect to the work of the motive power itself represented by the product PS, it admits of a maximum value;
for when the receiver to which P1 is applied is at rest, P1 will
act with its greatest intensity, but the velocity then being 0
the product PS, will also be 0; but as the velocity increases
39




610              EDITORIAL APPENDIX.
after the receiver begins to move the intensity of the action
of P, upon it decreases, until finally the velocity of the
applied point may receive such a value V that P1 will become
0, and the product PIS, in this case will then also be 0. As
the work PSS, thus becomes 0 in these two states of the velocity, it is evident that there is a certain value of the velocity which will make PS, a maximum. To attain this
maximum the mode of action of the motive power selected
on each form of receiver to which it is applicable will require
to be studied, and such an arrangement of its mechanism
adopted as will prevent any decompositions of the motive
power that would tend in any manner to increase the hurtful resistances and thus diminish the useful work.
It will be very easy to show that the laws of motion of all
machines, that is the relations between the times, spaces and
velocities of the motion of any one of the moving parts are
implicitly contained in the general equation of living forces
as applied to machines which has just been discussed.
Resuming (equation B) with this view, and representing by
dmn any elementary mass in motion whose velocity is v, at
any instant when it has described the path, or space s, if we
take any other elementary mass dm in a given position and
denote by u its velocity at the same instant, we shall have
vU=u2 (ps), and v,=u, ((sp); in which ps is a purely geometrical function, since, from the connection of the parts of a
machine, in which any motion given to one part is transmitted in an invariable manner to the other, the space passed
over by any one point can always be expressed in terms of
that passed over by any other assumed at pleasure.
From the relations v2=u% (ps), and v dt=ds, we obtain
u2(s)=-v2  and u%2 du2 (q8s)2  dv2 - - ds.
Substituting these values of vII and v2 dv2 in (equations B and
A), and letting m still represent the sum of the elementary
masses as dn, there obtain the two equations
2U m(8)2-   2 Um(Pi                 dS2 2RdS2    f2ds2~IS   wdh. (B')




EDITORIAL APPENDIX.             611
u2duYs(pS)2=m(s)    s s  = d8-P1d S-s RdzP,dS,2 ~zwd. (A'),
the first showing the relations between any two states of the
velocities ut and u1 for any definite interval, and the second
for the infinitely small interval dt. Now as the relations
between the quantities dS,, dS2, &c., or the elementary
paths described by the points of application of PP PP,, &c.,
and the elementary space ds, from the connection of the
parts of the machine, can be expressed in functions of s and
of the constants that determine the relative magnitudes and
positions of those parts; and as, moreover, P1, P2, &c., are
either constant, or vary according to certain laws by which
they are given in functions of the paths S1, S, &c., we see
that all the relations in question are implicitly contained in
the two preceding equations.
Let us examine the kinds of motion of which a machine is
susceptible and the conditions attendant upon them. We
observe, in the first place, supposing the machine to start
from a state of rest, that the elementary work P1dS, of the
motive power must be greater than that of the resistances
combined, or PdS — RdS-&c. >0, so long as the velocity
is on the increase. The living force is thus increased at
each instant by a quantity d (mv2)=2mvdv, or by an amount
which is equal to twice the elementary work of the motive
power and resistances combined; and this increase will go
on so long as the elementary work of the motive power is
greater than that of the resistances. But, from the very
nature of the question, this increase cannot go on indefinitely,
for the point of application of the motive power would in the
end acquire a velocity so great that P1 would exert no effort
on the receiver, whereas the resistances still act as at the
commencement, and some of them even increase in intensity
with the velocity. The living force therefore will, at some
period of the motion, attain a limit beyond which it will not
increase, a fact which the operation of all known machines
confirms, and, having thus reached this state, it must either
continue the same during the remainder of the time that the
machine continues in motion, or else it must commence to
decrease until the velocity attains some inferior limit from
which it will again commence to increase, and so on for each
successive period of motion during which the action of the
forces remains the same.




612               EDrORIAL APPENDIX.
Supposing the machine to continue its motion with the velocity it has attained at this maximum state of the living force,
we shall then have'd(mvu2)=PdS,-RdS-PdS, ~WdlH=O;
and
mv22-mv12=2 (P,S,-RS -P2S2:WH)=O;
inasmuch as the motion being now uniform the difference
between the living forces corresponding to any finite interval of time is 0. Considering the manner in which the parts
of machines are combined to transmit motion from point to
point, we infer that this condition with respect to the
increase of living force, and which constitutes uniform
motion, can only obtain when the velocities of all the different parts bear a constant ratio to each other. Representing
by v', V", v'", &c., these velocities which are respectively
equal to —, d-, -d-, &c., we see that the ratios of ds',
equal to — h t-, - dt'1 dt' &                     G7
sds", s"', &c., will also be constant when those of v', V", &c.,
are so; that is, this constancy of the ratio of the effective velocities and of the quantities ds', ds", &c., must subsist together
for all positions of the parts of machines to which they refer;
but as the latter, which are the virtual velocities, or elementary paths described, depend entirely on the geometrical
laws that govern the motion of the parts, a little consideration
of the various mechanical combinations by which motion is
transmitted will show that, in order that their ratios shall
respectively remain constant, no pieces having a reciprocating motion can enter into the composition of the machine,
as the velocities of such pieces evidently cannot be made to
bear a constant ratio to the others. This condition it will be
seen refers exclusively to the mechanism of the machine, or
the geometrical conditions by which the parts are connected,
and has nothing to do with the action of the forces themselves.
But when the condition of uniform motion is satisfied
there obtains also
P1dS1- RdS —P2S2~WH=O;
that is, according to the principle of virtual velocities, an
equilibrium obtains between the forces which act on the
machine irrespective of the inertia of the parts. As a general rule this condition requires that not only must the forces




EDITORIAL APPENDIX.             613
P1, i, &c., be constant both in intensity and direction and
act continuously, but that the term WdSI must be separately equal to 0, or the centre of gravity of each part must
preserve the same level during the motion; for were this
not so any piece whose weight is w would evidently impress
an elementary work represented by ~wldh which would be
variable in the different positions of the mechanism; unless
w, having itself a uniform velocity, formed, as might be the
case, a part of the motive power P,, or of the useful resistance P,.
It thus appears that to obtain uniform motion not only
must the mechanism used for transmitting the motion contain no reciprocating pieces, and therefore consist solely
of rotating parts, as wheels, &c., and parts moving continuously in the same direction, as endless bands, and chains, &c.;
but that the centres of gravity of these pieces shall remain
at the same level during the motion, which will require that
the wheels and other rotating pieces shall be accurately centered so as to turn truly about their axes.
The difficulty of obtaining a strictly uniform motion in
machines is thus apparent, for it involves conditions in themselves practically unattainable, that is, applied forces acting
continuously and with a constant intensity and direction, and
that the ratio of the virtual velocities of the different parts
should be constant and independent of the positions of the
mechanism, a condition which requires that the terms (ps)
and Idm((ps)' in the preceding equations shall also be con.
stant for all of these positions. But even were these condi.
tions satisfied, it can be shown that rigorously speaking a
machine starting from a state of rest will attain a uniform
velocity only in a time infinitely great. This will appear
from geometrical considerations of a very simple character,
or from the form taken by equation. By the first method,
let OT, OY be two co-ordinate?Tn~~ r           axes, along the one set off the.v —-  -abscissas Ot', Ot", &c., to re-,V' ~ Ipresent the times elapsed from
/                the commencement of the moI/I  i                tion, and the ordinates t'v', " v"
o   t'"        T         &c., the corresponding velocities, the curve Ov'v", &c., will
give the relation between the times and the velocities. Now,
from the circumstances of the motion, the increments of the
velocities will continually decrease, and the curve, from the
law of continuity, will approach more nearly to a right line




61               EDITORIAL APPENDIX.
as the time increases; having for its assymptote a right line
parallel to OT, drawn at a distance Ov from it, which is the
limit the velocity attains when the motion becomes uniform.
We moreover see from the -form the curve may assume that
this limit will be approached more or less rapidly.
From  (equation B'), representing by c the quantity
Sm(cps)2, we obtain
d2s    dv       S,       dS    dS,
-toc dt        1 ds     ds    2 ds
Now, from the preceding discussion, the forces being supposed to act continuously, and with a constant intensity and
direction, and the quantities,  - bein  onstant the
ds' ds
function expressed by the second member of this equation
has its greatest value when v,=0, or when the machine is
about to move, and that after motion begins it decreases
more or less rapidly as the velocity increases, until it becomes 0 for a certain finite value of the velocity. Hence it
follows that the function must be of the following, or some
equivalent form,
in which k is essentially positive and a function of v2 and
certain constants, and V is the limit of the velocity in question. We shall therefore obtain from (equation B'), by substituting this function for the second member,
v
e =~ -- k (V —v )"; and t = ejd(v _;
dtv   kY-()'; andt=f       cdv 2
The second member of this last equation, when integrated
between the limits v2=0, and v  Y, must contain, according
to the known rules applicable to it, at least one term of the
form of -a log. (V-v2) if the exponent n is odd; or
— a(V-2)-"+ 1, if n is even; either of which functions will
become infinite for Y-v -=0, or when 2 attains its limit.
From the conditions requisite to attain uniformity of motion in a machine, the advantages attendant upon it, so far
as it affects the mechanism are self-apparent; not only will
there be none of that jarring which attends abrupt transitions in the velocity, but, from the manner in which the




EDITORIAL APPENDIX.             615
forces act, the strains on all the parts will be equable, and
the respective form and strength of each can thus be regulated in accordance with the strain to be brought upon it,
thus reducing the bulk and weight of each to what is strictly
requisite for the safety of the machine. But advantages not
less important than these result from the use of mechanism
susceptible of uniform motion, owing to the fact that for
each receiver and operator there is a velocity for the applied
and working points with which the functions of the machine
are best performed as respects the products; and these
respective velocities can be readily secured in uniform
motion by a suitable arrangement of the mechanism intermediate between these two pieces.
The advantages resulting from uniform motion in machines
has led to the abandonment of mechanism that necessarily
causes irregularity of motion, in many processes where the
character of the operation admits of its being done; and
where, from the manner in which the motive power acts on
the receiver and is transmitted to the operator, parts with a
reciprocating motion have to be introduced, every possible
care is taken to so regulate the action of these parts and to
confine the working velocity within the narrowest limits that
the character of the operation may seem to demand. Many
ingenious contrivances have been resorted to for this purpose, but as they belong to the descriptive part of mechanism
rather than to the object of this discussion, and, to be understood, would require diagrams and explanations beyond the
limits of this work, they can only be here alluded to. There
is one however of general application, the fty wheel, the
general theory and application of which to one of the simplest cases of irregularity are given in (Arts. 75, 76, 265, &c.)
The functions of this piece are to confine the change of velocity, arising from irregularities caused either by the mechanism, or the mode of action of the motive power within certain
limits; absorbing, by the resistance offered by its inertia,
or accumulating work whilst the motion is accelerated, and
the work of the motive power is therefore greater than that
of the other resistances, and then yielding it when the reverse
obtains; thus performing in machinery like functions to
those of regulating reservoirs in the distribution of water.
It should however not be lost sight of that whatever resources
the fly wheel may offer in this respect they are accompanied
with drawbacks, inasmuch as the weight of the wheel, its
bulk and the great velocity with which it is frequently
required to revolve, add considerably to the prejudicial




616               EDITORIAL APPENDIX.
resistances, as friction and the resistance of the air, and thus
cause a useless consumption of a portion of the work of the
motive power. Whenever therefore, by a proper adjustment
of the motive power and the resistances, and a suitable
arrangement of the mechanism, a sufficient degree of regularity can be attained for the character of the operation, the
use of a fly wheel would be injudicious. In cases also where,
from the functions of the machine, its velocity is at times
rapidly diminished, or sudden stoppages are requisite, the
fly wheel might endanger the safety of the machine, or be
liable itself to rupture, it should either be left out, or else
the mass of the material should be concentrated as near as
practicable around the axis of rotation; thus supplying the
requisite energy of the fly wheel by an augmentation of its
mass. In all other cases the matter should be thrown as far
from the axis as safety will permit, as the same end will be
attained with less augmentation of the prejudicial resistances.
From this general discussion some idea may be gathered
of the relations between the work of the power and that of
the resistances in machines, and of the means by which the
latter may be so reduced as to secure the greatest amount of
the former being converted into useful work. It must not
however be concealed that the problem, as a practical one,
presents considerable difficulty, and requires, for its satisfactory solution, a knowledge of the various operators and
receivers of power, as to their forms and the best modes of
their action.  This knowledge it is hardly necessary to
observe must, for the most part, be the result of experiment;
theory serving to point out the best roads for the experimenter to follow. Both of these have shown that the work
of the motive power consumed by the resistances, caused by
the parts through which motion is communicated from the
receiver to the operator, is but a small fractional part of the
total work uselessly consumed, whenever the mechanism has
been arranged with proper attention to the functions required
of it; but that the principal loss takes place at the receiver
and operator, and this is owing to the difficulty of so arranging the receiver that the motive power shall expend upon it
all its work without loss from any cause; and in like manner
of causing the operator to act in the most advantageous way
upon the resistance opposed to it. Some of the general conditions to which these two pieces must be subjected, as to
uniformity and continuity of action of the motive power and
the resistances, and the avoidance of jarring and shocks have




EDITORIAL APPENDIX.            617
been pointed out. as well as the fact that to each corresponds
a certain velocity by which the greatest amount of useful
effect will be attained.
This discussion will make apparent that, comparatively
speaking, but a small amount of the work due to the motive
power is expended on the useful resistance, or the matter to
be operated on. In some of the best contrived receivers, as
the water wheel, for example, where the motive power can
be made to act with the greatest regularity, and the receiver
be brought to as near an approach to uniformity of motion
as attainable, the quantity of work it is capable of yielding
seldom exceeds eight tenths of that due to what the water
expends upon it, under the most careful arrangement of the
wheel and the velocity of its motion.
NOTE (u).
As an example under this head (Art. 149) equation (115),
and an illustration of the circumstances attending the attainment of uniformity of motion Note (t) in machines; suppose
the axle A carrying two arms B, B, to the
I n    extremities of which two thin rectangular
disks C, C, are attached, their planes passing through the axis of rotation, to be put
<   in motion by the descent of a weight P, atA.i  o   tached to a cord wound round the axle.
In this case the resistances to the moving
force during the acceleration will be
that of the air acting against the disks and
the two arms, the inertia of the parts in
motion, and the friction on the gudgeons
of the axle.
Represent by A the sum of the areas of
the two disks, a the distance of their centres from the axis,
dm an elementary mass of the machine at the distance r
from the axis, w the angular velocity of the system, a, the
radius of the axle measured to the axis of the cord, p the
radius of the gudgeon, p the limiting angle of resistance,
1, the total length of the cord, I the length of the part
unwound, w the weight of the unit in length of the cord, W
the total weight of the machine excepting P,.
From experiment we have for the resistance of the air to
the motion of the two discs cAv2=cA2a2, in which v=wca




618               EDITORIAL APPENDIX.
expresses the velocity of the centre of the disk and c a constant determined by experiment. The resistance offered by
the inertia of dm during the acceleration of the motion is
represented (Art. 95) equations (72) (73) by dmr d  in
which d) is the acceleration of the angular velocity in the
element of time dt, the resistances offered by the inertia of
the weight P, and that of the pendant portion of the cord
P~ +WI dw
represented by wl are, in like manner, expressed   al,
the total pressure upon the gudgeons will evidently be ex-P +wl    dw ~
pressed by P1+W — P ~      da,  since, during the acceleration of the motion, the resistance of the inertia of the
weights Pi and wl act in an opposite direction to these
weights.
In the state bordering upon motion at each instant there
obtains
(Piw +w/) = cAw        d o'd+  + dmr   a  +
cut    (     dt
(  ~+W' P+1w — dop sin..Representing by n2 the coefficient of o2, by m2 that of dd
dt
and by q2 the algebraic sum of the other terms, there obtains
+mdo -q 2=O.. dt= =m2-.
t=f  2 2=ql-og. s q+ )        (        )
0 =  __ 2  ~,)2   0g. ~2.:.. (a -    m n
2 2q  -n)2n(7              +i;1
From this last equation we se that w approaches rapidly the
limit - which it only attains when t= oo. As this limit corn
responds to that in which the motion would become uni



EDITORIAL APPENDIX.            619
form, it might have been deduced directly from the first of
these equations; for when o becomes constant  0,
dt.. *n2L='.. 2 -.
NOTE (v).
Manner of estimating the amount of work consumed by the
trip hammer.
The trip hammer is used in forging heavy iron work,
motion being given to it for this purpose by teeth, termed
cams, firmly i-xed in an axle A, termed the cam shaft,
cams, firmly fixed in an axle A, termed the cam shaft,
around which they are arranged at equal intervals apart.
The tail of the hammer is furnished with an iron band, the
upper surface of which receives a suitable form to work
truly with the surface of the cam whilst the two remain in
contact during the ascent of the head of the hammer, on the
same principle as the teeth are fashioned in other cases.
The interval between the cams is so calculated that each
cam shall take the band at rest at the point t on the horizontal line 0CA  joining the centres of rotation of the cam
shaft and hammer.
To estimate the work consumed in the play of this machine, it must be observed that it consists of three distinct
parts; the first is that consumed by the impact or shock;
the second that due to the period after the shock, in which
the cam and tail of the hammer remain in contact; the third
that consumed by the cam shaft in the interval between the
separation of the cam and hammer and the moment when
the succeeding cam takes the hammer.
Denote by R, the radius of the primitive circle 02t of the
cams; by 2 the angular velocity of the cam shaft at any
period of the shock; by p2 the radius of the gudgeon on




620              EDITORIAL APPENDIX.
which the shaft revolves; by p9 the limiting angle of resistance for the surfaces of the gudgeon and its bed; by m an
elementary mass of the shaft; by r the distance of m from.,; by R1= Ct, o,1 p1, (p ml, and rl the corresponding quantities for the hammer.
Now if we represent by P the mutual pressure between
the surfaces of the cam and band at any period of the impact,
there must be an equilibrium at each instant between P and
the forces of inertia and the passive resistances developed
in the play of the machine. Considering the equilibrium
around the axis of rotation C, of the hammer in the first
place, we have for the velocity of any element rn,, at any
instant, r,o,, and for the increment of velocity impressed
upon it by the cam rdnrc; the force of inertia therefore developed by this increment is expressed by
do,
dt'
and its moment with respect to the axis C0 is
and the sum of the moments of all the forces of inertia is
(Arts. 95, 106)
r d-1     dlMR,2
dt    dt
To obtain the friction on the trunnions of the hammer due
to P and the resultant of the forces of inertia zra,  we
dt
have for the resultant of the latter (Art. 108) equation (82)
dIMG;
dt
in which M represents the mass of the hammer, its handle,
&c., and G the distance of its centre of gravity from C, the
axis of rotation. Now, decomposing this resultant into two
components perpendicular and parallel to the line CC0, representing by a the angle between this line and the one C1G
through the centre of gravity of the hammer, &c., we have
for the perpendicular component




EDITORIAL APPENDIX.            621
dt MG cos. a
dt
and for the parallel one
-- MG sin. a.
The total pressure on the trunnions, from P and the forces
of inertia, will therefore be
/(P+  dMG cos. ) + (  M O sin.
As however, in most cases of practice, the angle a is either 0,
or very small, the value of the quantity under the radical
may be taken without sensible error
do
P + diMG.
The equation of equilibrium about the axis CO is therefore
PR = dt M1R, + (P + d MG) p, sin.,1.
- dp&, MR, + MG p sin. p.' —  dt  E1,-psin.p,... (A).
Now with respect to the cam shaft we have, to express the
sum of the moments of the forces of inertia with respect to
the axis C,,
diw, 2mr= -  M _ 2
dt     -di= M2R'2
As the pressure on the trunnions of this shaft is due to the
force P alone, the moment of the friction on them will be
expressed by P p sin. %p.
The equation of equilibrium of all the forces with respect
to C, will therefore be
dt - 2R, =PR, + PP2 sin. 9p.... (B).
Eliminating P between equations (A) and (B) there obtains




622              EDITORIAL APPENDIX.
dt   2 2 - 1R, -psin.q,                 dt
d% MR    = R+ p  sin. ( MRl +MG,     ^ -d  * * -n * (C).
the coefficient of  Wl can be written as follows,
dt
E, + P2 sin. c.
R - p sin. 1 (MIR1 + MG p, sin. -)=il- p, s2,
1 + P2 Sin. g
RP   (1. p+, sin., MG \ MR,=l KMIRR,
1    sin., 9MR,                         2
R
placing K for the coefficient of MRYR. Making these substitutions in equation (C) there obtains
do                  dw,
-Mdt M2R —KMRR      d'
G)2              G)=j =0
Q- 2Q         KM   221 —in which  represents thegreatest and  the least angular
in which Q2 represents the greatest, and (02 the least angnlar
velocity of the cam shaft; and w- 0,  =-  R 2, the angu1,
lar velocities of the hammer; since before the impact it is
at rest, and finally attains the same velocity as the cam has,
in which, from the circumstance of the mechanism, WIR-=
(02R2.
From the preceding equation there obtains
= 2     M2....(M
M2+ K1'M, 
Now, as a general rule, the quantities p. sin. 2p p sin. p and
E, E' R,
p, sin. %Mll\IG
p si. p,-MG are very small with respect to unity, and may
MIE1




EDITORIAL APPENDIX.             623
therefore be disregarded, and'the quantity K will differ but
very little from unity also. From this it will be seen that
(2 will differ the less from Q as M, is greater than M1. But,
as the mass of the cam shaft ordinarily very much exceeds
that of the hammer, we can assume, without liability to any
great error, that the mean angular velocity of the cam shafts
deduced from observing the number of revolutions made by
it in a given time, is sensibly the arithmetical mean of 2
and w2. Designating this mean by u, we have s2 Q-  + - o0
2.
From this relation and equation (D) there obtains
221i(M2+KM). and a       2Q1X2
2M+KM 2+             2M 2 2+ KMA'
From these two relations the living force destroyed by the
impact can be deduced as follows. Before the impact the
living force of the cam shaft was 2M2.R,; after the impact,
as the point of contact of the cam and band moved with the
same velocity, the living force of the whole machine is
22 M21R2 +  2 M,R,2 =  2R (M, + Mi).
The living force destroyed therefore is expressed by;2 M2-22R-  R2(M2 + iM);
or, substituting for &2 from equation (D), by
p212 { M    M2(M +M,) =
29         2:M2     (2KE-1)M2M, + K2M.
finally, substituting for. 2 and. 2 their values in Q,, there
obtains
2K —1 i.lIW
4M M, 2R 2 (2K- i)M +K2M,-41    2R2 12        M.
4M2M1 12  (2aM2+KM,)2       11 2- (2+ K     2)
It is now readily seen, from the form of this last expression
for the loss of living force by the impact, that, since K may




624              EDITORtAL APENDIX.
be assumed as sensibly equal to unity, the numerical value
of this expression will depend upon the ratio M  Taking
M-=M   the value of the expression becomes Qp2M R 2; and
for M=oo it becomes Q21MR22. Therefore between these
limits the difference is ~ only of the living force lost under
the supposition of M2=oo.
In the ordinary arrangement of this machine it rarely
occurs that M, is not less than vM2,. Assuming this as the
limit, and substituting in the preceding expression 10M for
M2, there obtains for the required loss of living force
0'97712nM1R2. It is therefore seen that, in all usual cases,
M2 may be assumed as infinite without causing any noticeable error in the result.
To estimate the accumulated work expended by the cam
shaft for each shock, Q, )2 and Q1 being the same as in the
preceding expression, this work is expressed by
411222M -R22K
— )}M --   2M,+2KM
As the cam shaft expends this amount of accumulated work
at each impact, a quantity of work equal to the half of this
must be yielded by the motive power at each impact, or
2Q22M\2MIR2 2K
2M2 + KM,
If therefore there are N cams on the shaft, and it makes n
revolutions in one minute, then the work consumed by the
number of shocks in one second will be expressed by
60N  2M2+KMM,.K
This then is the work consumed by the impact in one second
for the first period of the play of the machine; and it has
been calculated according to what was laid down in Note (t)
on the subject of shocks, by disregarding the work of the
other forces as inappreciable during the short interval of the
impact.
To estimate now the work expended during the second
period, or whilst the cam and band are in contact after the
shick, let CG, be any position of the line CG, during this




EDITORIAL APPENDIX.             625
period, making an angle GCG=-a with its position when
the hammer is at rest. Represent by P, the normal pressure
at the surface of contact of the cam and band which will
balance all the resistances developed in the motion of the
hammer, leaving out of consideration that of inertia, as the
change of velocity between the end of the impact and when
the cam disengages from the band is so small that the living
force due to this interval may be neglected in comparison
with the work of the other forces by W, the weight of the
hammer, its handle, &c.
When the line G-C is in the position G0C,, the line Clt will
oe in that Ct1 making the angle tCtl=a with its original
position. The force P, acting at t1 in this position and perpendicular to the line t,0,-since the surface of the band
produced passes through the axis C, the surface of the cam
being an epicycloid-has for its vertical and horizontal components P1 cos. a and P1 sin. a. The pressure on the trunnions of the hammer, which is the resultant of P1 and W,,
therefore will be expressed by
4/(W, + P, cos. a) + P,1 sin.2 a;
and since the first term of the radical is in all cases greater
than the second, the value of the radical itself may be
expressed by (NOTE B)
y(W1+Pl cos. <)+ SP, sin. a.
The equation of equilibrium between P1 and the other forces
will therefore be
PER, =W1G cos. (a + a) + y7(W, +P, cos. a) + /P, sin. a) p sin..
The moment of the friction at the point t, due to P1 with
respect to the point C0, in this case from the form of the cam
and band, being 0.
As the pressure P1 varies with the angle a, we can only
obtain its mean valie by first finding its quantity of work
for the angle a=cca described whilst the cam and band are in
contact.  Multiplying the last equation by da, and then
integrating between a=O and a=a, there obtains
fPIRdaW=W1G \ sin.( + a,)-sin. a} + 1yW  +yPm sin. aa=O          fP, cos. a + 3Pm} p, sin. p,;
40




626              EDITORIAL APPENDIX.
representing by Pm the mean value of P, or tne constant
force applied vertically at t, which multiplied by RoaC the
path described by the point of application, will give the
amount of work of the variable pressure P1 for the same
path; and introducing this mean value into the term of the
preceding equation that represents the moment of the friction on the trunnions, as this will not produce any sensible
error in the results.
Now observing that the quantity G sin. (a+a)-sin. oa is
the vertical height through which the centre of gravity of
the hammer, &c. is raised during the period in question, and
that PmRi,1 is the work of the mean force; calling this vertical height A, and substituting the work of the mean for
that of the variable force in the last equation; there obtains
PRa,=_, WI + {yWl, +7 P, sin., — PP cos. a, +
B/P,1 p, sin. p.
P. Wph-+-yW1M1p, sin. p,                  (Ep' E*   - c y sin. a, +  (1- cos.cca) pi, sin. p,
If we now multiply the second member of equation (E) by
REa, we shall obtain the approximate value of the work of
the variable force P1 during the period in question; or the
value of PmRa, as determined from equation (E).
To find now the work that the motive power must supply
to the cam shaft for this expenditure P,,Ra, due to the
motion given to the haTmmer during the period in question,
and also that arising from the resistances developed by the
motion of the cam shaft itself during this period, represent
by P, a force which, acting at a distance R. from the axis C,
of the cam shaft, will balance all the resistances around 02;
by W2 the weight of the cam shaft and its fixtures; by 0 any
angle described by the cam shaft during the period considered; and qp the limiting angle of resistance at the point
of contact of the cam and band.
The pressure on the trunnions of the cam shaft is evidently
expressed by
W2 +P-3-Pm;
and the equation that expresses the work of P, for the elemlentary angle dO is




EDITORIAL APPENDIX.                        627
PsRd0=PmRd0+Pm R +RR R+,a tan. p R2dO+
R2R,     2
(W +P3 P-)-Pi sin. 9 R2de.
Now representing by P. the mean value of P2, and substituting it for PS in the last term    of the second member of this
equation, which may be done without causing any sensible
error in the result;     observing, from     the conditions of the
mechanism that R20-=R-l; and integrating this equation
between the limits      =0 and 0=-       - R-M; there obtains, to
R,
express the total work of P, for the angle 0,,
R1ial
v    R2     p  R     p     R+RI       (RIa) 1tan.(p+
PAR61=fPShdO             Pm    a      m                    tan. p+ 2
0=0
(W2+P, -P)) p, sin. p, 0,.
* Omitting the work consumed by the friction of the axles in equation (251)
(Art. 220), that which is expended on the teeth in contact whilst the arc rIVP is
described is represented by the term of the equation
5^ ( 1 +~21) sin. 0 P2 a2).
Now if we suppose a2=r2, or that P2 acts at the point of contact and normal
to the surface, this term, modified to suit the supposition, becomes
ib (1-+' r2) sin.      r P2. r2 =  P2 (rlr  r22 tan. 0=
\ l       cos~           \   r  /
P2 (1+r2)     2   tan. r b.
\ rir2 /   2
Dividing this last expression by r2ai, there obtains,
P   r1 t2 r2. O tan. 0
\'r2  2
as the value of a mean or constant force which applied tangentially to the
circumference having the radius r2 will expend, whilst the point of application
describes the arc r'2b, the same quantity of work as that consumed by the friction of the teeth in contact whilst this arc is described. In this expression
the value of P2 is less than the true value.
The foregoing is the theorem of M. Poncelet referred to on page xii.
Author's Preface. The direct manner of deducing it is found on page 192
Navier. Resume des Leponss &c. Troisieme Partie. Paris, 1838.




628.            EDITORIAL APPENDIX.
R2t, + R, R:al       -P.)psin. q),
P (1 +    R, tan. + (W2-Pm)  sin. p
p2 E     2... (F).
R3- P2 sin. 2'
The work therefore that the motive power must supply to
the cam shaft during this period is found by multiplying the
second member of equation (F) by R30,,=R R  or the path
passed over by the point of application of the mean force PI
during this period.
Representing in like manner, by 60 the number of times
the hammer is raised per second, the quantity of work that
the motive power must supply for this expenditure will be
expressed by
Nl pR301 N=  n pR Ra i.. (2).
60         60      2,
During the last period, or whilst the hammer is down, the
motive power will only have to supply the expenditure of
work caused by the friction on the trunnions of the cam
shaft, arising from the weight of this shaft and its fixtures
and the power; any accumulation of work in this shaft
during this period being neglected as small in amount.
Representing by p=N the number of cams on the shaft, their
distance apart on the primitive circumference whose radius
is R is evidently — 2 and, as the arc described on this
circumference whilst the cam shaft and hammer are engaged
is R20,, that described whilst the hammer is down is  2
REa. Calling Pp the power which acting at the distance R,
will balance the friction arising from the weight W2 of the
cam shaft and fixtures and P2, the value of Pp will be found
according to the conditions stated as follows,
PPR= (W2 + Pp) P2 sin. %.
W2p, sin. (p
" *p  R3-p2 sin. p,'
The work of Pp is
PZR12~ZR2R )




EDI'ORIAL APPENDIX.             629
as the path passed over by its point of application is evidently the arc  (        j.
The work which the motive power must supply therefore
per second during this last period is expressed by
Nn PR    2 (p-r   ).... (3)
By taking the sum of the quantities expressed by the
formulas (1), (2), and (3) there obtains
Nn   22IMR          p       VR  2R -21,),
N  2l   Xt1a 1   p 2 o  p +     R 30a1 P\ 
60    2M12 +KM  -              -p- p      / 
to express the total work that the motive power must yield
to the cam shaft per second to supply the work consumed
by all the resistances.
That consumed by the useful resistances, which consist of
half the living force transmitted to the hammer and the
work consumed in raising the centre of gravity of the hammer, &c., through the vertical height h is represented by
iM'f,  =,   KMAR.) 3 Wh.
2        1   (2M2+KM 1)     1
From the preceding expressions, it is easy to deduce the
work which must be expended in producing a given depth
of indentation by the hammer upon the metal when brought
to a given state of heat. For this purpose, we observe that
to half the living force acquired by the hammer there corresponds a certain amount of work, estimated in terms of
the weight of the hammer and a certain height h, to which
its centre of gravity has been raised, and expressed by
-M   W It' 
2
the total work therefore expended by the hammer in
indenting the metal is expressed by W1h, + Wh; since, from
the state of the metal the molecules which are displaced by
the impact acquire velocities which are not appreciable from
their smallness; the resistances therefore offered by the
metal to indentation may be regarded as independent of the




630              EDITORIAL APPENDIX.
velocity and, fiom the laws of the penetration of solids into
different media, proportional simply to the area of the indentation. Representing then by a and b the sides of the area
of the indentation, supposed rectangular, at the surface of
the metal impinged on, d the depth of the indentation, and
C the constant ratio of the resistance and the area of the
indentation, the following relation obtains between the work
expended by the hammer in its fall and that offered by the
resistance of the metal
W, (A + hA) =Cabd;
an equation from which C may be determined by experiment in any particular case.
It will be readily seen that the preceding expressions will
be rendered applicable to the cases where the cam catches
the hammer on the same side of its axis of rotation as its
centre of gravity, by writing - dr] MG for + d- MG, and
dt           dt
moreover in this case when P- dOw MG- 0, there will be no
dt
shock on the trunnions (Arts. 108, 109), and there then
obtains, to find the point where the cam should catch the
hammer corresponding to this case,
R    SMG
* Morin, Suite des Nouvelles Experiences sur le Frottement, p. 67. Paris, 1835.




APPENDIX.
NOTE A.
THEOREM.-The definite integral ffxdx is the limit of the sums of the
a
valies severally assumed by the product fx. A x, as x is made to vary by
successive equal increments of Ax, from a to b, and as each such equal
increment is continually and infinitely diminished, and their number therefore continually and infinitely increased.
To prove this, let the general integral be represented by Fx; let us suppose that fx does not become infinite for any value of x between a and b,
and let any two such values be x and x + Ax; therefore, by Taylor's theorem, F (x + Ax) = Fa + Axfx + (Ax)' +iXM, where the exponent 1 + X is
given to the third term of the expansion instead of the exponent 2, that the
case may be included in which the second differential coefficient of Fx, dfx
is infinite, and in which the exponent of Ax in that term is therefore a
fraction less than 2.
Let the difference between a and b be divided into n equal parts; and
let each be represented by Ax, so that
b-a
n
Giving to x, then, the successive values a, a +  ax, a + 2 A x. a + (n-1)
Ax, and adding,
F(a + nAx)=Fa   Axtzrnf{a + (n-l) x} + (A )' + XM,,.'. Fb-Fa= Ax,"f{ a + (n —)Ax} + (A)' + X7M..
Now none of the values of M are infinite, since for none of these values is
fx infinite. If, therefore, M be the greatest of these values, then is 4M, less
than ZnM  and therefore
Fb   Fa- Ax If{+( a + (n-1) Ax} < (b-a) M (Ax)X.
The difference of the definite integral Fb -Fa, and the sum Xi (Aax)f{a+
(n- 1) Ax} is always, therefore, less than (b- a) M (Ax)x. Now  I is finite,
and (b -a) is given, and as n is increased Ax is diminished continually;
and therefore (Aa)X is diminished continually, x being positive.
Thus by increasing a indefinitely, the difference of the definite integral




682                         APPENDIX.
and the sum may be diminished indefinitely, and therefore, in the limit, the
definite integral is equal to the sum (i. e.)
Fb -Fa = limit ~, (Ax).f{a~+(n-1) Aix};
or, interpreting this formula, FI&-Fa is the sum of the values of Ax. fx
when x is made to pass by infinitesimal increments, each represented by
A;, from a to &.
NOTE B.
PONOELET'S FInST THEOrEM.
* The, values of a and b in the radical v/a2 + b being linear and rational,
let it be required to determine the values of two indeterminate quantities
a and p, such that the errors which result from assuming a2 + ib2= ca + ib,
through a given range of the values of the ratio  b ), may be the least possible in reference to the true value of the radical; or that a + ~b- _ - b
va+-b2
aacL- + 
or /-~   2 —-1 may be the least possible in respect to all that range of
values which this formula may be made to. assume between two given
extreme values of the ratio -.. Let these extreme values of the ratio 5
be represented by cot. i, and cot.,  nd any other value by cot. 4. Substituting cot. 4 for in the preceding formula, and observing that Va2+-b
=    b/2Cot.2; +a bS  cosec. 4, also that aa + b3 = ab cot. 4 + 13b= ( cos. 44+
sin, 4)b cosec. 4, the corresponding error is represented by
sa cos. 4+-3 sin. 4-l..e.   (1),;
which expression is evidently a maximum for that value A& of p which i
determined by the equation
Ch
cot. ==..... (2);
so that its. maximum value is....2-......  3)
Moreover, the function admits of no other maximum value, nor of any
minimum value. The values of a and p being arbitrary, let them  be
assumed tobe such that C or cot. t,3 may be less than cot. qi, and greater
* The method of this investigation is not the same as that adoptedl by MPoncelet; the principle is. the ame.




PONCELET'S THEOREM.                     633
than cot...  Now, so long as all the values of the error (formula 1)
remain positive, between the proposed limits, they are all manifestly diminished by diminishing a and i; but when by this diminution the error
is at length rendered negative in respect to one or both of the extreme
values 4,, or 42 of 4, and to others adjacent to them, then do these negative errors continually increase, as a and p are yet farther diminished,
whilst the positive maximum error (formula 3) continually diminishes.
Now the most favorable condition, in respect to the whole range of the
errors between the proposed limits of variation, will manifestly be attained
when, by thus diminishing the positive and thereby increasing the negative
errors, the greatest positive error is rendered equal to each of the two
negative errors; a condition which will be found to be consistent with
that before made in respect to the arbitrary values of a and p, and which
supposes that the values of the error (formula 1) corresponding to the
values 41 and 42 are each equal, when taken negatively, to the maximum
error represented by formula 3, or that the constants a and 3 are taken
so as to satisfy the two following equations.
1 —(a cos. W,+-  sin. -,)= 4/2-+p2-1.
l-(a cos.,+l/3 sin. i)=l —(a cos. q2-P sin. T2).
The last equation gives us by reduction
cos. (, —2)
a cos. 4,~13 sin. PW=Psin C' (+,t)
sil.    +' +
and a - 3 cot. ~(c1 + 2).
Substituting these values in the first equation, and reducing,
2 sin. -(, + T,) _ sin. -(,1 + v2)
=l+eos. (*;, —'*      -cos. 2_(i, —*d.....  (4);
1+COS. i(41-_'92)  COS. -1(t1 i 2)      ( * i
2 cos. i(V, + 2) _ cos. i(r — ) 2)
"'.=1 + cos.   5 —)cos. ) 
These values of a and 3 give for the maximum error (formula 3) the expression
tan. 2(q1,-q2)..... (6).
Thus, then, it appears that the value of the radical V/a + ba is represented,
in respect to all those values of - which are included between the limits
cot.', and cot. I',, by the formula
cos. ~(-,q + 2)  b sin. J(r, +2) 
cos.'2 (1 (- I'P) +cO.S2 _I(1 —  ) * * ()
with a degree of approximation which is determined by the value of
tan. 2 ( -,I)
If in the proposed radical the value of a admits of being increased infinitely in respect to b, or the value of b infinitely diminished in respect to
a, then cot.', = infinity; therefore r', =- 0. In this case the formula of
approximation becomes




634                         APPENDIX.
a (1 - tan.'  ) + 2b tan. iP2.... (8);
and the maximum error
tan. 2,2..... (9).
If the values of a and b are wholly unlimited, so that a may be infinitely
small or infinitely great as compared with b, then cot. T', = infinity,
cot. 42=, 0;
therefore'i=0, P-2=2. Substituting these values, the formula of approximation becomes
*8284a + 8284..... (10);
and the maximum error'1716, or Ith nearly.
If b is essentially less than a, but may be of any value less than it, so
that - is always greater than unity, but may be infinite, then cot.?p1 = infinity, cot. =2=1; therefore,==0, = 2=2. Substituting these values in the
formula of approximation, and reducing, it becomes
*96046a+'39783b..... (11);
and the maximum error
*03945, or -Ith nearly.
It is in its application to this case that the formula has been employed in
the preceding pages of this work.
The following table, calculated by M. Gosselin, contains the values of
the coefficients a and 3 for a series of values of the inferior limit cot. t.2, the
superior limit being in every case infinity.
Reato oato..                      Approximate Value
Relation of a to b. Ov  Value of a. Value of. Maximum Error.  of 4/a2+b2
of ~.
a and b any         0 82840 0'82840  0'17160 or     0-8284 (a +b)
a> b         1   0-96046 0'39783 0 03954 or'96046a + 39783b
a > 2b      2    0-98592 0,232'0 0-01408 or -~ 98592a+ 23270b
a > 3b       3   099350 0-16123 0'00650 or y4'99350a+-16123b
a> 4b       4    099625   12260   00375 or     99625a+ 12260b
a> 5b        5   0'99757 0-098(78 0-00243 or-,'99757a+ 09878b
a> 6b       6    0-99826 0'08261 0-00174 or I'99826a+ 08261b
a > 76      7    0'99875 0'07098 0-00125 or-8'99875a+ 07098b
a> 8b       8    0'99905 0-06220 0-00095 orl --'99905a + 06220b
a> 9b       9    099930 005535 000070 or -1'99930a + 05535b
a> lOb     10    0-99935 0-04984 0'00065or  1'99935a+'04984b
I                     T 5  0'005o




PO:NCELET'S SECOND THEOREM.                635
PONCELET'S SECOND THEOREM.
To approximate to the value of 4 —b2, let aa -  b be the formula of
approximation, then will the relative error be represented by
_                        (aX_______-_( )
A a2b2,or by 1- 
-4/&a —bb B
Now, let it be observed that a2 being essentially greater than b, > 1;
let i, therefore, be represented by cosec. 4, then will the relative error be
(a cosec. -3)P)
represented by 1-  --   -, or by
4/cose. 2-l'
1-a sec.4+   tan......(12),
which function attains its maximum when sin. 4 = -. Substituting this
a
value in the preceding formula, and observing that -a sec. 4 + p tan. 4 =
-sec.24 (a-[p sin4)= -        --  =  i-= 2, we obtain for the maximum
l1-2
error the expression
1 —   a2-i 2.... (13),
Assuming 4, and 42 to represent the values of 4, corresponding to the
greatest and least values of a, and observing that in this case, as in the
preceding, the values of c and 3, which satisfy the conditions of the
question, are those which render the values of the error corresponding to
these limits equal, when taken with contrary signs, to the maximum error,
we have
-l +   sec. 4, —   tan. 4, = 1 - 4/2-2.... (14).
1 — a sec. 4, +  tan.,=l — a sec. 42 + 3 tan. 42.... (15).
The latter equation gives, by reduction,
cos. ('-,I — (2)' sin. i(+, +  )..
3            cos. 2i (4'1-4)  1 4 _',CoS. -FI COS. #2
sin. -I (P, +                   Y-1  s=n. ('. +'P.
And a sec.,i + 3 tan. V,== 3 cot. i (,i + t2).... (17).
Substituting these values in equation (14), and solving in respect to 3




636                         APPENDIX.
______2 sin. ( + p2)             (18).
COS. i- (i, + %2) + 4/cOS.   P Cos. %2
2 cos. j (1, —.2) 
a_ -= —....0....................... 9).
cos. ~ (,1 + q2) +  /Cos. P1 cos. P2
The maximum error is represented by the formula
2,/cos. ~, cos...
1 -       24/cos. — O                (20).
cos. i (1p + 2) + Vcos., cos. o 2
These formula will be adapted to logarithmic calculation, if we assume
(+1'2)=\,  s and. ~('~  +2)= cosec. I2; we shall thus obtain from
equations (16) and (17) =     3 cosec. T2, Vo2 2-  = f3 cot. t2, and a sec. p
— 3 tan. qi = 3 cot.,I; therefore, by equation (14),
2         2 sin. P, sin,2 
= cot. rl + cot.',- sin. ('tI +         (21)
2 cosec. 2       2 sin.,   j 
= cot. P, + cot. "2 q   sin. (il + "' 2)
sin. (', -'I2)
Maximum errror = -
sin. (%, + T2).... (22).
The form under which this theorem has been given by M. Poncelet is
different from the above. Assuming, as in the previous case, the limiting
values of - to be represented by cot. l, and cot. P2, and proceeding by a
geometrical method of investigation, he has shown that if we assume
tan. qP = cos. co, tan. P2 - cos.,2, WI + ~2 = 2yT, Ao - (2 = 28, and cos. 72 
COS. yl; then
cos. 8
2 cos. y,        2 cos.2 yi                   sin. (y - 72)
a c -. -,=,-  --,- maximum error =         l —
sin. (71 +72)'  sin(. (y,+y 8 cos. m'        sin. (y, + 72)
If the.least possible value of a be 1-Lb, and its greatest possible value
be infinite as compared with b, M. Poncelet has shown the formula of
approximation to become
4/ a- b2 ='1319a-0-72636b....   (23),
with a possible error of 0-1319 r r  nearly.
If the least possible value of a be 2b, and its greatest possible value
infinite compared with b; then
a4/2-b2 = 1018623a-0-272944b.... (24),
with a possible error of'0186 or -d nearly.




ON THE ROLLING OF SHIPS.                    637
NOTE C.
ON THE ROLLING OF SHIPS.
(Pirst published by the Author in the Transactions of the Royal Society
for 1850, Part II.)
Let a body be conceived to float, acted upon by no other forces than its
weight W, and the upward pressure of the water (equal to its weight);
which forces may be conceived to be applied respectively to the centre of
gravity of the body and to the centre of gravity of the displaced fluid;
and let it be supposed to be subjected to the action of a third force whose
direction is parallel to the surface of the fluid. Let All represent the vertical displacement of the centre of gravity of the body thereby produced*,
and AH2 that of the centre of gravity of its immersed part. Let moreover the volume of the immersed part be conceived to remain unaltered t
whilst the body is in the act of displacement. If each centre of gravity
be assumed to ascend, the work of the weight of the body will be represented by -W.AH,, and that of the upward pressure of the fluid by +
W.AH2, the negative sign being taken in the former case because the force
acts in a direction opposite to that in which the point of application is
moved, and the positive sign in the latter, because it acts in the same direction, so that the aggregate work Xzu (see equation 1, p. 123.) of the forces
which constituted the equilibrium of the body in the state from which it
has been disturbed is represented by
- W.AH + W.AH2.
Moreover, the system put in motion includes, with the floating body, the
particles of the fluid displaced by it as it changes its position, so that if
the weight of any element of the floating body be represented by w,, and
of the fluid by o,, and if their velocities be 9v and v2, the whole sis viva is
represented by: When a floating body is so made to incline from any one position into any
other as that the volume of fluid displaced by it may in the one position be
equal to that in the other, its centre of gravity is also vertically displaced;
for if this be not the case, the perpendicular distance of the centre of gravity
of the body from its plane of flotation must remain unchanged, and the form
of that portion of its surface, which is subject to immersion, must be determined
geometrically by this condition; but by the supposition the form of the body
is undetermined. It is remarkable what currency has been given to the error,
that whilst a vessel is rolling or pitching, its centre of gravity remains at rest.
I should not otherwise have thought this note necessary.
r This supposition is only approximately true.
t If the centre of gravity of the body or of the displaced fluid descends (a
property which will be found to characterise a large class of vessels), AHl in
the one case, and iI{2 in the other, will of course take the negative sign.




638                          APPENDIx.
1      1
and we have by equation 1 (p. 123),
1
U(o)-W(AH - AH)=2,g   +     W 32....2 (25).
In the extreme position into which the body is made to roll and m
which  wlv=0O,
U(O)=W.(AHI- AH2)+ I     w,.... (26).
or if the inertia of the displaced fluid be neglected,
U(o)=W.(AH, —H)..... (27).
Whence it follows that the work necessary to incline a floating body
through any given angle is equal to that necessary to raise it bodily through
a height equal to the difference of the vertical displacements of its centre
of gravity and of that of its immersed part; so that other things being
the same, that ship is the most stable the product of whose weight by this
difference is the greatest.
In the case in which the centre of gravity of the displaced fluid descends,
the sum of the displacements is to be taken instead of the difference.
This conclusion is nevertheless in error in the following respects:1st. It supposes that throughout the motion the weight of the displaced
fluid remains equal to that of the floating body, which equality cannot
accurately have been preserved by reason of the inertia of the body and
of the displaced fluid.*
From this cause there cannot but result small vertical oscillations of the
body about those positions which, whilst it is in the act of inclining, correspond to this equality, which oscillations are independent of its principal
oscillation.
2ndly. It involves the hypothesis of absolute rigidity in the floating
body, so that the motion of every part and its vis viva may cease at once
when the principal oscillation terminates. The frame of a ship and its
masts are, however, elastic, and by reason of this elasticity there cannot
* The motion of the centre of gravity of the body being the same as though
all the disturbing forces were applied directly to it, it follows, that no elevation
of this point is caused in the beginning of the motion, by the application of a
horizontal disturbing force, or by a horizontal displacement of the weight of
the body, which, if it be a ship, may be effected by moving its ballast. The
motion of rotation thereby produced takes place therefore, in the first instance,
about the centre of gravity, but it cannot so take place without destroying the
equality of the weight of the displaced fluid to that of the body. From this
inequality there results a vertical motion of the centre of gravity, and another
axis of rotation.




ON THE ROLLING OF SHIPS.                  639
but result oscillations, which are independent of, and may not synchronise with, the principal oscillation of the ship as she rolls, so that the vis
viva of every part cannot be assumed to cease and determine at one and
the same instant, as it has been supposed to do.
3rdly. No account has been taken of the work expended in communicating motion to the displaced fluid, measured by half its vis viva and
represented by the term  -:w2v in equation 26.
From a careful consideration of these causes of error, the author was
led to conclude that they would not affect that practical application of the
formula which he had principally in view in investigating it, especially as
in certain respects they tended to neutralise one another. The question
appeared, however, of sufficient importance to be subjected to the test of
experiment, and on his application, the Lords Commissioners of the Admiralty were pleased to direct that such experiments should be made in Her
Majesty's Dockyard at Portsmouth, and Mr. FINCHAM, the eminent Master
Shipwright of that dockyard, and Mr. RAWsoN, were kind enough to
undertake them.
These experiments extended beyond the object originally contemplated'by him; and they claim to rank as authentic and important contributions
to the science of naval construction, whether regard be had to the practical importance of the question under discussion, the care and labor
bestowed upon them, or the many expedients by which these gentlemen
succeeded in giving to them an accuracy hitherto unknown in experiments
of this kind.
That it might be determined experimentally whether the work which
must be done upon a floating body to incline it through a given angle be
that represented by equation 27, it was necessary to do upon such a body
an amount of work which could be measured; and it was further necessary to ascertain what were the elevations of the centres of gravity of the
body and of its immersed part thus produced, and then to see whether
the amount of work done upon the body equalled the difference of these
elevations multiplied by its weight.
To effect this, the author proposed that a vessel should be constructed
of a simple geometrical form, such that the place of the centre of gravity
of its immersed part might readily be determined in every position into
which it might be inclined, that of its plane of flotation being supposed to
be known; and that a mast should be fixed to it, and a long yard to this
mast, and that when the body floated in a vertical position a weight
suspended from one extremity of the yard should suddenly be allowed to
act upon it causing it to roll over; that the position into which it thus
rolled should be ascertained, together with the corresponding elevations
of its centre of gravity and the centre of gravity of its immersed part,
and the vertical descent of the weight suspended from the extremity of
its arm. The product of this vertical descent by the weight suspended




640                        APPENDIX.
from the arm ought then, by the formula, to be found nearly equal to the
difference of the elevations of the two centres of gravity multiplied by
the weight of the body; and this was the test to which it was proposed
that the formula should be subjected, with a view to its adoption by practical men as a principle of naval construction.
To give to the deflecting weight that instantaneous action on the extremity of the arm which was necessary to the accuracy of the experiment,
a string was in the first place to be affixed to it and attached to a point
vertically above, in the ceiling. When the deflecting weight was first
applied this string would sustain its pressure, but this might be thrown
at once upon the extremity of the arm by cutting it. A transverse section of the vessel, with its mast and arm, was to be plotted on a large
scale on a board, and the extreme position into which the vessel rolled
being by some means observed, the water-line corresponding to this
position was to be drawn. The position of the yard, in respect to the
surface of the water in that position, would then be known, and the vertical
descent of the deflecting weight could be measured, and also the vertical
ascent of the centre of gravity of the immersed part or displacement.
To determine the position of the centre of gravity of the vessel, it was
to be allowed to rest in an inclined position under the action of the deflecting weight; and the water-line corresponding to this position being drawn
on the board, the corresponding position of the deflecting weight and of
the centre of gravity of the immersion were thence to be determined.
The determination of the position of the vertical passing through the
centre of gravity of the body would thus become an elementary question
of statics; and the intersection of this line, with that about which the
section was symmetrical, would mark the position of the centre of gravity.
This determination might be verified by a second similar experiment with
a different deflecting weight.
These suggestions received a great development at the hands of Mr.
RAwsoN, and he adopted many new and ingenious expedients in carrying
them out. Among these, that by which the position of the water-line
was determined in the extreme position into which the vessel rolls, is
specially worthy of observation. A strip of wood was fastened at right
angles to that extremity of the yard to which the deflecting weight was
attached, of sufficient length to dip into the water when the vessel rolled;
on this slip of wood, and also on the side of the vessel nearest to it, a
strip of glazed paper was fixed. The highest points at which these strips
of paper were wetted in the rolling of the vessel, were obviously points
in the water-line in its extreme position, and being plotted upon the board,
a line drawn through them determined that position with a degree of
accuracy which left nothing to be desired.
Two forms of vessels were used; one of them had a triangular and the
other a semicircular section. The following table contains the general
results of the experiments.




ON THE ROLLING OF SHIPS.                     641
h   No.o  Weight of  Disturb.-  t. 
experi-  model and  ing  Zj          q 
Imodbs.   lbs. 
Trian 1 r 1.   88-8626 -5485 *5161  661  23 80.....  12 80  8961
Triala  2.  36e8590      g 4887l  4  15 30..8'90  98 114
model.       87-3563 *537T 1-1724 1-4508  24 0.....  18 0  -88512
4.   38-2911 -5789 1-26  18460 2 0          13 0   9880
Circu ar  1.  19T-18  2-8225 7-T861 7-394  26 0  24 20  13 0
model   3.   55-48  1-96T0 1-27 t 1-766T to  0  10 0  4 80
In the experiments with the smaller triangular model the differences
between the results and those given by the formula are much greater than
in the experiments with the heavier cylindrical vessel
In explanation of this difference, it will be observed, first, that the conditions of the experiment with the cylindrical model more nearly approach
to those which are assumed in the formula than those with the other; the
disturbance of the water in the change of the position of the former being
less, and therefore the work expended upon the inertia of the water, of
which the formula takes no account, less in the one case than the other;
and, secondly, that the weight of the model being greater, this inertia
bears a less proportion to the amount of work required for inclining it
than in the other case,
The effect of this inertia adding itself to the buoyancy of the fluid,
cannot but be to lift the vessel out of the water and to cause the displacement to be less at the termination of each rolling oscillation than at its
commencement.* This variation in volume of the displacement was apparent in all the experiments. Its amount was measured and is recorded
in the last column of the Table; its tendency is to produce in the body
vertical oscillations which are so far independent of its rolling motion
that they will not probably synchronise with it. The body, displacing,
when rolling, less fluid than it would at rest, the effect of the weight
used in the experiments to incline it is the     reby increased  and thus is
explained the fact (apparent in the eighth and ninth columns of the Table)
that   nc the inclination by eeriment i s somewhat greater than th e forme  ula
would make it.
the dynamical stability of a vessel whose athwart sections (where they
This resulcon nects itself with the well-known fac t of the rise of a vessel
out of the water when propelled rapidly, which is so great in the case of fast
trak-heoats, as consider   abl  y to reduce the resistance upon them
41




6(42                         APPENDIX.
are subject to immersion and emersion) are circular, having their centres in
a common axis.
Fig. 1.                       Fig. 2.
}     K 
Let EDF, fig. 1. or 2., be an athwart section of such a vessel, the
parts of whose periphery ES and FR, subject to immersion and emiersion,
are parts of the same circular arc ETF, whose centre is C. Let G, represent the projection of the centre of gravity of the vessel on this section,
and G, that of the centre of gravity of the space whose section is SDRT,
supposing it filled with water. The space lies wholly within the vessel in
fig. 1. and without it in fig. 2. Let
Al = CGI, h  = CG.
W  = weight of vessel.
W, = weight of water occupying, or which would occupy, the space
whose section is STRD.
0 = the inclination from the vertical.
Since in the act of the inclination of the vessel the whole volume of
the displaced fluid remains constant, and also that volume of which STRD
is the section, it oos that thion,* it llows that the volume of that portion of which the
circular area PSRQ is the section remains also constant, and that the
water-line PQ, which is the chord of that area, remains at the same distance from C, so that the point C neither ascends nor descends. Now the
forces which constituted the equilibrium of the vessel in its vertical position were its weight and that of the fluid it displaced. Since the point C
is not vertically displaced, the work of the former force, as the body
inclines through the angle 0, is represented by - W1 h vers. e. The work
of the latter is equal to that of the upward pressure of the water which
would occupy the space of which the circular area PTQ is the section
increased, in the case represented in fig. 1., by that of the water which
would occupy STRD; and diminished by it in the case represented in
fig. 2.
But since the space, of which the circular area PTQ is the section,
remains similar and equal to itself, its centre of gravity remains always
at the same distance from the centre C, and therefore neither ascends
nor descends.  Whence it follows that the work of the water which
would occupy this space is zero; so that the work of the wehole displaced
fluid is equal to that of the part of it which occupies the space STRD,
* It will be observed that the space STRD is supposed always to be under
water.




ON THE ROLLING OF SHIPS.                     643
taken in the case represented in fig. 1. with the positive, and in that represented in fig. 2. with the negative sign. It is represented, therefore,
generally by the formula ~W2 h vers. e. On the whole, therefore, the
work E,2 of those forces, which in the vertical position of the body constituted its equilibrium, is represented by the formulaZu2 =        -Whi vers.  0 ~ W,,vers..,
Representing, therefore, the dynamical stability:uL by U (0), we have by
equation (2. p. 123.)
U (0) = (WI h, TF W2 h2) vers. 0,
in which expression the sign qF is to be taken according as the circular
area ATB lies wholly within the area ADB, as in fig. 1., or partially without it, as in fig. 2. Other things being the same, the latter is therefore a
more stable form than the other.
13. The work of the upward pressure of the water upon the vessel
represented in fig. 2. being a negative quantity, - W2h vers. 0, it follows
that the point of application of the pressure must be moved in a direction
opposite to that in which the pressure acts; but the pressure acts upwards,
therefore its point of application, i. e. the centre of gravity of the displaced
fluid, descends. This property may be considered to distinguish mechanically the class of vessels whose type is fig. 1., from that class whose type is
fig. 2.; as the property of including wholly or only partly, within the area
of any of their athwart sections, the corresponding circular area ETF, distinguishes them geometrically.
The dynamical stability of a vessel of any given form subjected to a rolling or pitching motion.
Conceive the vessel, after having completed an oscillation in any given
direction-being then about to return towards its vertical position-to
be for an instant at rest, and let RS represent the
intersection of its plane of flotation then, and PQ
of its flotation when in its vertical position, with  p
a section CAD of the vessel perpendicular to the                  D
mutual intersection O of these planes. The sec-     Lii            s
tion CAD will then be a vertical section of the          G
vessel.                                                 a
Let G be the projection upon it of the vessel's
centre of gravity when in its vertical position.
H, that of the centre of gravity of the fluid displaced by the vessel in the
vertical position.
g, that of the fluid displaced by the portion of the vessel of which QOS
is a section.
h, that of the fluid which would be displaced by the portion, of which
POR is a section, if it were immersed.
GM, HN, gin, An, KL, perpendiculars upon the. plane RS.
W = weight of vessel or of displaced fluid.
w = weight of water displaced by either of the equal portions of the
vessel of which POR and QOS are sections.




64F4                       APPENDIX.
H, = depth of centre of gravity of vessel in vertical position.
H2- = depth of centre of gravity of displaced water in vertical
position,
AH, = elevation of centre of gravity of vessel.
AH2 = elevation of centre of gravity of displaced water.
P = area of plane PQ.
0 = inclination of planes PQ and RS.
= inclination of line 0 in which planes PQ and RS intersect,
to that line about which the plane PQ is symmetrical.
h = perpendicular distance of line 0 from centre of gravity of
plane PQ.
r = inclination to horizon of line about which the plane PQ is
symmetrical.
x = distance of section CAD, measured along the line whose
projection is 0, from the point where that line intersects
the midship section.
Y = PQ.
y, = PQ.
2 = RS.
z = An + mg.
=KL.
I = moment of inertia of plane PQ about axis 0.
A and B = moments of inertia of PQ about its principal axes.
= weight of a cubic unit of water.
Suppose the water actually displaced by the vessel to be, on the contrary,
contained by it; and conceive that which occupies the space QOS to pass
into the space POR, the whole becoming solid. Let AII3 represent the
corresponding elevation of the centre of gravity of the whole contained
fluid. Then will AHl + AH3 represent the total elevation of the centre of
gravity of this fluid as it passes from the position it occupied when the
vessel was vertical into the position PAQ. But this elevation is obviously
the same as though the fluid had assumed the solid state in the vertical
position of the body, and the latter had revolved with it, in that state, into
its present position. It is therefore represented by KH - NH;.. AH2 +Al H   KH - NH and AH = KH -NH - aH.
Since, moreover, by the elevation of the fluid in QOS, whose weight is w,
into the space OPR, and of its centre of gravity through (gmr + hn), the
centre of gravity of mass of fluid of which it forms a part, and whose weight
is W, is raised through the space AIl; it follows, by a well-known property
of the centre of gravity of a system,* that
* The line joining the centres of gravity of the vessel and its immersed part,
in its vertical position, is parallel to the plane CAD, for it is perpendicular to
the plane PQ to whose intersection with the plane RS the plane CAD is perpendicular;., GK = HI and HK = H2.




ON TIE IOLLING OF SHIPS.                     645
W. a H3 = w (gm  t hn);.-. W(KH - NH - A H) = W (gm + hi).
But
NH = KH cos. o - KL= H2 cos. -;...KH- NH = Hvers. 0 +,
and
mg+nh =z;'. W (H2viers. 0 +  -A H2) = -  z;. W.; = W (H vers. 0 + ) —.... (28).
Also      A H, = KG     MG = I, - (H, cos. -- ) = H vers. 0 +;.. W (A H, - A H) = W (HI,- H) vers. 0 + wz;.'. (equation 27.) U (0, W) = W (HI - Hi) vers. 0 +-wz;... (29).
If cm3 be a vertical prismatic element of the space QOS, whose base
is dx dy cos. 0, and height y sin. 0 then will wu.mg be represented, in
1              1
respect to that element, by py sin. 0. dx dy cos. 0. 2 y sin. 0, or by  s sin.s 0
cos. 0 y'dx dy; and wz will be represented, in respect to the whole space
of which PrsQ is the section, by
2  sin. 0 cos. O  _ y'dx dy,
1
or by                   2, sin.2 0 cos. o. I.
If therefore we represent by p the value of cz, in respect to the spaces
of which the mixtilinear areas PRr and QSs are the sections, we have
1
wz = oA Isin.2 cos. 0+q.
But the axis 0, about which the moment of inertia of the place PQ is
I, is inclined to the principal axes of that plane at the angles q and  -,
about which principal axes the moments of inertia are A and B,.. I =A  cos.2 + B sin.2 V +Ph2,'. U(0,)=
W (Hi - H2) vers. 0 +  (A cos.1 +B sin. 2 +Ph2) sin2 0cos. 0+q... (30).
It has been shown by M. DuPeN* that when 0 is small the line in
* Sur la Stabilite des Corps Flottants, p. 32. In calculations having reference to the stability of ships, it is not allowable to consider 0 extremely small,
except in so far as they have reference to the form of the ship immediately
about the load-water line. The rolling of the ship often extends to 20~ or 30~,
and is therefore largely influenced by the form of the vessel beyond these
limits. Generally, therefore, equation 30. is to be taken as that applicable to
the rolling of ships, those which follow being approximations only applicable
to small oscillations, and not sufficiently near (excepting equation 37) for
practical purposes.




6 46                         APPENDIX.
which the planes PQ or RS intersect passes through the centre of gravity
of each; in this case.*. I = Acos.2 + B sin.2;
therefore by equation (30),
U (0o,) = W (H  - H,) vers. 0 + - ^ (A cos.2  B + B sin.2 )) sin.2 0 cos. 0 + q.
If 0 be so small that the spaces PrR and QsS are evanescent in comparison with POr and QOs, then, assuming  = 0 and cos. 0 = 1,
U (0, V) = W (Hi - -H,) vers. o + / ^ (A cos.2  B + B sin.2 ) sin.,... (31),
which may be put under the form
U(0,n) =    W W(Hl,-H2i)+ (Acos.2+Bsin.2) } vers.0.
Again, since
sin. = sin. 0 si..... (32),
and      (A cos.2 + B sin.) sin.2   A   (B   A) sin.2  sin. 0,. (A cos.2y + B sin.2 0) sin.2 =   sin.2 0 + (B - A) sin.2;.'. by equation 31,
U (0, ) = W (H,- H,) vers. + 2  {Asin. 0 +(B-A)sin.2?},.... (33),
by owhich formulc the dynamical stability of the ship is epresented, both
as it regards a pitching and a rolling motion.
If in equation 31. 7 = =, the line in which the plane PQ (parallel to the
2
deck of the ship) intersects its plane of flotation is at right angles to tlhe
length of the ship, and we have, since in this case 0 ==  (see equation 32.),
U (=W (Hi - H2) vers r + 2 B sin2..... (34),
which expression represents the dynamical stability, in regard to a pitching motion alone, as the equation
U (0) = W (H, — H) vers 0 + 2 t A sin2 0..... (35),
represents it in regard to a rolling motion alone.
16. If a given quantity of work represented by U (0) be supposed to
be done upon the vessel, the angle 0 through which it is thus made to
roll may be determined by solving equation 35. with respect to sin.-.
2
We thus obtain
0  W (H,-H) + PA — V      ( W (H, —H) +  A}2 —2A. U       ().  (36).
sin2= 2=A




ON THE IOLLING OF SHIPS.                   617
17. If PR and QS be conceived to be straight lines, so that POR and
QOS are triangles, then w. z, taken in respect to an element included
between the section CAD, and another parallel to it and distant by the
small space dx, is represented by
1yy1Y sin.odx(mg - nh);
4
12
12      ^
*r   sic 1_ sin.0,YTY2dX
12
and, equation 29
U(0,-=W(H1, —  H) vers.o+-1 sin. 20 y12dy,... (37),
24:      _
which formula may be considered an approximate measure ofthe stability
of the vessel under all circumstances.
If, as in the case of the experiments of Messrs. FINrHAM and RAwsoN,
the vessel be prismatic and the direction of the disturbance perpendicular
to its axis,
y = constant = a, and z = -a sin,;.'.    zs =- aw sin. 0, and
U(0)=W(, —Ha)vers.0+ l aw sin.0.
3
A rigid surface on which the vessel may be supposed to rest whilst in the
act of rolling.
If we imagine the position of the centre of gravity of a vessel afloat
to be continually changed by altering the positions of some of its contained weights without altering the weight of the whole, so as to cause
the vessel to incline into an infinite number of different positions displacing, in each, the same volume of water, then will the different planes
of flotation, corresponding to these different positions, envelope a curved
surface, called the surface of the planes of flotation (surface desflotaisons),
whose properties have been discussed at length by M. DUPIN in his excellent memoir, Sur la Stabilite des Corps Flottants, which forms part of
his Applications de Geometrie.*  So far as the properties of this surface
concern the conditions of the vessel's equilibrium, they have been exhausted in that memoir, but the following property, which has reference
* BAOHaLI, Paris, 1822




648                          APPENDIX.
rather to the conditions of its dynamical stability than its equilibrium, is
not stated by M. DTPIN:If we conceive the surface of the planes of flotation to become a rigid
surface, and also the surface of the fluid to become a rigid plane without
friction, so that the former surface may rest upon the latter and roll and
slide upon it, the other parts of the vessel being imagined to be so far immaterial as not to interfere with this motion, but not so as to take away
their weight or to interfere with the application of the upward pressure of
the fluid to them, then will the motion of the vessel, when resting by this
curved surface upon this rigid but pe)featly smooth horizontal plane, be
the same as it was when, acted upon by the same force, it rolled and pitched
in the fluid.
In this general case of the motion of a body resting by a curved surface upon a horizontal plane, that motion may be, and generally will be,
of a complicated character, including a sliding motion upon the plane,
and simultaneous motions round two axes passing through the point of
contact of the surface with the planes and corresponding with the rolling
and pitching motion of a ship. It being however possible to determine
these motions by the known laws of dynamics, when the form of the
surface of the planes of flotation is known, the complete solution of the
question is involved in the determination of the latter surface.
The following property*, proved by M. DuPri    in the memoir before
referred to (p. 32), effects this determination:" The intersection of any two planes of flotation, infinitely near to each
otber, passes through the centre of gravity of the area intercepted upon
either of these planes by the external surface of the vessel."
If, therefore, any plane of flotation be taken, and the centre of gravity
of the area here spoken of be determined with reference to that plane of
flotation, then that point will be one in the curved surface in question,
called the surface of the planes of flotation, and by this means any number
of such points may be found and the surface determined.
The axis about which a vessel rolls may be determined, t e direction M
which it is, rolling being given.
If, after the vessel has been inclined through any angle, it be left to
itself, the only forces acting upon it (the inertia of the fluid being neglected)
will be its weight and the upward pressure of the fluid it displaces; the
motion of its centre of gravity will therefore, by a well-known principle
of mechanics, be wholly in the same vertical line.
Let HK represent this vertical line, PQ the surface of the fluid, and
aMb the surface of the planes of flotation. As the centre of gravity G
traverses the vertical HK, this surface will partly roll and partly slide
by its point of contact M on the plane PQ.
If we suppose, therefore, PRQ to be a section of the vessel through
W This property appeaxs to have been fnrst given by EvLnm




ON THE IOLLING OF SHIPS.                  649
the point M, and perpendicular to the axis, about which it is rolling, and
if we draw a vertical line MO through the point M, and through G a
horizontal line GO parallel to the plane PRQ, then
the position of the axis will be determined by a line  Fig. 8.
perpendicular to these, whose projection on the plane  / ^  c
PRQ is 0.                                                    b
For since the motion of the point G is in the verti-       /Q
cal line HK, the axis about which the body is revolv-  \ ---
ing passes through GO, which is perpendicular to
HK; and since the point M of the vessel traverses    K
the line PQ, the axis passes also through MO, which
is perpendicular to PQ; and GO is drawn parallel to, and MO in the
plane PRQ, which, by supposition, is perpendicular to the axis, therefore
the axis is perpendicular to GO and MO.
If HK be in the plane PRQ, which is the case whenever the motion is
exclusively one of rolling or one of pitching, the point 0 is determined by
the intersection of GO and MO.
The time of the rolling through a small angle of a vessel whose athwart
sections are (in respect to the parts subject to immersion and emersion)
circular, and have their centres in the samq longitudinal axis.
Let EDF (fig. 1. or fig. 2, represent the midship section of such a
Fig. 1.                    Fig. 2.
E  H
/     /                        H
1'S  W Q:   j:       o
Sy    CO   B              ~e   fio     y 
vessel, in which section let the centre of gravity G, be supposed to be situated, and let HK be the vertical line traversed by G. as the vessel rolls.
Imagine it to have been inclined from its vertical position through a given
angle 0, and the forces which so inclined it then to have ceased to act
upon it, so as to have allowed it to roll freely back again towards its position of equilibrium until it had attained the inclination OCD to the vertical, which suppose to be represented by 0.
Referring to equation 1. page 123. let it be observed that in this case
u2==0, so that the motion is determined by the condition
~    2u= - SV.. (38).
But the forces which have displaced it from the position in which it
was, for an instant, at rest are its weight and the upward pressure of the




650                         APPENDIX.
water; and the work of these, U(0,)- UJ(0), done between the inclinations
0 and 0, when the vessel was in the act of receding from the vertical, was
shown to be represented by (W,h,:F Wh2  (vers. 0 -vers. 0,); therefore
the work, between the same inclinations, when the motion is in the
opposite direction, is represented by the same expression with the sign
changed;.'. U,=I(Wh,:F W2h2) (vers. 0, - vers. 0),
and since the axis about which the vessel is revolving is perpendicular to
the plane EDF, and passes through the point 0, if Wik' represents its
moment of inertia about an axis perpendicular to the plane EDF, and
passing through its centre of gravity G,,
Substituting in equation 38. and writing for OG, its value i, sin. 0, we
have
(WA,hTW,  A) (vers. 0- vers. 0)-2  (+9+h sin;o) (d);
-01
*'" tO  \       W=h/ /- -~ ~ ik  ~+ hi sin.0
t(o/)         -             I,,+h sin0           d       ()
\/2ah(,     Wh           \  vers.0,-vers. 0...    ).
+01 
+)-1                 1       1
^-  1i k /7+4,h2sin. 2 0 os. 12 
-0L
~01  1     1
_ 1          +0         2 k'sec.2 o+4h2 sin.2'0      1,             1'
/         WA       /      /         1w-  I    I. cos.- O.  2
1ghl  T     ), /S           sin.'2 0,-sin.2' 0
-01
or assuming 0 to be so small that the fourth and all higher powers of
sin.  0 may be neglected, and observing that, this being the case,
A/M sec.2 o0+4h sin.2 0 =  /(l+sin.2    0 +4h2 sin.2 0




ON TILE ROLLING OF SHIPS.                   651
- =- k  / 4 1         7 (.   4/+7.. +1  }2
- 1 + -/ — sin.2   0 =k { 1 + 4 — sin.21 
+01
4 1+      k2 
t(O,)=                  2f                    2 -     sin.10
1 +/ sIin.e 02 1-sin.2            0
-01
P. 1
d sin.- 0
But                                  /     -         1-,
_ sin.  0- sin. 20
-01
and
~ 02
zp        1       1
sin.2  0 dsin.2 
-~_'ti                  sin. i-s.
Sin.2     sin.''.*t(02) =;ghj=     7- j 1 + 4     sin. 2 01.... (40).
b      t     a        a  h                    h 
The sign + being taken accord;.ng as the centre of gravity of the displaced
fluid ascends or descends.
The time of a vessel's rolling or pitching through a small angle, its form
and dimensions being any whatever.
Let EDF (figs. 1. or 2.) represent the midship section of such a vessel,
supposed to be rolling about an axis whose projection is 0; and let C
represent the centre of the circle of curvature of the surface of its planes
of flotation at the point Mi where that surface is touched by the plane PQ,
being above the load water-line AB in fig. 1, and beneath it in fig. 2. Let
the radius of curvature CM be represented by p; then adopting the same
notation as in the last article, and observing that the axis 0 about which
the vessel is turning is perpendicular to EDF, we shall find its moment of
inertia to be represented by
w{+ (ll-p)2
W,{   + (H —p)' sin.'o t (  )
where H, represents the depth of the centre of gravity in the vertical position of the vessel.




652                       AIPPENDIX.
Also, by equation 35.
Elu,=U(o0)-U(0)=W,(H,1-H) (cos. 0-cos. 0,) +               2A(cos.O0-cos.'0,),. by equation 38.
0, H+1^A(cos.' -cos. )Y W  P k2+(H,-p_/sin.  (d
W1(E1-112)(eos. 0-cos. 0H) S + 22            2g 01)p) sin 0          dt
t(0 =  -/ /2    (H, -H2,)+ (cos.  0 - c oss.. +  c s  - cos. 0,)
— 0l
+0,
-0,
Assuming 0 and 0, to be so small that cos. 0 + cos. 0, = 2, and observing
that
cos. 0 - cos. 0, = vers. 0 - vers. 0,
+01
________________  _          /l2~ +(Hi -p)2 n 10,- 2g  1 -H27 + W tA        v +Vers. 0,-vers.  d
-01
Supposing, moreover, p to remain constant between the limits-0, and +0,,
and integrating as in equation 39.
t (01)_=  (;:        )     +.i1 +     +... (41).
g {(H,-H,2    ) +
Since the value of sin.2 0 is exceedingly small, the oscillations are
nearly tautochronous, and the period of each is nearly represented by the
formula
t (01)...(42.)
9 ~ ~  ~   ~~i. o,...2       ai).
WI~~




EQUILIBRIUM   OF PIESSURES.                 653
The following method is given by M. DUPIN for determining the value
of p*:" If the periphery of the plane of flotation be imagined to be loaded at
every point with a weight represented by the tangent of the inclination of
the sides of the vessel at that point to the vertical, then will the moments
of inertia of that curve, so loaded, about its two principal axes, when
divided by the area of the plane of flotation, represent the radii of greatest
and least curvature of the envelope of the planes of flotation."
If p be taken to represent the radius of greatest curvature, the formula
41. will represent the time of the vessel's rolling; if the radius of least
curvature (B being also substituted for A), it will represent the time of
pitching.
NOTE D.
On the conditions of the equilibrium of any number of pressures in the
same plane, applied to a body moveable about a cylindrical axis in the state
bordering upon motion. (From a memoir on the Theory of Mechanics,
printed in the second part of theTransactions of the Royal Society for 1841.)
LET P,, P2, P,, &c. represent these pressures, and R their resultant. Also
let a,, a2, a,, represent the perpendiculars let fall upon them severally from
the centre of the axis, those perpendiculars being taken with the positive
signs whose corresponding pressures tend to turn the system in the same
direction as the pressure Pa, and those negatively which tend to turn it in
the opposite direction. Also let X represent the perpendicular distance of
the direction of the resultant R from the centre of the axis, then, since R
is equal and opposite to the resistance of the axis, and that this resistance
and the pressures Pi, P,, P3; &c. are pressures in equilibrium, we have by
the principle of the equality of moments,
P,ca + Pa2 + Pa,3 + &c. =?R.
Representing, therefore, the inclinations of the directions of the pressures
P1, P2, P2, &c. to one another by 127, 3,7 2, 2, &c., &c., and substituting
for the value of R.1
* Applications de Geom6trie, p. 47.
t The inclination 1,.2 of the directions of any two pressures in the above expression is taken on the supposition that both the pressures act from, or both
towards the point in which they intersect, and not one towards, and the other
from, that point; so that in the case represented in the figure in the note at p.
171., the inclination i1l2 of the pressures P1 and P2, represented by the arrows,
is not the angle P IP2, but the angle PIQ, since IQ and IP, are directions of
these pressures, both tendingfrom this point of intersection, whilst the directions of P2I and IP, are one of them towards that point, and the other from it.:: POssoN, Mecanique, Art, 33.




654:                         APPENDIX.
Pla + P2a,+P3a....
1'P2+P 2+P'2.+...'
+ 2 P1P2 cos. t. + 2 PP3 COs. 113 +....
+ 2 P2P, cos.,.,+2 PP4 cos. C-2.4 +..
+ &c. &c.
(P2 + 2 P1 (P2 cos. t +Pg 2 os. 113 +. O L 1 +
P2a2 +P3a+3    +....   +P P   P3 2+P42 +...
a,-a              + 2PP + 22P4P+...
If the value of P, involved in this equation be expanded by Lagrange's
theorem *, in a series ascending by powers of x, and terms involving powers
above the first be omitted, we shall obtain the following value of that
quantity:p __P2ac2+Pa+...
~1-      ---- a, --
a-(Pa2 + P3a3+ P4a4+...)2
--  (P2a2 + Pa3 P4a....).
(\;'l j      (P cos. 1,2 +Pcos. 11.3+P4COS. 1.4+4..)
+ P  2 +P2  + 42 +....
+ 2 P2P3 cos. t2, 3+- 2 P2P4 cos. %34
+ 2 P3P4C os.,3.a+....
or reducing,
P  P,, A + PA ++...' P(a2 — 2ata.2 Cos. %1.2 + a22)
+ P2(a12 - 2aa3 cos. 1 1.3 + a32)?t.   -+ &c. &c.
a,2     +2 P2 P3{aa3 - a,(a, cos.,.,+ 4 a2 cos. 1.3 4 +a3 0os. 112)
2 PP4{aa4 -a(aeo,( 12 co.+c,., + 2 a cos.  os.'1.2)
+& +c. &e.
Now al-22a,a2 cos. 11.2,+ a represents the square of the line joining the
feet of the perpendiculars a, and a2 let fall from the centre of the axis
upon PI and P2; similarly a- -2  3a,a cos..3 4+ a represents the square of
the line joining the feet of the perpendiculars let fall upon PI and P3, and
* This expansion may be effected by squaring both sides of the equation,
solving the quadratic in respect to P,, neglecting powers of X above the first
and reducing; this method, however, is exceedingly laborious.




ROLLING MOTION OF A CYLINDER.              605
so of the rest. Let these lines be represented by L, 2, L.,, L1.4, &c., and let
the different values of the function
{a, a, - a, (a, cos..2-3 + a. cos. 1.3 + a cos.,,)}
be represented by M12., M,-4 M3.4, &c.,    P, a + P3 a3 +...  X (P.LI.22 + PaL332 + P42L.42 +... 
*'*la-,        aa,   + 2PP,+M t 2PP4M2.4+...j 
NOTE E.
ON THE ROLLING MOTION OF A CYLINDER.
(From a memoir printed in the Transactions of the Royal Society for
1851, part II.)
TIE oscillatory motion of a heterogeneous cylinder rolling on a horizontal
plane has been investigated by EULER.* He has determined the pressure
of the cylinder on the plane at any period of the oscillation, and the time
of completing an oscillation when the arcs of oscillation are small.
The forms under which the cylinder enters into the composition of
machinery are so various, and its uses so important, that I have thought it
desirable to extend this inquiry, and in the following paper I have sought
to include in the discussion the case of the continuous rolling of the cylinder, and to determine1st. The time occupied by a heterogeneous cylinder in rolling continuously through any given space.
2ndly. The time occupied in its oscillation through any given arc.
3rdly. Its pressure, when thus rolling continuously, on the horizontal
plane on which it rolls.
Under the second and third heads this discussion has a practical application to the theory of the pendulum; determining the time occupied in
the oscillations of a pendulum through any given arc, whether it rests
on a cylindrical axis or on knife-edges, and the circumstances under
which it will jump or slip on its bearings; and under the first and third,
to the stability and the lateral oscillations of locomotive engines in rapid
motion, whose driving-wheels are, by reason of their cranked axles, untruly
balanced.
* Nova Acta Acad. Petropol. 1788. " De motu oscillatorio circa axem cylindricum plano horizontali incumbentem."




656                         APPENDIX.
Let AMB represent the section of a heterogeneous cylinder through
its centre of gravity G and perpendicular to its axis 0; and let M be its
point of contact, at any time, with the horizontal plane BD on which it is rolling.
B         Assume
~  Jc              a = AC, hA CG, 0 = ACM.
Ak  l  y~ ~W = weight of cylinder. Wk = momentum of inertia of the cylinder about
~;sr   X          n             an axis passing through G and
parallel to the axis of the cylinder.
Q   given value of the angular velocity -d  when 0 has the given
value 01.,0 = given value of 0 when the angular velocity has the given value o.
I   given value of GM corresponding to the value 01 of 0.
Then W (7( 4G- GM2) = W(k2  + a2 + h2 - 2aa cos. 0) = moment of inertia
about M. Since moreover the cylinder may be considered to be in the act
of revolving about the point M by which it is in contact with the plane,
one-half of its vis viva is represented by the formula....                /do\2
1W (k2 + 2 - 2ah cos. 0 +  2)dt 
2g
and one-half of the vis viva acquired by it in rolling through the angle
0- 0, by
g    (P +a2-2ah os. O    2)(G ) -(P+ 2)
But the vertical descent of the centre of gravity while the cylinder is
passing from the one position into the other, is represented by
h (cos. 0 -  os. 0,).
Therefore, by the principle of vis viva,*. W  (k2 + 2 _ 2ah cos. 0 + h)() -   ( + 12)'\ = WA (cos. 0 - cos. o,),
2 g {                      \dt/            t
whence we obtain
rd- 2   2gh (cos. 0 - cos. 0,) + (72 + 15) c,
dt/        Po + a2 - 2ah cos. o + h2
2  ah     + a -- cos. 
et  t    + o +...
* POIesON, Dynamique, 2"m partie, 565.; PONCELET, Mecanique Industrielle,
or Art. (129.) of this Work.




ROLLING MOTION OF A CYLINDER.  657
= O 2gh ) (..t=  y01 os.   d,-..... ().
ak2 j \ co s.   -   - 2  
and.  + a2 h2>2ahcos.0 (+2),
Coand  s(a+ha)>cos.O -P -C +..2
0
where t represents the time of the body's passing from the inclination 0, to
zero.
Sow let it be observed that in this function a> 8 so long as a is less than
g, since
7T 2 + > -w-(h +/e)w, or cP a - _ 2a co. 0 + sec> — ( 1  )d 2,
and.'.+ k  +A~  + > 2ah cos.  _ -(ks + l),W2,
and    1f % + a+ -> cos. 0: —-1-.2
c2\( a   a2 kh
1+ac     1 — a   o-cos. 0
Let    _      -O. P -       -     se..
i +  ~ P  I —2 P  dos. o-P
also
When o =       o(1-, let =  2,ver,
irk-2  7   ax
(  )2            +-a + 2g- vers.' 0).,-.sec. a-COS.0\ -1  y  -os
Now      /Q (coso) ) df (cos.0-)   *
sec   cos.  0 -- C O+ 12
0             02
also
And since _  == 0  sec. 0,
2cos. - -( ++3) cos.2 +* (a-)   cos.2 0 + q2        h
C. (f+)(cos.2 0  + q) +2g  vers. o  + (-+ ('0.2  c os.  (cos. —— qcos. -   de
o o
a -- Cos. o
And since  -   2 see.' 2,
cos. o -- P
2 cos. o -- (0  + p) cos.2 -- q2
(a _.  )  -  cos.~ 2   + q2,
~-~+p)(cos.2 +~) +(m — )(cos.' 2p —O
(cos.' ~ +  2)
42




658                   APPENDIX.
c     cos. 0=  2.. (6).
in 0 = _ (cos.0 +. q)-2_ -(a cos0.2 ~, + 3q2)2
(COS.2 q + q2)2.
(cos.' + q -  o~ 0 c.2 -- 3q)(cos.2 0 + q2 + a co08.2 +3 g2)
(Cos.2 2  + q2)2
{(1-/3) + (1- a) cos.2 0}{(l +f)q2 +(1 + 1) c0.2 0}
(g2 + COS.2 0)2
=(1 - f2)q. sin.2 20 +p COS.2 );
(2+ COS.2 0)2
(q2 +p' COS.2 O)..sn.sin. 0=q(1 — + sin......(7).'Now
do   do   d cos. 0 d cos. 0  sin.  d cos. 0
d -d cos. 0' d cos.  d4  sin. 0 d cos. * * 
Also by equation (6.),
d cos. 0 2c(q + cos.2 ) cos. Q-2( cos.20 + 3q) cos. q_2(a-3)q2os. 9.
d cos. -          (q2 + COS.2 0)2        (2 + COS.2 q9)2..by equations (7.) and (8.),
do  2( -- 3)q2  q + o.2 os.2. 0
du - (1 — p2)q' (q2 +p2 COS.2 )  * (q2+ os.2 >)2
_2(a- p)g         cos. 0
- (1-2)    (q2+ tco.2 ) )(q2 +p2 COS.2 0), f cos. \dO 2( a-)q2            1' cos. — 0  dp- (1 -2)' (q2+cos.2 p) (q2 +2 cos.2 0) 
-    (1 —8('  (2+ 1-sin.' s)(q2 + p2 p2sin.2 )i
2(ma -  - )q2               1 
_(1 p2))(p2(+f2)(1+2)  (     n1 i)(1- p+q2 sin. 2
= l —y( m - p2q"            I 1( ( nin.. )(  -6 S..   ) 
-- (  1-p2) —2)- ( +   q 2) (1-n. sin.2 )(1 — sin. 
If
n= 1                      (9).
1+      1    1     -ac
- I+^="- i —i        * —'...
1 —
~u1a
+2 1-+-a     1 — +    2(a —g)      ()
1+#   1 —




ROLLING MOTION OF A CYLINDER.     959
f  -- cos. 0do
2f(aiS)q    rCIO
*= (1__2)i(p2+2)i(l q 2)(1  I  n Sn2 0)( -C2sin.2 0)k
2(ac - )q2
(1  2) 1 (p2+qf2) (+ q 2
where nr( —nc,) is that elliptic function of the third order whose parameter is -n and modulus c.
2)k                      1 1Now
P      { +  () — (/1 +  (12(-)
Ipflql(k1 -l —
q2     (1-c)    ___
-2          _ =,a — 1 --
2(a, — _)q2   2(, - 1)
(1'-2)-@2 + 2),(l + q q9-V/()  -),   — +- + -- 2
1k22 a h          (k2 + ~2)02
/ ah h+a  2 cos. 01 +
2 + (a — h)2
=/.h(k' + +2)(1 +... by equations 11. and 4.
k2 + (a _ h)2
t- — /- -2. I(-nC~,)...(13).
Jgh(k2 + l (1 +g)
where (9.) (2.) (3.)
1-/?     1- Cos, 01+ 2g     2hvers.0, ~++ g. 
= —--— ~l~2 a      2 -a               (14).
v+   aA312                      a~~~
* I cannot find that this function has before been integrated, except. in the
ease in which 0 is exceedingly small.




660                         APPENDIX.
and (10.) (2.) (8.)
(1  V     h  a                k 2 +   12+,
\ AM                            1 (a + h)2 a  vers.+j vers.1 ++  0, + 2  A
(+ (1 3        (k2 +t2)(1 + -
ka2+   1)- flvers.o+       - o + k2  (15).
2(2+2)(l + 2    )
The value of I(- — nc,) being determinable by known methods (LEGENDRE, Fonctions Elliptiques, vol. i. chap. xxxiii.), the time of rolling is given
by equation 13.
In the case in which the rolling motion is not continuous but oscillatory,
we have co   0; and therefore (equation 5.) q, = 2;  (- nc#,) becomes
2
therefore in this case a complete function.
To express the value of this complete elliptic function of the third order
in terms of functions of the first and second orders, let
n    2        2ah,
sin.2r   ==     -2    2a... (16).
Then*
nJ()    \)=r     ))I7E                            E    ) tanp- \
= (E            2 +        sin. 4 F    c 2  E(c+)-() Z) t
Representing therefore the time of a semi-oscillation by t,,
k2_+(a_ -          +    tan. p   Fi'A\1Ev2       h&
/ gh (2 + 12)i2                               -E     (           * * *i. 
2' 4(a+ h)'
where (15.)      c' =  2(k+ 1) vers. 0,..(18).
Since the values of elliptic functions of the first and second orders,
having given amplitudes and moduli, are given by the tables of LEGENDRE,
it follows that the value of t is given by this formula for all possible values
of o and,.
If the angle of oscillation 0, be very small c is very small, so that its
square may be neglected in comparison with unity. In this case
* LEGENDRE, Calcus des Fonctions Elliptiques, vol. i. chap. xxiii. Art. 116.




ROLLING MOTION OF A CYLINDER.                   661
Fc = E?4 = 4 and Fe -t = Ec -= -
2      22..Fe  Ec — E ec  Fc4 = 0.
2          2
For small oscillations therefore
t   7c2 +(a(   y )2....(19.
/gh(k2 +2) 2
If the pendulum oscillate on knife-edges a = 0, = hA, and we obtain the
well-known theorem of LEGENDRE (Fonctions Elliptiques, vol. i. chap.
viii.)
t  -    \..!k + h&2
____        F(e A),.. (20).
1              Q2
where (18.)       < = - vers. 01 =sin.2 0,
2              2:.    -C= sin. 0,.... (21).
In the case of the small oscillations of a pendulum resting on knife-edge,
equation 20. becomes
t= g   uh'",    (22).
which is the well-known formula applicable to that case.
If the pendulum be one which for small arcs beats seconds (21.),
/' + h.. (20.) 2...(2).
by which equation the time of the oscillation through any are, of a pendulum which oscillates through a small arc in one second, may be determined. I have caused the following table to be calculated from it.




662                         APPENDIX.
Table of the time occupied in oscillating through every two degrees of a
complete circle, by a pendulum which oscillates through a small arc in
one second.
Are. of oscillation  Time of one  Arc of oscillation  Time of one  Arc of oscillation  Time of one
on eachl side of  complete  on each side of  complete  on each side of  complete
the vertical  oscillation in  tie vertical sillion in  the vertical  oscillation in
in degrees.  seconds.  in degrees.  seconds.  in degrees.  seconds.
2       1.1899        32       1 3905       62        1 8032
4       1 2000        34       1'4089       64        1-8478
6       1.2123        36       1'4283       66        1'8963
8       1 2210        38       1-4486       68        1-9491
10       1-2322       40        1'4698       70        2'0075
12       1 2439        42       1'4922       72        2a0724
14       1 2560        44       1 5157       74        2'1453
16       1-2686        46       1 5405       76        2-2285
18       1 2817        48       1 5667'8        2'3248
20       1-2953        50       1-5944       80        2'4393
22       1-3099        52       1-6238       82        2-5801
24       1 3249        54       1-6551       84        2-7621
26       1-3400        56       1'6884       86        3-0193
28       1 3560        58   1'7240           88        3-4600
30       1-3729        60       1'7622        90      Infinite.
The pressure of the cylinder on its point of contact with the plane on which
it rolls.
Let A' be the point where the point A of the cylinder was in contact
with the plane.
Let A'N = x, NG = y.
f t            — X = horizontal pressures on M in direction
G, J                     A'M.
i/tA.[             Y = vertical pressure on M in direction MC.
3      M. N A" D       Since the centre of gravity G moves as it would
do if, the whole mass being collected there, all the impressed forces were
applied to it, we have, by the principle of D'ALEMBERT,
*W d2X
g dtP
Wd a22J Y   Tm i..... (28).
W d y_
But since CA = a, OG = h, MCA = 0,
a. o.   - a- h sin. 0,
y= a-hcos.0;
dx              d
( (A- cos. 0) dt
* Art. 96, equations (73.), (74.).




ROLLING MOTION OF A CYLINDER.     663
4       doe
= h sin. e0 -.. dt +A sin.01dti + (a-h cos. a-0....(29).
dty, =a       h.:odt2
by equation (29.),
dt - Msin.0 -N( a-    ) cos. 0)
dt --
-_y = MA cos. 0-NA sin. 0;
by equation (28.),
-      -.h s-in. o +N(-os. )0).
Y=W+ W- Mcos.o-Nsin.0o
But by equation (1.), substituting -0 and - 0 for 0 and 0,,
yi=(dM69x =- -3- t,..... (31).
do~ 2 gh(cos. -cos. o) + (.2+ 2).. (1).
tl /\dt -  2+ a2 2ah cos. 0 + h', 2ah(cos. 0 - cos. 0,) + (k + t2)  +e
a \     - k+ a-2ahcos.0+ h2
+a2+ hA- 2ah cos. 0 + (k' 2+ 2)-(+(k+ +h2-~-2ah cos. 0)
_a             P k2/i' + a' - 2ah cos. 0 + A
" _^D1 (I~')0(1k   ) - _?
a..) C-) +a.++h'-2ah cos.1 
Observing that a2 + h - 2ah cos. 0, = 12.
Differentiating this equation and dividing by( )
— dN- \V -        d p+,-+_2aA cos.e




664                          APPENDIX.
Substituting these values of M and N in equation (30.), and reducing,
W       sin. 0 1  (k   )(k+  - 2ah cos. 0)(g + ao2)
x na  0 1-    g(k+ +a +     h2 ah 2 cos, e0)2...
Y=-j4 ('la      (As + l2)(g + aco2){ah cos.2 0 -(A + a + h2) cos. 0 + ah} it (
a   \a   Cos.              g(A + a + h' - 2ah cos. 0)'
The rotation of a body about a cylindrical axis of small diameter,
Assuming a = 0 in equations (31.), (33.), and 01=0, we have
2gh(cos. 0-1).     N    gh sin. 0
=-.2         +            - h3. 2'
Therefore, by equation (30.),
X  WhA g2 —83 cos. 0) I2 tsin.... (40).
Wh gh(3 cos:Q o - 2 cosT  - 1)
=y _W+c                     o s     + WI cos. 0.... (41).
Y Z=W    ^   ^( co    ^  yos.^l)^ h'
The last equation may be placed under the form
~  -  3Wh2 e s 1/P+h2 2_,)2 1/2( h \ 3}
Y=W + t         icos.8 -0 +                        to  1   -
If.-(,2 (2'- 1) be numerically less than unity, whether it be positive
or negative, there will be some value of 0 between 0 and Xt for which this
expression will be equalled, with an opposite sign, by cos. 0, and for which
the first term under the bracket in the value of Y will vanish. This corresponds to a minimum value of Y represented by the formula
Y=W —    Wh J -  l-k     + nhe   a 1 2...g)   (42).
But if -2s    - 11) be numerically greater than unity, then the
3in    2g h
minimum of Y will be attained when 0 =- t, and when
^Y =      ^    " *-F 4gh.   (4^)
g        9+P A     0''




ROLLING MOTION OF A CYLINDER.                  665
The Jump of an Axis.
If Y be negative in any position of the body, the axis will obviously
jump from its bearings, unless it be retained by some mechanical expedient not taken account of in this calculation. But if Y be negative in
any position, it must be negative in that in which its value is a minimum.
If a jump take place at all, therefore, it will take place when Y is a minimum; and whether it will take place or not, is determined by finding
whether the minimum value of Y is negative. If therefore the expression
(42.) or (43.) be negative, the axis will jump in the corresponding case.
An axis of infinitely small diameter, such as we have here supposed,
becomes a fixed axis; and the pressure upon a fixed axis, supposed to
turn in cylindrical bearings without friction, is the same, whatever may
be its diameter; equations (40.) and (41.) determine therefore that pressure, and equation (42.) or (43.) determines the vertical strain upon the
collar when the tendency of the axis to jump from its bearings is the
greatest.
The Jump of a Rolling Cylinder.
Whether a jump will or will not take place, has been shown to be determined by finding whether the minimum value of Y be negative or not.
Substituting a for (\+- h -   and reducing, equation (35.) becomes
Y - T(1      cos\  W(h2 + l')(qg + 2a)  cos.2 0-2a cos.0 +1
\    a W   1 -                 \         COS.0)2
or
Y    -,,-h \s       W(~ - +) +))( f1f2  ) {  -G  92    **4
W ( — cos. 0 --            4         I         Cos. ).d  =WY   h, (P + P) ( +X a)(X9). - 1) osin....(45).
Wdo            2g.i'(- cos. 0)3'cos 2g(-           0    smi. 6,
dY                h   (P,  + L)(g +    a0n2d)(w —n) 
-0,       st when-, 2ndly, when 0 =-,
dit - I         aI    2gda.(a_-cos. 0)'
3rdly, when 0 = 0.
dd dd
The first condition evidently yields a positive value of dt-' since it




666                         APPENDIX.
causes the first term of the preceding equation to vanish; and the second
term is essentially positive, o being always greater than unity.
If, therefore, the first condition be possible, or if there be any value of
0 which satisfies it, that value corresponds to a position of minimum pressure. Solving, in respect to cos. 0, we obtain
( +1)(g          = + a )')(a —- 1)
a -/ ( —+ 12)(g + a)(c -1) = cos. 0.... (46).
-"v  ~ ~2gah
The first condition will therefore yield a position of minimum pressure, if
/('+ l2)(g     2)(a1) > >-1      /f 8(A + 12)( + a,-.)2)(a2 _-1) <(oa+l)
"-V           2gah <2         +    or if 02gah                  > (a- 1)
or if' - 2gah     > (a-1)3, or i    2ga (ag+ 1)      < 1
and (k +')(g + ac )(a+ ) >1.. (
2gahiZ       > (a'1)"         (ga) (a.+
and(k- + ) (g + a-o-) + —- >1
2gah(a-1)'
or if
2 2gah(a+ 1)'          2gh(a + 1)2.
g+C   < (f      )a-1 )' or o <     (   1) (a-l) -a
and
g 2gah(a- 1)2        2    2gh(ao - 1)'  g.
g+[a  > (k2+2F)( +l) or     > (k" +12)(a+t) -a'
whence, substituting for a and reducing, we obtain finally, the conditions
~a                                                    b i  + (a + h)g2 9G,  (a g  {V+(a-^)227  9 A
<       1 () 1 ( +(+     (-   ) ando >a (   2+)( 1T2)4..2 (a. h)  () ()1
Of these inequalities the second always obtains, because
lk2+(a- h^)21 2 < (2+ 12) {k2+(a+h21,
whatever be the values of k, a and h. And the first is always possible,
since
IV +(a + h)I? 2 > (k2 + 12) IkA + (a - h)'7  
If the first obtain, there are two corresponding positions of CA on either
side of the vertical, determined by equation (46.), in which the pressure Y
of the cylinder upon the plane is a minimum.




ROLLING MOTION OF A CYLINDER.                  667
IY
Substituting the other two values (A and 0) of 0 which cause - to
d2Y
vanish in the value of -o we obtain the values
7_(h ( + 12) (g+a-') (a-1)) and   (2 + 12) (g +  2) (  1) 
-  a     2gc2(a + 1)2   5       a      2ga2(a-1) 
or
ah: 1(k2+ l2)^ + aO2)(a-1) and  ( (k2 + 12)(g + a2)(c + 1)   9
a         2gh2a(a + 1)2 -  ad          2gh-a(2w-l1)2-.(49).
which expressions are both negative if the inequalities (47.) obtain. The
same conditions which yield minimum values of Y in two corresponding
oblique positions of CA, yield, therefore, maximum values in the two vertical positions; so that if the inequalities (48.) obtain, there are two positions of maximum and two of minimum pressure.
Substituting the values of cos. 0 (equation 46.) in equation (44.), and
reducing, we obtain for the minimum value of Y in the case in which the
inequalities (48.) obtain,
=W  l2 (a-2-7)-(1    + 12)( + a
+3 /(^++ 12) (+(a+h)21 {k2(a     +)2 (     ).
If this expression be negative the cylinder will jump.
In the case in which X = 0, which is that of a pendulum having a cylindrical axis of finite diameter, it becomes
_w l 2a_ —2h2- 38 —2+3 V(k2+t2){k2+(+h)+(a+-){        +    2} (a *...(50).
If the first of the inequalities (48.) do not obtain, no position of minimum pressure corresponds to equation (46.); and the inequalities (47.) do
d2Y
not obtain, so that the values (49.) of -2', given respectively by the substitution of Ct and 0 for 0, are no longer both negative, but the second only.
In this case the value A of 0 is that, therefore, which corresponds to a position of minimum pressure, which minimum pressure is determined by
substituting 7 for 0 in equation (35.), and is represented by
Y:W(1     a +    (M         +          ) —1 (k  ++ l)(+2
a X               1+ = 2SW   a -- h   ^h (I, + 12) (q +1 a- )
* When the pendulum oscillates on knife-edges a=0, and this expression
assumes the form of a vanishing fraction, whose value may be determined by
the known rules. See the next article




668                  APPENDIX.
=W   a    h A + (a+h) —4ah cos.21',  (g+ a'2)
gt  g   2
a + --   -----   - 2_h;
2 +(a(a+h)' 
W (, h.,,,1 )       }
4 ah l + —)cOS 2i4h cos.2,        4hcos. 2,.' Y=W {1 —   1  v-+ lc' /+2.... (61).h
g > +     7s2+(a+h        f'  &  +(a+A  >1 +)    *2
position ofthe centre of gravity,, and cos.  =  In this case,
4  I + a4)co8.2 1 ^
therefore (equation 51.)
Y=  w(1 —);... (52).
and there will be a jump if tb> h... (58).
Te Pendulum oscillating on Knife-edges.
In this case a is evanescent, and c=0. Equations (31.) and (33.)
become, therefore,
M  2g (cos. O-cos. o,) and = ghsin.
Ik( +             h       2+h2
Substituting these values of Md and N in equation (30.),
yWh-t-. +>2 (cos. 0- cos. 0) sin. 0 cos. 0 sin. 
Y=W t  o +  hJ  (cos. 0- cos. 0) cos. 0- sin.0;
*  x- 2 ( c -.   n.
Wh'.'. X  =r -w-b 2 (2 cos. 0,-  3 cos. ) sin.... (54).




ROLLING MOTION OF A CYLINDER.                   669
W A2 /                        2
Y   2+ h2 3 cos. 0-2 cos. cos.. 0 +. (55),
Y is a minimum when cos. o =  eos. 1,, in which case
3
Y      h f W  1 cos.'2o,)... (56).
There will therefore be a jump of the pendulum upon its bearings at
each oscillation if the amplitude 0, of the oscillation be such, that
-cos.2, > Vor cos.20,>.
The Jump of the falely-balanced Carriage-wheel.
The theory of the falsely-balanced carriage-wheel differs from that of
the rolling cylinder,-1-st, in that the inertia of the carriage applied at its
axle influences the acceleration produced by the weight of the wheel, as
its centre of gravity descends or ascends in rolling; and, 2ndly, in that
the wheel is retained in contact with the plane by the weight of the carriage. The first cause may be neglected, because the displacement of the
centre of gravity is always in the carriage-wheel very small, and because
the angular velocity is, compared with it, very great.
If W1 represent that portion of the weight of the carriage which must
be overcome in order that the wheel may jump (which weight is supposed
to be borne by the plane), and if Y, be taken to represent the pressure
upon the plane, then (equation 52.)
Yl =Wi+Y==W,+w        l      ).... (57).
In order that there may be a jump, this expression must be negative.
or
^    > W+W, or >2>     l+    )      (58).
or
7(i>^)...WI




670                        APPENDIX.
The Driving- Wheel of a Locomotive Engine.
The attention of engineers was some years since directed to the effects
which might result from the false balancing of a wheel by accidents on
railways, which appeared to be occasioned by a tendency to jump in the
driving-wheels of the engines. The cranked axle in all cases destroys the
balance of the driving-wheel unless a counterpoise be applied; at that time
there was no counterpoise, and the axle was so cranked as to displace the
centre of gravity more than it does now. Mr. GEORGE HEATON, of Birmingham, appears to have been principally instrumental in causing the
danger of this false-balancing of the driving-wheels to be understood. By
means of an ingenious apparatus*, which enabled him to roll a falselybalanced wheel round the circumference of a table with any given velocity,
and to make any required displacement of the centre of gravity, he showed
the tendency to jump, produced even by a very small displacement, to be
so great, as to leave no doubt on the minds of practical men as to the
danger of such displacement in the case of locomotive engines, and a counterpoise is now, I believe, always applied. To determine what is the
degree of accuracy required'in such a counterpoise, I have calculated from
the preceding formulae that displacement of the centre of gravity of a
driving-wheel of a locomotive-engine, which is necessary to cause it to
jump at the high velocities not unfrequently attained at some parts of the
journey of an express train; from such information as I have been able to
obtain as to the dimensions of such wheels, and their weights, and those
of the enginest. The weight of a pair of driving-wheels, six feet in
diameter, with a cranked axle, varies, I am told, from 21 to 3 tons; and
that of an engine on the London and Birmingham Railway, when filled with
water, from 20 to 25 tons. If n represent the number of miles per hour
at which the engine is travelling, it may be shown by a simple calculation,
22n
that the angular velocity, in feet, of a six-feet wheel is represented by ---
or by I n very nearly. In this case we have, therefore,-since W represents
2
the weight of a single wheel and its portion of the axle, and W, represents
the weight, exclusive of the driving-wheels, which must be raised that
* This apparatus was exhibited by the late Professor CowPER to illustrate his
Lectures on Machinery at King's College. It has also been placed by General
MORIN among the apparatus of the Conservatoire des Arts et M6tiers at Paris,
f I have not included in this calculation the inertia of the crank rods, of the
slide gearing, or of the piston and piston rods. The effect of these is to increase
the tendency to jump produced by the displacement of the centre of gravity
of the wheel; and the like effect is due to the thrust upon the piston rod.
The discussion of these subjects does not belong to my present paper.




ROLLING MOTION OF A CYLINDER.                  671
either side of the engine may jump*, that is, half the weight of the engine
exclusive of the driving-wheels,-W= li to 1~ tons, W, = 81 to 111 tons,
= -n, g = 8219084 whence I have made the following calculations from
formula (59.).
lj~~~~~ ~~~Displacement of the centre of gravity 
I'fl~~~~~~ ~~~~of a six-feet driving-wheel which
will cause a jump of the wheel on
Weight of             Formula (59)             the rail.
te engine in weight of a  reduced.
ud engie pair of wheels         WI________________________
with cranked'128'76 1 +
t ngs thenldv  t axle, in tons.  W    Rate of travelling in miles per hour.
eng wheels. ae, in tons. I>
50.      60.       70.
2-5        1030-08        412S8     2867      2106
20 
3                       385844 3344  2384   *1'51
1287.6
2'5'5150                 35'76     2628
25
1073. 3                      *4292'2908        2189
It appears, by formula (59.), that the displacement of the centre of
gravity necessary to produce a jump at any given speed, is not dependent
on the actual weight of the engine or the wheels, but on the ratio of their
weights; and, from the above table, that when the weight of the engine
and wheels is 61 times that of the driving-wheels, a displacement of 21
inches in the centre of gravity is enough to create a jump when the train
is travelling at sixty miles an hour, or of two inches when it is travelling
at seventy miles; this displacement varying inversely as the square of the
velocity is less, other things being the same, as the square of the diameter
of the wheel is less; for the radius of the wheel being represented by a,
22n
the angular velocity is represented byo = 1-5  and substituting this value,
-5a)
formula (59.) becomes
2 ga   1+
/15\)  _    +.
" It will be observed, that the cranks being placed on the axle at right angles
to one another, when the centre of gravity on the one side is in a favourable po



672                        APPENDIX.
If the weight W of the wheel be supposed to vary as the square of its
diameter and be represented by lia2, this formula will become
(15)2U(a3+W'
2  n2
still showing the displacement of the centre of gravity necessary to produce a jump to diminish with the diameter of the wheel. These conclusions are opposed to the use of light engines and small driving-wheels;
and they show the necessity of a careful attention to the true balancing of
the wheels of the carriages as well as the driving-wheels of the engine.
It does not follow that every jump of the wheel would be high enough to
lift the edge of the flange off the rail; the determination of the height of
the jump involves an independent investigation. Every jump nevertheless
creates an oscillation of the springs, which oscillation will not of necessity
be completed when the jump returns; but as the jumps are made alternately on opposite sides of the engine, it is probable that they may, and
that after a time they will, so synchronise with the times of the oscillations,
as that the amplitude of each oscillation shall be increased by every jump,
and a rocking motion be communicated to the engine attended with
danger.
Whilst every jump does not necessarily cause the wheel to run off the
rail, it nevertheless causes it to slip upon it, for before the wheel jumps
it is clear that it must have ceased to have any hold upon the rail or any
friction.
The Slip of the Wheel.
If f/be taken to represent the coefficient of friction between the surface
of the wheel and that of the rail, the actual friction in any position of the
wheel will be represented by Yf. But the friction which it is necessary
the rail should supply, in order that the rolling of the wheel may be maintained, is X. It is a condition therefore necessary to the wheel not slipping that
X
Yf > X, orf > y,.... (60).
If, therefore, taking the maximum value of  in any revolution, we find
Y1
thatf exceeds it, it is certain that the wheel cannot have slipped in that
revolution; whilst if, on the other hand,/ falls short of it, it must have
sition for jumping, it is in an unfavourable position on the other side, so that
it can only jump on one side at once, and the efforts on the two sides alternate'




ROLLING MOTION OF A CYLINDER.                   673
slipped.*  The positions between which the slipping will take place continually, are determined by solving, in respect to cos. 01, the equation
f=...(61).
The application of these principles to the slip of the carriage-wheel is
rendered less difficult by the fact, that the value of h is always in that case
so small, as compared with the values of k and a, that - may be neglected
a
in formule (34.) and (35.), as compared with unity. Those equations
then become
x   Wh sin.0 e   k (g +          (62).
X.+a 2) (6).
and
Y=a- acos. 0 + (g + C 2) cos. 0 }=w  l    2 
a     7           9       )
whence we obtain
Y=, —Wl1+    I + -.... (63).
and
X Wh    7cI (lg,,+ wt  sin. g0- 1      +    sin. 
- a          gikj   ) -1.      2        (a)g(... (64).
Y.7 W~+W 1+                      + (1 +)^.!, ( 1~+)      e~s.~c
Assume
W\ g             sin. 0
W 1 +    )2 and       3 + os. 0
du _    +  cos. 0  d2u {-   (3 + cos.0) + 2 (1 +  cos.0)} sin. 
do (P +cos. )2   d                 (P 0+cos.)3
Now if p>1, there will be some value of 0 for which  + cos. 0, and
therefore 1+-  cos. 0 =0; and since for this value of 0, d  o and* Of course, the slipping, in the case of the driving-wheels of a locomotive,
is diminished by the fact, that whilst one wheel is not biting upon the rail
the other is.
43




6T4:                       APPENDIX.
132
= —(   l)sit follows that it corresponds to a maximum value of u, and
therefore ofX.
Y,
But if a < 1, then there is some value of cos. 0 for which P + cos. 0 = 0,
and therefore for which u = infinity, which value corresponds therefore in
this case to the maximum ofX.
Y,
Thus then it appears that according as
por /3g <1or (    + W_)    ( g  > + W
x                                     1
the maximum value of  is attained when cos. 0=-    or= —; that
is, when
cos. =-( —-+g 1+    or=-        2.... (66).
(       +1)
In the one case the maximum value of y will be infinity,.... (67).
and in the other case it will be represented by the formula
_ (g     + 2) +
Y, -.,)_..
In the first case, i. e. when p < 1, the wheel will slip every time that it
revolves, whatever may be the value of f. In the second case, or when
X > 1, it will slip if f do not exceed the number represented by formula
(68.). The conditions (65.) are obviously the same with those (59.) which
determine whether there be a jump or not, which agrees with an observation in the preceding article, to the effect, that as the wheel must cease
to bite upon the rail before it can jump, it must always slip before it
can jump. When the conditions of slipping obtain, one of the wheels
always biting when the other is slipping, and the slips of the two wheels
alternating, it is evident that the engine will be impelled forwards, at
certain periods of each revolution, by one wheel only, and at others, by
the other wheel only; and that this is true irrespective of the action of
the two pistons on the crank, and would be true if the steam were thrown
off. Such alternate propulsions on the two sides of the train cannot but




DESCENT UPON INCLINED PLANE.                 675
communicate alternate oscillations to the buffer-springs, the intervals
between which will not be the same as those between the propulsions;
but they may so synchronise with a series of propulsions as that the
amplitude of each oscillation miay be increased by them until the train
attains that fish-tail motion with which railway travellers are familiar.
It is obvious that the results shown here to follow from a displacement of
the centres of gravity of the driving-wheels, cannot fail also to be produced by the alternate action of the connecting rods at the most favorable
driving points of the crank and at the dead points,* and that the operation
of these two causes may tend to neutralize or may exaggerate one another.
It is not the object of this paper to discuss the question under this point
of view.
NOTE F.
ON THE DESCENT UPON AN INCLINED PLANE OF A BODY SUBJECT TO VARIATIONS OF TEMPERATURE, AND ON THE MOTION OF GLACIERS.
IF we conceive two bodies of the same form and dimensions (cubes, for
instance), and of the same material, to be placed upon a uniform horizontal plane and connected by a substance which alternately extends and
contracts itself, as does a metallic rod when subjected to variations of
temperature, it is evident that by the extension of the intervening rod
each will be made to recede from the other by the same distance, and,
by its contraction, to approach it by the same distance. But if they be
placed on an inclined plane (one being lower than the other) then when
by the increased temperature of the rod its tendency to extend becomes
sufficient to push the lower of the two bodies downwards, it will not have
become sufficient to push the higher upwards. The effect of its extension will therefore be to cause the lower of the two bodies to descend
whilst the higher remains at rest. The converse of this will result from
contraction; for when the contractile force becomes sufficient to pull the
upper body down the plane it will not have become sufficient to pull
the lower up it. Thus, in the contraction of the substance which intervenes between the two bodies, the lower will remain at rest whilst the
upper descends. As often, then, as the expansion and contraction is
repeated the two bodies will descend the plane until, step by step, they
reach the bottom.
* A slip of the wheel may thus be, and probably is, produced at each revolution.




676                        APPENDIX.
Suppose the uniform bar AB placed on an inclined plane, and subject
to extension from increase of temperature, a portion XB will descend, and the rest XA will ascend;
the point X where they separate being determined.....by the condition that the force requisite to push
XA up the plane is equal to that required to push
XB down it.
Let AX = x, AB = L, weight of each linear unit =, ~ = inclination
of plane, = limiting angle of resistance.. ux = weight of AX.
p(L- ) = BX.
Now, the force acting parallel to an inclined plane which is necessary
to push a weight W up it, is represented by W   -s'(-); and that ne.
cos. q 
cessary to push it down the plane by Wsin(I. (Art. 241.)
cos. 0
sin. (p + )         sin. (0 - )..$.  --   ==(L-a) -— )
cos. 0              cos. *'. *. {sin. (a + ) 4+ sin. ( -  )} = L sin. (9 - 0. *.x sin. 9 cos. =L sin. (a- )
sin. (;0 —)
< X=IL -. —--      i
sin. ( cos. c
s     tan. At'.'.$_=~L 1 I- tan. 
When contraction takes place, the converse of
A</       the above will be true. The separating point X
*x^  ~ ~  will be such, that the force requisite to pull XB up
a -__ ~   the plane is equal to that required to pull AX
down it. BX is obviously in this case equal to AX
in the other.
Let X be the elongation per linear unit under any variation of temperature; then the distance which the point B (fig. 1.)will be made to descend
by this elongation = A. BX
=   (L -)
L(    tan. 
tan.




DESCENT UPON INCLINED PLANE.                 677
If we conceive the bar now to return to its former temperature, contracting by the same amount (X) per linear unit; then the point B
(fig. 2.) will by this contraction be made to ascend through the space
BX..=mX
tan.e'- 1ta-n           I
Total descent I of B by elongation and contraction is therefore determined
by the equation
tan. 
I  L,.... (2)
tan.p:s:         To determine the pressure upon a nail driven;7^^  ~    through the rod at any point P fastening it to the
plane.
It is evident, that in the act of extension the part BP of the rod will
descend the plane and the part AP ascend; and conversely in the act of
contraction; and that in the former case the nail B will sustain a pressure
upwards equal to that necessary to cause BP to descend, and a pressure
downwards equal to that necessary to cause PA to ascend; so that, assuming the pressure to be downwards, and adopting the same notation as
before, except that AP is represented by p, AB by a, and the pressure
upon the nail (assumed to be downwards) by P, we have in the case of
extension
sin. (  )        sin. (-)
= ^-   cos. -        -os.'
and in the case of contraction,
sin. ( +      sin. (-t)
pos. -  -O-OS..
Reducing, these formulae become respectively,
P=C os   { 2p sin. cos.- a sin.( b — ).  (8).
Cqs. p
P     os. a^ a sin. (0 +  2) - 2p sin. cos. ~..~ (4)
EXAMPLE OF THE DESCENT OF THE LEAD ON THE ROOF OF BRISTOL
CATHEDRAL.
My attention was first drawn to the influence of variations in temperature to cause the descent of a lamina of metal resting on an inclined plane




678                        APPENDIX.
by observing, in the autumn of 1853, that a portion of the lead which
covers the south side of the choir of Bristol Cathedral, which had been
renewed in the year 1851, but had not been properly fastened to the ridge
beam, had descended bodily eighteen inches into the gutter; so that if
plates of lead had not been inserted at the top, a strip of the roof of that
length would have been left exposed to the weather. The sheet of lead
which had so descended measured, from the ridge to the gutter, 19ft. 4in.,
and along the ridge 60ft. The descent had been continually going on
from the time the lead had been laid down. An attempt made to stop it
by driving nails through it into the rafters had failed. The force by
which the lead had been made to descend, whatever it was, had been
found sufficient to draw the nails.* As the pitch of the roof was only
16 ~ it was sufficiently evident that the weight of the lead alone could not
have caused it to descend. Sheet lead, whose surface is in the state of
that used in roofing, will stand firmly upon a surface of planed deal when
inclined at an angle of 30~t, if no'other force than its weight tends to
cause it to descend. The considerations which I have stated in the preceding articles, led me to the conclusion that the daily variations in the
temperature of the lead, exposed as it was to the action of the sun by its
southern aspect, could not but cause it to descend considerably, and the
only question which remained on my mind was, whether this descent
could be so great as was observed. To determine this I took the following data:Mean daily variation of temperature at Bristol in the
month of August; assumed to be the same as at Leith
(Kcemtz Meteorology, by Walker, p. 18.)  -  -   -   8~ 21' Cent.
Linear expansion of lead through 100~ Cent. -  -  -   -0028436.
Length of sheets of lead forming the roof from the ridge
to the gutter      -    -    -                      232 inches.
Inclination of roof  -    -    -16~ 32'.
Limiting angle of resistance between sheet lead and deal -  30~
Whence the mean daily descent of the lead, in inches, in the month of
August, is determined by equation (2.) to be
- The evil was remedied.by placing a beam across the rafters, near the ridge,
and doubling the sheets round it, and fixing their ends with spike-nails.
f This may easily be verified. I give it as the result of a rough experiment
of my own. I am not acquainted with any experiments on the friction of lead
made with sufficient care to be received as authority in this matter. The
friction of copper on oak has, however, been determined by General MORIN
(see a table in the preceding part of this work) to be 0-62, and its limiting angle
of resistance 31~ 48'; so that if the roof of Bristol Cathedral had been inclined
at 31~ instead of 16~, and had been covered with sheets of copper resting on
oak boards, instead of sheets of lead resting on deal, the sheeting would not
have slipped by its weight only.




DESCENT UPON INCLINED PLANE.                 679
8'21            tan. 160 32'
1=232 x     x'0028436X  tan. 80 
l='027848 inches.
This average daily descent gives for the whole month of August a descent
of'863288. If the average daily variation of temperature of the month
of August had continued throughout the year, the lead would have
descended 10-19148 inches every year.  And in the two years from
1851 to 1853 it would have descended 20-38296 inches. But the daily
variations of atmospheric temperature are less in the other months of the
year than in the month of August. For this reason, therefore, the calculation is in excess. For the following reasons it is in defect: -st.,
The daily variations in the temperature of the lead cannot but have been
greater than those of the surrounding atmosphere. It must have been
heated above the surrounding atmosphere by radiation from the sun in
the day-time, or cooled below it by radiation into space at night. 2ndly.,
One variation of temperature only has been assumed to take place every
twenty-four hours, viz. that from the extreme heat of the day to the
extreme cold of the night; whereas such variations are notoriously of
constant occurrence during the twenty-four hours. Each cannot but have
caused a corresponding descent of the lead, and their aggregate result
cannot but have been greater than though the temperature had passed
uniformly (without oscillations backwards and forwards) from one extreme
to the other.
These considerations show, I think, that the causes I have assigned are
sufficient to account for the fact observed. They suggest, moreover, the
possibility that results of importance in meteorology may be obtained
from observing with accuracy the descent of a metallic rod thus placed
upon an inclined plane. That descent would be a measure of the aggregate of the changes of temperature to which the metal was subjected
during the time of observation. As every such change of temperature is
associated with a corresponding development of mechanical action under
the form of work,* it would be a measure of the aggregate of such changes
and of the work so developed during that period. And relations might be
found between measurements so taken in different equal periods of time
-successive years for instance-tending to the development of new
meteorological laws.
* Mr. JOULE has shown (Phil. Trans., 1850, Part I.) that the quantity of heat
capable of raising a pound of water by 1~ Fah. requires for its evolution-'72
units of work.




680                        APPENDIX.
THE DExCENT OF GLACIERS.
The following are the results of recent experiments * on the expansion
of ice:Linear Expansion of Ice for an Interval of 100~ of the Centigrade
Thermometer.
0'00524 Schumacher.
0-00513  Pohrt.
0'00518 Moritz.
Ice, therefore, has nearly twice the expansibility of lead; so that a
sheet of ice would, under similar circumstances, have descended a plane
similarly inclined, twice the distance that the sheet of lead referred to in
the preceding article descended. Glaciers are, on an increased scale,
sheets of ice placed upon the slopes of mountains, and subjected to
atmospheric variations of temperature throughout their masses by variations in the quantity and the temperature of the water, which, flowing
from the surface, everywhere percolates them. That they must from this
cause descend into the valleys, is therefore certain. That portion of the
Mer de Glace of Chamouni which extends from Montanvert to very near
the origin of the Glacier de Lechaud has been accurately observed by
Professor James Forbes.t Its length is 22,600 feet, and its inclination
varies from 4~ 19' 22" to 5~ 5' 53". The Glacier du Geant, from the
Tacul to the Col du Geant, Professor Forbes estimates (but not from his
own observations, or with the same certainty) to be 24,700 feet in length,
and to have a mean inclination of 8~ 46' 40".
According to the observations of De Saussure, the mean daily range
of Reaumur's thermometer in the month of July, at the Col du Geant, is
4~'257t, and at Ohamouni 10~'092. The resistance opposed by the
rugged channel.of a glacier to its descent cannot but be different at different points, and in respect to different glaciers. The following passage
from Professor Forbes's work contains the most authentic information I
am able to find on this subject.  Speaking of the Glacier of la Brenva
he says:-" The ice removed, a layer of fine mud covered the rock, not
composed, however, alone of the clayey limestone mud, but of sharp sand
derived from the granitic moraines of the glacier, and brought down with
it from the opposite side of the valley. Upon examining the face of the
ice removed from contact with the rock, we found it set all over with
sharp angular fragments, from the size of grains of sand to that of a
cherry, or larger, of the same species of rock, and which were so firmly
e Vide Archir. f~ Wissenschaftl. Kunde v. Russland, Bd. vii. s. 338.
j Travels through the Alps of Savoy. Edinburgh, 1853.
4 Quoted by Professor FORBES, p. 231.




DESCENT OF GLACIERS..681
fixed in the ice as to demonstrate the impossibility of such a surface being
forcibly urged forwards without sawing any comparatively soft body
which might be below it. Accordingly, it was not difficult to discover in
the limestone the very grooves and scratches which were in the act of
being made at the time by the pressure of the ice and its contained fragments of stone." (Alps of the Savoy, pp. 203-4.) It is not difficult
from this description to account for the fact that small glaciers are sonme
times seen to lie on a slope of 30~ (p. 35.). The most probable supposition
would indeed fix the limiting angle of resistance between the rock and
the under surface of the ice set all over, as it is described to be, with
particles of sand and small fragments of stone, at about 30~; that being
nearly the slope at which smooth surfaces of calcareous stone will rest on
one another. If we take then 30~ to be the limiting angle of resistance
between the under surface of the Mer de Glace and the rock on which it
rests, and if we assume the same mean daily variation of temperature
(4-257 Reaumur, or 5-321 Centigrade) to obtain throughout the length
of the Glacier du Geant, which De Saussure observed in July, at the
Col du Geant; if, further, we take the linear expansion of ice at 100~
Centigrade to be that ('00524) which was determined by the experiments
of Schumacher, and, lastly, if we assume the Glacier de Geant to descend
as it would if its descent were unopposed by its confluence with the
Glacier de L6chaut; we shall obtain, by substitution in equation (2.)
for the mean daily descent of the Glacier du Geant at the Tacul, the
formula
~00524  tan- 8~ 46'
1= 24700 x 5-321x   - x  — 1   30
I= 1-8395 feet.
The actual descent of the glacier in the centre was 1-5 feet. If the
Glacier de L6chaut descended, at a mean slope of 5~, singly in a sheet of
uniform breadth to Montanvert without receiving the tributary glacier of
the Talefre, or uniting with the Glacier du Geant, its diurnal descent would
be given by the same formula, and would be found to be -95487 feet.
Reasoning similarly with reference to the Glacier du Geant; supposing it
to have continued its course singly from the Col du Geant to Montanvert
without confluence with the Glacier de Lechaut, its length being 40,420
feet, and its mean inclination 6~ 53', its mean diurnal motion I at Montanvert would, by formula (2.) have been 2-3564* feet. The actual mean
daily motion of the united glaciers, between the 1st and the 28th July, was
at Montanvert,t
* On the 1st of July the centre of the actual motion of the Mer de Glace at
Montanvert was 2'25 feet.
t Forbes' " Alps of Savoy," p. 140.




682                         APPENDIX.
Near the side of the glacier - r 1441 feet.
Between the side and the centre - 1750 
Near the centre - --            2'141  "
The motion of the Glacier de Lechaut was therefore accelerated by their
confluence, and that of the Glacier du Geant retarded.  The former is
dragged down by the latter.
I have had the less hesitation in offering this solution of the mechanical
problem of the motion of glaciers, as those hitherto proposed are confessedly imperfect. That of De Saussure, which attributes the descent of
the glacier simply to its weight, is contradicted by the fact that isolated
fragments of the glacier stand firmly on the slope on which the whole
nevertheless descends. It being obvious that if the parts would remain at
rest separately on the bed of the glacier, they would also remain at rest
when united.
That of Professor J. Forbes, which supposes a viscous or semi-fluid
structure of the glacier, is not consistent with the fact that no viscosity is
to be traced in its parts when separated. They appear as solid fragments,
and they cannot acquire in their union properties in this respect which
individually they have not.
Lastly, the theory of Charpentier, which attributes the descent of the
glacier to the daily congelation of the water which percolates it, and the
expansion of its mass consequent thereon, whilst it assigns a cause which,
so far as it operates, cannot, as I have shown, but cause the glacier to
descend, appears to assign one inadequate to the result; for the congelation
of the water which percolates the glacier does not, according to the observations of Professor Forbes,* take place at all in summer more than a few
inches from the surface. Nevertheless, it is in the summer that the daily
motion of the glacier is the greatest.
The following remarkable experiment of Mr. HIookins of Cambridge,t
which is considered by him to be confirmatory of the sliding theory of
De Saussure as opposed to De Charpentier's dilatation theory, receives
-a ready explanation on the principles which I have laid down in this
-note. It is indeed a necessary result of them.  Mr. Hopkins placed a
mass of rough ice, confined by a square frame or bottomless box, upon
a roughly chiselled flag-stone, which he then inclined at a small angle;
and found that a slow but uniform motion was produced, when even it
was placed at an inconsiderable slope. This motion, which Mr. Hopkins
attributed to the dissolution of the ice in contact with the stone, would,
t apprehend, have taken place if the mass had been of lead instead of ice;
* "Travels in the Alps," p. 413.
+ I have quoted the following account of it from Professor Forbes's book,
p. 419.




DIMENSIONS OF A BUTTRESS.                  683
and it would have been but about half as fast, because the linear expansion of lead is only about half that of ice.
NOTE G.
THE BEST DIMENSIONS OF A BUTTRESS.
IF mn (Art. 299.) represent the modulus of stability of the portion AG of
the wall, it may be shown, as before, that
P{(h, - h1) sin. a   (- (-a - ml)cos. a} = (l'  ) (hi - )a. P{(, - h,)sin. a - (I - a2)cos. a}
(=  (hL - h2)a',2^-m1{P cos. a + (h- h^2)a,}
If m-=m, the stability of the portion AG of the structure is the same
with that of the whole AC; an arrangement by which the greatest
strength is obtained with a given quantity of material (see Art. 388.).
This supposition being made, and m eliminated between the above equation and equation (392:), that relation between the dimensions of the
buttress and those of the wall which is consistent with the greatest
economy of the material used will be determined. The following is that
relation:-c(a12hL + 2ala2h, + -a2h2)- P (hA sin. a -  cos. a)
P cos. a + ~(a1h, + a2h2)
_   (h — h)ai -P{(h, - h2) sin. a - (l-a2) cos. a}
P cos. a + jal (hI -  2)
It is necessary to the greatest economy of the material of the Gothic
buttress (Art. 303.) that the stability of the portions Qa and Qb, upon
their respective bases ac and be, should be same with that of the whole
buttress on its base EC.  If, in the preceding equation, Ah-h3 be
substituted for h, and h2 — h for h2, the resulting equation, together
with that deduced as explained in the conclusion of Art. 301., will determine this condition, and will establish those relations between the dimensions of the several portions of the buttress which are consistent with the
greatest economy of the material, or which yield the greatest strength to
the structure from the use of a given quantity of material.




684                        APPENDIX.
NOTE H.
DIMENSIONS OF THE TEETH OF WHEELS.
THE following rules are extracted from the work of M. Morin, entitled
Aide Mbmoire de Mcanique Pratique:-If we represent by a the width
in parts of a foot of the tooth measured parallel to the axis of the wheel,
and by b its breadth or thickness measured parallel to the plane of
rotation upon the pitch circle; then, the teeth being constantly greased,
the relation of a and b should be expressed, when the velocity of the pitch
circle does not exceed 5 feet per second, by a =4b; when it exceeds
5 feet per second, by a= 5b: if the wheels are constantly exposed to wet,
by a= 6b.
These relations being established, the width or thickness of the tooth
will be determined by the formulse contained in the columns of the following table:Material.        French measures, cents  English measures, feet
and kils.            and pounds.
Cast iron -       -       b - 105 i/P         b =-002319 4/P
Brass         - -         b ='131 4/P         b  -002894 1/P
Hard wood       -         b =-145 4/P         b — =003203 VP
Assuming that when the teeth are carefully executed the space between
the teeth should be yltlh greater than their thickness, and -Tlth greater
when the least labor is bestowed on them, the values of the pitch T will
in these two cases be represented by b(2 + Iy) and b(2 +  ), or by 2 067b
and 2'1b. Substituting in these expressions the values of b given by the
formula of the preceding table, then determining from  the resulting
values of c (see equation 233.) the corresponding values of the coefficient
0 (see equation 234.), the following table is obtained:Value of c (equation 2388.).  Value of 0 (equation 234.).
Material.    For teeth of  For teeth of  For teeth of  For teeth of
the best work- inferior work- the best work- inferior workmanship.    manship.     manship.    manship.
Cast iron -  -    004795       004870       0'912       0'922
Brass    -   -    005982'006077       1 057       1 068
Hard wood    -    *006621'006726       1 131       1-143




TRACTION OF CARRIAGES.                  685
The following are the pitches commonly in use among mechanics:in.  in.  in.  in.  in.  in.  in.
1, 13, 1+,   11, 2, 21, 3.
Prof. Willis considers the following to be sufficient below inch pitch:in.  in.  in.  in.  in.
+,  7, +, f, 
Having, therefore, determined the proper pitch to be given to the tooth
from formula 234., the nearest pitch is to be taken from the above series
to that thus determined.
NOTE I.
EXPERIMENTS OF M. MORIM ON THE TRACTION OF CARRIAGES.
THE following are among the general results deduced by M. Morin from
his experiments:1. The traction is directly proportional to the load, and inversely proportional to the diameter of the wheel.
2. Upon a paved or a hard Macadamised road, the resistance is independent of the width of the tire when it exceeds from 3 to 4 inches.
3. At a walking pace the traction is the same, under the same circumstances, for carriages with springs and without them.
4. Upon hard Macadamised and upon paved roads the traction increases
with the velocity; the increments of traction being directly proportional
to the increments of'velocity above the velocity 3828 feet per second, or
about 2+ miles per hour. The equal increment of traction thus due to
each equal increment of velocity is less as the road is more smooth, and
the carriage less rigid or better hung.
4. Upon soft roads of earth, or sand or turf, or roads fresh and thickly
graveled, the traction is independent of the velocity.
5. Upon a well-made and compact pavement of hewn stones the traction
at a walking pace is not more than three-fourths of that upon the best
Macadamised road under similar circumstances; at a trotting pace it
is equal to it.
6. The destruction of the road is in all cases greater as the diameters
of the wheels are less, and it is greater in carriages without than with
springs.




G6 h3 APPENDIX.
NOTE K.
ON THE STRENGTH OF COLUMNS.
Me. HODGKINSON has obligingly communicated the following observations
on Art. 430.:
1. The reader must be made to understand that the rounding of the
ends of the pillars is to make them moveable there, as if they turned by
means of a universal joint; and the flat-ended pillars are conceived to be
supported in every part of the ends by means of flat surfaces, or otherwise
rendering the ends perfectly immoveable.
2. The coefficient (13) for hollow columns with rounded ends is deduced
from the whole of the experiments first made, including some which were
very defective on account of the difficulty experienced in the earlier
attempts to cast good hollow columns so small as were wanted. The
first castings were made lying on their side; and this, notwithstanding
every effort, prevented the core being in the middle; some of the columns
were reduced, too, in thickness, half way between the middle and the
ends, and near to the ends, and this slightly reduced the strength. These
causes of weakness existed much more among the pillars with rounded
ends than those with flat ones; they are alluded to in the paper (Art. 47.).
Had it not been for them, the coefficient (13) would, I conceive, have
been equal to that for solid pillars (or 14-9).
3. The fact of long pillars with fiat ends being about three times as
strong as those of the same dimensions with rounded ends is, I conceive,
well made out, in cast iron, wrought iron, and timber; you have, however, omitted it, being perhaps led to do it through the low value of the
coefficient (13) above mentioned.
The same may be mentioned with respect to the near approach in
strength of long pillars with flat ends, and those of half the length with
rounded ends. It may be said that the law of the 1'7 power of the length
would nearly indicate the latter; but this last, and the other powers 3 76
and 3'55, are only approximations, and not exactly constant, though
nearly so, and I do not know whether the other equal quantities are not,
with some slight modifications, physical facts.
4. The strength of pillars of similar form and of the same materials
varies as the 1'865 power, or near as the square of their like linear
dimensions, or as the area of their cross section.




COMPLETE ELLIPTIC FUNCTIONS.                       687
TABLE I.
The Numerical Values of COMPLETE Elliptic Functions of the FIRST and
SECOND Orders for Values of the Modulus k corresponding to each Degree
ofthe Angle sin. —'k.
Sin. —k.       F1.          SE.       Fin.-i.       F.           El.
0~       1-57079       1-57079        46       1886914      1'34180
1        1-57091       1-57067        47       1 88480      1133286
2        1 57127       157031         48       1-90108      1-32384
3        1'57187       1-56972        49       1917.99      1-31472
4         1-57271       156888        50       1 93558    1'30553
5         1-57379       1'56780     --' 1
51       1-95386       1'29627
6         1 57511       1-56649       52       1-97288     1P28695
7         1'57667      1P56494        53       1 99266      1-27757
8         157848        1-56316       54       2-01326      1-26814
9         1 58054       1-56114       55       2.03471   1.25867
10         1-58284,      155888                    --
10   1-_______   55888-56             2-05706       1,24918
11         1-58539       155639        57       2-08035      1-23966
12         1'58819       1-55368       58       2-10465      1-23012
13         1 59125       1-55073       59       2-13002      1-22058
14         1-59456       1-54755       60       2-15651      1P21105
15         1'59814      1-'54415
15   159814        1 —54415      61       2-18421       1-20153
16         1-60197       1.54052       62       2-21319      119204
17        1P60608        1-53666       63       2-24354      1-18258
18         1-61045       1-53259       64       2-27537      1-17317
19        1'61510        1 52830       65       2-30878  1'16382
20         1-62002       1-62379       66       234390        115454
21         1-62523       1-51907       67       2-38087       1-14534
22         163072        1-51414       68       2-41984       1-13624
23         1-63651       1-50900       69       2-46099       1-12724
24         1-64260       1'50366       70       2-50455       1-11837
25         1-64899       1-49811
25^_                          7 6489  1 4981-  ~71'  2 55073  1-10964
26         1-65569       1-49236       72       2-59981      1-10106
27         1'66271       1-48642       73       2-65213      1 09265
28         1-67005       1 48029       74       2-70806      1 08442
29         1-67773       147396        75       2-76806      1 07640
30.       168575        1-46746       7-    -2              1__"' —  7__6'                2'83267      1-06860
31         1-69411       1-46077       77       2-90256      1-06105
32         1-70283       1-45390       78       2-97856      1'05377
33         1-71192      1P44686        79       3-06172      1-04678
34         1-72139       1-43966       80       3-15338      1 04011
35         1-73124       1-43229        1    -    2   0 
81        3'25530      1-0337836         1 74149'      142476        82       3-36986      1 02784
37         1-75216       1-41707       83       3-50042      1 02231
3 8        1-76325.      140923        84       3-65185      1-01723
39         177478        1-40125       85       3-83174      1-01266
40         178676        1-39314
86       4,05275       1-00864
41         1-79922       1-38488       87       4-33865      1 00525
42         1-81215       137650        88       4-74271      1.00258
43         1-82560       1-36799       89       5-43490      1-00075
44         1-83956   1.35937
45         1-85407       1-35064 




688                       APPENDIX.
THE TABLES OF M. GARIDEL.
TABLE II.
Showing the Angle of Rupture' of an Arch whose Loading is of th same
Material with its Voussoirs, and whose Extrados is inclined at a gien
Angle to the Horizon. (See Art. 344.)
a = ratio of lengths of voussoirs to radius of intrados.
e=ratio of depth of load over crown to radius of intrados, so that
= 3(1 +a). (Art. 388.)
= inclination of extrados to horizon.
=0.a       e       =0O1    =0-2     =0=01  0  c=004  e=0-5  c-10
0-05    68-0~   59-19~   54'04~  51'150   49-350  48.20~   4574~
0-10    65-4     60-48   57-70   56-01    54-93   54-17    52-34
0.15    64-0     61.3    59.7    58-69    58-0    57'49    5621
0-20    63-1     617     60-88   60-30    59-90   59-60    58-80
0-25    62-24    61-76   61-44   61 22    61-05   60-94    60-59
0.30    61'3     61-42   61-54   61'60   61-66    61-67    61-81
0-35    60-17   60-80    61'21   61-54    61'78   61-98    62-56
0-40    58-8     59-8    60'52   61-05 i 6148     61-67    62-9
0-45    57^-32   68-53   59-45   60-19    60-80   61-28    62-85
0-50    55 63    56-97   58'09   58'98    59 72   60-34    62-40
= 70 30's
a  =     c=0 6=0  c=02     =0'8   o=0'4    c.0-5    c=-0
0-05    68.-3~   57.3~   51.690~  48.61~  47'84~  46-110   44-85'
0-10    64-3     58&68   55-95   54-52    53 64   53-0S    51-68
0-15    62-43    59667   58 33   5755     57 00   56-61    55-66;
0-20    61-48    60-42   59'72   59-35    59-07   58-87    58'29
0-25    60-75    60-55   60-44   60-38    60-33   60-21    60-17
0,30    60-09    60-49   60177   60-95    61-08   61-18    61-48
0-35    59'27    60-12   60-62   61-02   61 33    61-59    62-31
0-40    58-25    59'33   60-11   60-72   61-18    61-51   62-7 
0-45    57-11   58-35    59-29   60-05   60-67    61-16   62'78
0-50    55-82   57-13    58-21   59-08   59-81    60-41   62- 45!,,.                                                   ~ ~  -




ANiGLE OF RUPTURE OF AN ARClH.                 689
= -15'.
a       0=O     =O               =0,=  c=01  -==0'5 c=0e o=04   c
0-05    648~    50'5~    46'95~  4569~   4503~    44.67~  43-9~
0'10    59-3    55'07    53-34   52-47   51-99    51-69   5093
0'15    59'08   5732    56'65    5605    55-75    5555    5505
0-20    59-06   5860     58-35   58-20   5810     58-02   57 84
0-25    59'05   59-28    59'42   59-53   59-60    59'65   5979
0'30    58-90   5957     59-98   60-26   60-48    60-66   61'15
0'35    5853    59'41    60'09   60-57   60-93    61-17   62'0
0,40    57 99   59-08    59-87   6048    6095     6136    626
0-45    57-26   5843     59-34   60-06   60-67    61-15   62-7'050     56'38   57 61   58-58   5936     6006    60-64    62-5
1=220 30'.
a       e=O    C=01     e=0' 2  c=0-8   c==0'4  C=0'5    c=1'0
0'05    36-1~   41-2~    420~    42-3~   426~     42-7~   42'9~
0-10    505      50-3    5019    50-17    50'14   50'13    50-11
0-15    5425     54-31   54-35   543    554'36    54'36    54'38
0'20    56-17    5660    5682    56'95   5704     57'11    57-28
0'25    57-27    57-93   58-33   58'61    5879    58'95    5933
030     57'85   58'68    59-23   59-60    5993    60'16    60-83
0'35    58'07    5901    5970    60-21    60-61   6091     61-85
040     58-02    59-02   5979    60-38    6087    61-25    62.2
0-45    57 74    5878    59-60   60'26    60-82   61 27    62-7
0'50    57'30    58-31   59'16   5988    6047     61-00   62'9
X = 30o.
a   ~?=0    c=0-1    c=0-2   c=08    c=04     0=04 5 0 c=l-0
0'05    313~0    36-2~   384~    39-57~  40-28~   4077~   41-9~
0-10    43-3     4606    47'25   47 90   4830     48'59   49 24
0'15    5007     51-46   52-18   5263    52-94    53-14   53-68
0-20    5366     54'69   55'27   5567    55'96    56616   56-72
0-25    55-80    56-72   57.30   57-72   58-01    58-23   5889
0.30    57-13   58.01    5862    5906    5940     59-69   60-48
0.35    57.93    5880    59.43   59.94   6033     6066    61-64
040     58-33   59-20    5989    60-42   60-87    6123    62-39
0'45    5847     5933    6003    6061    61-08    61-48   6287
0150    58-38    5922    5993    6053    61-03    61-47   630
44




690                          APPENDIX.
=370 30'.
a       — =-O  e=01    c=0'2   c=0-3   c    =04   =5 =0l-0
0'05    31-1~   34-3~   36'28~  37-59~  38 48~  39160    40'82
010     4098    43 59   45'09   4601    46-67   47 14    48-35
0'15    4771    49'40   50'43   5112    51-61    51'96   5293
0-20    5201    52-23   54-01   54-54   5494    55'24    56'10
0-25    54-87   55-80   5645    56-94   57'29    5759    5841
030     56'77   57'8    58'16   58'62   58-98    59'26   6016
0-35    58'04   58'78   5934    59-81   6017     60-47   61-45
040    5889    59'58   60'13   60'60   6097     61-30   63-2
045    59-38   6006    60-62   61-07    61-47   6183    63-0
0'50    56-69   60-29   60-84   6126    6172     6207    633
= 45~.
a      c=O     c=0'1   c=02    c=0-8   c=04    c=O-  c-=O10
0'05    31-30   338680  35'460  36368 36722      38007   3989~
010     40 6    42 4    43-7    4464    45835   45 92    47 45
0.15    46-77   48'20   4918    4993     5047    5092    52-15
0-20    51-23   52-27   5305    5364    5407     5442    55'47
0-25    54'42   5522    5584    5631    56170    57'01   5797
030     56-72   57-38   5790    58-0    58-65   5894     59-85
035     58-35   5894    5940    5979    6011     6038    61-30
0-40    5956    60'09   6052    6089     61-19   6146    62'4
0-45    6040    60-89   61 29   6167     61 97   6224    63-2
050     60'99   6143    61 8    62-2     625     62-8    63-8




HORIZONTAL THRUST OF AN ARCH.                     691
THE TABLES OF M. GARIDEL.
TABLE III.
Showing the Horizontal Thrust of an Arch, the Radius of whose Intrados
is Unity, and the weight of each Cubic Foot of its Material and that of
its Loading, Unity.  (See Art. 344.)
N.B. To find the horizontal thrust of any other arch, multiply that given
in the table by the square of the radius of the intrados and by the weight
of a cubic foot of the material.
-= 0.
c=O     c=0-1    c=0-2   c =03    c=0'4    c=0-5    c=10
a      P       P        P        P       P        P        P
yr2     r2       r2]               I r     r2 r0 05    0o08174 0o14797 0.21762 0.28877    0,36060  0,43277  0-79541
0110    0'10279  0'16370 0"22588  0-28862  0,35164 0,41481  0-73161
0'15    0'11894 0-17480, 023111   0-28764 0-34429  0-40100  0-68504
0'20    0'13073 0'18191  0-23322 0'28460 0'33603 0"38747    0-64488
0'25    0 13871 0'18553 0-23237   0'27922  0-32607  037293  060727
0-30    0'14333  0'18604 0-'"-2874  0'27145 0-31416 0-35687  0-57041
0'35    0'14504 0'18379 0-22258   0-26140  0-30023 0-33907  0-53335
0'40    0'14422 0'17913 0-21415   0'24924 0'28437  0'31953  0,49560
0-45    0'14124  0'17240 0-20374  0'23520  0'26674 0'29835   0'45693
0'50    0'13649 0-16396 0-19168   0'21957  0,24760 0-27573   0-41728
=70 30'.
c=O     c=0o1    c=0o2   c=0o8    c=0 4    c=0O6    c 1-0
a        P       P        P        P       P        P        P
r2      ~r        8       r2 ~      2       r2      ~r2
0.05    0-06180 0o12867  0-19937 0-27125 0o34356 0-41606 0o77944
0.10    0-08514 0-14666  0.20930 0.27237  0.33561 0-39895   0-71618
0-15    0-10380  0 16001  0,216507 0.27326  0.33003 0.38683  0.67110
0'20    0-11813 0'16948 0'22089 0-27237    0'32384 0'37533  0'63286
0'25    0-12870 0-17557  0-22244 0-26932 0'31619 0'36306    0-59743
030     0-13598  0'17866 0'22134 0-26403 0030673 0'34943    0'56295
0835    0-14040 0-17909 0'21783 0-25661   0'29542 0 33424   0'52846
0-40    0-14234 0-17718  0'21215 0-24720   0-28230:031744  0 49344
0o45    0-14211 0-17323  020454 0-23598    0-26751 0229910  0.45763
0.50    0-14003  0-16753  0-19528  0-22319      4 25124  027938  042096




692                          APPENDIX.
1=150.
c=O     Co=01    c=0-2    c=0-8    c=04     c=0-5     c=10
a        P        P        P        P        P        P        P
r92     r72       7*       r2        2       r2        2
0 05    0-05310  0-12265  0-19488  0'26748  0-34018  041293   077681
0'10    0'07903 0'14170   0'20493  0-26832  0'33176  039524   071277
0.15    0-09990  015658   0'21336 0-27022 0'32708    0-38395  066840
0.20    0'11631 016781 0.21931 0.27083 0.32234       0-37386  063145
0'25    0'12894 0-17582   0'22268  0-26955  0'31643  036330  059767
0'30    0-13835  0-18096  0'23361  0-26627  0-30895  0'35163  056510
0.35    0-14494  0-18355  0'22224  0-26098  0'29976  0'33855  0 53271
0 40    0'14905  0'18384  0-21878 0-25380   0'28888  032399   0'49995
045    0-15097  0'18212  0'21344  0-24488  0'27641  030800   046652
0'50    0-15099  0,17860  0 20642 0-23439   0 26247  029065   043232
*=220 30'.
c=O     c=0-     c=0o2     =0-3    c=0-4     c=05     C=1-0
a        P        P        P        P        P        P        P
r2                         r        r2                 r
0 05    0     06102  013346  0-20621  0 278{9 0-35178  0-4248  078857
0'10    0-08700  0-15053  0-21407  0-27760 0-34113   0-40466  0-72233
0-15    0-10877  0'16567  0-22257  0'27947  0-33638  0 39328  067778
0'20    012635   0-17785  0-22936  0,28087  0-33239 0'38391   0,64150
0-25    0'14037 0'18716   0-23399  0-28082  032767   037453   0-60886
0'30    0'15129  0-19381  0-23640  0'27902  032166   0'36432  057773
035     0-15948  0-19804  0-23669  0-27540 0-31415   0'35292  0'54700
0'40    0'16525  0-20005  0-23497  0-26999  0'30506 034017    0-51608
0-45    0-16883  0'2)005  0-23141  0'26289  0'29444 032604    048460
0.50    017047   019824   0'22617  0)25423  0'28238  031060   0-45241, = 30~.? =O    c=01     c=0-2    c=038      =0-4   c=05      o-1 0
a'       P                 P        P        P        P        P
r2       r2                      - r  ~'     r        r2
0105    0-09355  0-16408  0'23605  0,30845  0-38101  045365  081731
0-10    0-11297  0 17592 0283922   0 30263  0-36609  0 42957  0-74711
0-15   013295   0'18962  0,24640  0-30323  0-36009  0 41696  0,70138 1
0-20    0 15038  0-20172 0-25314   0-30459 0-35606   0-40755  0,66506
0-25   016493   0 21160 0-25834   0 30513 0-35193   0-39876  0-63299
0-30   0O17673 0-21917 0O26170     0-30427 0-34688   0-38951  0-60282
0 -5    0,18599  0 22452  0-26314  0 30182 0-34055   0-37930  0-57332
0 40   019293   0 22777  0-26271  0-29773 033280   0 36791  0-54380
0-45    0,19774  0-22906 0-26050   0 29202 0 32361   0135624  0-51385
0,50    0,20060  0,22854  0,25661  0-28476  0-31299  0-34128  0-48327 1




HORIZONTAL THIUST OF AN ARCH.                    693,=37o 30'..rQ  i  rq   i           rsr2                      r2
0'05  0-14749  0-21733  0-28854 0-36038  043255  050490 0'86784
0'10  0'15949 022174 0-28457 0-34768 0'41093 0-47426 079141
0'15  0-17605  0-23233 0-28886 0-34553  0-40226  045904 9'74322
0-20  0-19209  0-24321 0-29448 0-34583  0-39722 0-44865 070598
0'25  020627  0-25282 0'29948 0-34619   0-39294  043972  067382
0'30  0'21827  0-26066 0-30314 0-34568  0'38825 0'43085 0-64406
035   022805  0-26659 030521   0-34388  038259 042133 0'61529
0O40  0-23570  0-27060 0-30558 0-34062 0-37571 0-41083 0-58673
0'45  0-24130  027275 0'30427 0-33586   0'36749 039916 0'55787
0'50  0-24499  0-27312 030132 032958    035789 0-38625 052845, —45~.
c=O     c —=O1   c=0.2   c=03     c=04    co=05    c=1l0
a     p       P        P       P        P       P        P
o   o Ire        r2I      r2       o       r2      r2
0 05  0-23105  0-30081  037 162 0-44305  0-51485 0-58688  0-94881
0'10  0'23318  0-29507  0-35754  0-42034 048333  054646  086300
0-15  0'24478 0-30079  0-35708 0-41355 0-47013 052678    081059'
0'20  0-25819 0930915  036028  0-41151  046281 051415    077124
0'25  0-27104  0'31752  036410  0'41074 045744 050417    073809
0'30  0-28248  0-32486  0-36731  0-40981  045235  049493  070803.
0'35  0-29216 0-33073 0'36935  0'40803 0 44674 048547    067939
0'40  0-29997  0-33494  0-36998  0'40506 044016 047530   065123
0'45  0-30589 033745   0-36907 0-40072 043240   046412 062294
0'50  0'30996  033824  0'36657  0'39494 042334 045177    059419




694                                   APPENDIX.
TABLE IV.
Mechanical Properties of the Materials of Construction.
Note.-The capitals affixed to the numbers in this table refer to the following authorities:
B.     Barlow, Report to the Commissioners of     La. Lame.
the Na/vy, &c.                          M. Muschenbroek, Introd. ad Phil. Nat. i.
Be.    Bevan.                                     Mi. Mitis.
Br.   Belidor, Arch. Hydr.                        Mt. Mushot.
Bru. Brunel.                                      Pa. Colonel Pasley.
C.    Couch.                                      I. Rondelet, L'Art de Batir, iv.
D. W. Daniell and Wheatstone, Report on the       Re. Rennie, Phil. Trans. &c.
Stonefor the Houses of Parliament.      T. Thomson, Chemistry.
F.     Fairbairn.                                 Te. Telford.
H.    Hodgkinson, Report to the British As-       Tr. Tredgold, Essay on the Strength of
sociation of Science, &c.                     Cast Iron.
K.     Kirwan.                                    W. Watson..     Tenacity    Crusig
Specific  Weight of 1  per       force per  Modullls of  Modulls of
Names of Materials.   gravity.  cubic foot  square inch  square inch  elasticity E.  rupture S.
inl bs.    in  Ibs.   in 11)..
Acacia (Eng. growth)      -710 B.   44 37     16000 Be...    1152000 B. 11202 B.
Air (atmospheric)..      -01228      0-0768
Alabaster (yel. Malta)   2-699      168-68
Do. (stained brown)     2-44      17150
Do. (oriental white)    2-730     17062
Alder....     800 M.    50-00    14186 M.     6895 H.
Antimony (cast).    4500 M.    281-25     1066 M.
Apple-tree..     793.   49-56    19500 Be.
Ash.                   90 to   8  4334812               881T B. H. }1644800 B. 12156 B.
to'845       58'81                 9363 H.
Bay-tree...     882 M.    51-87    12896        7158 H.
Bean (Tonquin).   1080        67-50..    2601600 B. 20886 B.
* 854        583873    15784 B.    7783 I.    185360o  B. 98 
Beech                                                                1353600'B. 9336 B.
Beech  *.   \ to   690       43-12    17850 B.   9863 dryH. 13600 B       96 B
Birch (common).     79 B.     49150    15000    { 6402 dry1. } 1562400 B. 10920 B.
Do. (American).      648 B.   40-50..   11668 H.    1257600 B. 9624 B.
Bismuth (cast)..   9810 M. 613887        8250 M.
Bone of an ox..   1656 M.    103-50     5265
Box (dry)...      960 B.    6000     19891 B.    10299 H.
Brass (cast).    8 899      525 00    17968 Re.   10304 Re.   8980000
Do. (wire-drawn).      8544       584'00
Brick (red)..   2168 Re. 185-50        280         807 Re.
Do. (pale red)..    2085 Re. 180 81         00         562 Re.
Brick-work.   1800       112-50
Bullet-tree (Berbice).   1-029 B.    6431...     2610600 B. 15636 B.
Cane..                  0 400      25 00      6300 Be.
Cedar (Canadianfresh)   0-909 C.   56 81     11400 Be     5674 H.
Do. (seasoned).    0753       47-06..    4912 H.
Chalk.                  2784       1 400..     334 Re.
to 1 869     116'81 
Chestnut...   0657 R.     4106     18800 R.
Clay (common)..   1919 Br. 119-93
Coal (Welsh furnace).    133887 Mt.  83856
Do. (coke)..      1-000 Mt.  62-50
Do. (Alfreton)..    1285 Mt.   77 18
Do. (Butterly).         1264 Mt.   79-00
Do. (coke)..    1'100 Mt.  68 75
Do. (Welsh stone).      1368 Mt.   85-50
Do. (coke)...    1-90 Mt.   86-87
Do. (Welsh slaty).    1409 Mt.   88-06
Do. (Derby. cannel).     1-278 Mt.  79-87
Do. (Kilkenny).    1602 Mt. 100-12
Do. (ccke)...    1657 Mt. 10356
Do. (slaty)...    1448 Mt.   90-18




PROPERTIES OF MATERIALS OF CONSTRUCTION.                    695
oolc  Wogh   Tenacity  Crushing
Specific  We'ght of  per   force per  Modulus of  Modulus of
Names of Mtaterials  gr ity.  I cubic foot  square inch  sq r h  elasticity E.  rupture S.
in Ibs.  in Ibs.   in Ibs.
Coal (Bonlavooneen).  1-436 Mt.  89-75
Do. (coke)...  1596 Mt.  99 75
Do. (Corgee)..  1403 Mt.  8768
Do. (coke)...  1656 Mt. 108-50
Do. (Staffordshire).  1240   78-12
Do. (Swansea)..  1357 K.   84-81
Do. (Wigan)..  1268 K.   7925
Do. (Glasgow)..  1290      80-62
Do. (Newcastle).  1257 K.   78-56
Do. (common cannel)  1232 K.  77-00
Do. (slaty cannel).  1426 K.  89-12
Copper (cast)..  8607     538793   19072
Do. (sheet)..  8785      549-06
Do. (wiredrawn).  8878     560-00   61228
Do. (in bolts)....                48000
Crab-tree...  0-765'4780..    6499 
Deal (Christiana middle)....  0698 B.    43'62   12400..   1672000 B. 9864 B.
Do. (Memel middle).  0-590 B.  36'87...    1535200 B. 10386 B.
Do. (Norway spruce)  0-340    21-25   17600
Do. (English)..  0470      2937      7000
Earth (rammed)      1'584 Pa.  99-00
Elder....  0-695 M.  43843    10230     8467 H.
Elm (seasoned)..  0-588 C.  86-75   13489 M. 10881 H.   699840 B. 6078 B.
Fir (New England)   0-553 B.  84-56...    2191200 B. 6612 B.
Do. (Riga).  0-53B.   406'11549 B.  5748 H.  1828800 B. 6648 B.
Do. (Eriga)...  0-758 B.  47-063 to 12857 B. to 6586H. I 869600 B. 7572 B.
Do. (Mar Forest).  0693 B.  43831
Flint....  2-60 T.  1643
Glass (plate).      2 48   153-31     9420
Gravel..  120      120-00
Granite (Aberdeen).  2'526  164-00
Do. (Cornish)..  2662    166 30
Do. (red Egyptian).  2 654  165-80
Hawthorn..  0'91 Be.  38-12    10500 Be.
Hazel...      086 Be.   58375    18000 Be.
olly...  0-76 Be.  47-5     16000 Be.
Horn of an ex..  1689 M.  105-56     8949
Hornbeam (dry).  0-760.   47'50    20240 Be. 7289 IH
Iron (wrought Eng.). 75700   481-20  25 tons, La.
Do. (in bars)     i    00    475'50
Do. (n bars)..  to7-800  487-00  25j tons, La.
Do. (hammered)...    80 tons, Bru.
Do (Russian) in bars.          27 tons, La.
I Do (Swedish) in bars..             32 tons, R.
Do. (English) in wire i               36 to 43
-lOth inch diameter -. tons, Te.
Do. (Russian) in wireo 
1-20th to 1-30th inch..           tons La.
diameter,       1~,     tons, La.
diameter..... 
Do. rolled in sheets
and cut length-wise.;..       14 tons, Mi.
Do. cut crosswise.....   18 tons, Mi.
Do. in chains, oval 
links, 6 inches clear,. iron Y inch diam....     2i tons, Br.
Do. (Brnnton's) with
stay across link...         25 tons, B.
Iron, cast (old Park)....           18014400 T. 48240 T
Do. (Adelphi)......  1885600 T. 4560 T.:Do. (Alfreton)..                       17686400 T.44046 T.
Do. (scrap).......        18082000 T. 45828 T.
Do. (Carron, No. 2
cold blast)..  7-066 H  44162     16683 H. 106875 H. 17270500 H. 88556 H.*
Do. (hot blast)..  7-046 H.  440387  13505 H. 108540 H. 16085000 H. 7508 H.*
t Tha umhers -marked hils * &ae aelc lated from the exoperiments of Messrs. Hodgkinson & Fairhaira




696                                 ~APPENDIX.
Tenacity    Crushing
Specific  Weight      pr        for p     Mod    of  Modulus of
Names of Materials.   gravity.  cubic foot  squareinch  square inch  elasticity E.  rupture S.
in Ibs.   in lbs.     in Ib3.
Iron, cast (do. No. 8,
Carron cold blast.   7094 F.    443-87   14200 H.   115442 H. 16246966 F. 35980 F.*
Do. (hotblast).        1.056 F.   441'00   17755 H. 133440 H. 17878100 F. 42120 F.
Do. Devon. No. 8
cold blast)             295 H. 9 5455983...     22907700 I. 36288 IH.*
Do. (hot blast)..      229 H.   451-81   2910T H.  145435 H. 22473650 H. 43497 H."
Do   (Buffery, No. 1,
cold blast)..   T079 H.    442-43   17466 H.    98866 H. 15381200 H  837503 H.*
Do. (hot blast).      6 998 H.   43T-37   1348 H.    86397 H. 18730500 H. 35316 II.*
Do. (Coed Talon, No.
2, cold blast)        6.  6955 F.  434-06  18855 11.  81770 1. 14818500 F. 8104 F *
Do. (hot blast)           6 F. 68 435-50    16676 H.   82739 1. 14322500 F. 8*145 F.*
Do. (Coed Talon, No.
3, cold blast)..     194 F.   449-62....   17102000 F. 4541 F.*
Do. (hot blast).   6-90.   485-62...  14707900 F. 4159 F.*
Do. (Elsicar, No. 1,
cold blast).   7-030 F.   489-387....      13981000 F. 34862 F.*
Do   (Milton, No. 1,
hot blast)...    6-976F.   436-00...   11974500 F. 28552 F.*
Do. (Muirkirk, No 1,
cold blast)            111..  711 F.  44456..   14008550 F. 85923 F.*
Do. (hot blast)..    6953 F.    48456...   13294400 F. 833850 F.4;
Ivory....   1826 P.    114-12   16-626
Laburnum..        0-92 Be.    5750    10500 Be.
Lance-wood..   1 022       68S     24696
L8201cH.      897600 B. 4992 B.
Larch..          0-522 B.    82'62   10229                   89760 B. 4992 B.
Do., (second specimen)  50-5  B.   8500    0     B..  5568 r      1052800 B. 6894 B.
Lead (cast English).  11446 M.   717 45    1824  e.               720000 Tr.
Do. (milled sheet).  11407 T.    712-93    8328 Tr.
Do. (wire)...   11-817 T.  C70512     2581 M.
Lignumn viter  [ous)     1-220      T6-'25  11800 M.
Limestone (arenace-.    2-742      171387
Do. (foliated).     2-887      177831
Do. (white fluor).   8156       197T25 
Do. (green)..   8182       198-87
Lime-tree..        0760        47-50   28500 Be. 
Lime (quick)...  0-488 B:r.  5268
Mahogany (Spanish).    0-800       50-00   1650         8198 H.
Maple (Norway).   0793        4956    10584
Marble (white Italian)  2-638 H.   164'87...    220000 T. 1062
Do. (black Galway).    2-695 H.   168-25.                                 2664
Mercury (at 82~).  18-619      851-18: 
Do. (at:60~)..   13-580      848-75
Marl.        1-600      100 00
M~ar~l..      to 2 877 T.  118-31
Mortar.   11751 By.  107-18      50
Mortar..
( 4684 H..
Oak (English)..   0-984 B.    58-687  17-00 M..,  9509 H.. 1451200 B. 10032 B.
f   (dry)
4281 H.,
Do. (Canadian)..   0-872 B.    5450    10-258M..  9509 H.    2148800 B. 10596 B.
(dry) 
Do. (Dantzic).        0-756 B.    424    12-780.. 1191200 B. 8742 B.
Do. (Adriatic).        0998 B.     62f06..               974400 B. 8298 B.
Do. (African middle)..  0-972 B..   6075..... 22883200 B. 18566 B.
Pear-tree               0661.    41-1           1    518 H.
Pine (pitch.            0-660 B.    4125     7818 M.. 1225600 B. 9792 B.
Do.red)                   65 B. (4106..   575 H.     1840000 B. 8946 B..
Do. (Amer; yellow).    0-461. C.   28'81..   5445.   1600000 Tr.
Plane-tree.          64 Be.    40-00   11T00 Be. 
9867 H.
Plum-tree...   0785.M.     49 06     11-851   8657 H..
(wet)
807'H.
Poplar....   0-88 M.     2393    7200 Be.   5124 H.
(dry)
Pozzolano...   26771 K.   169-1
Sand (river)..   1886       117187
Serpentine (green)      2574 K. 2  168-87




PROPERTIES OF MATERIALS OF CONSTRUCTION.                                 697
Tenacity   Crushing.Nssss sfsmtcs.s.a     Specsfis  Weight of     per      force per        Modlsf  ulus of
Names of mater'als.   psavity.   I cubic foot  squar e  inc   h inch elsticity E.  rupture S.
in ibs.     in  bs.    in Ibs.
Shingle  -          -    1 424 Pa       8900
Silver (standard)     -108312 T.       644-50     40902 M.
Slate (Welsh)      -     2-888         180-50     12800               15800000 Tr. 11766 Re.
do. (Westmoreland) - 2791 W.          17443     -                    12900000 Tr.
do. (Valentia) -     - 2880 Re.       180-00         -                             5226 Re.
do. (Scotch)   -            -  -9 -               9615000 Tr.
Steel (soft)    -     - 7780           48625     120000
do. (razor-tempered) - 7840           490-00    150000               29000000 Y.
Stone (Ancaster)      - 2182 D.W.      186-37
do. Barnack   -      - 2-090 D.W.     180 62
do. Binnie    -      - 2194 D.W.      137-12
do. Bolsover -       - 2816 D.W.      144-75
do. Box       -      - 1889 D.W       114-93
do. Bramham Moor - 12008 D.W.         125-50
do. Brodsworth       - 12093 D.W.     180-81
do. Cadeby     -     - 1951 D.W.      121-98
do. Caithness -      - 2764 Re.       172-75                 -5142 Re.
do. Craigleith-      - 2-266 D.W.     141-62
do. Chilmark (A)     - 2366 D.W.      147-87
do. Chilmark (B)     - 2388 D.W.      148-98
do. Chilmark (C)     - 2-481 D.W.     155-06
do. Darby Dale (Stancliffe)     -      - 262 D.W.      164-25
do. Giffneuk -       - 2 280 D.W.     13937
do. Gunbarrel (Stanley) 2 260 D.W.    141-25
do. I-am Hill- -        2260 D.W.     14125
do. Haydon    -      - 12040 D.W.     127-50
do. Heddon    -      - 2229 D.W.      139381
do. Hildenly -       - 2098 D.W.      138112
do. Hookstone        - 2253 D.W.      140 81
do. Huddlestone      - 2147 D.W.      134'18
do. Little Hulton    - 12857 H.       147-31                                        774  I.
do. Kenton    -      - 2-247 D.W.     140-48
do. Ketton    -      - 2045 D.W.      127'81
do. Ketton rag       - 2490 D.W.      155-62
do. Mansfield, or Lindleyes red   -      - 233888 D.W.   146-12
do. white     -      - 2-277 D.W.     14231
do. Morley Moor      - 2-053 D.W.     128381
do. Park Nook        - 2188 D.W.      138-62
do. Park Spring      - 28321 D.W.     145-06
do. Portland (Waycroft
Quarry         -     2-145 D.W.    184-06
do. Redgate    -     -2 289 D.W.      189-93
do. Roach Abbey      - 2184 D.W.      138837
do. Rochdale -       - 2577 H.        161-06.                                    2358 H.
do. Stanley    -     - 2227 D.W.      189'18
do. Taynton -        - 2103 D.W.      181-43
do. Totterwhoe       - 2'891 D.W.     118-18
do. Jackdaw crag     - 2-070 D.W.     129 87
do. Yorkshire flag   - 2820 H.        140 00...              1116 H.
do. Green Moor       - 2-534 Re.      158-37                                       2010 Re.
Sycamore        -      - 069 Be.        4312     13000 Be.
Teak (dry)      -      - 065T C.        41-06    15000 B.   12101 H.   2414400 B. 14772 B.
Tile (common) -        - 1815 Br.      118343
Tin (cast)             - - 7'291 Tr.   455-68     5322 M.   -      -   4608000 Tr.
Water (sea)           - 1 027 T.        6418
do. (rain)    -      - 1000           62 50
Walnut -        -        0671 M.        4198      8130 M.    6645 H.
Whalebone       -                                 7 -  -  7667  -       820000 Tr.
Willow (dry)    -        0-890          2437     14000 Be.
Yew (Spanish) -        - 6807 M.        50-48     8000 Be.
Zinc     -             - 1 7028 W.     489-25  1 -                 -  18680000 Tr.
Note.-The experiments of Mr. Hodgkinson, from which the moduli of rupture of stones contained
in the last column of the above table have been calculated, together with those detailed in Art. 410.,
form part of a more extended inquiry, which, when completed, will be laid before the Royal Society.
The crushing forces given in the above tables are in every case determined by the compression of
prisms of such a height as to allow the rupture to take place by the sliding of the upper portion freely
off, along its plane of separation from the lower (see Art. 406). The experiments of Mr. Hodgkinson
have shown that all results obtained without reference to this circumstance are erroneous. (See
Jllustrations of Mechanics, p. 402.)




698                APPENDIX.
TABLE V.
Useful Numbers.... -=31415927       1
/. =07071068
Log. z.. =0-4971499     4/2    
Log. 7. =1-1447299       -.. =4-4428829
1.   =0-3183099.  =22214415'2214415
2       9..  98696044 
1.. =01013212        -      =0-4501582
4g.. =1-7724538              =1.2533141
--.. =0-5641896 
4/  *. =*1-4142136      /-     -=0-7978846...........  =2-7182818
Log...........  =0-4342945
Modulus of common logarithms...'434294482
Log. of ditto.....=96377843
g..........       =32-19084
4/g........... =5-67363
Log. g.........  1-5077222
Inches in a French metre... =39-37079
Log. of ditto.......          =15951741
Feet in ditto.......          =32808992
Log. of ditto.........  =05159929
Square feet in the square metre.....=10764297
Acres in the Are........=0024711
Lbs. in a kilogramme......=2'20548
Log. of ditto......      =03435031
Imperial gallons in a litre......  =0-2200967
Lbs. per square inch in 1 kilogramme per square millimetre =1422
Cwts. ditto, ditto.....  =12-7
Volume of a sphere whose diameter is 1.  =05235988
Arc of 1~ to rad. 1.....          =0017453293
Arc of 1' to rad. 1.....          =0000290888
Arc of 1" to rad......... =0000004848
Degree in an arc whose length is 1.... =57295780~
Grains in 1 oz. avoirdupois...... =4374




USEFUL NUMBERS.             699
Grains in 1 lb. ditto...=7000
Grains in a cubic inch of distilled water, Bar. 30 in., Th. 62~ =252-458
Cubic inches in an ounce of water...=173298
Cubic inches in the imperial gallon..  =277-276
Feet in a geographical mile....=6075 6
Log. of ditto......           37835892
Feet in a statute mile... =5280
Log. of ditto......3-7226339
Length of seconds' pendulum in inches.. =3919084
Cubic inches in 1 cwt. of cast iron.....  =430'25
--      Bar iron..          397'60
-          Cast brass. =368-88
-       Cast copper.. =352-41
Cast lead..       =272-80
Cubic feet in 1 ton of paving stone..  =14-835
-       Granite....             =13505
-   Marble.                 =13-070
-   Chalk                    =12-874
-       Limestone.. =11'273
-   Elm...... =64-460
-   Honduras mahogany.=.64'000
-   Mar Forest fir... 51650
Beech.... =51-494
-       Riga fir..             =47-762
-   Ash and Dantzic oak. =47-158
-   Spanish mahogany. =42-066
-   English oak... =36205
To find the weight in lbs. of 1 foot of common rope, multiply the square of its circumference in inches by.  044 to -046
Ditto for a cable.....                 027
Note.-The numerical values of the function of r in this table were calculated by Mr. Goodwin. These, together with the numbers of cubic inches and
feet per cwt. or ton of different materials, are taken from the late Dr. Gregory's
excellent treatise, entitled Mechanics for Practical Men. The other numbers
of the table are principally taken from Mr. Babbage's Tables of Logarithms
and the Aide Memoire of M. Morin.
THE END.








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