QA
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P315

1.
ARTES
LIBRARY
181
VERITAS
SCIENTIA
OF THE
UNIVERSITY OF MICHIGAN
į
EL
ALUNIAUS UKUN
TUEBOR
SI QUERIS PENINSULAM·AMINAM
CIRCUMSPICE
PROF.
THE GIFT OF
ALEXANDER ZIWET
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3
QA
805
•P915

Puchard 13 Howes
to his Shot Friend


Me
( in memoria beneficiorum acceptorin
done didit)
Nov 8th 1837
1
4
:
:

3238
6.2
Mexander Ziwex
1
THE
MATHEMATICAL PRINCIPLES
OF
MECHANICAL PHILOSOPHY,
AND THEIR APPLICATION TO
THE THEORY
OF
UNIVERSAL GRAVITATION.
BY
JOHN HENRY PRATT, M.A. 1809-1871
FELLOW OF GONVILLE AND CAIUS COLLEGE, AND OF
THE CAMBRIDGE PHILOSOPHICAL SOCIETY.
CAMBRIDGE;
PRINTED BY JOHN W. PARKER, UNIVERSITY PRINTER.
PUBLISHED BY J. & J. J. DEIGHTON, CAMBRIDGE;
J. H. PARKER, OXFORD; MILLIKEN AND Co., DUBLIN;
MACLACHLAN & STEWART, EDINBURGH;
AND
JOHN W. PARKER, WEST STRAŃD, LONDON.
M.DCCC.XXXVI.

Prof, Aley. Givet
gt.
1-29-1923
$
10-1-26.FW?
PREFACE.
A LEADING object that I have had in view in pre-
paring the present Treatise has been to gather into one
uniform system the principles of mechanical science,
beginning with the most elementary and ascending to
the most general. In attempting to accomplish this
I have collected the fundamental principles into sepa-
rate Chapters, and placed after them Chapters of ap-
plication of these principles to the demonstration of
others of a second class, and have then added collections
of problems and, in some instances, hints to guide
to their solution.
An attachment, and that in most respects a laudable
attachment, to the geometry of the Principia had,
till of late years, led to the practice of retaining in
our course of University reading some parts of that im-
mortal work, rather for the beauty and elegance of
its demonstrations, than for the importance of the
theorems demonstrated. But this practice has been
gradually sinking into disuse, a result which we owe
to Professor Woodhouse's Physical Astronomy, to
M. Poisson's Traité de Mécanique, which has been
extensively used amongst us, and very largely to Mr
Whewell's Treatises on Statics and Dynamics and
Ɑ
416414
iv
PREFACE.
Mr Airy's Mathematical Tracts. But notwithstanding
the great and happy changes thus brought about we
still cling to the old methods, not as a whole, but just
so far as to derange our system and give to it the
ambiguous character of being neither strictly geometri-
cal, nor strictly analytical. But I wish not to be
misunderstood; I mean not to imply that geometry
should be discarded and banished from our academical
course of study; far from it; for the analyst will find
his analysis of little benefit if he have not the power
of gathering from his formulæ geometrical conceptions.
Neither would I have it for a moment conceived, that
I would in the least degree repudiate the profound
veneration, which is so justly due even to the letter of
of the Principia: my own admiration of the clearness
and conciseness of its demonstrations rather induces
me to invite others to participate of the pleasures they
may enjoy from its attentive and diligent study. But
this I desire, that we should pay more regard to
system than we hitherto have done; if our course is to
be geometrical let us adhere to geometry, if analytical
to analysis; if we are to admit both (the preferable
course) let us keep our systems well apart, and not
have our course of reading confused, here analysis and
there geometry.
My own experience has impressed me also with the
conviction, that many of our candidates for University
honours are debarred the high enjoyment of penetrating
into the sublimer investigations of Physical Astronomy
from the want of some treatise that would lead them by
PREFACE.
V
a clear and distinct path, and with an undivided at-
tention, through the train of reasoning which leads
from elementary mechanical principles to the demon-
stration of celestial phenomena. Some, it is true, of
our first rate students do attain this eminence; but
might not this few be considerably augmented, if their
path were well pointed out and disencumbered of many
of the obstacles which lie in their way and impede
their course?
Let it not be imagined, however, that I send
forth the present volume with the presumptuous con-
fidence, that the want of a complete analytical system
of mechanics is supplied by its appearance; though
I will so far commit myself as to confess, that to sup-
ply this want has been my earnest desire;—no, I
would rather use the experience of a distinguished
Author, whose name I have already used, who is a far
better judge, in such a case, than myself, and say in
his words, "a few years experience has a great tendency
to diminish the confidence of producing what shall
satisfy himself and others, with which a young author
sets out and he learns that the vivid impression of
fancied deficiencies and imperfections of preceding
works which at first induced him to write, is a very
insufficient warrant of his own skill and judgment."
But yet my object has been unique; and it has not
been till after much time and thought spent upon the
subject, that I have ventured to lay my work before
the public: how I have satisfied my own desire I leave
to the candour of my readers to determine.
vi
PREFACE.
In the first, second, and third Chapters of Statics
will be found the principles of the composition and
equilibrium of statical forces acting, first on a particle,
then on a rigid body, and lastly on a system of bodies
connected in any manner. From the conditions of
equilibrium the principle of Virtual Velocities has
been deduced. Afterwards this principle has been de-
monstrated independently after the method of Lagrange,
and from it are deduced the conditions of equilibrium.
Then follow Chapters of application. The fourth
Chapter is a collection of examples of finding the
centre of gravity of bodies, being an application of the
formulæ for the co-ordinates of the centre of parallel
forces. In this Chapter I have aimed at explaining
and illustrating integration between limits: see par-
ticularly Examples 8, 13, 14, 19, 25, 26. The fifth
Chapter contains the application of the principles to the
six mechanical powers, and concludes with some notices of
the laws of friction. The sixth Chapter is upon roofs,
arches, and bridges, which form interesting applications
of the principles of equilibrium. In this Chapter will
be found some remarks upon the roofs of Trinity
College Hall and Westminster Hall, the action of
buttresses, and the stone vaulting of King's College
Chapel, as well as notices of other noted edifices and
structures. An example is given of the method of
calculating the lengths and weights of the support-
ing rods, chains, and road-way of suspension bridges,
that the strain may in every part be proportional to
the strength of the chain. Then follows a Chapter
of statical problems, beginning with some general re-
PREFACE.
vii
marks on their solution. And the treatise on Statics
closes with a Chapter on Attractions. After calculating
the attraction of spherical and spheroidal bodies of
homogeneous mass I have proceeded to the more
general investigation of the attraction of a body differ-
ing but little from a sphere in form, with a view to
the calculation of the Figure of the Earth in a future
part of the work. This has led me to introduce
Laplace's Coefficients, a subject unknown in
University course, till introduced a few years since by
Mr Murphy in his Treatise on Electricity. I have
followed Laplace's course, and not the inverse method
of Mr Murphy. The frequent occurrence of the
equation
our
d² V
dx²
+
d² V
+
d² V
= 0
dve
dy
in physical investigations makes it highly desirable,
that a knowledge of the profound analysis of Laplace
should be made as familiar as possible to the higher
class of students in the University. For this reason
I have introduced, in as concise and at the same
time as clear a manner as I was able, the principal
properties of the Coefficients of that great analyst,
breaking up and arranging the subject in the form of
propositions.
The treatise on Dynamics opens with a Chapter
upon the fundamental principles of the motion of
bodies, which I gather wholly from experiment and
observation. After explaining the conventional method
viii
PREFACE.
of measuring motion I proceed to an enquiry into the
laws that regulate the motion of bodies when unin-
fluenced by external causes: a variety of experiments
and facts of ordinary occurrence point to the principle
called the First Law of Motion. This leads to an
explanation of the conventional method of measuring
force dynamically. An investigation is then made into
the laws which regulate the motion of bodies when acted
on simultaneously by different causes, and this leads to
the principle called the Second Law of Motion. This
law enables us to introduce a method of referring the
motion of a body to three rectangular axes.
The ne-
cessity is then shewn of obtaining a relation between the
two arbitrarily assumed measures of force, viz. pressure,
the statical, and velocity generated, the dynamical
measure: this leads to the principle called the Third
Law of Motion. The introductory Chapter of Dynamics
concludes with the enunciation of a self-evident prin-
ciple, analogous to that first introduced by D'Alembert,
whereby we deduce the equations for calculating the
motion of a system from the equations of equilibrium.
This principle is, in fact, the interpretation of the
Three Laws of Motion into analytical language. In
explaining the means of estimating force I have aimed
at giving a distinct idea of the nature of forces that
require a finite time to develop their effects, and
those which generate velocity in an indefinitely short
time.
The second Chapter is upon the motion of a single
particle.
In this I have entered fully into the pro-
PREFACE.
ix
perties of central forces, and calculated the motion
in various cases: and at the close of the Chapter
Kepler's Laws are made use of to guide us to the dis-
covery of the nature of the force acting on the planets;
and we thus catch a first glimpse of the theory of
gravitation: after shewing that there is sufficient
ground to justify us in undertaking the task of cal-
culating the consequences of this law, a large portion
of the remainder of the work is devoted to that
enquiry. In the third Chapter the motion of two
particles attracting each other according to the law of
gravitation is calculated. In the fourth the pertur-
bations in this motion by the introduction of a third
attracting body are explained upon the supposition
that the disturbing body is very distant. This Chapter
contains the Sixty-sixth Proposition of the First Book
of the Principia and its corollaries and some Pro-
positions of the Third Book put into an algebraical
form this is a digression from the chain of exact
reasoning which is the professed object of the work;
but being in a separate Chapter the student, if he
choose, may pass over this and proceed to the fifth
Chapter, in which the distance, longitude, and latitude
of the Moon are calculated to a second approximation
upon the theory of gravity. The sixth Chapter con-
tains the calculation of the perturbations of the planets.
In the Lunar Theory the only part peculiar to this
work is the way in which I have introduced the
constants c and g, which give rise to the motion of
the line of apsides and the line of nodes. In the
Planetary Theory I have used M. Pontecoulant's
X
PREFACE.
method of integrating the equations of motion of an
undisturbed planet and then applied Lagrange's prin-
ciple of the variation of parameters to calculate the
variations of the elliptic elements. This Chapter closes
with a demonstration of the Stability of the Planetary
System, retaining the squares of the eccentricities
and inclinations. The next is upon the motion of
a particle on curves and surfaces and also on the
oscillations and perturbations of pendulums. A col-
lection of problems on the motion of bodies considered
as particles follows.
The next two Chapters contain the calculation of
the motion of a rigid body acted on, first, by forces
of finite intensity, and secondly, by instantaneous or
impulsive forces. The subject of the first of these
Chapters leads to the demonstration and calculation
of the Precession of the Equinoxes and the Nutation
of the Earth's Axis: the formulæ for these are pre-
pared for numerical calculation; the reduction to
numbers will be found at the close of the Chapter on
the Figure of the Earth. After calculating the motion
of a flexible body in the simple case of a vibrating
cord, I close the treatise on Dynamics with the fifteenth
and sixteenth Chapters, in which are developed the
General Dynamical Principles of the Motion of a
Material System, and their application to the solution of
dynamical problems, of which a large collection is given.
In the treatises on Hydrostatics and Hydrody-
namics the general principles of those Sciences are
PREFACE.
xi
developed, and applied to the determination of the
Figure of the Earth upon the hypothesis of its mass
having been at some former epoch in a semi-fluid
state, the form of the atmospheres of the planets,
the tides, and the effect of a resisting medium upon
the elements of the planetary orbits.
Lastly, a summary of the arguments in favour of
the Theory of Universal Gravitation closes the Work.
A reference to the Table of Contents will give a
better view of the character of the Work than I have
deemed it necessary to give in this place.
The prevailing argument with me for using the
old differential and integral notation is the excellence
of Fourier's notation for definite integrals: I much
prefer that to any other that I have seen, and this
naturally led me back to the old form of differentials
and integrals. In case any
In case any of my readers are not
acquainted with Fourier's notation I now give it,
دام
a
uda represents the integral of the differential
coefficient u, or of the differential uda, with respect
to a taken between the limiting values a and b of x.
In successive integration the order of arrangement of
the integrals is the same as that of the differentials:
thus
[[ Paudw
0
Pdudo represents the double integral of P
with respect to μ and w, the limits of being - 1 and 1,
and the limits of w being 0 and 2 π.
b
段
​xii
PREFACE.
I repeat, that it is not with the expectation that
I have fully succeeded in satisfying even my own de-
sires that I present this volume to the students of the
University, but with the earnest wish that it may be
found useful and that my labours may not have been
altogether spent in vain. Should any of my readers
favour me with any suggestions of improvement I shall
receive them with the greatest thankfulness.
CAIUS COLLEGE,
Nov. 26, 1836.
ERRATA.
J. H. P.
PAGE LINE
20
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CORRECTION.
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afford
DIFINITIONS
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affords.
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DEFINITIONS
incompressible.
TABLE OF CONTENTS.
ARTICLE
1—10. INTRODUCTION and definitions, measure of force in Statics and
Dynamics
STATIC S.
CHAPTER I.
THE COMPOSITION AND EQUILIBRIUM OF FORCES ACTING UPON
A MATERIAL PARTICLE.
14. Resultant of forces acting in the same straight line
PAGE
1
6
16, 17. Resultant of two forces acting not in the same straight line.
Parallelogram of Forces
7
12
13
14
14
20. Resultant of three forces acting at right angles to each other
22. Resultant of any number of forces acting in any directions
23. Conditions of equilibrium of any number of forces
24. Principle of Virtual Velocities......
CHAPTER II.
THE COMPOSITION AND EQUILIBRIUM OF FORCES ACTING ON
A RIGID BODY.
28. The transmission of force through a rigid body...
17
29. Resultant of two forces acting in the same plane but not on the same
point
17
31. Moment of a force with respect to a point
19
32. Resultant of two parallel forces
19
35. Definition of a Couple
36. Resultant of any number of parallel forces ...
37, 38. Centre of parallel forces; centre of gravity
39. Moment of a force with respect to a plane
41-45. Properties of couples
20
21
223
23
xiv
CONTENTS.
ARTICLE
46. Resultant of couples in parallel planes
47. Resultant of two couples not in the same plane....
PAGE
24
•
25
26
.
27
48. Resultant of three couples acting at right angles to each other
50. Resultant of any number of forces acting in the same plane
58. Conditions of equilibrium of any number of forces acting in the same
plane
....
54. Two resultants of any number of forces acting in any directions on a
rigid body
56. The principal moment of the forces
58. The locus of the centres of the least principal moments
62. Condition when the forces have a single resultant
63, 64. Magnitude and direction of the single resultant
65. Six equations of equilibrium of a rigid body
67. One point of the body fixed …..
68. Two points, or an axis, in the body fixed
•
69. Conditions of equilibrium when a body rests on a plane
CHAPTER III.
29
30
32
33
35
36
37
38
40
40
THE EQUILIBRIUM OF A SYSTEM OF RIGID BODIES.
71. Conditions of equilibrium
42
72-77. Principle of Virtual Velocities
44
78. Centre of gravity highest or lowest
47
79. The equilibrium stable or unstable according as the centre of gravity
is in its lowest or highest position
81. Lagrange's proof of Virtual Velocities
48
49
83. Conditions of equilibrium deduced from the principle of Virtual
Velocities
51
CHAPTER IV.
CENTRE OF GRAVITY.
86. Triangle, pyramid, and frustrum of pyramid.
87. Co-ordinates of the centre of gravity, and the calculation of them in
various cases. Integration between limits explained and illus-
trated
88, 89. Guldinus' Properties
CHAPTER. V.
FRICTION.
MACHINES, AND
93. Equilibrium of the lever, pressure on the fulcrum
52
55
70
73
CONTENTS.
XV
ARTICLE
94. Three species of lever
+
95, 96. The balance: requisites for a good balance
98. Graduation of the steel-yard. 99. Roberval's Balance
100. Wheel and Axle. 102. Toothed Wheels
105.-111. Pullies. White's Pully. 113. Inclined Plane. 114. Wedge.
116. Screw
118. Friction
CHAPTER VI.
ROOFS, ARCHES AND BRIDGES.
PAGE
75
77
888
80
78
82
90
121. Tension of the tie-beam of a simple truss-roof
122. Explanation of the action of buttresses
Roofs of Trinity College Hall and Westminster Hall
93
94
96
123. Equilibrium of a symmetrical roof of any number of beams
96
128. Conditions of equilibrium of an arch.
124. Arch, explanation of terms....
126. The effect of friction in supporting an arch...
129. A form of arch that will bear any weight in any part.
99
100
101
103
130. Blackfriars Bridge......
104
131. The Gothic Arch. St Dunstan's Church.
St Dunstan's Church. Flying-buttresses. An
arch built in a wall.
132. Equilibrium of a dome. St Paul's Cathedral. Steeples. Salisbury
Cathedral. Stone roof of King's College Chapel....
104
105
136-139. Common Catenary. Property of its centre of gravity. Tension. 108
140. Catenary of equal strength. Suspension Bridges....
113
141. Formulæ reduced to Tables. Examples..
116
CHAPTER VII.
STATICAL PROBLEMS.
142–145. Remarks on the solution of statical problems...
146. Problems. Prob. 9, Roberval's Balance. Prob. 10, Application of
Virtual Velocities. Prob. 11, stable and unstable equilibrium.
Probs. 12, 13, 14, examples of friction......
CHAPTER VIII.
ATTRACTIONS.
148, 149. Attraction of a spherical shell and sphere on an external part-
icle, law of attraction being the inverse square
150. Ditto for an internal particle....
#4
120
122
137
139
xvi
ARTICLE
CONTENTS.
151, 152. Attraction of a spherical shell, law of attraction being any func-
tion of the distance. Examples…………..
PAGE
139
153, 154. The laws for which a shell attracts as if condensed into its centre. 141
155, 156. Attraction of an oblate spheroid of small ellipticity on a particle
at its pole, and also another at its equator.....
143
157. Ditto for a prolate spheroid of small ellipticity…….
145
158. Attraction of a homogeneous oblate spheroid of any ellipticity on a
particle in its mass..
145
159. Attraction of a spheroidal shell on an internal particle..
148
162. Formulæ for the attraction of an ellipsoid
149
163, 164. Attraction on an external particle. Ivory's Theorem..
149
165. Attraction of a body on a very distant particle the same nearly as if
condensed into its centre of gravity
153
167. Formulæ for the attraction of any heterogeneous mass..
154
168.
+
2
d² V
df²
d² V
d² V
+
dg² d h²
0, or Απρί
-
according as the particle is external or internal
169. Transformation of the equation in V to polar co-ordinates
155
157
170, 171. Method of expanding Vin a series. Laplace's Coefficients...... 159
172. Calculation of V for a homogeneous sphere
173, 174. Attraction of a nearly spherical homogeneous body
2
176. A function of µ, √1–µ³ cos w, √1-µ³ sin w can be expanded in
ω
a series of Laplace's Coefficients, provided that the function do
not become infinite between the values -1 and 1 of μ, and 0
and 2 of w
27
1
2
•
161
162
164
179. Expansion of r in the series a 1+ a Y₂+a Y₁+ a Y₂+ ......... 169
180. If Q; and R be two of Laplace's Coefficients, then if i and i' be
different integers ₁₂ Q R dµ dw = 0.
i'
2 T
181. A function of μ, √√√1-μ² cos w,
2
√1-μ³ cos w, √1-2 sin w can be expanded in
only one series of Laplace's Coefficients..
182. Values of V for an external and internal particle in terms of Yo,
Y₁, Y₂, &c.... ..
169
171
172
183, 184. If a=the radius of the sphere of which the volume equals the
volume of the attracting body, then Yo=0: also if the origin of r
be placed at the centre of gravity of the mass, then Y₁ =0........... 173
185, 186. Attraction of a heterogeneous body consisting of thin strata
nearly spherical, homogeneous in themselves, but differing from
one another in density..
174
CONTENTS.
xvii
ARTICLE
DYNAMICS.
CHAPTER I.
DEFINITIONS. LAWS OF MOTION.
187. The measure of the position of a body in space...
188. Necessity of choosing units of measure
PAGE
176
177
189–191. Means of measuring motion. Velocity, uniform and variable. 177
193–196. Motion of a body uninfluenced by external causes. First Law
of Motion. Nature of the proof of the laws of motion. Inertia... 180
198–206. Dynamical measure of force. Accelerating force, impulsive
and finite
207-210. Motion of a body moving under the combined influence of two
or more causes. Second Law of Motion. The use of this law in
referring the motion of a body to rectangular axes. Resolution
of velocity...
211-225. The object of the Science of Dynamics. The necessity of
discovering the relation between the statical and dynamical mea-
sures of force. Experiments for obtaining this relation. Atwood's
Machine. Moving force and Momentum. Experimental laws
of the elasticity of bodies. Third Law of Motion. Origin of the
term vis viva. Units of statical force and mass
226, 227. Interpretation of the Three Laws of Motion into a Principle
whereby the calculation of the motion of a system is made to
depend upon the conditions of equilibrium of forces
CHAPTER II.
THE MOTION OF A MATERIAL PARTICLE.
182
188
191
203
228-230. Equations of motion: determination of the arbitrary constants.. 205
231–236. Rectilinear motion, under the action of gravity, and central
forces, and on a plane ..
237-239. Curvilinear motion under the action of gravity..
240-245. Properties of central orbits …..
246. Differential equation of central orbits
249–251. Orbit when the force varies as the distance..
207
213
216
220
222
252. Orbit when the force varies as the inverse square of the distance:
conditions of projection that it shall be an ellipse, hyperbola, or
parabola......
253,254. The law of force found when the orbit is known. Force in the
conic sections about the focus: in the ellipse about the centre.
Force in a circle about the centre. Centrifugal force and centri-
petal force defined...
*
•
226
228
xviii
ARTICLE
CONTENTS.
255. The centrifugal force of a particle moving in space equals the
square of the velocity divided by the radius of absolute curva-
ture, and acts in the osculating plane……
256--263. Kepler's Laws, and their application to determine the nature
of the force acting on the planets: first conception of the exist-
ence of the Law of Universal Gravitation: the evidence in its
favour sufficient to induce us to enter upon the calculation of its
consequences.....
PACE
230
232
CHAPTER III.
MOTION OF TWO MATERIAL PARTICLES ATTRACTING EACH OTHER.
264. Motion of the centre of gravity, law of force the inverse square……………
265, 266. The orbits the bodies describe about each other, and about their
centre of gravity
238
240
267. The orbits about the centre of gravity are similar
241
268, 269. Remarks on the errors in Kepler's Laws..
243
270. Method of determining the elements of an orbit from observation... 244
273. Time of motion in an elliptic orbit about the focus..
274-280. Expansion of the eccentric, and true anomalies, and the distance
246
in terms of the mean anomaly..
248
281. Time in a parabolic orbit about the focus...
251
282, 283. Place of a body in a parabolic orbit, and also in a very eccentric
ellipse, at a given time
251
CHAPTER IV.
EXPLANATION OF THE LUNAR PERTURBATIONS.
284-287. Introductory remarks..
254
288. Principle of the superposition of small motions...
256
289–293. Brief history of lunar inequalities....
257
294–298. Calculation of the disturbing forces and explanation of terms... 258
299–301. Effect on the periodic time of the Moon……..
262
302. Effect on the velocity in the circular orbit.
263
303. Effect on the form of the circular orbit..
264
304. The ratio of the axes of the oval orbit..
265
305, 306. The velocity in the oval orbit. Moon's Variation
266
307. Instantaneous elliptic orbit….
268
309. Effect of the mean central disturbing force on the line of apsides
270
311. Effect of the tangential force on the line of apsides......
272
312-314. Effect of the disturbing forces on the eccentricity.
317. Effect on the inclination and line of nodes....
274
278
•
CONTENTS.
ARTICLE
318-320. Calculation of the motion of the nodes, after Newton's method.
xix
PAGE
Mean motion....
279
321. Calculation of the inclination of the Moon's orbit, after Newton's
method.....
283
CHAPTER V.
LUNAR THEORY.
285
323. The equations of motion of a body attracted by other bodies..
325, 326. Equations for calculating the distance of the Moon, and the incli-
nation of the lunar orbit to the elliptic in terms of her longitude….. 289
327. Comparison of small quantities....
292
328. Expansions of P, T, S to the third order...
293
329. Integration of the equations, first approximation....
295
330. Reason why some terms must be calculated to the third order for the
second approximation.......
296
331–333. Calculations for a second approximation………
297
334, 335. Differential equations for second approximation: introduction
g.....
of the constants c and g.
336, 337. Integration of the equations, second approximation...
338. Distance of the Moon from the Earth.....
339. Longitude in terms of the time.......
341. Geometrical interpretation of the terms in the formulæ for the dist-
ance and longitude of the Moon. Progression of the line of
apsides. The Variation. The Evection. The Annual Equation.
The Reduction......
342. Geometrical interpretation of the terms in the formula for the in-
clination of the Moon's orbit to the ecliptic. Regression of the
line of nodes..
305
299
302
303
304
308
343. Error in the calculated progression of the apsides...
309
345. Perturbations in the motion of a satellite are deducible from those
of the Moon...
310
346. Notices of inequalities of the third and higher orders.....
310
347. The centre of gravity of the Earth and Moon nearly describes an
ellipse about the Sun..
311
348. Means of determining the mass of the Moon....
314
CHAPTER VI.
PLANETARY THEORY.
349. Comparison of the Lunar and Planetary Theories…….
350. Instantaneous ellipse, or ellipse of curvature.
315
315
•
C
XX
CONTENTS.
ARTICLE
352. Explanation of the process of integration....
353. Integration of the equations for an undisturbed planet..
354. Elements of the orbit in terms of the arbitrary constants
355-357. Integration of the equations for a disturbed planet...
358-362. Transformation of the differential coefficients of R..
363-366. Variations of the elements....
PAGE
317
319
323
325
328
332
367. Method of expansion of R…
338
368–372. The form of the terms and the order of magnitude of the co-
efficients in this expansion. The constant part of R………
340
374. Effect of the terms of R (after the first) periodical..
348
375. Some terms in R much increased by integration: Jupiter and
Saturn, the Earth and Venus......
349
350
377. Difference between periodic and secular variations..
378. Equations for calculating the secular variations..
379–383. Stability of the mean distances, and the mean motions: and of
the eccentricities and inclinations. The fact that the planets re-
volve about the Sun in the same direction ensures the Stability
of the System...
384-389. Secular variations of the eccentricity, longitude of the peri-
helion, inclination, and longitude of the node..
391, 392. The masses and elements of the heavenly bodies..
351
352
355
362
CHAPTER VII.
MOTION OF A PARTICLE ON CURVES AND SURFACES.
SIMPLE PENDULUM.
393. Motion on a plane curve, gravity acting...
394. Pressure on the curve.....
395. Motion on a cycloid: oscillations are isochronous...
396. Motion on a circular arc……..
397-399. Pendulum: measuring of heights and depths..
401. Oscillation of a pendulum in a cycloid....
368
369
370
372
374
376
402. Table of the lengths of the seconds pendulum in various places....... 377
403-406. Perturbations of the motion of a pendulum. Examples. Pen-
dulum escapements..
378
408, 409. Motion on a spherical surface...
407. Velocity on a surface independent of the path....
410. Pressure on a curve surface..
411. Pressure when the particle moves in a bent tube..
387
388
390
391
1
CONTENTS.
xxi
CHAPTER VIII.
ARTICLE
PAGE
PROBLEMS ON THE MOTION OF BODIES CONSIDERED AS PARTICLES. 393
CHAPTER IX.
PRELIMINARY ANALYSIS.
412. Object of this Chapter stated...
415, 416. Formulæ for the transformation of co-ordinates.
398
399
417. There is one, and in general only one system of principal axes in
a body...
400
418. To find the principal axes when one axis is known. Examples....... 402
419–421. Moment of Inertia: its property with respect to the centre of
gravity. Radius of Gyration. Examples....
422. Moment of inertia about any axis...
423-426. Properties of the principal moments of inertia…
CHAPTER X.
403
406
407
MOTION OF A RIGID BODY ACTED ON BY FORCES OF FINITE INTENSITY.
427. General equations of motion...
410
428. The motion of the centre of gravity the same as if all the forces
acted at that point.......
411
429. The motion of rotation the same as if the centre of gravity were fixed. 412
430. The use of these two principles....
413
CHAPTER XI.
MOTION OF A RIGID BODY ABOUT A FIXED AXIS: FINITE FORCES.
431. The angular accelerating force.....
414
415
432. The time of a small oscillation, gravity. The Compound Pendulum. 415
433. The centres of oscillation and suspension are reciprocal...
434-436. The length of a seconds pendulum determined experimentally.
Captain Kater's pendulum. The effect in shifting the sliding
weight. The axes of support at the centres of oscillation may be
knife edges with the edges passing through those points, or equal
cylinders with the circumferences passing through the points of
suspension and oscillation.......
416
438. Pressure on the fixed axis..
421
439. The principal axes through the centre of gravity are permanent axes
when no forces act..
422
xxii
CONTENTS.
CHAPTER XII.
MOTION OF A RIGID BODY ABOUT A FIXED POINT: FINITE FORCES.
ARTICLE
PAGE
441. The linear velocities of a particle parallel to the axes in terms of the
angular velocities about the axes. The Instantaneous Axis......... 424
442-445. The position of this axis in space, the angular velocity of the
body about it, and its position with respect to the principal axes... 425
446. Equations for calculating the angular velocities about the principal
axes.
427
447-449. The position of the body in space in terms of these angular
velocities..
429
450-456. Motion about a fixed point when no forces act..
431
457, 458. Stability of the Earth's rotation.....
439
459. Equations for calculating the effect of the Sun and Moon on the
rotatory motion of the Earth....
441
460, 461. The length of the mean day is invariable...
443
462-465. The Precession of the Equinoxes and the Nutation of the
Earth's Axis. The cause of these...
446
466. The inclination of the Earth's axis to the lunar orbit is nearly in-
variable. The cause of Lunar Nutation is the regression of the
Moon's line of nodes....
467, 468. Calculation of Lunar Precession and Nutation…………….
450
452
469, 470. Solar and Lunar Precession, (calculated finally in Art. 551.)...... 454
CHAPTER XIII.
MOTION OF A RIGID BODY ACTED ON BY IMPULSIVE FORCES.
472. Object of this Chapter..
456
473. General equations of motion.....
457
474. The motion of the centre of gravity the same as if the forces all
acted at that point......
458
475. The motion of rotation the same as if the centre of gravity were fixed. 459
477. Motion about a fixed axis...
460
•
478. Motion about a fixed point..
460
479. The position of the Axis of Spontaneous Rotation...
480. The position of the Centre of Percussion....
461
A
463
.....
CHAPTER XIV.
THE MOTION OF A FLEXIBLE BODY.
482. Equations of motion of a vibrating cord..
466
483, 484. Integration and interpretation of the equations......
468
CONTENTS.
xxiii
CHAPTER XV.
GENERAL DYNAMICAL PRINCIPLES.
ARTICLE
486. Conservation of the motion of the centre of gravity, whether the
forces be impulsive or of finite intensity…….
487, 488. Conservation of areas, whether the forces be impulsive or of
finite intensity. Case in which this principle is true when the
forces are not all internal....
489-495. The Invariable Plane: also the plane of Greatest Moments.
The action of the Comets on the Planetary System is insensible.
Remarks upon the hypothesis of the change in the Earth's tem-
perature, adopted by geologists. Effect of earthquakes, volcanic
explosions, winds, friction and pressure of the Ocean, &c. on the
position of the Earth's axis of rotation...
PAGE
474
477
481
496-503. Principle of Vis Viva: and remarks. Conservation of vis viva.
The vis viva of a system equals the vis viva owing to the motion.
of translation + vis viva owing to rotation. Vis viva is lost or
gained in a system by the action of internal impulsive forces
according as they are of the nature of collision or explosion: a
perfectly elastic system sustains no loss. The effect of the degra-
dation of rocks, breakers on the sea-shore, volcanic explosions, &c.
the descent of rivers, vapour and cloud in the form of rain, boul-
ders and avalanches, evaporation, &c. on the length of the day.... 487
504. Principle of Least Action (or, of Stationary Action)..
493
506, 507. Professor Hamilton's Principle of Varying Action..
495
508-511. The Laws of Small Oscillations: Co-existence of Small Vibra-
tions; and the Superposition of Small Motions..
499
CHAPTER XVI.
PROBLEMS ON THE MOTION OF RIGID BODIES, AND ANY MATERIAL
SYSTEM.
512. Observations on the solution of dynamical problems: applications
of the general principles of Chap. xv. Prob. 7, Atwood's Machine.
Probs. 20-22, examples of the action of friction. Probs. 23-25,
pressure on a fixed axis. Probs, 32-37, impact of balls. Probs.
38-41, illustrations of the action of springs in removing shocks.
Probs. 42-58, examples of the action of impulsive forces. Prob.
62, Robins' Ballistic Pendulum...
504
xxiv
CONTENTS.
HYDROSTATICS.
CHAPTER I.
ARTICLE
513, 514. Definitions.
515. Equable transmission of pressure…..
DEFINITIONS AND PRINCIPLES.
PAGE
528
529
530
531
532
533
516. Pressure referred to a unit of surface....
518. Pressure in the interior of a fluid………..
519, 520. Conditions of equilibrium of a fluid mass..
521. These conditions satisfied by central forces...
522, 523. Internal arrangement of a mass of fluid in equilibrium. Sur-
faces et couches de niveau………..
534
CHAPTER II.
FIGURE OF THE EARTH.
536
524. Previous remarks.
525-529. Spheroid the form of equilibrium of a homogeneous mass of
fluid: ellipticity too small for the figure of the Earth. A limiting
angular velocity beyond which the equilibrium is impossible……………... 537
530. The figure of a mass of fluid revolving about an axis through a
centre of force varying inversely as the square of the distance...
531-535. Heterogeneous figure of the Earth. The equation for calcu-
lating the form of the strata. Simplification of the equation. Sim-
plification of the radius vector of the strata. The strata are all
spheroidal, and increase in ellipticity from the centre to the surface 541
536. Equations for calculating the ellipticity..
538-541. Clairaut's Theorem, Value of gravity. Length of the seconds
pendulum. Length of a degree of latitude. An independent
proof of the correctness of the approximation..
540
549
551
•
543. Principal moments of inertia of the Earth...
554
545. Law of density of the strata found..
556
546, 547. Calculation of the ellipticity..
557
548. Moments of inertia calculated with this law..
559
549. Determination of an unknown constant q in the formula for the
density in the case of the Earth..
560
550, 551. Numerical calculation of the ellipticity, and the Precession of
the Equinoxes
561
552-556. Additional theorems..
562
CONTENTS.
XXV
CHAPTER III.
FORM OF EQUILIBRIUM OF THE OCEAN UNDER THE MOON'S ATTRACTION,
AND THE FORM OF THE ATMOSPHERE.
ARTICLE
PAGE
557, 558. The form of the Ocean a prolate spheroid of small ellipticity;
if we neglect the Earth's motion and ellipticity: remarks....
559. Form of the atmosphere: only one form of equilibrium. Zodiacal
Light does not arise from the Sun's atmosphere....
567
570
HYDRODYNAMICS.
CHAPTER I.
EQUATIONS OF MOTION.
573
561. The transmission of pressure through a fluid in motion.....
562-565. Equations of motion: fluid incompressible: fluid compressible 574
566–568. The pressure at any point of a homogeneous incompressible
fluid mass in motion when udx+vdy+wdz is a perfect differen-
tial: if this be a perfect differential at any instant it is at every
instant of the motion......
578
569, 570. Equations for calculating the motion of an elastic fluid, the
excursions of the molecules being very small, and no forces acting 579
CHAPTER II.
TIDES AND STABILITY OF THE OCEAN.
572. The difficulty of the subject of the Tides requires some hypothesis
to be made in addition to that of gravitation. Laplace's and
Daniel Bernoulli's Theories..
573–576. Calculation of the height of the tide on Bernoulli's hypothesis.
Time of high tide at a given place. Establishment of the port.
Comparison with observations. Tide at a port where the tidal
wave arrives by two distinct routs: Interference of tides......
577, 578. Transformation of the equations to polar co-ordinates..
579, 580. Stability of the Ocean...
581
583
589
595
CHAPTER III.
MOTION OF THE HEAVENLY BODIES IN A RESISTING MEDIUM.
581. Law of resistance
597
xxvi
CONTENTS.
ARTICLE
582. Effect of a resisting medium on a planet moving in an orbit nearly
circular is to diminish the mean distance, increase the mean mo-
tion, diminish the eccentricity, and to leave the position of the
axis major unchanged. These effects not sensible in our plane-
tary system; but may be so with the comets.
be so with the comets. Method of calcu-
lation for comets....
CONCLUSION.
SUMMARY OF ARGUMENTS IN FAVOUR OF UNIVERSAL
GRAVITATION. 602.
PAGE
598
.
INTRODUCTION AND DEFINITIONS.
1. THE uniformity which characterises the operations of
nature leads us to conjecture that the phenomena of the mate-
rial world are regulated by certain fixed laws. The never-
ceasing alternation of light and darkness, the unvarying suc-
cession of the Seasons, the periodical flux and reflux of the
Ocean, the constant tendency of bodies downwards, and num-
berless such like appearances, mutually strengthen the sus-
picion that they are the necessary consequences of some
universal principles with which matter has been endued by
the Creator of the World. It is the object of Mechanical
Philosophy to search out these Principles.
But at the very outset we are overwhelmed with such
a variety of causes, all in simultaneous action, that it becomes
no easy task to disentangle the simple laws from the maze in
which they are involved.
The whole universe is in perpetual fluctuation, changes are
incessantly taking place, and while we are occupying ourselves
with the investigation of present appearances, the appear-
ances themselves are in the act of transition from one state to
another, and new phenomena press themselves upon our notice.
It is only by a careful and attentive examination of the
phenomena perpetually presenting themselves to our view,
arranging them in groups, selecting, re-examining and re-
arranging, that we are able to rise, by a process of induction.
and generalisation from the mass of facts accumulated by ob-
servation, to the laws from which they flow. Having once
reached this summit, we descend, making these laws our guides,
and follow out, by a deductive process, the phenomena which
must naturally result from their operation. A comparison of
A
INTRODUCTION AND DEFINITIONS.
the calculated results with the phenomena observed determines,
by their agreement or disagreement, whether the laws, to
which our investigations have conducted us, are laws of nature
or not. It is by a process of this nature that we are con-
vinced of the truth of the Law of Universal Gravitation: and
the chief object which we have in view in the present under-
taking is to lead the student step by step up to this great
Principle, and then shew him the real foundation on which we
rest our belief of its truth by displaying its power of explain-
ing accurately every astronomical phenomenon with which we
are acquainted.
2. We give the name Matter to everything that affects
our senses in any manner whatever. Bodies are portions of
matter limited in every direction, and are consequently of a de-
terminate form and volume. The Mass of a body is the
quantity of matter of which it is composed. A material par-
ticle is a body infinitely small in every dimension.
3. We may consider a body of finite dimensions to be an
assemblage of an infinite number of material particles, and its
mass to be the sum of all their infinitely small masses.
The mass of a body is said to be homogeneous when the
same quantity of matter is contained in equal volumes of the
body. When this is not the case, the mass is said to be he-
terogeneous.
Bodies of different material have different quantities of
matter comprised in the same volume. The term density is
used to indicate the quantity of matter contained in a given
volume of a mass, and serves to measure the quantities of
matter in different bodies. The density of a homogeneous
mass is measured by the quantity of matter in a unit of a vo-
lume: when the mass is heterogeneous, the density at any
point is measured by the quantity of matter of the same
nature as that at the given point that would occupy a unit
of volume.
4. A body is in motion when the body or its parts
occupy successively different positions in space. But since
space is infinite in extent and in every part identical, we
cannot judge of the state of rest or motion of a body without
comparing it with other bodies: and, for this reason, all mo-
INTRODUCTION AND DEFINITIONS.
3
tions which come under our observation are necessarily relative
motions.
All bodies are capable of motion; but experience shews
us that matter will not move spontaneously. Also it is a
matter of experiment, as it is indeed of ordinary experience,
that when a body is passing from a state of rest to a state
of motion, we can always attribute the change to the action
of a foreign cause.
5. Any cause which produces or tends to produce motion
in a body or to change its motion is termed Force.
6. MECHANICS is the Science which treats of the Laws
of Rest and Motion of Bodies, whether Solid or Fluid.
We divide this science into four branches.
(1) STATICS, which treats of the laws of the equilibrium
of solid bodies.
(2)
DYNAMICS, of the laws of motion of solid bodies.
(3) HYDROSTATICS, of the laws of the equilibrium of
fluid bodies, and
(4) HYDRODYNAMICS, which treats of the laws of motion
of fluid bodies.
7. In Statics force is estimated by the pressure it causes
a body when at rest to exert against another with which it
is in contact and is said to be estimated statically. In Dy-
namics, however, the estimate used is the space through which
the force causes a body to move in a given time, and the
measure is said to be dynamical. We shall endeavour to
make this more intelligible.
8. Let us begin with the consideration of the ordinary
phenomenon of a falling body. Experience teaches us that
if a body be let free from the hand, it will fall downwards
in a certain determinate direction: however frequently the
experiment be made, the result is the same, the body strikes
the same spot on the ground in each trial, provided the place
from which it is dropped remain the same. Now this un-
deviating effect must be the result of some cause equally
undeviating. The cause is assumed to be an affinity which
4
INTRODUCTION AND DEFINITIONS.
all bodies have for the earth, and is termed the force of
Attraction. It is found to prevail in all parts of the globe.
The direction in which the body falls is called the vertical
line of the place where the experiment is made: and a plane
perpendicular to this is called the horizontal plane of the
place. If the motion be prevented by interposing the hand,
the body exerts a pressure, and it requires a muscular effort
to keep the body from falling.
In one case then the attraction of the Earth produces
a pressure, in the other motion: now of these, viz., the pres-
sure exerted by the body when at rest, and the space through
which the body falls when in motion, either may be taken
as a means of estimating the intensity of the force of attrac-
tion at different places on the Earth, at different elevations
above or depressions below its surface.
9. And the same may be said of any force: as another
instance let us consider the force exerted by a constrained
spring. If the force of the spring be estimated by the pres-
sure it produces on a body holding it in its constrained posi-
tion, the estimate is said to be statical. But if the force be
estimated by the magnitude of the motion generated in a body
which it causes to move, the estimate is dynamical.
10. Weight is the name given to the pressure which
the attraction of the Earth causes a body to exert on another
with which it is in contact. Since the gravitation of bodies
downwards is unceasing, weight becomes a very useful means
of estimating all statical forces. Thus the force of a con-
strained spring, may be measured by the weight which will
just hold the spring in its constrained position. The force
of attraction of a magnet may be measured by the weight
it will sustain; and so of other forces.
STATIC S.
CHAPTER I.
THE COMPOSITION AND EQUILIBRIUM OF FORCES ACTING UPON
A MATERIAL PARTICLE.
11. WHEN a single force acts upon a particle, it is clear,
from the meaning we attach to the term force, that the particle
cannot be at rest.
Experience shews us, however, that two forces may coun-
teract each other's effects in producing motion. In such a
case the forces, even though they originate from different
causes, are said to be equal; since they are measured by their
effects. Experience likewise shews us that three or more
forces may be in equilibrium with each other, if their directions
and magnitudes are properly adjusted. The object of the
Science of Statics is to determine the relations which must
exist among the forces, in magnitude, direction, and points of
application, that they may produce equilibrium when acting
on a body.
12. Now any given number of forces acting upon a par-
ticle must either be in equilibrium, or else produce an effect
on the particle which some single but unknown force would
produce. For if the forces be not in equilibrium, the particle
will begin to move in some determinate curve line immediately
the particle is abandoned to the action of the forces. It is
clear, then, that a single force may be found of such a mag-
nitude, that if it act along the tangent at the commencement of
6
STATICS.
the curve, and in a direction opposite to that in which the mo-
tion would take place, this force would prevent the motion,
and would consequently be in equilibrium with the other forces
which act upon the particle. If, then, we were to remove the
original forces, and replace them by a single force, equal in
magnitude to that described above, but acting in an opposite
direction, the particle would still remain at rest. This force,
which is equivalent in its effect to the combined effect of the
original forces, is called their resultant, and the original forces.
are called the components of the resultant.
13. It will be necessary, then, to begin by deducing
rules for the composition of forces; that is, for finding their
resultant force. After we have determined these, it will be an
easy matter to deduce the analytical relations which forces in
equilibrium must satisfy, by equating the expression which
gives the magnitude of their resultant to zero.
PROP.
To find the resultant of a given number of forces
acting upon a particle in the same straight line: and to find
the condition that they must satisfy, that they may be in equi-
librium.
14. When two or more forces act on a particle in the
same direction, it is evident that the resulting force is equal to
their sum, and acts in the same direction.
When two forces act in opposite directions on a particle,
it is equally clear that their resultant force is equal to their
difference, and acts in the direction of the greater component.
When several forces act in different directions, but in the
same straight line, on a particle, the resultant of the forces
acting in one direction equals the sum of these forces, and acts
in the same direction: and so of the forces acting in the op-
posite direction. The resultant, therefore, of all the forces
equals the difference of these sums, and acts in the direction of
the greater.
If the forces acting in one direction are reckoned positive,
and those in the opposite direction negative, then their resultant
equals their algebraical sum; its sign determining the direc-
tion in which it acts.
PARALLELOGRAM OF FORCES.
7
15. In order that the forces may be in equilibrium,
their resultant, and therefore their algebraical sum must equal
zero.
PROP. To find the resultant of two forces acting upon
a particle not in the same straight line*.
16. Let P and Q represent the magnitudes of the two
forces: A the particle (fig. 1.), AP, AQ the directions in which
* The following proof of the Parallelogram of Forces given by M. Duchayla,
is well worthy of attention for the simplicity of its demonstration.
1. To find the direction of the resultant of two forces acting upon a point.
When the forces are equal, it is clear that the direction of the resultant will bisect
the angle between the directions of the forces: or, if we represent the forces in mag-
nitude and direction by two lines drawn from the point where they act, the diagonal
of the parallelogram described on these lines will be the direction of the resultant.
Let us assume that this is true for two unequal forces, p and m: and also for
p and n. We can prove that it must then necessarily be true for two forces, p and
m+n.
Let A (fig. 2.) be the point on which the forces p and m act, AB, AC their
directions and proportional to them in magnitude: complete the parallelogram BC, and
draw the diagonal AD: then by hypothesis, the resultant of p and m acts along AD.
Again, take CE in the same ratio to AC that n bears to m. Since it is an ex-
perimental fact that the point of application of a force may be transferred to any point
of its direction, without disturbing the equilibrium, so long as the two points of appli-
cation are invariably connected, we may suppose the force n to act at A or C: and
therefore the forces p, m and n, in the lines AB, AC, and CE are the same as p and
m+n in the lines AB and AE.
Now, replace p and m by their resultant, and transfer its point of application from
A to D then resolve this force at D into two, parallel to AB and AC; these resolved
parts must evidently be p and m, p acting in the direction DF, and m in the direction
DG. Transfer these two forces, p to C and m to G.
But by the hypothesis, p and n acting at C have a resultant in the direction CG;
let then p and n be replaced by their resultant, and transfer its point of application
to G. But m acts at G.
Hence by this process we have, without disturbing the equilibrium, removed the
forces p and m+n which acted at A to the point G.
Therefore the resultant of p and m+n acts in the direction of the diagonal AG,
provided our hypothesis is correct.
But the hypothesis is correct for equal forces, as p, p, and therefore it is true for
forces p, 2p; consequently for p, 3p, and so it is true for p, r. p.
Hence it is true for p,r.p and p, r. p, and consequently for 2p, r.p, and so
forth; and it is finally true for s.p and r.p, r and s being positive integers.
We have still to shew that the Proposition is true for incommensurable forces.
Let AB, AC (fig. 3.) represent two such forces. Complete the parallelogram BC.
Then if their resultant do not act along AD, suppose it to act along AE; draw EF
parallel to BD. Divide AB into a number of equal portions, each less than DE;
mark
8
STATICS.
the forces act a the angle between these directions. Let R
represent the magnitude of the resultant, and suppose AR is
the direction in which it acts, this line being in the same plane
as AP, AQ, and lying between them: let & be the angle be-
tween AP and AR. Draw a line PAQ₂ in the plane of the
forces through the point A, and perpendicular to AR.
2
2
Now let us imagine that P is the resultant of two forces
P₁ and P₂ acting in the directions AR, AP; and that Q is the
resultant of two forces Q and Q₂, acting in the directions AR
and AQ. Then (Art. 14.)
R = P₁ + Q₁
a
1
2
and 0 – P₂ – Q₂
=
1
2
(1),
P₁ and P₂ are functions of P and 0; and Q₁ and Q are similar
functions of Q and a 0. Since P, P₁, P₂ are merely the
numerical ratios which the corresponding forces bear to the
unit of force, and since the relation they bear to one another
must manifestly be independent of the unit we choose to adopt,
the relation between P and P, must be of the form
and ..
P₁
P
P₂
= function of 0 = ƒ (0) suppose;
P=S (5-0).
2
f
We have, then, to determine the form of ƒ (0).
mark off from CD portions equal to these, and let G be the last division, this evi-
dently falls between D and F; draw GK parallel to AC. Then two forces repre-
sented by AC, AG have a resultant in the direction AK, because they are commen-
surable and this is nearer to AG than the resultant of the forces represented by AC,
AB, which is absurd, since AB is greater than AG,
:
In the same manner we may shew that every direction besides AD leads to an
absurdity, and therefore the resultant must act along AC, whether the forces be com-
mensurable or incommensurable.
2. To find the magnitude of the resultant.
Let AB, AC be the directions of the given forces, AD that of their resultant:
(fig. 4.) take AE opposite to AD, and of such a length as to represent the magnitude
of the resultant. Then the forces represented by AB, AC, AE balance each other.
Complete the parallelogram BE.
Hence AC is in the same straight line with AF: hence FD is a parallelogram :
and therefore AE FB = AD.
Or the resultant is represented in magnitude as well as in direction by the diagonal
of the parallelogram.
PARALLELOGRAM OF FORCES.
9
We assume that a force can produce no effect in a direc-
tion perpendicular to its own direction.
2
This principle points out to us two general conditions
which P₁ and P₂ must fulfil; for since P can produce no
effect in a direction at right angles to its own, it follows that
the sum of the resolved parts of P₁ and P, in a direction at
right angles to that of P must equal zero; and the sum of
their resolved parts in the direction of P must equal P.
These conditions furnish the equations
Ꮎ
P₁ƒ (0) - P, ƒ (0) = 0,
2
2
1
P₁ƒ (0) + P₂ ƒ ( − 0)
π
P.
2
Then, by putting for P. and P₂ their values Pf(0) and
π
2
Pƒ (1-0), and dividing by P, we have the first equation
identical, and the second gives
π
2
=
1.
9
{ƒ (0)}² + {ƒ (==
(2).
This is the equation which ƒ (0) is to satisfy; but it admits of
an infinite variety of solutions, and we assume (as the result of
experiments allows us) that P, bears a determinate ratio to P,
or ƒ (0) has a determinate value, for every value of 0. There
must consequently be some other conditions, arising from the
nature of the question, which f(0) must satisfy; and which
are to be our guides in selecting the proper solution of the
equation just deduced.
The direct process would be, first to obtain the general
solution of our equation, and then to determine the values of
the arbitrary quantities involved in the general solution by the
particular values of ƒ (0) for particular values of given by
the nature of our problem. We may, however, reverse the
process, and first search for the particular values of ƒ (0), and
use these as our guides in detecting the proper solution.
B
•
10
STATICS.
Now the principle which has hitherto guided us-viz. that
a force produces no effect in a direction at right angles to its
own-furnishes us with new conditions which point out which
of the solutions of equation (2) is to be chosen.
For whenever the direction of P₁ is at right angles to the
direction of P, then P₁ = 0 and P₂ = P or - P; and when-
ever the direction of P₂ is at right angles to that of P, then
Por
P as exhibited below;
= 0 and P₁ = P or
P 2
when 0 = 0,
P₂ = 0,
P₁ = P;
.. ƒ (0),
= 1,
π
0 =
P₁ = 0,
2
P₂ = P ;
f
1914
0,
Ꮎ
П,
2
P₂ = 0,
1
P₁ = - P;
ƒ (π)
19
3π
3π
A
P₁ = 0,
P₂ = - P ;
f
0,
10
2
0 = 2п,
2
P₁ = 0,
P₁ = P;
..
ƒ (2π) = 1.
and all these cases are comprised in the formula
f(n.) = cos (n.)
2
n being an integer.
2
…………. (3),
These equations (2) and (3) are the only conditions which
f() is to satisfy and since, as we have observed, ƒ (0) must,
from the nature of the question, have a determinate form, it
follows that there is only one form of ƒ (0) which satisfies both
equations (2) and (3); consequently if we can find one, this
is the solution we are seeking.
Now equation (3) suggests ƒ(0) = cos; this fully satisfies
both (2) and (3), and is consequently the required solution.
Hence equations (1) become
R = P cos 0 + Q cos (a − 0)
0
0 = P sin - Q sin (a 0)...... (4);
-
PARALLELOGRAM OF FORCES.
11
adding the squares of these,
R² =
P² + Q² + 2 P Q cos a…………………
a. ……..(5).
Equation (4) determines the direction of the resultant, and
(5) its magnitude.
17. These equations point out the following geometrical
construction, (fig. 1.)
Take AB, AC in the ratio of P to Q, through B draw
BD parallel to AC and cutting AR in D: join CD. Then
by Trigonometry,
BD = AB
sin
sin (a - 0)
AB by (4) = AC
P
by the construction.
Hence BC is a parallelogram: and its diagonal is the
direction in which the resultant of P and Q acts.
Again, by Trigonometry,
AD2 AB² + AC² + 2 ABAC cos a.
=
Compare this with equation (5), and we see that the diagonal
represents the magnitude of the resultant on the same scale
that the sides of the parallelogram represent the forces P
and Q.
This Proposition is, in consequence of the property just
proved, called The Proposition of the Parallelogram of Forces.
18. COR. 1. Any force acting on a particle may be re-
placed by two others, if the sides of a triangle, drawn parallel
to the directions of the forces, have the same relative propor-
tion that the forces have.
This is called the resolution of a force.
19. COR. 2. When three forces acting on a particle are
in equilibrium, they are respectively in the same proportion as
the sines of the angles included by the directions of the other
two.
12
STATICS.
For if we refer to fig. 4, we have
P: Q R :: AB: AC (or BD) AD
PROP.
:: sin ADB : sin BAD: sin ABD
:: sin CAE : sin BAE: sin BAC.
Three forces act upon a particle in directions
making right angles with each other: required to find the
magnitude and direction of their resultant.
20. Let AB, AC, AD represent the three forces X, Y, Z
in magnitude and direction: fig. 5.
Complete the parallelogram BC, and draw AE: then AE
represents the resultant of X and Y in magnitude and direc-
tion, by Art. 17. Now the resultant of this force and Z,
which are represented by AE, AD, is represented in magnitude
and direction by AF, the diagonal of the parallelogram DE.
Hence the resultant of XYZ is represented in magnitude.
and direction by AF.
Let R be the magnitude of the resultant, and abc the
angles the direction of R makes with those of XYZ.
Then, since AF2 = AE2 + AD°
= AB² + AC² + AD²;
R2
X² Y²
X + Y + Z.
AB
X
Also, cos a =
AF R
AC Y
cos b
AF
R
AD
Z
COS C
AF
R
Whence the magnitude and direction of the resultant are de-
termined.
21. COR. Any force R, the direction of which makes
the angles abe with three rectangular axes fixed in space,
ANY FORCES ACTING ON A PARTICLE.
13
may be replaced by the three forces R cos a, R cos b, R cos c,
acting simultaneously on the particle on which R acts, and
having their directions parallel to the axes of co-ordinates re-
spectively.
PROP. Any number of forces act upon a particle in any
directions required to find the magnitude and direction of
their resultant.
22. Let PP,....
α
ין
be the forces, and aẞy, a,ß,y,,
the angles their directions make with three rectangular axes
drawn through the proposed point.
Then the component parts of P in the directions of the
axes are, by Art. 21,
P cos a,
P cos ß, P cos y,
or X, Y, Z, suppose.
Resolving each of the other forces in the same way, we
reduce the system to three forces, by adding those which act
in the same lines (Art. 14.), we thus have
P cos a + P, cos
cos a,
+
or . P cos a or Σ. Χ,
P cos B+ P, cos ß, +
or . P cos ẞ or Σ. 1,
and P cos y + P, cos y, +
or Σ. P cos y or Σ. Z,
acting in the directions of the axes of æ, y and ≈.
The symbol indicates that we are to take the sum of all
the quantities in the system which are symmetrical with that
before which it is placed.
If we call the resultant R, and the angles which the di-
rection of R makes with the axes a, b, c, we have, by Art. 20.
R² = (Σ ⋅ X)² + (Σ . Y)² + (Σ. Z)²,
and cos α =
Σ. Χ
R
cos b
=
Σ. Γ
R
Σ. Ζ
cos c
COS C =
R
to determine the resultant.
14
STATICS.
1
PROP. To find the conditions of equilibrium when any
number of forces act upon a material particle.
23.
R = 0;
When the forces are in equilibrium, we must have
·· (E. X)² +
.. Z. X=0,
Σ.Χ
(2. Y)² + (2. Z)² = 0;
Σ.
Σ.Υ
E. Y=0, Σ. Z = 0,
and these are the conditions among the forces, that they may be
in equilibrium. It follows, then, that if a material particle at
rest be acted on by forces whose intensities and directions
satisfy these three equations it will remain at rest.
These conditions may be expressed under another form:
and lead to a principle denominated the Principle of Virtual
Velocities.
PROP.
To prove the Principle of Virtual Velocities when
forces acting on a particle are in equilibrium.
24. Let xyz be the co-ordinates to the point of appli-
cation of the forces, and x+dx, y+dy, ≈+d≈ the co-ordinates
to a point near the former. Draw perpendiculars from this
latter point upon the directions of the forces P, P,...and let
Sr, Sr...be the distances of these perpendiculars from the
point xyz: hence
Sr = dx. cosa + dy. cos ß + ds.cos y
dy.cos
Sr, = Sx ⋅ cos a, + dy ⋅ cos ß,
.
·
If then we multiply the equations
Σ. P cos a = 0, Σ. P cos ẞ = 0,
+
Sz.cos y,
Σ. P cos γ = 0
by da, dy, dz, and add the equations, bearing in mind that
Sa, Sy, Sz are independent of the forces, and may therefore
be written inside the symbol Σ, we have
Σ. P (Sx cos a + dy cos B + d≈ cos y) = 0,
or Σ. Pdr = 0,
VIRTUAL VELOCITIES SINGLE PARTICLE.
15
which proves the following principle:-That if any number
of forces acting upon a particle be in equilibrium, and the
point of application be moved geometrically through any small
space, then the sum of the products of the forces and the spaces
described by the point of application relatively to the direc-
tions of the forces will vanish; these spaces being reckoned
positive when drawn in the direction in which the force acts,
and vice versa.
This is termed the Principle of Virtual Velocities, since
the spaces above mentioned measure the relative velocity of the
geometric motion in the direction of the forces. We shall
see in the next Chapter that this Principle is true for any
system of forces.
酱
​CHAPTER II.
THE COMPOSITION AND EQUILIBRIUM OF FORCES ACTING ON
A RIGID BODY.
25. A SOLID or fluid body is conceived to be an aggre-
gation of indefinitely small material particles or molecules,
which are held together by their mutual affinities. This ap-
pears to be a safe hypothesis, since experiments shew that any
body is divisible into successively smaller and smaller portions
without limit, if sufficient force be exerted to overcome the
mutual action of the parts of the body.
26. By the term rigid we mean to express that the mole-
cules of the body are held together in an invariable form; so
that the intensity of the molecular forces is infinitely greater
than that of the other forces which act upon the body. Were
this not the case, the figure of the body would depend upon
the forces which act upon it.
Now, in matter of fact, no body is perfectly rigid; every
body yields more or less to the forces by which it is acted on.
If, then, in any case this compressibility is of a sensible mag-
nitude, we shall suppose that the body has assumed its figure
of equilibrium, and then consider the points of application of
the forces as a system of invariable form.
27. We are quite unacquainted at present with the laws
according to which the molecules of a mass of matter act upon
each other. In consequence of this, we must look for some
principle which will enable us to calculate the effect of forces
acting upon a rigid body, without bringing the molecular
forces into the calculation.
And now we fall upon a case of the action of force totally
different from anything we have yet met with. In considering
TRANSMISSION OF FORCE.
17
its action on a single particle, the force was supposed to act
on the whole of the particle: but now we have to consider
the effect of forces acting on individual particles of an as-
semblage held rigidly together by their mutual affinities. The
force which acts upon any particle of the body must in some
way have its effect propagated through the whole system of
particles, in consequence of their invariable connexion. Sundry
experiments have led philosophers to the following principle;
which, as will be seen, exactly answers our purpose.
28. When a force, acting in combination with others,
holds a solid body in equilibrium, the equilibrium of the body
will not be disturbed if we transfer the point of application
of the force to any other point whatever in the line in which
the force is acting.
We shall commence with the simplest case of a rigid body
acted on by forces, and so ascend to the most general.
PROP. Two forces act upon a rigid body in the same
plane but not at the same point: required to find their re-
sultant.
29. Let A, B (fig. 6.) be the points upon which the forces
P and Q act: AP, BQ their directions: join AB, and pro-
duce PA, QB to cut each other in C, and let aß be the
angles which AP and BQ make with AB produced: then, by
Art. 28, we may suppose P and Q to act at C, the point C
being rigidly connected with AB.
Take Ca, Cb along CA, CB, in the ratio of P to Q: and
on these describe the parallelogram ab, and draw the diagonal
Cd: and let it be produced to cut AB in D.
Then, by Art. 17, Cd represents the resultant of P and Q
in magnitude and direction: let R be the resultant: then
since
Cd2
=
Ca² + Cb² - 2 Ca. Cb cos (a + B) ;
.. R
=
P² + Q³ − 2 PQ cos (a + ß),
this gives the magnitude of the resultant.
Let be the angle which the direction of R makes with
AB: then
C
18
RIGID BODY.
STATICS.
P
Ca Ca
sin Cda
sin DCB
Q
Cb
ad
sin a Cd
sin DCA
sin (3 + 0)
sin (0 - a)
sin ẞ+ cos ẞ tan 0
;
tan cos a
-
sin a
.. tan cos
{cos
tan
α
P
Q
cos ẞ} = sin ß + sin a
P sin a + Q sin ß
P cos a Q cos B
-
this determines the direction of the resultant.
Let AD = ∞: AB
= a: then
a
a
ᎠᏴ
DB DC
Ꮳ
DA DC DA
sin (B+0)
sin a
sin B
sin (0 − a)
P
Q
20
11
sin a P
sin ß`Q'
Q sin B
α P sin a + Q sin ß'
this determines the point of application of the resultant.
α 0
30.
Cor. 1. From the value of
we obtain
20
Р
(ax) sin B
Q
∞ sin a
perpendicular from D on Q's direction
perpendicular from D on P's direction
This shews us that if the point D be a fixed fulcrum, about
which the body can turn, then, in order that P and Q may
be in equilibrium about this fulcrum, in which case their
resultant must pass through D, they must be inversely pro-
COMPOSITION OF TWO PARALLEL FORCES.
19
portional in magnitude to the perpendiculars drawn from the
fulcrum on their directions.
31. COR. 2. The product of a force and the distance of
its direction from a given point is called the moment of the force
with respect to the point.
If through the point an axis be drawn at right angles to
the plane, passing through the point and the direction of the
force, this product is called the moment of the force with
respect to the axis.
Hence when P and Q, acting in the plane through D, are
in equilibrium about D, we learn by Cor. 1. that their moments
with respect to D, or the axis through D, must be equal and
opposite.
If the two forces P and Q are parallel to each other, their
directions will not meet when produced: and therefore the
demonstration of the last article must not be received; but, by
a simple artifice, we can easily remedy this difficulty.
PROP.
To find the resultant of two parallel forces acting
in the same plane on a rigid body.
:
32. Let P and Q be the forces; A, B (fig. 7.) their points
of application let P and Q act in the same direction, making
angles a with AB. The state of equilibrium of the body will
not be altered if we apply two equal and opposite forces, each
equal to S, at the points A, B, acting in the line AB.
Then P and S acting at A, are equivalent to some force
P' acting in some direction AP'; and Q and S acting at B,
are equivalent to some force Q' acting in some direction BQ'
inclined to AP.
Produce P'A, Q'B to cut each other in C, and draw CD
parallel to AP and BQ, and cutting AB in D.
Transfer P' and Q' to C, C being rigidly connected with
AB, and resolve them along CD and parallel to AB; the latter
parts will be S and S acting in opposite directions, and the
sum of the former is Q + P.
Hence R, the resultant of P and Q, = Q + P. Also since
the sides of the triangle ACD are parallel to the directions of
the forces P, S, P'; (sce Art. 18.)
20
STATICS. RIGID BODY.
P CD
S
DA
S
DB
and similarly,
Q
CD
P
DB a X
Q
DA
X
if AB = a and AD: = x;
X
Q
α
Q + P '
this determines the point of application of the resultant Q
33. If the force P act in a direction opposite to that of
Q, (fig. 8.) a similar process will lead us to
R = Q − P,
21
-
Q
a Q - P'
but these are included in the formulæ of last article by putting
- P for P.
34. We also observe that the formulæ of Art. 29. com-
prehend those for parallel forces, although we thought it best
not to assume this. We must put ẞ = π α or 2π a ac-
cording as P and Q act in the same or opposite directions.
35. If P = Q in Art. 33., then R = 0 and x = ∞
∞, a re-
sult perfectly nugatory. It shews us that two equal and op-
posite parallel forces do not admit of a resultant. In fact the
addition of the forces S, S still gives, in this case, two equal
forces parallel and opposite in their directions.
Such a system of forces is called a Couple: the tendency
of a couple is to twist the body upon which it acts.
We shall return to this subject, and investigate the laws
of the composition and resolution of couples; since to these
we shall hereafter reduce the composition and resolution of
forces of every description acting upon a rigid body. Pre-
vious to this, however, we proceed to determine the resultant
of any number of parallel forces.
COMPOSITION OF PARALLEL FORCES.
21
PROP. To find the resultant of any number of parallel
forces acting upon a rigid body.
36. Let the points of application of the forces be re-
ferred to a system of rectangular co-ordinate axes (fig. 9.)
mm.......... the points of application: y11, X2 Y2≈2,
their co-ordinates. P₁ P₂ ...... the forces acting at these points,
those being reckoned positive which act in the direction of P,
and those negative which act in the opposite direction.
mm»: and take the point n, on m₁ m₂ such that
Join
P₁₂
ՊՈՆԴԻ
•
M1 M 29
P₁ + P₂
then n is the point of application of the resultant of P, and P2,
that is, of P₁ + P₂: see Arts. 32, 33.
2
Draw m₁a, nb, mc perpendicular to the axis of a, and
m₁de parallel to the axis of x, cutting nb, m₂c in d and e.
Then, by similar triangles,
ՈՆ, Պ
m₁d
ab Ab - X1
;
Mr M 2
m₂e
ac
-
X V1
P
•
(x − x\) ;
1
P₁ + P
.. Ab X₁ =
.. Ab, the abscissa to n₁,
P₁æ₁ + P₂x¿
P₁+ P2
1
Then, supposing P, and P to be replaced by (P₁ + P₂) acting
at n₁, the abscissa to the point of application of the resultant
of (P₁ + P:), P3
(P₁ + P.) · Ab + P3 · №3
(P₁ + P₂) + PÅ
P₁₁ + P₂. x 2 + P3. x3
2
P₁ + P₂+ P3
1
2
Let R be the resultant of all the forces, and x, y, the
co-ordinates to its point of application ;
X, Z
22
STATICS.
RIGID BODY.
Ρ
.. RΣ. P₁1,
18
Σ.Ρ101
Σ. Ρ
Σ.Ριψι
Similarly y
Σ. Ρ.
1 22
Σ. Ρ1*1
Σ. Ρ
These determine the magnitude and point of application of the
resultant.
37. These co-ordinates are independent of the angle
which the directions of the forces make with the co-ordinate
axes. Hence if these directions be turned about the points of
application of the forces, at the same time preserving their pa-
rallelism, the point of application of the resultant will not move.
For this reason that point is called the centre of the
parallel forces.
38. A heavy body consists of an aggregation of material
particles, each of which, in consequence of the Earth's at-
traction, tends towards the Earth's centre.
The weight, then, of a body may be considered as the
resultant of the weights of the different elementary portions
of the body acting in parallel and vertical lines.
In this case the centre of parallel forces is termed the
centre of gravity of the body. The obvious property of this
point is, that if it be fixed, the body will rest in any position;
no forces but the body's weight being supposed to act.
39. The expression P.x is denominated the moment of
the force P with respect to the plane yz. This must be care-
fully distinguished from the moment of a force with respect to
a point mentioned in Art. 31.
In consequence of the above definition, the equations for
determining the position of the centre of parallel forces shew
that the sum of the moments of any number of parallel forces
with respect to any plane equals the moment of their resultant.
40. We shall now investigate the laws of composition
of couples, since we shall hereafter reduce to these the com-
position of forces of every description acting on a rigid body.
We have already (Art. 35.) mentioned that by a couple wo
mean a system consisting of two equal parallel and opposite
PROPERTIES OF COUPLES.
23
forces acting on a body not on the same point. This system
does not admit of a single resultant force, as we have shewn :
but two or more couples acting upon a body may be replaced
by a single couple: this we proceed to demonstrate, after
proving some of the properties of couples.
41. DEFINITIONS. The arm of a couple is the distance
between the directions of its forces.
The moment of a couple is the product of the force at
either extremity and the arm: (see Art. 31).
The axis of a couple is a straight line perpendicular to
the plane of the couple and proportional in length to the mo-
ment.
PROP. The effect of a couple upon the equilibrium of a
body is not altered, if its arm be turned through any angle
about one extremity in the plane of the couple.
42. Let the plane of the paper be the plane of the couple
(fig. 10.) and AB the arm: AB' its new position: the forces
P₁ P₂ are equal, and act on the arm AB.
1
At A and B' let the two pair of equal and opposite forces.
P3 P, P₁ P, each P₁ or P₂ be applied, acting perpendi-
cular to AB': this will not affect the equilibrium.
=
1
2
Let BP2, B'P3 cut in C: join AC: AC manifestly bisects
the angle BAB'.
Now P2 and P3 are equivalent to some force in direction CA,
P₁ and P
same force
AC:
…. P、 P½ P3 P¸ are in equilibrium with each other;
1
2
therefore the remaining forces P5, P acting at B'A pro-
duce the same effect as P₁ and P, acting on AB. Hence the
proposition is true.
2
PROP. The effect of a couple on the equilibrium of a
body is not altered if we transfer the couple to any plane
parallel to its own, the arm remaining parallel to itself.
43. See fig. 11. AB the arm: A'B' the new position
parallel to AB. Join AB', A'B bisecting each other in G.
24
RIGID BODY.
STATICS.
+
At A' B' apply two equal and opposite forces each = P1
or P2: and let these forces be P3 P4 P5 P6: this will not alter
the effect of the couple.
But P₁ and P₁ are equivalent to 2 P₁ acting at G in direction Ga,
1
and P2 and P3
4
1
5
Gb.
Hence P₁ P2 P3 P4 are in equilibrium with each other,
and may be removed; therefore the remaining forces P, P6
acting at A' and B' produce the same effect as P₁ and P₂ acting
on AB.
Hence the proposition is true.
44. COR. Combining these two propositions, we see that a
couple may be any how transferred so long as its plane remains
parallel to itself.
PROP. The effect of a couple on a body at rest will not
be altered if we replace it by another whose moment is the
same: the plane remaining the same, and the arms being in
the same line, and having a common extremity.
45. Let AB be the arm, (fig. 12.): P, P the forces: and
suppose P = Q + R : let AB
Q+R: let AB = a: and make AC, a new arm,
b: at Capply two equal and opposite forces Q, Q₂ each = Q:
this will not alter the effect of the couple.
1
Now R at A and Q, at C will balance Q + R or P at B,
if AB BC: Q R (Art. 32.),
:
1
:
or if AB AC :: Q₁:
Q₁ + R
=
P,
P.α,
or if Q.b
we then have remaining the couple Q, Q₂ acting on the arm AC.
Hence the couple P, P acting on AB, may be replaced
by the couple Q Q acting on AC, if Q. b = P. a; that is, if
their moments are the same.
PROP. To find the resultant of any number of couples
acting upon a body, the planes of the couples being parallel
to each other.
46.
First suppose the couples all transferred to the
same plane (Art. 43.): next let them all be transferred so as
COMPOSITION OF COUPLES.
25
to have their arms in the same straight line, and one extremity
common (Art. 42.) and lastly let them all be replaced by
others having the same arm (Art. 45).
?
:
Thus if P, Q, R, S, ……………. be the forces, and
a, b, c, d, …………… be their arms,
we shall have replaced them by the following forces, (sup-
posing a the length of the common arm)
α
P.
α
b
Q. R. C
α
α
acting on the arm ɑ.
a.
Hence their resultant will be a couple whose force
b
Q. - + R.
α
= P.
+ Q.
a
α
C
+
a
and arm ɑ,
or whose moment
P.a + Q.b + R.c +
Hence the moment of the resultant couple is equal to the
sum of the moments of the original couples.
If one of the couples, as (S, S), act in a direction oppo-
site to the couple (P, P), then the force at each extremity of
the arm of the resultant couple will be
a
P.9
α
b
+ Q . − + R .
a
d
C
- S.
+
a
α
and the moment of the resultant couple will be
P. a + Q. b + R. c - S. d +
or the algebraical sum of the moments of the original couples ;
the moments of those couples which tend in the direction
opposite to the couple (P, P) being reckoned negative.
PROP. To find the resultant of two couples not acting
in the same plane.
47. Let the planes of the couples intersect in the line AB,
which is perpendicular to the plane of the paper (fig. 13.), and
let the couples be referred to the common arm AB, and let
their forces, thus altered, be P and Q.
D
26
RIGID BODY.
STATICS.
In the plane of the paper draw Aa, Ab perpendicular to
the planes of the couples (P, P) and (Q, Q): and equal in
length to their axes, (Art. 41).
Let R be the resultant of the forces P, Q at A; and P, Q
at B.
Since AP, AQ are parallel to BP, BQ respectively, there-
fore AR is parallel to BR.
Hence the two couples are equivalent to the single couple
(R, R) acting on the arm AB.
Draw Ac perpendicular to the plane of (R, R), and in
the same proportion to Aa, Ab that the moment of the couple
(R, R) has to those of (P, P), (Q, Q).
Then Ac is the axis of (R, R).
·
Now the three lines Aa, Ac, Ab make the same angles with
each other that AP, AR, AQ make with each other; also they
are in the same proportion in which
AB. P, AB.R, AB. Q are,
or in which P, R, Q are.
But R is the resultant of P and Q;
therefore Ac is the diagonal of the parallelogram on Aa, Ab
(see Art. 17).
Hence if two straight lines, having a common extremity,
represent the axes of two couples, that diagonal of the paral-
lelogram described on these lines, which passes through their
common extremity is equal in magnitude and direction to the
axis of the resultant couple.
PROP. To find the magnitude and position of the couple
which is the resultant of three couples which act in planes at
right angles to each other.
48. Let AB, AC, AD be the axes of the given couples,
(fig. 5). Complete the parallelogram CB: and draw AE the
diagonal. Then AE is the axis of the couple which is the
resultant of the two couples whose axes are AB, AC.
Complete the parallelogram DE, and draw AF the dia-
gonal. Then AF is the axis of the couple which is the re-
COMPOSITION OF COUPLES.
27
sultant of the couples whose axes are AE, AD, or of those
whose axes are AB, AC, AD.
Now AF
AE + AD
=
AB+ AC² + AD².
Let G be the moment of the resultant couple, L, M, N those
of the given couples;
.. G²
Ꮐ
L² + M² + N² ;
and if λ, µ, v be the angles the axis of the resultant makes with
those of the components
cos λ
AB L
AF G
M
N
COS μ
COS V
G
G
49. COR.
Hence conversely any couple may be replaced.
by three couples acting in planes at right angles to each other,
their moments being
G cos λ, G cos μ,
G cos v,
where G is the moment of the given couple, and λ, µ, v the
angles its axis makes with the axes of the three couples.
PROP. To find the resultant of any number of forces
acting on a rigid body in the same plane.
50. Let the system be referred to any pair of rectangular
co-ordinate axes Ax, Ay in the given plane: (fig. 14).
Let P, P1, P2,
α,
x Y, X1 Y1 X g Y Z
2
be the forces,
the angles which their directions
make with the axis of x.
the co-ordinates to their points of
application.
Let B be the point of application of P: join BA: the
points B and A are rigidly connected. At A apply two equal
and opposite forces, each equal and parallel to P. This will
not affect the equilibrium. Draw Ap perpendicular to PB
produced if necessary.
J
28
RIGID BODY.
STATICS.
Hence P acting at B is replaced by P acting at A, to-
gether with a couple
couple whose moment
(P, P) acting on the arm Ap, or a
P. Ap, and tending to turn the body
from the axis of x to the axis of y.
Now Apa sin ɑ y cos a.
Hence the moment of the couple (P, P)
= P. (∞ sin a
y cos a).
The moments of those couples are reckoned positive that tend
to turn the body from the axis of a to the axis of y: and
those negative that tend the other way.
The other forces may be similarly replaced.
Hence our system is reduced to the forces
P, P₁, P2
P₂ ...
acting at A
in directions parallel to those of the original forces; and the
couples whose moments are
P {x sin a y cos a
},
P₁ {x, sin a, - y₁ cos a₁},
P₂ {x, sin a₂ - y₂ cos a₂},
{x₂
2
S
απ Y2
acting in the plane of the paper.
Let R be the resultant of the forces acting at A, a the
angle which R makes with the axis of x; G the moment of the
resultant couple: then, by Art. 22,
R cos a = Σ. P cos a,
R sin a = Σ. P sin a,
and, by Art. 46, GE. P (x sin ay cos a),
and if P cos a Land P sin a = Y, these may be written
R² = (Σ. X)² + (Σ. Y)², tan a =
and G. {Y.x - X. y}.
Σ.Υ
Σ. Χ
و
;
COMPOSITION OF FORCES IN SAME PLANE.
29
51. Let the arm of the resultant couple be turned in the
plane of the forces and about its extremity A, till it is per-
pendicular to the direction of R. Art. 42: (fig. 15).
Let AR be the direction of
R: AB = α, the arm of the
resultant couple, and, consequently,
tremity let this = R'.
G
a
the force at each ex-
Hence the forces are all reduced to a force R + R' acting
at A in the direction AR, and R' acting at B in the direction
BR', parallel to AR. The resultant of these is R, acting at a
point C in the direction CR parallel to AR, the distance AC
R'
being AB (by Art. 32.)
R
G
R
Wherefore the resultant of all the forces P, P₁,
force R acting in the straight line whose equation is
y + AC cos a = tan a (x − AC sin a),
..... is a
which simplified becomes
AC
æ tan a
M
Y
cos a
or a sin a
y cos a
AC,
Σ.
G,
or x. Z. Yy. Z. X =
the direction in which R acts will be determined by the sign of
tan a.
52.
COR. If it should happen that the forces are such
that R 0, then we are left with the couple whose moment is
G, and there is not a single resultant force.
PROP. To find the conditions of equilibrium of any
number of forces acting on a rigid body in the same plane.
53. We have shewn in the last Prop. that the resultant
of any number of forces acting in the same plane on a rigid
body equals a force R acting about the origin of co-ordinates,
at a distance
G
where R² = (2. X)² + (2 . Y)²,
R
and G = 2. {la - Xy}.
30
STATICS. RIGID BODY.
Now when the forces are in equilibrium their resultant
must vanish, therefore R = 0; also G = 0, since the distance
G
must be indeterminate and not infinite; for if it were in-
R
finite, then the resultant of the forces would be a couple :
see Art. 35.
Hence the conditions of equilibrium are
Σ Χ = 0,
and E. {Y.
These may be written
Σ. P cos a = 0,
X
Y
Σ. Υ = 0,
— -
X. y} = 0.
Σ. P sin a = 0,
a y cos a) = 0.
and Σ. P(x sin ɑ
PROP. To find the two resultants of any number of
forces acting upon a rigid body in any directions.
54.
Let the forces be referred to three rectangular axes
Ax, Ay, Az; and suppose PP, P... are the forces, ayz,
X1Y1Z1, X2Y2Z2... the co-ordinates to their points of application,
and aẞy, aẞy, aẞy... the angles their directions make
with the axes: fig. 16.
19
Let m be the point of application of P, mP its direction
Ar=x, rn=y, nm = ≈: An in the plane xy, also ns parallel
to Ax.
Now P may be replaced by its three components
Pcosa, Pcos ß, Pcosy, (or X, Y, Z suppose)
parallel to the axes, (Art. 21.)
Z produces the same effect if it be transferred to n. Now
the equilibrium of the body will not be disturbed if we apply
at A and also at two opposite forces, each equal and parallel
to Z.
Then Z at m is equivalent to Z at A, and the two
couples of which the moments are Z.rn and Z. Ar and the
axes coincide respectively with the co-ordinate axes of x and y.
Hence Z at m is replaced by Z at A, and the two couples
Z.y and -Z.a acting in the planes perpendicular to w and y
COMPOSITION OF FORCES IN ANY DIRECTIONS.
31
respectively: the moments of those couples which tend to turn
the body from the axis of ≈ to that of y about the axis of ≈,
from y to≈ about x, and from ≈ to x about y, are reckoned
positive, and those in the opposite direction negative.
In the same manner we may substitute for Y and X.
Wherefore the force P acting at m may be replaced by
X, Y, Z acting at A along the axes, together with the
couples
Z.y and Y. in plane perpendicular to axis of v
X.≈ and
Z.x
Y.x and -X.y
Y
રર
or, by adding the moments of the couples acting in the same
or parallel planes (Art. 46.)
P is replaced by X, Y, Z acting at A and the couples
whose moments are
Z.y - Y.≈ in plane perpendicular to axis of x
X.z-Z.x
Y.x - X. y
Y
By a similar resolution of all the forces, we shall have
them replaced by the forces
Σ.Χ, Σ.Υ, Σ. Ζ,
acting at A along the axes: and the couples
Σ. {Z . y − Y. ≈} = L acting in the plane perpendicular to axis of a
A
Σ. {X.≈ - Z.x} = M
…..
Σ. {Y. x − X. y} = N ...
·
Y
Let R be the resultant of the forces acting at A; a, b, c
the angles its direction makes with the axes of co-ordinates:
then (Art. 22.)
R² = {Σ. X}² + {Σ. Y}² + {Σ. Z}²
3.
cos a
α=
Σ.Χ
R
cos b
Σ.Γ
R
Σ.Ζ
COS C =
R
32
STATICS. RIGID BODY.
Let G be the moment of the couple which is the resultant
of the three couples above mentioned; A, u, v the angles its
axis makes with the axes of co-ordinates; then (Art. 48.)
G² = L² + M² + N²,
L
M
N
cos λ
COS u
COS V
G'
G'
G
55. We may still further reduce the forces in the fol-
lowing manner.
Let the plane of the couple be turned round its axis till
the projection of the direction of R on this plane is perpen-
dicular to the arm. Let a be the length of the arm chosen
arbitrarily.
let
G
Then is the force at each extremity.
α
Also
be the angle between the directions of R and the axis
of G. Hence the whole force at A is

G2
2 RG
A
R² +
+
sin 0, (Art. 17.)
a²
a
where cose = cos a cos λ + cos b cos µ + cos c cos y : 8:
and the second force is
the arm.
G
,
acting at the other extremity of
a
These two forces cannot in general be reduced to a single
force, since their directions do not meet.
In one case, viz., when the directions of these two forces
meet, they can be reduced to a single force.
PROP. To prove that G is the principal Moment of the
Forces.
56. The Principal Moment means the Moment of greatest
Magnitude.
The quantities L, M, N are the sums of the moments of
the forces with respect to the axes of x, y, ≈ respectively
(Art. 31), and they are equivalent to √ L² + M + N² (= G),
PRINCIPAL MOMENT.
33
the moment of the forces with respect to an axis which makes
L M N
G'G'G'
angles with the axes whose cosines are
Now G, the resultant moment of the forces, must be in-
dependent of the directions of the axes of co-ordinates*: but
L, M, N depend upon these directions. But since L² + M² +
N² = G³, it shews that the greatest value of L is G, in which
case M = 0 and N = 0. Consequently G is the principal mo-
ment about the given centre.
57. The values of L, M, N in the general case (Art. 54.)
shew that the moment about an axis through the given centre,
and making an angle with the axis of principal moments,
equals G cos .
PROP. To find the locus of the centres which give the
least principal moments; the magnitude of these moments and
the position of their axes.
1
1
1
58. Let ₁₁, be the co-ordinates to a centre which gives
a minimum principal moment: let L, M, N, G₁ be the values of
LMNG at that point: then these are found by putting ≈ − ∞₁,
for xyz in LMNG;
Y - Y1, Z 21
... L₁ = L − y₁ Z. Z + ≈₁ Z. Y,
Σ
M₁ = M - ≈₁ Σ. X + x₂ Z. Z,
Σ.
31
N₁ = N − x₁ Σ. Y + y₁ £. X,
2
G₁² = L²² + M₁° + N₁²
* We might prove this in the following manner :
', ....
Let r, r', be the distances of the points of application of the forces P, P',
from the origin of co-ordinates; that is, from the centre of moments: Imn, l''m'n',.....
the angles that r, r',...... make with the axes: also let (PP'), (Pr),
the angles between straight lines drawn through the origin parallel to the directions of
P and P', of P and r, and so on. Then
L=2.Pr
M
represent
Σ. Pr (cos m cos y — cos n cos ß),
- cos n cos ẞ), M= Σ. Pr (cos n cos acos l cos y),
N = ß
Σ. Pr (cos / cos ẞ – cos m cos a);
•. G² = L² + M² + N² (after reduction)
Σ. P² ² sin² (Pr) +2Σ. PP'rr' { cos (P P') cos (rr') – cos (Pr') cos (P'r)},
and this is independent of the directions of the axes of co-ordinates, though it does
depend on the situation of the centre of moments.
E
34
STATICS. RIGID BODY.
= (L − y₁ Σ. Z + ≈₁Σ . Y)² + (M − ≈₁ Z. X + ∞₁ Σ. Z)²
+ (N − ∞₁ Σ. Y + y₁ Σ. X)².
When this is a minimum, its three partial differential coeffi-
cients with respect to x,y,z, must vanish: hence three equa-
tions which may easily be written in the form
x
R. 1 = (@.Σ. Χ + y Σ . Υ +
R. y = (Σ. Χ + y Σ . Υ +
1
1
1
Σ. Ζ) Σ. Χ + ΝΣ. Υ - ΜΣ. Ζ,
Σ. Ζ) Σ . Υ + LΣ . Ζ – ΝΣ. Χ,
R². %₁ = (x₁ Σ. X + y₁ Z. Y + ≈₁ Σ. Z) Σ. Z + MΣ.X - LΣ . Y.
%1
If we multiply these respectively by E. X, Z. Y, and
Σ. Z, we find an identical equation: which shews that these
three are equivalent to only two equations: and since they are
simple equations in my₁₁, we learn that the centres of mini-
mum principal moments lie in a straight line, any two of the
above equations being the equations to this line.
19
It is evident that G, increases indefinitely with x₁Y₁≈1,
therefore does not admit of a maximum value.
and
equa-
59. If we eliminate the second terms of the above
tions, they become of the ordinary form of equations to a line:
we have
(ΙΣ. Χ +- ΜΣ . Υ + ΝΣ . Ζ) Σ. Χ
Ly, Z. Z+ ≈₁Σ. Y=
R²
Μ - Σ. Χ + Σ. Ζ
=
1
(LZ. X + MZ. Y+NΣ. Z) Σ. Y
R2
N − ∞₁E. Y + y₁ Ɛ . X _ (L£. X + MƐ. Y + NE. Z) 2. X
Yi E.
R2
60. The minimum principal moment is
ΙΣ.Χ + ΜΣ. Υ + ΝΣ. Ζ
G₁ =
R
β, γι
61. Let a₁ ẞ₁₁ be the angles which the axis of G₁ makes
with lines parallel to the co-ordinate axes: then
M₁
N₁
L₁
COS a1
cos Bi
COS YI
G1
G₁
G₁₁
SINGLE RESULTANT.
35
and these become, by the above equations,
Σ.Χ
cos al
cos B₁
R
Σ.Υ
R
Σ.Ζ
cos Yi
R'
which shew that the axes of all the minima principal moments
are parallel to each other, and to the direction of the resultant.
PROP. Required to find the condition among the forces
that they may have a single resultant.
62. In order that this may be the case, it is clear that the
force R must be in the plane of the couple G. For then the
force resulting from the composition of R with one of the
forces of the couple will, when produced, meet the other force
of the couple, and, being compounded, will thus produce a
single resultant. Now this condition is satisfied when the angle
between R and the axis of G equals 90°: or when the cosine
of this angle equals zero: that is, when
cos a cos λ + cos b cos µ + cos c cos v = 0 ;
therefore the condition is that
(E. X)L + (2. Y) M + (≥ . Z) N
R.G
0,
or (E. X) L + (2. Y) M + (2. Z) N = 0,
unless R or G vanishes.
This is no condition when R = 0: that is, when Σ . X = 0,
Σ. Y=0, Σ. Z = 0, for the above equation is then identical.
In fact we then have only the couple G: which does not
admit of a single resultant.
Also this is no condition when G = 0, for then L = 0,
M = 0, N = 0, and the equation is again identical.
But in this case it is evident we have a single resultant R.
PROP. When the forces are reducible to a single resultant,
required the magnitude of this force and the equations to the
line in which it acts.
63. In this case the force R is in the plane of the couple
of which the moment is G.
36
STATICS. RIGID BODY.
Let the arm of the couple be turned about its extremity A
(see fig. 15), and in the plane of the couple, till it is perpen-
dicular to the force R: and let AB = a be the arm of the
couple: then the force of the couple (R')
G
α
; and the single
resultant equals R acting at C in the direction CR parallel to
AR, C being in BA, produced and determined by the equation
AC
R'
R
G
AB
R
64.
We must now find the equations to the line in which
this resultant acts.
1
Let x₁y₁₁ be the co-ordinates to some point in this line;
then, transferring the origin to this point, it is clear that the
body must have no tendency to revolve about the origin.
Therefore the new values of L M N when we put a₁ + X9
Y₁ + Y, ≈1 + ≈ for xyz must = 0;
.. 0 = Σ. P. { (y₁ + y), cos y − (≈1 + ≈) cos ẞ},
1
و
or 0 = L + y₁ Σ. Z – ≈₁ Σ. Y
Similarly,
(1).
ω.Σ.Ζ ..
(2),
(3).
0 = Μ + *Σ. Χ - Σ. Ζ
0 = N + x₁ Σ. Y − y₁ Σ. X. ...
X 1
These three equations are equivalent to only two: for if we
eliminate ≈ from (1) and (2), we have
0 = ΙΣ. Χ + ΜΣ. Υ - Σ. Ζ. Σ . Υ + y Σ.Ζ.Σ.Χ.
ΝΣ. Ζ
But LZ. X + ME. Y + NE. Z= 0, by Art. 62;
· ·. 0 = N + ∞₁ EΣ . Y − y₁ Σ . X;
1
and therefore equation (3) is a necessary consequence of (1)
and (2) wherefore any two of equations (1), (2), (3) are the
equations to the line in which the single resultant acts.
PROP. To find the conditions of equilibrium of any
number of forces acting upon a rigid body in any directions.
EQUILIBRIUM OF FORCES IN ANY DIRECTIONS.
37
65. We have shewn that the forces are in the general
case reducible to two acting in different planes. These forces,
then, must each vanish when there is equilibrium.
Hence (Art. 55.)
2
G
2 RG
R² +
+
sin 0 = 0,
a
α
and
G
a
.. R2
=
0, a being arbitrary;
R² = 0, and G²
=
0, or
(2. X)² + (2. Y)² + (Σ. Z)² = 0 and L2+ M² + N² = 0,
and these lead to the six conditions
Σ.Χ = 0, Σ.Υ = 0,
Σ.Ζ = 0,
Σ. (Zy - Y≈) = 0,
Σ. (X≈ - Zx) = 0, 2. (Yx - Xy) = 0.
ẞ=
These may be thus written:
Σ. P cos a = 0, E. Pcos ß= 0, Z. Pcos y = 0,
Σ. P (y cos y
≈ cos B) 0, Z. P (≈ cos a x cos y) = 0,
Σ. P(x cos ß - y cos a) = 0.
66. If we derive the conditions of equilibrium from the
case where the forces admit of a single resultant, we shall
arrive at the same conclusion.
R = 0, and also the distance
For we must have the force
G
at which it acts must be
R
arbitrary and not necessarily infinite: hence also G = 0, and
the conclusions are the same as before*.
* We have remarked in Art. 25, that the property of the divisibility of matter,
leads us to the supposition that every body consists of an assemblage of material par-
ticles, or molecules, which are held together by their mutual attraction. Now we are
totally unacquainted with the nature of these molecular forces: if, however, we assume
the two hypotheses, that the action of any two molecules on each other is the same, and
also that it acts in the line joining their centres, two suppositions which appear to be
perfectly legitimate, then we shall be able to deduce the conditions of equilibrium of a
rigid body from those of a single particle.
PROP. To find the conditions of equilibrium of a rigid body from those of a
single molecule.
Let
38
STATICS. RIGID BODY.
PROP. To find the conditions of equilibrium of forces
acting upon a rigid body when one point is fixed.
67. Let the fixed point be taken as the origin of co-
ordinates.
Let the body be referred to three rectangular co-ordinate axes: and let ryx be the
co-ordinates to one of its constituent particles: XYZ the sums of the resolved parts
parallel to the axes of the forces which act upon this particle, neglecting the molecular
forces: P, P' ... the molecular forces acting on this particle; aẞy, a'ß'y', ... the
angles their respective directions make with the three axes of co-ordinates.
Then we may suppose the rest of the body to be removed, and this particle held in
equilibrium by the above forces. Hence by Art. 23.
X + P cos a + P' cos a' +
Y+ P cos ẞ + P' cos ẞ' +
Z+P cos y + P' cos y' +
= 0
(a).
We shall have a similar system of equations for each particle in the body: if there
be n particles, we shall have 3n equations. These 3n equations will be connected one
with another, since any molecular force which enters into one system of equations must
enter into a second system; this is in consequence of the mutual action of the molecules.
There are two considerations which will enable us to deduce from these 3n equa-
tions, six equations of condition, independent of the molecular forces. These will be
the equations which the other forces must satisfy, in order that the equilibrium may be
established.
The first consideration is this, that the molecular actions are mutual; and that,
consequently, if P cos a represent the resolved part parallel to the axis of r of any one
of the molecular forces involved in the 3n equations, we shall likewise meet with the
term - P cos a in another of those equations which have reference to the axis of x.
Consequently, if we add all those equations together which have reference to the same
axis, we have the three following equations of condition independent of the molecular
forces
Σ.Χ = 0, Σ. Y=0,
Σ.Ζ=0.
The second consideration is this:-that the straight lines joining the different parti-
cles are the directions in which the molecular forces act.
Thus let P be the molecular action between the particles whose co-ordinates are
(xyz) and (x₁ Y1 Z1) :
P cos a,
-
- P cos a,
-
P cos ẞ,
P cos B,
P cos Y,
P cos Y,
the corresponding resolved parts of P for the two particles.
Then
COS α =
cos B
=
cos Y =
X1 Ꮖ
'(x₁ − x)² + (Y₁ − y)² + (≈1 − ∞)º¹
−
31-y
У
√√ (x¸ − x)² + (Y₁ − y)² + (≈1 − 2)²
31
√(x, − x)² + (y₁ − y)² + (≈1 − ≈)º
-
These
ONE POINT OF THE BODY FIXED.
39
Now the action of the forces on the body will produce
a pressure on the fixed point, and this will act in some definite
direction.
Let X'Y'Z' be the resolved parts of this pressure parallel
to the axes.
-
-
-
If then we consider the forces · X', - Y', Z' in con-
nexion with the given forces, we may suppose the body to be
free, and the equations of equilibrium give
Σ. X-X' = 0, Z. Y-Y' = 0,
Σ.
E.Z-Z' = 0,
L = 0, M
M = 0,
0, N = 0.
These enable us to obtain three more equations free from molecular forces: for if
we multiply the first and second of equations (a) by y and x respectively, and then
subtract them, we have
Yx - Xy+
+ P { x cos ẞ − y cos a } +
=
0,
and by the same process, we obtain from the system of equations which refer to the
particle (₁₁ ≈1),
X1
Y₁ x₁ - X₁Y₁ +
1
P { x cos ẞ - Y₁ cos α } +
0.
But the values of cos a and cos ß, given above, lead to the condition
(x, − x) cos ẞ — (y₁ − y) cos a = 0.
Wherefore the equation
Y.x
+ Y₁ • x 1
S
1
X.y +
= 0,
will not involve P, the molecular action between the particles whose co-ordinates are
xyz and x₁₁₁, respectively.
It follows readily from what we have shewn, that if we form all the equations
Y₂. X3-X3. Y3 +
Y3, X3 X3. Y3 +
and add them to those above, we shall have a final equation
Σ . (Y. x - X. y) = 0,
independent of the molecular forces.
In like manner we should obtain
0,
0,
Σ . (X . ≈ – Z . x) = 0,
2. (Z.y-Y. z) = 0.
Σ.
These six equations are the only conditions which can be obtained independent of
the molecular forces: they must he satisfied by the forces which hold in equilibrium
the assemblage of molecules, whatever be the laws of their molecular action.
Now in the case of a rigid body, the molecular forces are supposed to be themselves
in equilibrium independently of the extraneous forces; hence the above six equations
express the conditions of equilibrium of a rigid body.
40
RIGID BODY.
STATICS.
The first three equations give the resolved parts of the
pressure on the fixed point: and the last three are the only
conditions to be satisfied by the given forces.
PROP. To find the conditions of equilibrium of a body
which has two points in it fixed.
68. Let the axis of a pass through the two fixed points:
and let the distances of the points from the origin be ≈′ and x".
Also let X'Y'Z', X"Y"Z" be the resolved parts of the pres-
sures on these points.
Then, as in the last Prop. the equations of equilibrium
will be
Σ.X-X-X" =0, Z. Y-Y'-Y" =0, Z. Z-Z-Z"=0,
L-Y'.'-Y"." =0, M+ X'.≈′ + X" ≈″ = 0,
N = 0.
The first, second, fourth and fifth of these equations will
determine X'X"Y'Y": and the third equation gives Z' + Z",
shewing that the pressures on the fixed points in the direction
of the line joining them are indeterminate, being connected
by one equation only.
The last is the only condition of equilibrium, viz. N = 0.
PROP. To find the conditions of equilibrium of a rigid
body resting on a plane.
69. Let this plane be the plane of xy: and let x'y' be
the co-ordinates to one of the points of contact, R' the pressure
which the body exerts against the plane at that point. Then
the force - R', and similar forces for the other points of con-
tact, taken in connection with the given forces ought to satisfy
the equations of equilibrium.
Hence Σ. X=0, Z. Y= 0, Z. Z-E. R' = 0,
L+Z. R'y' = 0, M-Z. R'x'= 0, N=0.
If only one point be in contact with the plane, then the
third equation gives the pressure, and we have five equations
of condition,
1
THE BODY RESTING ON A FIXED PLANE.
41
Σ. Χ = 0, Σ.Υ = 0, L + y Σ. Ζ = 0,
M - x'Σ. Z = 0,
N = 0.
If two points be in contact, then
R'y' + R"y" - L, R'x' + R"x"= M,
·
My
give R'
Lx" + My
Ꮮ a' -- Ꮇ y
R"
>
y'x" — x'y″
y' x″ – x'y'
-
and the equations of condition are
L. (x"
Σ.Χ = 0, Σ.Υ = 0, Σ.Ζ +
x') + M (y' + y')
y' x" — x'y"
0,
and N = 0.
If three points be in contact, then the pressures are deter-
mined from the equations
R' + R" + R"" = Σ. Z
"//
R'y' + R"y" + R""' y" :- L
R' x' + R" x" + R""' x'" =
and the conditions of equilibrium are
Σ.X=0, Σ. Y=0,
M
N= 0.
If more than three points are in contact, then the pres-
sures are indeterminate, since they are connected by only three
equations: but the conditions of equilibrium are still
2. X=0, Z. Y= 0, N = 0.
F
CHAPTER III.
THE EQUILIBRIUM OF A SYSTEM OF RIGID BODIES.
70. In order to obtain the conditions of equilibrium of
two or more rigid bodies connected together in any way what-
ever, we must substitute unknown forces in the place of the
mutual actions at the points of connexion, and then write down
the equations of equilibrium of each body. These systems of
equations will be connected together, since a force depending on
the mutual actions of any two of the bodies must enter both the
systems of equations, which correspond to those bodies.
PROP. To find the conditions of equilibrium of a system
of bodies acted on by given forces.
1 1
71. Let any one of the bodies A be acted on by the
given forces X₁ Y₁ Z₁, ...... at the points (x, y₁₁),
also suppose that in consequence of the connexion of the system
that a mutual force P acts between the bodies A and B, making
angles aẞy with the axes: and let y≈ be its point of
øj 1
application in A, and way's the point of application in B.
2 2
Now if we suppose the force P to act on A and analogous
forces for all the other mutual actions arising from the con-
nexion of the system, the body A may be supposed to be in
equilibrium under the action of these forces and the forces
X, Y, Z₁ .... Hence, by Art. 65,
1
Σ. X₁+ P cos a +
0, Σ. Y₁+ P cos ẞ +
1
1
Σ. Z₁ + P cos y +
=
0,
Σ. (Z₁ y₁ − Y₁≈₁) + P(y,' cos y ≈' cos ẞ) + ...... = 0,
0,
EQUILIBRIUM OF A SYSTEM OF BODIES.
43
sa − x cos y) +
Σ. (X₁≈₁ - Z₁∞₁) + P(x cosa -
Σ. (Y₁∞₁ — X₁y₁) + P(x,' cos ß − y cos a) +
In the same manner for the body B,
0, Σ. Y₂ - P cos ẞ +
2
0,
0,
=
= 0.
Σ. X₂ - P cos a +
Σ.
Ζ. - P cos
P cos y +
Σ. (Z2Y2 - Y½≈½) − P (y½ cos y
· P (y½ cos y −
Σ. (X2≈ 2 – Z₂∞) - P (≈ cos a
Z2X2)
z½ cos ß) +
0,
x½ cos y) +
0,
2
Σ. (Y₂x₂ - X₂Y₂) - P (x2 cos ẞ - y½' cos a) + ...... = 0.
If we add the second set of equations to the first set, each to
each, we shall have six equations free from P: for P evidently
vanishes from the first three; and it enters the fourth in the
form P{(y-y₂) cos y - (≈1 - ) cos B: and this va-
{(yí' – ≈½') ß}:
nishes when the bodies are in contact, because then y'
2
=
up
and x = x': also it vanishes when the bodies are not in con-
1
tact, because then P must act in the line passing through the
points (xy), (x'y' ); and then, r being the distance.
between these points,
1 COS
Sy y =
21 – Z2, r cos ẞ = y;' - Y2' ;
.. (yı' — y₂') cos y - (
(≈ý
-
≈2) cos ẞ = 0.
and in the same way P disappears from the fifth and sixth
equations.
Hence the six final equations are free from P: and, by
adding together the equations referring to all the bodies, each
to each, we shall have finally
Y
Σ.Χ = 0, Σ. Υ = 0, Σ.Ζ = 0,
Σ.
Z. (Zy - Y2) = 0, E. (X≈ - Zx) = 0, 2. (Yx – Xy) = 0,
Σ.
free from all the mutual actions of the bodies of the system.
We might have been led to this conclusion by remembering
that the equilibrium of the system would not be disturbed by
supposing the bodies, when at rest, to become united rigidly at
the points of mutual action, and so considering the system as
one rigid body.
44
STATICS. SYSTEM OF BODIES.
PROP.
To prove that the Principle of Virtual Velocities
is true of any system of forces which keep any material system
in equilibrium.
72. We shall first enunciate this Principle.
Suppose a material system is held in equilibrium by the
action of a system of forces: suppose the points of application
of the forces are geometrically moved through very small spaces
in a manner consistent with the connexion of the parts of the
system one with another. Suppose perpendiculars drawn from
the new positions of the points upon the directions of the forces
acting at the points in their positions of equilibrium. The
distances of any perpendicular from the original point of appli-
cation of the corresponding force is called the virtual velocity
of the point with respect to that force, and is estimated positive
or negative, according as the perpendicular falls on the side of
the point towards which the force acts or the opposite side:
then the Principle is this,
The algebraical sum of the products of each force of the
system and the corresponding virtual velocity vanishes.
73. I. Suppose the system consists of only one rigid body.
We must cause the different particles to describe small
spaces consistent with their connection; this will, in the case of
a rigid body, be as well accomplished by supposing the co-
ordinate axes to receive a slight alteration of position.
Suppose the axes to revolve round through a small
angle : then
x, y, z become x + yo, y − x0, 29
neglecting small quantities of the second and higher orders.
Next, suppose these new axes to revolve through a small
angle about the new axis of y: by these means the original
values x, y, z become
(x + y0) − xP, y - æ Ꮎ,
-
≈ + (x + y0) $,
or x + y0 −z0, y − x0, z + x P.
Next, suppose the axes to revolve about the new axis of x,
through a small angle, and the co-ordinates become
-
x+yə-zp, y − x0 +≈ 4, ≈+x4 −y4,
omitting small quantities of the second and higher orders.
VIRTUAL VELOCITIES.
45
Lastly, let the origin be shifted to a point whose co-ordi-
nates are a, b, c: hence, if da, dy, dx be the total changes in
x, y, ≈ produced by these changes of axes,
dx = a + y✪ ≈0
– zp
(1),
Sy = b + x 4
− x✪
(2),
Sz
= c + x − y 4
Q
-
(3).
Now multiply the equations of equilibrium
Σ.Χ = 0, Σ.Υ = 0, Σ.Ζ = 0,
Σ. (Xy - Yx) = 0, E. (Zx - X≈) = 0, Σ. (Y≈ – Zy) = 0,
by a, b, c, 0, 0, ↓ respectively, and add;
·· Σ. {X (a + y0 −≈p)+Y (b+≈↓ − x0)+Z (c+xp−y¥)} = 0;
and, consequently,
£.{Xô? +Yông+Z8}=0.
74. Let R be the force, of which XYZ are the compo-
nents: a, b, c the angles which the direction of R makes with
the axes;
· X X= R cos ɑ,
Y = R cos b, Z = R cos c.
Also let ds be the small geometric displacement of the point
of application of R, of which dx, dy, dx are the resolved parts:
a', b', c' the angles ds makes with the axes: then
бос
ds.cos
Sx = ds. cos a', dy = ds. cos b', d≈ = ds. cos c';
.'.
Xdx + Ydy + Zd≈ = Rds (cos a cos a'
α
+ cos b cos b' + cos c cos c')
=
R&r
where dr is the resolved part of ds in the direction of R's
action; that is, the virtual velocity of the point (v y ≈) with
respect to R.
Hence . R&r = 0,
and the Principle of Virtual Velocities is true of a system of
forces holding a rigid body in equilibrium.
46
SYSTEM OF BODIES.
STATICS.
75. II. Suppose the system consists of any number of
rigid bodies.
Let P be the mutual action of any two of the rigid bodies,
whether by contact or by any means of connexion whatever :
let aẞy be the angles which its direction makes with the
axes, and let a yz, x'y' z' be the co-ordinates to the points.
where the force P acts.
α
Now each of these bodies is in equilibrium under the action
of its own forces, together with the force P, and the mutual
actions it has with the other bodies of the system. Hence, by
the first case,
Σ. Rdr + P (dx cos a + dy cos ẞ + dx cos y) +
also for the other body on which P acts,
= 0 ... (1) ;
Σ. R'dr' - P (dx' cos a + dy' cos ẞ + dx' cos y) + ... = 0 ... (2) ;
and P will not occur in any of the equations that have reference
to the other bodies.
Adding equations (1) and (2), will give
Σ. Rdr + E. R'dr' + P {(dx - dx') cos a
+ (dy - dy') cos ẞ+ (dz - dx') cosy} + ...... = 0.
Now in consequence of the geometric displacement of the
system, suppose the points (xyz) and (x'y'') describe the
small spaces St and St', making respectively with the axes the
angles m, m', m" and n, n', n": hence
Sx = St cosm,
Sy = St cosm',
dx = St cos m",
=
Sa' St' cos n,
Sy' = dt' cos n',
St' cos n',
dx' = dt' cos n".
Hence (Sada) cosa + (dy-dy') cos B + (dz - dx') cos y
St (cos m cos a + cos m' cos ß + cos m″ cos y)
St' (cos n cos a + cos n' cos ß + cos n" cos y)
= resolved part of St - resolved part of St in direction of P
= sum of virtual veloc. of the pnts. (xyz), (x'y's') with resp. to P
Sp+dp' suppose.
VIRTUAL VELOCITIES.
47
Wherefore if we form the equations analogous to equations
(1) and (2) for all the bodies, and add them together, we shall
have, supposing Σ now to extend through the whole system,
Σ. Rdr + Σ. P (dp + dp′) = 0.
and the Principle of Virtual Velocities is still true.
76. COR. 1. If the force P be the mutual normal pres-
sure of two surfaces in contact, then by giving the system such
a geometric motion that these surfaces shall remain in contact,
we shall cause P to disappear from this equation, because then
Sp + Sp′ = 0.
77. COR. 2. If the points (xy≈) and (x'y'x') are con-
nected invariably, as for instance by an inextensible string,
then P disappears from the equation of Virtual Velocities, since
Sp + Sp' = 0.
PROP. When a system of rigid bodies is in equilibrium
under the action of no forces but their weights, mutual forces,
and pressures upon smooth immoveable surfaces, then the
centre of gravity is in the lowest or highest position it can
possibly attain by moving the system consistently with the
connexion of its parts one with another.
78. For let the axis of ≈ be taken vertical: and let P₁,
P,...be the vertical forces with which the different particles.
tend downwards by reason of the attraction of the Earth:
₁₂...the vertical ordinates to their points of application, ≈
the vertical ordinate to the centre of gravity (see Art. 36.)
P₁~~₁ + P½Ñ³½ + P3~3 +
P₁ + P₂ + P3 +
Now suppose the system to receive a slight displacement
of its parts consistent with their connexion, and let d≈≈≈3...
be the vertical displacement of the points of application of
P₁P₂ P¸...(these are the virtual velocities of the points); and
let become+d;
2
ི
··· 3+ d =
P₁ ~₁ + P₂ ≈2 + P3~3 +
P₁ + P₂ + P3 +
2
48
SYSTEM OF BODIES.
STATICS.
+
.. dz=
P₁ d≈₁ + P₂d≈₂ + P38%3 +
2
P₁ + P₂ + P3 +
1
2
P₁d≈₁ + P₂d ≈₂ + P38≈3 +
P₁ + P₂+ P3 +
1
2
But by the principle of Virtual Velocities, the numerator
of this fraction vanishes when there is equilibrium;
... d z = 0,
б
and is a maximum or minimum: and the centre of gravity
is in its highest or lowest position.
PROP. To prove that when the centre of gravity has
its lowest position the equilibrium is stable, and when it
has its highest position the equilibrium is unstable.
79. When a system of bodies is in equilibrium and an
indefinitely small motion is given to the parts of the system
so as to disturb the state of rest, the equilibrium is said to be
stable or unstable according as the parts of the system do or
do not return to their original positions of rest.
Now suppose the pressures (mentioned in the last Prop.)
and the weight of the parts of the system are not in equili-
brium. We shall prove that the centre of gravity cannot
ascend, but must descend.
The resultant of the weights of the different parts of the
system passes through the centre of gravity of the system.
Let W be the weight of the whole system: and suppose the
centre of gravity would move in the direction GG₁ (fig. 17.)
making an angle with the vertical drawn downwards from
G, if not prevented by a force P acting in the opposite
direction and combining with the pressures to preserve equi-
librium: GG, a: then by Virtual Velocities,
=
W. a cos 0 - P. a = 0;
P
... cose
W
LAGRANGE'S PROOF OF VIRTUAL VELOCITIES.
49
therefore cos cannot be negative, or cannot lie between
101
2
and
3 п
2
; that is, G cannot move upwards but must move
downwards when the system is not in equilibrium.
Now if the system be in equilibrium with its centre
of gravity as high as possible, any slight disturbance must
bring it lower; and since, by what we have just proved, it
can never rise again, it follows that the equilibrium will be
unstable. But if the equilibrium be such that the centre of
gravity is in its lowest position, any disturbance must raise it
higher; and since when left to itself it must fall, it follows
that the centre of gravity will return to its former position, or
the equilibrium is stable.
80. We have in the foregoing part of this work deduced
the conditions of equilibrium of a material system from the
simplest principles, commencing with the equilibrium of a
single material particle: and we have from these conditions
proved the Principle of Virtual Velocities. But we might have
pursued an inverse course and commenced with proving the
Principle of Virtual Velocities, and thence deducing the con-
ditions of equilibrium of a material system.
PROP. To prove the Principle of Virtual Velocities
independently of the Parallelogram of Forces.
81. The following is Lagrange's Proof of this Principle.
Let us suppose that the forces are P1, P2, P3... and that they
are commensurable and in the proportion of the numbers
N1, N2, N3,.... let A1, A2, A3,...(fig. 18.) be their points of
application: A¸α₁, А‚α2, Аžαз,... their directions.
1
Now imagine a and b, to be two blocks consisting each of
n, wheels of equal size, the wheels in the same block turning
freely about the same axis: and let the centres of these blocks
be in the straight line A₁a, produced. Let a₁ be connected
with 4 by an inextensible string: and suppose b, is firmly
fixed to an immoveable beam B₁; and a,, b, connected by an
inextensible string passing round their wheels alternately, one
end of the string being attached to a fixed point M any where
in the plane of the first wheel of b, over which it passes; and
G
50
STATICS. SYSTEM OF BODIES,
1
the other end being carried (as represented in the figure) to
another system of blocks corresponding to the force P₂, each
block having no wheels; and so on: and lastly, let the string
be passed over a simple wheel at C and be stretched by a
weight W hanging by it.
The string is imagined to be perfectly flexible, and the
wheels perfectly smooth: consequently the string will be
stretched uniformly throughout, with a tension equal to the
weight W. It is very evident, then, that since the wheels of
ɑ₁ and b₁ are all equal, the portions of string connecting them
are parallel, and (they being 2n, in number) the tension of
A₁a₁ equals the weight 2n, W; in the same manner the tension
of A2a2 is 2n2 W; and so on.
1
Consequently by this imaginary contrivance the weight W
produces forces at the points A₁₂ ……………………. in the directions
A₁α1, А₂α2 ... and in the proportion of n₁ n₂
in the proportion of P₁P½
1
; that is,
But P₁ P½ ……………. are in equilibrium and since the unit of
force may be any force, a system of forces in the same propor-
tion as P₁P₂ ... acting at the same points and in the same
directions as P₁ P₂ .... will be in equilibrium.
1
1
Hence if we remove the forces P₁ P₂ ...... and replace
them in the manner described above, W will be at rest and
this will be the case of whatever magnitude W be, since by
increasing or diminishing W, the forces P, P
so as to retain their proportion unchanged.
are altered
Wherefore, however much we alter W, we cannot thereby
cause the moveable block (a) of any of the systems (as a b₁)
to move.
1
This shews that the relation of the magnitudes of the
forces P₁ P..., their directions, and points of application is
such, that if we forcibly make the block a₁, or any other block,
to approach or recede from the other block b₁ of the system by
an indefinitely small space, then the other moveable blocks will
so shift, that on the whole the length of string given off from
the blocks which approach will exactly equal the length of
string taken in by the blocks which separate. If this were not
the case, this indefinitely small displacement of the system
would give W an indefinitely small motion, and this would
LAGRANGE'S PROOF OF VIRTUAL VELOCITIES.
51
shew conversely, that it is possible to move W, which (as we
have proved) cannot be done, however much we alter W in
magnitude.
Sp₁
Hence, if Sp, Sp₂ be the spaces through which
dag...... move in consequence of the indefinitely small dis-
placement, those being reckoned positive when the blocks ap-
proach, or string is given off, and the others negative. Then
n₁dp₁, n₂dp₂, will be the lengths of string given off or taken on
the wheels, according as they are positive or negative ;
... пор, + порг +
or P₁dp₁ + P₂dp₂ +
which is the Principle of Virtual Velocities.
0,
=
0,
82. The displacements Sp₁, Sp2 ...... must be taken
indefinitely small, otherwise the equilibrium will be sensibly
disturbed, and W will not remain at rest. In fact the
best way of representing the principle is this; that when any
part of the system is moved through a space less than any
assignable quantity, then W will move through a small space
which varies as the square or some higher power of the dis-
turbance, so that it vanishes in the limit.
PROP. To obtain the equations of equilibrium of a rigid
body from the Principle of Virtual Velocities.
83. By this principle we have 2. Pdp = 0. Let XYZ
be the resolved parts of P: and dx, dy, dx the virtual velocities
of the point (xyz) with respect to P;
…. 2.(Xô + Yông + Z8s) = 0.
Now, by Art. 73, we must put
ба
Ꮎ, dx = c + x − y&,
Sx = a + yo≈p, dy = b+≈4 − x0,
y0
in which a, b, c, 0, 0, & are arbitrary small quantities: hence
a. X+b. Y+cE. Z
+ YΣ. (Y≈ − Zy) + pΣ. (Zx − X≈) + 0Σ. (Xy – Yx) = 0,
and because a, b, c, 0, p, & are arbitrary,
Σ.Χ = 0, Σ.Υ = 0, Σ. Ζ = 0,
Σ. (Ys - Zy) = 0, E. (Zx - X≈) = 0,
=
E. (Xy - Yx) = 0,
which are the six equations of equilibrium deduced in Art. 65.
CHAPTER IV.
CENTRE OF GRAVITY.
84. Ir was shewn in Art. 37. that there is a point in
every body such that, if the particles of the body be acted on
by parallel forces and this point be fixed, the body will rest
in whatever position it be placed.
85. Now the weight of a body may be considered as the
resultant of the weights of the different elementary portions of
the body acting in parallel and vertical lines. In this case the
point above described, the centre of parallel forces, is called
the centre of gravity of the body. We intend to devote the
present Chapter to the determination of this point in bodies of
various forms.
86. We shall first give a few geometrical calculations of
the position of the centre of gravity.
Ex. 1. To find the centre of gravity of a triangular
figure of uniform thickness and density.
Let ABC be one surface of the triangular figure: fig. 19.
Bisect AC in D; join BD: draw adc parallel to ADC cutting
BD in d. Then by similar triangles
ad: AD :: Bd: BD
and de: DC :: Bd: BD
.. ad : AD :: de: DC
but AD = DC; ... ad dc.
Hence BD bisects every line parallel to the side AC: and
therefore each of these lines will balance on BD, and conse-
quently the whole triangle will balance on BD: and therefore
the centre of gravity must be in the line BD.
CENTRE OF GRAVITY.
53
PYRAMID.
Bisect AB in E and join CE; let this cut BD in F.
Then, as before, the centre of gravity must be in CE: but
it must be in BD: and therefore F is the centre of gravity.
Join DE.
Then AD DC and AE EB;
=
.. DE is parallel to BC and BC = 2. DE,
and by similar triangles
DF
BF
... DF = BF;
DE
BC
... DF = DB.
Hence to find the centre of gravity of a triangle, bisect
any side, join the point of bisection with the opposite angle,
and the centre of gravity lies a third of the way up this line.
Ex. 2. To find the centre of gravity of a pyramid on
a triangular base.
Let ABC be the base; V the vertex: fig 20, bisect AC
in D; join BD, DV: take DF = 1. DB, then F is the centre
of gravity of ABC. Join FV: and draw abc parallel to ABC
cutting VF in f: join bf; and produce it to meet DV in d.
Then by similar triangles, we easily see that ad dc: also
bf Vf df
BF VF DF
: but DF = BF;
... df = }} bf ;
=
therefore ƒ is the centre of gravity of the triangle abc: and
if we suppose the pyramid to be made up of an infinitely great
number of infinitely thin triangular figures parallel to the base,
each of these has its centre of gravity in VF. Hence the
centre of gravity of the pyramid is in VF.
3
Again, take DH = DV: join HB cutting VF in G.
Then as before, the centre of gravity of the pyramid must.
be in BH: but it is in VF: hence G, the point of intersection
of these lines, is the centre of gravity.
54
STATICS.
Join FH: then FH is parallel to VB:
also ··· DF = 1 DB
;
·. FH=1VB:
FG VG
and
but FH = 1 VB ;
FH VB
.. FG = | GV = 1 FV.
Hence the centre of gravity is found to be one-fourth of
the way up the line joining the centre of gravity of the base
with the vertex.
Ex. 3. To find the centre of gravity of any pyramid
having a plane base.
Divide the base into triangles: if any part of the base is
curvilinear, then suppose the curve to be divided into an
indefinitely great number of indefinitely short straight lines.
Join the vertex of the pyramid with the centres of gravity of
all the triangles, and also with all their angles. Draw a plane
parallel to the base at a distance from the base equal to one-
fourth of the distance of the vertex from the base: then this
plane cuts every line drawn from the vertex to the base in
parts, having the same ratio 3: 1; and therefore the tri-
angular pyramids have their centres of gravity in this plane,
and therefore the whole pyramid has its centre of gravity in
this plane.
Again, join the vertex with the centre of gravity of the
base: then every section of the pyramid parallel to the base
will be similar to the base, and will have its centre of gravity
in this line. Hence the whole pyramid has its centre of gra-
vity in this line.
Wherefore the centre of gravity is one-fourth of the way
up the line joining the centre of gravity of the base with the
vertex.
Ex. 4. To find the centre of gravity of the frustrum
of a pyramid, formed by parallel planes.
Let ABC abc be the frustrum, fig. 20: G, g the centres
of gravity of the pyramids VABC, Vabc: it is clear that the
CENTRE OF GRAVITY. FRUSTRUM OF A PYRAMID. 55
centre of gravity of the frustrum must be in gG produced;
at G' suppose.
Let GF=x; Ff = c; AB = a, ab=b.
Ff=c;
Now the smaller pyramid and the frustrum supposed to
act at their centres of gravity are in equilibrium about G:
hence by Art. 32.
GG'
Gg
weight of smaller pyd.
weight of frustrum
vol. of small pyd.
vol. of large-vol. of small pyd.
Gg = VG - Vg = 2 (VF − Vƒ)
b3
a³ - b³ ³
b3
3c
4
3 c b3
.. GG'
4 a³ - b³
Also GF = 1VF = } (VF − Vƒ)
-
a
a
-
b
с
α
4 a - b
by similar figures,
C
a
3b3
.. FG′ = FG – G'G
4
a - b
a³ b3
Cons
c a² + 2ab+ 3b²
11
4 a² + ab + b²
This is true of a frustrum of a pyramid on any base, a and
b being homologous sides in the two ends.
We proceed now to the analytical calculations.
PROP. To obtain formula for the calculation of the
co-ordinates of the centre of gravity of a body.
87. Let ays be the rectangular co-ordinates to an elemen-
tary parallelopiped of the body, the mass of the element being
dm: then if g be the constant ratio of the mass of a body to
its weight, gdm is the weight of this element: or the force with
56
STATICS.
which it presses downwards in a vertical line: gdm.x is the
moment of this force with respect to the plane of y≈ (see
Art. 39.), and ſ.gxdm is the sum of the moments of the forces
which the parts of the body exert downwards in vertical lines:
also fgdm is the sum of the forces. Hence if ≈ be that co-
ordinate of the centre of gravity of the body which is parallel
to the axis of x, (Art. 36.)
f.gxdm
f.xdm
XC
f.gdm
f.dm
Similarly, y =
f.ydm
21
f.zdm
f.dm
f.dm
the limits of integration being determined by the form of the
body.
When the body is not bounded by continuous surfaces,
these formulæ cannot be used, except in some particular cases,
as we shall see when we come to apply them to examples.
When these formulæ will not apply we must divide the body
into distinct portions, of which the respective centres of gra-
vity can be calculated by the above formulæ; and must finally
find the centre of gravity of the whole body by considering
these constituent portions as condensed, each into its centre of
gravity, and so forming an assemblage of particles to which
the formulæ of Art. 36. can be applied.
Ex. 1. A straight rod of uniform thickness and den-
sity: (fig. 21.)
=
X,
AB the rod: P, Q two transverse sections, AP
PQ = dx, M the mass of the whole rod and its length:
then the mass of PQ M
=
dx
Z
*. X
f. xdx
.dx
M
since
divides out
12
=
½ 1 = AG.
CENTRE OF GRAVITY OF A LINE.
57
Ex. 2. A curved line of uniform density and thickness,
the curvature lying in one plane: (fig. 22.)
Let APs, PQ = ds;
ds
the mass of PQ = M
and ds
√1+ dy da
dx,
dx²
x and y being co-ordinates to P;
X
Jads
fds
Jyds
Y
fds
}
between the proper limits. The two following examples are
applications of these formulæ.
Ex. 3. A quadrant of a circle: (fig. 23.)
Here y² = a² – x², the centre B being origin: BA axis of x.
dy
dx
ac
ds
a
a²
x²
dx
a² - x²
a;
the limiting values of x are 0 and AB
. X
a
a
a x dx
Ja
a²
adx
So Ja
a
for
ay dx
√ a²
a d x
202
شرج
Y
Jo
Va²
x2
။
a²
2 a
BH,
Επα
fcdx
dx
So Ja² = x²
2 a
|=
Ex. 4. The arc of a semi-cycloid: (fig. 23.)
HG.
The axis AB being the axis of a and the vertex A the origin,
X
- 1
y = a vers.
+ √2ax x².
α
H
58
STATICS.
dy
2α- x
2 a
X
ds
2 a
dx √
2αx x02
20
dx
20
the limits of x are 0 and AB or 2a,
f2ax N
2α
dx
2 a
20
४
2a
2 a
dx
X
со
=
AH,
2a
dy
Say N
dx 2 x¹y - 2√xt
dx
dx
20
dx
2 a
α
Sza
a
dx
between the proper limits
√x
(2a)$π- 2
2 f2a
2a - xdx
4 a
= πα
HG.
2√2a
3
Ex. 5. A curve line of double curvature.
dy2 dz2
In this case.ds √
1 +
dx²
+ da, xyz being co-
dx²
ordinates to the variable extremity of s: this value of ds put
will give the required co-ordinates.
in
x y z
Ex. 6. Any portion of a helix, or the curve of the thread
of a screw: (fig. 24.)
2
The equations are y = √ a²
a² - x²,
x², Z
Z = na cos
a cos-
a
dy
a
dz
·na
dx
da
√ a²
dy²
dx2
a² (4 + n²)
.. 1 +
+
dx²
dx²
a² - x²
and the limits of r are a and x;
CENTRE OF GRAVITY OF AN AREA.
59
18
11
x d x
2
x² Va²
Så Ja² - a²
a
dx
2
Så Ja
fx dx
dx
Va²
202
x²
X
cos - 1
a
α
X
20
f
x
cos-1
x²
α
X dx
fa na cos
Sa
1
α
Va
a²
x²
na
X
cos-1
dx
Sa
√ a²
x²
1
y
| 22
Ex. 7.
A body of uniform thickness and density bounded
by a plane curve and its ordinate.
Let the plane parallel to the plane faces of the body and
bisecting it be the plane of xy: the centre of gravity is evi-
dently in this plane: M the mass of the body, and A its area:
then the mass of an elementary portion of the area at the point
M
P, of which the co-ordinates are wy, is Mdady; and since
A
A
divides both numerator and denominator, the co-ordinates of the
centre of gravity become
x =
ffxdx dy
fj d x d y
ffy d x d y
Y
ffdxdy
between proper limits.
We shall sometimes find it convenient to use polar co-
ordinates: (fig. 25.)
Let AP = 1, x AP = 0:
0: dr, rde the sides of the ele-
mentary portion of the area at P; then M
rdrdo
A
is the mass
of the element at P: and x = r cos 0, y = r sin 0;
x =
[² cos Odrde
Jrdrde
ffre sin Odrde
frdrde
60
STATICS.
between the proper limits: the following examples are appli-
cations of these; we shall sometimes use rectangular and some-
times polar co-ordinates.
Ex. 8. Let the curve be the semi-parabola AC, (fig. 23.)
AM = x, {MP=y: QM² = 4mx, the equation to AC.
Now x and y are independent variables, we may consequently
integrate our expressions, first considering y variable and a
constant, and then with regard to a. This admits of an easy
explanation. Integrating our expressions on the supposition
that is constant and y variable is the same as calculating the
expressions only for the elementary masses which lie in a strip
of the area, like QM in the figure, in which PM or y is different
for each element, but AM or x is the same: the limiting values
of y in this integration will be y = 0 and y = MQ = 2√mx,
and the result will therefore be a function of a only: then in-
tegrating this result with respect to a is the same as adding
together the expressions for all the strips like QM, of which
the area consists: the limits of x are 0 and AB or a;
a
.. ffadady between proper limits = fª x (y + X) dx
(X a function of x) fª 2 √ m x ³ d x
4
√mas,
5
[[dædy between limits = ſoª (y + X') dx = √/ma³ ;
3
.. X
3 a
5
AH,
in like manner y
3 √ ma
3 BC
or
=
GH.
4.
8
+
If we had taken the double area CAC', the limits of Y would
have been y MQ'
we should have found
2 √mx, and Y
3 a
XC
5
=
MQ = 2√mx, and
= AH, ÿ= 0, and therefore H
is the centre of gravity of the whole.
Y
CENTRE OF GRAVITY OF AN AREA
61
Ex. 9.
Let CAC' be a circular area: (fig. 23.)
y² = 2ax-x²: and if we take a portion ACB, the limits of
y are 0, and
2ax - x², and those of x are 0 and AB or a
La fy x d x d y
a
6ª x√2ax
x² dx
X
X 13
fx [ d x d y
£* √2ax
x² dx
2ax² 003
Soa
11
Soa
√2ax²-x3
2αx x²
√2ax
-
dx
4 a
AH*.
3 П
dx
x²
La
6ª by dx dy
α
Also
Ja joy d x d y
fª y³ da
joa y dx
dx
1 foª (2 ax − x²) dx
a
foª √2ax − x² dx
2
1 ( a³ − 1 a³)
4 a
GH.
219
3 п
a²
Ex. 10.
Let CAC' be an ellipse.
Then if we take the quadrant ACB, AB = a, BC = b,
Ex. 11.
4a
4b
X
, Y
3 п
3 п
Let CAC' be a cycloid: AB = 2a,
7 a
T
8
it =
AH
Y
=
HG=
= α
6
Ωπ
Ex. 12. A triangle: (fig. 26.)
Draw AD perpendicular to BC; A the origin, AD the
axis of x: DAB = a, DAC = ẞ(− A − a), AD
· ẞ (= A − a), AD = e; x tan a
= y', a tan ẞ = y", the limits of
The general form is
Y ;
Σ
xndx
2n-1
ľ
x²-1dx
2-1
2ax-x²
。 V2ax — x²
12
。 √2ax − x²
N.
62
STATICS.
18
fox² (tan a + tan ẞ) dx
Sex (tan a + tan ẞ) dæ
e
1 Soc a² (tan² a - tan² B) dx
e
fox (tan a + tan ß) dæ
.. AH = AD,
and
&
2e
AH.
3
e
(tan a
tan ẞ) = HG.
3
GH = }} (BD – CD),
DE = GH;
.. BD – CD = 2 DE;
.. BD – DE = CD + DE;
and AG =
and BE = CE,
AE, as in Ex. 1. Art. 86.
As an instance of the application of polar co-ordinates, we
will take the following.
Ex. 13. A semi-ellipse CBC': (fig. 25.)
Let H be its centre of gravity: AP=r, BAP = 0.
In this case we must integrate first with respect to › and
then with respect to 0; but not first with respect to 0 and then
with respect to r.
For if we first integrate with respect to r,
we take the sum of the elements in AQ, and the whole area can
be divided into strips like AQ: but if we begin by integrating
with respect to 0, we take the elements in an annular strip
through P: and the area cannot be divided into strips described
after the same law, hence we should be unable to integrate
again with respect to r.
The limits of r are 0 and AQ or
of 0 are
... xc
π
| 02
!!
and
π
2
ffr² cos 0d0dr
ffrdedr
2b
03
ja
α
sa
b
the limits
1- e cos² 0
between these limits
cos Ꮎ dᎾ
(1 − e² cos² )
do
d Ꮎ
a 1 - e² cos² 0
r
2014
CENTRE OF GRAVITY OF SURFACE OF REVOLUTION. 63
||
2b
3
ja
α
fa
1
α
d. tan e
(1 − e² + tan² ()}
d. tan
1-e² + tan² 0
tan a
2b 1 − e² √1 − e² + tan² a
α
П
။
214
26
l
4 a
3 П
GO
3
1
tan a
3
π
tan-
1
1 – eª
1 - e²
Ex. 14.
1
1 - e²
The sector of a circle: (fig. 25.)
Let BP'A be the sector: ▲ BAP' = a
It matters not in this example whether we integrate with
respect to r first or first; since the area may be made
up of
either strips like AQ, or of annular strips like that passing
through P.
४।
Y
...
α
La far cos ededr
α
α α
La far de d r
Lafar sin ededr
a
Lordedr
· ▲ GAB = tan-¹
α
3 fa de
α
2 a foª cos de
2a (sin a)
3 a
2a (1
cos a)
3 a
α
2a fa sin ede
α
3 fa do
1
-1
Y
cos a
= tan-
1
x
sin a
α
8102
4a sin
2 a
812
a
AG = √ x² + ÿ
2
·2 cos a =
3 a
3 a
Ex. 15. A surface of revolution: (fig. 27.)
Let AM = x, MP = y; MM' = da: through M and M'
draw two planes at right angles to the axis of the figure; that
is, the axis of x. Now every portion of the surface between
these planes is equally distant from the axis, and therefore the
centre of gravity of the surface PQQ'P' is at M ultimately: let
M be the mass of the whole surface (the thickness and density
being uniform) and S the whole surface;
ལ་བྱོལ་
STATICS.
.. mass of the surface PQQ'P'
*
M
2πуds
S
ds
α
La xy
d x
dx
=
AG, AB ɑ.
= a.
X
ds
α
Lay
dx
dx
It is evident that
y
= 0.
Ex. 16. The surface of a portion of a sphere.
y² = 2 ax
a
2
ds
a
dx √2ax
ນໍ າ
―
x²
x
४ ।
X
fr a x d x
La adx
a x²
2
X
AG.
2ax 2
Ex. 17.
The surface of a cone.
ds
Y
= ax,
√1 + a²
dx
.. x =
£* x² d x
Loa x d x
200
= AG.
3
Ex. 18.
Let the body be any surface of uniform thick-
ness and density: (fig. 28.)
Let xyz be co-ordinates to any point of the surface: the
area of a small portion of the surface at that point is
1 +
d&2 dx2
+
dx dy;
dx² dy²
therefore mass of the corresponding element
11
M
S
1 +
dz2
dx2
+ dady;
dx² dy²
CENTRE OF GRAVITY OF ANY SURFACE.
65
18
11
dx² dx2
SS x
1 +
+
dx dy
dx²
dy²
ss
dz2 dx2
SS √ 1 + + dx dy
dxx2 dy
between proper limits, and similar expressions for y, z.
Ex. 19.
The surface of an eighth part of a sphere.
The origin being at the centre, x² + y² + ≈² = a² (fig. 28.)
1 +
de dx2
+
dx2 dy²
ɑ
a
Va²
x² – y²
We shall consider y to vary, a remaining constant: that
is, we shall take all the elements in the strip QQ': hence the
limits of y are 0 and Q'M or √a² – x², which is obtained
from the equation to the surface by putting ≈ = 0: then the
limits of x are 0 and AB or a;
x =
La fu
ary'
x dx dy
√ a² - x² - y²
dx dy
√
a² — x² — y²
π
fr = x d x
π
dx
α
a
19
y' = √ i
a
2
-
x²
in the same manner
=
У
α
>
22 |
α
1012
Ex. 20. Let the body be a solid formed by the revolu-
tion of a plane curve about the axis of x: (fig. 27.)
The centre of gravity of the slice PQ' is evidently at M
when the thickness of the slice is diminished indefinitely and
M
the mass of this slice
Tydx, V = whole volume;
V
I
66
Ex. 21.
Ex. 22.
STATICS.
•*. x =
2
foxxy² dx
£* y² d x
Let the body be a hemisphere.
y² = 2 ax
x²
X =
Só
ƒª (2 a x − x²) dx
ƒª (2 a x² − x³) dx
α
5 a
8
Let the body be a paraboloid.
18
y2
Ꮖ
= 4mx
fox x² d x
2x
x
£* x d x
3
Ex. 23. The frustrum of a paraboloid.
Let a and b be the radii of the larger and smaller ends:
a and ẞ the values of a measured from the vertex to the ends:
then a
a²
4m
2 a³
3
α B3
B
b2
4 m
>
2 a² + aß + B2
x² d x
dx
X
fo x d x
3 a² - B²
3
a + ß
therefore the distance from smaller end
2a² - aß - ß²
X
β
=
3 (a + B)
2a + B
c 2 a² + b²
(a - B)
3 (a + B)
3 a² + b²
c = length of the frustrum.
Ex. 24. Frustrum of a cone.
Distance from smaller end
=
c b²+2ab+3a²
4 b² + ab + a²
as in Ex. 4. Art. 86.
CENTRE OF GRAVITY OF ANY SOLID.
67
Ex. 25. Let the body be any solid.
We shall first suppose the body referred to rectangular co-
ordinates, as in fig. 29.
Let the body be divided into slices, like Q'N"M, by planes
parallel to the plane yx: let these slices be divided into prisms,
like QN, by planes parallel to the plane za: and let these
prisms be divided into parallelopipeds, like PP', by planes
parallel to the plane xy. In this manner the body is divided
into a number of elementary parallelopipeds: those at the ex-
tremities of the prisms will not be perfect; but when the dist-
ance of the cutting planes is diminished indefinitely, the sum
of these imperfect portions vanishes.
Let x y z be co-ordinates to P; dx, dy, dx the sides of the
parallelopiped at P: then dadydz is the volume of this
figure, and V being the volume and M the mass of the whole
body, supposed homogeneous, the mass of the element at P
M
dx dydz;
V
fff x d x d y d z
SSS
X
JJ dx dy dz
Y
ffy d x d y d z
jjj d x d y d z
fff x d x d y d z
and
J J J d x d y d z
between the proper limits.
We shall now suppose the body is referred to polar co-
ordinates as in fig. 30.
:
Let the body be divided into slices, such as CN'NA, by
planes passing through AC: let these slices be divided into
pyramids having their vertices in A, like AQ, by the rotation
of rays like AQ about AC, preserving a constant inclination
to AC during the rotation: lastly, let each of these pyramids
be divided into six-sided figures, like PP', by planes per-
pendicular to its length. In this manner the body is di-
vided into a number of six-sided figures which become paral-
lelopipeds ultimately when the distance of the cutting planes is
diminished indefinitely.
Let CAP A,
AP = "₂
1'?
BAN = 0 ;
68
STATICS.
therefore the sides of the figure at P are dr, rde, r sin edo,
and its volume ultimately equals the product of these
= r² sin 0 drdedp.
Also x = r sin 0 cos 0,
y = r sin 0 sin &,
ર
x = r cos 0 ;
0;
therefore, supposing the body homogeneous,
X
fffr³ sin² 0 cos pdrd@do
2
fr² sin @drdedo
y =
SSS
3
ƒfƒ³ sin² 0 sin
drd0do
tv |
fr² sin @drded p
fe sin Ꮎ cos Ꮎ d d Ꮎ d p
♫♫fr² sin Odrded p
between the proper limits.
Ex. 26. The eighth part of a sphere: (fig. 29).
L
We
Now xyz, being co-ordinates to any point P in the body,
are independent variables: we may therefore integrate with
respect to ≈, considering x and y not to vary that is the
same as taking all the elements of the mass in a given prism
QN, since although ≈ or PN is different for each element, yet
x and Y remain the same: the limiting values of ≈ are 0 and
QN=√ a² − x* — y² (= x' suppose) obtained from the equation
to the surface. This integration with respect to ≈ between
limits will leave a result a function of x and Y without ≈.
shall then integrate with respect to y, considering a constant;
this is the same as taking all the prisms in the same slice as
Q'N"; since, although MN or y is different for each prism,
yet AM or x is the same. The limits of y are 0 and MN' or
a² (=y suppose) obtained from the equation to the
line BN'. We shall finally integrate with respect to a from
x = 0 to x = AB or a, which is the same as taking all the slices,
and therefore the whole body;
20 =
a fy
ƒª f³´ f˜ x d x d y d z
a fy
ご
​fffdx dy dz
α
ƒªɓ” x √ a² — x² — y³ dx dy
k
2
Soª£%" √ a² — x² - y² d x d y
EXAMPLE OF VARIABLE DENSITY.
69
α
Soª. = x (a²
fa
a
4
π
.
x²) dx
√ª — (a² - x²) d x
4
$ a²
a³ - 1 a³
//
3 a
со
3 a
in like manner we shall find
Y
8
૨૨ ।
3 a
8
Ex. 27. The same as last example, but referred to polar
co-ordinates: (fig. 30.)
We shall integrate first with respect to r, then 0, and
lastly . The limits of " are 0 and AQ or a; those of ◊ are
π
π
o and ; those of are 0 and ;
2
.. x =
α
2
ƒªƒªƒª r³ sin² 0 cos død 0 dr
ra α τα
fafafa² sin dpd0dr
α
3a faf sin cos do de
Ꮎ афак
4
ra
faf sine do de
α
2019
3 a
8
00
La
π
cos o do
2
За
со
8
Гоаф
За
So also
y
3 a
12 1
8
со
Ex. 28. A hemisphere in which the density varies as
the nth power of the distance from the centre.
We shall use polar co-ordinates.
The volume of an element at P = 1² sin død0dr; and if
the density at a distance a the density at a distance = p
Pp+2
... mass of element at P = + sin død @dr.
P
an
(
a
P be
??
;
70
STATICS.
П
π
of 4,
and
2
2
The limits of r are 0 and a; of 0, 0 and π
*. X =
α
ƒª a ƒ ƒª p²±³ sin² 0 cos p dp d0 dr
α
T
a
•n+2
ƒ“a √ √ª²+² sin 0 d q də dr
0
n + 3 a
n + 42
2
α
214
π
GULDINUS's PROPERTIES.
PROP. To prove that if any plane figure revolve about
an axis lying in its own plane, the content of the solid gene-
rated by this figure in revolving through any angle is equal
to a prism, of which the base is the revolving figure and
height the length of the path described by the centre of gra-
vity of the area of the plane figure.
88. Let the axis of revolution be the axis of x, and the
plane of the revolving figure in its initial position to be the
plane of xy; we shall suppose the figure to be wholly on one
side of the axis of x: the angle through which the figure
revolves.
Then the elementary area da dy of the plane figure in
revolving through an angle de, generates the elementary solid
whose volume is yd0dxdy; therefore whole solid
Θ
= Lº ffyde dx dy,
the limits of x and y depend upon the equation to the curve
= 0 ffydx dy
between the proper limits.
But if be the ordinate to the centre of gravity of the
Y
plane figure, then by Art. 87. Ex. 7.
Y
ffy dx dy
ff d x d y
the limits the same as before;
GULDINUS'S PROPERTIES.
71
therefore whole solid = 0 ffydx dy = ÿ0. ffdx dy
= arc descd. by centre of gravity × area of figure.
Hence the Prop. is true.
PROP. To prove that the surface of the solid generated
is equal in area to the rectangle of which the sides are the
length of the perimeter of the generating figure and the
length of the path of the centre of gravity of the perimeter.
89. The surface generated by the arc ds of the figure
revolving through an angle de equals ydēds;
Ꮎ
.. whole surface = ƒª ſyd0ds = 0 fyds between proper limits.
But y = ordinate to centre of gravity of perimeter
Jyds
fds
between same limits as before;
therefore whole surface y0. fds
= arc descd. by centre of gravity × length of perimeter.
Hence the Prop. is true.
90. It is evident that these theorems are true also when
the generating figure is bounded by a line not of continuous
curvature.
Ex. 1. To find the solid content and the surface of
the ring of an anchor.
Let the radius of the axis be a, and the radius of a
transverse section be b: then the length of the path of the
centre of gravity of the area of the generating figure
and the area of the figure = π b²;
.. content of solid = 2π² a b².
=
2a,
Also path of centre of gravity of the perimeter = 2πa,
and the length of the perimeter = 2πb;
.. surface = 2
= 4π² ab.
Oi
72
STATICS.
Ex. 2. To find the centre of gravity of the area and
also of the arc of a semi-circle.
A semi-circle by revolving about its diameter generates
a sphere: the content of the sphere
the surface = 4πα²:
π
4π
3
a³, a the radius :
the area of the semi-circle = a²; the perimeter = πα;
therefore distance of centre of gravity of area from diameter
content of sphere
4 a
•
2π area of circle
3 п
;
and distance of centre of gravity of arc from diameter
surface of sphere
1
2π. arc of circle
2 a
π
CHAPTER V.
MACHINES. FRICTION.
91. A MACHINE is an instrument, or a system of solid
bodies, for the purpose of transmitting force from one part to
another of the system.
It would be endless to describe all the machines that have
been invented; we shall consequently confine ourselves to
those of simple construction. The most simple species of
machines are denominated the Mechanical Powers. These
we shall explain, and also a few combinations of them.
92. A Lever is an inflexible rod moveable only about a fixed
axis; which is called the fulcrum. The portions of the lever
into which the fulcrum divides it are called the arms of the
lever: when the arms are in the same straight line, it is called
a straight lever; in other cases a bent lever.
Two forces act upon the lever about the fulcrum, called
the power and the weight: the power is the force applied by
the hand (or other means) to sustain or overcome the other
force, or the weight. There are three species of levers: the
first has the fulcrum between the power and weight; in the
second the weight acts between the fulcrum and the power; and
in the third the power acts between the fulcrum and the weight.
PROP.
To find the conditions of equilibrium of two forces
acting in the same plane on a lever.
93. Let the plane of the paper be the plane in which the
forces act, and also be perpendicular to the axis, of which C is
the projection, and about which the lever can move (fig. 31.),
A, B the points of application of the forces P, W; a, ẞ the
angles which the directions of the forces make with any line.
a Cb drawn through C on the paper.
Let R be the pressure
K
74
STATICS.
upon the fulcrum, and the angle which it makes with the line
a Cb; then if we apply a force R in the direction CR, we may
suppose the fulcrum removed, and the body to be held in equi-
librium by the forces P, W, R.
We shall resolve these forces in directions parallel and
perpendicular to a Cb; and also take their moments about C: 0
then, by Art. 53, we have the following equations of condition :
P cos a
W cos ẞ - R cos 0 = 0 ..
P sin a + W sin ß – R sin 0 = 0 ...
and P. CD – W.CE 0 ..
(1),
0 ......... (2),
..... (3).
CD and CE being drawn perpendicular to the directions of P
and W.
These three equations determine the ratio of P to W when
there is equilibrium; and the magnitude and direction of the
pressure on the fulcrum.
For equation (3) gives
P CE_ perpendicular on direction of W
W
CD perpendicular on direction of P
Also by transposing the last terms of (1) and (2), we have
R cos 0 = P cos a W cos B,
R sin 0 = P sin a + W sin ß.
Add their squares;
.. R² = P² + W2 - 2 PW cos (a + ß),
which gives the magnitude of R.
Take the ratio of the above equations;
.. tan
P sin a + W sin ß
P cos a W cos B'
which gives the direction of the pressure.
If we suppose B to be the fulcrum and take the moments
about B instead of C, we have instead of equation (3) the
following*
:
*This is not a new equation of condition; but is a consequence of the three
already given, (1), (2), (3). To shew this imagine AD and BE produced to meet
CR:
MACHINES. LEVER.
75
P
perpendicular on direction of R
R
perpendicular on direction of P'
It follows, then, that the condition of equilibrium in a lever
of any species is that the two forces must be inversely as
the perpendiculars drawn upon their directions from the
fulcrum.
94. This property of the lever renders it a useful instru-
ment in multiplying the power of a force. For any two forces,
however unequal in magnitude, may be made to balance each
other simply by fixing the fulcrum so that the ratio of its
distances from the directions of the forces shall be equal to the
ratio of the forces; an adjustment which can always be made.
If the fulcrum be moved from this position, then that force will
preponderate from which the fulcrum is moved and the equi-
librium will be destroyed. We are thus led to understand how
mechanical advantage is gained by using a crow-bar to move
heavy bodies, as large blocks of stone: a poker to raise the
coals in a grate: scissors, shears, nippers, and pincers; these
last consisting of two levers of the first kind. The brake of a
pump is a lever of the first kind. In the Stanhope printing-
press we have a remarkable illustration of the mechanical
advantage that can be gained by levers. The frame-work in
which the paper to be printed is fixed, is acted upon by the
shorter arm of a lever, the other arm being connected to a
second lever, the longer arm of which is worked by the pressman.
These levers are so adjusted that at the instant the
paper comes
in contact with the types, the perpendiculars from the fulcra
upon the directions of the forces acting at the shorter arms are
exceedingly short, and consequently the levers multiply the
force exerted by the pressman to an enormous extent.
CR: they will meet this line in the same point, since the distances by these two con-
structions are CD cosec (0− a) and CE cosec (0+ẞ); and these are made equal, by
equations (1), (2), (3), if we eliminate P, Q, W. Suppose, then, F to be the point
in which these lines meet. By multiplying (1), (2), respectively by sin ẞ and cos ß,
and adding, we have
;
P sin (0+ẞ) FB sin (0 + ß) __ perpendicular on direction of R
R sin (a+ẞ) FB sin (a +ß) ~ perpendicular on direction of P
therefore this equation is a consequence of the equations (1), (2), (3), as might have
been anticipated.
!
76
STATICS.
As examples of levers of the second kind, we may mention.
a wheelbarrow, an oar, a chipping-knife, a pair of nutcrackers.
It must be observed, however, that as the lever moves about
the fulcrum the space through which the weight is moved is,
in the first and second species of lever, smaller than the space
passed through by the power: and therefore what is gained in
power is lost in despatch. For example in the case of the
crow-bar to raise a block of stone through a given space by
applying the hand at the further extremity of the lever, we
must move the hand through a greater space than that which
the weight describes.
But in the third species of lever the reverse is the case.
The power is nearer the fulcrum than the weight, and is con-
sequently greater; but the motion of the weight is greater than
that of the power. In this kind of lever despatch is gained at
the expense of power. An excellent example is the treddle of
a turning lathe. But the most striking example of levers of the
third kind is found in the animal frame, in the construction of
which it seems to be a prevailing principle to sacrifice power to
readiness and quickness of action. The limbs of animals are
generally levers of this description. The condile of the bone
rests in its socket as the fulcrum; a strong muscle attached to
the bone near the condile is the power, and the weight of the
limb together with any resistance opposed to its motion is the
weight. A slight contraction of the muscle gives a considerable
motion to the limb. A drawing of the human arm is given as
an illustration of these remarks: (fig. 32.)
95. The lever is applied to determine the weight of sub-
stances. Under this character it is called a Balance. The
Common Balance has its two arms equal, with a scale suspended
from each extremity; the fulcrum being above the line joining
the extremities of the arms. The substance to be weighed is
placed in one scale, and weights placed in the other till the beam
remains in equilibrium in a perfectly horizontal position; in
which case the weight of the substance is indicated by the
weights by which it is balanced. If the weights differ ever so
slightly, the horizontality of the beam will be disturbed, and
after oscillating for some time (in consequence of the fulcrum
being placed above the line joining the points of support of the
MACHINES. BALANCE,
77
scales) it will, on attaining a state of rest, form an angle
with the horizon, the extent of which is a measure of the
sensibility of the balance.
In the construction of a balance the following requisites
should be attended to. 1. When loaded with equal weights
the beam should be perfectly horizontal. 2. When the weights
differ, even by a slight quantity, the sensibility should be such
as to detect this difference. 3. When the balance is disturbed
it should readily return to its state of rest, or it should have
stability. We shall now consider how these may be fulfilled.
PROP.
To find how the requisites of a good balance
may be satisfied.
96. Let P and Q be the weights in the scales (fig. 33.):
AB =2a: C the fulcrum: h its distance from the line joining
A, B: W the weight of the beam and scales: k the distance
of the centre of gravity of these from C measured downwards:
the angle the beam makes with the horizon when there is
equilibrium.
Let us take the moments of P, Q, W about C: their sum
equals zero since there is equilibrium (Art. 53.) Then
since the distance of P's direc. from C = a cos 0 − h sin
we have
Q's....
= a cos 0 + h sin ✪
WV's.....
= k sin 0,
P (a cos 0 – h sin () − Q (a cos ( + h sin () – Wk sin 0 = 0 ;
.. tan
(P − Q) a
(P+Q) h + Wk
+
The first
This determines the position of equilibrium.
requisite the horizontality when P and Q are equal-is satis-
fied by making the arms equal.
For the second we observe that for a given difference of
P and Q the sensibility is greater the greater tan is; and for
a given value of tan 0, the sensibility is greater the smaller the
78
STATICS.
difference of P and Q is: hence
tan
is a correct measure
P - Q
of the sensibility: and therefore the second requisite is fulfilled
by making (P + Q)
h
a
k
+ W
a
as small as possible.
The stability is greater the greater the moment of the
forces which tend to restore the equilibrium when it is de-
stroyed. Suppose P = Q, then P and Q may be placed at the
mid-point between A and B: and the moment of the forces
tending to restore equilibrium equals {(P+Q)h+Wk} sin 0.
Hence to satisfy the third requisite, this must be made as large
as possible. This is, in part, at variance with the second re-
quisite. They may, however, both be satisfied by making
(P+Q)h + Wk large, and a large also: that is, by increasing
the distances of the fulcrum from the beam and from the centre
of gravity of the beam and scales, and by lengthening the
arms.
It must be remarked that the sensibility of a balance is
of more importance than the stability, since the eye can judge
pretty accurately whether the index of the beam makes equal
oscillations on each side of the vertical line; that is, whether
the position of rest would be horizontal: if this be not the
case, then the weights must be altered till the oscillations are
nearly equal.
97. Another kind of balance is that in which the arms
are unequal, and the same weight is used to weigh different
substances by varying its point of support, and observing its
distance from the fulcrum by means of a graduated scale.
The common steelyard is of this description.
PROP. To shew how to graduate the common steelyard.
98. Let AB be the beam of the steelyard (fig. 34.) A
the fixed point from which the substance to be weighed is
suspended, Q being its weight: C the fulcrum: W the weight
of the beam together with the hook or scale-pan suspended
from A; G the centre of gravity of these.
Suppose that P suspended at N balances Q suspended fron
A, then taking the moments of P, Q, W about C, we have
MACHINES. ROBERVAL'S BALANCE.
79
Q. ACW.CGP. CN= 0;
W
CN +
CG
Р
:: Q
. . P.
AC
W
Take the point D, so that CD=
CG;
P
· Q
CN + CD
AC
DN
. P:
P.
AC
Now let the arm DB be graduated by taking Da₁, Dα,
Dag,......equal respectively to AC, 2 AC, 3AC,......let the
figures 1, 2, 3, 4,......be placed over the points of graduation,
and let subdivisions be made between these. Then by ob-
serving the graduation at N we know the ratio of Q to P;
and this latter being a given weight we know the weight of Q.
In this way any substance may be weighed.
99. There is a remarkable balance called after its inventor
Roberval's Balance: a representation of it is given in fig. 35.
DC' is a frame of which the opposite sides are equal, and the
extremities are connected by pins at D, C, D', C' so as to
allow of free motion: this frame is supported by a stand
EE'A, being connected to it by pins at E and E' so as to allow
of free motion about those points: EE' must be parallel to DC
and D'C', but not necessarily equi-distant from them: arms
are fixed at right angles to the sides DD', CC to support
weights Q and P. The peculiarity of this machine is, that if
P and Q balance in any given position on the horizontal arms,
the equilibrium will remain undisturbed if we shift P or Qor
both of them along their arms in either direction: also if we
push one arm down and consequently raise the other the whole
will remain at rest in the position in which it is left.
We
shall prove these facts, and explained the paradoxical cha-
racter of the machine in the Chapter of Problems. We may
however easily prove by the Principle of Virtual Velocities
the facts mentioned above, though the paradox will not be
removed.
80
STATICS.
If we lower the arm on which P acts through a space a,
then D' sinks through a space x, and D, and therefore the
arm on which Q acts, rises through a space
a'x
а
>
which is
independent of the distances of P and Q along their arms:
a and a' are the lengths DE and ED'.
ax
Then P.x
Q.
= 0 by Virtual Velocities,
a
Pa
or
Q
a
for all positions of the frame and of the weights.
It will be seen upon referring to the Chapter of Problems
that although the equilibrium remains undisturbed when P
and Q have different positions, yet the strains at the joints
D, D', C, C', E, E' and the point of application (B in figure)
of the downward-pressure undergo changes.
It is on the principle of this machine that the balances
used of late years in shops are constructed: the scales rest
each by one point upon the extremities of a lever below them,
and the only motion they are capable of is in a vertical
direction.
100. The second of the Mechanical Powers is the Wheel
and Axle. This machine consists of two cylinders fixed together
with their axes in the same line: the larger is called the wheel
and the smaller the axle: the axis of the axle is generally much
larger than that of the wheel. The cord by which the weight
is suspended is fastened to the axle, and then coiled round it,
while the power which supports the weight acts by a cord
coiled round the circumference of the wheel; by spokes acted
on by the hand, as in the capstan; or by the hand acting on
a handle as in the windlass.
PROP. To find the ratio of the power and weight in
the Wheel and Axle when in equilibrium.
101. Let AD be the wheel and CC'B the axle (fig. 36.)
P the power represented by a weight suspended from the cir
MACHINES.
81
TOOTHED WHEELS.
cumference of the wheel at A: W the weight hanging from
the axle at B.
Then since the axis of the machine is fixed, the condition
of equilibrium is that the sum of the moments of the forces
about this axis vanishes (Art. 68.);
.. P. AC – W. C'B = 0 ;
W AC
rad. of wheel
Р BC'
rad. of axle
It will be seen that this machine is only a modification of
the lever. In short it is an assemblage of levers all having
the same axis: and as soon as one has been in action the next
comes into play; and in this way an endless leverage is ob-
tained. In this respect, then, the wheel and axle surpasses
the common lever in mechanical advantage. It is much used
in docks, and in shipping.
102. The third Mechanical Power is the Toothed Wheel.
It is extensively applied in all machinery; in cranes, steam-
engines, and particularly in clock and watch work. If two
circular hoops of metal or wood having their outer circum-
ferences indented, or cut into equal teeth all the way round,
be so placed that their edges touch, one tooth of one circum-
ference lying between two of the other (as represented in the
figure $7.); then if one of them be turned round by any means,
the other will be turned round also. This is the simple
construction of a pair of toothed wheels.
PROP. To find the relation of the power and weight
in Toothed Wheels.
103. Let A and B be the fixed centres of the toothed
wheels on the circumferences of which the teeth are arranged.
C the point of contact of two teeth: QCQ a normal to the
surfaces in contact at C. Suppose an axle is fixed on the
wheel B, and the weight IV suspended from it at E by a cord:
also suppose the power P acts by an arm AD: draw Aa, Bb
perpendicular to QCQ. Let the mutual pressure at C be Q.
Then since the wheel A is in equilibrium about the fixed axis
A, the sum of the moments about A equals zero:
L
82
STATICS.
.. P . AD – Q. A a
= 0.
Also since the wheel B is in equilibrium about B, the sum of
the moments about B equals zero:
·. Q. Bb – W. BE = 0.
Then by eliminating Q from these two equations,
P
P Q
W
Q W
Aa BE
AD Bb
moment of P
A a
ΟΙ
moment of W
Bb
when the teeth are small this ratio
rad. of wheel A
rad. of wheel B
very nearly.
104. Wheels are in some cases turned by means of
straps passing over their circumferences. In such cases the
minute protuberances of the surfaces prevent the sliding of the
straps, and a mutual action takes place such as to render the
calculation exactly analogous to that in the Proposition.
For the calculation of the best forms for the teeth, the
reader is referred to a Paper of Mr Airy's, in the Camb. Phil.
Trans. Vol. II.
p. 277.
105. The fourth Mechanical Power is the Pully. There
are several species of pullies: we shall mention them in order.
The simple pully is a small wheel moveable about its axis:
a cord passes over part of its circumference. If the axis is
fixed the effect of the pully is only to change the direction of
the cord passing over it: if, however, the axis be moveable
then, as will be presently seen, a mechanical advantage may
be gained.
MACHINES.
83
PULLIES.
PROP. To find the ratio of the power and weight in
the single moveable Pully.
106. I. Suppose the parts of the cord divided by the
pully are parallel (fig. 38.)
Let the cord ABP have one extremity fixed at A, and
after passing under the pully at B suppose it held by the hand
exerting a force P. The weight W is suspended by a cord
from the centre C of the pully.
Now the tension of the cord ABP is the same throughout.
Hence the pully is acted on by three parallel forces, P, P,
and W: hence
2 P – W = 0;
IV
= 2.
P
II. Suppose the portions of cord are not parallel (fig. 39.)
Let a and a be the angles which Aa and Pb make with
the vertical.
Now the pully is held in equilibrium by W in CW, P
in a A, P in b P. Hence by Art. 53,
=
horizontal forces, Psina - P sin a 0...(1)
and vertical forces, P cos a + P cos a IV 0...(2)
the equation of moments is identical.
=
By (1), sin a =
sin a' and a =
= a' ;
W
... by (2),
2 cos a which is the relation required.
P
PROP.
To find the ratio of the power and weight in
a system of pullies, in which each pully hangs by a separate
string, one end being fastened in the pully above it and the
other end on a fixed beam: all the strings being parallel.
I.
107. Let n be the number of pullies (fig. 40.)
Let us neglect the weights of the pullies themselves.
1
Then the tension of b¸ IV = IV; .. the tension of a,b,b, = ¦ W;
|
84
STATICS.
1
22
.. tension of ɑ¿b₂bз = W, tension of a3b3c=
1
{
W
23
and so on, and the tension of the string passing under the n
1
pully W, and this =
Qn
P;
W
2".
P
th
II. Let us suppose the weights of the pullies to be con-
sidered: and let www3...w, be these weights.
Then if P1 P2 P3...Pn be the weights which they would
sustain at P and P, the weight W would sustain at P, we
1
have
Ş|
W2
Wn
W
P1
P2
2n
Qn-
1
.Pn
P₁ =
2
2n
+ Pn + P1,
1
or P =
2n
·. P = P₁ + P₂ +.
{ W + w] + 2 w2 + 2² w; + ... ... + 2¹² - ¹ wn
If w₁ = w₂ = W3 =
Wn3
1
Р
{ W + (2″ − 1) w₁}.
Qn
PROP. To find the ratio of the power and weight when
the system is the same as in the last Proposition; but the
strings are not parallel.
108. We shall neglect the weights of the blocks. The
pullies will evidently so adjust themselves that the string at
their centre will bisect the angle between the strings touching
their circumference.
(3
Let a₁ a₂ aз ...... a, be the angles included between the
strings touching the first, second, third, ...... nth pullies re-
spectively (fig. 41.)
Then, by Art. 106, tension of a₁ b₁ b₂
W
2 cos a
MACHINES. PULLIES.
W
therefore tension of a b₂ b3
22 cos a₁ cos α2
W
85
tension of a b3 c =
23 cos a₁ cos a½ cos az
W
tension of the last string
2" cos al cos a cos az
Cos an
and this = P;
W
2" cos a₂ cos a½ cos az
cos an
P
PROP.
To find the relation of the power and weight in
a system of pullies where the same string passes round all the
pullies.
109. This system consists of two blocks; each containing
a number of the pullies with their axes coincident.
The weight
is suspended from the lower block, which is moveable, and the
power acts at the loose extremity of the string, which passes
round the respective pullies of the upper and lower block
alternately.
Since the same string passes round all the pullies, its ten-
sion will be everywhere the same, and equal to the power P.
Let n be the number of portions of string at the lower block;
then n . P will be the sum of their tensions;
.. W = n. P.
If we take into account the weight of the lower block, and call
it B, then
W + B = n. P.
If the strings at the lower block are not vertical, we must
take the sum of the parts resolved vertically, and equate it to W.
But, in general, this deviation from the vertical is so slight,
that it is neglected.
110. As the weight is rising or falling, it will be observed
that in general the pullies move with different angular motions.
The degree of angular motion of each pully depends upon the
86
STATICS.
magnitude of its radius.
Mr James White took advantage of
this to choose the radii of the pullies in such a manner as to
give them the same angular motion, and so prevented the wear
and resistance caused by the friction of the pullies against each
other. This being the case, the pullies might be fastened to-
gether. Instead of this, however, the pullies were cut in the
same block.
It will be seen without much difficulty, that the angular
motions of the pullies of the upper block will be as the series of
odd integers 1, 3, 5, ...... and those of the pullies of the lower
block as the series of even integers 2, 4, 6,
These series, then, determine the relative sizes of the
pullies in the two blocks.
PROP. To find the ratio of the power to the weight when
all the strings are attached to the weight.
111. If we neglect the weight of the pullies (fig. 42.) the
tension of the strings b, a, P; the tension of a, b = 2 P; and
so on: if there be n pullies, then the sum of the tensions of
the strings attached to the weight
= P + 2 P ÷ 2² P +
-
1
+ 2^~¹ P = (2″ – 1) P;
W
2" 1.
Р
If we suppose the weights of the pullies are w₁wy wz
reckoning from the lowest, and w' w" w""
the portions of
W which they respectively support (since they evidently
assist P), and W' the portion of W supported by P; then
W' = (2″— 1) P,
w' = (2″− 1 − 1) w1,
W
(2n−2 — 1) w₂,
(2
w(n-1) = (2-1) w₂-15
.. W = W'+w'+
(2ª − 1) P + (2”~1—1) wi
+ (2n −2 − 1.) w½ +
.....
+ (2 − 1) w₂−1 ·
+(
MACHINES. INCLINED PLANE,
87
If w₁ = w₂ = W3
W =
(2″ − 1) P + {2”−1 + 2-2 +
+ 2 − (n − 1)} wi
-
w₁
=
112.
(2” − 1) P + (2″ — n − 1) w₁ ⋅
The fourth Mechanical Power is the Inclined Plane.
By an inclined plane we mean a plane inclined to the
horizon. A weight W may be supported on an inclined plane
by a power P less than IV.
PROP. To find the ratio of the power and weight in the
inclined plane.
113. Let AB be the inclined plane (fig. 43.): a the angle
it makes with the horizon. Let the power P act on the weight
in the direction CP, making an angle e with the plane. Now
the weight at C is held at rest by P in CP, WV in the vertical
CW, and a pressure R in CR, at right angles to the plane.
Hence, by Art. 23, if we resolve these forces perpendicular
and parallel to the plane, we have
R + P sin e
W cos a = 0)
P cos e IV sin a = 0 ..
(1),
(2).
P sin a
α
The second gives the required relation
II
COS E
equation gives the magnitude of the pressure R.
α,
; and the first
COR. 1. If P act horizontally, e =- and P = IV tan a.
COR. 2. If P act parallel to the plane, e = 0, P
0, P = W sin a
COR. 3. If P act vertically, e=
π
2
P = IV.
114. The fifth mechanical power is the Wedge. This is
a triangular prism, and is used to separate obstacles by intro-
ducing its edge between them and then thrusting the wedge
forward. This is effected by the blow of a hammer or other
such means, which produces a violent pressure for a short time,
sufficient to overcome the greatest forces.
88
STATICs.
PROP. An isosceles wedge being introduced between two
obstacles, required to find its tendency to separate the obstacles
when the wedge is prevented from being thrust back by a given
force.
115. Let 2 P be the force acting at the back of the wedge
(fig. 44.) In the figure we suppose the obstacles to be the two
halves of a tree. The portions of the tree we suppose similarly
situated on the two sides of the wedge: let A and A' be the
points of contact between the wedge and the obstacles: AN,
A'N' normals to the wedge at A and A': R, R the mutual
resistances of the wedge and obstacles at A and A.
Now if the wedge were to move backwards or to be thrust
forwards the points A and A' would move in some unknown
curve line the nature of this curve would depend upon the
elasticity and strength of the material of the obstacles and upon
other circumstances. Draw AT, A'T' tangents to these curves
at the points A and A'. Then it will be seen that the parts of
the pressures at A and A' which measure the tendency of the
obstacles to separate will be their resolved parts along these
tangent lines; since if they separate it must be by A and A'
moving along these lines. The resolved parts perpendicular to
these tangents are counteracted by the resistance of the ground
at E.
Let W be the resolved part of R along the tangent on
either side and suppose the angle NAT = i. Also let the
angle of the wedge be 2a.
Then the wedge being sustained by the forces 2P, R and
R; we have by resolving them vertically
2P-2R sin a = 0
(1);
the horizontal parts counteract each other of necessity, also the
equation of moments is an identical equation.
Again W
W = R
R cos i
P
sin a
W
cos i
(2);
If, then, we know the angle i we shall know W: but we have
MACHINES. THE SCREW.
89
no means of ascertaining the value of i, and consequently the
preceding calculation is of little importance.
When i is very small, then W =
P
sin a
nearly.
116. The last Mechanical Power is the Screw.
This machine in its simple construction consists of a cy-
linder (fig. 45.) AB with a uniform projecting thread abcd ...
traced round its surface, and making a constant angle a with
lines parallel to the axis of the cylinder. This cylinder fits
into a block D pierced with an equal cylindrical aperture, on
the inner surface of which is cut a groove the exact counterpart
of the projecting thread abcd.
It is easily seen from this description, that when the cylinder
is introduced into the block, the only manner in which it can
move is backwards or forwards by revolving about its axis, the
thread sliding in the groove. Suppose IV is the weight acting
on the cylinder (including the weight of the cylinder itself)
and P is the power acting at the end of an arm AC at right
angles to the axis of the cylinder: the block D is supposed to
be firmly fixed, and the axis of the cylinder to be vertical.
PROP. To find the ratio of the power and weight in the
Screw when they are in equilibrium.
117. Let AC a: rad. of cylinder = b.
=
Now the forces which hold the cylinder in equilibrium are
IV, P and the reactions of the pressures of the various portions
of the thread on the corresponding portions of the lower surface
of the groove in which the thread rests: these reactions are in-
determinate in their number; but they all act in directions
perpendicularly to the surface of the groove, and therefore
their directions make a constant angle a with a horizontal plane.
If, then, R be one of these reactions, R sin a, R cos a are the
resolved parts vertically and horizontally: the horizontal por-
tions of the reactions act each at right angles to a radius of the
cylinder. Hence resolving the forces vertically, and also taking
the moments of the forces in horizontal planes, we have
IV. R sin a
= = 0..
(1),
PaΣ. R cos a b = 0.
(2),
M
90
STATICS.
we might write down the other four equations of equilibrium ;
but they introduce unknown quantities with which we are un-
concerned in our question.
01
W
a sin a Σ. R
Hence
because b and a are constant:
P
b cos a Σ. R'
a sin a
b cos a
Σπα
2πb cot a
circumference of circle whose rad. is AC
vertical dist. between two successive winds of the thread
The screw is used to gain mechanical power in many ways.
In excavating the Thames Tunnel the heavy iron frame-work
which supported the workmen was gradually advanced by
means of large screws.
FRICTION.
118. In the investigations of this Chapter we have sup-
posed that the surfaces of the bodies in contact are perfectly
smooth. Now in practice this is not the fact; for no surface
can be so entirely freed from roughness and asperities as to be
perfectly smooth, although their effect may in many cases be
greatly diminished. By a smooth surface is meant a surface
which opposes no resistance whatever to the motion of a body
upon it, and therefore the resistance is wholly perpendicular to
the surface. A surface which does oppose a resistance to the
motion of a body upon it is said to be rough.
The friction of a body on a surface is measured by the
least force which will put the body in motion along the surface.
Coulomb made a series of experiments upon the friction of
bodies against each other and deduced the following laws:
Mémoires des Savans Etrangers, Tom. x.
(1) The friction varies as the pressure, when the ma-
terials of the surfaces in contact remain the same. When the
pressures are very great indeed it is found that the friction is
somewhat less than this law would give.
FRICTION.
91
(2) The friction is independent of the extent of the sur-
faces in contact so long as the pressure remains the same.
When the surfaces in contact are extremely small, as for in-
stance a cylinder resting on a surface, this law gives the
friction much too great.
These two laws are true when the body is on the point of
moving and also when it is actually in motion: but in the case
of motion the magnitude of the friction is much less than when
the body is in a state bordering upon motion.
(3)
The friction is independent of the velocity when
the body is in motion.
u
It follows from these laws that if P be the normal pressure
of the body upon the surface then the friction = μ. P, where μ
is a constant quantity for the same materials, and is called the
coefficient of friction.
In the state bordering on motion and when the surfaces in
contact are of finite extent we have the following results from
experiment :
1
μsurfaces wood, the grain being in same direction.
1
4
metallic surfaces.
opposite
11
1.
one surface wood and the other metal.
Oil and grease considerably diminish friction; fresh tallow re-
duces it to half its value.
In the state bordering on motion and when the surfaces in
1
ль
for wood.
12
contact are single lines, then
When the sur-
face in contact is a physical point the statical friction is incon-
siderable.
But for full particulars on this subject we refer the feader
to Coulomb's papers, and also to two Memoirs recently pub-
lished in the Mémoires de l'Institut. by M. Morin.
PROP. To find the greatest angle which the direction of
the mutual pressure of two surfaces in contact may make with
their common normal at the point where the pressure acts
without sliding; the coefficient of friction being given.
92
STATICS.
119. Let P be the mutual pressure, its direction making
an angle ẞ with the normal. Then P cos 3 is the direct or
normal pressure of the surfaces, and P sin ẞ is the force
balanced by the friction acting wholly or in part.
P sin B
Hence the greatest value of the ratio
is μi
P cos B
..ẞ tan-¹µ is the greatest value of ß.
=
CHAPTER VI.
ROOFS, ARCHES AND BRIDGES.
120. In the present Chapter we shall apply the principles
of equilibrium to explain in what manner the thrusts, strains,
and pressures in general act in roofs and arches. We refer the
reader to Robison's Mechanical Philosophy, Vol. 1. for two
Articles on Roofs and Arches, which contain many interesting
details that would be entirely out of place in these pages.
PROP. In a simple isosceles truss-roof required to calcu-
late the tension of the tie-beam.
121. Let AB, BC be two beams of the roofing connected
by the tie-beam AC (fig. 46.): the truss resting on walls, as
drawn in the figure. Let B be the weight of each sloping
beam and the portion of the tiling or thatching supported by
the beam: let G be the point at which this weight acts. Also
suppose that the weight on the vertex arising from other ap-
pendages equals W: let a be the angle the roof makes with the
horizon: AG = b, AB = a.
Now the forces acting on AB are a pressure at A perpen-
dicular to the wall, = R suppose: the tension of the tie-beam
acting at A in the direction AC, = T suppose the weight B
acting vertically at G and lastly, some force P acting at B in
direction BP making an unknown angle with the beam in the
plane of the paper, and arising from the weight W and the
action of the beam BC.
:
In order to find the connexion between P and IV, we re-
mark that the point B is held at rest by the force IV downwards,
and the two reactions P, P acting along the dotted lines.
94
STATICS.
For the equilibrium of AB we have
vertical forces RBP sin (a 0) = 0
=
−
moments about A = Bb cos a Pa sin 0
horizontal forces
=
T – P cos (a – 0)
= 0...
(1),
(2),
= 0.
(3).
vertical forces
=
2 P sin (a – 0) – W = 0 ..
(4).
For the equilibrium of the point B,
Here, then, we have four equations and four unknown quantities
T, P, R, 0: and therefore we can determine the unknown
quantities, and therefore T.
By (1) (4) T = 1 W cot (a – 0), eliminating P:
and by (3) (4), eliminating P, we have
Bb cos a =
1 Wa
sin e
sin (a - 0)
sin {a - (a-0)}
1 Wa
sin (a -0)
Wa {sin a cot (a - 0)
-
cos a};
2 Bb + Wa
.. cot (a - 0)
cot a
Wa
2Bb + Wa
.. T =
cot a.
2 a
This measures the horizontal thrust of the roofing against the
supporting walls supposing the tie-beams to give way: and we
learn that this will be less the larger a is, or the steeper the
roof is, the other quantities remaining the same. Also the
smaller b is in proportion to a, or the nearer G is to A,
smaller is this thrust.
the
PROP. To explain the manner in which buttresses act
in supporting a roof: and to calculate their angle of elevation.
122. Let (as before) AB, BC (fig. 47.) be two beams of
the roofing: AD a piece of timber firmly attached to AB run-
ning down the inside of the wall, and resting on a corbel E:
FH a beam to strengthen the attachment of AD, AB. Let
R be the pressure on the top of the wall and corbel; N the
point at which the resultant of the reactions of all the hori
ROOFS.
95
zontal pressures on the wall acts; T this resultant; AN = x;
G the point through which the weight on the slanting timbers
acts; AG = b, AB = a.
Now the action of the beams on each other at B must be
in a horizontal direction, since we suppose there is no extra
weight acting at B, as in the last Proposition; let P be this
2
mutual pressure.
For the equilibrium of BAD,
horizontal forces
T-P = 0
vertical forces = R − B = 0...
moments about A = Ta Bb cos a + P a sin a = 0
-
(1),
(2),
(3).
Here we have three equations and four unknown quantities
T, P, R, x: and another relation connecting these quantities
cannot be found; hence the problem is indeterminate: we shall
see in the solution what quantities are indeterminate.
By (2) R = B, and is therefore not indeterminate.
By (1) (3), eliminating P, we have
T
Bb cos a
x + a sin a
= P by (1).
Hence P, T, and a are indeterminate: but when a value is
given to one of them, then, that the equilibrium may subsist,
the other two must satisfy the two conditions just deduced.
Let be the angle which the resultant of R and T makes
with the vertical; then
T
b cos a
tan
R
x + a sin a
Now our object is to find the least angle at which a buttress
need be built to support the roof. If the roof be on the point
of sinking it must be so by tending to turn about the extremi-
ties D, D of the framework: in which case the force T acts at
D, and a will then have its greatest value: also we perceive
that both T and also are smaller the greater v is: hence the
least angle at which the buttress need be built is given by the
equation
96
STATICS.
tan o
A'G'
A'G
AD + AA A'D
BA' being horizontal; AA', GG' vertical.
Hence the dotted line G'D represents the limiting angle
which the buttress must make with the vertical in order that
the roof may not fall.
This calculation shews the great use of the part of the
framework which runs down the wall.
We have drawn in our figure but one connecting beam
HF: but there might have been more, and the calculation
would have been precisely the same, supposing that ABD is a
rigid framework and GG' the vertical line in which the weight
of this framework and the superincumbent tiling or other
covering acts. The simple rule is to draw a horizontal BG'
from the vertex of the roof, cutting GG' in G' and join G' with
the lowest point D of the framework: G'D gives the least
inclination of the buttress. Also the buttress need not extend
higher up the wall than the level of D.
The roofs of Westminster Hall and of Trinity College
Hall, Cambridge, are good illustrations of this kind of roof.
Before quitting this subject we will investigate the fol-
lowing Proposition.
PROP. To calculate the conditions of equilibrium of any
number of beams forming a framework in a vertical plane,
symmetrical with respect to a vertical line through the highest
point.
2
123. Let the lengths of the beams be a₁ɑɑ...reckoning
from the lowest G GG...the points at which the weights
of the beams and the weights with which they may be loaded
act; b₁b₂b...the distances of these points from the lower ex-
tremities of the beams; aaa...the angles which the beams
make with the horizon; R the vertical pressure on the walls
of support of each of the two lowest beams; T the horizontal
thrust of these beams on the walls; or the tension of the tie-
beam connecting them if there be one.
ROOFS.
97
Now the actions of any two beams on each other at the
points of junction must be in the same line, since there is no
third force to keep them in equilibrium. Let then P, P2...
be these mutual actions between the first and second, the
second and third beams, and so on; ₁02...the angles which
the directions of these forces make with the horizon.
For the equilibrium of the lowest beam
T-P₁ cos 0₁ = 0......
1
·
B₁ - R+ P₁ sin ₁ = 0
1
.........
...(1)
..(2)
B₁b₁ cos a₁- P₁a, sin (a, - 0,) = 0......(3).
For the equilibrium of the second beam
1
P₁ cos 0₁ - P₂ cos 0₂ = 0 …..
2
..(4
(4)
2
B₂ - P₁ sin 0, + P₂ sin 0₂ = 0.........
02
……..(5)
1
2
B₂b₂ cos a₂ - P₂a, sin (α₂ − 0₂) = 0…………….(6),
and so on, till we come to the highest beam in which the angle
0, since the two highest beams which form the vertex
have no third force at their point of junction to keep P₂ and
P₁ in equilibrium. Hence for the last (the nth) beam
Өт
must
=
Pn-1 cos 0-1 - P₂ = 0
..(3n − 2)
(3 n − 1)
B₁ bn cos an
- Pra, sin
an
0 ………..(3n).
B₂ – Pr-1 sin („-1=0.
Also we have the following analytical relation connecting
a1a2..., ajɑ²...
а₁ cos α1 + α₂ cos a² + ... + An
+ an cos an= D...(3n + 1)
2D being the distance of the opposite walls from each other.
We have then 3n+1 equations from which to eliminate
the quantities P1 P2... Pn 0102...On-1, R, T, which are 2n+1
in number and we have n equations remaining to determine the
n angles a₁a2a3...a,; and these being known we know the
position of equilibrium of the beams.
α α
N
98
STATICS.
If we add together all the second equations we have
R = B₁ + B₂ +……………..+ B₂
wherefore R is known.
Now by (2) (3) eliminating P₁ we have
1
sin (a,- 0₁)
sin 0₁
B₁b₁ cos a
1
(R −B₁) a₁
;
.. sin a₁ cot 0₁ = {1
+
B₁b₁
(R-B₁) a
cos a
.. cot ₁
Ꮎ,
cot a ;
1
α 01
(R − B₁) a₁ + B₁b₁
(R − B₁) a₁
.. T = (R − B₁) cot 0, by (1) (2)
(R − B₁) α₁ + B₁b₁
a1
whence T is known.
1
cot a₁ = B² + ... + B₂ + B₁
ก
cot ai
Again, by adding equations (1), (4); (2) (5) respectively
we have
T - P₂ cos 0₂ = 0
2
B₁ + B₂ − R + P₂ sin 0₂ = 0)
2
2
also by (6) B₂b₂ cos a₂ – P₂a, sin (a½ − 0½) = 0.
Hence, as by solving (1) (2) (3), we have
b2
T
= {B₂+
B-
Bз + ... + B₂ + B2
cot a2,
and in the same manner we should obtain
B¸ +...+ B½ + B3
T= {B
T = {B. })
ხ,,
ก
cot an
b3
b) cot
cot as
ROOFS.
99
These n values of T, being equated, give n − 1 relations
connecting the angles a, a... a; and these combined with
equation (3n + 1) determine these angles.
The relation of one angle am to the preceding am-1 is
given by
bm
B₂+1 + ... + B₂+ Bm
a 172
tan an-1°
tan am
bm-1
Am-1
Bm +...+ B₂+ Bm−1
This shews that every one of the angles a₁a...a, is greater
or every one less than 90° and equation (3n + 1) shews that
they must be all less than 90°, otherwise we should have D =
a sum of negative quantities.
Also the beams must be less and less inclined to the horizon
as we ascend, since tan a, is less than tan am-1
•
124. By an Arch is meant an assemblage of bodies sup-
ported, as represented in fig. 48, by their mutual pressures and
the pressures of the two extreme bodies against fixed obstacles.
We shall suppose the bodies to have the usual form, that
of truncated wedges; and to be placed so as to have their sides
which are in contact perpendicular to the same vertical plane.
These bodies are then called voussoirs: the highest voussoir
is called the key-stone of the arch: the surfaces which separate
the voussoirs are called the joints: the external curve of the
arch is called the extrados; the internal curve the intrados:
the solid mass against which the lowest voussoir on each side
rests is called the pier or abutment.
125. It is found in practice that the friction of the vous-
soirs against each other is so great that they are incapable of
sliding past each other; and in many cases all possibility of
sliding is prevented by the voussoirs being joggled; that is,
being united by a piece of stone or iron which is partly im-
bedded in one voussoir and partly in the voussoir in contact
with it.
In consequence of this the conditions of equilibrium of an
arch reduce themselves to the condition that the arch shall not
break at any part by the rotation of one voussoir upon another:
or, which is the same thing, by the opening of any of the joints,
100
STATICS.
PROP. To explain how an arch is supported; and in
what manner friction tends to preserve the equilibrium.
126. When two bodies with smooth surfaces in contact are
pressing against each other, then, in order that they may not
slide upon each other, the mutual pressure must act in a line
which is perpendicular to the two surfaces. But if the sur-
faces in contact be rough, the mutual pressure need not act
in a direction perpendicular to the surfaces to prevent sliding,
but may act in any line making an angle with the perpendi-
cular less than a certain finite angle, the magnitude of which
depends on the degree of roughness of the surfaces.
Art. 119.
See
Suppose fig. 48. represents an arch in equilibrium, the
extreme voussoirs G₁ and G resting on the piers A and B :
G, G₂ G.......are the centres of gravity of the voussoirs: the
weights of the voussoirs act in the vertical lines G₁b, G₁d,
ƒ G3.
Now any voussoir is held in equilibrium by the action of
its weight in a vertical direction downwards, and by the pres-
sures of the contiguous voussoirs or abutment on its two
joints. The mutual pressures of any two voussoirs at a joint.
must evidently be equal to some single force acting at some
unknown point.
Suppose the pressure of the pier A acts on the voussoir
G₁ at the point a in the direction ab cutting G₁b in some point.
b: then the third force which supports G, must act through
b in some determinate direction dcb; let this cut the vertical
through G in d: this is the direction of the mutual pressures
of the voussoirs G₁ and G, and c is the point at which they
act: then the third force which supports G, must act through
d in some direction fed; and so on: and it follows that the
mutual pressures of the voussoirs act in some determinate
(though unknown) line abdfhjlno made up of portions of
straight line. This is called the line of pressure.
Now if the surfaces of the voussoirs were smooth it would
be necessary for the equilibrium, that this line of pressure
should be perpendicular to the joints at the points a, c, e, g,
i, k, m, o.
This condition establishes certain relations between.
ARCHES.
101
the weights of the voussoirs and the angles which the joints
make with the horizon. These relations may be easily cal-
culated, but since the voussoirs never are smooth in practice
the calculation would be useless. But when the surfaces of
the voussoirs are rough, the only conditions for the equili-
brium are that the angles which the line of pressure makes.
with the perpendiculars to the joints at the points a, c, e, g,
i, k, m, o should not exceed a certain finite angle, which can be
made as large as we please by having the stones rough-hewn
or by joggling them (Art. 125.)
We see, then, that in consequence of friction the weights.
of the voussoirs and the angles of inclination of their joints
need not fulfil those exact relations, which would be necessary
if the surfaces were smooth.
127. But the great advantage of friction in the support
of an arch is yet to appear. For in order that an arch in a
bridge may be of service, it must be able to sustain weights
(not immoderate in their magnitude) placed on different parts.
without breaking.
Let us now suppose a weight W to be placed on the vous-
soir G. This weight adds to the weight of G₂, and con-
sequently disturbs the line of pressure and shifts it to some
new position a'b' d'f'h'j' l'n'o' as represented in fig. 49. But
the arch will still stand if the angles at a', c', e', g', i', k', m', o'
do not exceed the finite limit. Whereas the equilibrium of
the arch would certainly be disturbed by IV if the surfaces.
were smooth.
In this way we see, then, the important aid that friction
affords in the support of an arch.
PROP. To find the conditions of equilibrium of an arch.
128. Since we suppose the friction of the voussoirs
against each other to be so great that they cannot slide upon
each other, it follows that the arch can fall only in consequence
of its breaking at the upper or lower extremities of some of
the joints. And, since we suppose the piers A and B to be
immoveable, simple geometrical considerations shew, that if the
arch break it must break in at least three pieces, four joints
102
STATICS.
at least opening, the points where they open being alternately
in the extrados and intrados of the arch. So long as the line
of
pressure cuts all the joints (as is represented in figures 48, 49.)
the arch must stand, because in that case the joints are pre-
vented from opening by the pressure acting along that line.
But by continually loading the arch in the same part we
may gradually shift the line of pressure till it passes through
the extremity of one of the joints, as g' in fig. 50: and now the
pressure acting through g' will not prevent the joint g'H
opening at H, although other circumstances may.
From what we have already said about the arch not falling
till two joints at least in the extrados and two joints at least
in the intrados open, it follows, that the arch will certainly
stand as we continually load it till the line of pressure passes
through the extremities of at least four joints, the extremities
being alternately in the intrados and extrados.
If, then, our arch be such, that by loading it we cannot
shift the line of pressure into this position, the arch will sus-
tain any load without falling. Nevertheless when the arch is
much loaded, and the line of pressure passes through the ex-
tremity of any joint, there will be a great strain at that point.
This explains the fact observed by Professor Robison, who
constructed some chalk models and found that chips fell off
from three or four of the extremities of the joints.
If, however, the arch be of such a form that we can place
on it a sufficient load to cause the line of pressure to pass
through at least four extremities of joints situated alternately
in the intrados and extrados, there is a possibility of the arch
breaking and falling by the opening of the joints.
Let D, K, L, M (fig. 51.) be the points through which
the line of pressure passes in this case: join them by straight
lines. Then the arch may be supposed to be a system of
heavy beams DK, KL, LM. In order to determine the con-
ditions that the equilibrium of the arch shall be stable, suppose
the joints are forcibly opened through very small angles, the
parts of the arch being sustained in the position represented in
fig. 51. That the equilibrium of the arch may be stable, the
joints, when the arch thus sustained in a broken form is left
to itself, ought to collapse and not open wider; a condition
ARCHES.
103
which is satisfied if the system of beams DK, KL, LM be
such that when left to themselves the point K shall ascend
and the point L descend.
Hence to ascertain whether an arch of certain dimensions
and figure will sustain any weight placed on it, we must con-
sider all the ways in which the arch can break and find whether
in each case the system of beams is of the nature just described.
If this be the case we may be assured that the arch will sup-
port any weight.
PROP. To prove that an arch, in which a tangent line
drawn at the highest point of the intrados and produced to
the abutments lies wholly within the voussoirs, will sustain
any weight placed on any part of the extrados without break-
ing: also if the arch be of greater span than this, it will
bear any weight placed on those parts of the extrados from
which straight lines can be drawn through the voussoirs to
both abutments.
129. We will take the arch of greatest length under the
first conditions mentioned in the enunciation. Let F be the
highest point of the intrados (fig. 52.); then the tangent at F
passes through the highest point C and C' of the extreme
joints. Hence from any point G in the extrados a straight
line can be drawn to each pier lying wholly within the mass of
the voussoirs. This would not be the case if the arch were
the least portion longer without having the voussoirs propor-
tionably lengthened. For suppose the left hand abutment
(in the figure) had the position of the dotted line, then no
straight line can be drawn from a point c through the vous-
soirs to the right hand abutment.
It appears, then, that if two straight lines Ga, Gb can
be drawn from G through the voussoirs to the piers, the
tangent at F must be wholly in the voussoirs: and, this
being the case, any weight placed on G will be sustained, since
the portions of the arch on the right and left of G will act like
beams Gb, Ga, of which the points b and a cannot slip, be-
cause we suppose the friction of the voussoirs, or at any rate
the joggling, sufficient to prevent sliding.
104
STATICS.
The second part of the Proposition is evidently true after
what has been already written.
=
130. COR. 1. The principle of this Proposition seems to
have been used by Mylne in the construction of Blackfriars
Bridge, London, one of the arches and piers of which we have.
represented in fig. 53. AVY is a circular arc of which C is the
centre, and the radius 56 feet: OV height above low-water
40 feet: VK = 6 feet 7 inches. AB, YE are circular arcs
of radius 35 feet: ab 19 feet: YI about 8 or 9 feet. All
the joints are joggled: and a line from K to the middle point
of the joint YI lies wholly within the masonry; and does not
even pass near the extremities of the joints, so that chipping
of the voussoirs cannot take place. Therefore the portion YVL
cannot break however great the load on or near the crown of
the bridge, except by the crushing of the materials: and it
would require an enormous pressure on the haunches (near Y)
to raise the crown, since the weight of KY is about 2000 tons.
The tangent at Y falls within the foot of the pier F:
and the pier itself is like one solid mass by having the stones
and oaken planks below ab (low-water mark) well joggled, and
by having each of the voussoirs between Y and a projecting
over the one below it, and so giving each a firm hold of the
rubble-work in the centre of the pier (as represented in the
figure). The rubble-work itself is held down in its place by
the small inverted arch IG. Since, then, the tangent at Y
falls within the foot F of the pier, and the pier is as one solid
mass, the arch BAVE would stand of itself even were the other
arches to fall, since if KY were on the point of falling the
pressure would act through Y. This gives additional security
to the bridge.
131. COR. 2. From this Proposition we learn how it is
that the Gothic Arch will sustain such enormous weights upon
its crown, as we see is the case in many of our ecclesiastical
buildings. The stone steeple of St Dunstan's in the East,
London, is supported by four semi-pointed arches. In fact, it
is a principle that a pointed arch must have a great pressure
upon its crown to prevent its falling; for we may consider it as
consisting of the two extreme portions of a very large circular
arch brought together, so that the pressure on the crown must
DOME OR CUPOLA.
105
at least equal the pressure of the portion of the circular arch
which is removed. Flying buttresses always have a great
pressure upon their highest part.
But besides this the pointed arch, for the reason explained
in the Proposition, will sustain almost any weight on its crown
provided the lowest stones do not give way: and consequently
the Gothic arch is stronger for lofty buildings than the cir-
cular but the circular arch is far better adapted than the
Gothic arch for bridges, since the pressure of weights passing
over may act upon any part of the arch, not only on the
crown.
:
An arch built in a wall is almost sure to stand of whatever
form it be, so long as its foundation is firm: for suppose the
haunches were about to fall in, the crown rising; then, in order
that the crown may rise, the whole of the masonry or brick
work above the black line (in fig. 54.) must be moved upwards,
a weight sufficient to prevent the crown from rising. More-
over, suppose the crown would rise; then directly it had risen
and thrust the masonry above it through a small space, the
pressure which caused the haunches to sink will cease to act
in consequence of the dove-tailing together (so to speak) of
the stones or bricks, which lie above the haunches. In the
the same manner we see that the crown could not sink.
PROP. To explain the manner in which a dome is sup-
ported.
132. A Dome or Cupola is an assemblage of stones,
bricks, or other materials in equilibrium, of which the intrados
and extrados are surfaces of revolution having a common ver-
tical axis.
If we consider any given horizontal course of stones, it is
evident that this course cannot fall inwards, since all the stones
tend equally towards the center, and consequently wedge each
other in. But the form of the dome might be such that the
weight of the superincumbent courses should thrust out the
course under consideration and the courses below. In this
way, and in this way only, can the dome fall.
It is very easily seen that a conical dome is secure, and
will bear any weight on its upper course, provided the lowest
0
106
STATICS.
course is kept from bursting outwards. A dome with its con-
vexity inwards would be still more secure: for every stone is
pressed inwards, since it forms part of an arch with its con-
vexity inwards, and extremities in the highest and lowest
horizontal courses: consequently the stones of each horizontal
course are more firmly held together than in the conical dome.
133. The stone lanthern on the top of St. Paul's Cathe-
dral weighs several hundred tons, and is supported by a brick
cone, which is concealed between the outer and inner domes.
The lowest course of this cone is above the stone gallery at the
bottom of the outer dome, and is held from bursting outwards
by an iron chain.
The pyramids, which form the steeples of Gothic archi-
tecture, are for the same reasons as the cone stable in their
equilibrium. The enormous weight of these steeples is sup-
ported by very pointed arches, which spring from the square
tower at a considerable distance below the top, and are of such
a slight curvature that a straight line can be drawn from the
key-stone through all the stones of each leg (Art. 129, 131.):
and the thrust down these arches is counteracted by the massive
masonry of the tower and the buttresses (as explained in
Art. 122.)
In the walls and tower of Salisbury Cathedral are to be
seen some fearful cracks which seem to indicate a want of
sufficient support for the stupendous steeple which forms so
striking a feature of this edifice. The foundation remains
firm. Sir Christopher Wren examined these defects, and
found that the steeple was braced in different parts with iron
bars: and added more for greater security. A little less than
a century ago Price, the author of the British Carpenter, nar-
rowly inspected the whole, and seems to have proved, that the
cathedral was erected under two architects, one completing the
work that the other commenced; but that the first architect
never contemplated the erection of the steeple nor so lofty a
tower as the second architect was bold enough to add to the
low tower built by his predecessor; and in consequence of the
insufficiency of the supports the cracks now to be seen warned
the architect to resort to the expedient of bracers to hold the
base of the steeple from spreading. For a very interesting ac-
KING'S COLLEGE CHAPEL.
107
count of this building we refer the reader to Price's Series of
Observations upon the Cathedral-church of Salisbury. We
have mentioned these particulars because this building is a
very remarkable illustration of the necessity of attending to the
connexion between the weights and pressures in a building, and
the walls arches and buttresses by which they are to be sus-
tained.
134. As an example of the support of a stone vaulting,
we shall explain the manner in which the roof of King's College
Chapel, Cambridge, is supported.
Figure 55 represents a projection upon a horizontal plane
of one compartment of the roof included between the four but-
tresses f, g, h, k; and figure 56 represents the projection of
half this compartment upon the vertical plane of one of the
windows on the south side: the same letters in the two figures
refer to the same points.
The rib be runs from the east to the west end of the
Chapel, the stones which form it lie in the same horizontal
line and at a greater elevation from the ground than any other
part of the roof: K is the central stone of the compartment,
and is the upper part of one of the ornamented drops seen
hanging from the roof in the interior. The stones in a K´d lie in
an arch of which K is the key-stone: it is clear that the ten-
dency of this arch is to sink at the crown K, and thrust down
the walls at a and d. We shall proceed, then, to explain how
the stones in this arch are supported; and also the stones in
the rib be: and in the course of the explanation it will be seen
that we shew how every stone in the compartment fghk is
supported.
On examining the roof carefully it will be found that the
stones are placed in semi-arches in vertical planes through the
buttresses; the spring of all the semi-arches in the space ba
being at ƒ, and their crowns or key-stones in the courses bK
or Ka: this is best seen in figure 56. Now any stone s in
the arch a Kd is the key-stone of the two semi-arches sf and
sg and the thrust of the stones in K's is propagated down.
the semi-arches sf and sg, and ultimately acts upon the but-
tresses at ƒ and g; the same is true of every stone in Ka:
likewise on the other side of be the stones in Kd are supported
108
STATICS.
by the semi-arches, of which they are the key-stones, and which
spring from the buttresses h and k. Again, any stone r in Kb
is the key-stone of two semi-arches rk and rf, and is held in
its place by the thrust of the stones in Kr; and this thrust
is propagated down the semi-arches rk and rf, and acts ulti-
mately upon the buttresses k and f: the masonry of the rib be
is sufficiently heavy to prevent these semi-arches from sinking
by their key-stone rising. It will be clearly seen, then, how
every stone in be and a Kd is supported: it will also be
seen that every other stone in the roof is sustained by being a
member of a semi-arch springing from one of the buttresses,
and having its key-stone in be or a Kd. The pressure of the
compartment fghk upon the buttresses acts obliquely for
instance, that on f will act downwards in a line whose pro-
jection on the horizontal plane will lie towards the south-east.
But the compartment east of fghk will press upon the buttress
ƒ in a line whose horizontal projection lies towards the south-
west and consequently the resultant of these pressures will
act in a line whose horizontal projection runs due south: let
fF be this line (fig. 57.); this figure represents one of the but-
tresses. The dimensions of the buttress are so arranged that
ƒF shall lie within the masonry and pass into the foundation
within the foot of the buttress.
:
The resultant pressure of the roof on the walls at each of
the four angles acts obliquely; consequently instead of but-
tresses of the ordinary form at the four angles of the building,
towers crowned with lofty turrets are erected of such a weight
as to deflect the line of pressure of the roof, and cause it to
pass into the ground through the masonry.
135. We proceed now to find the position of equilibrium
of a chain suspended from two fixed points, and briefly to
explain the construction of Suspension Bridges.
A chain is an assemblage of rigid pieces of iron linked
together, or connected by pivots, as in the chains of suspen-
sion bridges. We may therefore apply the principles of
Chapter III. to determine the position of equilibrium of the
chain. The length of the chain is generally so great in com-
parison with the length of each link, that we shall 'suppose the
polygonal figure in which the chain hangs to be a continuous
COMMON CATENARY.
109
curve. Also we suppose that the motion of the links about
their points of connexion is perfectly free; or, in other words,
that the mutual action of any two links acts in a tangent line
to the curve in which the chain hangs. The curve in which
the chain hangs when in equilibrium is called the Catenary.
PROP. A chain of uniform density and thickness is
suspended from two given points: required to find the equa-
tion to the curve in which the chain hangs when it is in
equilibrium *.
136. Let A and B (in the plane of the paper, which is
supposed to be vertical) be the two points of support: fig. 58.
After the chain has ceased oscillating and has attained its
position of permanent rest, suppose ACB is the curve which
it forms, C being the lowest point: take this as the origin of
co-ordinates, CM vertical = x; MP horizontal= y; CP = 8;
P being any point in the curve.
Now the equilibrium of any portion CP will not be dis-
turbed if we suppose this part of the chain to become rigid:
this appears from Art. 71. Let and t be the lengths of
portions of the chain of which the weights equal the tensions
* We may calculate the form of the curve in the following manner.
Let us suppose the chain to consist of an infinitely great number of rigid and
straight portions, each equal to ds in length: and let of these portions lie between
C and P: fig. 58. then s=rds: also a, and a,-, being the angles which these por-
tions make with the axis of a, we have by Art. 123.
·
λ +
tan α, =
r + a
Hence d. tan PTM = tan a,
-
tan ar-1
1
tan ar
r + 3
dy
ds
dy
dx
or d.
;
dx
s + 33 5 s
tan ar-1.
dy
loge;
C
dey I d y d s
.*.
dx2
s dx dæ
dy
dx
=
ނ
S
C
loge dx
as in the text.
110
STATICS.
at C and P. Then CP is a rigid body acted on by three
forces which are proportional to c, t, s, and act respectively
in the directions Ce, Pt, Gs.
Draw PT the tangent at P cutting the axis of x in T.
Then the forces holding CP in equilibrium have their direc-
tions parallel to the sides of the triangle PMT, and therefore
bear the same proportion one to another that these sides do;
(see Art. 18.)
tension at lowest point
weight of the portion CP
PM
MT
dy
C
or
dx
S
ds
dy
2
1 +
dx
dx
dx
es
S
ds
c² + s²
... x + c =
=
√ c² + s²
√ c² + s²
ហ
S
(1),
the constant added being such that when ≈ = 0 then s =
the origin of co-ordinates is taken on the curve at C;
0, since
Also
s² = x² + 2cx
(2).
C
dy
dx S
C
;
√x² + 20x
20²
x + c + √ x² + 2 cx
.. y = c loge
(3),
с
the constant being so chosen that a and
so chosen that a and y vanish together.
This last equation may be put under another form
x + c
x + c
e
+
- 1,
C
C
x + c
then transposing
and squaring both sides of the equation
C
COMMON CATENARY,
111
2y
1 x + c
- 1;
с
··· x + c =
Also s =
012
y
{ec
+e
(4).
√√(x + c)² – c² by equation (2), - *-, ~14)
–
=
C
Y
{eč
e
(5).
Any one of these five equations may be taken as the equation
to the curve.
When the chain is uniform in density and thickness, (as in
the present instance) the curve is called the Common Catenary.
137. COR. 1. Of all curves of a given length drawn
between two fixed points in a horizontal line, the common
catenary is that which has its centre of gravity furthest from
the line joining the points.
For since the chain is in equilibrium the depth of its
centre of gravity from the horizontal line is a maximum or
minimum (Art. 78.) and it is clear that it is a maximum and not
a minimum, because if you displace the chain slightly it will
return to its position of equilibrium, or its equilibrium is stable
(Art. 79). Hence in any other position of the chain than that
of equilibrium the centre of gravity will be nearer the given
horizontal line. But the chain which hangs in the common
catenary is of uniform density and thickness, and therefore its
centre of gravity coincides with that of the curve: and con-
sequently the common catenary is the curve of the nature
described.
COR. 2. By means of the formulæ of Art. 87. Ex. 2. we
shall find that the co-ordinates to the centre of gravity from the
lowest point are
cy
X C
Cx
+
y = y
28
10
2
S
138. COR. 3.
We might have taken the origin of co-
ordinates at any other point than the lowest; as C', fig. 59.
112
STATICS.
༑'.
}
Let the tangent at C' make an angle a with the vertical.
We shall then readily get, if c' be used instead of c in the
Proposition,
$ C'T
C
é¯ C'R
sin (a - PTM)
sin PTM
sin a cot PTM - cos a
dx
=
sin a
cos a;
dy
dx
ds
11
1
1 +
dy
dx
2
s + c cos a
(s + c′ cos a)²+ (c′ sin a)²'
x + c'′ = √ s² + 2sc′ cos a + c′
We shall also find that
y = c' sin a loge
PROP.
19
x + c² + √ x² + 2c′ x + c²² cos² a
c' (1 + cos a)
To find the tension of the chain at any point.
139. Let t be the tension at P acting in the direction of
the tangent at P and estimated in terms of the length of chain
of which the weight equals the tension: then, by what was
mentioned in the last Proposition, (fig. 58.)
tension at P
PT
weight of CP
MT
t ds
1 0
S dx
But s² = x²+2cx, by equation (2) of Art. 136;
.. t = x + c.
This shews that the lengths of chain of which the weights equal
the tensions at the various points of the common catenary are
such, that if they were suspended from those points their lower
extremies would lie in a horizontal line.
For draw CE and PQ vertically downwards and equal to
c and a + c respectively: these then are the lengths of chain
which measure the tensions at C and P. But PQ = x + c
MC + CE, and PM is horizontal: therefore Q and E are in
the same horizontal line.
CATENARY OF EQUAL STRENGTH.
113
COR. 1. If a uniform chain hang freely over any two
points, the extremities of the chain will lie in the same horizontal
line when the chain is in equilibrium.
PROP. A chain of variable thickness, but of the same
material throughout, is suspended from two points: required
to find the law of the thickness that the tension at different
parts of the chain may vary as the strength of the chain at
those parts.
140. Let S be the length of a uniform chain of which the
thickness equals that at the lowest point, and weight equals the
weight of the length s of the chain to be suspended.
Let, as before, C be the lowest point (fig. 58.): CM= ∞,
MP = y, CP =s: c the length of uniform chain of the thick-
ness at C, of which the weight equals the tension at C. The
portion CP when it has assumed its form of equilibrium may
be supposed to become rigid. The forces which retain it in
equilibrium are its weight and the tensions at C and P, and
these are parallel to the sides of the triangle MTP;
and · PT
PT = √/PM² + MT² ;
.. tension at P =
C² + S
C
tension at C.
But the thickness of the chain at P varies ultimately as the
quantity of material in a given short length ds of the chain,
since the density is constant: it therefore varies as
dS
ds
But
by the hypothesis the tension must vary as the thickness of the
chain;
dS
ds
dS
c²² + S
varies as √²+ S² or
ds
C
since S and s are ultimately equal ;
S + √ S²² + c²
.'. s = c log,
(6).
Ր
P
114
STATICS.
C
MP
dy
ds
√ c² + S²
Also
S
MT
dx
dx
S
d S
c² + S¹²
But
ds
C
d S
c² + S²
dx
CS
or
dx
CS
d S
c²+ S¹²
.. x = c loge
√ c² + S¹²
C
(7).
Also
dy
dy dx
d S dx dS
c²
c² + S
ર
S
У
= c tan
- 1
or S = c tan
Y
с
(8).
C
141. These formulæ have been reduced to Tables by
Sir Davies Gilbert in the Philosophical Transactions for 1826.
We give the following extracts from them to elucidate the
application of the equations to the construction of Suspension
Bridges.

Y
TABLE I. The Common Catenary.
100.
e
2
18
S
t
Angle.
1000 5.004 100.166
980 5.106 100.173
1005.004
84° 16' 48"
985.106
84 9 49
420 11.961 100.947
400 12.565 101.045
380 13.234 101.158
431.961
412.565
76 29 6
75 49 22
393.234 75 5 35
SUSPENSION BRIDGES.
115

TABLE II. The Common Catenary.
c = 100.
Y
يم
ނ
$
t
Angle.
1
.005
1.000
100.005
89° 25′ 39″
10
.020
2.000
100.020
88 51 15
20
2.007
20.134
21
2.213
21.155
102.007 78 36 59
102.213 78 3 19

TABLE III. The Catenary of equal strength.
y = 100.
с
V
S
S
t
Angle.
1000
5.008 100.167 100.334
1005.021
81° 16′ 13″
980 5.111 100.174 100.348
985.124
8.1. 9 12
420
12.019 100.958 101.933
132.193
76 21 29
400
12.631 101.057 102.137
412.832
75 10 33
116
STATICS.

TABLE IV. The Catenary of equal strength.
c = 100.
y
X
is
S
t
Anglc.
1
.005
1.000
1.000
100.005
89º 25′ 37″
2
.020
2.000
2.000
100.020
88 51 14
20
2.013
20.135
21
2.221 21.156
20.271 102.034
21.314 102.246
78 32 23
77 58 4
To explain the use of these Tables we shall take an
example of each species of Catenary.
Ex. 1.
Let the span proposed for a Suspension Bridge
be 800 feet, and let the adjunct weight of suspension rods,
road-way.... be taken at one half of the weight of the chains:
and let it be determined to load the chains at the point of their
greatest strain, that is at the points of suspension, with one-
sixth part of the weight they are theoretically capable of
sustaining.
The modulus which measures the full tenacity of iron is
shewn by numerous experiments to be 14800 feet: this being
the greatest length of iron bar which another iron bar of equal
transverse dimensions will support without sensibly stretching.
Now this modulus must be reduced in the ratio 3 : 2,
since we have supposed the weight of the rods road-way... to be
equal to half the weight of the chains, and consequently we add
to the weight of the chains without adding to their strength.
The virtual modulus is therefore 9867 feet: and the tension of
the chain at the points of support is by hypothesis to
98676 feet 1611.5 feet.
SUSPENSION BRIDGES.
117
The semi-span is 400 feet.
In Table I. y is taken = 100
measures; therefore each of these measures is 4 feet: and the
tension at the points of support expressed in these measures
But by Table I. when t = 412,
=
1644.5÷ 4 = 411.124.
C =
400 measures =
1600 feet,
x =
12.565
50.260 ....
S = 101.045
404.180 ....
The angle of suspension = 75° 49′.
Having found the value of e we may make use of Table II. to
find the lengths of the rods for the different ordinates of the
In this Table c is taken at 100 measures, consequently
each measure equals 16 feet.
curve.
Each gradation of y in that Table will therefore be 16 feet;
and the second column gives the number of measures by which
the suspending rods corresponding to the respective values of
y must exceed the length of the suspending rod at the apex or
centre of the bridge.
Let the following Table be formed from Table II. by
taking the successive differences of the values of s :

1st measure of
Y.
length of arc of catenary
= 1.000 measures.
2nd ..
= 1.000
21st
= 1.021 ..
The last column of numbers gives the proportional part
of the adjunct weights which must be suspended from the
successive portions of the catenary, in order to distribute them
equally throughout.
118
STATICS.
In this example we have supposed the adjunct weights to
be equally distributed along the chain, so as virtually merely
to increase its uniform thickness. We shall now in
Ex. 2. Suppose the catenary to be one of equal strength:
i. e. the tension at every part proportional to the strength :
the other data the same as before.
In this case c represents the uniform tension on each
portion of iron throughout the chains whose transverse section.
equals that at the lowest point. In the uniform catenary the
greatest tension (that at the points of support) was found
equal 411.125 measures of 4 feet each: we shall take this then
for the value of c in the case of a catenary of equal strength.
Turning then to Table III. (in which, as before, each
measure is 4 feet) and taking the proportional part between
400 and 420, we have
x = 12.290 measures or
49.161 feet,
S = 101.002
S = 102.024
t = 423.602
angle = 76° 3′ 17″.
=
404.008
408.096
1694.408
We have taken c at 411.125 measures or 1644.5 feet, but
Table IV. is calculated for c 100: and therefore each mea-
sure of this table is 16.445 feet: and the second column deter-
mines the excess of length of the respective rods over that
at the apex for every gradation of y.
Let us form a Table, as before, of the differences of s
and S.
SUSPENSION BRIDGES.
119

Differences of s. Differences of S. Ratios of these Dif.
1st measure of y
2nd
1.000
1.000
1.000
1.001
1.000
.999
21st
1.021
1.043
1.002
The fourth column gives the quantity of matter of which
the chain must be composed at the various ordinates of
which the values are in the first column. Also the adjunct
weights of rods, road-way...should be distributed in portions
proportional to the numbers of the third column.
CHAPTER VII.
PROBLEMS.
142. In the last two Chapters we have illustrated the
principles of equilibrium by applying them to the solution of
various questions. Our object in the present Chapter is to
make some general remarks upon the solution of Statical
Problems, and to give a few more applications.
143. The conditions of equilibrium of a single particle
acted upon by forces which act in any directions, are three
in number,
Σ.Χ = 0, Σ.Υ = 0, Σ.Ζ = 0,
X, Y, Z being the resolved parts parallel to three rectangular
co-ordinate axes of any one of the forces: Art. 23.
If the directions of these forces all lie in the same plane,
and this plane be taken for that of xy, then the third equation
becomes identical and there are only two conditions. If the
forces all act in the same line and this line be taken for the
axis of x, then the last two equations are identical and there
is only one condition.
The conditions of equilibrium of a rigid body, or of a
system of rigid bodies, acted on by forces which act in any
directions, are six in number,
Σ.Χ = 0, Σ.Υ = 0, Σ.Ζ = 0,
Σ. (Zy - Y≈) = 0, E. (X≈ - Zx) = 0, E. (Yx - Xy) = 0,
X, Y, Z being the resolved parts parallel to three rectangular
co-ordinate axes of any one of the forces, and xy the co-
ordinates to the point of application of that force.
PROBLEMS.
121
If the forces all act in the same plane and this plane be
taken for the plane of xy, the third, fifth and sixth equations
become identical, and there are only three equations of
condition. Art. 65, 53.
:
144. When we wish to solve a statical problem we must
consider what forces act upon the body that is to be in equi-
librium for unknown pressures and reactions we must sub-
stitute unknown forces, which we shall call mechanical quan-
tities: also for unknown distances, angles of position, and so
on, we must use unknown quantities; these we shall term
geometrical quantities. After this we must write down the
equations of equilibrium, the number of which will depend
upon the nature of the problem, as mentioned in the last
article. We must next write down the equations (if there
be any) which connect the geometrical quantities. Lastly,
we must count the unknown quantities involved in the equa-
tions; and if their number exceed the number of equations,
it shews that the problem is indeterminate, or else that we
have not written down all the equations of condition: we
must therefore search for more; they must be equations con-
necting the geometrical quantities, since we know, by the
principles of equilibrium, that there cannot be any more
mechanical equations.
If in the end the number of equations be less than the
number of unknown quantities, then equilibrium will subsist
under several circumstances, and is said to be indetermi-
nate; it does not follow that all the unknown quantities are
indeterminate. If the number of unknown quantities equal
the number of equations, then equilibrium will subsist in one
way only. If it be found that there are more equations than
unknown quantities, then the equilibrium will not subsist
unless the known quantities fulfil the conditions at which we
arrive by eliminating the unknown quantities from the equa-
tions.
145. It will often happen that we can materially diminish
the labour of solving the equations by properly choosing the
centre of moments, and the lines parallel to which we resolve
the forces. Also by having regard to the object of the pro-
blem, whether it be to find the position of equilibrium of a
Q
122
STATICS.
body, the magnitude and direction of an unknown pressure,
and so on, we may frequently set aside some of the equations
as having no reference to the particular point of enquiry.
Thus in Art. 121. the object is to find T, the tension of the
tie-beam. Upon examining the four equations we see imme-
diately that (2) may be set aside, because it contains an un-
known quantity R, which does not enter any of the other
equations, and therefore (2) is of use solely to determine R,
a quantity which it is not the immediate object of the problem
to discover. Equation (1) gives T when P and are known,
and these are found from (3) and (4). Again, Art. 122. gives
a good illustration of an indeterminate problem. For (1) (2)
(3) are the only mechanical equations that can possibly exist,
and these contain only one unknown geometrical quantity a,
and consequently a fourth equation does not exist, or the
problem is indeterminate as we might easily have foreseen
from the nature of the case. It does not follow that every
unknown quantity in the equations is indeterminate, as we see
in this instance.
146. We shall now add a few Problems.
PROB. 1. A given weight W is held at rest on a known
curve AP lying in a vertical plane by means of a given weight
Q acting over the pully B: required the position of rest:
fig. 60.
The vertical BM through B is the axis of a, B the origin,
BM= x, MP = y, P being the position of the weight; angle
B = 0. Now the weight is held in equilibrium by Q acting
in PB, Win PW, and the reaction of the curve, or R, acting
in GR a normal to the curve at P: hence, resolving these
forces vertically and horizontally, Art. 23. gives
W - Q cos - R cos PGB = 0,
Q sin - R sin PGB = 0 ;
or, since tan PGB
dx
=
dy
dy
W - Q cos 0 – R
-
ds
= 0......(1),
PROBLEMS.
123
dx
Q sin - R
O
ds
two equations and five unknown quantities R, 0, x, y, s: since
equations (1) (2) are the only conditions of equilibrium, the
other three equations must be among the quantities 0, x, y, s:
they are
y
ds
tan 0 = 2/ ...... (3),
Vi
dy2
1 +
(4),
X
dx
dx²
and † (x, y) = 0…………….(5)
the equation to the curve. These five equations will solve
the problem when we select any particular curve.
The elimination of R from (1) and (2) gives
dy
W - Q (cos 0 +
sin 0) = 0,
dx
y dy
or W - Q
+
0 by (8): 1² = x² + y²,
r da
or Wda
Qdr = 0
the equation of virtual velocities which we should have ob-
tained from Art. 24.
Suppose the curve is a circle the centre being at a vertical
distance c from the point B: then a being the radius
y² + (x − c)² = aª ;
.`. r² = x² + y² = a² − c² + 2 c x ;
dr
C
•
dx
Q
༡.༠
Q⁹
C
W
C²
W?
.. X =
a² + c
j² −
2c
(Q² + W²) c² - W² a²
and the position of IV is known.
2 c W2
124
STATICS.
PROB. 2. A cord AAA...a is held at rest by forces
acting at its extremities and at the knots A¿Â½Ã……….in given
directions having given the form of the polygonal figure of
the cord required to find the relations of the forces; also to
find the tensions of the portions of cord: fig. 61.
1
1
The portions of cord need not be in the same plane; but
the force which acts at any knot, as P₁ at Д₁, must have its
direction in the plane of the portions of cord which join in A₁.
Let P, P...be the forces acting at the knots 442...: TT₁T2
...T, the tensions of the portions of cord: a₁ß₁, a2ß2,...the
angles which the directions of PP2...make respectively with
the portions of cord at the knots.
1
1
1
Then A₁ is held at rest by the three forces P₁T₁T;
hence, resolving these forces in the direction of P, and at right
angles to this, we have by Art. 23.
P₁- T cos a₁ T₁ cos ẞ₁ = 0.
1
T' sin a₁- T₁ sin ß₁ = 0 ..
Again, A, is held at rest by T₁ P₂ T₂; hence
1 2 2
P₂-T₁ cos a₂- T₂ cos ẞ₂ = 0.
2
1
1
T₁ sin a- T₂ sin B2
= 0
0 ..
(1),
(2).
(3),
(4),
2
P₁
and so on: if there be n knots we shall have 2n equations,
involving 2n + 1 unknown forces P1 P2 ... P₂ T T₁...... T₂:
we shall therefore have an equation of condition connecting
these forces, we shall suppose T to be known.
By equations (2) (4)
sin an
we have
T
T
sin an
sin ẞ₁'
T
2
sin az
T
ก
T₁
sin B₂
Tn-1
sin Bn
.. T₁ =
sin ai
sin Bi
sin αι sin az
T, T₂
T₂ =
T, and so on,
sin ẞ, sin Ba
2
and the tensions are all known in terms of T.
Also by (1) (2) eliminating T₁,
T sin ẞ₁
1
and in like manner
P₁
sin (a₁+ ẞ₁)
PROBLEMS.
125
1
T₁ sin B₂
2
P2
sin (a₂+ ẞ₂)
Hence all the forces P, P,
2
T sin a, sin Ba
B₂
sin ẞ, sin (a₂+ ẞ₂)
and so on.
are known in terms of T.
We shall now solve a few problems of forces acting on a
rigid body in the same plane: see Art. 52, 53. When the
system consists of more than one rigid body, we shall consider
each body separately.
PROB. 3. A uniform beam passing freely through a hole
H in a wall rests with one end on an inclined plane: find the
position of equilibrium: fig. 62.
AH horizontal=h,
A=a, PH=x, PG=a: AHP=0,
pressure at P=R perpendicular to the plane, pressure at H per-
pendicular to beam and = Q: resolving the forces vertically and
horizontally
IV – R cos a – Q cos 0 = 0 ..
(1),
R sin a - Q sin 0 = 0 ..
(2),
taking the centre of moments at P,
Wa cos 0 – Qx = 0 ..
(3),
these equations involve four unknown quantities R, Q, 0, x,
we must search for a relation between x and 0: this is
212
h
sin a
sin (a + 0)
(4).
:
Our object is to determine the position of equilibrium that
is, to find a and 0: we have one equation (4), we must there-
fore obtain another between a and by eliminating R and Q
IV sin (a + 0)
from (1) (2) (3).
By (1) (2), elim³. R,
Q
IV
: by (3)
sin a
Q
a cos
sin (a + 9)
sin a
V
a cos A
126
STATICS.
Eliminating a from this by (4) we have
h
cos → sin² (a + 0)
sin a,
a
from which 0, and therefore the position of the beam, is to be
determined.
If a = 90º, cos 0 =
3/h
a
PROB. 4. A sphere and cone in contact rest, as in fig. 63,
on two inclined planes, the intersection of which is a horizontal
line required the angle of the cone and the position of equi-
librium.
W, W' the weights of the sphere and cone: R the reaction
at B: P the mutual action at E: the resultant of the reactions
of the plane on the base of the cone must act at some point D,
let Q be this resultant: CD =x: G the centre of gravity of
the cone: rad. of sphere a, Ge≈, e being the point where
the normal at E cuts the axis of the cone: 20 = the angle of
the cone: a, ẞ the angles the planes make with the horizon.
=
For the sphere, W - R cos ẞ + P sin (a − 0) = 0 ………….. (1),
R sin BP cos (a - 0)
identical equation.
0.. (2).
The equation of moments is an
For the cone, W'- Q cos a
P sin (a - 0)
==
0 ... .... (3),
Q sin a
P cos (a - 0)
==
0.
(4),
0 = 0 ..
(5).
moments about G, Qa - P≈ cos
These five equations involve six unknown quantities: if there
be a sixth equation it must be a relation connecting the geome-
trical quantities involved in these five equations: but a little
consideration will shew us that no necessary connexion exists
between any two of a, e, ≈ hence the problem is indeterminate.
By examining the equations we perceive that the first four
involve only the four unknown quantities P, R, Q, 0: hence
PROBLEMS,
127
these are determinate; but a and ≈ are indeterminate since they
are connected only by (5): for any given position, however,
of the bodies is known by geometry, and consequently a
becomes known by (5).
We learn from this that if be chosen so as to satisfy
equations (1) (2) (3) (4), the bodies will remain at rest in
whatever position they are placed, their centres of gravity re-
maining in the plane of the paper and as we give the bodies.
different positions ≈ varies, consequently and therefore the
point of application of Q changes.
:
P
By (1) (2)
IV
sin B
cos (a + ß - 0)
P
sin a
; by (3) (4)
IV'
cos
W
sin a cos (a+ß-0) sin a
{cos (a+B)+sin (a+ß) tan 0};
W
sin ẞ cos 0
sin B
I
B
sin ẞ - IV' sin a cos (a + ß)
.. tan
sin (2a + B)
sin a sin (a + B)
X
By (4) (5)
sin a
| ટૈર
The value of tan
IV' sin a sin (a + ß)
(I + IV') sin ß
W' sin a sin (a + ß)
sin a cos
cos (a - 0)
cos a + sin a tan ↔
II' sin a sin (a + B)
I'{cos a sin (a+ẞ) − sin a cos (a+ß) } + IV sin ß
WV' sin a sin (a + B)
(I + IV') sin B
gives the angle of the cone necessary for
equilibrium, and the value of a gives the point of application
of Q for any given position of the bodies.
PROB. 5. A person suspended in a balance of which the
arms are equal thrusts his centre of gravity out of the vertical
by means of a rod fixed to the furthest extremity of the beam
of the balance, the direction of the rod passing through his
centre of gravity: given that the rod and the line from the
nearer end of the beam of the balance to his centre of gravity
128
STATICS.
make angles a, ẞ with the vertical, shew that his apparent and
true weights are in the ratio sin (a + ß): sin (a – ß).
PROB. 6. A uniform beam placed in a hemispherical
bowl is in equilibrium, find its position.
PROB. 7. A cylinder with its axis horizontal is supported
on an inclined plane by a beam which rests upon it and has its
lower extremity fastened to the plane by a hinge: find the
conditions of equilibrium.
PROB. 8. Two uniform beams of equal length are loosely
connected, each by one extremity, to the extremities of another
uniform beam, they are then placed on a sphere; find the
pressures on the sphere at the three points of contact, the
length of the middle beam being less than the diameter of the
sphere.
PROB. 9. To determine the conditions of equilibrium on
Roberval's Balance; see Art. 99. and fig. 35.
This machine consists of five rigid bodies; and since the
forces all act in the same (the vertical) plane we shall have
fifteen equations: the figure will point out the meaning of the
various unknown quantities, the description of which we omit
here to save room.
The equilibrium of the part supporting Q gives
Q - R cos - R′ cos 0′ = 0.
(1),
R sin - R' sin O' : 0 ..
(2),
Qr – Rb sin 0 = 0 .
(3).
The equilibrium of the bar CC' gives
V cos - R cos
S' cos & = 0 ...
(4),
V sin + R sin ◊ – S sin 0.
Ra sin (a + 0) — Sa′ sin (a − q) = 0
The equilibrium of the bar DD' gives
V'cos - R' cos e'- S' cos d' ' = 0
V' sin '+R' sin '- S′ sin p' = 0
R'a sin (a − 0′) – S'a' sin (a + p')
(5),
.... (6).
0 ..
(7),
0.
(8),
(9).
PROBLEMS.
129
The equilibrium of the part supporting P gives
P - S cos o -S' cos q′ = 0 ..
S sin o S' sin d'= 0..
Ps - Sb sin = 0
(10),
........ (11),
(12).
The equilibrium of the stem and stand gives
W - T + V cos y + V' cos y′ = 0.
(13),
V sin
V' sin y' = 0..
(14),
Tx - V' (h + b) sin '+ Vh sin y
(15).
These equations contain 15 unknown quantities, namely, RR'
V V' SS' T☺ ☺'† 'oo'x and the ratio of Q to P.
Some
Q
of these must be indeterminate since (as we might have foreseen)
(14) is a consequence of (2) (5) (8) (11).
To obtain the ratio
P
Q
By (1) (2)
R
sin (0 +0')
sin o'
by (10) (11)
S
P_sin (p+g')
sin o'
Q sin (4 + 0')
R sin o'
a' sin (a − p) sin o'
by (6).
P sin (0+0')
S sin O'
a sin (a + 0) sin ('
If we had eliminated R and S first and then R' S',
Q sin (p + p')
R' sin
P sin (0 + 0′)
S' sin 0
a' sin (a + p′) sin o
a sin (a – 0′) sin ✪
by (9).
Adding these equations after multiplying them respectively by
the denominators of the right-hand sides we have
Q sin (+0)
{sin (a + 0) sin &' + sin (a -') sin 03 P sin (0+0')
a {sin (a − p) sin q'+ sin (a + p') sin p} ;
a
…. sin a sin (0 + 0′).
Q sin (p + p')
P sin (0+0')
a'
sin a sin (p + ');
በ
R
130
STATICS.
:
Q
a'
,
P
a
that is, the weights must always be inversely as the arms DE',
D'E', and do not depend on r and s.
To find T.
Add together (1) (4) (7) (10) (13) after changing the signs
in (4) (7), we have
T W+P + Q.
To find a.
By (14) (15) Tx Vb sin
=
=
P ( s − 2 ) ;
Ps - Qr by (5) (3) (12) = P (s
a
P
W+P + Q
α
(s
r).
α
This shews that as we shift the weights P and Q the point
B, at which the reaction and consequently the resultant down-
ward-pressure acts, shifts also. If the ratio of r and s be
such that B is at C, then if P be shifted outwards or Q inwards
the balance will fall moving about the point C. If the stem be
fixed of course the balance will not fall; but then the strain
upon the stem will change as we shift P and Q. The strains
at the pivots are indeterminate, nevertheless they alter as P
and Q are shifted.
In this way the paradoxical character of the balance is
explained.
We shall illustrate the Principle of Virtual Velocities in
the solution of the following problem.
PROB. 10. A beam in a vertical plane rests
and against a wall at A, as represented in fig. 64
circumstances of equilibrium.
on a post B
required the
Distance of B from the wall b: AG = a:
-b: AG = a: ▲ GAD = 0.
The reaction (P) of the post at B is perpendicular to the
surfaces in contact, and therefore to the beam: the reaction
(R) of the wall is perpendicular to the wall for the same
reason: W the weight of the beam. We may consider the
PROBLEMS.
131
beam in equilibrium under the action of P, R, W, and suppose
the post and wall removed.
Now the object of the problem might be solely to deter-
mine the position of equilibrium, or also to determine P and
not R, or R and not P, or to determine both P and R and also
the position of equilibrium. We shall solve the problem by
the Principle of Virtual Velocities under these four suppositions
in order to explain the method of proceeding so as to avoid
as much trouble as possible according to the nature of the
question.
1. Suppose the position of equilibrium only required.
We must then give the beam a small arbitrary geometric
motion such that the unknown pressures P and R shall not
occur in the equation of virtual velocities: the beam must
therefore remain in contact with the wall and the post: as in
fig. 64.
Let de be the increase of owing to the displacement.
Then height of G above the horizontal through B (or h)
=
GB cos 0 = (a –− b cosec 0) cos s Ꮎ = a cos 0 -b cot 0;
.. vertical space described by G = dh = (
b
2
sin Ꮎ
and by virtual velocities W&h= 0;
.. b − a sin³ 0 = 0, sin 0
3
b
α
and this determines the position of equilibrium.
a sin Ꮎ | dᎾ,
e)
2. But suppose we wished to find the pressure P as well
as the position of equilibrium.
We ought in this case to have moved the beam off the
post, as in fig. 65, in order that the virtual velocity of B with
respect to P may not vanish, and consequently P not disappear
as in case (1).
Let AA'= c, and let, as before, de be the change of 0.
Then the space described by B in direction of P's action,
(since BP is perpendicular to AB) equals the difference of
the resolved parts of AA' and A'B' in the direction of P
AA′ sin ◊ – A'B' cos (90° – d0), A'B' = AB = b cosec
= c sin - b cosec 0 80.
132
STATICS.
Also space described by G in direction of W
= AG cos 0 — AA' – A'G' cos (0 + SO)
= a cos 0) — c — a cos ✪ + a sin 0 d0 = a sin 0 80 − c ;
therefore by the equation of virtual velocities
W (a sin 080-c) + P(e sin - b cosec 080) = 0;
.. 80 (Wa sin ✪ – Pb cosec 0) — c (W – P sin 0) = 0 ;
and since c and d may be any independent small quantities
Wa sin 0 - Pb cosec 0
=
0, W - P sin ◊) = 0 ;
3
b
P
3
... sin
and
√
a
a
W
b
3. Suppose we wished to know R and the position of
equilibrium, and not P.
Then we should give the beam such an arbitrary motion
(fig. 66.) as to give A a virtual velocity with respect to R, but
not one to B with respect to P. Let AA' = c, BAA' = a;
c sin a
AB - c cos a
с
b
sin a sin ; and the virtual vel. of G
= AG cos 0 − c cos (0 − a) –
A'G' cos (0 + 80)
2
= ( sin³ > – sin 0)
c sin a
-
c cos a cos 0;
Ꮎ
and virtual velocity of A= c sin (0 − a);
a
{( sin² 0 – sin ) e sin a
.. W
WC sin
c cos a cOS
Cose}
+ R (c cos a sin ◊ − c sin a cos 0) = 0 ;
a
. W
sin² ✪ – sin ✪
sin e
•)
R cos 0 = 0,
Ꮎ
W cos 0-R sin 0 = 0;
3
... sin 0
band
R
að - bš
a
W
b3
4. Lastly, suppose we wished to determine P and R and
the position of equilibrium.
}
PROBLEMS.
133
Then we must give the beam the most general disturbance ·
possible in the plane of the forces: fig. 67.
AA′ = c: BAA′ = a: and 80 the increase of 0;
=a:
. vir. vel. of A with respect to R = c sin (0 − a),
Ᏼ ...
G..
W {a sin 0.80 – c cos (0 − a)}
b80
P = c sin a
sin
Wasino. 80 — c cos (0 − a);
+ P (e sin a
b80
+ Rc sin (0 − a) = 0 ;
sin
.. c sin a (P - W sin - R cos 0) — c cos a (W cos 0 - R sin ()
い
​b
-de Wa sin ◊ – P
= 0;
sin Ꮎ
and c sin a, c cos a, and 80 are independent;
.. P
P - W sin – R cos 0 = 0…………(1),
W cos e
R sin 0
= 0......(2),
Wa sin - Pb cosec 0 = 0…………..(3).
These three equations are the equations which we should
have obtained by the principles of Art 53. they give by
elimination
sin
=
اشد
3
P a
R
√ a
as - b3
,
a
W
IV
b 3
We have thus illustrated the method of application of
this principle: and we observe, in general, that when the
object of the problem does not require certain unknown forces
we must give the body the most arbitrary geometrical motion
possible without giving the points of application of these forces
any motion in their direction.
The first case of the four just solved is an application of
the principle proved in Art. 78. and which was deduced from
the principle of virtual velocities. We may determine whether
134
STATICS.
the equilibrium be stable or unstable (Art. 79.) by differen-
tiating h a second time:
dh
b
d2 h
2b
a sin 0;
d Ꮎ sin20
d Ꮎ?
sin Ꮎ
3
+a).
cos 0,
dh
which is negative when
0: hence h is a maximum and
de
the equilibrium is unstable.
We may frequently make use of this method to discover
the nature of the equilibrium.
PROB. 11. A body with a convex surface rests on a fixed
body with a convex surface: required whether the equilibrium
is stable or unstable: fig. 68.
Let CAO be a normal to the two surfaces at the point of
contact A of the two bodies when the upper body is at rest:
then the centre of gravity of the upper body is in that line: let
C be its distance from O the centre of curvature at A: let a
and b be the radii of curvature at A of the curves in which the
plane of the paper (supposed vertical) cuts the bodies: dis-
place the upper body through a very small angle as in the
figure: angle C=0:
... h dist. of cen. of grav. from horizontal through C,
=
= (a + b) cos 0-c cos (0+ A'O'B), A'O'B
A'B ав
Ꮎ
b
b
=
(a + b) cos 0 − c cos (1 +
-
(1 + 1/1)
α
Ꮎ
=
=
(a + b) (1
(1-9)
02
b
b?
C
{ (a + b) − 1 = (a + b) ² } /*
2
Hence h is a maximum or minimum, or the equilibrium is
unstable or stable, according as c is < or
or as AG is > or < (b − c or)
b2
a + b
ab
a + b
We shall close this Chapter with a few examples of
Problems in which Friction is considered. The only change
PROBLEMS.
135
will be that we must substitute some unknown force for the
friction acting at right angles to the pressure; if we suppose
the parts acted on by friction to be on the point of slipping,
this force. P, where P is pressure of the rough surfaces
and μ a constant known by experiment: see Art. 118.
PROB. 12. A cylinder with its axis horizontal is held at
rest on an inclined plane by a string coiled round its middle
and then fastened on the plane; fig. 69: find the conditions of
equilibrium friction being considered. The forces act as drawn
in the figure.
The conditions of equilibrium are
W – R sin a – F cos a
R cos a F sin a
Tcos (9 + a) = 0…………….(1),
T' sin (0 + a) = 0…………….(2),
moments about the axis, Ta - Fa = 0…………….(3),
these are the only equations; and they contain four unknown
quantities R, T, F, : but we know that F cannot be greater
than μ.R: this limits the indeterminateness of the problem.
COS a
F
Eliminate T from (2) (3);
R
sin a + sin (0+ a)
cos a
sin
a + sin (0 + a) cannot be less than
M
COS a
μ
sin a
ль
+ a cannot be less than sin-¹
but it may be greater.
PROB. 13. A cylinder lies upon two equal cylinders all
in contact and having their axes parallel: and the lower
cylinders rest on a horizontal plane: u' the coefficients of
friction respectively between the cylinders and each cylinder
and the plane find the conditions of equilibrium, and the
relation of μ and ' that all the points of contact may begin
to slip at the same instant: fig. 70.
u
M
The forces as in the figure.
The upper cylinder, W-2R cosa - 2 F sin a = 0......(1),
the other two equations of this cylinder are identical.
136
STATICS.
One of the lower cylinders,
W' – R' + R cos a + F sin a = 0......(2),
-
F" – R sin a + F cosa = 0………
(3),
(4),
F' - F = 0 ...
these are all the equations.
F
sin a
α
By (3) (4)
= tan
,
not greater than
μ.
R
1 + cos a
By (1) (2) 2R' = 2 W' + W,
by (1) (3) (4) F'
W sin a
W
α
tan
(1 + cos a)
2
૪ ૦૫
α
F"
W tan
2
R'
2 W' + W'
not greater than μ'.
If
ль
=
' then since W is less than W+2 W' the lower
cylinders will slip first as we continually increase the weight of
the upper cylinder. In order that the points of contact may
all slip together, we must have
α
W tan
α
2
tan
2
M
μ and
2 W' + W
μ
W'
M
W
PROB. 14.
2 μ
Three equal rough rods are loosely connected
together by one extremity of each, and placed on a rough
horizontal plane. Shew how to graduate one of the rods so
that by noting the position of a smooth ring resting in a
horizontal position on the rods and just in equilibrium we may
know the coefficient of friction between the rods and the plane.
CHAPTER VIII.
ATTRACTIONS.
147. THE phenomena of the motion of the heavenly
bodies lead us to conjecture, as we shall hereafter perceive,
that the various particles of matter in the universe attract each
other with a force which varies directly as the mass of the
attracting particle and inversely as the square of the distance
of the attracted from the attracting particle. Now in antici-
pation of this it will be an interesting and useful enquiry to
calculate the resultant attraction of an assemblage of molecules
which constitute a mass such as the Earth, the Sun, or any of
the heavenly bodies. We shall commence with the calculations
of the attraction of homogeneous bodies bounded by surfaces
of the second order, and then of any homogeneous bodies
differing but little in figure from a sphere, and lastly of hete-
rogeneous bodies consisting of homogeneous strata all differing
but little from spherical shells in their form. Also in the course
of these calculations we shall introduce a few Propositions
which we shall find of use hereafter.
PROP. To find the resultant attraction of an assemblage
of particles constituting a homogeneous spherical shell of very
small thickness upon a particle outside the shell: the law of
attraction of the particles being that of the inverse square
of the distance.
148. Let O be the centre of the shell (fig. 71), P any
particle of it, dr its thickness: C the attracted particle OC=c,
¿ POC = 0. OP=r: m PMn a plane perpendicular to OC,
≤ m MP = 0, PC = y.
ОР
The attraction of the whole shell C acts in CO.
S
138
STATICS.
Let OP revolve about O through a small angle de in the
plane MOP: then rde is the space described by P. Again,
let OPM revolve about OC through a small angle do, then
r sin edo is the space described by P. Likewise the thickness
of the shell equals dr. Hence the volume of the elementary
solid at P equals drrdersindo ultimately, since its sides
are ultimately at right angles to each other.
Then, if the unit of attraction be chosen to be the attrac-
tion of a unit of mass at a unit of distance, the attraction of
the elementary mass at P on C in the direction CP
pr² sin Odrdedo
y²
, p the density of the shell;
... attraction of P on C in CO
=
pr² sin @drded cr cos 0
Ꮎ
Ꮎ
y²
We shall eliminate from this equation by means of
Y
y² = c² + p²
2cr cos 0,
do
Y
y² + c² — p²
... sin
c - r cos e
s Ꮎ
dy Cr
20
prdr
2 c²
c²
2.2.
+
dydo.
y²
.'. attrac. of P on C in direct. CO:
To obtain the attraction of all
tegrate this with respect to
and 2π, those of y being c
-
... att". of shell on C in CO =
the particles of the shell we in-
and y, the limits of
r and c + r;
prdr
c + r 2π
202
πρrdr
C²
προdr
ㅠ
​c²
C² -
being o
S" (1+= "") dydo
c - r 0
c+r
C²
y²
S" (1 + 2 =) dy
C
(2r + 2r)
mass of the shell
=
y²
4 π pr² dr
T
૨
c²
c²
ATTRACTION OF SPHERICAL BODIES.
139
This result shews that the shell attracts the particle at C
in the same manner as if the mass of the shell were condensed
into its centre.
149. It follows also that a sphere which is either homo-
geneous or consists of concentric spherical shells of uniform
density will attract the particle at C in the same manner as
if the whole mass were collected at its centre.
PROP. To find the attraction of a homogeneous spherical
shell of small thickness on a particle placed within it.
150. We must proceed as in the last Proposition: but
the limits of y are in this case r c and r+c: hence
p² – C²·
attraction of shell - prd (1~-~~) d
πρrdr
c²
y²
dy
x + c
π p r d r
c²
(2c - 2c) = 0;
therefore a particle within the shell is equally attracted in every
direction.
PROP. To find the attraction of a homogeneous spherical
shell on a particle without it; the law of attraction being re-
presented by (y), y being the distance.
151.
The calculation is exactly analogous to that of
Art. 148 we have only to alter the law of attraction: then
attraction on C in CO
r
π îṛ d ¹ ƒ© + ' (y² + c² − r²) $ (y) dy, (integrated by parts)
c²
πprdr
c²
πрrdr
c²
{ (y² + c² − r²) ƒ¢ (y) dy – 2 [[y fp (y) dy] dy}
·
{(y²+ c² − r²) Pı (y) − 2¥ (y) + const.} suppose
between the specified limits
Je+r
1
= 2 = pr dr {C++" qp, (c+r) — — 4 (c+r)
C
d
−
= p, (e−r) + 4 (e−r)}
C
r) = 4 (c − r)}.
− −
= 2 w prdr & { 4 (c +
de
C
1/2
140
STATICS.
this latter form being introduced merely as an analytical arti-
fice to simplify the expression.
PROP. To find the attraction of the shell on an internal
particle.
152. The calculation is the same as in the last article
except that the limits of y are r
c and r + c;
r + c
1
... attraction = 2π prdr
+
C
C
C
+ " = ° ₁ (r− e) + 4 (r - c)}
P₁ (r + c ) − — 1 ¥ (r + c)
C
1
C²
= 2π prdr
d ff (r + c)
= =
− 4 (r − c)
dc
C
The formulæ of these two Articles will give the attraction when
the law of attraction is known.
Ex. 1. Let (r) ==
4
·´`· P1 (~)
+ A
A
↓ (r) = -
r+
p² + B: A and B arbitrary constants;
therefore attraction on an external particle
= 2πprdr
d {-
dcl
4r + A {(c + r)² – (c − r)²}
d 2r
Ar
= 2 = p r dr = { = = " + 24r}
dcl c
2c
- =
4πpr²dr
(see Art. 148.)
C²
Attraction on an internal particle
=
= 2nprdr
d c
{
d (− 4c + A {(r + c)² − (v − e)
>
2c
ATTRACTION OF SPHERICAL BODIES.
141
d
2 π pr dr
dc
{−2+2 Ar}
= 0, (see Art. 150.)
Ex. 2.
Let $(r) = r ;
-· · •; (1) = + 4, 4 () = + = r² + B.
P1 22²
2
Attraction on an external particle
8
d (c + r)¹ − (c − 1º)² + 4 A { (c + r)² − (c − r)²}
= 2 x prdr = {(c+r)²
dc
d
=
2 x pr dr de
=
dc
{c²r + m² + 2 Ar}
4πpr²drc = mass of shell × c.
8c
The attraction is the same as if the shell were collected at its
centre. This property we discovered for the law of the inverse
square. We shall now ascertain whether there are any other
laws which give the same property.
PROP. To find what laws of attraction allow us to
suppose a spherical shell condensed into its centre when
attracting an external particle.
153. Let (") be the law of force: then if c be the
distance of the centre of the shell from the attracted point and
r the radius of the shell, and (r) = f{r fp (r) dr} dr, then
the attraction of the shell
= 2 w p r d r = {4 (0 + 1 ) = 4 (0 - 1)).
d (c − (c − r
dc
C
But if the shell be condensed into its centre this attraction
4 π r² drpp (c);
d (c − − r)
... del
• 2rp (0) = 2 {4 (0 + 1) = ↓ (c = 1)}}
C
142
STATICS.
2rp(c)
.. 2rp (c) = 2 dc
d
f
d f c r
d³ & c p³
1
+
de c
dc3 c 1.2.3
· 2 rp (c) + 2
d f1 ď³ (c)
&
de e
cr
d (ď³ ↓ (c) p³ 1
dc dc³ c 1.2.3
(+)=0
de³
c
d f1 f c
}
+
=0 whatever r be;
d c
2 [1ove) = 0, 1 (1040) -
de³
de
= 0...
d ↓ c
But
= cfp (c) dc,
ď² & c
/ф (c) dc + сфо
dc
dc²
афс
dc³
3
2pc + c
dc
therefore by the first of the above equations of condition for
† (c)
2
афс
фот = 3A,
de
and multiplying by c² and integrating
c³p (c) = Ac³ + B: A and B being independent of c
B
$ (c) = Ac + ă,
and this satisfies all the other equations of condition for † (c);
therefore the required laws of attraction are those of the direct
distance, the inverse square, and a law compounded of these.
PROP. To find for what laws the shell attracts an in-
ternal point equally in every direction.
154.
When this is the case
d ff (r + c) ~ † (r – c)
= 4
C
=
de
d↓ (r) d³ ↓ (1)
C²
+
+
A
dr
d 23
1.2.3
whatever c is, A being a constant independent of c;
ATTRACTION OF SPHEROIDS.
143
dy (r)
A,
dr
ď³ & (1)
dp3
= 0......
.......
These conditions are all satisfied if the first is: this gives
−
A
rfp (r) dr = − A, 4 (r) = =—,
و 200
and therefore the inverse square is the only law which satisfies
the condition.
PROP. To find the attraction of a homogeneous oblate
spheroid on a particle at its pole: the law being the inverse
square of the distance.
155. Let APBp, AQB q be sections of the spheroid and
the sphere touching it made by a plane through the axis of
the spheroid: fig. 72. AM = x, MP=y, AC = c, CD = ɑ,
c = a(1-e), e very small. The mass of the annulus Pp
between the sphere and spheroid and of thickness da
a²
πpy'da (1-2):
dx
: also AQ=√2cx; y² =
(2cx − x²),
-
c²
α
and if we consider every particle of the annulus Pp equidistant
from A, the attraction of this annulus on A in direction AB
C
1
X
==py'dz (1-0) 208 √201
пру
a²
2cx
πρε
(2c) 3
(2 c – x)dx ;
therefore attraction of whole difference of sphere and spheroid
2 π р € ( ²° ( 2 c x š
2πρε
4c
(2 c xì – x4) dx
-
(2c)*
{
4 c
16πρε
3 51
15
4 прс
(Art. 149.)
3
Απρ
the attraction of the sphere on A =
therefore attraction of spheroid on A
=
CO
1 +
46
5
C.
$
144
STATICS.
PROP. To find the attraction on a particle at the
equator.
156. Let DC be the axis of revolution (fig. 73), APBp
and AQBq sections of the spheroid and circumscribing sphere
by the plane of the paper passing through the axis of revo-
c²
lution: AM = x, MP=y, AC = a, CD=c, y²=
AC=a, → (2ax − x²).
α
Let an elementary slice of the spheroid and sphere be made
by planes perpendicular to the axis of x, one passing through
P and the other at a distance da from it; therefore mass of the
part of this slice between the sphere and spheroid
= πp(QM² – MN.PM) dx
= πρ
пр
2
y².
a
C
yda; because MN = QM
(2 ax - x²) dx.
(1-2) (2 ax
Now the distance of each portion of this from A nearly = AQ
√2ax; therefore attraction of the part between the sphere
and spheroid in the direction AC
=
2a
(1 − 2 ) 2 (2 ax - x²)
ƒª²ºª
a
0
2α 1
x d x
(2 a x) *
= πρ
=
πρ
=пр
0
(2a) *
(1-2) (4a - 10)
3
5
and the attraction of the sphere
- xp (1-2), (80)
(2 ax✈ − x*) d x
dx
4 a
πρα 6
15
Απρ
α a;
3
Απρ
therefore attraction of the spheroid
3
Απρ
3
(1-20) 0
5
(+)
5
α
C.
ATTRACTION OF SPHEROIDS.
145
157. In the same manner it might be shewn that the
attractions of a homogeneous prolate spheroid of small ellip-
ticity on particles at the pole and equator are respectively
Апр
3
(1 - 4€)
c and
5
АПР
3
3 €)
C₂
5
2c being the axis of revolution of the spheroid.
PROP. To find the attraction of a homogeneous oblate
spheroid upon a particle within its mass: the law of attrac-
tion being that of the inverse square of the distance.
158. Let a, c be the semi-axes, the minor-axis of 2c
coinciding with the axis of : then the equation to the spheroid
from the centre is
x² + y²
a²
+
1,
2
fgh the co-ordinates to the attracted particle: we shall take
this as the origin of polar co-ordinates, A in fig. 30.
↑=
radius vector of any particle of the attracting mass :
→ = angle which makes with a line parallel to ≈ :
4 = angle which the plane r≈ makes with the plane ≈≈:
.. x = ƒ + r sin 0 cos p, y = g + r sin 0 sin 0, ≈ = h + r cos 0,
and the equation to the spheroid becomes
(ƒ + r sin @ cos p)² + (g + r sin@ sin ()²
+
(h + r cose)™
C²
1,
f sin cos + g sine sin hcos
a
sin20
cos20
or 2.2
+
+2r
C²²
a²
f + g
h
= 1
a
+
C²
cos e}
sin20
cos Ꮎ
put
= K,
a²
f'sine cos +g sine sin
a²
T
h cos
+
F
146
STATICS.
=
H, then
h2
and F + K (1-+-) -
a²
K²² + 2 KFr + F² = H
and the values of are
F+√H
K
-F-H
and "
K
ρ
Volume of element at Prsin@drded as in p. 68.
let p be the density of the spheroid: then the attraction of
this element on the attracted particle is p sin@drdėdė; and
the resolved parts of this parallel to the axes of xyz are
р
2
sin² e cosparded, psin esingdrd0dp
and psine cos@drd@dp.
Let A, B, C be the attractions of the whole spheroid in
the directions of the axes estimated positive towards the centre
of the spheroid: then these equal the integrals of the attrac-
tions of the element; the limits of r being – 7' and ''", of 0
being 0 and, and of being 0 and hence.
π:
π
П
A:
ρ
p sin²0 cos drdedp,
B =
-LIF
Π
Π
p sine sin drdedp
and C =
///
0
0
Then A
=
р
π
Π
π
P
sine cose drdedp.
(?" + r'´) sin² 0 cospd0dp
2p
Π
0
0
ПF
0
K
sin²0 cos&d0d¶.
Now it is easily seen that if R (sina, cos²a) be a rational
π
function of sina and cos² a then SR (sina, cos² a) cosa da 0.
0
Wherefore by substituting for F and K we have
ATTRACTION OF SPHEROIDS.
147
A = 2 fpc"
= πfpc²
Π
0
0
[
0
T
π
" fpe³ [*
c²
0
2.
Tsin³0 cos³odedo
c² sin² + a² cos20
3
sin Ꮎ d Ꮎ
c² sin² + a² cos² 0
(1 - cos²0) sin edə
Ꮎ
c² + (a² c) cos20
a² sin
*
>
==fp === √. {~ + (-) cose - sine de
πήρ
= π
περι
a²
C²
2
C
a² - c²
a²
√ a² - c²
between specified limits,
C
tan
-
.1
(a² c²)
a²
C
C²
cose)
cose + cose +
C²
a²
a² – c²
√ a²
= 2πfp
a² - c² \c √ a²
tan-1
= 1 − e²
a² - c²
C
a²
1 −e²
e
1 - e²
2 пfp
tan-1
e³
3
1 - e²
e²
1 - e²
e
ПЃР
sin-¹e -
e3
€²
2
In the same manner we should find that
B = 2x8p VT ==
e²
1 − e
sin-¹e
e³
e
πρ
F
Also C = 2p
sin cos ded
K
π
Tsin cos 0d0d p
0
c'sin*0 + a cos² 0
*If the spheroid be prolate c is >a and the denominator of this must be written.
c² – (c² — a²) cose, and the integral would involve logarithms instead of circular arcs.
148
STATICS.
=
a²
π
c² sin 0
- 2x ph. {sine - + (-e) cos'
a² c²
c² (a² c²)
&
do
a²
C
- 4πρη
tan-1
√ a²
a²
C²
√a
a²
C²
C
=
приве
1
e³
-
- e²
sin - 1
159. COR. 1. We see from these expressions that the
attraction is independent of the magnitude of the spheroid,
and depends solely upon the eccentricity.
Hence the attraction of the spheroid similar to the given
one and passing through the attracted particle is the same as
of any other similar concentric spheroid comprising the at-
tracted particle in its mass. Hence a spheroidal shell the
surfaces of which are similar and concentric, attracts a point
within it equally in all directions.
160. COR. 2. If we put the ellipticity of the spheroid
= e and suppose e very small so that we may neglect its
c²
square, we have e² = 1
1 −
€)²
(1 − e)² = 2€ ;
a²
4πρ
2 €
. Á
f,
B
Απρ
2 €
g,
3
3
5
4€
161. COR. 3.
h.
C = ATP (1 + 2010) M.
Απρ
3
5
By the values of A, B, C after integrating
with respect to » we have
+
(c² sin³ + a² sin e cos 0) dedo
c² sin" + a² cos²
B
C
+
2p S* S***
h
Ꮎ Ꮎ #про
A
f
But if V = ſ
= 2p ff sin dedo = 2πpf sin 0d0 = 4πр.
element of mass
distance from attracted point
dm
{(x − ƒ)² + (y − g)² + (x − h)² }
(≈ ! *
ATTRACTION OF SPHEROIDS.
dv
df
d² V
d A A
f
dm (x −ƒ)
{(x −ƒ)² + (y − g)ε + (≈ − h)²}
by the form of A.
df2 df
df f
dev
B
de V
C
In the same manner
do
g
dh²
h
Hence for an internal particle
2
d² V
d² V
d² V
+
+
πρ.
df
dg
dh²
A ;
149
162. COR. 4. If we had taken an ellipsoid instead of a
spheroid we should have had
3ƒM
A
L. B
a³
a³
3 g M d (XL)
αλ
3h M d (XL)
C =
a❞
dx
a² - b²
a² - c²
where M is the mass of the ellipsoid, X²
12
a²
a²
and L = √
J
x² dx
1 - X
the integration of this depends upon the properties of elliptic
transcendants: see Legendre's Traité des Fonctions Ellip-
tiques, Tome 1, p. 545.
163. COR. 5. If we wished to find the attraction on an
external particle we should have the same integrals for A, B, C
as in the Proposition, but the limits of would be r' and '”
(and not - ' and r"), since the point from which is measured,
the attracted particle, is outside the spheroid;
.. A
pf
π
sin² cos pdrd0dp
2
= pƒƒ„" (r'-7″) sin cos pd0dp
=
2p S** S*
√H
K
sin" e cos &ded,
and this cannot be integrated by any known method.
150
STATICS.
Mr Ivory has, however, discovered a relation between the
attractions of ellipsoids on external and internal particles: so
that by means of this relation we can calculate the attraction
on an external particle.
PROP. To enunciate and prove Ivory's Theorem.
164. Let +
22
yo
a² b2
C²
1 and +
α B²
x² y²
2
+
2
r
be the equations to the surfaces bounding two homogeneous
ellipsoids having the same centre and foci: then
a² - b² = a² -ß², a³-c² = a² - z²
(1).
Let fgh, f'g'h' be the co-ordinates to two particles so
situated on the surfaces of these ellipsoids that
f
a
f'
। ૪
a
,
مة أمة
b
h
C
B
h
Y
(2).
Also since (fgh) and (f'g'h') are points in the surfaces of the
first and second ellipsoids respectively, we have
ƒ¹² g
a
h2
+
b2 C
1,
-2
に
​+
h'2
+
B
1 ………….. (3).
Y
Then the attraction of the first ellipsoid parallel to the axis
of z on the particle situated at the point (f'g'h') on the sur-
face of the second is to the attraction of the second ellipsoid on
the particle situated at the point (fgh) on the surface of the
first in the same direction as ab: aß the law of attraction
being any function of the distance: and similarly with re-
spect to the axes of y and z. This is Ivory's Theorem.
We shall, for convenience, represent the law of attraction by
the function ro (~²), ↑ being the distance.
The attraction of the first ellipsoid on the particle (ƒ'g'h')
parallel to the axis of ≈
p Sff (h' −≈) ¢ { (f" − a)² + (g' − y)² + (h' − 2)²} dx dydz,
ATTRACTION OF SPHEROIDS. IVORY'S THEOREM. 151
the limits of x are
CA
√1-2-2 and e√1-2-3
y²
a²
b2
cA
x²
y²
a²
29
the limits
the limits of y are
X
C
b
and b
a²
of x are
a and a
℗ SS {4 [(ƒ' − x)² + (g′ − y)² + (h′+ z)²]
− † [(ƒ' ~ x)² + (g' − y)²+ (h' — ≈)']} dxdy
between the specified limits: (r) = 2 fp (r) dr: it must be
remembered that in this expression = c√ 1
J
XC²
y²
a²
be,
we
do not substitute this value merely for preserving the functions
under as simple a form as possible.
== ct, then the attraction
Now put a
ar, y = bs,
= pab ſſ {† [(f' – ar)²+ (g'− bs)²+ (h'+ ct)³]
− † [(ƒ − ar)² + (g′' — bs)² + (h'−ct)²]} drds,
the limits of s being - √√√1 – 7² and √1 - 7², and those of r
being - 1 and 1: also t = 、 √1 − 1² − s².
=
=
=
12
Now (f' - ar)²+ (g' − bs)² + (h'±ct)²
12
· ƒ¹² + g²² + h²² — 2 (far+g'bsh'ct) + a²² + b²s²+ c©t°,
substituting for h' by (3) and putting 1-s for t
= f¹²
12
- ƒ³ ( 1 − 2 ) + 8'' ( 1 − 3 ²) + ~ ²- 2 (far + g'ba±k'et)
a²
y²− g′bs
+ (a³ − c°) r² + (b² − c²) s² + c² :
;
eliminating f'g'h' by (2) and making use of (1)
a²
(a³ — c²) +
62
(bº − c²) + c² − 2 (far + gßs ±hyt)
+ (a² — y³) ‚¹² + (B² — y³) s² + y²
152
STATICS.
=
· ƒ² + g²² + h² − 2 (far + gßs ± hyt) + a²r¹² + ß² s² + y°ť² by (3)
−
· (ƒ − ar)² + (g − ẞs)² + (h ±yt)².
Hence the attraction of the first ellipsoid on (f'g'h') parallel to ≈
= pab ff {↓ [(ƒ- ar)² + (g - ẞs)² + (h + yt)²]
- [(far)² + (g − ẞs)² + (h - yt)"]} drds,
—
the limits of s being -√1-2, 1-; of being - 1, 1
r², r
ab
αβ
× attraction of second ellipsoid on (ƒgh) parallel to ≈:
the same may be proved for the attractions parallel to the other
axes and consequently the Theorem, as enunciated, is true.
We observe that one of these ellipsoids lies wholly within
the other for if not the points in which they cut each other
lie in the line of which the equations are
y² སྣ་
x²
a²
+
b2 C²
1 and +
y
+
1.
2
α B2
We shall suppose a less than a the points of intersection must
therefore satisfy the equation
x²
1
y²
~ ( - ) + ( − 1 ) +
2
α
and this by (1) becomes
b2
Y
2
(-)
= 0,
2
2
(-)² + (₁₂)² + (-)²-
αα bß
0, an equa-
tion which can be satisfied solely by x 0, y = 0, ≈ = 0: but
these do not satisfy the equations above, and therefore the
surfaces do not intersect in any point.
Hence to find the attraction of an ellipsoid of which the
semi-axes are a, b, c on an external particle of which the co-
ordinates are f'g'h', we must first calculate the attraction of an
ellipsoid of which the semi-axes are a, ß, y parallel to the
axes on an internal particle of which the co-ordinates are f, g, h,
these six quantities being determined by the equations
a² - B² = a² - b²,
a² — y² = a² — c²,
12 h'2
2-4-0-6, 5 + 6 + 5 -1,
a²
B2
ATTRACTIONS.
153
f
ɑf"
a
bo'
ch'
h
В
γ
сто
and then the attractions required will be these three calculated
attractions multiplied respectively by
b c
a c
ab
αγ
ав
By' ay
The following Proposition we shall find of use in a sub-
sequent part of this work.
PROP. To prove that the resultant attraction of the par-
ticles of a body of any figure upon a body of which the distance
is very great in comparison with the greatest diameter of the
attracting body, is very nearly the same, as if the particles
were condensed into their centre of gravity and attracted
according to the same law, whatever that law be.
165. Let the origin of co-ordinates be taken at the centre
of gravity of the attracting body, the axis of a through the
attracted particle; let c be its abscissa and ay≈ the co-ordinates
of any particle of the body, p the density of that particle.
',
Then the distance between these two particles, or ",
√ (c − x)° + y² + ~.
Let ro() be the law of attraction: then the whole at-
traction parallel to the axis of x, or A
SSS p (c − x) & (c² − 2cx + x² + y²+ z²) dx dydz,
the limits being obtained from the equation to the surface of the
body
= [[] p c − x) { † (c²) − (2 c x − x² − y³ — *) 4′ (c²) + ... } dxdydz
$
x
− — ...'}
$ (c²)
=c$(c) []]p{ 1 − = { 1 +
2 c² p´(c²)
1+
+ (y² + ≈² − x²)
C
p' (c²³)
$ (c²)
...
+ dxdydz
y²+ x²x²
dx d y d z +
Mc¢ (c²) + c³q'(c) [[[e
M being the mass of the body: also fpxdxdydz = 0 since a
is measured from the centre of gravity of the body (p. 67).
U
154
STATICS.
Now suppose xyz to be exceedingly small in comparison
with c; then all the terms of A after the first are extremely
small in comparison with that term, it being observed that
c³p' (c²) is of the same order as cop (c²) in terms of c. Hence
the Proposition is true.
COR. It appears also that to produce a given resultant
law, the law of attraction of the constituent molecules must be
the same.
166. We shall now proceed to the calculation of the at-
traction of bodies differing but little from a sphere in figure.
The object of these calculations will be seen when we come to
the higher branches of Physical Astronomy. The reader may
therefore, if he please, omit the remainder of this Chapter till
he enters upon those investigations. We shall suppose that the
law of attraction is that of the inverse square of the distance.
PROP. To obtain formulce for the calculation of the
attraction of a heterogeneous mass upon any particle.
167. Let p be the density of the body at the point (xyz):
fgh the co-ordinates of the attracted particle: and, as before,
suppose A, B, C are the attractions parallel to the axes of
Then
X, Y, Z.
A
B =
= SSS
SSS
C=
=
SSS
p (fx) d x d y d≈
{ (ƒ − x)² + (g − y)² + (k − ≈)² } } ?
p (g − y) dx dydz
{(ƒ − x)² + (g − y)² + (h − ≈)²? *'
−
p (h− z) dx dydz
{(ƒ − x)² + (g − y)² + (h − 2)²? * '
the limits being determined by the equation to the surface of
the body.
p d x d y d z
Let V = SSS {(ƒ − a)² + (8 − y)² + (k − ≈)*} ¹ '
(g
(h
d V
d V
d V
A
B =
C
df'
dg'
dh
ATTRACTION OF BODIES NEARLY SPHERICAL. 155
It follows, then, that the calculation of the attractions A, B, C
depends upon that of V. This function cannot be calculated
except when expanded into a series: it satisfies a differential
equation, which leads to some remarkable properties of the
coefficients of the terms of the series into which V is deve-
loped. We proceed to determine this equation.
PROP. To prove that +
d2V d2V
+
0, or
4 πρʹ,
d² V
df d g² dh²
according as the attracted particle is not or is part of the mass
itself: p' being the density of the attracted particle in the
latter case.
168. By differentiating V we have
d V
df
d2 V
df"
-p(fx) dxdydz
SSS = (@~ =
{(ƒ − x)² + (g − y)² + (h − ≈)² } } '
SSSP
{2 (f
{
2 (ƒ − x)² − (g − y)² – (h − ≈)²} dx dydz
{ (ƒ − x)² + (g − y)²+ (h − x)²} ?
In the same manner we should have
d² V
dg2
dev
SSSP
{2 (g − y)² — (ƒ − x)² – (h − z)²} d x d y d z
− — — —
(f
{(ƒ − x)²+ (g − y)° + (h − ≈)²}§
ρ {2 (h − ≈)² − (ƒ − x)² – (g − y)²} dvdydz
ƒfƒ p
{(ƒ − x)² + (g − y)² + (h − ≈)² } ?
5
dh
−
-
−
—
−
d² V
d² V
+
ď v
+
df
dg
dh
SSS
0 × d x d y d z
{(ƒ − x)² + (g − y)² + (h − 2)²
。
When the attracted particle is not a portion of the attracting
mass itself then ays will never equal fgh respectively and
consequently the expression under the signs of integration
vanishes for every particle of the mass :
de V
df
ď v
+
d Ꮩ
+
0.
dg dh
This equation was first given by Laplace and Poisson was
the first who shewed that it was not true when the attracted
156
STATICS.
The error arises in
particle is part of the attracting mass.
consequence of the expression under the signs of integration
not vanishing for all values of xyz; since it equals when
x = ƒ, y = g, z = h.
=f,
d² V
d² V
To determine the value of
+
df²
dg²
d² V
0
+ in this case,
dh2
suppose a sphere described in the body so that it shall include.
the attracted particle: and let VU+ U', U referring to the
sphere, and U' to the excess of the body over the sphere.
Then, by what is already proved,
d² U'
d U'
d U'
+
+
= 0.
df2
d ove
dh²
de V
d² V
d² V
d² U
d² U
d² U
Hence
+
+
+
+
df²
dg
dh²
df²
dg2
dh2
The centre of the sphere may be chosen as near the attracted
particle as we please, and therefore the radius of the sphere
may be taken so small that its density may be considered ulti-
mately uniform and equal to that at the particle (fgh), which
we shall call p'.
Let f'g'h' be the co-ordinates to the centre of the sphere;
then the attractions of the sphere on the attracted particle pa-
rallel to the axes are, by Art. 149, 150,
Απρ
3
Απρ
Απρ
(ƒ −ƒ'),
(g — g'),
(h - h'),
3
3
or
-
d U
df
,
de U
d U
d U
by Art. 167.;
dg
dh
d'U
d2 U
+
4 πρ' ;
df2
dg²
dh2
d² V
+
ď V
+
df² + dg²
when the attracted particle is within the attracting mass.
Since the attracting body is supposed to be nearly spherical
we shall find it most convenient to transform our rectangular
dev
Απρ,
dh
ATTRACTION OF BODIES NEARLY SPHERICAL. 157
to polar co-ordinates, the origin of the radius vector of the
surface being near the centre.
PROP. To transform the partial differential equation in
V to polar co-ordinates.
169. Let r, 0, w be the co-ordinates to the point (fgh),
r', O', w'
(xyz),
the angles and 'being measured from the axis of : w and
w' being the angles which the planes on which and 0' are
measured make with the plane za; as in fig. 30. and at p. 67,
being replaced by w;
.. ƒ = r sin 0 cos w,
g= = 7' r sin & sin w,
h = r cos 0,
X
r' cos e'.
x = r′ sin l′ cos w', y = r' sin e' sin w',
Also the volume of an element of the mass-dr'r' de'r' sin 'do';
therefore V = the sum of the elements of the mass divided re-
spectively by their distances from the attracted particle
=S"S" S
0
0
pr'² sin 'dr'de'dw'
{r²+r²²−2rr' [cos e cos 0'+ sin e sin e' cos (w-w')]}
r being the value of r at the surface of the body.
,
h
Now p² · ƒ² + g² + h²,
h²,
cos
s Ꮎ
tan w =
,
ƒ²² + g² + h²
2014
& ... (1)
f
dV
dV dr dV do
d V dw
+
+
df
de V
df
dr df de df
d dV dr d dV do
+
df dr df df de
+
d dV dw
df df dw df
,
dw df
dV ď²r
dV ď² 0
d V ď² w
+
+
+
dr df²
do df
dw df"
ď² V dr²
dev de
d² V dw²
+
+
dr² df²' do² dƒ˜²
dw² d f
158
STATICS.
de V dr do
+ 2
+2
+2
drde df df
d² V dr dw
drdw df df dede df df
d² V do dw
dV d²r
dV ď² 0
d V ď² w
+
+
+
dr df²
də df²
dw df2
d² V
d² V
The expresssions for
and
are of the same form.
dg²
dh²
These must all be added together and equated to zero or
- 4p. When this is effected the formulæ (1) make
d² V
dr²
dr² dr²
the coefficient of
+
+
1,
dr²
df² dg² dh2
d² V
d Ꮎ
d 02
d02
1
the coefficient of
+
+
d02
df²
ď² V
dw² dw2
2
dg²
dh2 ༡•?
>
dw²
1
the coefficient of
+
+
dw²
df²
dg
dh²
² sin² '
d² V
dr de
dr do
dr de
the coefficient of
2
+2
+2
0,
drdo
df df
dg dg
dh dh
d² V
dr dw
dr dw dr dw
the coefficient of
2
+2
+2
0,
drdw
df df
dg dg
dh dh
d² V
do dw
do dw
do dw
the coefficient of
+2
+2
0,
d0d w
df df
dg dg
dh dh
d V
d² r
ď² r
dr
the coefficient of
120
+
+
dr
df"
dg²
dh2
J
d V
d Ꮎ
d² 0 d Ꮎ
cos
the coefficient of
+
d Ꮎ
df²
dg
do²
+
?
dhⓇ
, sin Đ
r
d V
the coefficient of
dw
Hence the equation in V becomes
ď w
dg²
+
dh2
Ζω
+
0.
df²
1
d² V
인
​Q
dw
= 0 or -4π p';
ď² V
2 d V
1 dᏙ
cose dv
+
+
+
+
dr²
r dr
12 do²
sin e do r² sin
ATTRACTIONS.
159
LAPLACE'S COEFFICIENTS.
d²r V
d² V
cos o dv
1 d² V
+
+
+
dr²
d 02
= 0 or -4π pr².
sin Ꮎ ᏧᎾ sin2 0 dw2
Put cos 0 = μ and cos 0' =µ: then
V=
SLS
1 2π
0
predr'du'da
r²+r'² −2rr'[µu'+√√1−µ³ √1−u”²cos (w−w')]}§ '
dr V
d
dv
1
d² V
and r
+
-
(1 − µ²)
+
= 0 or 4π pr².
- Απρ.
dr²
du
du
2
1 - μ² dwⓇ
By integrating this equation we should determine the value
of V. But this has never been accomplished, and we are con-
sequently obliged to resort to approximation by series.
PROP. To explain the method of expanding V in a series.
170. The expression
12
J
{x² + r²² – 2rr' [µµ' + √1 - μ² √1 - μ
-
may be expanded into either of the series
12
cos (w - w')]},
1
༡་
Po
+ P
19
+
1
or P
P₁ = +
+ P₁
i |
where Po, P₁,
functions of μ,
201
+ P
+
pl¿ + 1
(1),
+ P₂
+
jo? +1
Pi,
are all rational and entire
µ,
I-2 cos w, and 1-2 sin w, and the
12
same functions of μ', √1-μ cos w', and
VI
the general coefficient P; is of i dimensions in u,
and √1 - μ sin w.
12
1- µª² sin w' :
1-μ² cos w
2
The greatest value of P; (disregarding its sign) is unity.
For if we put
2
12
µµ' + √ 1 − µ³ √ 1 − µ³ cos (w – w') = cos 4 = ½) (≈ + 1),
- – 1
ર
160
STATICS.
then P
=
coefficient of c in
C
(1 + c² – 2c cos p)−¹ or (1 − cx) − (1 − )
coefficient of c² in
Z
1.3
+ 3/2 c x² +
2.4
-3
ଓ
2
+
1 + + 3/1/20
2.4
1.3 c²
+
......
A
1
− 1 ( ~ ' + 1 ) + B (x²-² + -—-—-)
=
+
2 A cos i +2 B cos (i − 2) † +
A, B...... being all positive and finite: the greatest value of
this is when = 0: hence P; is greatest when = 0.
But then P = coefficient of c in (1 + c²− 2c)¯½ or (1 − c) −¹
2
1.
(1 + C + c² + ... + C² ...)
1
Hence 1 is the greatest value of P. It follows that the first
or second of the series (1) will be convergent according as r is
less than or greater than »'.
Using the first series we have
r
1
2π
2°
1
V
pr' {P₂+P₁+P₂]
1
+
20
+ P¿ ¿¡+ …... } dr'dµ'dw'.
1 0
We substitute this in the equation in V of Art. 169, and then
remove the powers of outside the signs of integration and
equating the coefficients of the same powers of " on each side
of the equation we have a series of equations of which the
general one is
1 2π
d
1 ď²P;
dr'dµ'd w
+
du 1-μ" dw
2
2
+i (i+1) P;
1) P;
=0,
SSS Plan (1-4) dP)
0
du
excepting the single case when i2 and the particle is in-
ternal, in which case the second side is - 4πp'.
ATTRACTION OF BODIES NEARLY SPHERICAL. 161
It follows, then, that in every case
d
du
{
dP
1
d² P;
i
(1
μ²)
+
du
1
-
2 dw²
μ²
+ i (i + 1) P; = 0.
2
For this is evidently true in every case but the excepted
one mentioned above; and in that case it is equally true, for
if the multiplier of did not equal zero, the definite integral
ጥ
would be infinitely great (instead of equalling -4πp) since
can become infinitely small. When P, is determined from
this equation then V will be known.
171. This equation we meet with very frequently in the
higher branches of Physical Science. It has never yet been
integrated except by series. Laplace has demonstrated various
properties of the integral, and Mr Murphy has effected the
same by a new analysis: Treatise on Electricity. We shall
call the functions which satisfy this equation Laplace's Coeffi-
cients of the first, second,...orders according as i = 1, 2,...:
and the equation itself the Equation of Laplace's Coefficients.
PROP. To calculate the value of V for a homogeneous
sphere.
172. Let the sphere be referred to polar co-ordinates the
centre being the pole (fig. 71): C the attracted particle ;
OC=r; P a particle in a shell of the sphere of which the radius.
OP
=r₁, ≤POC=0, ≤PMm = w; a the radius of the sphere:
then PC = √² + r₁² − 2rr¸ cos§; and the mass of the element
at P = pr₁² sin@dr, dedw, the limits of w are 0 and 2π; of
are 0 and π of " are 0 and a;
2
2
ri
1
1°1
2
2π
a
π
pr² sine dr,ded w
sindr₁
.. V
0
0
- 2 πρ
=
= 2 пр
・ログ
​a
0
p² + r² = 2rr, cose
2
r² sindr₁ də
√²+1,2 − 2rr, cose
2
›² + r¸² − 2rr, cose + const.} dr₁
X
162
STATICS.
a
=
Σπρ
{(r + r₁) = (r − r) } dr₁
when C is without, and + when C is within the shell,
4π P
α
2
r² dr、
4πρα
3r
2
0
when C is without the sphere.
And when C is within the sphere, the part of V for the
shells which enclose C = 2πp2r₁dr₁ = 2πp (a² — r²): and
the part of V for the other shells of the sphere
Απρ
4πρη
r₁dr₁
3
2°
0
4 πρα
Hence V
for an external particle
3r
2 προ
V = 2 πρα
for an internal particle.
3
We shall find the use of these in the next two Pro-
positions.
PROP. To find the attraction of a homogeneous body,
differing little from a sphere in form, upon a particle
without it.
173. Since the attracted particle is without the attracting
mass we must expand V in a descending series of powers of r:
let V = +
0
1
V2
+ +
Vi
+
jo? +1
+....
But by Art. 170. taking the second of expansions (1),
2′2
r
1
V
- SS S P { P
2π
2013
p{P + P₁
0
0
202
µ¿+2
P;
+ ... + P₁₂+1 + ... } dr'd µ'dw';
2π
0 ρι
i+
ATTRACTION OF BODIES NEARLY SPHERICAL.
163
Let the mean radius of the body a: and let a (1+ay')
be the variable radius, y' being a function of ' and w', and a
being a very small numerical quantity whose square and higher
powers are to be neglected.
Then, for the excess of the attracting mass over the sphere
of which the radius = a, the value of v¿
+3
2
i+3
2π
= aa²+³ pƒ ƒ²,ƒ³™ P¿dý dµ'd' = aa²+³µ‚ƒ P₁y'dµ'dw'.
Let f₁f P¡ý'du'dw' = U¿,
hence for the excess over the sphere we have
3
a
√ =
αρ
a¹
Vo+ U₁+...
༡༧
ai +3
+
U; +
x² + 1
But for the sphere of which the radius is a, V
by Art. 172.
Hence for the whole mass
V
4πρα αρα
3r
2°
a
+ {U。 + − U₁ + ...
d V
グ
​(see Art. 167.)
+
1
U; + ...}
3
4πρα
31
and the attraction
>
dr
Απρα αρα
2 a
+
Здо?
{Vo+
(i + 1) ai
0
U₁ +......+
U₂ + ... } .
2°
PROP. To find the attraction of a homogeneous body,
differing but little from a sphere in form, upon an internal
particle.
174. We must in this case expand V in an ascending
series of powers of r let
V = v。 + v₁r + v₂r² +...
+......+v₂n² +...
But by Art. 170.
V=
v = [ [ [ [ " p { Por² + P }
p{Por'+P₁r+P₂¬,+...+P P₁ =
ipi-
み
​ן
+…..} dr'dµ'dw';
164
STATICS.
p P
i
.. Vi
-
SS SO R
dr'dµ'dw',
L L
-1
zoʻ ¿ — 1
and as in the preceding Proposition the value of v; for the
excess of the attracting mass over the sphere of which the
radius is (a)
αρ
2π
P₁y'du'd w
αρ
- [ [ " = "0.
a
i – 2
a
Also for the sphere of which the radius = a the value of V
is 2π ρа²
2 π pr²
П
V = 2 пра².
3
(Art. 172). Hence for the whole mass
2 π pr²
and the attraction
3
+ apa² { U¸ + − U₁ + ... + √ į U₂ + ... }
a
a
d V
dr
4πρη
2r
ip-1
3
apa {U₁ +
a
a²
U₂ + ... + -i-1 U₂ + ...}.
175. The calculation of the functions U, U₁,...U¿‚…..
can be effected without integration when we know the equation.
to the surface of the body. We proceed to demonstrate this
in the three following Propositions.
PROP. To prove that a function of μ,
u
2
2
1- u² cos w
and √1-μ² sin w, as F(u, w), can be expanded in a series
of Laplace's Coefficients: provided that F(µ, w) do not be-
come infinite between the values - 1 and 1 of μ, and 0 and
2π of w.
176. Let μ'u + √ 1
by Art. 170.
12
2
µ² √1 - µ² cos (w' – w) = p: then
ль
(1 + c² − 2cp) − ¹ = P₂ + P₁c + P₂c² +...+P¿c² +...
c being any quantity not greater than unity.
Differentiating with respect to c,
p-c
(1 + c² − 2cp)*
----
1
P₁ + 2 P₂c + ... + ¿P¿ci−¹ +...
EXPANSION IN A SERIES OF LAPLACE'S COEFFICIENTS. 165
Multiply the latter equation by 2c and add it to the former,
1 - c²
(1 + c² − 2cp)³
-1 JO
= Po + 3P₁c + 5 P₂c² + ... + (2i + 1)P¿c² +...
0
~2π (1 − c²) F' (u', w') du' dw'
(1 + c² − 2cp);
2
0
2 π
{P₂+3 P₂c+...
+ (2i + 1) P¿c² + ...} F' (u', w') du' dw'.
Now c being arbitrary we may put it
= 1. Then the
fraction under the symbols of integration on the left-hand
side of this equation vanishes, except when p= 1, in which
case the fraction equals - We proceed, then, to determine
the value of the integral, we shall call it X.
When p=1 then

1- μ' μ
cos (w' - w)
´(1 − µ'²) (1 − µ³)
10
ૐ
1-2μμ + μ μ
12
2
1
µ² + μ²² µ²
12. 2
μμ
and that this may
2
μ
= w.
not be greater than unity we must take
µ²² + µ³ not greater than 2u'u, or (u' ~µ)² not greater than
zero: hence μ' = µ, and therefore cos (w' — w) = 1 and w'
We shall therefore put uμ+v and ww+ and 1-c=g:
v, being small quantities which vanish when p = 1, and g
a small quantity which vanishes when c =
= 1. Then we have
1-c² = 2g, p = 1 − (1 − u²)
2 (
(1 − µ³) ³
in which we retain only the lowest powers of g, ≈, v which
occur, since the higher powers must ultimately vanish in
comparison with the lower:
1 - c²
1 – c²
(1 + c² − 2cp)³
{ (1 − c)² + 2c(1 − p)}}
2g
{² +
1
u²
2
+ (1 − µ³) ≈² } }
Also since, by hypothesis, F(u', w') does not become
infinite between the specified limits of u' and w', then
166
where
STATICS.
F (u', w') = F (u + v, w + x) = F (u, w) + §
is a very small quantity which vanishes with and ≈:
then after the substitutions
ע
X = 2F (μ, w)
{g² +
1
ν
-
gd v d z
2
+ ≈² (1 − µ³) } §
+2
e /
Egdvdz
L2
{0.2 +
+ ≈² (1 − μ² ) } }
1 μ
between proper limits.
or g
V
Now, since when c = 1
1 or g = 0 the fraction under the
symbols of integration in X vanishes for all values of and ≈
that are not indefinitely small, it follows, that we may choose
limiting values for v and ≈ of any magnitude that we please,
if we put g = 0 after the integration.
O after the integration. We shall take - h and
h for the limits of and
−
ע
co and co for the limits of x.
Put s² (1 − x²) = (g² + ~~,;) a²º ;
1 μ
therefore the first term of X
2
h
2F (u, w)
g √ 1 – u² dv
da
·h g² (1 − µ³) +
D²
∞ (1 + x²)};
h
= 4 F (u, w)
& Vi
u2 dv
h
-7
g2 1
μ²) + v²
8F (µ, w) tan¯¹
g
=
= 4πF (µ, w), when g= 0, or c = 1.
Now being a very small quantity, let ẞ be its greatest
value within given small values of v and then the second
term of X is less than
В
--
- h
✔
Z:
gdvdz
2
μ2
2
+ ≈² (1 − µ²) }
EXPANSION IN A SERIES OF LAPLACE'S COEFFICIENTS.
167
or is less than 2ß: but ẞ ultimately vanishes and therefore
Thus we have determined the value of
X = 4π F (μ, w).
the integral;
1
.. F(μ, w) =
1
2π
[ [ ...
4. π
1
{P + 3 P₁ + 5 P₂ +
+ (2i + 1) P; + ...}F (u', w') du' dw'.
1
2 i + 1
2 π
Now the general term
P₁F (u', w') du'dw'
Ап
0
W
is a function of μ and w which satisfies the equation of Laplace's
Coefficients (see Art. 171.)
Hence F(u, w) can be expanded in a series of Laplace's
Coefficients.
177. If w or u have one of its limiting values 0 and 2π
2″
1 and 1, we must not take the limits of integration which
we have used in the last Article.
or
ω
ле
For it is only for the values of w' and ' which differ by
indefinitely small quantities from w and that the fraction
under the symbols of integration in X does not vanish. Now
if w = O then w' 0 and 2π; and therefore it is for indefinitely
small positive or negative values of w' and for values a very
little less and a very little greater than 2 that X does not
vanish: but the negative values and the values greater than
2π are not included between the specified limits (0 and 2π)
of and must therefore be left out of consideration, though
this was not necessary in the general case, because the fraction
X then vanished for these values.
Hence if w = O the value
of X is found by putting = 0 and integrating with respect
to ≈ from 0 to ∞, and adding to it the result of putting
2π and integrating from x = ∞ to ≈ = 0: we then have
w
w'
X = 2 π { F (µ, 0) + F (µ, 2π)} = 4π F (µ, 0) and 4πF (µ, 2π)
for F(u, 0) = F (μ, 2π)
since F(u, w) is a function of μ,
1-cosw, and 1-u'sin w.
The same will be the result if w have its other limiting
value 2π.
168
v
STATICS.
Again, suppose μ 1; then u' 1, and we must not
take the limits of negative since cannot be less than -1:
but since in this case w' is quite indeterminate we shall refer to
the original form of X. Since F(u', w) is a function of ',
√1-u cos' and 1-2sino' which does not become infinite
between the specified limits of u' and w', it follows, that F(-1, w')
is independent of w': hence, since pu' when u=-1, we have
12
X
-
1
2π
(1 − c²) F' (u', w') du' d w'
SS.
1 0
(1 + c² + 2cμ') }
1
= F (- 1, w') [ [.
T
-
= 2 π F ( − 1, w')
[
=
2 п
{const.
π F(-1, w') const.
2π (1
(1 − c²) du' d w'
(1 + c² + 2cμ') }
(1 − c²) dµ'
C
(1 + c² + 2cμ² )½
c²)
= c(146-1204')}}
= 2 π F ( − 1, w') — { (1 + c) − (1 − e) } = 4 π F (− 1, w').
- −
C
In the same way, when u = 1, X = 4πF(1, w): and con-
sequently the result of the last Article is correct even in these
limiting cases.
CC
178. The demonstration of this Proposition is the sub-
stance of that given by Poisson in several memoirs: the last
place in which it appeared is in the Théorie Mathématique
de la Chaleur, Chap. VIII. He intoduces this Theorem with
these words, "La démonstration que j'en ai donnée dans plu-
"sieurs mémoires, et que je vais reproduire ici, me semble
propre à dissiper tous les doutes que l'on avait élevés sur
"sa généralité." If the reader be inclined to enquire into
the merits of the controversy hinted at in this passage, he
may consult the Mécanique Céleste, Livre III. Chap. II., two
Papers by Mr Ivory, in the Philosophical Transactions for
1812 and 1822, a Memoir by Lagrange in the Journal de
l'Ecole Polytechnique, Cahier 15, Poisson's Papers, and a Paper
in the Cambridge Philosphical Transactions, Vol. II. by Mr
(Professor) Airy, Astronomer Royal.
EXPANSION IN A SERIES OF LAPLACE'S COEFFICIENTS. 169
ų
2
It will be observed, that in the enunciation of the Proposi-
tion we have restricted the nature of the function F(u, w) to
such forms as fulfil the conditions of Art. 107. of the Théorie
de la Chaleur, by supposing it a function of μ, √1- µ³ cosw,
1- μ sin w. It is the opinion of those mathematicians who
most restrict the generality of this theorem, that it is demon-
strated only in such cases as F (µ, w) is a rational and entire
function of μ, I-ucos w, and 1 μ sin w.
に
​-
The doubt
attending this question will not affect any results to which we
come in this work, since it will be seen that we apply the
theorem only to such functions as prove in the end to be
rational and entire functions of μ, √1-μcos w, and
√1-μ²
1 - μ² sin w.
2
ω.
2
179. It results from this Proposition that in spheroids
of which the radius vector can be expressed in terms of u,
√1-µ³cos w, √1-μsino we can expand y in a series of
Laplace's Coefficients
Yo
Y₂ + Y₁ + Y₂ +......... + I; +......
We proceed to demonstrate in the course of the next two
Propositions that y can be expanded in only one such series,
a result of the utmost importance in the future calculations.
We must first prove the following property of Laplace's
Coefficients.
i
PROP. To prove that if Q; and R; be two of Laplace's
Coefficients, then f₁ ƒ« Q¡R; dµdw= 0, i and i' being dif-
ferent integers.
180. By the equation of Laplace's Coefficients, (Art. 170.)
d
i (i + 1) Q;= -
(1 − µ³)
du
d Qi
аль
1 dⓇ Q;
1 - µ² dw²
1
i (i + 1)
Эп
1
2ㅠ
​LL
d
1
0
Q; R ; dµd w
- μ²)
d QA
dul
Y
+
Ridµdw.
ω.
1- µ² dw
170
STATICS.
Now by a double integration by parts
d Ꭱ ;
d Qi
d
d Qi
(1 − μ²)
R¿dµ = (1 − µ²)
R; − (1 − µ³)
Qi
du
αμ
-
du
ам
d Ri
+
(1 − u²)
Q;du;
du
αμ
dR
d
i'
du
{ (1 − µ³)
Q₂dμ.
αμ
d Ꭱ ;
dw;
2
d
Lal
1
da} Redu
ď Qi
dw²
dw = R;
Again, fR, Tidw
2π
dQ;
Ri dw
ď² Q;
R
dw
i
dw²
2
=
i
1
-1
0
Qi
2T
d Ri
dw
Qi
+ Qi
d Ꭱ ;
ď Ri_dwr
2
d w²
άω,
dw²
since when w = 0 and 2π, each of the functions Q, R¿,
dQ dR
dw dw
of
M₂
11
has the same values, because they are functions
2
2
√1- µ³ cosw, √1 – µ³ sin w.
1
1
i (i + 1)
-1
1
Hence
LS
2π
0
0
2π
d
ам
(1
¿' (i' + 1)
QRrdudw
dR
du
i
1 dR
+
Q; dμd w
1 - µ² dw
2
Qq Rr du dw
1
0
LS
2π
(+3)
i (i + 1)
by the equation of Laplace's Coefficients.
2π
Hence QR, dudw = 0, when i and i' are unequal.
ƒ ƒ³™ Q¿R;dud
If ii' the above equation becomes identical and there-
fore gives no condition.
We are enabled now to prove the following very important
Proposition.
PROP.
To prove that F (u, w) can be expanded in only
one series of Laplace's Coefficients.
EXPANSION IN A SERIES OF LAPLACE'S COEFFICIENTS. 171
181. We have shewn in Art. 176. that
F (i, w) =
1
1
2π
{Po+ 3 P₁ +
1
4π
-1
0
+ (2i + 1) P¿ +
} F' (u', w') du'd w'
....... (1)
=
R。 + R₁ +
+ R¿ +
suppose,
1
this being a determinate series of Laplace's Coefficients, since
P; ……… are determinate (Art. 170).
Po, P₁
We have now to shew, that use what artifice of develop-
ment we may instead of that used in Art. 176, it will be im-
possible to expand F (u, w) in a series of Laplace's Coefficients
differing from R。, R₁ R... respectively.
For, if possible, let Qo + Q₁ +
be another
development of F (u, w) in which Q, differs from R;;
1
+ Q; +
(Q。 − R) + (Q1 − R₁) +
+ (Q; − R¿) +
= 0.
Multiply by P, then by Art. 180.
2π
ƒ₁²™ P; (Q; — R¡) dµdw = 0
(2).
But, by hypothesis, Q; R; does not = 0, but equals some
function of μ and w which does not become infinite between the
μ
ω
limits of μ and w: hence interchanging
in (1) and putting Q-R; for F (u, w)
1
and u', w and w
and Q - R for
F(u', w') and observing that PP₁.... are not altered by this
substitution (Art. 170), we have by the general principle ex-
pressed in the formula (1)
1
Q; – R{
2.π
= S_S²*
4 π
2i+1
4π
- 1 0
1
(P₁+ ... + (2i+1) P; + ...) (Q;- R₁) dudw
Σπ
–
[_] [** P. (Q; – R.) dµdw (Art. 180.)
= 0 by (2);
- 1 0
.. Q= R and therefore Q;= R; and our hypothesis is false,
i
and F (µ, w) admits of only one expansion.
182. We are now able to shew, as we promised in Art.
175, that the functions U U₁ U₂ can be calculated
1
172
STATICS.
without integration where the equation of the surface of the
attracting body is known.
For y can be expanded in a series of Laplace's Coefficients
1
2π
{ P₁ + 3 P₁ + ...... + (2i + 1) P¿ + ....} y'd µ'd w'
1
АП
[ √ ²* { P. + 3 P₁ +
– 1
Y₂+ Y₁+
+ Y; +
0
1
and this series admits of only one form by what has just been
proved. Hence when the equation to the surface, and therefore
y, is known the functions Yo, Y₁, ... ………. Y¡,……. are determinate,
and we may equate terms of the same order in the two series
for Y written above:
U₁
1
· SS
1
1
0
2π
0
(2 i + 1) P¡y'du'd w' = 4 π Y ; ;
Py'du'do' (Art. 173.):
and consequently when the equation to the
U U₁ U½ …………... U;, ... are also known, as
2
Art. 175.
0
4π
2 i + 1
Yi'
surface is known
was mentioned in
By substituting for U, U, ... in the expressions of Arts.
173, 174. we have for an external particle
3
4πρα πρα
3
ai
a
V
=
+
{Y +
Y₁ +
+
31
2
31.
(2 i + 1)²
Y; + ... } :
and for an internal particle
~
pi
V = 2 пра3
πρη
3
+4πpaa² {Y+
1
Y₁ + ... +
Y₁+...}.
3 a
(2i+1) a
The property of Laplace's Coefficients proved in Art. 180,
enables us to prove that Y, and Y, may be made to disappear
from the expression for y by properly choosing the value of (a)
and the origin of the radius-vector of the surface.
PROP. By choosing a equal to the radius of the sphere of
which the mass equals that of the attracting body we cause Y,
to vanish from the series Yo+ Y₁ + ... + Y¡ + ... ; and by
SIMPLIFICATION OF THF RADIUS-VECTOR.
173
taking the centre of gravity of the body as the origin of the
radius-vector we cause Y₁ to vanish.
183. If r,
1
t
If r, 0, w be the co-ordinates to any point in the
body, an element of the mass
=
=
pdrrder sin 0dw = - pr៰drdudw;
therefore the mass of the body
-
- [ [ [ Pardudo - G [ [
1
P
3
-
0
ρ
0
then putting r = a (1 + ay) we have
2π
r³dudw,
mass of body=mass of sphere (rad.=a)+pa³aƒ_ƒ³™ y dµdw
2π
=mass of sphere + pa³ a f₁² Ydµd w by Art. 180.
3
=mass of sphere +4πpa³a Yo, since I is constant.
0
If, then, a be taken equal to the radius of the sphere of which
the mass equals the mass of the body Y 0, as was stated.
0
184. Again: let a y≈ be the co-ordinates to the centre of
gravity of the body, M its mass: the co-ordinates to the
element, of which the mass is – pr²drdude, are
グ ​√1-μ² cos w,
グ
​2 A
usin w, and rµ;
√1 – µ³ sin
に
​•. Mx = ƒ«ß-1ƒ³™ prô Vi
Sjö s₁s™ pr³ √1 - µ³ cos w drdµdw
Jo
2π
= √₁₁ pr² √1 μcos w dµdw,
-
My = √√√√3" pr³ √1-2 sin w drdudw
= = √ √ √
2π
pr² √1 - usin wdµdw,
T
pr³ udrudw=f₁f prududw,
= a (1 + a Y + a Y₁ +
1
+ aŸ¿ + ...),
α
M = = √ √ √
putting r = a (1 + ay)
and observing that I cos w, 1-
Laplace's Equation (Art. 170), and
(Art. 171), we have by Art. 180,
µ²
μ
sin w, u satisfy
are of the
first order
174
STATICS.
2π
2
Mã = pa¹a f¹¹ƒ²™ Y₁ √1 - µ³ cos wdµdw,
1/0
2π
2
Mỹ = pa¹a f¹¹ƒ³™ Y₁√1-μ² sin wddw,
2π
Mz = pa'a f₁₂ Y₁ududw.
1 µ²
2
1u2 ω
But Y₁, being a function of u, VI- μ² cos w, √1- µ³ sin w
of the first order, is of the form
4πpa¹
.. Mx
a* a
3
2
μ² cos w + B√1-2 sin w + Cμ ;
A, My:
4πρα α
3
B. Mz
Апрача
C.
3
Hence if we take the origin of co-ordinates at the centre of
gravity x = 0, y = 0, 0, and consequently A= 0, B = 0,
C = 0 and therefore Y₁ = 0, as stated in the enunciation.
1
We shall in future parts of this work require to know the
attraction of a body consisting of strata nearly spherical and
varying in density according to any law. We shall therefore
proceed to the calculation of these attractions.
PROF.
To find the attraction of a heterogeneous body
upon a particle without it: the body consisting of thin strata
nearly spherical, homogeneous in themselves, but differing one
from another in density.
185. Let a' (1 + ay') be the radius of the external surface
of any stratum, a' being chosen so that
y' = Y; + Y½ +
+ Y + ... (Art. 183).
Let p' be the den-
Since the strata are supposed not to be similar to each other y'
is a function of a' as well as of μ' and '.
sity of the stratum of which the mean radius is a'. Now the
value of V for this stratum equals the difference between the
values of V for two homogeneous bodies of the density p' and
mean radii a' and a- da'. But for the body of which the
mean radius is a' (Art. 182.)
V
3
Απρα παρα" (α'
3r
+
a'i
Yi+
+
1
3r
(2 i + 1) pi
Y + ...}
ATTRACTION OF BODIES NEARLY SPHERICAL.
175
hence for the stratum of which the external mean radius is a'
12
4 πρα
V
da +
παρ'
4πар' d fa'¹
Y₂+......+
ጥ
ጥ da 3r
i
a'i + 3
Y+
..Jdá',
+(24+1) + ... da',
ri
and therefore for the whole body
71
a
4π
d
a'i + 3
V
+a
Y+
+
Y; +
da
3r
(2 i + 1) pi
..) } da'.
d V
dr
The attraction
PROP.
and is easily found by differentiation.
To find the attraction of the same body on an
internal particle.
186. Let r = a (1 + ay) be the radius of the attracted
particle. Then for the strata within the surface of which the
radius is a (1 + ay) we have (Art. 185.)
a
4π
V
0
{a's
d
α
a'i+3
a
12
a²² + a
Y₁ +
+
Y
da 3r
pi
(2+1) +...
da'.
But for a stratum external to the attracted particle we obtain
by Art. 182.
d γα
П
V = 4π p'a'da' + 4πp'а
a
Y+
+
da' 3
(2 i + 1) a'i - 2
Y' '+...) da',
and therefore for all the strata external to the particle
a
V=4π
+a
d ra
da' 3
Y₂+......+
α
and consequently for the whole body
4 π
V
2°
(2i+1) a'¹-ë ¹'/' + ... ) } da',
a'i+3
(2 i + 1) ri
d a
1
12
+ a
Y₁+
+
Y+
da' 3r
+...) } d.
}
da
a
d α
a'r
pi
+4π
ta
Y₁+...
+
da
3
a
(2 i + 1) a'i−
Y'{' + …..
da'.
d V
From this the attraction, or
"
is easily obtained.
dr
DYNAMICS.
CHAPTER I.
DEFINITIONS. LAWS OF MOTION.
187. IN this part of our Work we are engaged with the
laws which regulate the motion of bodies. We shall proceed
therefore to explain the means we use for measuring the
motion of a body algebraically.
The position of a body in space, considering the body as
a material particle, is determined at any instant by its distances
from three fixed planes at right angles to each other: these
distances are called the co-ordinates of the particle; and the
position of a rigid body in space is determined at any instant
by the co-ordinates of a given point of the body and the angles
which three fixed lines in the body make with three fixed lines
in space.
If the body be in motion the co-ordinates will be continually
changing in magnitude: and one of the chief objects of the
Science of Dynamics is to find the analytical relation between
each co-ordinate and the time of motion.
188. We shall pause, however, a little to make a few re-
marks which cannot be too carefully remembered.
All our ideas of the magnitude of quantities (such as space,
time, and so on) are ideas of comparative and not absolute
magnitude for a quantity may be great when compared with
one standard, and small when compared with another. In
consequence of this it is necessary, in order to avoid ambiguity,
to choose for quantities of the same kind a certain standard to
NECESSITY OF CHOOSING UNITS OF MEASURE.
177
which they may be referred.
This standard is called the
unit of these quantities. Thus we speak of the unit of time,
and the unit of space; by which we mean the duration of
time and the extent of space which we choose as standards to
which all other quantities of these species are to be severally
referred.
It is by this means that quantities are made the subjects of
numerical calculation. For instance, when we say that a body
describes a spacer in a time t, we mean, that x and t repre-
sent the ratios which the space described and the time of de-
scribing it bear to their respective units: and so of all other
quantities. We forbear choosing these units at once because
it generally happens, as we shall see, that by a judicious
selection our formulæ may be materially simplified. Before
closing these remarks we will observe, that though in the
same calculation we must have only one standard of quantities
of the same kind, yet in different calculations we need not
retain the same unit, so long as we bear in mind what unit
is chosen in each calculation. Thus in one calculation we
might take the length of the mean day as the unit to which
we should refer all portions of time; while in another calcu-
lation we might take a year as the unit of time. We return
now to the consideration of the means of measuring the motion
of a body.
189. Velocity is a term used to indicate the degree of
quickness or slowness with which a body moves. Velocity may
be uniform or variable.
190. Uniform Velocity. Velocity is said to be uniform
when the body passes through equal spaces in equal times.
It appears, then, that the magnitude of the velocity of a
body moving uniformly depends conjointly upon the space
described and the time of describing the space; and is greater
or less exactly in the proportion in which the space described
in any given time is greater or less, and the time of describing
any given space is less or greater.
Consequently when bodies move with different uniform
velocities, these velocities are in the proportion of the ratios
which the spaces described in any times bear respectively to
the times of describing them.
Z
178
DYNAMICS.
Suppose, then, that a body moving uniformly with the
velocity describes a space s in the time t: also suppose that
a body moving uniformly with the unit of velocity describes
a space S in the time T: then by what precedes
v: 1: ::
S
S
:
t T
Ts
St
In this formula the only arbitrary quantities are S and T; we
shall choose them so as to simplify the formula as much as
possible: in choosing their values we fix the unit of velocity.
We shall take S = 1 and T=1; we then have
V
S
t
the unit of uniform velocity being the velocity of a body moving
uniformly through the unit of space in the unit of time.
It will be seen that the units of space and time are as yet
quite arbitrary.
191.
Variable Velocity. Velocity is said to be variable
when the body in motion does not describe equal spaces in
equal times.
Suppose a body moves uniformly and at the time t we wish
to estimate its velocity. Let s' be the space described in any
portion of time t', this time either terminating or commencing
with the instant of expiration of the time t. Then, by what
s'
I'
precedes, the velocity will equal however large or small ť
be taken.
s'
But when the velocity is not uniform the ratio is not the
ť
same for different values of t, and therefore cannot be taken as
a measure of the velocity at the time t: unless we select some
particular value of t' always to be used.
Now if the time t terminate with the time t, then the
space ss is described by the body in the time t-t'; and
therefore by Taylor's Theorem, since s is a function of t,
MEASURE OF VELOCITY.
179
ds
d's t²
- 8′ =
ť +
dt
dt 1.2
ds
d2s t
17
dt
dť' 1.2
If t' commence with the expiration of t, then s+ s' is the
space described in the time t+t', and
ds
d²s t'²
s + s' =
= 8+
ť +
+
dt
dť 1.2
s' ds
d2s
+
ť
+
ť
dt dť² 1.2
dt
We gather from these expressions that when t' is taken in-
definitely small the values of the ratio
s'
ť
are the same, and each
equal to
ds
dt
We shall therefore select this particular value as
the measure of variable velocity.
It will be observed that in selecting this as the measure of
variable velocity we do not violate any conditions previously
established in reference to uniform velocity: we only restrict
those conditions, inasmuch as t' may be of any value in the case
of uniform velocity, but we take it indefinitely small in that of
variable velocity. But, notwithstanding this, the formula
ds
dt
includes the case of uniform motion: for if v be constant
we have by integration vts (the constant of integration
vanishes, since when t 0, s = 0), and this is the formula
already adopted for uniform motion.
Hence, then, if v be the velocity of a body moving uni-
formly or not at the time t and s be the space described in
that time, the quantities v, s, t are connected by the formula
V
ds
dt'
192. Having thus explained the means of measuring
algebraically the motion of a body we shall enter upon an
180
DYNAMICS.
enquiry into the laws which regulate this motion. Since, as
far as we know, it might have pleased the Author of the Uni-
verse to endue matter with laws and properties different from
those which He has chosen to impress, it is evident that these
laws can be discovered by no process of abstract reasoning,
but solely by an appeal to experiment.
193. As the simplest case we shall first consider the
motion of a body uninfluenced by external forces. We have
already defined force to be any cause which produces or tends
to produce motion in a body; see Art. 5.
Throughout the whole universe it is impossible to find a
single spot free from the action of force. It is consequently
beyond our power to determine by direct experiment the nature
of the motion of a body uninfluenced by external causes. But
by combining the results of various experiments we shall be
able to eliminate, so to speak, the principles which are foreign
to our enquiry, and in that way ascertain the laws we are
seeking.
Experience teaches us that the more external causes are
removed the more nearly uniform is the motion of a body.
A bowl thrown along a bowling green is observed to move
slower and slower till it finally stops: but the smoother the
green is made, the longer does the motion continue. If the
bowl be thrown with the same velocity along a pavement the
motion is of longer duration; and still longer when the motion
takes place on a sheet of ice. One cause of the diminution of
velocity is the friction of the body on the plane: this is inferred
from the fact, that the retardation is less the smoother the plane.
on which the motion takes place. Also any change in the
uniformity of the decrease of the velocity can always be attri-
buted to some disturbing cause; as the greater roughness of
the surface and the deficiency in perfect horizontality.
The experiment shews likewise that the motion is in a
straight line, unless some assignable cause produce a deviation.
Steam-carriages moving on horizontal rail-roads, when once
in motion, require a constant power of the engine to maintain a
uniform velocity: and since, when the motion is uniform, the
retarding effect of friction and the resistance of the air may be
assumed to be constant, we infer (after what we have said in the
FIRST LAW OF MOTION.
181
case of the bowl) that the constant power of the engine exactly
counterbalances the constant retarding force, and that therefore
supposing them both removed the result would be a uniform
motion.
The reader is referred to Desaguliers' Course of Experi-
mental Philosophy, 4to. 1734, Vol. I. Lecture V. for more
experiments upon the motion of bodies.
194. Philosophers have assumed, then, as a fundamental
principle of the motion of matter that
A body in motion, not acted on by any external force,
will move uniformly and in a straight line.
This is called the First Law of Motion.
195. It must not be imagined that these experiments prove
the truth of the law here enunciated: for the law embraces
an infinite variety of cases, and many in which it would be
impracticable to make experiments. Also the roughness of the
experiments prevents our supposing it proved even for the
cases we have mentioned. The truth is, that the law is only
suggested by the facts we have detailed; and it remains to be
seen whether or no this, in conjunction with other laws (which
we shall soon consider), satisfies the tests we shall hereafter
have to submit them to; whether, combined in endless variety,
they will account for the numerous phenomena continually
coming under our observation. It is found that they do lead
to results which precisely accord with observation. Of the
more obvious phenomena, the explanation of which depends on
the truth of these laws, we may mention the prediction of the
time of an eclipse and the certainty of its fulfilment. Results
of this nature are the only satisfactory proofs.
196. It appears from the First Law of Motion that a
body has no internal forces residing in it that influence its
motion; for when all external forces are removed the velocity
is uniform and in a straight line. In other words, matter has
no inherent property of changing its state of motion. It is
equally a result of experiment and observation that matter has
no inherent property of changing its state of rest (Art. 4).
This property of matter, that when not acted on by any
external force it continues in the same state whether of rest
or uniform rectilinear motion, is called its Inertia.
182
DYNAMICS.
197. We proceed now to discover the laws which regu-
late the motion of a body when acted on by external forces.
198. But previous to this we must explain the means
we use for measuring forces in terms of the magnitude of the
motion they generate in a body subjected to their influence.
We infer from the First Law of Motion, that when a body
moves with a variable velocity force is acting on the body:
and, conversely, when force acts upon a body its velocity is
continually changing. Now we take the magnitude of the
change of velocity during a given time as the measure of the
magnitude of the force which acts upon the body: and, for
the sake of distinction, when force is measured in this manner
it is termed Accelerating Force.* We have already men-
tioned that when force is measured statically, it is called
Pressure (Art. 7).
199. Although the sources of force are very various yet
its effect in accelerating the motion is always measured in
Dynamics by the change in velocity in a given time. Thus
when a body is dropped from the hand, the accelerating force
of the Earth's attraction at any instant is estimated dynamically
by the velocity generated in a given time after that instant.
Suppose a body placed on a smooth horizontal table is drawn
along by means of a thread passing over the edge of the table
and attached to a falling body. The magnitude of the accele-
rating force which causes the body to move on the table, is
measured by the change in velocity in a given time. When
a body is moved along a smooth horizontal table by means of
a constrained spring, the accelerating force which causes the
body to move, though differing in its source from the force.
mentioned in the last case, is measured in the same way.
a body resting on a smooth horizontal table be set in motion
by the sudden blow of another body upon it, the accelerating
force which causes the motion is measured as before. When
lf
* When a force retards the velocity of a body, it is called a retarding force; but
still it is of exactly the same nature as an accelerating force since it is measured by
the decrements instead of the increments of velocity in a given time. In short, a re-
tarding force is an accelerating force when estimated in the direction of its action, and
if the body were moving in the direction in which the force acts instead of the opposite
direction, the force would become an accelerating force. Thus it will be seen that
retarding force is merely a relative term and is included in the term accelerating force.
1
THE DYNAMICAL MEASURE OF FORCE.
183
a ball is fired from a cannon the accelerating force which
causes the motion is still measured by the velocity generated.
It will be seen in the first two of these cases (especially
in the second if the descending body be small), that the motion
is gradually communicated, the velocity increasing continu-
ously. But in the last two cases it may perhaps be thought
that the motion is instantaneously communicated: this is not,
however, true: for the time occupied in generating the velo-
city is of finite duration, although, to our senses, it is of inap-
preciable magnitude. That it is of finite duration appears in
the case of the collision of the bodies from the fact, that if a
small spot of ink be put upon the point of contact of either of
the bodies before the motion takes place, then after the col-
lision the ink is found spread over a larger surface than it
occupied before, and on both bodies; shewing that the bodies
suffered mutual compression and then separated, and this must
have occupied time. In the case of the cannon ball, the
expansive force of the ignited powder acts during the time that
the ball takes to move along the bore of the cannon.
both these instances, as well as in the others, the velocity of
the body commences from zero and passes through successive
and continuous gradations of magnitude, the only difference
being that the intensity of the force originating from the colli-
sion and from the explosion is very far greater than the inten-
sity of the force arising from the Earth's attraction; and con-
sequently the velocity which is generated in a falling body, in
a few seconds by the attraction of the Earth may be generated
by impact, or other such means, in an extremely short portion
of time.
In
200. When a body moves under the action of a force a
continual change of velocity takes place; and if the force cease
to act the body will move uniformly in a straight line with its
last acquired velocity, as the First Law of Motion teaches us.
If the force act for a finite time, then our object is to discover
such laws of nature and to establish such conventional rules
as shall enable us to determine the velocity acquired and the
space described by the body during any portion of the time
that the force is in action. If, however, the force act for only
an indefinitely short time, we are concerned only with the velo-
184
DYNAMICS.
city and position after the action of the force ceases, since the
changes that take place during the action of the force are
so rapid that the whole process of the action appears to our
senses to be instantaneous.
We have a popular illustration of the effects of forces
which act for a finite time and for an indefinitely short time
in the game of cricket. The bowler rotates his arm in order
to give the ball velocity, he opens his hand and the ball
flies from him with the velocity acquired, and (supposing he
delivers the ball full pitch) after moving in a curve slightly
deflected downwards by the Earth's attraction is received upon
the bat. Now this velocity was generated by the muscular
effort of the bowler's arm acting on the ball during the finite
time that he retained it in his grasp. While this is going on
the batter swings his bat that it may acquire a great velocity ;
and the ball and bat come in collision: and what is the conse-
quence? the ball flies back; not only is its original motion
destroyed, but new motion is given to it, as if instantaneously,
in an opposite direction.
We explain the phenomenon of this sudden recoil in the
following manner. When the ball and bat come in contact
their particles are moving in opposite directions, and tend to
penetrate each other: but the molecular forces by which the
particles of each of the bodies are bound together are too
powerful to allow of this separation; nevertheless the relative
positions of the particles are slightly changed by the yielding
of the bodies, and in consequence of their unnatural restraint
a mutual resultant pressure is exerted by the bat on the ball
and by the ball on the bat, till their relative motion is de-
stroyed: but the particles of the two bodies are still under
restraint when the motion is destroyed, and the mutual pres-
sure of the bodies now acts to effect their separation, and new
velocity is generated: this process, which we conceive repre-
sents the actual process in nature, goes on with inconceivable
rapidity in consequence of the great intensity of the molecular
forces which bind the particles of each body together. If the
bat split or the ball burst, then the molecular forces which
held together those particles which separate were not powerful
enough to resist the separation.
It is evident that we are
FINITE ACCELERATING FORCE.
185
concerned not with the changes which take place during the
collision but the whole change produced.
201. We have made these remarks in this place in order
to shew that it is necessary in explaining the means of mea-
suring force dynamically to consider two cases: first when the
force acts for a finite time; and, secondly, when the force acts
for an indefinitely short time.
In the second case the accelerating force is measured by
the whole velocity generated during the action of the force.
And such forces we shall term, for the sake of distinction,
Impulsive Accelerating Forces: and in contradistinction ac-
celerating forces which require an appreciable duration of time.
to manifest their effects may be termed Finite Accelerating
Forces. We shall, however, generally drop the term Finite:
and it must therefore be remembered that when we speak of
accelerating forces we mean finite accelerating forces, and never
impulsive accelerating forces unless the term impulsive be
prefixed.
202. We proceed now to explain more fully how accele-
rating forces which require an appreciable duration of time to
manifest their effects are measured.
Accelerating force may be uniform or variable.
203. Uniform Accelerating Force. When equal velocities.
are generated in equal times the force is said to be uniform.
It appears, then, that the magnitude of the force depends.
conjointly upon the velocity generated by the action of the
force and the time in which this velocity is generated: and is
greater or less exactly in the proportion in which the velocity
generated in a given time is greater or less, and the time in
which a given velocity is generated is less or greater.
Consequently when bodies are acted upon by different
uniform accelerating forces, these forces are in the proportion
of the ratios which the velocities generated in any times bear
respectively to the times in which they are generated.
Suppose, then, that a body acted on by the constant
accelerating force ƒ has the velocity v generated in the time
t: also suppose that a body acted on by the unit of uniform
accelerating force has a velocity V generated in the time T:
then by what precedes
A A
186
DYNAMICS.
ช V
Tv
ƒ: 1
f
t
T
V t
In this formula the only arbitrary quantities are V and T:
we shall choose them so as to simplify the formula as much as
possible in choosing their values we fix the unit of uniform
accelerating force.
=
V
t
We shall take V = 1 and T = 1, we then have ƒ
the unit of uniform accelerating force being the force which
generates in a body a unit of velocity in a unit of time.
We have already chosen the unit of velocity (Art. 190); we
may consequently say that the unit of uniform accelerating
force is the force that causes a body during each successive
unit of time in its motion to describe a space greater by the
unit of space than it did during the unit of time immediately
preceding.
204. Hence, in uniformly accelerated motion, s the space
described from rest, t the time of describing it, v the velocity
acquired during that time, and ƒ the constant force are con-
nected by the equations.
v =
ds
dt
and f
ย
t
the units of v and ƒ being given in Arts. 190. and 203. By
means of these equations we can obtain four equations differing
from each other, and each containing three of the quantities
s, t, v, f. Thus, if we eliminate v we have
ds
ft²
dt
ft; '. S
2
By eliminating t we have
ds ข
v2
. S
dv f
2f
(1), f is constant.
(2).
s =
Also v = ft………….. (3),
ft......(3), 28 vt...... (4), by (2) (3).
205.
Variable Accelerating Force. Accelerating force is
said to be variable when equal degrees of velocity are not
generated in equal times.
FINITE ACCELERATING FORCE.
187
Suppose a body is moving under the action of a uniform
accelerating force and at the time t we wish to estimate the
magnitude of the force. Let v' be the velocity generated in
any portion of time t', this time either terminating or com-
mencing with the instant of expiration of the time t. Then,
by what precedes, the uniform force will equal
large or small ť be taken.
But when the force is not uniform the ratio
t
V
however
is not the
same for all values of t, and therefore cannot be taken as a
measure of the force at the time t, unless we select some par-
ticular value of t' always to be taken.
Now if the time t' terminate with the time t, then the velo-
city vv is generated in the time t-t', and therefore by
Taylor's Theorem, since v is a function of t,
d v
d2v t'2
v – v′ = v
ť +
dt
dť² 1.2
v'
ť
dv
d² v ť
+
17
dt
dt² 2
If t' commence with the expiration of t then v + v′ is the velo-
city generated in the time t + t';
d v
d² v t'²
:: v + v'
= 1 +
ť +
+
dt
dť² 1.2
v dv
d² v ť
+
ť'
dt
d t°
201
+
We gather from these expressions that when t' is taken inde-
v'
finitely small the values of the ratio are the same and each
equal to
dv
dt
ť
We shall therefore select this particular value as
the measure of variable accelerating force.
It will be observed (as in the case of variable velocity)
that in selecting this as the measure of variable accelerating
188
DYNAMICS.
force we do not violate any conditions previously established in
reference to uniform accelerating force: we only restrict these
conditions, inasmuch as t may be of any value in the case of
uniform force, but we take it indefinitely small in the case of
d v
dt
variable force. But, notwithstanding this, the formula ƒ =
includes the case of uniform motion: for if ƒ be constant we have
by integration ft = v(the constant of integration vanishes since
when t=0, then v = 0), and this is the formula already adopted
for uniform accelerating force. Hence, if f be the accelerating
force, uniform or variable, which generates the velocity v
in a body in the time t, then f, v, t are connected by the
dv
equation f
dt
ds
206.
We have seen (Art. 191.) that v =
Hence the
d t
equations connecting ƒ, v, s,
t are
ds
d v
d2s
ช
f
=
dt
dt
dt2
in which it must be observed that the unit of velocity is the
velocity of a body moving uniformly through a unit of space
in a unit of time: and the unit of accelerating force is the
uniform force which generates a unit of velocity in a unit of
time.
We have thus explained the methods of estimating the
magnitude of forces dynamically.
207. The next enquiry we shall make into the laws
which regulate the motion of bodies is, how to calculate the
combined effect of two or more causes acting simultaneously
on a body.
We must, as before, appeal to experimental facts for the
solution of this question. And first we will take the case of
two causes acting upon the body, each of which would by itself
make the body move uniformly for instance, suppose the
body projected at the same instant by impulsive forces acting
in different directions.
:
SECOND LAW OF MOTION.
189
A ball rolled along the horizontal deck of a vessel moving
equably will move on the deck as it would if the vessel were
at rest; this is proved by experiment. Suppose S is the deck
of a boat moving uniformly on a sheet of water, fig. 74: and
in a given time suppose it moves to S'. Let A be the place of
the ball at the beginning of the time of motion: and B' its
place in space at the end. Draw AA' in the direction of the
boat's motion, and equal to the distance through which the
boat has moved; and join A'B'. Suppose AB is the space
the ball would have described if the vessel had not moved.
Now, as we have already stated, experiment shews that A'B',
the space actually described on the deck, is the same in refer-
ence to the vessel as if the vessel had been stationary. Hence
A'B' is equal and parallel to AB. From this we gather, that
if two causes act simultaneously on a body to produce uniform
motions, each cause will have its full effect in its own direction;
and the body will be found at the extremity of the diagonal
of the parallelogram described on the linear spaces, which the
body would have passed through under the action of the
causes separately.
This principle is found to be true if one or both of the
separate motions be not uniform. For a ball dropped from
the top of the vertical mast of a vessel sailing uniformly, falls
at the foot of the mast, although the vertical motion is not
uniform.
The following experiment well illustrates this principle.
Two balls are placed at the same height above the ground:
one is projected horizontally, the other suffered to fall of itself:
it is so contrived that the motions shall commence at the same
instant. The result is that they are heard to strike the ground
at the same time, although they describe very different paths,
one ball having moved in a straight line, the other in a curve.
This experiment shews that although one ball had a horizontal
motion, still the attraction of the Earth produced the same
effect on the two balls in a vertical direction.
The muscular efforts necessary to raise the arm, move the
head, or raise the body are the same on board a vessel sailing
equably, or in a steam-carriage moving uniformly on a rail-
road, as when the ship or carriage is at rest.
190
DYNAMICS.
It can be proved independently of any mechanical princi-
ples that the Earth revolves round its axis from east to west;
but the effort of moving a body from one place to another does
not depend, cæteris paribus, on the point of the compass.
towards which the motion is directed. To bring to our aid,
however, more delicate tests, it is found that the motion of a
pendulum is presisely the same in whatever vertical plane it
vibrates, whether east and west, or north and south, or in any
other direction.
For more facts and experiments we refer again to Lec-
ture V. of Desaguliers' Experimental Philosophy.
208. These facts point out to us the following general
principle:
When a force acts upon a body in motion, the change
of motion in magnitude and direction is the same as if the
force acted on the body at rest.
This is called the Second Law of Motion.
For the full elucidation and proof of this Law we ought
to make experiments with forces of all degrees of magnitude
and motions combined in all directions; since, however, this
can never be accomplished, we must have recourse to the
expedient spoken of in Art. 195, to satisfy ourselves of the truth
of this as well as the First Law.
209. We shall now shew the importance of this Law in
enabling us to refer the motion of a particle to three rect-
angular axes.
Since the various positions of a material particle in space
are generally determined by means of co-ordinate axes, it
becomes necessary to refer the motion to these lines. At any
proposed instant of the motion the particle is moving with a
definite velocity and in a definite direction. Now this motion
may be supposed to be the result of three motions taking place
simultaneously parallel to the three axes of co-ordinates. Imagine
the particle, in the first place, to have only its motions parallel
to the axes of y and ≈ combined. Then, in the second place,
by combining with these the motion parallel to the axis of a,
we have the actual motion of the particle in space: and the
change in the motion by this last step is, that the particle has
moved to a distance a parallel to the axis of a in the time t.
SECOND LAW OF MOTION.
191
But by the Second Law of Motion this change is the same as if
the other motions did not exist. Hence the velocity and acce-
lerating force of the particle parallel to the axis of x are the
same as if the particle described the space a in the time t: and
dx
dt
we have proved in Arts. 191, 205, that these are and
dť
and in a similar way it may be shewn that those parallel to the
dy ď³y d z ď² z
axes of y and ≈ are
dt
and
d t
dt' dť
210. It follows then that when a particle is moving in
space, and ayz are its co-ordinates at the expiration of the
time t, the velocities of the particle parallel to the axes are
dx dy dz
,
dt
dt dt
and the accelerating forces parallel to the three axes are
ď x
dt ' dť²'
d²y
ď z
dť
COR. By the Differential Calculus.
ds2
dx²
dy² dx2
+
+
d to
dt dt² df
s being the length of the curve described.
If we compare this
with the formula R² X² + Y² + Z³ taken from Arts. 20, 21,
it follows that velocities may be resolved and compounded
in exactly the same way that we resolve and compound statical
forces.
211. The grand Problem of Dynamics is to find the
relation which exists between the motion of a system of bodies
and the forces which act upon them: so that when the forces
are known the motion may be determined, and vice versâ.
We have seen that if no forces act upon any part of the
system, each part will move uniformly in a straight line, when
once put in motion. This will also happen if the forces acting
upon each particle of the system are in equilibrium with each
other.
192
DYNAMICS.
In the general case, however, each particle will move in a
determinate curvilinear path, and the acceleration (or retard-
ation) of its motion will take place whenever the forces acting
on the particle are not in equilibrium, i. e. whenever they have
a resultant. This resultant is measured at every instant by
the change of velocity produced in a given time, as explained
in Art. 198. Let xyz be the co-ordinates of position of any
particle of the system at the expiration of the time t, then the
resultant is measured dynamically by the accelerating forces
d²x d²y d² z
acting parallel to the fixed axes of co-ordinates.
dt' dt²' dť
dť
These are termed the effective accelerating forces of the par-
ticle at the time t parallel to the axes of co-ordinates. The
forces which act upon the particle to produce the motion, not
including the molecular actions of the particles on each other
(if there be any), are termed the impressed forces by way of
distinction.
212. Now it is immediately evident that if at any instant
of the motion we were to apply to each particle of the system
forces equal in magnitude but opposite in direction to the
effective forces of that particle, these would at that instant
check the acceleration of the motion, or, in other words, would
be in equilibrium with the impressed and molecular forces
which act upon the system: and will therefore together with
them satisfy the equations of condition we have deduced in
the former part of this Work for the equilibrium of forces.
By this principle, the truth of which is self-evident, we
shall obtain equations which connect together the forces that
dx dy dz
act upon the system and the analytical quantities
dť²' dť²' dť
and all similar quantities for the other particles. If the
question be to determine the motion when the forces are given
in terms of xyz and t, the solution is effected by integrating
these equations. If, on the other hand, the question be to
determine the forces which will cause the system of particles
to move in given curves, we must differentiate the equations to
d² x d'y d²≈
the curves with respect to t, and substitute for
dt' de' df²**
EQUATIONS OF MOTION OF A MATERIAL SYSTEM.
193
in the equations resulting from the application of the above
principle in this way the forces will become known.
213. But we have been supposing that the forces which
act upon the system are of the nature described in Art. 201.
as requiring time to manifest their effect. We shall now
consider the case of impulsive forces.
Let
dx dy dz
dt' dt' dt
be the resolved parts parallel to the
axes of the velocity of any particle of the system arising from
the action of the impulsive forces. Then the effect of the
impulsive forces is the same as three impulsive accelerating
dx dy dz
forces acting parallel to the axes and equal to dt' dt' dt'
(Art. 201); these are termed the effective impulsive accelerat-
ing forces and the original forces are termed the impressed
impulsive forces.
Wherefore it is immediately evident that if, at the instant
of the action of the impulsive forces on the system, we
were to apply to each particle impulsive forces equal but op-
posite to the effective impulsive forces of that particle, these
would check the effect of the impulsive forces actually im-
pressed on the system and would consequently with them
satisfy the equations of condition for the equilibrium of forces.
As before, then, we obtain equations by means of which
the motion of the system may be calculated.
214. Now in the calculations of the conditions of equili-
brium of forces acting upon a single particle, a rigid body, or
any material system given in Chapters I, II, and III of
Statics we have considered the magnitudes of the forces to be
estimated statically; in other words, we have supposed them.
to be pressures. Wherefore before we can make use of the
results of those Chapters for determining the equations of
motion of a system, in the manner explained in Arts. 212, 213,
we must discover the relation that connects the dynamical
and statical measures of force; so that when we know the
degree of acceleration of a force, we may be able to determine
the magnitude of the pressure that the force causes the body
on which it acts to exert; and vice versâ. It is manifest that
BB
194
DYNAMICS.
some relation between these two measures of force must exist,
since the cause of the pressure and the cause of the motion and
of the change of motion are the same. But since causing
pressure and causing motion are two properties of force which,
abstractedly speaking, have no common character, we cannot
discover the relation they bear to each other by reasoning
à priori; but must again appeal to experiment.
215. When two balls of the same size and substance
are dropped at the same instant from the same altitude they
move downwards in exactly the same manner; having the same
velocity at every instant and having moved through the same
spaces during any given time. If the bodies be connected the
motion is the same. And the same would be the case what-
ever number of balls were connected. Hence it appears that
although the weight of a body of homogeneous structure, or the
statical measure of the Earth's attraction, varies as the mass
of the body, yet the accelerating force, or the dynamical
measure, is invariable when the experiments are made at the
same place. Newton made a variety of experiments with
gold, silver, lead, glass, sand, common salt, wood, water, and
wheat, and arrived at the same result. Principia, Vol. III.
Prop. vi. It is found by experiments made under a receiver
exhausted of air that a guinea and a feather fall in exactly the
same manner and strike the plate of the air pump at the same
instant, if they are set at liberty together and from the same
altitude. These experiments shew that the accelerating force
of all falling bodies at the same place is the same: and, after
what has been proved of a homogeneous mass of matter, these
experiments lead us to conclude that bodies differ in weight
at the same place in the same proportion as their masses
differ. We therefore infer that at the same place on the
Earth's surface the weight of a body varies as its mass. Let
M be the mass of a body of which the weight is W; then
W = Mg, g being some arbitrary quantity which is constant
for the same place and depends upon the units of weight and
It remains to be determined what change the weight
undergoes if the body be removed to a place where the ac-
celerating force is different: or if by any contrivance the ac-
celerating force of a body be changed without changing the
mass.
CONNEXION BETWEEN PRESSURE AND MOTION.
195
place of the experiment.
We shall for this purpose describe
a machine invented by Atwood: see Atwood on Rectilinear
Motion for a full explanation.
216. Four wheels, two of which A and B are represented
in figure 75, the other two being hid by these, are placed
parallel to each other, their centres being fixed so as to allow
of rotation with as little friction as possible: A and B are
placed as near as possible without touching: and so are the
other two wheels. Upon these four rests the axle of another
wheel C placed midway between A and B and the other two
wheels: a fine string as flexible and inextensible as possible
is passed over the circumference of C and two weights P and
Q are attached to its extremities. When P and Q are left to
themselves the heavier will descend and draw up the lighter
of the two. It will be readily understood that the object of
the four wheels is to diminish the friction on the axle of C;
which it does very considerably, since the friction of rolling
is far less than that of rubbing. Suppose that P descends,
then P - Q is the weight or pressure which causes the motion
and P+Q is the weight put in motion. It is found by
experiment that the inertia of the wheels produces the effect
of adding to the weight moved without adding to the pressure
producing motion. Atwood determines by experiment what
this weight is, we shall call it . Hence P-Q is the
weight causing the motion and P+Q+ IV is the weight put
in motion. A graduated scale of inches is placed behind the
thread supporting P in order to mark the motion of P. The
excellence of this machine consists in this, that we can have
bodies falling with various degrees of acceleration and as
slowly as we please by altering P and Q. The time of motion
is marked by a seconds pendulum.
Now suppose P is placed with its lowest surface level with
the zero point of the scale and set at liberty at any tick of the
pendulum: it is always found, however much P and Q are
altered, that in each experiment the spaces described by P in
successive seconds form an arithmetic progression, and there-
fore that the accelerating force in each case is uniform. Also
it is found that the common differences of the series in the
various experiments are proportional to the respective values
196
DYNAMICS.
of the ratio
P-Q
P + Q + W®
This is proved by numerous ex-
periments, for the details of which we refer to the work of
Atwood.
We gather, then, from this that the accelerating force
P - Q
P+Q + W
varies as
in the different experiments, and therefore
the pressure producing motion (or P-Q) varies as the product
of the accelerating force and the weight moved (or P+Q+W),
and therefore as the product of the accelerating force and the
mass moved, since the weight of a body at the same place
varies as the mass (Art. 215), and these experiments were made
at the same place.
217. The product of the mass of a body and the accele-
rating force is called by Newton the Moving Force of
the body: and the product of the velocity and mass of a body
he calls its Momentum, or quantity of motion.
These ex-
periments therefore shew that the pressure communicating
the motion varies as the moving force, or as the momen-
tum generated in the body: for moving force must be
measured by the momentum generated in a given time, since
accelerating force is measured by the velocity generated in
a given time, Art. 198.
218. We shall now give the results of experiments with
pendulums. It is found by numberless trials that the time.
of oscillation of a leaden ball suspended by a fine thread and
moving through any very small angle is constant for the same
length of thread, but for different lengths it varies as the
square root of the length. Now let s be the small circular
arc measuring the distance of the centre of the ball from its
point of rest, fig. 76, t the time of describing s, l the length
of the pendulum: then 7 varies as t, by experiment. M the
mass of the ball, and Mg its weight, Art. 215. Pt a tangent
to the arcs at P: then the weight Mg is employed partly in
producing the motion and partly in stretching the thread: the
part producing motion Mg cos t PW = Mg sin 4;
... the pressure
=
S
Mg sin A = Mg nearly ∞ M
S
だ
​CONNEXION BETWEEN PRESSURE AND MOTION.
197
But if ƒ be a uniform accelerating force causing a body to
describe a space s in the time t (and the acceleration in this
case is ultimately uniform) then 2s = ft²;
.. the
the pressure Mƒ
× Mf∞ moving force, as before.
219. We must now enquire into the connexion between
pressure and the motion generated or destroyed when impulsive
forces act.
We have already explained the nature of impulsive
forces; and have shewn that they differ from finite forces
solely in intensity, and that we measure them dynamically
by the velocity generated during the action, and not by the
velocity generated in a unit of time*. The result of the
last four Articles must therefore be true for impulsive
forces; and we shall assume that impulsive pressure is pro-
portional to the momentum generated or destroyed. In
fact, the pressure is so enormous that to make its measure
a matter of experiment would be very difficult.
We can
nevertheless mention some experiments which illustrate, and
in part prove, that impulsive pressure varies as the momentum
generated. We shall, however, first speak of the elasticity of
bodies.
220. It is found that all rigid bodies rebound more or
less when struck together: this property is termed their elas-
ticity. Consequently no bodies are totally devoid of this
property yet some have it more eminently than others; balls
of clay have little elasticity, but ivory balls and balls of glass
are considerably elastic. The degree of elasticity is measured
The following experimental fact seems to shew that impulsive forces are of the
same nature as finite forces, generating or destroying velocity by continuous gradations.
Robins' experiments on the velocity of bullets and cannon balls lead to the fol-
lowing result. If bullets of the same diameter and density impinge on the same solid
substance with different velocities they will penetrate that substance to different depths,
which will be in the duplicate ratio of those velocities nearly; Robins' Mathematical
Tracts, edited by Wilson, Vol. I. p. 152.
This was proved by various experiments. Now a property of uniformly accele-
rating (or retarding) forces is, that the squares of the velocities generated (or destroyed)
are proportional to the spaces described: Art. 204. Hence the retarding force of the
solid substance used in each experiment was a uniform force. But the duration of its
action was so short and its intensity so great, that although the changes effected by the
force were continuous, yet they were so rapid, that the force comes under the denonți-
nation of what we term impulsive forces.
198
DYNAMICS.
by the ratio which the velocity of rebound bears to the velocity
at the first contact. The elasticity is perfect when these two
velocities are the same, but this is a limit which no bodies
actually attain. The cause of this property of matter is of
course conjectural, and our conclusions as to its laws are
deduced solely from experiment.
Tables of the results of a series of experiments made by
Mr Hodgkinson, of Manchester, on the elasticity of bodies will
be found in Vol. III. p. 534. of the Reports of the British
Association for the Advancement of Science. The following
are the Conclusions deduced.
(1). All rigid bodies are possessed of some degree of elas-
ticity and among bodies of the same nature, the hardest are
generally the most elastic.
(2). There are no perfectly hard inelastic bodies, as
assumed by the earlier, and some modern writers on Mechanics.
(3). The elasticity as measured by the velocity of recoil
divided by the velocity of impact is a ratio, which, though
decreasing as the velocity increases, is nearly constant, when
the same rigid bodies are struck together with considerably
different velocities.
(4). The elasticity, as defined in (3), is the same whether
the impinging bodies be great or small.
(5). The elasticity is the same, whatever be the relative
weights of the impinging bodies.
(6). In impacts between bodies differing very much in
hardness, the common elasticity is nearly that of the softer
body.
(7). In impacts between bodies of which the hardness
differs in any degree the resulting elasticity is made up of the
elasticities of both; each body contributing a part of its own
elasticity in proportion to its relative softness or compressibility.
221. Hence when one body impinges on another a mutual
pressure takes place, which by incessantly acting as the compres-
sion of the bodies goes on finally checks their relative motion;
after this new velocities are generated and the balls separate: and
our object now is to enquire what the connexion is between the
velocity destroyed during the compression of the balls and the
pressure that destroyed it, and also between the velocity gene-
CONNEXION BETWEEN PRESSURE AND MOTION. 199
rated and the pressure by which it is generated. The difficulty
of discovering this from experiment arises partly from the
immediate action of the forces of restitution after the force of
compression ceases to act: in consequence of which any expe-
riment upon impinging balls is sure to involve the action of
two sets of impulsive forces; first, those that act while the
compression of the figures of the bodies is taking place, and
secondly, those that act during the restitution of figure. We
may call the first kind impulsive forces of the nature of colli-
sion, and the second kind impulsive forces of the nature of
explosion: by the first velocity is destroyed, by the second
velocity is generated. By certain artifices, however, we may
overcome this difficulty.
222. We shall mention a few experiments which will lead
us to a satisfactory conclusion.
Let A and B be two balls (fig. 77.) suspended by threads
from two points C and D, so that they may just touch when
at rest and have their centres in the same horizontal line:
FAE, ƒBe circular arcs with centres C, D: now the velocities
of a ball in falling through different arcs of a circle to the
lowest point are in the proportion of the chords of those arcs,
as is proved in the note*. Let therefore a scale be placed
=
* The pressure producing motion W cos r Pt Wsin 0 (fig. 76): therefore the
moving force (Art. 218), and also the accelerating force (since the mass of the body is
invariable) varies as sin 0. Let a be the greatest value of 0: 7 the length of the
thread: then (a–0) is the space described by the body in the time t; and, supposing
that at each instant the body is moving in the tangent line to the arc, the accelerating
force = 1
तू
Į
ď² (a−0)
dt2
d° 0
ī
d t²
d20
.*.
d t²
varies as sin 0, and 2 c² sin & suppose;
= 0
do 2
...
= 4 c² (cos + const.)
Ꮎ
d Ꮎ
When = a, velocity = 0,
=
dt
d Ꮎ
4 c² (cos 0 cos a),
dt
0;
angular velocity at lowest point = 2 c√1 – cos a = 4c sin
=2c chord a.
Hence the velocity varies as the chord of the arc.
2
200
DYNAMICS.
below A and B so graduated as to mark the velocities of the
balls A and B when at the lowest positions by knowing the
arcs through which they move.
Now suppose a small steel point is fixed in A so that when
A and B come in contact separation is prevented. It is found
that if A and B are drawn through arcs of which the chords
are inversely as the masses of the bodies, and then left to
themselves, they will impinge and exactly destroy each others
velocity, a small allowance being made for the resistance of the
air. If one of the balls be moved through a greater arc, then
when the balls come in contact they will not be at rest, but
move in the direction in which that ball was moving before
impact. This shews, that when the bodies impinge on each
other with equal momenta, their mutual pressures exactly
balance the momenta; but if the momentum of one ball be
greater than the momentum of the other the mutual pressure is
not sufficient to overcome the momentum of the first, but not
only overcomes the momentum of the second but generates new
momentum. This is found to be true for masses and velocities
of all finite magnitudes.
Desaguliers mentions an experiment (Experimental Philo-
sophy, Vol. II. Lecture vI. p. 62.) in which he replaced 4 and
B by two cylinders closed at the outer extremities; one was
introduced a short way into the other, and the cavity filled
with gunpowder: it was found that after the explosion the
cylinders rose through arcs varying inversely as their masses.
Consequently the momenta generated by the action of the im-
pulsive force of the explosion were the same.
In these experiments suppose that the mass and velocity of
the body A remains the same: then if we vary the mass or
velocity of B we must change them so that the pressure on A
shall be the same: and this condition is that their product
shall be constant. Hence, then, a given impulsive pressure
This is
generates in different bodies the same momentum.
all that these experiments prove: they do not shew that the
pressure varies directly as the momentum generated. This,
however, we infer as in Art. 219.
223. Wherefore the results of the last nine Articles lead
us to the following Principle.
THIRD LAW OF MOTION.
201
When pressure communicates motion to a body, the mo-
mentum generated in a given short time is proportional to
the pressure.
This is called the Third Law of Motion.
Newton has given this Law under the more general form,
that Action and Reaction are equal and opposite.
If action
and reaction in dynamics be measured by the quantity of
motion gained and lost, this is an immediate deduction from
our Third Law of Motion.
224. Leibnitz in the Acta Eruditorum 1695, p. 149,
and after him Jean Bernoulli and others raised objections to
Newton's measure of force, contending that it ought to be
proportional to the product of the mass and the square of the
velocity.
In their own words, "A force is said to be dead (vis
mortua) which consists in nothing but the endeavour, or the
tendency to motion. Such is gravity" it was said "as long
as a heavy body hung by a thread endeavours to descend, but
cannot actually descend. A force is said to be alive or quick
(vis viva) which always accompanies actual motion, and
tends to produce a local motion. There is such a force in
a body falling by gravity when it has already acquired some
degrees of velocity." Professor Wolfius, quoted by Desa-
guliers; Exp. Phil. Vol. 11. p. 72, 80.
Our object in making this quotation is to shew the origin
of the term vis viva, which, as a term only, is still in use
among us. The incorrectness of the above notion appears
from the fact that it implies that matter has some inherent
power of exerting force when in motion which it has not
when at rest.
The reasoning by which these philosophers were led to
the idea that pressure should be measured by the product of
mass and the square of the velocity generated appears from
the nature of the experiments from which they argued.
was found that when balls of equal size and density impinged
upon clay they penetrated the clay by spaces which are as the
squares of the velocities of impact: as in the example of the
note in page 197. It was reasoned (as in that note) that
when balls are projected against different solid substances so
Cc
202
DYNAMICS.
as to penetrate to the same depth the forces will be as the
squares of the velocities and hence arises the mistake, for
this supposes that we measure force by the velocity generated
or destroyed in moving through a given space irrespective of
the time of motion: but we measure force by the velocity
generated in a given time irrespective of the space described.
If then we retain our definition of force estimated dynamically
by the velocity generated in a given time, the force must vary
as the product of the mass and the velocity generated in a
given time but if we were to adopt the second measure of
force estimated dynamically by the velocity generated in moving
through a given space we should find that the force varies as
the product of the mass and the square of the velocity generated.
The term vis viva is still used to express the product of
the mass and square of the velocity.
225. We shall now choose the units of pressure, or
statical force, and mass.
Let P be the pressure, ƒ the accelerating force and M the
mass, then P varies as Mf. Let the unit of pressure be that
of a body of which the mass is M' and the accelerating
force f' then
P: 1 Mf: M'f';
· P
Mf
M'f
we shall choose M' and f' so as to simplify this formula as
much as possible: let M'= 1, ƒ' = 1; then
P = Mƒ .....
(1),
the unit of pressure being the pressure of a body of a unit
of mass and acted on by the unit of accelerating force.
When the pressure is impulsive its unit is that of a body
of mass unity moving with a unit of velocity: if we, as
above, suppose
P = Mv
(2).
Let W be the weight of a body of which the mass is M,
and let the accelerating force of the Earth's attraction, or
gravity, equal g: then
UNITS OF PRESSURE AND MASS.
W = Mg
(3).
203
Also suppose that the body is homogeneous, of density p, and
volume V: let p' and V' be the density and volume of a body
of which the mass equals the unit of mass: then
M: 1 :: pV: p' V
.. M
pP
V'
=
1: then
to simplify this formula as
(4),
we shall choose p' and V' so as
much as possible: let p= 1,
M = pV
the unit of mass being the mass of a body of a unit of
volume and a unit of density.
By (3) (4) we have
W = pVg
(5).
Now by experiments made by Atwood's Machine described
in Art. 216, it is found that the spaces described by a body
falling freely from rest are 16.1, 3 × 16.1, 5 × 16.1,.... feet in
the first, second, third,.... seconds of time. Hence gravity
is a constant force and generates a velocity of 2 × 16.1 or
32.2 feet in a second of time. Wherefore if we take a foot
as the unit of length and a second as the unit of time we have
g = 32.2
IV = 32.2 Vp
P.....
1
and when p= 1 and IV = 1, V
(6),
(7);
; hence the relation
32.2
among the units chosen gives this result, that the unit of
weight is the weight of a body of the unit of density and
volume equal the 32.2th part of the unit of volume. The
density of distilled water is taken generally as the unit of
density; and a cubic foot as the unit of volume.
226. Having discovered the relation between the statical
and dynamical measures of force, (which was the desideratum
in Art. 214), we may now enunciate the Principles mentioned
in Arts. 212, 213.
204
DYNAMICS.
The Third Law of Motion and the units of measure
chosen in the last Article shew, that finite statical force equals
the moving force of the body resulting from its action; and
impulsive statical force equals the momentum of the body
resulting from its action. We shall suppose in what follows
that statical forces are all replaced by these dynamical measures.
Let m be the mass of any particle of a material system,
xyx its rectangular co-ordinates, then,
I. If the system be in motion under the action of finite
forces, the forces
m
d2x
dt2
m
d² y
d t²
d² z
m
d t²
acting on m parallel to the axes of x y z respectively, and similar
forces acting on each of the other particles of the system,
must, together with the impressed moving forces (Art. 211),
satisfy the conditions of equilibrium.
II. If the system be acted on by impulsive forces, the
forces
dx
dy
dz
m
m
m
,
dt
dt
dt
acting on m parallel to the axes, and similar forces acting
on each of the other particles of the system, must, together
with the impressed impulsive forces or momenta, satisfy the
conditions of equilibrium.
227. These Principles are the interpretation of the Three
Laws of Motion into mathematical language. The Laws
themselves are the results solely of observation and experi-
ment. But these Principles are the results not only of the
Laws, but also of certain conventional rules for measuring the
quantities treated of; without which indeed we could not make
the phenomena resulting from the Laws subjects of calculation.
We must therefore be careful to interpret all results to which
they lead us in conformity to these conventional rules.
CHAPTER II
THE MOTION OF A MATERIAL PARTICLE.
228. LET xyz be the co-ordinates to the particle at the
end of the time t, and m its mass.
Suppose the accelerating forces acting on the particle are
resolved parallel to the axes and compounded into three X,
Y, Z in these directions.
Then by the first of the Principles enunciated in Art. 226,
the moving forces
mX, mY,
m Z
M
d2 x
dt
d²y
d² z
m
m
dt
dt2
will be in equilibrium with each other at the time t.
Hence by the conditions of equilibrium of a particle acted
on by any forces given in Art. 23, we have the equations
dt2
0,
m X
m
d² x
0,
dť
do y
m Y - m
=
m Z-m
d² z
d t
0,
d² x
d² y
d² z
or
X,
Y,
Z.
d t²
d t²
dť
These are called the equations of motion of the material par-
ticle and by integration we shall have three equations in-
volving a, y, z, t and constant quantities.
206
SINGLE PARTICLE.
DYNAMICS.
By eliminating t we have two equations involving x, y, z
without t. These are the equations to the curve described by
the particle.
229. In the course of the integration six arbitrary con-
stants will be introduced: these are determined by the initial
circumstances of the motion: by the term initial we mean at
the epoch from which t is measured*. The general integrals
determine the nature only and not the dimensions of the curve
described. The dimensions depend upon the initial conditions.
These are, first, the three co-ordinates which give the position
of the particle at the commencement of the motion. By sub-
stituting these in the three integrals and putting t = 0 we have
three equations involving the six arbitrary constants and known
quantities. The other initial quantities are the velocity and
direction of projection, or, which amounts to the same, the
initial velocities* parallel to the three axes.
By differentiating the three integrals with respect to t, we
dx dy dz
shall have three equations involving x, y, ≈,
dt' dt
d t d t
,
and
the arbitrary constants: and giving the variable quantities their
initial values we have three more equations involving the arbi-
trary constants and known quantities.
From these six equations, then, we can determine the six
arbitrary constants and the problem is completely solved.
230. Suppose, on the other hand, the problem to be
solved be the converse of the one already considered, namely, to
determine the forces which will make a body describe a given
curve.
* If any particle of the system commence its motion with a finite velocity, this is
imparted to it by an impulsive force, which acts for so short a time as to produce its
effect instantaneously: for this reason it is evidently indifferent whether we measure the
time from the commencement or termination of the action of the impulsive force: and
the term initial velocity, though there is no velocity, rigorously speaking, at the
commencement of the motion, is perfectly allowable.
In short, when a system of material particles is projected into space and submitted
to the action of surrounding bodies, two entirely different systems of forces act upon
the particles. The first is a system of impulsive forces, of the nature described in
Arts. 199, 200: these produce their effect in an indefinitely short time, after which
they cease to act. The second system consists of forces of the nature described in the
same Articles: these require a length of time of sensible duration to produce their effect.
This latter system differs from the former merely in intensity.
MOTION OF A MATERIAL PARTICLE.
207
We shall in this case have given two equations involving
xyx, from which we are to obtain the three quantities
d²x d³y d²z
dt' dť
2
dt dť
indeterminate.
or X, Y, Z: this shews that the problem is
The following is the way to proceed.
The two equations involving x, y and x must be differen-
tiated twice with respect to t: by this means we have two
equations involving the four quantities X, Y, Z, and velocity (v).
ds²
dx² dy
dz
But v2
=
+
+
d f dt
d t
dt
d. v²
dx
dy
dz
X
+ Y
+2
dt
d t
dt
dt
d. v²
dy
dz
X + Y
+ Z
dx
dx dx
This is a third equation involving X, Y, Z, v. By assuming
a value of any one of these four quantities the other three may
be determined, in terms of xyz.
RECTILINEAR MOTION.
231. PROP. A body is acted on by a uniform force (that
of gravity for instance) the motion being in the line of action
of the force required to determine the motion.
Let a be the distance of the body at the time t from a fixed
point in its course, measured in the direction of the force: and
let g be the force.
Then the equation of motion is
ď² x
=
g.
df
By integration we have
dx
dt
= gt + C, C being an arbitrary constant.
208
SINGLE PARTICLE.
DYNAMICS.
To determine C we must refer to the initial circumstances
of the motion.
Suppose the body is projected with a velocity u in the di-
rection in which the force acts.
dx
Then when t = 0,
= U ;
.. u = C';
dt
dx
.*.
gt + u.
dt
Integrating again
X = gť² + ut + C'.
Let a be the distance of the body from the origin of x at
the commencement of the motion: then the initial circumstances
are that when t = 0, x = ɑ ; .. C' = a;
·· » = a + ut + 1 g ť²,
or the space described in the time t is ut + 1gt.
This is a necessary consequence of the second law of
motion.
If the body be not projected then u = 0 and x = a + 1 gť².
If the body be projected with a velocity u in a direction
opposite to that in which a is measured, then when t=0,
dx
= u since x is diminished as t increases: * and
dt
x = α
a
ut + ½ gť³.
* In Art. 191, it was shewn that if s be the space described in the time t by a body,
ds
and v its velocity at the end of that time, then ข.
dt
But if the space be measured in a direction opposite to that in which the motion
takes place, then, b and s' being the distances of the point from which the space is
measured at the commencement of the motion and at the end of the time t, then
sb-s' and
=
d t
ds'
dt
=V.
Also in Art. 205, it was shewn that if ƒ be the magnitude of the force at the end of
d² s
the time t, then=f. If, as before, the space be measured in the direction opposite
to that of the action of the force, then
d² s'
d t2
=f.
RECTILINEAR MOTION: CENTRAL FORCES.
209
PROP. A body falls towards a centre of force the in-
tensity of which varies directly as the distance of the body
from the centre: required to determine the motion.
232. Let μ be the magnitude of the force at a distance
unity from the centre of force: this is called the absolute force
of the centre: a the distance of the body from the centre at
the commencement of the motion, r the distance at the time t.
Then ua is the magnitude of the force at the distance :
and the equation of motion is
μα
d² x
thương
df
the negative sign being taken because the tendency of the
force is to diminish x;
2
dx d² x
dt dť²
dx
-
dt
dx²
integrating,
C-μα,
dť²
C being an arbitrary constant to be determined by the initial
circumstances of the motion: these are that when t=0, x=a, and
dx
the velocity, or
,
dt
= 0; .. C = μα;
dx²
= µ (a² − x²) ;
dt²
dt
1
1
dx
√ a² - x²
the negative sign being taken in extracting the square root
because a diminishes as t increases.
1
V
Integrating, t =
COS
- 1 ²² + C'
(b
μ
when t = 0, x
0, x = α,
.. C' = 0;
D D
210
DYNAMICS. SINGLE PARTICLE.
1
X
:. t =
- 1
COS
ль
a
when x = a, the body arrives at the centre;
.. time of falling into the centre =
The velocity is zero when
dx
dt
=
π
2 √ μ
0, or when x = a and
a: hence the body passes through the centre and stops at a
distance on the other side equal to the original distance. From
this point it will return to its original position and continually
oscillate over the same space: the time of oscillation from rest
It is remarkable that this is independent of
to rest is
π
Ju
the initial distance of the body from the centre of force.
The expression for the time shews, that the body will
oscillate backwards and forwards: for suppose a is the least
positive value of cos-¹
X
for any given value of x, then
a
α
2 п α
2π + α
t =
or
or
Tū
μ
2ηπτα
or, generally,
,
n being any integer.
Vμ
This proves that the body will periodically arrive at any
given point of its path: the intervals of time between the
2π
· 2a
2 a
successive arrivals being
and
alternately.
Vu
PROP. Suppose the body in the last Proposition is pro-
jected with a velocity u in the line in which the force acts.
233. As before we have
d t
dx²
= C
μ
RECTILINEAR MOTION: CENTRAL FORCES.
211
dx
when x = α,
= u or
-
u according as the direction of
pro-
dt
jection is from or towards the centre: in both cases
u² = C - μa²
dx²
u² + µ (a² − x²).
dt2
Considering the motion towards the centre
dt
1
dx
u2
μ
a² +
x²
ле
1
x
1
t
COS
+ C
μ
a² +
u2
u
when t = 0, x = a;
1
a
α
1
COS
cos-1
μ
u²
u2
a² +
The greatest distance to which the body goes from the
centre is
a² +
u2
>
μ
and the time of a complete oscillation
from rest to rest is as before
π
Tu
PROP. A body falls towards a centre of force the inten-
sity of which varies inversely as the square of the distance
of the body: required to determine the motion.
234. Let be the absolute force of the centre as before:
μ
u
then the force at distance a is equal to and the equation
of motion is
u
dt
212
DYNAMICS.
SINGLE PARTICLE.
dx d² x
2μ dx
2
dt dt2
x² dt
da²
2 μ
integrating,
+ C
dt2
Ꮖ
dx
2μ
when x = a,
0; .. 0
... =
+ C
dt
00
dar² = 2x (---)
dť
м
a
dt
a
al
dx
2 μ
a
20
α
20
2μ Vax
a
a
X
dt
a
2
2
dx
2μ Vax
a
t =
a x
x²
212
a
vers
x²
x²
2
2x
1
+ C
a
2 м
when t = 0, x = a; .. 0
а ап
+ C
2µμ 2
t =
√la-vers-
2 м 12
2
2x
1 2 + √ax - a²
a
when the body arrives at the centre x = 0, therefore time of
falling to the centre
π
a
2
24
235. In a subsequent part of this work we shall see, that
the attraction of the Earth on external bodies varies inversely
as the square of the distance from its centre, supposing the
Earth a sphere. And that the attraction on any bodies within
the Earth varies directly as the distance from the centre.
It is for this reason that in the foregoing Propositions we
have selected these particular laws of force. No other laws
are known to exist in the universe.
RECTILINEAR MOTION ON AN INCLINED PLANE.
213
PROP. A body acted on by the constant force of gravity
moves down an inclined plane: required to calculate the
motion.
236. Let the plane of the paper be the vertical plane
in which the motion takes place: AB (fig. 78.) the intersection
of this with the inclined plane: P the position of the body at
the time t, A being its place when t = 0: a the angle the plane
makes with the horizon. Now the forces which are acting
upon the body at the time t are the force of gravity g, which
acts vertically and the pressure of the plane on the body. If
we resolve the forces in the direction of the motion we shall not
introduce the pressure.
Let AP = x.
Now the part of g resolved along the line AP is g sin a,
hence the equation of motion is
ď x
g sin a,
dt²
and the results will be precisely the same as those in Art. 231,
if we there substitute g sin a instead of g.
If we wish to know the pressure P upon the plane, by
resolving the forces perpendicularly to the line of motion we
have, since no space is described by the body in that direction,
0 mg cos a
P (Art. 225.)
Hence P = mg cosa, and is constant and is in proportion
to the weight of the body in the ratio cos a
: 1.
CURVILINEAR MOTION OF A PARTICLE.
PROP. A body is acted on by the constant force of
gravity, which acts in parallel lines: required to determine
the motion of the body when it is projected in a direction not
vertical.
The
237. Let the axis of y be vertical and reckoned positive
upwards and drawn through the point of projection.
motion will evidently take place wholly in a vertical plane.
214
DYNAMICS.
SINGLE PARTICLE.
Let the axis of a be drawn in this plane the origin being the
point of projection A, (fig. 79.) Let also g be the accelerating
force of gravity.
Then the equations of motion are
dx
d²y
0,
dt
dt2
By integration
dx
dy
C,
d' - gt,
dt
dt
c and c' being constants to be determined by the circumstances
of projection.
Let u be the velocity of projection, a the angle its direc-
tion makes with the axis of x.
dx
dy
Then when t = 0,
u cos a,
u sin a;
dt
dt
... u cos a = c, u sin a c
c' ;
dx
dy
= u cos a,
= u sin a
gt.
dt
dt
Integrating again
x = ut cosa, y = ut sina - gt...... (1),
no constants are added after integration because when t = 0,
x = 0 and y = 0 by the circumstances of the problem.
These two equations determine the position of the body
at any time.
238. To find the curve described we eliminate t from
equations (1); Art. 228;
80
y = x tan a
gx²
2u² cos² a
This is the equation to a parabola.
For it may be written
u2
Qu²
(
cosa sin a)²
cos" a (y
85
sin" a).
2g
CURVILINEAR MOTION: UNIFORM FORCE OF GRAVITY. 215
And by transferring the origin to a point of which the
co-ordinates are
u²
U²
cos a sin a and
g
sin a
2g
the equation becomes
2 u²
x²
cos² ay,
g
which is the equation to a parabola with its axis vertical and
measured downwards,
Qu²
and latus rectum
cos² a.
g
The range is the distance between the point of projection
and the point where the body strikes the ground.
described is called the projectile.
The curve
PROP. To find the range of the projectile, the time of
flight, and the greatest height the body reaches.
239. When y = 0, x tan a
gx²
Qu² cos² a
: 0;
Qu²
u²
.. x = 0 and a =
cos² a tan a =
sin 2 a,
g
g
this latter value of x is the range on a horizontal plane.
If the body be projected from an inclined plane perpen-
dicular to the plane of the projectile, then, if i be the angle
of inclination of the plane to the horizon, y = a tani is the
equation to the intersection of this plane and the plane of
motion: and the value of a when the body strikes the plane is
found from
x tan i = ∞ tan a
gx²
2 u² cos² a
QuⓇ
.. =0, and x =
cos a (tana - tan i)
Qu² cosa sin (a−i)
=
g
g
COS &
this latter value of is the range on the inclined plane.
216
DYNAMICS. SINGLE PARTICLE.
By (1) x = ut cos a ;
therefore time of flight on the inclined plane
2u sin (a — i)
X
2 и
;
sin
ɑ,
if i = 0.
u cos a
g
cos i
g
When the body reaches its greatest height
dy
u²
2
U²
0; .. x
.*. X=
tan a cos² a =
sin a cos a
dx
g
g
u²
u²
.. greatest height
g
180
{sin² a - sina}
sin2 α.
2g
CENTRAL FORCES.
240. Forces which continually tend towards a given point,
and the intensity of which depends upon the distance from that
point, whether fixed or in motion, are called Central Forces.
All the forces with which we are acquainted in nature are of
this description, as will appear in the sequel. For this reason
we shall devote a large portion of these pages to the con-
sideration of their action.
We shall, in the first place, investigate the most important
general properties of orbits described by bodies moving under
the influence of central forces, and in the next place determine
the nature of the orbits when the law and intensity of the forces
are given, and, conversely, determine the forces requisite to
cause a body to describe given orbits.
PROP. When a body is acted on by one central force the
motion is wholly in one plane.
241. Suppose xyz are the co-ordinates at the time t to a
material particle moving about a centre of force, the origin of
co-ordinates being at this centre: the distance of the particle
from the centre: and let P, some function of r, represent the
intensity of the force at the distance r.
The resolved parts of this force parallel to the three axes
of co-ordinates are
CENTRAL ORBITS: GENERAL PROPERTIES.
217
X
P
ру
and P,
Z
ጥ
and since these tend to diminish the co-ordinates the equations
of motion are
ď² x
dt2
20
d²y
P
ру
d
૨૨
d t2
2°
d t2
**
P
22
…….. (1).
Multiplying the first by y and the second by a and sub-
tracting the equations we have
d² y
ď² x
0;
XC
У
dt
dt²
dy
dx
*. X
y
h,
dt
dt
being an arbitrary constant.
dx
dz
In like manner z
Ꮳ
h₁,
dt
dt
dz dy
Y
hes
dt
dt
h₁ and h₂ being arbitrary constants, which, as well as h, are to
be determined by the circumstances of the motion at any given
time.
Now multiply these last three equations by ≈, y, x respect-
ively and add them together;
. 0 = hã+ky+hử,
This is the equation to an invariable plane passing through the
origin of co-ordinates, its position depending on the values of
h h₁ h₂.
1
Hence the motion takes place wholly in a plane passing
through the centre of force, the position depending upon the
initial (or any other given) circumstances of the motion.
PROP.
The areas described by the body about the centre
of force are proportional to the time.
E E
218
DYNAMICS. SINGLE PARTICLE.
242. In consequence of the property proved in the last
Proposition we shall refer the body's motion to two co-ordinates
instead of three. Let the plane of motion be the plane xy.
Then the equations of motion are
d² x
a
dεy
· P.
(1),
-
Р
- Py
(2),
d t²
d t
and, as before, we obtain
dy
d x
X
y
h,
dt
dt
and let A be the sectorial area swept out during the time t by
the radius vector;
d A
dt
= 32 ( ~
dy
dx
h
8
J
Y
by Diff. Calc.
dt
d t
2
ht
.. A
2
if t and A be both measured from the commencement of the
motion. This proves that the area swept out by the radius-
vector is proportional to the time of describing it.
When polar co-ordinates are used let O be the angle that
the radius-vector makes with the axis of x; then x = r cos e
and y = r sin : and by substitution
d A
dt
= 글
​d Ꮎ
d t
202
22
d Ꮎ
h.
dt
a
The following is an immediate consequence of this property.
PROP. To prove that the velocity of the body at different
parts of its path is inversely proportional to the perpendicular
on the tangent.
ds
ds do
243. Velocity = v
dt
do dt
202 do
,
by the Differential Calculus,
P
dt
J
CENTRAL ORBITS: GENERAL PROPERTIES.
219
p is the perpendicular on the tangent at the distance r,
z² h
h
by last Art.
Ρ
p
PROP.
To prove that the velocity is independent of the
path described.
244. Multiply equations (1) (2) of Art. 242. by 2
dx
dt
dy
2
respectively and add them, then
dt
dx d²x dy d²y
2 P
2
+2
XC
dx
+ y
dy
dt dt²
dt dť
gr
dt
dt
dr
2 P
,
x² + y²
=
p²;
dt
ds2
da
dy
but v²
+
d t
d t²
dť
dr
—
2 P
;
d. v°
dt
dt
.. v² = V² − 2 ƒ½ Pdr, r = R when v = V ;
f
R
and since this, when integrated from one position of the body
to another, will be a function only of the corresponding dist-
ances, it follows, that the velocity is independent of the orbit
described, and at any given distance depends solely on the
magnitude and law of the force and the velocity and distance
of projection.
COR. This is true also when the body is acted on by any
number of central forces tending to fixed centres.
There is one more property of central orbits which we shall
demonstrate owing to its utility in determining the velocity
whenever the force and orbit are known.
PROP. To prove that the velocity at any point of a
central orbit is that due to a body falling through one fourth
of the chord of curvature at that point through the centre of
force under the action of the force at that point supposed to
remain constant.
220
DYNAMICS. SINGLE PARTICLE.
dv
245. By last Article v
P.
dr
h
Also by Art. 243. v =
differentiate the logarithm of each side of this equation;
p
1 dv
1 dp
v dr
p dr
divide the first equation by this;
dr
... v² = PP
dp
of force at dist. r.
2P 1 chord of curvature through the centre
4
Hence the Proposition is true.
Having demonstrated these Properties of Central Orbits
we shall proceed to the determination of the nature of the
orbits themselves.
PROP. A body being acted on by a central force: re-
quired to find the polar equation to its path.
246. The equations of motion are
d² x
dt2
20
d² y
- P-
(1),
-py
P
(2);
T
dt
2'
ď² y
ď² x
x
Y
0,
d t2
d t²
dy
dx
00
y
= constant = hg
dt
dt
cos✪
putting a = r cos and y = r sin 0; we have
do
212
h.
dt
dx
dy
Again, multiplying (1) and (2) by 2
and 2
and adding,
dt
dt
DIFFERENTIAL EQUATION TO CENTRAL ORBITS.
221
dx d²x
2
dt dť
P
dx
+ 2 dy dy -- 21 (ie die + y dy);
dt dť²
dt dt
9
d fdx² dy²
dt dt + dt
dr
2 P
x² + y² = r²,
dt
and introducing polar co-ordinates
d
d 22
+
dt
d Ꮎ
)
d Ꮎ
dr
2 P
dť
dt
do
h
But
dt
d
1 dr2
2 P dr
+
Ꮎ
み
​dont do ²
h2 de
Put
2°
1
= 2:
1 dr
du
and
22 do
d Ꮎ
d [du²
ᏧᎾ ] de
2 P du
+ u²
h² u² do
and then performing the differentiation on the left-hand side
du
and dividing by 2
d Ꮎ
ď² u
+ u =
d 02
P
h² u²
This is the differential equation to the orbit described. The
force P being given in terms of r, we must integrate this
equation and the solution will be the equation to the orbit
described.
:
The integral will contain three arbitrary constants, two
introduced in the process of integration and the other, h, exist-
ing in the differential equation. These are determined by the
initial (or any other given) circumstances of the motion: viz.
the velocity, distance, and direction of projection.
The general integral determines only the nature of the
orbit described: but the circumstances of the motion at any
given time determine the species and dimensions of the orbit.
222
DYNAMICS. SINGLE PARTICLE.
247. The differential equation P = h²u²
ď³ u
+ и
I
dᎾ
do²
u}
may
be used to ascertain the law of force which must act upon
a body to cause it to describe a given curve. To effect this
we must determine the relation between u and 0 from the
equation to the orbit: we must then differentiate u twice with
respect to and substitute the result in the expression for P,
eliminating 0, if it occur, by means of the relation between
u and 0. In this way we shall obtain P in terms of u alone,
and therefore of r alone.
248. When we know the relation between r and 0, we
de
dt
make use of the equation ² h to determine the time of
=
describing a given portion of the orbit: or, conversely, to find
the position of the body in its orbit at any time.
We proceed now to exemplify these principles by various
applications.
PROP. A body moves about a centre of force varying
directly as the distance: required to determine the motion.
249. Let be the absolute force: then P = μr
u
м
U
In order to simplify the calculation we shall first suppose
the body projected perpendicular to the radius vector.
Let V, R be the velocity and distance of projection :
.. h = 2 area described in 1″ VR by Art. 243.
ď u
+ W
м
d 02
V2R2 us
du
multiplying by 2
and integrating
de
du²
th
+ u² = C ·
-
d02
when
1
и
=
dr
R.
= 0
0 and
VR u² 9
du
0%
d Ꮎ
d Ꮎ
1
C
+
R
CENTRAL ORBITS. FORCE VARYING AS THE DISTANCE. 223
du²
d02
V² + R² μ
R2V2
ль
u²;
V2R2u2
d. u² 2
V² – R² µ
2
d Ꮎ
2 R2 V2
- (2²
V² + R² μ
2
;
2 R2V2
extracting the square root, inverting, and integrating
2 0 + C = sin-¹
2 R² V ² u² − ( V² + R³µ)
V² – R³µ
T
when = 0, u =
•. C = sin¹ 1
R'
1
༡༠
1 =
(V² + R³µ) + (V² – R² µ) cos 20
2 R2 V2
V² cos² + R2u sin² 0
R2 V 2
r cos 2
("Das 0) + (√μr sin @)"
R
V
2
Hence the orbit is an ellipse, the force being in the centre.
The semiaxes are R and
V
Vu
250. The periodic time may be found by integrating the
equation
dt
202
>
do h
more simple.
Periodic time
(Art. 242). But the following method is
=
2 area of ellipse
h
2пR
V
Ju
VR
2 п
,
(see Art. 242.)
μ
This result is remarkable; for it shews that the period is
independent of the dimensions of the ellipse and depends
solely on the intensity of the force.
*
224
SINGLE PARTICLE.
DYNAMICS.
π
2
251. COR. 1.
it will be found,
If the angle of projection be ẞ instead of
that the orbit is still an ellipse, the force
being in the centre; and if a, b be the semi-axes*,
* This may be demonstrated with greater facility by using the equations of
motion, which are, in this case,
d² x
dt2
d² y
μπ
μy.
dt2
dx
dy
Multiplying them respectively by 2
2
dt'
dt
and integrating we have
dx2
dy2
dt2
= µ (h² — x²),
=
d t2 =µ (k² — y²) ....
(1),
h and k being arbitrary constants, introduced in the above form for the sake of
symmetry;
dy2
dx²
У
k
=
k² — y²
h² - x²
..sin-1 = sin-1
c being an arbitrary constant ;
1
√k² - y² dx
X
h
+ sin-1 c
dy
1
=***
√h² -- x²
(2),
Y
k
=
h
√1 − c² + c
1
ww
by transposing and squaring and transposing again
n2
h2
y2
y²
x² 2√1-c² xy
=
c².
k² + h²
hk
This is the equation to an ellipse from the centre: since B² – 4 AC
-
4(1-c²)
4
h² k² h² k²
4 c2
is essentially negative; A, B, C being the coefficients of y², xy, x² re-
h2 k²
spectively.
be the velocity of projection,
with the axis of a, a, b, the co-
In order to determine the constants h, k, c, let
a the angle which the direction of projection makes
ordinates to the point of projection: then equations (1) give
V² sin² a = µ (k² – b²),
V² cos² a = µ (h² — a²),
万
​a
and (2) gives sin-1
-
k
sin-1
h
=
sin-1 c
by which h, k, c are known.
If, as in Art. 249, we suppose the body projected from the axis of at right
angles to that line, then b=0, a =
V²,
= 90°;
.. h² = a²,
uk2
µ k² = V ²,
Ꮮ
a
k
sin-1 c sin-1 sin-1
=
and the equation to the orbit becomes
sin-1, therefore c² = 1,
μ
bride-
1
1.
a2
The equation to an ellipse of which the semi-axes are a and
V
FORCE VARYING AS THE SQUARE OF THE DIST. INVERS. 225
2 V*R* sin² ß
1
1
and -
V²+ μR²±√ (V²+µR²)2−4µ VR sin² ß
respectively.
a²
b
2πab
2 п
Hence periodic time =
the same result as before.
(h =) VR sin ߨ¯ Vu
μ
COR. 2. The result of this Proposition is of great im-
portance in Physical Optics. For the forces which act upon
the disturbed molecules of the vibrating medium of light all
vary as the distance so long as the displacements are not very
great. Now the colour of the light is assumed to depend upon
the time of vibration of the molecules: and the intensity of the
light upon the extent and magnitude of the vibrations, that is,
upon the quantity of motion. The preceding Proposition shews,
then, that light may alter in intensity without changing in
colour, since the time of vibration is independent of the mag-
nitude of the motion, when the law of force is that of the di-
rect distance.
PROP. A body is acted on by a central force varying
inversely as the square of the distance required to deter-
mine the orbit described.
Many Propositions of this description may be solved in the following manner.
By Arts. 243, 244, v² 12 - 2 2 [[ Par, v = "";
=
f
h
=
[2-2
R
Ρ
p2
he
Ex. 1. Let P = µr;
p²
+ µ R² — µ ²²²,
2 S
Pdr.
R
which is the equation to an ellipse about the centre, the axes being given by the
equations
ho
a² + b²
+ R², a²b²
14
μ
he 2 μ
Ex. 2.
Let P =
μ
+ 12.
7.2
R
This is the equation to a conic section about the focus.
1
2 α
1
The equation to the ellipse is
-
p2
br
1 2 (1
1
hyperbola is
12
+
b²r
1
1
parabola is
#
p²
་
Fr
In
226
DYNAMICS. SINGLE PARTICLE.
252. Let
Let μ be the absolute force: then P =
и чег
V, R, ẞ the velocity, distance, and angle of projection: then
h = VR sin ẞ (Art. 243.)
d² u
ре
+ u = >
d02
h
du
multiplying by 2
and integrating
dᎾ
du2
2 м
+ u²
u + C,
dᎾ
h²
1
ᏧᎾ
when
R, r
U
dr'
or the tangent of the angle between the
radius vector and the tangent line,
ta
tan ß:
d Ꮎ
... U
tan ß;
du
1
1
2 м
1
2 м
V2R-2μ
.. C
+
;
R2 tan B
R2
h2 R
R2 sin² ß
2
h² R
h² R
du²
V2R
-
d02
h² R
+
2
x-(-x):
extracting the square root, inverting, and integrating
2 pe
;
μ
h2
0 + C' = cos−1
V²R - 2 μ
h² R
+
h¹
1
C' is found by the condition, that when u
0 = 0.
R
In the case of the ellipse
a
1
V2
ре
=
b2 h²
b2
Rh²
h2
Rμ
=D '.'
b
2μ- V²R'
Rh2
2μ- V2R
The path is an ellipse, hyperbola, or parabola according as V2 is less than, greater
than, or equal to
2μ
R
FORCE VARYING AS THE SQUARE OF THE DIST. INVERS. 227

Then
1
μ
+
7 h2
is the equation to the
V²R - 2 μ
h2 R
Ав
2
+ cos (0 + C')
h¹
path: it is the equation to a conic
and may be written
section from the focus,
1
1 + e cos (0+C')
;
ጥ
a (1 ~ e²)
the angle 0 + C' being measured from the shorter length of the
axis major, and 2a and 2a√1 – e² being the axes:
V2R-2μ
Then e
h² + 1, subs. for h
Rμ
V2R
Ωμ
RV2 sin ẞ+1... (1);
u²
h²
V2R2 sin² B
and a (1
e²)
(2).
u
Now the conic section is an ellipse, parabola, or hyperbola
according as e is less than, equal to, or greater than unity.
Hence, from equation (1), the orbit described is an ellipse,
parabola, or hyperbola about the focus according as V² is less
than, equal to, or greater than
2 μ
Ꭱ
This proves the remark-
able property, that the species of the conic section described
is independent of the direction of projection.
= 2 a
In the case of the ellipse and hyperbola the axis major
2 μ R
V2R
of projection.
u
2 μ
and this is also independent of the direction
In the case of the parabola, the
the focus, or D = a (1 − e) (e = 1)
distance of the vertex from
V2R2 sin² ß
The position of the axis major with respect to the radius
vector R, is determined by C', which is the angle between these
two lines.
Put = 0 and r = Rin the value of
ין
228
SINGLE PARTICLE.
DYNAMICS.
..cos C'
a (1 ~
e²)
V2R sin² ß - μ
Re
e
ме
By referring to Art. 234, we see that the velocity of
a body falling from an infinite distance to a distance R from
м
a centre of force is equal to A
2.2
2 μ
R
Hence the orbit
described about this centre of force will be an ellipse, parabola,
or hyperbola according as the velocity is less than, equal to,
or greater than that from infinity.
253. We might make use of the equation
P = h²u²
ď² u
dᎾ?
+ и
to discover the law of force when the orbit is given.
Thus if the orbit be a conic section with the force in one
of the foci, and m be the distance of the pole from the nearest
vertex, then the equation to the orbit is
1 + e cos 0
h² u²
P =
m (1 + e)
m (1 + e)
h2 1
m(1+e) p²
or the only law of force is that of the inverse square of the
distance.
If the orbit be the ellipse the centre being the centre of
force, then, a and b being the semi-axes,
u²
cos² 0
+
sin? Ꮎ
du
; u
cos Ꮎ sin Ꮎ,
a²
b2
d Ꮎ b2
a²
.. P =
h²
и
h²
{
ď² u
+
dᎾ
zet + чез
+23
do
du²
d02
a²
(~ ~ - -—-) (cos° 0 – sin³ 0);
-
d² u
h²
И
{
du
U¹
u2
+ u² u
d02
(u
ď³ u
du²
+
d Ꮎ
d0
U
{
cos 0
sin² 0
2
+
a²
b?
cos² sin² 0
+(cos + sin²) (-) (cos² 0 — sin²
a²
- 0
b
a
1
CENTRIFUGAL FORCE.
229
h² (cos¹0 + 2 cos² 0 sin² + sin¹
✪ Ø
h
ว
a² b²
a²b²
2
and therefore the only law is that of the direct distance.
254. If the orbit be a circle the centre of force being in
the centre of the circle: then a being the radius r = a is its
equation and
h²
P = h'u³
α
dt
Also by Art. 242,
11
d Ꮎ
702
h
11
a²
h
in this case, and therefore the
velocity is constant;
:. ht = a² 0 + const.
when t 0, suppose ◊ = 0; and when t T, the time of re-
volution,
.. hT=2a".
=
2π;
Va
Let V be the velocity, then h = l'a Art. 243;
12
Ωπα
·、 P =
and T
Q
:
Since the velocity is uniform it follows that the force produces
no effect upon the velocity in short, the only effect of the
force is to deflect the body from the rectilinear path which it
would describe with the uniform velocity V if no force acted.
Consequently the central force is a measure of the tendency
that the body has at every instant to preserve a rectilinear
course. This tendency is sometimes called the Centrifugal
Force; and the central force is then called in reference to this
the Centripetal Force.
When a particle describes a curve in space the force which
acts upon it is employed partly in changing the velocity and
partly in deflecting the course of the body. A force equal
and opposite to the part of the force which deflects the course
of the body is called the centrifugal force in this general case
as well as in that specified above.
230
DYNAMICS. SINGLE PARTICLE.
PROP. To prove that the centrifugal force of a particle
moving in space at any point of its course equals the square
of the velocity divided by the radius of absolute curvature at
that point, and acts in the osculating plane.
255. If X, Y, Z be the accelerating forces acting on the
particle, the equations of motion are
ď²x
ď³y
d2
d² z
X
Y
Z.
dt2
dt2
dt2
Now if s, the arc described, be the independent variable
the absolute radius of curvature (p) is given by the equations
1
ď²x
ds2
+
d² y
ds
ď² z
2
+
ds²
(1).
Hence if we change the independent variable in the equations
of motion from t to s, we have
dt d²x dx d² t
ds ds2
ds ds2
d² x
dx d²s
ds
X
v2
+
dt³
ds2
ds dť²'
,
dt
ds3
in like manner
Y = v²
2
dy
dy d²s
ď² z
dz d² s
+
Z = v²
+
ds2
ds dt:
ds²
ds dt
If, then, P be the resultant of X, Y, Z we have
P² = x² + Y² + Z²
= v¹
1
& x
ď² y
d² ≈
+
+
ds
ds²
ds™
+ 2 v²
d's fda d'a
dy d³y
dz d²z
+
+
dfds ds?
ds ds² ds ds
+
Jd
(dx² dy³
dz
d's
2
+
+
ds ds ds
dt²
d.
dy™
d
But
+
-1, and therefore by equation (1)
ds
ds
ds²
CENTRIFUGAL FORCE.
231
p?
des
Now
d t²
2
to
+
d's
dť
2
(2).
is the part of the force P that produces the
change in velocity (Art. 206) and the other part acts at
?y°
P
right angles to the former, as the form of equation (2) shews,
and consequently is what we have termed the centrifugal force:
this expression proves the first part of the Proposition.
The force P acts through the point (xyz), let
X1 x = A (≈1 − ≈), Y₁ − y = B (≈1 − ≈)
—
1
be the equations to its direction: then the cosines of the angles
which this line makes with the axes are
A
B
1
1 + A2 + B²
√ 1 + Ų + B²
1 + A + B²
but these cosines are also
X
Y
Z Ꮓ
√ x² + Y² + Z²
√x²+ Y² + Z
√ X² + Y² + Z®°
X
I
Hence A
B
Ζ'
Z
and the equations to the direction of the resultant are
−
Z (x, x) = X (≈, − ≈), Z (y₁ − y) = Y (≈, − ≈),
the equations to the tangent line, or the line in which the
des
force
acts, are
d t²
(x, − x)
dz
ds
dx
dz
ds,
(༧ - ~) (Y₁ − y) d x = (~, − ≈)
dy
ds
ds
Hence the equation to the plane passing through these lines,
or the plane in which the centrifugal force acts, is
(≈1 − ≈) [X Y
dy
ds
dx
dx
dz
+(y₁y) (Z
+ (Y₁ − y)
Ꮧ
ds
ds
ds
dz
+ (№1
x) { v
Z
Ꮓ
ds
ds
dy)
0,
232
DYNAMICS. SINGLE PARTICLE.
and substituting in this equation the values of X, Y, Z it
becomes
'dy d2x dx d'y
ds ds ds ds
+ (1₁ − y) (
dx d² z
dz d²x
ds ds2
ds ds2
+ (x2 − ∞) (
dz d² y
dy do z
=
0,
ds ds²
ds ds)
which is the equation to the osculating plane at the point
(xy), the arc s being the independent variable.
Hence the second part of the Proposition is true.
After this digression respecting centrifugal force we shall
return to the subject of central forces.
256. Kepler discovered by calculation depending on ob-
servations, that the planet Mars moves in an ellipse having
the Sun in the focus. He also discovered that the areas de-
scribed by the planet when near its perihelion and aphelion
distances (that is, the nearest and farthest distances from the
Sun) were proportional to the times of describing them. These
two empiric laws have since been proved to hold for the other
planets and also for every part of their course. Kepler like-
wise discovered that the squares of the periodic times of the
planets about the Sun were in the same proportion as the
cubes of their mean distances. These three laws are known by
the name of Kepler's Laws and may be thus enunciated.
I. The planets move in ellipses, each having one of
its foci in the Sun's centre.
II. The areas swept out by each planet about the Sun
are, in the same orbit, proportional to the time of describing
them.
III. The squares of the periodic times of the planets
about the Sun are proportional to the cubes of the mean
distances.
We shall shew how we are led by these empiric laws to
conjecture respecting the nature of the force which acts upon
the planetary system.
PROP. To determine the nature of the force which acts
upon the planetary system.
LAW OF GRAVITATION FROM KEPLER'S LAWS.
233
257. Let XY be the forces which act on a planet
parallel to two co-ordinate axes drawn through the Sun in the
plane of motion of the planet: then the equations of motion
are
dx
ď y
X,
Y;
dt
dť
ď y
ď²x
=
*. x
Y
x Y − y X.
- yX.
d t²
dt
But by Kepler's Second Law the area is proportional to
the time: therefore area = c.t, c being the area described in
a unit of time:
d. area
dt
dy
dx
or
20
Y
= c;
dt
dt
d²y
dť
ď²x
Y
= 0;
dť
X XxX
Y Y
x Y − y X = 0;
This shews that the resolved parts of the force acting upon
the planet are proportional to the co-ordinates from the Sun's
centre: and therefore, by the composition of forces, the force
itself must pass through the Sun's centre.
Hence the forces acting on the planets all pass through
the Sun's centre.
Let
1
2⁰
or u =
1 + e cos (0 − a) be the equation to the
a (1-e²)
elliptic orbit: Kepler's First Law.
Then the force P, since we have shewn it to be central,
= h² u²
= h² u² [d² u
d02
+ u Art. 246.
e cos (0 - a)
a (1 − e²)
h2
"}
+
1 + e cos (0 - a)
1
a (1 − e³) µ‚²
G G
a (1 − e²)
234
DYNAMICS. SINGLE PARTICLE.
Hence the law of the force acting upon the planets is that
of the inverse square of the distance.
Let T be the periodic time of a planet and a the semi-axis
major of its orbit, and therefore the mean distance from the
Sun's centre.
Then T
2 area of ellipse 2πα Vi-e
h
h
h2
4π² a³
a (1 − e³)
T¹2
4π
2 a² 1
and P
=
T2
a³
T¹2
But by Kepler's Third Law is the same for all the
planets. Hence not only is the law of force the same for all
the planets; but the absolute force is the same: and conse-
quently the same cause seems to act on all the planets.
From this calculation, then, we conclude that the Sun
attracts the planets, and that with a force which varies as the
inverse square of their distances from his centre.
258. The elliptical orbits of the planets are nearly
circular: since, then, in a circle there is no variation of
distance, it may at first sight be a matter of doubt whether
the calculations which prove Kepler's Laws are sufficiently
accurate to allow us to believe the law of variation of the
Sun's attraction to be correctly determined.
For in the case
This doubt is, however, easily removed.
here contemplated the Third of Kepler's Laws determines both
the law and intensity of the Sun's attraction.
1
In this case u is constant and
;
a
h2
4π² a
472 a³ 1
..
·. P =
a³
T¹2
T2
T¹² a²
But T2 varies as a³, for different planets;
1
.. P varies as
for different planets,
a²
LAW OF GRAVITATION FROM KEPLER'S LAWS.
235
and therefore the law of attraction is that of the inverse square
as before and the magnitude is the same.
259. Now the greatest diameters of the planets are
proved by observation to be exceedingly small when com-
But in Art. 165,
pared with their distances from the Sun.
Cor. we have shewn that the constituent particles of bodies
of this description, if they attract, will attract according to
the same law as that according to which the bodies themselves
attract. And we have just shewn (Art. 257.) that the Sun
attracts the planets with a force varying inversely as the square
of the distance from his centre. It is therefore highly probable
that the particles of the Sun attract the particles of the planets,
and vice versâ, with a force varying directly as the mass of
the attracting particle and inversely as the square of the
distance.
260. These consequences to which we have been led by
Kepler's Laws are equally satisfied whether we suppose the
centre alone of each body to have an inherent property of
attraction, or each particle of the system to attract. But this
ambiguity is removed by Dr. Maskelyne's observations on the
stars from stations near the mountain Shehallien in Scotland.
By these it was proved that the mountain produced a sensible
effect in drawing the plumb line out of the vertical: see the
Philosophical Transactions, 1775. Also some beautiful ex-
periments by Cavendish on the attraction of leaden balls, re-
corded in the Philosophical Transactions, 1798, shew the same
thing; that the property of attraction does not reside only in
the centres of the heavenly bodies but in every portion of their
mass. We are therefore led to conjecture that matter is en-
dowed with a general gravitating principle by which every
particle attracts every other particle according to the law before
mentioned.
261. Were, however, this principle universally true, not
only would the Sun attract the planets, but the planets would
attract the Sun (which we have imagined immoveable*) and
likewise one another: and our calculations are erroneous, but
these depend on Kepler's Laws. Wherefore it follows, that
*
See Arts. 240. 246. In these the centre of force is fired.
236
DYNAMICS. SINGLE PARTICLE.
either Kepler's Laws are not true, or that Universal Gravita-
tion is not a Principle of Nature.
Now in point of fact observations of greater nicety than
those made by Kepler prove that his laws are not accurately
true, though they differ but slightly from the reality.
Here then is an additional argument (as far as it goes) in.
favour of Universal Gravitation. For since the magnitudes of
the planets are very small in comparison with that of the Sun,
we should anticipate that the perturbations of their elliptic
motion about the Sun and of the position of the Sun in space
by the action of the heavenly bodies would be small; and, con-
sequently, that the deviation from Kepler's Laws would not be
considerable.
262. Our investigations thus far are only a first approx-
imation to the truth: it yet remains to be determined whether
the perturbations actually experienced agree, both in their
nature and magnitude, with those which are calculated on this
hypothesis of Universal Gravitation. These are the real tests.
of the existence of such a principle. Probably many imaginary
laws would explain the ordinary phenomena of the motion of
the heavenly bodies; but that alone is the law of nature which
will stand the test of the more refined calculations of the per-
turbations.
It is by the complete harmony which is found to subsist
between the numerical results deduced from theory and obser-
vation, that we become convinced of the truth of the Law of
Universal Gravitation. To prove this complete accordance is
the object of Physical Astronomy.
263. Having stated the main arguments which lead us
to conjecture that the motions of the heavenly bodies are re-
gulated by a universal principle of attraction with which all
matter is endued, we proceed to a more strict investigation of
the consequences of this principle, and shall now enter upon the
consideration of the motion of a given number of material par-
ticles attracting each other with forces varying directly as the
mass of the attracting body and inversely as the square of the
distance. This Problem is one of insuperable difficulty when
considered in a general point of view, and has baffled the com-
bined exertions of mathematicians from the days of Newton to
LAW OF GRAVITATION FROM KEPLER'S LAWS.
237
For
the present time. In our Solar System, however, the masses
of the planets are so small in comparison with that of the Sun,
and the inclinations of the planes of their orbits to one another
is also so small that the Problem is rendered capable of solution
by methods of approximation. But it must not be imagined,
that the results are for this reason not to be relied upon.
by the process of successive approximation in which we begin
by obtaining a first approximation, thence proceeding to a
second, and so on, we can by extending our calculations ap-
proach as near the truth as we please: and although the num-
ber of calculations must, strictly speaking, be infinite in order.
to arrive, by this method, at an exact result, yet the error in
stopping at the third or fourth approximation is so slight as in
fact to be inappreciable to our senses. Suppose, for example,
that the longitude of the Sun's centre is calculated to be
134°. o'. 1" at some given time and that the real longitude is
134º: what difference does this make in a practical point of
view? But even if we were able to obtain an exact solution of
the Problem, yet in calculating numerical results we are
obliged to reduce the whole to decimals; and though the la-
bour in this case would be perhaps diminished, yet the result
would still be only approximate.
We shall first calculate the motions of two bodies, con-
sidered as particles, attracting each other, and then proceed to
the more general question.
CHAPTER III.
MOTION OF TWO MATERIAL PARTICLES ATTRACTING EACH OTHER.
PROP.
Two material particles attract each other with
forces varying inversely as the square of their distance and
directly as the mass of the attracting body: required to
determine the motion of their centre of gravity.
264. Let M and m be the masses of the two particles:
r their distance at the time t: then, if the unit of attraction
be the attraction of a unit of mass at a unit of distance, the
accelerating force produced in M by the attraction of m
and that produced in m by M's attraction
M
2º
Let xyz be co-ordinates to M at time t,
x'y's'
m
m
202
Then resolving the attractions parallel to the axes, and at-
tending to the directions in which the resolved parts act, the
equations of motion of M are
& x
dt
m (x-x')
203
dy
m (y-y') d² z
m (x − x')
d t
203
d t²
203
and those of m are
Y
d² x'
M (x − x')
d² y
M (y − y')
d²
M (x − x')
dt
703
df
2.3
d t
203
Multiply the first three equations by M and the last three
by m, and add the first, second, and third of the first set to the
first, second, and third of the second set respectively;
MOTION OF TWO MATERIAL PARTICLES, &c. 239
ď²x
ď² x'
d²y
d²y'
M
+ m
0, M
+ m
0,
dt2
dť
df
dt
.(1).
ď² z
M
ď z
+ m
= 0.
dt
dt
xy 2
Let zy be the co-ordinates to the centre of gravity of
the two bodies at the time t: then
(M + m) x = Mx +mx', (M+m) y = My+my',
(M + m) ≈ M≈ +mz'.
Differentiating these twice with respect to t and making
use of equations (1), we obtain
d²x
ďy
ď
0,
0,
0....
(2);
dt²
dť
df
dx
dy
do
b,
dt
dt
dt
a, b, c being constants to be determined by the initial cir-
cumstances of the motion of the bodies.
Hence the velocity of the centre of gravity = √ a² + b² +c²,
(Art. 210. Cor.) and is therefore uniform.
da α
dy b
Also
18
X
a
dz C dz C
b
===+a', y = =≈ + b',
C
≈+a
C
a', b' being constants to be determined as before.
These are the equations to the path of the centre of
gravity; and, since they are the equations to a straight line
in space, they prove that that point will move in a straight
line.
If a, b, c each = 0, then the expression for the velocity
of the centre of gravity vanishes: and the general conclusion
is, That the centre of gravity of the two bodies will either
remain at rest during the motion of the bodies, or move
240
TWO PARTICLES.
DYNAMICS.
uniformly in a straight line.
is determined by the initial
the bodies.
Which of these will be the case
circumstances of the motion of
PROP. To determine the orbits the bodies describe about
each other, and about their centre of gravity.
265. Let us subtract the equations of motion for m from
those of M respectively, and we obtain
+
d² (x-x')
(M + m) (x − x')
-
d t²
2.3
d² (y-y')
dt²
(M +m) (y — y')
203
ď²
d² (≈ - x')
(M + m) (≈ − ≈')
dt²
20:3
These are the equations we should obtain by supposing
either of the bodies at rest, and the force acting on the other
to be the sum of the masses divided by the square of the
distance.
Hence (Art. 252) each will describe relatively to the other
a conic section, the nature of the path being determined by the
circumstances of projection of the bodies.
266. To determine their paths about their centre of
gravity, let r, and r' be the distances of M and m from that
point at the time t: then
m
M
2°
プ
​2'.
M+ m
M + m
Also, if P and Q be the two particles (fig. 80), G their
centre of gravity,
PN
PN'
x − x'
X
х
PQ
PG'
r
1*
and in the same way
y - y'
Y - Y
and
グ
​プ
​Now subtract equations (2) of Art. 264. from the equa-
tions of motion of m in that Article respectively :
MOTION OF TWO MATERIAL PARTICLES, &c. 241
ď² (x − x)
dt
m (x − x')
20:3
m³
Ꮳ X
(M+m)² r³
d² (y − y)
m³
y-y
ď³ (≈ − 2)
m³
Z
and
dt
(M+m)² r³
3
d t³
(M+m)² r³
3
These are the equations of motion of M relatively to the centre
of gravity of M and m, which as we have seen is at rest, or
is moving uniformly in a straight line. They prove that the
path about the centre of gravity is such as would be de-
scribed about a force
m³
(M + m) r
1
2
residing in that point.
Hence the orbits of M and m relatively to the centre of
gravity are conic sections, their nature and magnitude being
determined by the circumstances of projection relatively to the
centre of gravity of M and m.
PROP. To compare the relative orbits of M and m about
their centre of gravity.
267. Let v, v' be the absolute velocities of projection
of M and m: aßy, a'B'y' the angles the directions of these
velocities make with the axes.
α
V and 'the relative vels. of project. about centre of gravity,
R and R' the initial distances from the centre of gravity,
Sand & the relative angles of projection,
a and a' the semi-axes major of the orbits,
e and e' the eccentricities of the orbits,
u and the absolute forces.
μ
Then by equations (1) (2) of Art. 252,
and
1
1 − e²²
a (1 ~ e²)
a' ( 1
Also
e²²)
R m
R'
M'
2 μ
V2R RV sin &
2 µ' - V²²R' R'V' sin d'u
V2R2 sin du
V'R' sin² d'
and
u
м
H H
M3
اوه بر در در
μ
by Art. 266.
242
DYNAMICS. TWO PARTICLES.
To find V, V', d, & we proceed as follows:
The velocities of the centre of gravity parallel to the axes
are at first and therefore during the motion respectively (Art.
264.)
Mv cos a+mv'cos a Mv cos ẞ+mv' cos B' Mv cos y+mv' cos y'
M+m
M+m
M+m
Also the absolute velocities of projection of M parallel to
the axes are v cos a, v cos ß, v cosy and therefore the rela-
tive velocities of projection of M about the centre of gravity
parallel to the axes are
m (v cos a—v′ cos a') m (v cos ß-v' cos ẞ′) m (v cos y-v'cos y´)
a-v' B')
M+m
M+m
M+m
Adding the squares of these, (Art. 210. Cor.) the square of
the relative velocity of M about the centre of gravity (V²) =
m²
(M+m) ³ { (vcosa¬v'cos a')² + (v cosß-v'cos ß')² + (v cosy —v'cos y')?}
= {
m²
(M + m)²
(v² + v²² – 2vv′ cos 4),
where A is the angle between the directions of projection of
M and m: and therefore determined by the equation
= cos a cos a' + cos ß cos ß' + cos y cos y'.
cos A
Similarly V¹² =
M2
(M + m)²
12
(v² + v′² − 2vv′ cos A).
Let the line joining M and m at the commencement of the
motion be the axis of : then the cosine of the angle which
the direction of V makes with the distance of projection (which
coincides with the axis of x), or cos d, equals the relative velo-
city parallel to the axis of a divided by the whole relative.
velocity (V) =
m
v cos a
M+ m
v' cos a
V
MOTION OF TWO MATERIAL PARTICLES, &C.
243
M
v' cos a'
Similarly cos &
s a' — v cos a
M+ m
V'
1 − e²
Substituting these in the expression given above for
we find
1 − e
1 - e²
12
1; .. e=e,
1 e
or the orbits are similar to each other.
3
m² M³
a
m
Also
a = M¹
α M¹ m³
3
M'
or the linear dimensions of the orbits of M and m are in the
ratio of m to M.
268. COR. 1. It follows from this that the perturbation of
the Sun by any planet is very small, because his mass is so
much the greater of the two masses.
In the same way it will be shewn that the combined effect
of the heavenly bodies in moving the Sun is very slight; and
therefore the error in Kepler's Laws, anticipated in Art. 261,
owing to the supposed immobility of the Sun, is not very
great. Thus far, then, we are confirmed in our hypothesis of
Universal Gravitation.
М
269. COR. 2. We have seen (Art. 257.) that if be the
absolute force of a centre of which the law is that of the inverse
square, and a the semi-axis major of the orbit described, the
periodic time
3
πα
Ωπα
(T)
=
(Art. 265.)
√ μ
√ M
M+ m
M and m being the masses of the Sun and a planet.
Let m' be the mass of another planet: and a' the semi-axis
major of its orbit, T'its period ;
.. T':
Ωπα
√ M + m²
T2
a³ M + m'
and ..
T/2
a
13 M + m
This shews that Kepler's Third Law would not be true
even if we suppose that the planets do not attract each other,
244
DYNAMICS. TWO PARTICLES.
unless their masses were equal to each other.
however, from the truth is extremely small.
The deviation,
270. The investigations in Arts. 252, 265, shew us that
if our law of gravitation be true, the only orbits which a
heavenly body will describe, supposed to be acted on only by
the Sun, are an ellipse, a parabola, or a hyperbola with the
Sun's centre in the focus.
The manner in which the magnitude and position of the
orbit of a heavenly body is determined by actual observation
will be found in Works on Plane Astronomy. We shall here
briefly explain the process. There are six quantities which de-
termine the position and magnitude of an elliptic or hyperbolic
orbit, and the place of the body in its orbit: these are called
the elements of the body's orbit, and are (1) the inclination
of the orbit to the ecliptic, and (2) the longitude of the ascend-
ing node, these determine the position of the plane of the
orbit in space: next (3) the longitude of the perihelion, (or
point of the orbit nearest the Sun), which determines the
position of the orbit itself: then (4) the mean distance, and
(5) eccentricity, which determine the magnitude of the orbit,
and lastly (6) the epoch, or the time of the planet's being in
the perihelion, this determines the position of the body itself
in its orbit.
The elements of a parabolic orbit are five in number, being
the same as the above, if we replace the mean distance and
eccentricity by the perihelion distance.
The elements of a circular orbit are only four in number,
the eccentricity and longitude of the perihelion not being
required.
In order to determine the numerical values of the elements
of any heavenly body (supposed to move in a conic section with
the Sun in the focus) two Trigonometrical equations* are
deduced connecting the elements with the right ascension and
* For a parabolic and circular orbit see Maddy's Plane Astronomy, Chap. XIV.
Woodhouse's Plane Astronomy, Chap. XXIV.
But for other orbits the reader may consult the Work of Lalande; Gauss's Theoria
Motus Corporum Caelestium; the Mécanique Céleste, Vol. I.; Lagrange's Mec.
Analytique; Pontécoulant's Théorie Anal. du Système du Monde, and Mr Lubbock's
Mathematical Tracts and various Papers in the Transactions of the Philosophical and
Astronomical Societies,
ELEMENTS OF A PLANET'S OR COMET'S ORBIT. 245
declination of the body and the distance of the Earth from
the Sun.
Since there are five or six quantities to be determined three
independent observations must be made on the declination and
right ascension of the body: when these are substituted
successively in the two equations mentioned above we shall
have six equations involving the elements: by means of which
we shall be able to calculate the magnitude and position of the
orbit.
271. By methods of this nature Kepler discovered his
three planetary Laws.
Also Astronomers have in this way proved, that comets
move in orbits most of which are parabolic, some elliptic, and
others probably hyperbolic. In consequence of the vast dis-
tances to which comets penetrate into space, they are invisible
except when near the Sun. During their appearance numerous
observations are made, in order that the elements may be de-
termined with the greatest possible accuracy. The calculations
for parabolic motion are less laborious than for elliptic or
hyperbolic motion. The elements are therefore first calculated
on the supposition that the orbit is a parabola. If the
elements thus calculated shew that the comet has passed so
near any of the planets as to have experienced a sensible per-
turbation the elements must be corrected in a manner to be
explained hereafter.
If a parabola will not coincide with the orbit calculations.
must be made for an ellipse or hyperbola. It is thus found
that "three or four comets describe very long ellipses: and
nearly all the others that have been observed are found to move
in curves which cannot be distinguished from parabolas. There
is reason to think that two or three comets move in hyper-
bolas." (Airy's Gravitation, page 15.)
272. Our calculations have been hitherto respecting the
nature of the orbits described. We now proceed to deduce
formulæ for determining the time that the body occupies in
moving through a given angle; and conversely the angle described
in a given time: by the former we know the time of the body
being at a given place, and by the latter we know the place of
the body at a given time.
246
DYNAMICS. TWO PARTICLES.
PROP. To find the time of motion of a planet or comet
through any portion of an elliptic orbit, the Sun's centre
being in the focus.
273. Let 0 and be the longitudes of the body and the
perihelion, that is, the point of the orbit nearest the Sun: a
the semi-axis major of the orbit: e the eccentricity: μ the sum
of the masses of the Sun and the body (Art. 265): then the
equation to the orbit is
જે ।
1 1 + e cos (0-0)
a (1 − e²)
dt
22
Also
Art. 242.
do
h
Now h must be determined in terms of the quantities above
given, since the orbit to be described is known and not the
original circumstances of projection. The following method,
which we here apply to the ellipse, will answer our purpose
in every case. By Art. 243, h = vp at every point of a central
orbit; v being the velocity and p the perpendicular from the
centre of force on the tangent at that point: also by Art. 245,
the velocity is that due to one-fourth the chord of curvature
through the centre of force;
b²r
2 м
dr
... v2
22
1 / 1 p
: but p²
dp
from the focus;
2 a
7
dr
лв
:. h = vp
3
Vub
µa (1 − e³).
dp
a
Then the time of moving from the perihelion through the
angle - ☎ =
t =
Ꮎ - d Ꮎ
h
a³ (1
e²) ?
do
{1 + e cos (8) − ∞)}%
·
d Ꮎ
al(1-e) (1+e) cos³ ³½ (0) + (1 − e) sin² 1 (0–₪) } *
vi
−
{ −
sec² 1(0
W
d tan 1 (0 – w)
d Ꮎ
d Ꮎ
2a (10) | ((1 + e) + (1 − e) tan² 1 (0 – ☎)}
TIME OF MOTION IN AN ELLIPTIC ORBIT.
247
To simplify this let
:. t:
tan 1 (0 – ☎) =
2 at (1-e²)*
the
αξ
["
น
0
U
1+
1 + e
1-e
1+e
U
tan
(1);
1 - e
2
| R
427
d
1+e
tan
du
du
1 e
tan2
(1+e)² sec¹
W
10
2
- " {(1 − e) cos" " " + (1 + e) sin' " du
az
Ju
ль
αξ
0
2
(1 − e cos u) du
(u – e sin u), let
.. nt = u e sin u.
11
αξ
1
Ju
;
N
(2).
When is given we calculate u by (1), and substituting
in (2) we know t.
The angle -, or the excess of the longitude of the
body over the longitude of the perihelion, is called the true
anomaly: and nt is called the mean anomaly, since it varies
uniformly with the time and coincides with the true anomaly
at the end of each revolution, as the formulæ (1) (2) shew.
Also the angle u is called the eccentric anomaly, since it equals
the angle QCA (fig. 81), as may easily be proved: P is the
body, AP a the ellipse, S the focus, AQɑ a circle on Aɑ.
COR. 1. If t be not measured from the epoch of passing
the perihelion, but from the time when u u,, then
t =
=
ał
✓ {(u — u,) − e (sin u — sin u)}.
ль
COR. 2. Whenever u increases by 2π, 0 increases by 2π,
and t by
πα
Ju
μ
This, then, is the periodic time of the
248
DYNAMICS. TWO PARTICLES.
--...
planet: it is remarkable that it is independent of the eccen-
tricity of the orbit.
To solve the converse of this Proposition, that is, to find
the position of a heavenly body in its elliptic orbit at any time.
in terms of the time and the elements of the orbit, we must
effect several expansions.
PROP. To expand the true anomaly in terms of the
eccentric anomaly.
274. By last Article tan
Ө-чо
19
2
1+ e
Ղ
tan
1 e
10!
Substituting the exponential expressions for the tangents,
¿(0–∞) √ — 1 — 1
U
દ
1
1+ e
1 + 1-^(-A)3
+1
= m
૬૫
= m²,
M,
+1
1
e
in which is the base of Naperian logarithms.
Eu
•·.·• ε (0 - @) √=I__ (m +1) εu√−¹ — (m−1)
_
(m+1)-(m-1) εu√=1
1-λε-κνι
εu V=I
m-1
1 − λ ε« √ = T ›
m+1
.. (0_w) √ −1 = u√-1+ log, (1-λe="√-1) — log, (1 −λ ε" √−T)
(0−☎)
= u √√√ = 1 + λ (ε× √ =¹ _ &- "VFT) +
10
12
=¹_
(ε2× √F1 — ε−2u √FI) +
212
213
@ = u + 2 λ sin u +
sin 2u +
sin 3u +
10
Co
√1+ e −√1 − e
1 - √ 1 − e²
in which
1+e+ √1
e
PROP. To expand the eccentric anomaly in terms of
the mean anomaly.
275. By Art. 273, u = nt + e sin u.
Hence by Lagrange's Theorem, putting nt = x,
THE PLACE OF A BODY IN AN ELLIPTIC ORBIT. 249
e²
d sin²z
e³
d² sin³ x
u = x + e sin x +
+
+
1.2
dz
1.2.3
dx2
z z
1
=≈ + e sin ≈ + ½ e² sin 2x + e³ (2 sin ≈ − 3 sin³ ≈) + ...
3
= nt + e sin nt + ½ e² sin 2nt + 1 e³ (3 sin 3nt – sin nt) +
PROP. To expand sin u, sin 2u,...in terms of the mean
anomaly.
276. By Lagrange's Theorem,
sin u =
sin z + e sin g
+
d sin z e² d
dz 1.2 dz
d sin x
sin² x
+
dz
sin³ ≈) + ..
=
sin ≈ + e sin x cos x + e² (2 cos² ≈ sin ≈
z
sin ≈ + e sin 2≈ + e² (2 sin 3x - sin x) +
sin nt + 1 e sin 2nt + } e² (3 sin 3 nt − sin nt) +
d sin 2x
Again, sin 2u
sin 2x + e sin z
+
dz
sin 2nt + 2e sin nt cos 2nt +
sin 2nt + e (sin 3nt - sin nt) +
and so on.
sin 3u = sin 3nt +
PROP. To expand the true anomaly in terms of the
mean anomaly.
277. By Art. 274 we have
212
2λ3
-
w = u + 2λ sin u +
sin 2u +
sin 3u+
2
3
I - √1 - e²
e
where
e
0122
+
´%s |
8
Then substituting for u, sin u, sin 2u... the values ob-
tained in the last two Articles, and retaining powers of e as
far as the cube,
I I
250
DYNAMICS. TWO PARTICLES.
0-w=nt + e sin nt + e sin 2nt + e³ (3 sin 3nt - sin nt) + ...
+2λ{sinnt + e sin 2nt+e² (3 sin 3nt-sin nt) + ...}
2
+λ² {sin 2nt + e (sin 3nt - sin nt) +
3
+ & λ³ sin 3nt +
}
= nt + 2e +
(2e + 212)
5 e²
13e³
sin nt +
sin 2nt +
sin 3nt + ...
4
12
4
which is true as far as terms involving e³.
278. COR. If the time t be not measured from the time
of perihelion passage, suppose e is the mean longitude of the
body when t = 0; then the mean longitude at the time t is
nt+e; and the mean anomaly is nt + e-☎: in this case, then,
O - w = nt + €
5 e²
+
4
e is called the epoch.
€
+ a
(ze
(26 + )
2e+
4
sin (nt +ew)
sin 2 (nt + e − ☎) + ...
PROP. To expand the radius vector r in terms of the
mean anomaly.
279. The radius vector
a (1-e²)
a (1-e²)
2=
1+e cos (0−∞)
(1+e) cos² 1 (0–₪) + (1−e) sin² 1 (0–∞)
a (1 − e²) sec² 12 (0–☎o)
1+ e
―
U
2
2
2
1-24
1+e+(1-e) tan (0-w)
a (1 − e) { 1 + 1 = tan" }
2
+ 2}
a {(1-e) cos' " + (1+e) sin²
2
2
(1 + e) sec²
But unt+ e sin u; putting nt = x,
e
И
= a (1-e cos u).
d cos z
e²
d
cos u = cos z + e sin x
+
sin²
d cos x
X
+ .
dz
1.2 dz
dz
e2
e²
=
1 +
e cos nt
cos 2nt
૭
2
Q?
2
2°
a
= cos ≈ − 1 e (1
e (1 - cos 2≈) - e² (3 cos ≈ - 3 cos 3 ≈) + ...
—
3 e³
(cos 3nt - cos nt) + ...
8
THE PLACE OF A BODY IN A PARABOLIC ORBIT. 251
280. COR. If t be measured as in Art. 278, then
r=a { 1 + 1 e² · e cos (nt + ε-w) – ½ e² cos 2 (nt + E
@) — ....}.
The time of describing a given portion of an elliptic hyper-
bolic or parabolic orbit may be found in terms of the radius
vectors at the extremities of the arc and the chord of the arc.
These expressions are useful in determining the elements
of a heavenly body. They will be found in Maddy's Plane
Astronomy, Chapter XIII. New Edition: and in the Système
du Monde of M. Pontécoulant, Tom. I. Liv. 11. Chap. v.
PROP. To find the time of describing a given portion of
a parabolic orbit about the Sun in the focus.
281. We have r²
d Ꮎ
dt
=h: h=√2μD, and r=
Ꭰ
cos² 1 (0-w)
is the equation to the parabola, and being the longitude
of the comet and of its perihelion measured from the Sun, and
D the perihelion distance;
D
Ꮎ
d Ꮎ
.. t
√2μ
@ cost
H - wo
2
2D2
3
d tan (0)
dᎾ
{1 + tan² (0-w) } do
= √√√² D³ {tan } (0 − ∞) + } tan³‍½ (0 – w)},
"
–
t being measured from the time of the perihelion passage.
By this equation it is easy to calculate the time of de-
scribing a given angle.
PROP. To find the position of the comet in a parabolic
orbit at a given time.
282. This would require the solution of the cubic equation
in the last Article. This is, however, obviated in the following
manner.
Let V
"
N;
2 D3
..nt tan (0 ≈) + ¦ tan³ ! (0) — ☎).
= ½½ − }}
252
DYNAMICS. TWO PARTICLES.
@
A Table is formed consisting of two columns: one with values
of t and the other with the corresponding values of - calcu-
lated from this formula for an orbit in which n = 1. Suppose,
then, that we wish to find the position of a comet in a given
parabolic orbit (the mean motion in which is n) at a given time t.
We must multiply t by n and look for the value of w
opposite the value of nt in the first column. This gives the
position of the comet.
PROP. To find the place of a comet at a given time in a
very eccentric elliptic orbit.
dt
a³ (1 − e²)³
1
283. By Art. 273.
d Ꮎ
μ
{ 1 + e cos (0 − ∞) } *
2
Let D be the perihelion distance; .. Da (1 − e);
dt
D$ (1 + e) }
do
D³
sec¹ 1 (0 – ₪)
2
{ (1 + e) + (1 − e) tan² 1 (0 – ∞)}
2
√ μ (1 + e)
1
-
e
sec¹ 1 (0 − w) { 1 +
1 + e
tan" 1 (0 – ) } ~*.
Expanding in powers of 1e, and neglecting powers of 1-e
higher than the first, because e = 1 nearly;
... nt = (1-
Ꮎ
1
2
2
ро
|_ sec' 1 (0–π) { 1 − (1 − e) tan² 1 (0–₪) } dł
d tan 1 (0 – ₪)
= ['d tan
dᎾ
{1 + tan² 1 (0 - w)
tan³ 1 (0 – ₪) – tan' ½ (0 – ☎)]} d0 ;
+ (1 − e) [4 - 3 tan" 1 (0 – )
[¦
... nt = tan 1 (0
w) + }} tan" 1 (0 – ∞)
+ (1 − e) { ¦ tan ↓ (0 – ≈) - | tan" 1 (0 – ) - ' tan' ½) (0 - w)}{
};
5
The following is a convenient method for calculating the
value of
-
w for a given value of t.
PLACE OF A COMET IN A VERY ECCENTRIC ORBIT.
253
Suppose is the true anomaly of a comet at the time.
t moving in a parabolic orbit of which D is the perihelion dis-
tance; then by Art. 282.
nt
-
tan ½-½ (0′ – w) + }} tan³ 1½ (0′ – w).
Let 0 - w=0'-☎ +x: then putting this for – w in
the first expression for nt, and neglecting the squares and
products of a and e, we have by Taylor's Theorem
nt = tan 1 (0' − ∞) + 1 tan³ 1 (0'— ∞)
+
2018
-
sec¹
2
10
+
1
4
e
4
-
w)
.
tan (0') {1 – tan² 1 (0′ – ∞)
tan¹ 1 (0' — w)},
and eliminating nt from these last two equations
X =
1 (1–e) tan 11 (0′ – w) { 4–3 cos² 1 (0′ —₪) — 6 cos³ 1 (0′ —@)}·
A third column must now be added to the Table men-
tioned in Art. 282. consisting of values of
responding values of t and e-w.
the manner of using it is as follows.
X
1 е
for the cor-
When this is constructed
Suppose
u
2 D
= n in
our orbit then in the first column look for the time nt; and
X
take the corresponding values of ✪ – w and : multiply the
1 e
latter by 1e, which will depend upon the form of the orbit,
and then the true anomaly at the time t will be this quantity
added to the value of 8 - thus found.
CHAPTER IV.
EXPLANATION OF THE LUNAR PERTURBATIONS.
284. In the last Chapter we have calculated completely
the motion of two bodies, considered as particles, attracting
each other according to the assumed law of gravitation. When
the various formulæ there obtained are applied to calculate the
motion of the planets about the Sun, and for that purpose are
reduced to tables, they manifest an agreement with observation
so far complete as to leave no doubt of the correctness of the
principles, which form the basis of the calculation; provided
that observations be made at times separated by moderately
long intervals. If, however, we proceed to a more rigorous
nicety, and especially if we compare together observations which
embrace a very long series of years, it is found that the
ment is not so perfect. Minute irregularities are detected, and
the planets are found sometimes a little in advance, sometimes
a little falling short, sometimes a little above or below, to the
right or left, of their places, calculated on the theory of elliptic
motion.
agree-
Now this is exactly what was to be anticipated. For if
the principle of gravitation be universal the heavenly bodies.
disturb each other in their motion about the Sun, and so de-
range the elliptic form of the orbits and the equable description
of areas by the radius vector.
285. It is our object in Chapters V and VI to deduce
formulæ by which the mutual perturbations of the heavenly
bodies may be calculated.
be calculated. The equations of motion for three
or more particles attracting each other according to the law of
gravitation have never yet been integrated. In fact, their in-
tegrals depend upon the integration of a function analogous
PERTURBATIONS OF ELLIPTIC MOTION.
255
to the function V in Art. 169. See note to Art. 323. We
must therefore have recourse to methods of approximation.
286. The peculiar configuration of the Solar System
renders this approximation practicable, though under most
other arrangements it would not be so; the bodies of our
system are arranged either singly as the planets Mercury,
Venus and Mars, or in groups as the Earth and the Moon,
Jupiter, Saturn, and Herschel each with their Satellites, the
central masses of the groups being much greater than that of
the attending bodies: likewise the single bodies and the groups
are always at considerable distances from each other, and de-
scribe orbits about the Sun very nearly circular and in planes
nearly coinciding. There is, however, an exception in the
case of the four asteroids Ceres, Vesta, Pallas, and Juno, the
orbits of these being not very far different from each other in
magnitude; but their masses are so small that in this way
a compensation takes place. The mass of the Sun is of enor-
mous magnitude in comparison with that of the other bodies
since, as we have remarked in Art. 284, the calculations made
on the supposition that the Sun is the only attracting body
nearly coincide with observation. Again the mass of the Earth
is large when compared with that of the Moon; because the
Earth moves about the Sun and the Moon about the Earth,
nearly as if the Moon did not disturb the Earth's elliptic motion
about the Sun, and as if the Sun did not disturb the Moon's
elliptic motion about the Earth. In the same way we argue
that the mass of Jupiter is much larger than that of his satel-
lites by observing that Kepler's Laws are nearly verified, and
so of the other bodies.
;
It is in consequence of this peculiar configuration of the
Solar System that we are able to approximate to the solutions.
of our equations of motion by converging series.
287. In the present Chapter we intend to explain the
nature of the perturbations of the Moon's motion about the
Earth by the attraction of the Sun. We shall introduce a few
calculations as interpretations of Newton's geometry into ana-
lytical language (Principia Book I. Prop. 66. and Book III);
but shall reserve for the next Chapter the solution of the
problem by systematic approximation.
256
DYNAMICS. PROBLEM OF THREE BODIES.
We shall first explain a principle of great importance in
calculating the combined effect of several small perturbing
causes, which we shall find important throughout this and the
two following Chapters.
PROP. To explain the principle of the superposition of
small motions.
288. Let xyz be the co-ordinates of a body at the time t
when undisturbed by any other body: a a very small nume-
rical quantity which depends upon the disturbing force, of
which the square and higher powers may therefore be always
neglected.
x + ax', y + ay', ≈ + a' the co-ordinates of the body at
time t when disturbed by the body (m') only.
x + ax", y + ay", ≈+az" the co-ordinates of the body at
time t when disturbed by the body (m") only, and so on.
Now suppose the planets m', m", ...
the planets m', m”, ... all to disturb together.
In this case the alterations in x, y, z arising from the
several planets will not be the same as before; but they will
themselves suffer perturbations, since the action of each planet
is now modified by that of all the others.
Thus the value of a will not become
x + a (x' + x' + ...) :
for each of the terms after the first will be modified, but since
this modification arises from the disturbing forces, it follows,
that the quantities to be added will be multiplied by a, a,...
and may therefore be neglected and, under this restriction, the
co-ordinates of the body subjected to the combined perturba-
tions of all the others will be
a
x + ɑ (x' + x' +
..),
Y + a (y' + y'' +
·),
≈ + a (x² + x″ +
..), at the time t.
Hence the perturbation in any quantities
(x, y, x) = (u)
which depends upon the co-ordinates of the planet will
HISTORY OF LUNAR INEQUALITIES.
257
du
du
a
(x′ + x″
x"
Ꮖ + ......) +
a (y' + y' + ......)
dx
dy
du
+
a (z′ + z″+ ......).
dz
du
du
du
And this
=
ax +
ay +
a v
dx
dy
dz
du
du
du
+
ax" +
ay" +
a="
dx
dy
dz
+
But this latter form shews that the perturbation in p (x, y, ≈) is
equal to the sum of the separate perturbations of each planet
supposing the others not to exist. Hence the Principle we
enunciated is true. Its great use in our calculations is this,
that it reduces the problem from one of several bodies to that
of only three bodies. Hence the famous Problem of Three
Bodies.
By ob-
289. At an early period it was observed that the apparent
motion of the Sun and Moon round the Earth was not uniform.
This had been remarked by the Greek Astronomers.
serving the motion of the shadow of the gnomon they discovered
a considerable difference in the intervals of time between the
equinoxes and the solstices.
Hipparchus was the first who endeavoured to explain this:
he supposed the orbits of the Sun and Moon described about
the Earth to be eccentric circles, or circles of which the centres
do not coincide with that of the Earth.
290. After a lapse of three centuries Ptolemy discovered
that there was an error in the Moon's place in the heavens,
which could not be accounted for on the hypothesis of Hip-
parchus and he shewed that the magnitude of the error de-
pended upon the position of the line of apsides, or axis major
of the lunar orbit. This inequality was called the Evection of
the Moon: we shall hereafter explain from what cause it arises
291. The next remarkable inequality of the Moon's
motion was discovered by Tycho Brahe in the sixteenth
century. This was proved to depend upon the angular
distance of the Sun and Moon: and was greatest when the
K K
258
PROBLEM OF THREE BODIES.
DYNAMICS.
1
Moon was about 45° or 135° from the Sun. In this respect
this inequality, which was called the Variation, differed from
the Evection, which Ptolemy found to be greatest when the
Moon was ninety degrees from the Sun.
292. Tycho Brahe was the discoverer of one more in-
equality, which was called the Annual Equation, since it
depends on the distance of the Sun from the Earth, and
therefore goes through its changes in a year.
Not many years after these discoveries of Tycho Brahe,
Kepler published to the world his Three Laws, which he had
calculated with almost incredible labour and perseverance.
Theory has led to the discovery of many other inequalities in
the Moon's motion, but the above have been specified for their
historical interest and because they are more sensible than the
others.
293. All these, however, were merely bare facts, the re-
sults of continued and indefatigable observations and calcu-
lations. No common law appeared to connect them, no one
cause was known of which they were necessary consequences.
It was the glory of Newton, that he unravelled the mystery
and demonstrated that these were all results of a universal
principle with which matter is endowed by the Creator of the
World.
Kepler and other Astronomers had conceived the notion of
a universal gravitating principle: but it needed the master
genius of Newton to demonstrate its existence.
We now proceed to explain the causes of these perturbations.
PROP. To calculate the disturbing forces of the Sun on
the Moon.
294. The disturbing forces are the differences of the
forces of the Sun on the Moon and Earth.
Let E, M, S
represent the masses of the Earth, Moon, and Sun. The law
of force we assume to be that of the inverse square of the
distance between the centres of the bodies. We shall consider
the orbits of the Sun and Moon about the Earth nearly circu-
lar, since this is proved to be the case by observation*.
* The slight variations in the apparent magnitudes of the Sun and Moon con-
vince us of this.
CALCULATION OF THE DISTURBING FORCES.
259
Let be the distance between M and E (fig. 82),
*
Y
S and E,
S and M,
w the angle SEM,
measured in the direction in which the hands of a watch
move.
We must consider the motion of M about E as if E were
fixed in order that we may discover the apparent perturbations :
S
but the accelerating forces acting on E are and in the
M
go
directions EM and ES respectively and in order that the
relative motion may not be affected by supposing E fixed we
must apply forces equal to these upon each body of the system
in an opposite direction: the second law of motion shews the
legitimacy of this process.
Hence the forces acting on M, E being considered fixed, are
M+ E
ж
2.2
S
y²
S
in the direction ME,
MS,
MK, MK being parallel to SE.
Then, resolving the second of these in the directions ME
and ML (Art. 18), and combining the resolved parts of this
with the other forces, the disturbing forces of S on M are
Sr
S'r'
S
in the direction ME, and
in the direction ML.
y³
y³
'2
グ
​Again resolve these in the directions ME and perpen-
dicular to this line: then the whole forces which act upon M
about E at rest, are
M+ E
+
Sr
S 23
ド
​2
y³
1)
W
cos o in the direction ME,
S
12
y³
1) sin w in the direction perpendicular to ME,
(-1)
pl2 \ y³
and acting towards the nearest syzygy.
260
PROBLEM OF THREE BODIES.
DYNAMICS.
The Moon is said to be in syzygy when it is new or full;
and in quadrature when ninety degrees from syzygy.
The first of the above forces deprived of its first term is
called the central disturbing force: and the second is called
the tangential disturbing force.
295. We proceed to obtain approximate expressions for
these. Since the orbit of the Moon is nearly circular; and
γ
1
since is a very small fraction, being about equal to we
shall neglect its square and higher powers.
12
By the figure, y² = p²² + p¹² − 2r′r cos w;
400
20'3
13
2 r
y³
(1 - 237)
31
cos w) − = 1 +
cos w;
... central dist. force
Sr
3 Sr
Sr
2013
cos2 W
2'3
r
23(1+3 cos 2w)
tangential dist. force =
3 Sr
3 Sr
2013
sin w cos w =
27'3
sin 2 w.
In order still further to simplify these expressions; let ƒ be
the mean force of E upon M, E being supposed fixed: m the
ratio of the periodic times of the Moon and Sun about the
Earth (m=nearly): a and a' the mean distances of the
Moon and Sun from the Earth: therefore by Art. 273. Cor. 2,
m²
=
4 π² a³
Π
M+ E
M+ E
a²
S
a³
M+ Ea's nearly;
4 π² a
S+E
2 13
S'r
13
J'
=f: and
/3
m²ƒ
13
a
m²
13
2°
f
/3
(1 + 3 cos 2 w),
2
J
a
.. central disturbing force
a
3m²
13
α
tangential disturbing force =
f
21
13
2
α
sin 2 w.
CALCULATION OF THE DISTURBING FORCES.
261
296. If we suppose the orbit of the Earth about the Sun
and the undisturbed orbit of the Moon about the Earth to be
circular, then r' = a' and r
= a; and
central disturbing force = - 1 m² ƒ (1 + 3 cos 2 w),
=
tangential disturbing force 3m² f sin 2w.
297. The central disturbing force vanishes when
W =
1
1 cos¯¹ (− 1) = 55º, 125º, 235º, 305º nearly.
The points in the Moon's orbit determined by these angles
are called Octants.
The central disturbing force is said to be addititious at
points between those octants between which the quadratures
lie, because at those points the above expression for the force
is positive and consequently adds to the force of M to E. For
a like reason the central disturbing force is said to be ablatitious
between those octants between which the syzygies lie.
298. We proceed to examine the effects which the dis-
turbing forces have upon the form and position of the Moon's
orbit. We shall neglect quantities which depend upon the
square and higher powers of the disturbing force. Whenever
the undisturbed orbit is supposed to be nearly circular we may
dr²
d 0²'
neglect all such terms as
Thus
dr² | 2
+
d 0°
༡༧
radius of curvature
dr²
d² r
22² + 2
"'
d 02
Also (velocity)² = (ds)*
Again central force = huu+
h²
d Ꮎ
+
t
d t
(
ď u
d 02
d t
I'
2
-
dr
d02
by Art. 246.
do 2
dt
•
d02
2 dre
1 d²r
h²
I d²r
||
+
23 do
q²² do s
p²
d Ꮎ
do
262
PROBLEM OF THREE BODIES.
DYNAMICS.
༡༠
2
d²r
(d)(-1) by Art. 242.
dt
(vel.)²
rad. of curvature
We have introduced these expressions since we shall find
them useful hereafter.
PROP. To find the effect of the Sun's disturbing forces
on the periodic time of the Moon.
•
299. Since the tangential force passes through all its
degrees of magnitude positive and negative during half a revo-
lution of the Moon, it will compensate during one quarter of
the revolution for any loss or gain that the angular velocity
of the Moon may have experienced in the preceding quarter.
In fact the mean* tangential force equals zero. For this
reason the tangential force has no effect on the periodic time.
For the same reason we neglect the periodic term of the
central force. The mean central force = f(1-m²).
2
Since
this is less than f, it follows that the mean distance is increased
by the disturbing forces.
The absolute force = fa² (1
m²
;
2 π al
.. the periodic time
nearly, (Art. 273. Cor. 2.)
abs. force
m²
2π
1 +
nearly.
4
a
f
This is greater than if there were no disturbing force
(or m = 0); especially when we remember that a is greater
than in the undisturbed orbit.
*
By the mean value of a function of a variable angle we mean, the part of the
function which is independent of periodical terms which pass through all their changes
positive and negative as is increased by certain equal increments. Thus a is the
mean value of a + b sinn 0.
PERTURBATION IN PERIODIC TIME.
263
Hence the periodic time is increased by the disturbing
force. (Principia, Lib. I. Prop. LXVI. Cor. 6.)
300. COR. 1. If we suppose the Earth's orbit about the
Sun not circular, then by Art. 295, the mean central force
m² a
13
and the length of the month
= 2π
314
f
m² a
13
1 +
13
4
Hence the months are longest when the Earth is in peri-
helion, and shortest when in aphelion. This accounts for the
Winter Months at this epoch being longer than the Summer
Months.
301. COR. 2.
The mean velocity (V) of the Moon
af (1-
2 па
per. time
a
a
4
m² ) = √ at (1
m²
2
4
PROP. To find the effect of the Sun's disturbing force
on the velocity of the Moon, supposing the undisturbed orbit
of the Moon to be circular.
302. Since the angle w is referred to a line moveable in
space we must adopt another means of measuring the position
of the Moon. Let be the longitude of the Moon at the time.
t: then me is the longitude of the Sun, supposing that is
measured from the time when the Sun and Moon were in Aries
together; and supposing the orbits nearly circular: and there-
fore w =
(1 − m) 0:
tangential disturbing force = 3 m* f sin 2 (1 − m) 0,
and this is the only disturbing force which directly affects the
velocity. We shall see in Art. 305, that the velocity is affected
indirectly by the central disturbing force.
Now the space described by the Moon in the time t is a9;
and the tangential force is the only force which acts in the
line of the Moon's motion :
264
PROBLEM OF THREE BODIES.
DYNAMICS.
d Ꮎ
3m²
a
fsin 2 (1 m) 0,
dt²
2
the negative sign being taken because the tangential force acts
always towards the nearest syzygy, Art. 294, and consequently
tends to diminish when the Moon is in the first and third
quadrants, through which angles sin 2 (1 − m)◊ is positive,
and to increase in the second and fourth quadrants, through
which angles sin 2 (1 m) is negative.
ᏧᎾ
Multiplying by 2a-
dt
and integrating
d02
3 m²
dt
vº or a² = const. +
Let V be the mean velocity; then
fa cos 2 (1 m) 0.
2 (1 − m)
3m²
v² =
V² +
fa cos 2 (1 - m) 0
2 (1 − m)
3m²
{
V² { 1 +
2 (1 − m)
cos 2 (1 m) 0} neg. m',
since V² = ƒa (1 − 1 m²) by Art. 301.
The effect, then, of the disturbing force is to increase the
velocity above what it would be in the circular orbit, when the
Moon is not more than 45° from syzygy; and in other positions
to diminish it. This follows directly from the fact proved in
Art. 294, that the tangential force always acts towards the
nearest syzygy.
The velocity is greatest in syzygies and
3m²
=
V1+
4 (1
m)
3m²
4 (1 − m
least in quadratures and V1 -
(Principia, Lib. I. Prop. LXVI. Cors. 2, 3.)
PROP.
To find the effect of the Sun's disturbing force
on the form of the Moon's orbit; supposing the undisturbed
orbit to be circular.
PERTURBATION IN THE FORM OF THE CIRCULAR ORBIT. 265
303.
The curvature at any point of the orbit is measured
by the reciprocal of the radius of curvature: hence, by Art. 298,
central force
the curvature equals
(velocity)2
m²
Now the central force = f-
·ƒ (1 + 3 cos 2 w), Art. 296,
also (velocity) = V² {1+
cos2w}, Art. 302.
√at (1
m²
1
V = mean vel. = √ af
.. curvature
=
f
{
1
3m²
{1
a
10
3m²
2 (1 − m)
4
m² - 3m² (1.
2
1 +
I
1 M
by Art. 301;
+
1
1 m
)
cos 2 w}.
cos 2 w}
This is greatest in quadrature, when w= 90° and 270°;
and is least in syzygy, when w = 0 and 180º.
This shews that the orbit will assume an oval form with
respect to the Sun, having its minor axis in syzygy, (Principia,
Lib. I. Prop. LXVI. Cor. 4). Its form in space will be an irre-
gular curve, nearly circular, but not re-entering. Also the
expression for the curvature shews that the equation to the
orbit is ra (1 x cos 2w), the major and minor axes being
2a (1+a) and 2a (1-x).
PROP.
To find the ratio of the axes of the oval orbit.
304. The equation to the orbit is r = a (1 a cos 2w);
but since w is measured from a moveable line we must put
W = (1 m) as in Art. 302;
.. r = a {1
{1-a cos 2 (1 m) 0} ;
−
a
1 1 d²r
.. curvature
by Art. 298.
I'
2² do
1 + æ [1 − 4 (1 − m)°] cos 2 (1 − m) 0 }
L L
266
PROBLEM OF THREE BODIES.
DYNAMICS.
S
Equating this to the expression found in the last Article,
1
a
{ 1 + x [1 − 4 (1 − m)²] cos 2 (1 − m) 0}
1
3m²
1
2
(14
+
1
1 m
cos 2 (1 − m) 0};
1
1 +
3m²
.. x =
X
2
1 m
4 (1 − m)² – 1
1
1
If we put m =
X
9
nearly;
13.3
139
1 + x
70
.'.
9.
1
-
X
69
the ratio in the Principia Lib. III. Prop. xxvIII.
This is the ratio of the axes of the oval orbit which moves
round with the Sun while the Moon moves in it.
In Art. 302, we found the effect of the tangential disturbing
force on the velocity of the Moon; and we have just shewn that
the tangential and central disturbing forces draw the orbit of
the Moon into an oval figure with respect to the Sun. We
proceed, then, to calculate the velocity of the Moon when both
disturbing forces are considered.
PROP. To find the velocity of the Moon in the oval orbit.
305. In Art. 302, we found the effect of the tangential
force on the velocity: but the velocity will be affected by the
change in form of the orbit: and thus we see the indirect
effect of the central disturbing force upon the velocity.
Let v be the velocity of the Moon at any distance,
VI
the mean distance,
then v²
d 02
2.3
dr²
dt, neglecting
dt
which depends on the square of
the disturbing force: Art. 298.
VELOCITY IN THE OVAL ORBIT.
267
MOON'S VARIATION.
v2
ᏧᎾ
do ?
h2
202
h2
dt
احلام
V, 2
2
α
a²
شوح
V
1
By substituting for
a²
2.2
we shall correct v2 for the oval form of
1
2
the orbit: and by substituting for v² we shall correct for the
change in velocity in the circular orbit: and in this way the
complete velocity in the oval orbit is found.
a²
2,2
v₁² = V ²
1 + 2x cos 2 (1 − m) 0,
3 m²
2 (1 − m)
1 +
• v²= V² {1 + (2x +
(2x
cos 2 (1 - m) 0}. See Art. 302;
cos 2 (1 m) 0}.
3m²
2 (1 − m)
306. Let 80 be the error in longitude owing to this change
in the velocity: then
V_d (0 + SO)
dt
d Ꮎ
dt
a
d.So
v a
}}
nearly;
?'
1
{1 + (x +
60
d Ꮎ
3m²
4 (1-m.)
2x
cos 2 (1−m) 0} {1+æ cos 2 (1−m) 0 } − 1
3m²
-(2.8 + + 1))
x
4 (1 m)
+
3m"
cos 2 (1 − m) 0 ;
sin 2 (1 − m) 0.
1 M 8 (1 − m)²
This error in longitude is greatest (disregarding its sign) when
the Moon is 45° or 135° from the Sun on either side of syzygy:
and therefore explains the cause of the error in the Moon's
268
DYNAMICS. PROBLEM OF THREE BODIES.
place discovered by Tycho Brahe, and called the Variation
(Art. 291).
1
1
If we put x
and m²
139
179
Variation = m² (1 + 3)
(1. sin 2 (1 − m) 0 nearly
11m²
8
8
sin 2 (1 − m) 0,
which accords with the rigorous approximation of the second
order in the next Chapter, Art. 341.
We shall now suppose the undisturbed orbit of the Moon
to be an ellipse of very small eccentricity; the Earth's centre
being in the focus.
Our calculations will receive a remarkable degree of sim-
plification by considering the perturbations of the Moon to
affect, not the Moon itself directly, but the elements of its
orbit, and so the Moon indirectly. The legitimacy of this hy-
pothesis will appear from the following Proposition.
PROP. To prove that the motion of the Moon may bẻ
represented by supposing it to move in an ellipse, the elements
of which are continually changing.
307. We have to shew that at every instant an ellipse can
be drawn with one of its foci in the Earth's centre; its circum-
ference passing through the Moon's centre; its tangent at the
Moon in the direction of the Moon's motion at that instant ;
and the velocity in this ellipse calculated according to the prin-
ciples of elliptic motion (see Art. 252.) equal to the velocity of
the Moon at that time.
If an ellipse can be found which satisfies these conditions
it is clear that we may suppose the Moon to be moving in its
circumference at the proposed instant: and the perturbations.
of the Moon's motion will be found by calculating the change
in the elements of this instantaneous orbit, as it is termed.
Two of the elements, viz. the inclination and longitude of
the node, are fixed by the condition that the plane of the
}
269
INSTANTANEOUS ELLIPTIC ORBIT.
ellipse must pass through the direction of the Moon's motion
and the centre of the Earth.
α
Let 1 + e cos (0 − ∞)
7°
1
be the equation to the required ellipse, neglecting powers of e
higher than the first: also let r₁0, be the co-ordinates to the
Moon in the plane of its orbit at the instant under consideration;
v its velocity; the angle between the radius vector of the
Moon and the direction of her motion; then the above con-
ditions give
a
== 1 + e cos (01 − w),
d Ꮎ
can 4 = ( r 14 = ) -/-
1
e
dr
v² = 2 (E + M)
−
cosec (0₁ - ),
1
2 a
From these three equations a, e, w may be found.
The sixth
element, the epoch, determines the position of the Moon in the
instantaneous ellipse.
Hence an ellipse can always be drawn as described in the
enunciation.
308. We shall proceed to explain the nature of the alter-
ations in the elements. Since these variations during a revo-
lution of the Moon are small we shall, in accordance with the
Principle proved in Art. 288, consider the variation of each
element supposing all the others to remain invariable; and then
add their effects together.
At any proposed instant the equation to the orbit is
a
= 1
7°
1 + e cos (8 − ∞),
where a, e, ☎ have values depending upon the proposed instant.
Hence also if v be the velocity
1
" = 2 ਵਜੇ
が​=2μ
270
DYNAMICS. PROBLEM OF THREE BODIES.
ль
{1 + 2e cos (0 - @)}
α
= fa{1 + 2e cos (0 - )}, since
W
u
a²
· f.
We shall use the symbol & to indicate differentiation, not
with respect to the motion of the Moon, but with respect to
the geometric deviations of the actual orbit of the Moon from
the undisturbed orbit. Thus, if we wish to ascertain the
change in the eccentricity in consequence of a change dv in the
velocity, we have by differentiating the above equation
vdv = fa cos (0 – ₪) de,
which gives de in terms of dv: and so of other perturbations.
If we wish to ascertain the change in eccentricity arising from
a change of in the force f, we have the equation
0 = df (1 + e cos (0 - w) + ƒde cos (9 – ☎),
which gives de in terms of df.
We repeat the remark, that the direct tendency of the tan-
gential disturbing force is to change the velocity but the
central disturbing force has not this direct effect. Therefore
when we consider the effect of the central force on the elements
ɑ, e, w, we must take the variation of the equation
v² = fa {1 + 2e cos (0 – w)}
considering v constant: and when we consider the effect of the
tangential force we must take the variation considering f
constant.
We shall first find the effect of the disturbing forces on
the position of the line of apsides.
An apse is a point in a polar orbit where the radius vector
is perpendicular to the tangent.
PROP. To find the effect of the mean central disturbing
force upon the position of the line of apsides.
309. The mean central force
f
ド
​m²
f-
ƒ², Art. 295.
2 a
EFFECT OF MEAN CENTRAL FORCE ON LINE OF APSIDES. 271
We shall find it most convenient, in this case, to use the equa-
P
h²u²
proved in Art. 246, for central orbits.
tion
d² u
d 02
+ 2 =
af
− 1 m²) ;
1m²);
m² 1
2a± u³
In this case h = a V = a √ aƒ (1
Now u
—
1
d² u
d02
+ U =
-
1
a
α
· ( 1 + 1 m² )
-
varies as the eccentricity of the orbit; and
therefore we neglect its square ;
1
= = { + ( ~ - }) }
a
1
α
1
= a³ - 3 a¹
?
u³
dᵒu
+21
+ 24 =
d02
l
(1
m²
3m²
1
1 +
+
>
2 a
a
du
3m²
a
+
(au − 1) = 0,
do²
2
the integral of this is
a
au or = 1 + e cos
-
グ
​(1-
sm²) (→ − a).
-
+
e and a being arbitrary constants.
This is the equation to the path of the Moon, supposing
the mean central disturbing force to be the only disturbing
force. If the coefficient of a were unity it would be an
ellipse. The following is the geometrical construction for the
orbit.
-
Let A'M'E be the ellipse supposing the coefficient of 0
were unity (fig. 83). When 0 − a = 0 the values of in the
real curve and the ellipse are the same. Let, then, A'M be the
real path of the Moon. Take the angle A'EM = 0 a: and
the angle A'EM' = (1-3m²) angle A'EM: and let EM' cut
the ellipse in M'.
a
−
Then = 1 + e cos A'EM' by the equation to the ellipse
EM
= 1 + e cos
い
​Sm²
1
(a) by construction
4
272
DYNAMICS. PROBLEM OF THREE BODIES.
α
EM
by equation to Moon's path;
.. EM' = EM.
Draw EA so that angle MEA = angle M'EA': and make
EA = EA': we may then draw an ellipse about E as focus and
through A and M exactly equal to A'M'E.
The motion of the Moon may therefore be geometrically
represented by supposing it to move in an ellipse, the ellipse
itself revolving in its own plane about the focus with an an-
dA'EA 3m² do
gular velocity
or which is to the angular
4 dt
dt
velocity of the Moon as m² 1 in the direction of the Moon's
motion.
3
4
Hence the tendency of the mean central disturbing force is
to make the line of apsides progress.
310. Observation shews that the above result is only half
the amount of the progression. The reason will be explained
in the next Chapter.
We cannot extend the approximation without introducing
the tangential force, and therefore refer the reader to the
Luner theory in the next Chapter, where the whole Problem is
accurately solved to a second approximation. All that we
shall here attempt is an explanation of the tendency of the
disturbing forces in altering the motion of the Moon in her
eccentric orbit.
PROP. To explain the effect of the tangential disturbing
force of the Sun on the line of apsides.
311. Let the equation to the ellipse in which the Moon is
moving at any instant be
ጎ
a
= 1 + e cos (0) neglecting e²
Then by the principles of elliptic motion,
1
0² = 2μ {-2}) = = {1 + 2e cos (0 – ☎)}.
a
α
But the tangential force is continually causing v to differ from
what it would be in terms of on the supposition of elliptic
EFFECT OF TANGENTIAL FORCE ON LINE OF APSIDES.
273
still
motion. But, as we have shewn in Art. 307, we may sup-
pose the velocity to be the same as in the ellipse by altering ☎,
the longitude of the line of apsides, in such a way as to make
a compensation. By taking the variation of the above expres-
sion for v² in terms of w, we have
аби
avdvue sin (0
= µe sin (0 – ₪) dw;
...бо =
a v
δυ
ue sin (0 – ₪)
-
Now the effect of the tangential force is to diminish the velocity
of the Moon as it is moving from syzygy to quadrature, and
to increase the velocity as the Moon is moving from quadrature
to syzygy; see Art. 302. Hence dv is negative as the Moon
moves from syzygy to quadrature, and positive from quadrature
to syzygy.
Also sin (9) is positive or negative according as the
Moon is moving from perigee to apogee, or from apogee to
perigee.
Now the motion of the line of apsides during a revolution
of the Moon is very small, since the disturbing forces are small.
We may therefore suppose, in examining the effect that the
particular position of the line of apsides with respect to syzygies
and quadratures has upon their motion, that they remain sta-
tionary during a revolution of the Moon: otherwise we shall
be introducing quantities which depend upon the square of the
disturbing force, quantities which we have purposed to neglect.
We shall consider the two cases when the line of apsides is in
syzygies, and in quadratures: and then take the average effect.
I. Suppose the line of apsides is in syzygies: fig. 84.
Then as the Moon moves from quadrature to quadrature
through perigee, (that is, through the arc qYQ), dv and
sin (0) are of different signs and therefore do is negative,
or the line of apsides is regressing: but as the Moon moves
from quadrature to quadrature through apogee, (that is, through
the arc Qyq) Sv and sin (0 – ₪) have the same sign and there-
fore d is positive, or the line of apsides is progressing. Now
the time of moving from quadrature to quadrature through
apogee is greater than that of moving through perigee, Art. 242.
M M
!
274
DYNAMICS. PROBLEM OF THREE BODIES.
Also the alteration in velocity by the tangential force is greater
in the latter case than in the former.
Hence, when the line of apsides is in syzygies, it progresses
on the whole during one revolution of the Moon.
II. Suppose the line of apsides is in quadratures: fig. 85.
Then as the Moon is moving from syzygy to syzygy through
apogee, (that is, through the arc yqY) dw is negative, or the
line of apsides is regressing, and as the Moon is moving from
syzygy to syzygy through perigee d is positive, or the line
of apsides progressing. Now the time of moving through pe-
rigee is less than that of moving through apogee. And the
alteration in velocity in the former case is less than in the latter.
Hence when the line of apsides is in quadratures, it on the
whole regresses during a revolution of the Moon.
If we attentively examine the steps of the above investiga-
tion, and observe that the circumstances of regression in one
position of the line of apsides closely resemble those of progres-
sion in the other, and vice versa, it will easily be seen that in
these two positions of the apsidal line the progression is very
nearly the same as the regression.
When, however, intermediate positions of the apsidal line
are considered the progression decidedly has the preponderance,
and for this reason: regression takes place when the line of
apsides is distant from the Sun; but progression takes place
when the line of apsides is near the Sun. Hence regression,
which, from its nature, causes the line of apsides to meet the Sun,
moves it towards a progressing position: while progression,
from its nature, causes the line of apsides to linger in a progress-
ing situation. Hence, on the whole, the tangential force causes
the line of apsides to progress. See Airy's Gravitation, p. 99.
PROP. To explain the effect of the tangential force on the
eccentricity of the Moon's orbit.
312. We shall resume the equation v² = fa {1 + 2e cos
(0-w)} by which we can represent the real motion, if we
suppose the elements to alter.
Now vdv = fa cos (0 – w) de ;
...бе =
V
Ev
fa cos (0 – w)
EFFECT OF DISTURBING FORCES ON THE ECCENTRICITY.
275
Now, in consequence of the tangential force, dv has the same
sign and nearly the same magnitude for opposite positions of the
Moon in her orbit, Art. 305: and cos (0) has the same
magnitude with different signs. Hence de has different signs,
but nearly the same magnitude for opposite positions of the
Moon in her orbit: and therefore the tangential force has little
effect on the eccentricity, since a partial compensation takes
place in one part of the orbit for errors caused in the opposite
part.
:
But we must examine this a little more accurately for
although compensation will take place pretty accurately when
the Moon is about 90' from perigee and apogee, since the
distance of the Moon from the Earth is then about the same,
yet it will not be so complete when the Moon is near perigee
and apogee, since at those points the Moon's distances differ
more than in any other parts of her orbit. We shall therefore
consider the case in which the Moon is near perigee and
apogee.
1. When the line of apsides is before syzygies: fig. 86.
Then du is negative near perigee and apogee; and cos (§ – w)
is positive near perigee and negative near apogee. Hence de
is negative when the Moon is near perigee and positive when
near apogee.
And therefore in this position of the line of
apsides e is increasing when the greatest change takes place,
and consequently on the whole the tendency of the eccentricity
is to increase during each revolution of the Moon.
2. When the line of apsides is behind syzygies: fig. 87.
Then it will easily be seen that de is positive when the Moon
is near perigee and negative when near apogee. Hence in
this position of the line of apsides the tendency of the eccen-
tricity is to decrease during each revolution of the Moon.
PROF. To explain the effect of the central disturbing
force on the eccentricity of the Moon's orbit.
313. Now fa = v² { 1 - 2 e cos (0 )}, and we have to
find the variation of e corresponding to any variation in ƒ:
we have
A C
#1
በ
Ef
2 vi cos (0 - w)
276
PROBLEM OF THREE BODIES.
DYNAMICS.
Now of has the same sign, and nearly the same magnitude
in opposite parts of the Moon's orbit: Art. 295. Also cos (-)
has different signs in opposite parts of the orbit. Hence Se
has different signs and nearly the same magnitude in opposite
parts of the orbit, and a partial compensation will take place
in the changes in eccentricity.
But, as in the last Article, we must examine this a little
more accurately. The values of Sf differ more at apogee and
perigee than at any other parts of the orbit, since the distances
of the Moon from the Earth differ most at these points. We
shall therefore consider the changes in eccentricity when the
Moon is near perigee and apogee.
1. Suppose the line of apsides is near octants: figs. 86, 87.
When the Moon is near perigee and apogee, and therefore near
octants, of nearly vanishes and the eccentricity does not undergo
any material change; and therefore the compensation of eccen-
tricity is pretty accurate during a revolution of the Moon.
2. Suppose the line of apsides is near syzygies: fig. 84.
When the Moon is near apogee and perigee, and therefore near
syzygies, of is negative (Art. 297); and cos (0–₪) is negative
near apogee and positive near perigee. Hence de is positive
when the Moon is in perigee, and negative when in apogee.
Wherefore e is decreasing when the change is greatest and on
the whole the eccentricity decreases when the line of apsides is
near syzygies.
3.
Suppose the line of apsides is near quadratures: fig. 85.
Then &f is positive when the Moon is near perigee and apogee;
and we shall find that the eccentricity is on the whole increas-
ing during a revolution of the Moon.
314. The result of the last two Articles is as follows.
The tangential force tends to increase or diminish the eccen-
tricity according as the line of apsides is before or behind the
Sun. And the central disturbing force tends to increase or
diminish the eccentricity according as the line of apsides is
nearly 90° from the Sun or near the Sun: in other parts a
compensation takes place.
Hence, then, during the Sun's motion from the Moon's
line of apsides through 90" the eccentricity of the Moon's orbit
is decreasing and during the Sun's motion towards the line
EFFECT ON THE INCLINATION AND LINE OF NODES.
277
of apsides the eccentricity is increasing. The eccentricity is
greatest when the line of apsides is in syzygies and least when
the line of apsides is in quadratures.
315. We shall now proceed to explain the effect of the
disturbing forces on the inclination and the motion of the line
of nodes.
Hitherto we have supposed the Moon to move in the plane
of the ecliptic: let us now suppose that the planes of the
ecliptic and the Moon's orbit are slightly inclined to each other,
as is the case in nature. Let the disturbing force of the Sun
on the Moon be resolved into two parts, one in the plane of
the Moon's orbit, and the other perpendicular to this plane:
this latter is the part which affects the inclination and the
position of the line of nodes.
If we bear in mind that the Sun's disturbing force always
acts towards the Sun when the Moon is nearer the Sun than
the Earth, and from the Sun in the contrary case, we shall
easily see, by referring to fig. 88, that the part of the dis-
turbing force which is perpendicular to the plane of the Moon's
orbit always acts towards the ecliptic, except when the Moon
is between quadrature and the nearest node, in which case it
acts from the ecliptic.
By the plane of the Moon's orbit we mean the plane drawn
through the centres of the Moon and Earth, and through the
direction of the Moon's motion at any instant. And since in
consequence of the Sun's disturbing force the Moon is con-
tinually drawn out of the plane in which it is moving, the
plane of the orbit is continually shifting its position by revolv-
ing about the Moon's radius vector as an instantaneous axis.
316. Before we begin to explain the effect on the incli-
nation and line of nodes we shall enunciate the following
Lemma, the truth of which is self-evident.
LEMMA. When a body is moving from or towards a
plane and a force acts upon it in a direction from or towards
the plane, then the inclination of the body's resulting motion
will be increased or diminished according as the original motion
and the force act in the same or opposite directions with respect
to the plane.
278
PROBLEM OF THREE BODIES.
DYNAMICS.
PROP. To explain the effect of the Sun's disturbing
force upon the inclination of the Moon's orbit to the ecliptic
and on the position of the line of nodes.
317. I. Suppose the line of nodes is in syzygies.
It is clear that in this case the inclination and line of
nodes will not be affected; since no part of the force acts per-
pendicularly to the plane of the orbit. The line of nodes
would remain in this position were it
motion.
not for the Sun's
II. Suppose the line of nodes is in advance of the Sun :
fig. 89.
Let Nn be the line of nodes: take Nm = 90° on the orbit:
let Qq be the quadratures. Then as the Moon moves from
N to Q she moves from the ecliptic, and the disturbing force
acts from the ecliptic. Hence the inclination of the Moon's
path (Art. 316), and therefore of the plane of her orbit, is
increasing, and therefore in revolving about the radius vector
EM the node N must move towards quadratures, or the line
of nodes Nn must progress.
As the Moon moves from Q to m, her motion and the
disturbing force act in opposite directions, and therefore the
inclination is decreasing (Art. 316), and the line of nodes
regressing.
As the Moon moves from m to n her motion and the dis-
turbing force both tend towards the ecliptic and therefore the
inclination of the plane of her orbit is increasing, and therefore
in revolving about the radius vector EM' causes the point n
to move back, or the line of nodes to regress.
In the other half of the orbit the effect will be exactly
the same.
Hence, if o be the angular distance of the line of nodes
from syzygies ( being less than 90°), the inclination is in-
creasing as the Moon is moving through an angle
2 (NEQ + m En) = 360° – 24 :
and is decreasing as the Moon is moving through the remaining
angle 24. And the line of nodes regresses while the Moon is
EFFECT ON THE INCLINATION AND LINE OF NODES. 279
moving through an angle 180° + 2; and progresses while the
Moon is moving through the remaining angle 180º – 20.
III. Suppose the line of nodes is in quadratures.
Then as the Moon moves from quadrature to syzygy the
disturbing force and motion tend in different directions, and
therefore the inclination is decreasing and the line of nodes
is regressing. And as the Moon moves from syzygy to quad-
rature the inclination is increasing and therefore the line of
nodes still regresses.
Wherefore the increase and decrease of inclination coun-
teract each other; but the motion of the nodes is wholly
regressive.
IV. Suppose the line of nodes is behind the Sun:
fig. 90.
Then as the Moon moves from N to m the inclination is
decreasing and the line of nodes regressing: and as she moves
from m to q the inclination is increasing and the nodes re-
gressing. As the Moon moves from q to n the inclination is
decreasing and the nodes progressing.
Hence, if as before be the angular distance of the line
of nodes from syzygy, the inclination is increasing as the Moon
moves through an angle 24, and decreasing as she moves.
through the angle 360º – 20. But the nodes regress and
progress respectively while the Moon is moving through the
angles 180º +2 and 1800 - 20.
:
It appears, then, that on the whole the nodes regress
pretty steadily but the inclination is much more fluctuating
and on the whole is not affected during a revolution of the
line of nodes.
We introduce the two following Propositions as examples
of the method used by Newton in the Third Book of the
Principia: they will be found in Props. 30 and 31.
geometry is translated into analysis.
Newton's
PROP. To calculate the motion of the nodes of the
Moon's orbit considered circular.
318. Let MM' be the arc described by the Moon in a
unit of time, fig. 91: M'L2 space through which the dis-
280
DYNAMICS. PROBLEM OF THREE BODIES.
turbing force would draw the Moon in the same time: Nn
the line of nodes, Qq the line of quadratures, AB of syzygies:
Mm is a tangent to the Moon's orbit at M meeting the ecliptic
in m: join LM and produce it to meet the ecliptic in 7: this
gives the position of the tangent at M after the small time of
describing MM' and therefore mEl represents the motion
of the node.
Now LM' is parallel to the ecliptic and therefore can meet
no line in the ecliptic: but it is in the same plane with Im,
therefore LM' is parallel to lm.
Hence
motion of Node
motion of Moon
LlEm
L MEM'
sin mlE Im
L MEM' Em
nearly
sin AEn
L MEM' MM'
sin AEn LM' Mm
L MEM' MM' Em
Now the disturbing force in the direction M'L
LM'
sin MEm.
Sr S'
S
13
(see Art. 294.) =
y³
༡
y³
1)
3 Sr
213
cos MEA, ··· y = r' — r cos MEA;
3 Sr
LM'
COS MEA.
j′3
MM¹2
12
(vel.)
2
MM'. MEM'
=
force of Moon to E
1°
rad.
E + M
2.2
LM'
3S
20:3
MM'. MEM
M+ Er
13
cos MEA =3m² cos MEA,
Art. 295;
... motion of Node
3m² cos MEA sin MEN sin AEn. motion of Moon.
Let N = longitude of the Node,
Ꮎ
т. Ө
Moon,
Sun,
MOTION OF NODE CALCULATED.
281
supposing that the Sun, Moon, and Node were all in the first
point of Aries when = 0.
Hence the above equation gives
d N
3m² cos (0 – m0) sin (0 – N) sin (m0 – N),
d Ꮎ
the negative sign being taken because the mean motion of the
node is regressive.
319. We shall now transform this by the ordinary trigo-
nometrical formula
we have
d N 3 m²
2 sin a cos b = sin (a + b) + sin (a - b),
+cos 2 (0−m 0)−cos 2 (0–N)−cos 2 (m0–N)}.
d0
4.
For a first approximation we neglect the periodical terms and
take the mean values:
d N 3m²
ᏧᎾ
i suppose;
4
d N
.. N=-10, constant = 0.
For a second approximation we shall put this value of N
in the periodical terms;
i { 1 + cos 2 (1 − m) ( − cos 2 (1 + i) 0 − cos 2 (m + i) 0};
d Ꮎ
i
~'. N = − i 0
sin 2 (1 − m) 0 +
2 (1 − m)
i
2 (1 + i)
sin 2 (1 + i) 0
+
sin 2 (m + i) 0.
in
2 (m + i)
For a third approximation we shall put this value of N
d N
do
Ꮷ Ꮎ
after neglecting the terms divided by 1 - m and 1 + i,
because they are smaller than the term divided by m+i; ther
i
N = 10 +
- iA+
sin 2 (m + i) 0:
2 (m + i)
N x
282
DYNAMICS. PROBLEM OF THREE BODIES.
d N
3
=-i-i cos 2 (1-m) 0+i cos {2 (1 + i) 0
d Ꮎ
m + i
sin 2 (m+i) 0}
i
+ i cos {2 (m + i) |
sin 2 (m + i)0}.
m + i
If we expand these the last term gives
i²
22
sin² 2 (m + i) 0 or
m + i
2 (m + i) 2 (m + i)
cos 4 (m + i) &.
Hence we obtain
d N
22
i +
+ periodic functions of 0,
d Ꮎ
2 (m + i)
and therefore the mean value of N is
3m
N:
Ꮎ
3m²
Ꮎ .
2 (m + i)
8 (1
3m
4
1 +
4
If we expand in powers of m we have
3m² 9m³ 27m²
N
+
+
4
32
128
..) 0,
320.
In this calculation we have supposed the Moon's
angular velocity to be uniform. To correct for the oval orbit
let N₁ be the corrected value of N.
Now the motion of the node varies as the magnitude of
the disturbing force, which varies as the square of the time of
the Moon's describing MM', and therefore as the square of
the velocity at M inversely;
d N₁
(vel.) in octants
1
d N
(vel.) at M
3m²
2 (1-m)
cos 2 (1 − m) 0;
d N₁
i
¿ { 1
+ cos 2 (1 − m) 0...}
de
2 (m + i)
CHANGE IN INCLINATION CALCULATED.
283
2 i
X
{1
1
m
i
cos 2 (1 − m) 0 + ...}
¿ { 1
2 (m + i)
i
2 (m + i)
1
1
2 i
-
m
m
Ꮎ
cos² 2 (1 − m) & +
m) 0 + ... }
+ periodical terms,
and the mean value of N₁ is
N₁
i
R
m
Ө
2 (m + i)
1
3 m²
3m
4
8 (1-
3 m
+
4
3m²
9m
45
met +
4
32
128
3m²
4 (1 − m
-
...) 0,
Ꮎ Ꮎ
this correction does not affect the first and second terms.
PROP. To calculate the inclination of the Moon's orbit
to the ecliptic at any time.
321. Let ENm be the line of nodes (fig. 92), El its
position after a unit of time: Mp perpendicular to the plane
of the ecliptic, pG perpendicular to the line of nodes EN;
produce pG to cut El in g; join MG, Mg and draw Gr per-
pendicular to Mg: I the inclination of the plane of the Moon's
orbit to the ecliptic.
Now 81 MGp Mgp = GMg=
= L - ≤ ‹
= <
Gr
GM
Gg
Also N GEg=
= ▲
GE
Gr GE
=
SN = sin I. cot MEN. 8N;
MEN.SN;
Gg GM
284
PROBLEM OF THREE BODIES.
DYNAMICS.
dI
3 m² sin I cos MEA cos MEN sin AEn
d9
3 m² sin I cos (0 – m0) cos (0 – N) şin (m0 – N)
2
3m²
4
sin I {sin 2 (0-N)-sin 2 (0-m0) + sin 2 (m0-N)}.
This expression shews that I will always be small: and
therefore sin I = I nearly; let y be the mean value of I: also
3m²
N
Ꮎ
0=-10, (see Art. 319.)
4
dI
3m²
do
4
y {sin 2 (1 + i) 0 - sin 2 (1 m) + sin 2 (m + i)0};
3m²
..
·. I
=
1
Υ
· (1 + i)
cos 2 (1 + i) 0
Ꮎ
1
-
2 (1 − m)
cos 2 (1 — m) 0
1
+
COS
2 (m +
2 (m + i)
i) }
+ const.
The constant=y, the mean value of I. Therefore, neglecting
the first and second terms because they are of an order higher
than the third,
I = y {1+
3m
cos 2 (m + i) 0}
3m
8
I +
4+
3m
{1+ cos 2 (Sun's longitude-Node's longitude)},
8
neglecting ym², &c.
This accords with Chapter V. Art. 342.
CHAPTER V.
LUNAR THEORY.
322.
We now enter upon the calculation of the pertur-
bations of the Moon by a process of systematic approximation;
and shall proceed in the next Chapter to calculate those of the
planets. In the Lunar and Planetary Theories we use different
methods of calculation for this reason. The perturbations of
the Moon are much greater than those of any planet, because
the Sun, the mass of which is so enormous (Art. 286), is one
of the disturbing bodies. Likewise the ratio of the distances
of the disturbed and disturbing bodies from the central body
about which they move is very different in the two theories;
being about in that of the Moon, and sometimes so large as
in that of the planets. The difference of the methods of
approximation will be seen in the calculations of this and the
following Chapter.
4
1
400
Before entering upon the immediate subject of the present
Chapter we must investigate the following Proposition.
PROP. A number of bodies considered as material par-
ticles attract each other with forces which vary inversely as the
square of their distances, and directly as the mass of the
attracting body: required the equations of motion of any one
of the bodies relatively to a second.
323. Let M, m, m', m" .... be the masses of the bodies;
M being that of the body about which the motion is to be
calculated: and m the mass of that body of which the equations
of motion are to be determined,
286
DYNAMICS.
LUNAR THEORY.
Let X, Y, Z be the co-ordinates of M,
X + x, Y + Y, Z +
X +
Y +, Z+
m,
m'
Then the distance between m and m' is
{(x' − x)² + (y' − y)² + (≈′ − ≈)²} ³,
and the attraction of m' on m is
m'
(x' − x)² + (y' − y)² + (≈′ — ≈)³
Let this be resolved into three parts parallel to the axes: that
parallel to the axis of x is
or
1
m' (x' - x)
{(x' − x)² + (y' − y)² + (x′ − x)²}
d
mm'
m dx {{(x − x)² + (y'′ − y)* + (≈′ ·
and so of the other bodies m"
Now assume λ = >
Σ.
mm'
− ≈)
≈) } }
}
{(x′ − x)² + (y' − y)² + (≈' − ≈)²} } '
which expression is the sum of the quantities found by dividing
the product of every two of the masses m, m', m'
by
their respective distances.
Then the sum of the attractions of m',
m"
on m re-
solved parallel to the axis of a
1 dλ
m dx
1 dλ
1 αλ
In like manner
and
m dy
m dz
are the attractions of m', m"....
y
and s
on m parallel to the axes of
Let rr''".... be the distances of m m'm".... from M.
Then the attraction of M on m parallel to a is
Mc
and con-
"
3,3
GENERAL EQUATIONS OF MOTION.
287
sequently the equation of motion of m in space parallel to a is
d² (X + x)
df
1 dλ Μα
m dx ༡3
m x
m'x'
But
2+3
13
are the attractions of mm' ….. on M parallel
to ~: and therefore the equation of motion of M parallel to a is
d² X
m x
Σ
d t
2.3
and by subtracting this from the equation above we have the
equation of motion of m relatively to M
d² x Μα
M X 1 dλ
+
+ Σ.
0.
d t
2015
7.3
m dx
And in like manner
ď² y
My
my
1 dλ
+
+ Σ.
0,
dt
༡?
m dy
d²≈ Με
+
df 2013
MZ
I dλ
= 0.
2.3 m dz
Now assume
R*
m"
m' (xx' + yy+zz') + w″ (xa'"+ yy" +=="),
2′3
112
*The function R satisfies Laplace's Equation (Art. 168).
+
M
and so
For
dx
d R m'x
+
7'3
I dλ
m dx
m'x'
+
1
Σ
M
d²R
Σ
dx2
M
d² R
1
dy"
de R
dz?
772
m
Σ.
Σ
mm' (x' — x)
{(xx)²+(y' - y)²+(≈≈)
2
mm' { (y' − y)² + (≈′ — ≈ )º — 2 ( x' — x )º }
{(x-x)²+(y'-y) (≈'-x)}
3
mm' { (x'— x)² + ( ≈′ — ≈)² − 2 (y' − y)² }
{ (x' — x')² + ( y' − y)² + ( ≈′ — ≈)² ¦ F
mm' { (x' − x)² + (y' − y)² + 2 (≈
— ≈)}
5
{ (x' — x)² + (y' − y)² + ( ≈' — ≈)² ¦ 5
d²R
d² R
+
d² R
+
0.
days
dy
dze
288
LUNAR THEORY.
DYNAMICS.
d R
m'x'
then
-
+
m"x'
//
I dλ
+
dxx
13
113
m dx
งาน
m x
1 αλ
Σ.
,
20:3
203
m dx
and the first equation of motion becomes
ď²x
+
d t²
(M + m) x
20:3
d R
+
0
dx
and similarly
day
(M + m) y
d R
(1).
+
+
dť
2.3
dy
d² z
(M + m) ≈
d R
+
+
dť
dz
These are the equations by which the motion of the Moon
about the Earth, or of a planet about the Sun, is determined
when under the action of all the other bodies of the Solar
System. They have never yet been completely integrated.
For this reason we must resort to approximation. To effect
this R, which is called the disturbing function, must be de-
veloped in a converging series. The difference of the methods
adopted in the Lunar and Planetary Theories depends upon
the different modes of expanding the function R. In the
Lunar Theory R is expanded in powers of the ratio of the
distances of the Sun and Moon, a very small fraction nearly
equal but in the Planetary Theory R is expanded in
;
400
powers of the eccentricities and inclinations of the planetary
orbits, all of which are very small, with the exception of those
of Juno and Pallas; the eccentricities of these being about
and the inclination of the orbit of Pallas to the ecliptic being
about 35º: but the masses of these planets are very small.
1
4
324. We intend throughout our calculations in this and
the following Chapter to neglect quantities which depend upon
the square and higher powers of the disturbing forces. In
consequence of this we may calculate separately the perturba-
tions caused by the Sun or a planet on the supposition that the
TRANSFORMN. OF EQUATIONS TO POLAR CO-ORDINATES. 289
other heavenly bodies do not attract, and then add together
the separate perturbations: this follows from the Principle
explained in Art. 288.
PROP. To obtain equations for calculating the radius
vector of the Moon; and the inclination of the lunar orbit
to the ecliptic.
325. The equations of motion are by the last Article
ď x
MX
d R
+
+
dt
2013
dx
d² y
му
d R
+
+
(1),
dt
dy
d² z
+
dt2 2.3
dR
M Z
+
dz
where μ
= mass of the Earth + mass of the Moon.
Let the plane of the ecliptic be the plane of xy: also let p
be the projection of on the ecliptic: s the tangent of incli-
nation of p to the same plane: the longitude, the axis of x
passing through Aries: then
x² + y² = p² ;
que + p²
p² = p²
x =
p
cos 0,
y
=
p sin 0,
≈ = ps.
Multiply the first equation by y, and the second by a, and
subtract;
d R
d R
dx
X
d R
dy
cos
... 20
d² y
d t²
ď²x
У
= Y
df
dr
d
dy
dx
ac
-y
dt
dt
dt
= p {sin
d
d Ꮎ
or
pT,
dt
dt
d R
d R
if we put sin
COS
s Ꮎ
T.
dx.
dy
Оо
d R
dy
290
DYNAMICS. LUNAR THEORY.
!
Multiply each side by p²
d Ꮎ
;
dt
do d
ર
de
2
ᏧᎾ
dt dt
dt
= p³ T
dt
(p²
d Ꮎ
2
dt
= h² + 2 √ p³ Tdə;
h² being a constant introduced by integration;
d 02
h²
2
d t2
4
μ
=
4
3
Sprde:
Sp³ Td0; p=
h² u² + 2u* f
1
и
Tde
(2).
из
Again, multiply the first and second equations (1) re-
d x
spectively by 2
2 dy and add,
dt
dt
d (dx² dy²
dt dť
+
+
d t
20.3
putting a = p cos 0,
2
d fd p²
dt dt
Jdp²
+
p³
dᎾ
dx dy
00 + y
dt dt
+2
dx dR dy dR
+2
dt dx dt dy
y = p sin 0, x² + y² = p²;
2μp dp
ᏧᎾ
+
->
dt2
203
2P
dt
dt
(sin
d R
dR
cos
dx
dy,
dp
dR
dR
+ 2
dt
cos e
+ sin
= 0.
dx
dy
Let MP.
d fd p²
dt \d t
d fd p²
do dť²
+
+
up
d R
d R
+ cos
+ sin
dx
dy
=
P;
d02
2
ρ
d t²
+ 2 P
dp
-
2 P
dt
d02
2
ρ
dt
d Ꮎ
dt
+ 2 pdp
do
·2pT=0, p
И
T = 0 ;
= 0,
TRANSFORMN. OF EQUATIONS TO POLAR CO-ORDINATES. 291
d
1 du²
1 do²)
2 P du
2 T
+
= 0.
do u+ dť
Ꮎ
u² dť²
u² do
И
d02
Now
h2
2 u$ + 2 u*
dť
Qu+ f
Td0
by (2),
by this equation we can eliminate t, and we have
d
du²
+ u²
2
dė
d 0²
u²) (h² + 2 ƒ I do)}
dᎾ
2 P du
2 T
0,
u² do
U
performing the differentiation.
du
ď² u
2
+ и
de d Ꮎ
1) (n²
Ꭲ dᎾ
h² + 2
+2
из
2 T
du²
2 P du
2 T
+
+ 2²²
= 0;
u³
do
u²
do
น
P
T du
ď² u
u²
u³ do
+2
0 ...
d02
(3).
Td0
h² + 2
23
326.
To obtain an equation for calculating the inclination.
of the radius vector to the ecliptic, we have by the last of
equations (1),
d²
με
d R
S suppose,
dť
2.3
dz
S
dz
dz do
1
but ≈ =
26
d t
do dt
U²
(
ds
du do
dᎾ
W
S
ᏧᎾ
10 dt
ገ
ds
do
du
Td0
S
V h² + 2
by (2);
d Ꮎ
U³
ď z
d
ds
du
น
S
dt
do
de
do
d Ꮎ
√ h² + 2
Tdo de
чез dt
292
LUNAR THEORY.
DYNAMICS.
!
+
(
s
- {(u d-du)
ds
du
√
h² + 2
٢٠
Tde
из
T
d Ꮎ
и
S
d Ꮎ
dᎾ
Tde dt
чез
h² + 2
из
des
ď² u
Tde
ds
du T
S
u² h² + 2
+1 u
s
S;
dᎾ
d 02
из
dᎾ
d Ꮎ U
S
Tds
s du
+
d² s
s d² u
U³
u³ \ də
u do
0 ;
dᎾ?
u do
Td0
h² + 2
гиз
but by the last Article
Ps
T's du
s d² u
из
ut do
+ $
0,
u do²
Ꭲ d Ꮎ
h² + 2
из
adding these last equations we have
S-Ps
T ds
+
d² s
u³
u³ do
+8+
0...
(4).
d 02
h² + 2
୮
Td0
U³
327. It is necessary that we estimate the comparative
magnitude of the various small quantities involved in our
calculations.
Let e, e' be the eccentricities of the lunar and solar orbits;
k the tangent of the mean inclination of the lunar orbit to the
ecliptic; m the ratio of the Sun's mean motion to the Moon's
mean motion, a and a' the mean distances of the Moon and Sun
from the Earth: the values of these quantities are nearly
e
1
é
1
60"
= 20, e² = 0, k = 12, m =
志​,
1
139
these we shall reckon of the first order of small quantities.
COMPARISON OF SMALL QUANTITIES.
293
a
1
nearly is a quantity of the second order of
But
α 400
magnitude, since it e² nearly. The Sun's disturbing force is
=
greatest when the Moon is between the Sun and Earth: in
which case it
m'
(a' — a)²
m'
: the ratio which this bears to
a
12
the action of the Earth on the Moon
m'
((a' − a)³
m
m' 2a³
a'z
12
a²
a'3
nearly = 2m²
by Kepler's third law. Hence the disturbing force is of the
second order.
We proceed to expand the values of T', P, S.
PROP.
To expand the values of T, P, S neglecting
small quantities of the fourth order.
328. By Arts. 325, 326,
d R
d R
T = sin 0
cos
dx
dy
up
d R
d R
P =
=
+ cos
+ sin
2013
dx
dy
d R
+
dz
S
με
R = 'm' (xx' + yy' + 22') .
13
m':
and by Arts. 323, 324,
m'
{ (x − x′)² + (y − y')² + (≈ − x')² } } '
= mass of Sun; 'y's' co-ordinates of Sun: ' dist. of Sun.
=
Let x ρ cos e,
Y p sin 0,
x' = r' cos 0', y' r' sin e',
ps as before;
'=0,
since the plane of xy is the plane of the ecliptic.
m'p
.. R
12
cos (0 – 0')
-
m'
{p² + №² + r²² − 2 pr′ cos (0 – 0')}³
·
294
LUNAR THEORY.
DYNAMICS.
12
}
-}
2
R =
m'e
m'
= cos(
cos (0-0')
2p
* 1 - 2
cos (0 − 0′) + (1 + s²)
m'
m'
11
+
201
2} {1 + s°
12
m'
m'
42
2
ρ
7'2
pl
{ 1 + sº − 3 cos“ (0 – 0')}
– –
{1 − 2s² + 3 cos 2 (0 – 0′)
a
− } //2.
Z
S=
Also @ = tan-1%, p = √/s²+ y², 1= √o+y'
X
dR dR do dR dp
||
x² y²
d R ds
Hence
+
dxx do dx
dp dx
+
ds dx
3 m'p
13
27'3
sin 2 (00') sin
m' p
2r
||
m'p
Q p'3
smp
ρ
cos
2p'3
27'3
{1 + 3 cos 2 (0 - 0')} cos 0
cos (0 – 20');
d R
dR do
+
dy
do dy
d R dp
dp dy
d R ds
+
ds dy
3 m'p
m'p
27'3
27'3
sin 2 (0 – 0') cos 0 -
||
-
3 m'p
2p/3
+ 3 cos 2 (0 − 0') } sin 0
sin (0-20');
m'p
sin 0 +
2 p′3
d R dR do
dR dp
dR ds
m's p
+
+
dz
de dz
dp dx
ds dz
/3
2'
d R
Hence T = sin
cos
dx
dy
d R
3 m'p
27.13
sin 2 (0-0');
d R
d R
up
P:
=
+ cos e
Ꮎ
+ sin
2013
dx
dy
INTEGRATION OF THE EQUATIONS; FIRST APPROXIMATION. 295
up
m'p
3 m'p
{p² + p²
27'3
cos 2 (0 – 0')
273
3s2
m'p 3 m'p
1
cos 2 (0 − 8′);
2
2r'3
2p3
μα
d R
ups
S
m's p
+
+
203
dz
4
{p² + p² s² } }
13
Τ
35.3
m's p
S
+
13
PROP. To integrate the differential equations, first
approximation.
329. We here neglect all small quantities of an order
higher than the first, and therefore the disturbing force
(Art. 327): hence by last Article
T= 0, P
=
p
=
µÙ³, S
льб
2
= μsu²,
で​。
and the differential equations (3) (4) of Arts. 325, 326 become
d² u
+ U
d02
u
0:
h
d's
and
+ s
0.
d0°
The solutions of these equations are
и
W { 1 + e cos (0 − a) } = b { 1 + e cos (0 − a)}, h
h²
and s =
k sin (0 − y): e, a, k, y are constants.
(0-2):
h
The first of these proves that the orbit of the Moon is an
ellipse and the second proves that the tangent of the latitude
bears a constant ratio to the sine of the longitude reckoned from
the node, and therefore the Moon moves in a constant plane.
296
LUNAR THEORY.
DYNAMICS.
PROP. To shew that to integrate the differential equa-
tions to a second approximation we must introduce all terms
of the third order, in which the coefficient of 0 is either nearly
equal to unity, or is small.
330. By approximating to the values of the small quan-
tities we shall arrive at a differential equation in u of the form
ď u
d Ꮎ
+ u + a + a' cos (n0 + n') +
the integral of which is of the form
0,
u = − a + A cos (0 + B) + C cos (n 8 + n') +
A, B being arbitrary constants, and C...constants to be deter-
mined by putting this value of u in the differential equation.
This gives
C' (1 − n²) = — a' ;
- a'
.. C
1
from which we learn that if n nearly 1, then C will be large.
Wherefore when the coefficient of the argument of a cosine or
sine is nearly unity we must retain coefficients of the third
order, since these terms rise into importance by the process of
integration.
Again, the function R and therefore the differential equa-
tion in u contains terms depending on ': and the reciprocal of r
е
= b' {1 + e' cos (0' - a')} = b' {1 + e' cos (me + B - a) +... }
the accented letters refer to the Sun: m = ratio of the Moon's
period to the Sun's period: B = longitude of the Sun when
the Moon is in Aries. Hence
dt
dt
I
Ꭲ dᎾ -
calculated from
1 + 2
ᏧᎾ
ᏧᎾ
hu²
h2u3
(Art. 325, equation (2)) will contain a term C'cos (m0+ß−a'),
and therefore t contains a term
C
sin (m0+ẞ- a'):
ጎን
EXPANSIONS FOR SECOND APPROXIMATION.
297
Wherefore
hence C must be calculated to the third order.
all terms in which the coefficient of 0 is small must be cal-
culated to the third order; as well as those in which the
coefficient of is nearly equal to unity.
PROP. To calculate sin 2 (0 – 0′) and cos 2 (0 – 0′) to
the first order.
331. We need calculate these only to the first order
because they occur only in terms multiplied by quantities of
the second order.
dt
1
1+2
d A
hu²
·Td0) -
h² u³ S
-
1
first order:
3
hu??
2
>
1
b² h
{
$1
2e cos (0 − a)},
a)}, b³h = n = Moon's mean motion;
... nt = 0 − 2e sin (0 − a),
t = 0 when the Moon's mean longitude = 0; also let the Sun's
mean longitude then = ß:
n' t + ß = 0′ − 2 e′sin (0′ – a′), n'
n'
=
Sun's mean motion.
Now =m: hence multiplying the first equation by m
n
and subtracting, we have
O' - 2e' sin (0' a') = m0 + ß; neg. me, of second order;
- -
·. 0′ = m0 + ß + 2e′ sin (m0 + ß − a') ;
-
.. sin 2 (0 – 0′)=sin {[(2 −2 m)0 − 2 ß] – 4e′ sin (m0 +ß − a ) }
sin {(2-2m)0-2B-4e'cos {(2-2m) 0-2ẞ} sin(m0+ß-a')
= sin {(2-2m)0-2B-2e'sin {(2-m) 0 - ß-a'}
+ 2e' sin {(2 − 3 m) 0 − 3 ß + a'
cos 2 (0 – 0') : = cos {(2 - 2m) 0 - 2ẞ}
+ 4e′ sin { (2 − 2m) 0 – 2 ß} sin (m0 + ß − a')
= cos {(2 -
·
2m) 0 −2ẞ} + 2e' cos { (2 – 3m) 0 − 3 ß + a' }
2ß}
2é' cos {(2-m) 0 - ẞ-a'}.
PP
298
DYNAMICS. LUNAR THEORY.
T
Td0
T du
PROP. To calculate
to the
h2u3
h² u³ ›
h2u3
3
h2u³ do
third order.
T
3m'
332. By Art. 328,
h2u³
2uth2'3
sin 2 (0-0')
3m'b'3
4
2h2b4
+ e cos (0 − a)} −¹ { 1 + e' cos (0′ – a')}³ sin 2 (0–0′)
3 m² {1 - 4e cos (0 - a) + 3 e' cos (m0 + B − a)}
×{sin [(2 - 2m) 0 - 2ẞ] - 2e'sin [(2
·
m) 0 - ẞ-a']
+ 2e'sin [(2 – 3m) 0 − 3 B + a']}
-
α
= - 3 ß]
m² {sin [(2 − 2m)0 −2ẞ] − 2e sin [(1 − 2m) 0 − 2 ß + a
Again u=b{1+ e cos (0 − a) } ;
T du
h² u³ do
Td0
h² u³
3
du
..
do
=-be sin (0-a);
3 bm² e cos {(1 - 2m) 0 − 2 ß + a}.
f
Again d = 3
1
m²
2
2m
2e
cos[(2 - 2m) 0 – 2 ß]
-
cos [(1 − 2m) 0 - 2ẞ + a
1 - 2m
= 3 m² cos { (2 – 2 m) 0 – 2 ẞ} − 3 m² e cos { (1 − 2 m) 0 − 2 ß + a}
P
PROP. To calculate
to the third order.
h2u2
333. By Art. 328.
P
=
b (1 - 3 s²) -
h² u³
m'
2u³h2r's
– 3
{1 + 3 cos 2 (0 − 0')}.
3
First. b (1-3 s²) = b {1 − 2 k² + 2 k² cos 2 (0 − y)}·
Secondly.
m'
2u³ h2p/3
m'b'3
2h2b3
3
{1+e cos (0 − a)} −³ {1 + e' cos (0' - a')}³
- 1 bm² {1 - 3e cos (0 − a) + 3 e' cos (m0 + B − a
ß a')};
EXPANSIONS FOR SECOND APPROXIMATION.
299
both terms must be retained, since in the first the coefficient
of = 1, and in the second it is small.
Thirdly.
3 m'
Qu³ h2
2 2013
cos 2 (0 – 0')
=
- 3 bm² {1-3 e cos ( − a) + 3 e' cos (m0 + B − a )}
(0 ß
× {cos[(2 − 2m) 0 − 2 ß] + 2e'cos [(2 – 3m) 0 − sß + a']
ß
- 2e'cos [(2m) 0 - ẞ- a']}·
Multiplying these together by the formula 2 cos a cos b =
cos (a - b) + cos(a+b), neglecting quantities of the third order,
except those in which the coefficient of 0 is small or nearly
unity, we have this third part of
P
h² u²
- 3 bm² { cos [(2 - 2m) 0 - 2ẞ] - 3 e cos [(1 − 2m)0-2ẞ+a]}·
Hence the value of
2ß]
P
h² u²
is
b { 1 - 2 k² + 3 k² (0)}-bm² -
3
cos 2 (0 − y)} − 1 b m² { 1 – 3 e cos (0 − a)
+ se'cos (m0 + ẞ − a') + 3 cos [ (2 − 2m)0 – 2ẞ]
PROP.
-ecos [(1-2 m) 0 -2B+a]}.
α
To form the differential equation for u.
334. By Art. 325, equation (3),
PT du
ď² u
+26
U²
u³ do
0.
do²
Ꭲ d Ꮎ
h² + ?
U³
By expanding the reciprocal of the denominator of
of the
fractional part and neglecting the square of the disturbing
force which is of the fourth order, and neglecting all other
quantities of the fourth order, and observing that P contains.
a term uu, or by Art. 329 bhu, which is not small, we have
300
LUNAR THEORY.
DYNAMICS.
ď² u
P
T du 26
Ꭲ dᎾ
+ U
+
+
0.
ᏧᎾ
h2u2
h² u³ do h2
By the last two Articles, we have
ď² u
d O
+ u − b ( 1 − 3 k² - 4 m²) — 3 bm² e cos (0 - a)
- 3 bk² cos 2 (0 - y) + 3bm² cos {(2 - 2m) 0 - 2ẞ}
15 b m² e cos {(1−2m) 0 −2 ß + a } + ½ bm² e' cos (m0+ẞ−a')=0.
Now this equation cannot be integrated, as it stands, ac-
cording to the method mentioned in Art. 330; because the
term 3 bm³e cos (0 - a) would introduce an infinite coefficient
into the expression of u since the coefficient of 0 = unity. But
this may be remedied by putting for be cos (a) in the
term bm² e cos (-a), which is of the third order, its first.
approximate value ub: then the equation becomes.
d² u
ᏧᎾ
2
+ (1 − & m²) (u - b) + 4 b (3 k² + 2m²) - bk cos 2 (0-7)
15
+ 3bm² cos {(2−2m) 0-2B} - bm² e cos {(1-2m) 0-2ß+a}
2
+ 3 bm² e' cos (m0 + B − a') = 0.
Let 1
3 m² = c²; then if we neglect all coefficients of the
second order, we have
d² u
dᎾ
+ c² (u − b) = 0,
.. u = b {1 + e cos (co- a)}.
Now although e differs from unity only by a quantity of the
second order, yet cos (c0 − a) will differ very sensibly from
cos (0 − a) after several revolutions of the Moon. Wherefore
the peculiarity of the differential equation in u (mentioned in
the last page) when we proceed to a second approximation
teaches us, that the value of u in Art. 329 will cease to be
a first approximate value after several revolutions of the Moon;
the true first approximate value being b{1 + e cos (c0- a)}
We must therefore carefully retrace our steps, and replace 0
by c in every place where @ is introduced in consequence of
INTEGRATION OF EQUATIONS; SECOND APPROXIMATION. 301
its depending immediately on the first approximate value of u.
This may very easily be accomplished by putting a+ (1−c) 0
instead of a in every place where it occurs: for a enters the
equations solely in consequence of its dependence on 0 and u by
the equation u = b {1 + cos (0 - a)}.
The same will be the case with the value of s, as we shall
shew in the next Proposition. We shall write down the
corrected equations of u and s in Art. 336.
PROP. To form the differential equation for s.
335. By Art. 326, equation (4),
S - Ps
T ds
+
d's
do³
203
.3
u³ do
+5+
= 0.
Tde
h² + 2
u³
S - Ps
3m's
Now
3
1
Qu¹h²r'
2 13
{1 + cos 2 (0 - 0')}
h² u³
3 m'kb's
2h bi
sin (0 − y) {1 + cos [2 (1 − m) ( − 2 ẞ]}
1
3 m² k { sin (0 − y) − sin [(1 − 2m) 0 - 2 B + y]},
retaining those terms of the third order which have the multi-
plier of nearly = 1.
Again,
ds
d Ꮎ
=
k cos (0 - y);
T ds
3 m'k
cos (0 - y) sin 2 (0 – 9′)
h² u³ də
Qu¹h²r
d's
do
=-3 m² k cos (0 - y) sin (2(1 m) 0 - 2ẞ}
<
=-3 m² k sin {(1 − 2m) 0 − 2ẞ + y}.
Then the equation in s becomes
+ s + 3 m³ k { sin (0 − y) − sin [(1 − 2 m) 0 − 2ß + y ]} = 0.
302
LUNAR THEORY.
DYNAMICS.
This (as in the case of the equation in u) cannot be inte-
grated by the method explained in Art. 330, because the term
3m² k sin (0) would introduce an infinite coefficient into the
value of s. But by putting for k sin (0) its first approxi-
mate values in the term m'k sin (y), which is of the
third order, this difficulty is overcome; the differential equation
then becomes
d² s
d 02
+ (1 + — m²) s − 3 m² k sin {(1 − 2m) 0-2B+7}
− ß y} = 0.
Let 1 + 32 m² = g; then if we neglect the coefficient of the
second order, we have
d's
+ g³s = 0; .'. s
s = k sin (g) − 2).
d02
Hence (as in the last Article) although g differs from
unity only by a quantity of the second order, yet sin (g0 − y)
will differ sensibly from sin (y) after several revolutions.
of the Moon. Therefore k sin (0 − y) ceases to be a first ap-
proximation of s after several revolutions of the Moon: and
we must retrace our steps and put go for 0 in every place
where enters in consequence of its immediately depending
on s. This may be done by putting y + (1 - g) for y in
0
every place where Υ occurs.
PROP. To integrate the differential equations in u
and s.
336. After replacing ◊ by c◊ and go in all such places
as enters the equations in consequence of its immediate
dependence on u and s respectively, the equations of Arts.
334, 335, become
d2 u
d02
+ c² (u − b) + 4 b (3 k² + 2m²) − 3 b k² cos 2 (g) − y)
+ 3bm³ cos {(2 − 2 m) 0
{(2−2m) 0 - 2ẞ}-bm² e cos
2ẞ} – 15 bm² e cos { (2−2m −c) 0 −2ẞ+a}
d² s
and
d02
+ 3bm² e' cos (m0 + ß − a')
=
0 ;
+ g²s - & m² k sin { (2 − 2m - g) 0 – 2 B + y} = 0.
INTEGRATION OF EQUATIONS; SECOND APPROXIMATION. 303
To integrate the first assume u =
b { A+ e cos (c0 − a) + B cos 2 (g0 − y) +Ccos [(2 −2m) 0–2ß]
+ Dcos [(2-2m-c) 0-2ẞ+a] + E cos (m0+ẞ-a)},
α
A, B, C, D, E being indeterminate coefficients: to find these
substitute this value of u in the differential equation, and equate
the coefficients to zero: then, remembering that c² = 1 − 3 m²
and g² = 1 + 2 m³, and neglecting small quantities of orders
higher than the second, we have
-
c² A = c² — ³ k² — 1m³,
-
A
3
···
:.
▲ = 1 − ³ k² − 1 m²
- 3
(c² − 4g²) B = 3 k³,
..
B = −
- 4
k²
{c² - (2-2m)² } C = −3 m²,
.. C = m²
—
•. D= 1/5 me
.. E = - 3 m² é'.
-3
{ c² − ( 2 − 2 m − c )² } D= /m²e,
-
(c² - m³) E = - 3 m² e',
Hence u=b{1-3 k² — — m² + e cos (c0 − a) - 4k² cos2 (g0− y)
15
+m³cos [(2 −2 m) 0 − 2 ẞ] + me cos [(2 − 2 m −c) 0 −2ẞ+a]
- 3 m² e' cos (m0+ẞ-a)}·
337. Again, to integrate the equation in & assume
s=ksin (go-y) + Fsin { (2-2m-g)0−2ẞ+y},
F being an indeterminate coefficient.
differential equation, and we have
... s = k sin (g0 − y) + mk sin {(2
Substitute this in the
{g² - (2 - 2m - g)² } F = { m² k, .. F = 3 mk;
J
m k sin {(2 − 2m − g) 0 − 2 ß + y }
We shall now make use of these values of u and s to calculate
the distance and longitude of the Moon.
PROP. To find the distance of the Moon from the Earth.
338.
1
1*
u
Let be this distance; .'. 1' =
· = p √ 1 + s² ;
(1
- s), neglecting quantities of the fourth order,
304
LUNAR THEORY.
DYNAMICS.
= 4
u { 1 - 1 k² + 4 k² cos 2 (g0 - y)}, (Art. 337.)
= b { 1 −
− 4
k² – ½ m² + e cos (c0 − a) + m² cos [(2 − 2m) 0 – 2 ß]
-
+ 15 me cos [(2 - 2m - c) 0 - 2 B +
ß
+a]
- 3 m² e cos (m0 + ß − a')}. (Art. 336.)
· b {1 + e cos (c0 − a) + m² cos [(2 − 2 m) 0 − 2ẞ]
+ 15 me cos [(2 - 2 m −c) 0 - 2ẞ + a] - 3 m² e cos (m0+ẞ− a') },
8
a')},
PROP.
the time.
where bb (1 − k² - 1 m²).
To find the longitude of the Moon in terms of
339. By Art. 325. equation (2) we have
de
dt 1
hu2
1 + 2
Tdo - ž
h2u³
1
Ꭲ ᏧᎾ
Ꭲ
hu²
h² u³
2 3
Td0
Then substituting for
and u by Arts. 332, 336, and
h2u3
retaining those terms of the third order in which the coefficient
of is small (Art. 330),
1
dt
de
hb
2
{ 1 + 3 k² + m² + } { e² 2e cos (c0 - a)
-
+ ½ e² cos 2 (c0 − a) + ½ k² cos 2 (g0 − y)
1 m² cos [(2 - 2m) 0 - 2 ẞ] — 15 me cos [(2 - 2m - c) 0 −2ẞ+a]
+ 3m² e' cos (m0 + B − a')}·
Then putting hb² (1 − & k² - m² - 3 e²)
‡
and integrating,
11
2
2 e²)
-
= n, multiplying by n
nt=0−2e sin (ce—a) +3 e² sin 2 (c0−a) + k² sin 2 (g0−y)
- m² sin {(2-2m) 0-2ẞ} - 15 me sin {(2 - 2m-c) 0-2ẞ+a}
0—2ß}
4
+ 3 me' sin (m0 + B- a').
To obtain in terms of t we proceed as follows. Transpose
all the terms but to one side of the equation. If all small
MEANING OF THE TERMS IN THE RAD. VECTOR AND LONG. 305
quantities are neglected = nt; then for a first approximation
we neglect small quantities of the second order, and put = nt
in the small terms;
.. 0
=
nt + 2e sin (cnt – a).
For a second approximation we put this value of in small
terms and neglect small quantities of the third order;
.. 0=nt+2e sin (cnt-a) + e² sin 2 (cnt - a) - 4k² sin 2 (gnt-y)
11
15
+ m² sin { (2−2m) nt-2ẞ} + me sin {(2-2m −c) nt-2B+a}
8
- 3 me' sin (mnt + B − a).
340. These expressions for the radius vector and the lon-
gitude of the Moon shew, that her distance from the Earth
preserves nearly a constant value, fluctuating between very
small limits: and that her longitude varies nearly as the time
of motion, departing from this law only by small quantities.
It will be an interesting enquiry to examine these formulæ
for the radius vector and the longitude, and see whether they
will enable us to explain the various inequalities that observa-
tions have pointed out in the motion of the Moon.
The prin-
ciple of the superposition of small motions (Art. 288.) allows us
to examine the cause of each small term upon the supposition
that all the other small terms do not exist.
PROP. To interpret the physical meaning of the various
terms in the analytical expressions for the radius vector and
the longitude of the Moon.
341.
The first variable term of the reciprocal of the
radius vector, or be cos (ce - a), may be thus interpreted.
If c = 1 this term would be the ordinary variation from
circular motion when a body moves in an ellipse: but c does
Let E be the focus and a EA' the axis major (fig. 83.)
not=1.
of the ellipse of which the equation is
α
then
C
1
a
―
=
b {1+e cos
(0 - 25 ) }
C
is measured from the line EA': A' is a point in the
Q ૨
306
DYNAMICS. LUNAR THEORY.
Moon's orbit; let A'M be her orbit; and let
/
also let A'EM' = c 4 A'EM, and let EM'
above mentioned in M': then
1
EM
A'EM = 0
a
C
cut the ellipse
= b {1+e cos c 2 A'EM} = b {1+e cos A'EM'}
.. EM = EM'.
:
1
EM'
Draw EA equal to EA' making an angle equal to ▲ MEM'
with EA' then through the variable points A and M an ellipse.
can always be drawn having its focus in E and equal in dimen-
sions to the ellipse on a A'. Hence this inequality shews, that,
if we neglect all the other terms, the Moon's motion may be
represented by supposing that it moves in an ellipse, the perigee
of which revolves about the Earth with an angular velocity
(1 − c)
Cor. 7.
3m² do
nearly.
d Ꮎ
dt 4 dt
Principia, Lib. 1. Prop. 66.
The two terms of the longitude
2e sin (ent - a) + e² sin 2 (cnt − a)
correspond to the above term in the reciprocal of the radius
vector; as may be seen by comparing the form of the terms
with those in the expansion of 0 w in Art. 277.
The second term in the reciprocal of the radius vector is
bm² cos {(2 − 2m) 0 − 2ß}.
Now (1-m)0-ß=0−(m0+ß)=long. of Moon-longº. of Sun
angular distance of Moon from the Sun.
Hence this inequality has its greatest positive value when the
Moon is in syzygies, and its greatest negative value when the
Moon is in quadratures. This agrees with the Principia,
Prop. 66. Cor. 5. and also with Art. 303.
11
The term m² sin {(2 – 2m) nt - 28 in the longitude
corresponds to the above term: and is the inequality called the
Variation discovered by Tycho Brahe (Art. 291, 305).
The third term in the reciprocal of the radius vector is
15 bme cos {(2 - 2 m −c) 0 - 2 ẞ + a}.
8
MEANING OF THE TERMS IN THE RAD. VECTOR AND LONG. 307
Since 2-2m-c nearly equals unity it will be seen that this
term is nearly analogous to the first term, though of much less
importance because of the smallness of its coefficient. We shall
take it in conjunction with that term (see Airy's Tracts, Lunar
Theory), neglecting the motion of the perigee and other small
quantities.
1
Then b{
-
2°
{1 + e cos (0 − a) + 15 me cos (0 − 2ß + a)},
neglecting the other terms
=
8
= b { 1 + e cos (0 − a) + me cos [0 − a + 2 (a − ẞ)] }
15
8
= b { 1 + [e + me cos 2 (a - ẞ)] cos (8 − a)
15
15
8
15 me sin 2 (a – ẞ) sin (0 − a)}
15
b{1+e[1+m cos 2 (a - B)] cos [0-a+m sin 2 (a−ẞ)]},
as will easily be seen upon expanding this latter expression and
neglecting small quantities of the third order.
Hence the effect of this third term in the reciprocal of the
Moon's distance is to increase the eccentricity of the elliptic
orbit by me cos 2 (a-3); and to diminish the longitude of
the perigee by 15 m sin 2 (a – ß).
15
8
8
If we suppose the Sun to be stationary during one revo-
lution of the Moon, ẞ = longitude of the Sun: therefore
eccentricity=e{1+mcos2 (long. perigee-long. Sun)}
15
long. perigee (corrected) = a -m sin 2 (long. perigee-long. Sun).
The term me sin {(2
2 m c) nt − 2ẞ + a} in the
longitude exactly corresponds with the term above. It is
called the Evection, and was discovered by Ptolemy (Art. 290).
When the perigee is in syzygies, then a ẞ= 0 or π, and the
В
eccentricity is increased by me: and when the perigee is in
quadratures the eccentricity is diminished by that quantity:
Principia, Lib. 1. Prop. 66. Cor. 9. and Art. 314.
15
The last term in the reciprocal of the radius vector is
- 3 b m² e cos (m0 + ß − a').
308
DYNAMICS. LUNAR THEORY,
α
This is of the third order: but the corresponding term in the
longitude, viz. – 3me' sin (mnt + B - a), is of the second
order. This inequality in the longitude depends upon the
Sun's mean anomaly: when the Sun is in perigee and apogee
then (mnt + ẞ). — a′ = 0 and π, and this term vanishes: when
the Sun is moving from perigee to apogee the term is negative,
and positive as the Sun moves from apogee to perigee hence
the Moon is behind or before her mean place (in consequence of
this inequality) according as the Sun is moving from perigee
to apogee or from apogee to perigee. This is the Annual
Equation: (Art. 292, 300). Also see Principia, Lib. 1. Prop.
66. Cor. 6. and Lib. III. Scholium to Lunar Theory.
There is another term - ² sin (2gnt - 2y) in the lon-
gitude. This depends upon the Moon's distance from the
mean place of her node, and nearly equals the difference be-
tween her longitudes measured on the ecliptic and her orbit:
hence it is called the Reduction.
PROP.
To explain the physical meaning of the terms in
the analytical expression for the inclination of the Moon's
orbit to the ecliptic.
342. The first term is k sin (g07).
Y
Let N be the ascending node when 0-
= 0, fig. 93.
Take
g
▲ NEM' = = 0 -
γ
:
and ▲ M'En = g. ▲ M'EN: also let
g
M be the Moon, tan MEM's: then is the node, moving
backwards. For in the right-angled triangle MM'n, we have
sin M'n
=
= tan MM' cot Mn M', tan Mn M' = tan AB k;
k sin (gð − y).
..stan MM' k sin g 4 M'EN
=
Hence the meaning of this term is that the node regresses with
=
d Ꮎ
an angular velocity (g- 1) dt
3m² do
4 dt
−2ẞ+ y} is
The second term 3 m k sin {(2 − 2 m − g) 0 – 2 B + y
best considered in connexion with the first, as we did the
Evection (Airy's Tracts).
MEANING OF THE TERMS IN THE INCLINATION.
309
Neglecting the motion of the Node
s=k {sin (0-2)+m sin (0-y+27-2ẞ)}
=k
1+ 3 m cos 2 (y−ẞ)} sin (0−y)+3m k sin 2 (~7−ß) cos (8 − y)
= k { 1 + 3 m cos 2 (y−ß)} sin {0−y+3 m sin 2 (y−ẞ)}.
Hence the effect of the second term in s is to increase the tan-
gent of inclination of the lunar orbit by 3mk cos 2 (y−ß) or
3mk cos 2 (long. node - long. Sun), and to diminish the lon-
gitude of the node, calculated on the supposition of its uni-
formly regressing, by the angle 3m sin 2 (7 – ẞ) or 3m sin 2
(long. node long. Sun). Principia, Lib. III. Props. 33.
and 35.
The inclination of the orbit is greatest when the node is
in syzygies, and least when in quadratures: see Art. 321,
and Principia, Lib. 1. Prop. 66. Cor. 10.
343. The angle described by the perigee during a revo-
lution of the Moon, as calculated in Art. 341, equals 3m². 2 π
= 3m² = 1º. 30″ nearly: but its true value as proved by
observation is about twice this. This apparent discrepancy
between theory and observation shook Clairaut's belief in
Newton's law of gravitation, and induced him to propose a
new and more complicated law; pamphlets were already printed
and about to be circulated by Clairaut, when he discovered
that by extending the approximation the value of C is
1 - 4 m² - 23235 m³,
32
the third term of which, owing to the largeness of the coeffi-
cient, nearly equals the second term and therefore reconciles
the apparent difference.
344.
3
9
The value of g is 1+ m² - 3m, and therefore the
ratio of the motion of the perigee to that of the node
75
· (3 m² + 2325 m³) ÷ († m² − 3 m³) = (1 + m) (1 + ÷ m) = 2 nearly.
32
This ratio is much larger than for one of Jupiter's satellites,
because for that system m is very small indeed. Principia,
Lib. 111. Prop. 25.
310
LUNAR THEORY.
DYNAMICS.
345. If m, be the ratio of the mean motion of Jupiter
to that of one of his satellites; then the progression of the
perijove and regression of the node during a revolution of
the satellite each m². Hence the regression of the node
of this satellite during a given time equals the regression of the
Moon's node × (m,² ÷ per. time of satellite) ÷ (m² ÷ per. time
of Moon)
1
2
mean motion of Jupiter) 2 mean motion of Moon
mean motion of Earth
mean motion of satellite regress". of Moon's node.
The same formula is true for the satellites of Saturn.
The progression of the perijove = ½ πm
2
πm, and that of the
Moon nearly 3πm² (Art. 343): hence the progression of the
perijove during a given time
== 1/1
mean motion of Jupiter) 2
mean motion of Earth
mean motion of Moon
mean motion of satellite regress". of Moon's perigee.
The same is true for the satellites of Saturn.
If the series for e were more converging (Art. 343), then
the which multiplies this expression would be 1. Principia,
Lib. 111. Prop. 23. Newton omits the and says "diminui.
tamen debet motus angis sic inventus in ratione 9 ad 5 vel 2
ad 1 circiter, ob causam hîc exponere non vacat."
So it seems
that Newton had some way of accounting for this apparent
anomaly.
I:
The reader that wishes to enter more deeply into the
calculation of the lunar inequalities must consult a memoir by
Baron Damoiseau in the Mémoires présentés par divers savans
à l'Académie Royale des Sciences; Tom. 1: the Lunar
Theory of Messrs. Plana and Carlini; and that of Mr Lubbock.
In these works the approximation is carried so far as to enable
us to deduce all the inequalities from theory alone.
346. In this Chapter we have given the inequalities of
the first and second order: those of the third order are fifteen
in number, these and some of the inequalities of the fourth and
higher orders will be found in the Méc. Cél. Liv. vII.
We
shall mention some of the more interesting results.
Among the periodical inequalities of the Moon's motion
in longitude, that which depends on the simple angular distance
of the Sun and Moon is important on account of the great
light it throws upon the Sun's parallax. The parallax is found
INEQUALITIES OF HIGHER ORDERS THAN THE SECOND. 311
to be 8.56 seconds, being the same as several astronomers have
found from the last transit but one of Venus over the Sun:
Méc. Cél. Liv. VIII. § 24.
An inequality, which is not less important, is that which
depends upon the longitude of the Moon's node: as it did not
appear to depend on the theory of gravity, it was neglected
by most astronomers; till a more thorough examination led
Laplace to discover that its cause is the oblateness of the
Earth it gives an oblateness = 0.05: Méc. Cél. Liv. VII. § 24.
There is also an inequality in the Moon's latitude, which
Laplace discovered by theory: he shewed that it arises from
the oblateness of the Earth's figure: it gives the oblateness
: Méc. Cél. Liv. VII. § 25.
1
304.6
1
These two inequalities prove that the Moon's gravity to
the Earth arises from the attraction of all the particles of the
Earth, and not of the centre alone. (Art. 260.)
By examining the records of ancient eclipses of the Moon
it was found that the Moon's mean motion was continually
accelerated. The cause of this was long sought for in vain;
till Laplace discovered by theory that it depends upon the
variation (the secular variation, see Art. 377.) of the eccentricity
of the Earth's orbit. All the observations which have been
made during the last century and a half, have put beyond
a doubt this result of analysis. When the acceleration of the
Moon's mean motion was known, but not accounted for, con-
jectures were started as to its depending on the resistance of
a medium, or the transmission of gravity; but analysis shews
that neither of these causes produces any sensible alteration.
Méc. Cél. Liv. VII. § 23.
PROP. To prove that the centre of gravity of the Earth
and Moon very nearly describes an ellipse about the Sun.
347. Let x'y's' be the co-or. of the Earth from the Sun,
X1 Y1 1
X Y Z
X Y Z
Moon from the Sun,
Moon from the Earth,
centre of grav. of Earth
and Moon from the Sun.
312
LUNAR THEORY.
DYNAMICS.
m', E, M the masses of the Sun, Earth, and Moon.
r'r, the distances of the Earth and Moon from the Sun.
r the distance of the Moon from the Earth.
Then x₁ - x' = x, Y₁ — y' = y, z₁
Z
Z.
The ratio of E to m' equals 1 354936 and may therefore
be neglected.
The equations of motion of the Earth about the Sun, the
Moon being the disturbing body, are
+
m'x'
+
2'3
ď² x'
d t²
γ
d² y
dť
+
m'y'
20
/3
+
dR'
dx'
d R'
dy
=
(1);
d²x'
d R'
+
+
dť
13
dz
m'z'
R'
M (x'x₁ + y'y₁ + ≈′≈1)
+
2
{ x 2 + y ² + z 2 } {}
2
M
√ (x, − x')² + (y₁ − y')² + (≈1 − x')²
1
-
The equations of motion of the Moon about the Earth, the
Sun being the disturbing body, are
ď² x
dt2
+
(E+M) x
2013
d R
+
dx
ď² y
+
dt²
(E + M ) y
21.3
d R
+
0
(2);
dy
ď² z
(E + M ) ≈
d R
+
+
dť
2.3
dz
R
=
m' (x'x + y'y + x′≈)
12
12
x²² + y² ² + x ¹² } }
m'
(x' + x)² + (y' + y)² + (≈′ + x)²
y','are the co-ordinates of the Sun from the
since – x', - y',
Earth.
MOTION OF THE CEN. OF GRAV. of THE EARTH AND MOON. 313
Ex + Mx₁
Now x =
E+M
Mx
E+M'
Ex
also
= X1
E + M
ď²x
d²x'
M
d² x
+
df²
d t2
E+M dr² by equations (1) (2),
m'x'
Mx
d R'
M d R
2/3
203
dx'
E+M dx
m'x'
Mx
Mx₁
M(x,-x')
M Jm
[m'x' __ m' (x+x')
+
}}
12
3
203
2+3
E+M \ ‚'3
3
ペン
​E
m'x'
M m'x,
Mx1
3 "
neglecting
3
E+ M 7'3
E+M ri
ri
substituting for a' and x, in terms of x and x,
m'
E
13
r'³ E+M
1
Now
13
Also
||
Mx
m'
M
x +
E+M
3
r₂³ E + M
Ex
E+M
X
Mx
E+M
My
+ Y
+
E+M
(=
M&
2
3
E+M
M
1
xx
= = {1 + 3 x + y + z = +.....
E + M
Ex
2
7°
Ey
E+M
2
Ez
1 = {( ~ + - ) + ( ~ + B) ) + ( = + + M)"}
3
x
E+M
3 E
xx + YY + ≈≈
+
E+ M
....}
dť
3EMm xx + y y + z z
(E + M)²
グ
​2,5
M X
M X
3
X
Mx
E + M
18
Ex
E+M
}
+
3 EMm' x²x + xyy + xzz
+
3
(E + M)²
Ꭱ Ꭱ
r
5
+.
314
DYNAMICS. LUNAR THEORY.
Y
=
and
=
r
M X
3
I
X
+ terms multiplied by the products and powers of
}
higher than the first.
r
Դ
1
Now
nearly, and x, y, ≈ cannot be greater than r:
j
400
hence if we neglect small quantities of the fourth order, we
have
ď x
dx
m'x
+
dt2
3
m'y
and similarly
dt2
3
ď² z
m'z
+
d t
3
r
These equations shew that the path of the centre of gravity is
a conic section in one plane the Sun being in the focus, (Arts.
241, 246, 252); it evidently must be an ellipse.
348. Mr Airy, Astronomer Royal, has proposed a method
for determining the mass of the Moon, which depends upon
this Proposition. Since the centre of gravity of the Earth
and Moon describes an ellipse about the Sun, it follows that
the Earth does not describe an ellipse about the Sun: this
deviation from elliptic motion depends upon the mass of the
Moon, and can easily be calculated by theory: and thence can
be determined the error in the Sun's right ascension and de-
clination on the supposition of the orbit of the Earth being an
ellipse. Now when Venus is near inferior conjunction she is
only a third of the distance of the Earth from the Sun, and
consequently the errors in her right ascension and declination
will be much greater than in the Sun's. Some observations for
this end will be found in the Memoirs of the Royal Astrono-
mical Society, Vol. V. p. 223.
CHAPTER VI.
PLANETARY THEORY.
349. WE have already stated that the perturbations of
the Moon are far larger than those of the planets, because the
Sun, the mass of which is enormous and distance not propor-
tionably great, is one of the disturbing bodies.
The perturbations of the planets, on the other hand, are
very minute; and are not detected in short periods of time.
These might, however, be calculated in the manner pursued
in calculating the longitude, latitude, and radius vector of the
Moon but since the approximation is made by means of series
which proceed by powers of the ratio of the distances of the
disturbed and disturbing bodies from the central one, and since
this fraction is much smaller in the Lunar Theory than in the
Planetary Theory, it is necessary to retain many more terms
in the calculation of the perturbations of the planets than in
that of the perturbations of the Moon; and consequently the
process is much slower in the former than in the latter calcula-
tion. For this reason R is expanded in powers of the eccen-
tricities and inclinations of the orbits of the planets instead of
the ratio of the distances of the disturbed and disturbing
bodies and the calculation then conducted as in the last
Chapter.
350. But we shall make use of an entirely different mode
of calculation. It is to Lagrange that we are indebted for
the method we are about to lay before our readers of calculating
the Planetary Perturbations.
If at any instant the disturbing forces were to cease acting,
the planet would move in an exact ellipse; and this ellipse
and the actual orbit of the planet would manifestly have a
316
DYNAMICS. PLANETARY THEORY.
common tangent, and the actual velocity of the planet and
that calculated for the motion in this ellipse according to the
elliptic theory would be the same. For this reason this ellipse
is called the ellipse of curvature to the orbit at that instant:
it is also denominated the instantaneous ellipse of the planet.*
* That this ellipse is an ellipse of curvature to the actual orbit, and that the
contact is of the first order only may also be thus shewn.
The most general equations to an ellipse in space are the equations to a plane
and to a surface of the second order. These contain twelve arbitrary constants.
Now these constants are in our case, connected by the following relations.
1. The plane of the ellipse must pass through the Sun's centre.
one relation connecting the constants.
This gives
2. The focus of the ellipse must lie in the Sun's centre. This gives three
more relations.
3. The co-ordinates of the planet at the instant under consideration must satisfy
the equations of the ellipse.
This gives two more relations.
4. The velocities of the planet in the two orbits must be the same and their
directions also at the instant under consideration: or, which comes to the same thing,
the three velocities parallel to the axes of co-ordinates must be the same.
This gives three more relations.
5. The velocity in the ellipse must be equal to that which results from the
1
2 a
theory of elliptic motion, viz: 2u
This gives one more relation.
(ਕ)
These five considerations give ten relations among the constants of the equations
to the ellipse. Hence these equations in our case involve only two arbitrary constants.
Now let xyz be the co-ordinates to the common point of the orbits: x + dx',
y + dy, z + d z the co-ordinates of a point near this in the path of the planet and
x + d x', y + ¿y', z+dz' the co-ordinates of a point in the instantaneous orbit cor-
responding to the above: also let become t +7:
dx
d²x' +
then da =
++
+
dt
dt 2
da
d² x' +²
ôx'
T-+
+
dt
d t² 2
.. ô dx'
J
(
d2 x
d² x
+
by condition (4).
d t
d to
2
d2
y
d² Y
+
dt 2
===
(
d2
d² z
d²z'
+
dt2
dt
dt²
2
Similarly ôg – ô ý = (a
ô z'
Hence the distance between the new points in the curves
1
-
•
- {(x-2)+(-2)+(-Z) 7+&c...
d² x'
x'
d³y day'
z
z
d
dt
d
In
METHOD OF CALCULATION.
317
From what precedes it is evident that the motion of the
planet may be represented by supposing it to move in an
ellipse of which the elements are continually and slowly
changing. If we know the elements of the instantaneous
ellipse at any proposed instant, we have nothing to do but to
calculate the position of the planet in this ellipse by the
ordinary formulæ in Chap. III.
351. Since the perturbations of the planetary motions.
are very small, it follows from the Principle of the super-
position of small motions, that the perturbations will be the
sum of the perturbations produced by the several disturbing
bodies considered separately: (Art. 288). We shall therefore
in the following calculations consider only one disturbing body.
PROP. To explain the process of integrating the equa-
tions of motion of the disturbed planet.
352. The equations of motion of a disturbed planet are
by Art. 323.
d²x
dt
MX
d R
+
+
0,
7.3
dx
d³y
му
d R
+
+
0,
df
2.3
dy
d²
M
d R
+
+
0,
d t²
2013
dz
where R=
{ x²² + y² + ~´~}
m' (xx' + yy' + ≈≈′)
/2
m'
(x − x')ˆ + (y − y')² + (≈ − ~') *
μ
= mass of Sun + mass of disturbed planet
m'
= mass of the disturbing planet.
=
0:
In order to make the ellipse have a contact nearer than that of the first order
[which it has by condition (4)] we must make the first term of this expression
but this we cannot do since it requires three conditions to be satisfied and we have
only two disposcable constants.
Hence the ellipse described in the text is an ellipse of contact to the real orbit.
the contact being of the first order.
318
PLANETARY THEORY.
DYNAMICS.
We shall first integrate these equations of motion omitting
the disturbing forces: by this process we shall obtain six inte-
grals of the first order, containing six arbitrary constants.
These six constants must be determined in terms of the six
elements of the planet's orbit (Art. 270.); the inclination, the
longitude of the node, the mean distance, the eccentricity, the
longitude of the perihelion, and the epoch. By eliminating
dx dy dz
from these six integrals we
dt' dt
the three quantities
dt,
have the three final integrals of the equations of motion.
We shall then proceed to the integration of the equations
of motion taking into consideration the disturbing forces.
The six integrals of the first order obtained on the sup-
position that there were no disturbing forces, contain the six
dx dy dz
arbitrary constants and also the quantities x, y, z, at' at dť
≈,
dt
Now since the actual orbit and the instantaneous orbit touch
each other in the point (xy≈), and since the velocities in these
two orbits at that point are the same (Art. 350), it follows
dx dy dz
are the same in the actual orbit
dt' dt' dt
that x, y, 9
and in the instantaneous orbit. Consequently we shall con-
sider these six integrals to be still the integrals of the equations.
of motion when the disturbing forces are not neglected; with
this difference, that now we must consider the six constants as
variable quantities. To determine their values we must dif-
ferentiate the six integrals with respect to t and eliminate
d² x d'y d² z
by the equations of motion. In this way six
dt' dt²' dť
equations will be obtained for calculating the six variable
quantities.
We shall then use the equations which connect the elements
of the instantaneous orbit with these six quantities (which are
the six arbitrary constants when the disturbing forces are neg-
lected), and in this manner obtain equations for calculating
the variations which the elements of the instantaneous orbit
undergo: so that if at any epoch these elements are known,
they may be calculated for any other epoch. Then these
MOTION OF AN UNDISTURBED PLANET.
319
elements being put in the series of Arts. 278, 280, we know
the position of the planet at any given time.
We proceed now to the investigation of the several Pro-
positions necessary for these results.
PROP. To integrate the equations of motion of an
undisturbed planet.
353. These equations are
ď²x
+
MX
d ť
2.3
ď² y
му
+
(1),
dť
21.3
ď z
M B
+
dť
2.3
where
μπ M + m = mass of Sun + mass of planet.
Multiply the first by y and the second by a and take their
difference; then
ď y
ď²x
2
Y
0;
dt
dt2
dy
dx
... 20
Y
= const. = h
dt
dt
dx
*
ર
dt
x
dz
dt
=
h
in like manner
dz
Y
dy
*
h₂
dt
dt
(2).
These are three of the first integrals and they contain the three
arbitrary constants h, h₁, h.
Again, multiply the equations of motion by 2
dx
dy
dt
dt
20
dz
respectively, add them, and integrate: then
dt
dx2 dy² doe
2 м
+
dt
+
dt dt
= const. = c..
(3).
320
PLANETARY THEORY.
DYNAMICS.
This is a fourth integral of the first order and contains the
arbitrary constant c.
Again, multiply the first and second of the equations of
motion by the second and third of (2) respectively: then by
subtraction we have
ď² x
h₁
h₂
1 dt
ď² y
dt2
MX
dx
x
2.3
dt
dz
dt
му
+
2.3
(
dz
dy
Y
Z
dt dt
dz με
- * (x² + y² +2°) dx = = (x dx + y dy + zde)
y²+x²) dt
Joi
dt dt
dt
u dz
uz dr
d()
♦ | ર
;
r dt
p² dt
d t
dx
dy
μ Z
hi
h2
+ f
dt
dt
2.
dz
dx
му
so also h₂
h
+ fi
(4).
dt
dt
go
dy
dz
ŀl X
h
h₁
+f2
dt
dt
Դ
Thus we have three more integrals of the first order, containing
the arbitrary constants f, fi, f2.
It would appear, then, that we have seven, and not only
six, integrals of the first order: but we can shew that any one
of these seven is a consequence of the other six: and the con-
stants h, ha, ha, f, fi, fe are connected by an equation.
For by the last of equations (2) and (4) we have
2
h₂ f₂ = hh₂ h₁h₂
dy
dz μx
h₂
dt
dt 2
dt
dx
h1
dt
μ
m
dx
dt
dz
h2
dt
= h ( h₂ dy - h₁ da) + h (hda - h, d) - 4 h
r
= − h ƒ − h, f₁ − = (x h₂ + yh₁ + ≈h) by equations (4)
-
fï
1
до
2
= - hf - h₁f₁ by equations (2)
MOTION OF AN UNDISTURBED PLANET.
321
...
hf + h₂f1 + h₂f2 = 0 ……….
.....
(5)
or the arbitrary constants have a necessary relation, and therefore
the seven integrals found above are not independent integrals.
And moreover, since the seven integrals contained in (2)
(3) (4) do not involve the time t explicitly it would appear that
dx dy dz
we should obtain three final in-
dt dt' dt
by eliminating
tegrals functions of xyz without t: but this evidently cannot
be the case.
It follows, then, that the seven integrals must
be equivalent to only five independent integrals: and the con-
stants h, h₁, h, c, f, fi, f½ are connected by another relation.
This relation is found
Add the squares of
as
follows.
equations (4);
··· ƒ² + fi² + ƒ²² +
(f≈ + fr¥ + ƒ2«x) + µ³
1*
= (h² + h² + h₂²) +
Jdx² dy
dv²
d:
dy
d.x) 2
dt
+
dt dť
+h,
d t
dt
+ h₂
dt
(h² + h* + h₂³) (~~ + c) by equations (3) and (2).
But by equations (4) f+fiy + ƒɔx + µ r
dy dx
dx
= h ( x d − y de) + h ( z da - 8 da) + h, (ydz - z dy
dt
dt!
dt
X
dt
= h² + h₁₂² + h²;
dt
·· ƒ² + ƒï + ƒ¸² = µ² + (h² + h¦² + h₂³) c ………….. (6),
this is the relation sought for.
dt
We are unable to obtain a sixth integral of the first order
by direct integration: and must therefore integrate the integrals
already obtained to get a relation involving the time: this will
be one of the final integrals.
To obtain the other two final integrals we must eliminate the
dx dy dz
differential coefficients
from the integrals of the
dt' dt' dt
:
first order to effect this multiply the equations (2) respectively
by ≈, y, x and add
S s
322
PLANETARY THEORY.
DYNAMICS.
h z + h₁y + h₂ x = 0 ..
(7),
this proves that the undisturbed planet moves in a plane.
Again, multiply equations (4) by ≈, y, a respectively and
add: then
ƒ≈ +ƒ½∞
+ ƒïY + ƒ₂x + µ √ x² + y²+
00
fz
hx
(~
dy
dx
dx
Y
+ h₁z
d t
=
d t
2
d t
h² + h‚² + h₂² by (2) ………..
Xx
dz
x²
ar) + he
dz
dy
+ h₂ y
at
d t
(8).
dt
This is the equation to a surface of revolution of the second
order, the origin of co-ordinates being in the focus: the
equation to the plane generated by the directrix of the gene-
rating conic section being
2
ƒ≈' + ƒ₁Y' + ƒ2x′ = h² + h²² + h₂².
For the perpendicular from any point (xyz) of the surface
on this plane
2
ƒ≈ + ƒ₁Y + ƒ₂x — h² — h₂² — h₂²²
·
√ƒ² + fi² + ƒ²²
2
μη
√ ƒ² + ƒn² +ƒ½º
Now is the same for all points equally distant from the
origin. Hence the surface must be one of revolution about an
axis perpendicular to the plane of which the equation is
2
ƒ≈' + ƒ₁ý +ƒ½æ′ = h² + h₂² + h₂².
グ
​Also the ratio of the perpendicular to the distance › is constant:
and this is a property peculiar to the focus of conic sections.
Hence the surface is a surface of the second order from the
focus and by combining this with the equation (7) we learn,
that the planet moves in a conic section the Sun being in the
focus: (Art. 252).
To obtain the third integral add the squares of equations
(2): then
(x² + y²+≈²)
2
[dx²
d x²
dy²
dx2)
+
+
dt2
{
d.r
dy
d
?
+ Y
+ ≈
= h² + h₂² + h₂²,
2
2
dt
dt
dt
dt dt²
and r² = x² + y² + ≈²;
THE ELEMENTS IN TERMS OF THE ARBITRARY QUANTITIES. 323
dx²
dt2
dy2
شیج d
dr²
h² + h₂² + h₂²
2
2
+
+
(9).
d t
d t²
at
за?
Let be the longitude of the planet at the time t measured
on the plane of its orbit: then
dx² dy² dx2
+
+
dt dť dt2
dt
ᏧᎾ
r
d 02
+
dr²
dt2
df;
2
h² + h₂² + h₂²
In this we must substitute for in terms of by means of the
two other integrals: and in integrating we shall introduce the
sixth independent arbitrary constant: this constant is called the
epoch, since it depends upon the epoch of the planet's perihelion
passage.
Having integrated the equations of motion for an undis-
turbed planet we proceed to the following Proposition.
PROP. To calculate the elements of the orbit in terms
of the arbitrary constants introduced by the integration.
354. Let i be the inclination of the plane of the orbit to
the plane of the ecliptic; the ecliptic we shall take to be the
plane of xy,
the longitude of the node, the axis of a being drawn
through the first point of aries,
w the longitude of the perihelion projected on the ecliptic,
2a the axis-major of the orbit,
e the eccentricity,
€ the epoch.
The equation to the plane of the orbit is zh+y h₂+x h₂ = 0;
h
2
.. cos i
and tan i
=
>
√ h² + h₂² + h₂
2
h₁² + h₂²
h²
2
324
PLANETARY THEORY.
DYNAMICS.
By putting ≈ = 0 in the equation to the plane of the orbit,
we have h₁y+hx = 0, the equation to the line of nodes:
ho
... tan
h₁
dx
dy
dz
At the perihelion r is a minimum; ..æ
+ Y
+ z
0,
dt
dt
dt
Y
also tan =
at that point:
x
му
dz
dx
h₂
h
dt
dt
dx²
dx2 dy
Y
+
d ť²
dt2
dx2
dy2
da²
y
+
+
d t²
dt²
d t²
we must therefore find the value of this ratio at the perihelion :
for this end we have
=
-fi by (4) of last Article
ર
dz
dt
d x
+ x -fi by (2)
dt
-f₁ at the perihelion ;
dt
MX
dz2
dy²
d∞2
so also
= x
+
+
γ
d t²
dt
dť
-f2 at the perihelion ;
y
fr
...tan at perihelion
At the extremities of the axis-major
equations (3) and (9) of last Article give
३
.. 1' =
C
2
2
h² + h₂² + h₂²
H
22
2
グ
​2
+ C;
√ µ² + (h² + h₂² + h₂²) c
C
@
=
a
f2
dr
0, and therefore
dt
М
.. a
C
2
and e =
2
u² + (h² + h₂² + h₂²) c _ √ ƒ² + ƒ²² + ƒ²²°
м
by equation (6) of last Article.
M
MOTION OF A DISTURBED PLANET,
325
dt
d Ꮎ
Lastly to find the epoch (e) we must integrate the equation
p2
2
√ h² + hi² + h₂²
2
after having substituted for r.
Having thus obtained the elements of the undisturbed
orbit in terms of the constants we will proceed to shew the
importance of these expressions in determining the perturba-
tions of a disturbed planet.
PROP.
To integrate the equations of motion of a dis-
turbed planet.
355. The equations of motion are
ď² x
MX
d R
+
+
0,
dt2
203
dx
d³y му
d R
+
+
0,
d f
2.3
dy
d2z
με
dR
+
+
0.
df
23
do
In conformity with the method of the variation of para-
meters invented by Lagrange, and explained in Art. 352, we
shall assume that the following integrals (taken from Art. 353.)
satisfy these equations, h h h₂ cƒƒiƒ½ being variables,
dy
dx
h
= x
Y
dt
dt
dx
dz
h₁
=
2
dt
dt
d t
dz
dy
h₂ = y
dt
dt
c+
Ωμ
グ
​dx
+
dt² dt
dy²
d2
+
dt,
f+
μπ
2*
dx
dy
h₁
h₂
dt
dt
326
PLANETARY THEORY.
DYNAMICS.
му
dz
dx
fi +
h₂
h
1°
dt
dt
M 20
dy
dz
₤2 +
h
h₁
2'
dt
dt
and we now proceed to shew how to determine the values of
the variables h h h cffif in order that this may be the case.
Differentiate all these equations with respect to t and
dx d'y d³z
eliminate
dt²' dť²' dť²
by means of the equations of motion:
d R
dy
>
dh
d R
we have
Y
W
d t
dx
dhr
d R
d R
= x
dt
dz
da
dhe
d R
d R
-y
dt
dy
dz
dc
2
Jd R dx
dR dy
d R dz
d (R)
+
+
2
d t
dx dt
dy dt
dz dt
dt
the brackets surrounding R implying that the total differential
coefficient with respect to t is to be taken, but this only in so
far as R is a function of xyz*.
df
dh, da
dhy dy
d R
d R
hr
+ h₂
d t
d t d t
dt dt
dx
dy
df
dh₂ dz
dh dx
d R
dR
-
...
+ h
dt
dt dt
dt dt
dz
dx
df2
dh dy
dh dz
dR
d R
- h
+ hr
dt
dt dt dt dt
dy
do
*R is also a function of t in consequence of being a function of 'y's', but
the bracket is meant to imply that R is to be differentiated only in so far as it is
a function of xyz.
MOTION OF A DISTURBED PLANET.
327
356. The inclinations of the planes of the planetary orbits
to the ecliptic are very small; the asteroids (of which the
masses, however, are very small) being excepted. This is the
case also with the eccentricities. We shall consequently neg-
lect powers of these quantities higher than the square.
that
dx dy
dy dz
By referring to the value of R (Art. 352.) it will be seen
dR dR d R
all vary as m' the mass of the disturbing
planet, which in our system is always extremely small in com-
parison with that of the Sun: we shall therefore neglect these
quantities when they have small multipliers, and also their
squares.
The difference between all angles and distances measured
on the plane of the orbit and their projections on the ecliptic
varies as the versine of the orbit's inclination, and therefore as
the square of the angle of inclination nearly. This shews that
in calculating the perturbations of the mean distance, the ec-
centricity, the longitude of the perihelion, and the epoch, we
may neglect i and therefore h₁+h, and consequently h₁ and he,
and also f, Art. 353, equation (5): hence the equations of last
Article become
2
dh
d R
d R
Y
v
dt
dx
dy
dc
d (R)
2
d.t
dt
dx
dfi
dt
dt
{
X
d R
ข
d R
d R
+ h
dy
da
dæ
df2
dy
{
dR
d R
d R
h
У
dy
dx
dy
357.
dt dt
When we have expanded the function R then we
must calculate the terms of these equations which involve the
partial differential coefficients of R. After this we shall obtain
the variations of the elements of the instantaneous orbit of the
planet in terms of these variations of the arbitrary quantities
hhh₂ c f f f 2. Then by integrating these we shall know the
328
PLANETARY THEORY.
DYNAMICS.
elements of the instantaneous orbit. Let a,e,a,e,i,, be these
elements at the time t; the subscript accents being used to
denote that the elements are variable. Then by substituting
these in
2
2*
e
1 +
a
e, cos (n¸t + €, − @)
૭
2
∞) - ..
e²
cos 2 (n¸t + e − ∞₁) -
and 0 = n¸t + €, +
(20,
(20, -214) sin (2, t + e,
(n¸t+ − ∞₁)
5e2
+
sin 2 (n,t + €, — w₁) +
4
we know the position of the planet in its orbit; the position of
the orbit being known by i, and Q.
At present, however, we shall proceed to the transforma-
tion of R to polar co-ordinates.
PROP. To determine
d R d R
dx' dy
in terms of
dR dR
de' dr
358. In calculating these disturbing forces we may sup-
poser and the same as their projections on the plane of xy:
for otherwise we should be retaining quantities varying as the
product of the square of the inclination and disturbing force;
Y
X
;
.. x = r cos 0,
Y
1'
= 7 sin 0, r² = x² + y², tan 0
dR
dR dr
dR do dR
d R sin 0
+
cos A
dx
dr dx
de dx
dr
d Ꮎ
2
d R d R dr
d R do
d R
dR cos 0
+
sin 0 +
dy
dr dy
do dy
dr
ᏧᎾ
グ
​359. But since R is to be expanded in terms of t and the
elements we must still further transform these partial differ-
ential coefficients.
€,
Upon examining the expansions of r and we see that
and ☎, are remarkably connected with nt: r is a function
of nt +, -,, and 0 equals n,t+e,+ a function of nt
W and €, and occur in no other way in 7 and 8
+ €,
در
DIF. COEFS. OF R WITH RESPECT TO THE ELEMENTS. 329
and consequently in R. Hence by an analytical artifice we may
consider R as a function of e, and, in consequence of its
being a function of and 0, and may change the variables
from and to e and this will be better understood by
reading the next Proposition.
/
PROP. To obtain
W:
d (R) dR dR
dt de' dr
in terms of the par-
و
tial differential coefficients of R with respect to the elements.
d (R)
dt
360. In
R is supposed to be differentiated only
inasmuch as it depends on the co-ordinates of the disturbed
planet; viz. r and 0. Now by examining the expansions of
r and ✪ we see that wherever t occurs e, is connected with it
in the expression n,t +e,, and e,
€, occurs in no other place in
7 and 8: hence
d R
d (n¸t + €)
d (R)
d t
d R
= n
n
de
d R
d R
Again, to obtain
and
we observe, as before, that R
do
dr
is a function of
and
solely because it is a function of
→ and 0;
dR
dR do
dR dr
+
de
Ꮎ
do de
dr de
d R
d R de
dR dr
+
do dw dr dw
dw,
Now by referring to the expansions of r and we have
dr
dr
d Ꮎ
d Ꮎ
05
+
= 1 and
+
de
do
de,
dw.
in consequence of these the above equations give by addition
d R
d R
d R
+
d Ꮎ
de,
dw
Tr
330
PLANETARY THEORY.
DYNAMICS.
dR
361. Again, to obtain
we observe that r is a function
dr
1
in the equation
jo
of e, solely because it is a function of 0; for e, does not occur
1 + e, cos (0 – ☎)
a, (1 − e²)
dr
dr
de
r²e, sin (0 – ☎,) də
de,
ᏧᎾ de
a, (1 - e²)
de,
ᏧᎾ
d Ꮎ
1 de
√a, μ (1 − e²)
(Art 273.)
and
de
d (nt + €) n dt
п. 302
Substitute these in the formula
2
1
e
Jo2
2
d R dR de
d R dr
+
de
de de
dr de
dr
transposing and dividing by
de
2
d R
1
e
(dR
2
dr
a e sin (0
w) [de,
we have
a√1-e dR
2
2.2
d Ꮎ
des
We shall find the following Proposition of use hereafter
in reducing our formulæ.
d R
d R
d R
PROP. To obtain
in terms of
and
de
de,
dw
362. Since R is a function of e, solely because it is a
function of r and 0;
d R dR de dR dr
+
de
do de, dr de
ᏧᎾ
dr
We must therefore calculate
and
de,
de
3
Now 0 = n,t + €, + (2 e, − 4 e,³) sin (n,t + e − @) + ...
and r = a, { 1 + 1/2 e 2
-e, cos (n,t + e, w₁) ......},
DIF. COEFS. OF R WITH RESPECT TO THE ELEMENTS.
331
•
dr
ᏧᎾ
and from these we should obtain
and : but since we
de, de
do not know the law of these series we wust refer to the
functions from which they were developed, viz: (Arts. 273,
279.)
dt
(1 ·e²) }
N
de
n, {1+e, cos (0 - ∞) } ²
2
α
and r =
a (1 − e,²)
1 + e¸ cos (0 − ☎)
O is calculated by the first, and substituted in the value of r,
and then r is expanded.
By integrating the first we shall have t = † (0, e);
dt
dt
.. dt
=
de +
de,;
do
de
do
transposing and multiplying by
we have
dt
d Ꮎ
dt de
ᏧᎾ
dt-
de,;
dt
de dt
d Ꮎ
dt de
de
de, dt
1
-
(1 − e²)³ do
* Ꮎ
Now t
–
N 1 + e, cos (0 – ☎,)
2
1 − e ² { 3e, + (2 + e²) cos (0 − ☎,)}d0
-
@,)};
3
d t
1
de
ՊՆ,
{ 1 + e¸cos (0 – )
छ
2
1.
e
n
1 - e, cos (0 - )}²
+
2
1 + e, cos (0 – wo
sin (0 – ☎,)
sin (0 - ∞)
*This integral is obtained in the following manner.
Assume
fåe
3e + (2 + e²) cos x
dx
(1 + e cos x)³
=
this is evidently the form of the integral.
A sin a'
(1 + e cos x)² + 1 + e cos x
B sin a
Cdx
+
+ e cos x
Differentiating
?
332
PLANETARY THEORY.
DYNAMICS.
d Ꮎ
Also
n
2
dt
(1
e
ᏧᎾ
sin (0 – ☎
@o)
de
{1 + e, cos (0 - w)}²;
2
1 - e²
{2 + e, cos (0 - w)}.
Also, since the series for r is obtained by developing the
function
a, (1 − e,²)
-
1 + e cos (0 – w
dr
after substituting for 0,
dr
(ap)
dr do
+
de
de do de
a, { − 2 e, − (1 +e,²) cos (0—w‚)} a,e,sin² (0—w‚){2+e,cos(0—w‚)}
+
{1+e, cos (0-w) } ²
2
{1+e,cos (0-)}?
2
a cos (0 - w);
d R dR sin (0–₪) {2+e,cos (0 −@o ;)}
de,
d Ꮎ
2
1 - e
d R
a
cos (0–☎).
dr
We now proceed to obtain the formulæ for calculating
the variations of the elements.
PROP. To calculate the variations of the mean distance,
the eccentricity, and the longitude of the perihelion of the
instantaneous orbit of the disturbed planet.
Then
363. Let a, e,, w, be these elements; the subscript
accents indicating that the elements are functions of t.
by Art. 354
Differentiating and multiplying by (1+e cos x)³
(cosx+6
3e + (2 + e²) cos x = A { cos x (1+ecos x) + 2e sin² x } + B (1 + e cos x) (cos x + e)
+ C (1 + e cos x)².
Let x=0; 3e + (2 + e² ) = A (l + e) + ( B + C) (1 + e )²
or 2+ e = A + (B + C') (1 + e).........(1)
π
x =
2;
3e=2e1+c B+ C
......
40
(2)
x = π ; 3e − 2 – e² — — ↓ ( 1 − e ) − ( B − C) (1 − e)²
−
or 2 − c = A + ( B -- C')
(1 — €
( 1 − e)………………………………….. (3),
From these 4=1, B=1, C=0.
Hence the integral in the text.
VARIATIONS OF THE ELEMENTS.
333
:
1
C
μ
1.
The mean distance a
a
С
a
α,
М
da
a² de
dc
2a2 d (R)
by Art. 356.
d t
μ
dt
Ад
dt
2n, a² dR
by Art. 360.
ле
de,
2.
1
e₁ =
The eccentricity
ƒ² + ƒï² + ƒ½
1
2
2
2
√ fi² + f (Art. 356);
M
de
1
df
+ ₤2
dt
fi² + ƒ²²
dt
dt
2
d fl
1
df,
d f
sin w
+ COS w
sw
·.· tan w₁ =
· dt
'dt
fi
-
fo
But by Arts. 356, 358,
dfi
Sd r
de dR
cos - sin ✪
hfdR sin
0
dR
COS
dt dt
dt de
d Ꮎ グ
​dr
{bu
but in small terms h
= √ a, µ (1 − e,²)
(1 − e) { (
= √ a, µ (1 − e²), and
μ
-
√a,µ
dor
dt
cos dr
2 sin dR
d R
+ cos 0
22 de
"'
d Ꮎ
dr
d f2
Jd r
dė】 dR
sino+r cose
- h
[dR
Jd R cos 0
d R
+
sin ✪
dt
dt
dt de
d Ꮎ
dr
sin e dr
2 cos 0
d R
dR
- √α,µ (1
а, м
(1 − e
+
+ sin
1 do
Ꮎ
7°
do
dr
de,
dt
a, (1 − e)
√a,(1
u
-
sin (9 – w,) dr
xf- (sir
+
do
2 cos (0
ľ
–
- @
)) dR
do
- sin (0 - )
d R
dr
334
PLANETARY THEORY.
DYNAMICS.
in small terms, and using
{putting
1 1 + e, cos (0 – ∞o,)
2
a (1 − e,²)
the properties proved in Arts. 360, 361.}
X
X
√a, (1
a (1 − e
e,²)
ль
e,² sin³ (0 – ☎,)—e,² cos² (0 − ∞₁) + 1 dR
1- e
2 dR
е
a e, (1 − e,²)
-
ᏧᎾ
a e
de
2
e dR
а е
2
u
ae²μ \de, dw
3. For the longitude of the perihelion tan w
dR
1
2
€² dR
+
ма
e, √μa de
.
= 1,
f2
d w
dt
1
ƒ²² + ƒ²²
df
£2
-fi
2
dt
d fal
dt
2
1
dfi
d fa
2
COS D
ме,
• dt
a (1 − e²).
√a
мег
2
dt
sin w
ದ
cos (0 – ☎)
@) dr
200
2 sin (0 – w
d R
d Ꮎ
2
d Ꮎ
√a, (1
2
με
a, (1 − e,²)
-
sin (0,) {2 +e, cos (0 - w)} dR
사
​+ cos (0 − ☎,)
cos (0 – w
+
a (1 − e,²)
ᏧᎾ
1
e² dR
a µe² de,
by Art. 362.
2
d R
dr
dR
;)
dr
PROP. To find the variations of the inclination and
the longitude of the node.
VARIATIONS OF THE ELEMENTS.
335
364. We have by Art. 354 the formulæ
2
√ h₂² + h₂²
2
h₂
tan i
tan 2, =
h
h₁
h2
h₁
…. tani, sin Q,
tan i, cos 2,
همه احمد
h
Hence
d (tan i, sin )
dt
1
dh
dh₂
h₂
h
h² dt
dt
dz
≈ and
Substituting by the equations of Art. 355, and neglecting
as being small,
dt
d tan i
ΦΩ
y d R
sin
+cos, tani,
dt
idt
h dz
In a similar manner by differentiating
h₂
tan i, cos
=
we have
h'
d tan i
ΦΩ
x d R
cos -
sin Q, tan ¿,
dt
dt
h dz
Multiplying these equations by sin 2, and cos, respect-
ively and adding
d tan i
dt
1
(y sin N, + x cos N)
h
d R
dz
Multiplying by cos N,, sin N, and subtracting
dQ,
d R
1
tan i
· dt
h
(y cos N, − x sin Q,)
dz
d R d R dx
dR dy
d R dz
d R dz
Now
+
+
nearly;
di
dx di
dy di
dz di
dz di
dR
d R dz
and similarly
nearly.
ΦΩ
dz do,
336
PLANETARY THEORY.
DYNAMICS.
since for a given alteration in the inclination or longitude
of the node the alteration in ≈ is greater than in a or y.
Also ≈ =
F
h₂
h
h
m
00 y by Art. 353, equation (7).
h
= tan i, (~ sin N¸ — y cos N,),
dz
di
sec² i, (∞ sin N, - y cos N,) ;
dz
tan i, (x cos N, + y sin N,).
ΦΩ
d tan i
Hence
dt
1 d R
htani, do,
;
ΦΩ
1 d R
,
neglecting tan³ i, ...
dt
h tani, di,
Since
the squares of i,
u μ√ T-e
2
h
na
di
d R
dt
u
С
dQ,'
d R
are neglected and since
these may be written
ΦΩ
dt
na
2
μ sini, √1-e ds),
na
usini,√1-e² di,
which agree with those given by M. Pontècoulant Théorie
Analytique du Système du Monde, Tom. 1. p. 330.
PROP. To find the variation of the epoch.
365. Now R is a function of a, e, w, n,t + e, i, ,, and
since the instantaneous ellipse is an ellipse of curvature to the
orbit described of the first order (Art. 350), it follows that the
first differential coefficients of and with respect to t will be
the same in the real orbit and in the instantaneous ellipse. The
same will be the case with the first differential coefficient of any
function of and 0, as R.
VARIATIONS OF THE
337
ELEMENTS.
d R
d (R)
Now in the ellipse
クレ
​dt
de
; and in the real orbit,
since R is a function of the variable quantities a,, e,, w,, nt
+ e, i,, and Q;
d (R)
dR da
dR de
dR dw,
+
+
dt
da, dt
de, dt
dw, dt
d R
dn,
de
dR di
d R dQ,
+
n, + t
-+
+
+
de,
d t
dt
di, dt.
ds, dt
Equating these values of
d (R)
da,
and substituting for
dt
d t
de, do
dn
3n, da
di
d, the values found
dt.
dt
d t
2a, dt
dt'
dt
de
d R
in Arts. 363, 364, and transposing
and dividing by
We
dt
de,
have
de
3 na, dR
2 na² dR
t +
d R
n a
{√1-e;-(1-e;)}
d t
Ц
de,
μ
da
ме
de,
366. We shall now bring together the variations of the
elliptic elements obtained in the last three Articles, and present
them under one point of view.
2an dR
de,
da
(1)
dt
dR
na¸ (1−e²) dR
de,
+
(2)
dt
μ, e
ла
de,
dw,
ue,
de.
na¸√1—e; ¡dR
(3)
dw,
dt
de,
(4)
dt
na√1-e² dR
3 nža, d R
de
na” dR
t +
μ
de
da,
dR
na
{√ T− e² − (1 − e)}
de
με
338
PLANETARY THEORY.
DYNAMICS.
d tan i
(5)
dt
dQ,
na
d R
u tani,√1-e do,
na
п, а
d R
2
1
e
di,
(6) dt μ tani, Vi
Before we can make use of these formulæ we must explain
how R is to be developed.
PROP. To explain the manner in which R is to be
developed.
R =
367. If we recur to Art. 352 we see that
m' (xx' + yy + xx′)
12
(x²² + y²² + x²²) ³·
m'
√ (x' − x)² + (y' − y)² + (x′ − x)²
2
Let r and be the radius vector and longitude of m
measured on the plane of xy as far as the node and then on
the plane of m's orbit.
r, and 0, the rad. vect. and longitude projected on plane xy.
2, and i, the longitude of the node and inclination of m's
orbit to the plane xy.
λ the latitude of m.
Then we have
x = r, cose,, y=r, sin 0,, xr, tanλ = r,
tanλ = r, sin (0, – N,) tan i,,
Similar expressions are true for m'.
Hence R
m'{r¸r'cos (0,–0,')+xx'}
m'
(~/² + ~¹²²) +
(~12
12
√r²+r)²-2rr'cos(0,−0')+(x−x')²'
Now r¸ = r cos λ = r { 1 − 1 tan³ \} very nearly
= r {1 - tan² i, sin² (0, − N,)}
= r {1 − 4 tan² ¿, +tan² i, cos 2 (0, - §,)}.
METHOD OF EXPANSION OF R.
339
·
Also tan (0,2) = cos i, tan (0 – ) ;
i
0, − 2, = 0 -2, - tan² - sin 2 (0 – N), (see Art. 274).
2
Substitute in these the values of r and 0 given by Art.
357, and we have
2
r₁ = a, {1+e-e, cos (n, t + e, − ) - e² cos 2 (nt + e − @₁)
↓ w,)
- tan² i, +tan² i, cos 2 (n,t + €, − 2) + ... }
4
a, {1 + u}
}
and 0¸ = n¸t + €, +2e, sin (n,t +e, − ) + ½ e² sin 2 (n¸t + e, − w¸)
ན
=
· tan² i, sin 2 (n,t + €, − Q,) +
= n t + €, + v suppose.
-
Now let R' be the value of R when a, and a, are put for
r r
, and and supposer, = a, (1+u) and r = a(1 + u')
u and u' are small quantities because the orbits of the planets
are nearly circular: then by Taylor's Theorem
dR'
R = R'+
a, u +
da
d R'
a'u' + ..
daa'u'
also R
=
+
m' { a¸a,' cos ( 0, − 0,') + a, a,' tan i, tan i, sin (0, — Q,) sin (0,' — Q,') }
{ a,'°+a, tani,' sin² (0,-,') } *
m'
√a¸²+aƒª—2a¸à¦' cos (0,−0,' )+ { a, tan i̟¸ sin (0,–Q,)—a,' tan i' sin (0,'-Q;')
m'a, cos (0,– 0,')
+
α
-
12
m'
√ a²+ a²² — 2a,a', cos (0,– 0,′)
α
m'a, tan i, tan i, sin (0,– Q,) sin (0, – Q;)
10
α
3 m'a, tan² i sin² (0,'— Q') cos (0,– 0,')
12
2a
m' {a, tan i, sin (0,– §,) – a, tan i sin (0,−9,')}
2 { a² + a^ — 2a, a' cos (0,– 0,')}
+....
>
340
DYNAMICS. PLANETARY THEORY.
Let
1
√ a² + a* - 2a, a' cos (0,– 0,)
C。 + C₁ cos (0¸– 0') + C₂ cos 2 (0,−0') +…….....
0
12
1
{ a² + a
a* - 2a,a, cos (0,-0,')}*
= ½ D¸ + D₁ cos (0, − 0') + D₂ cos 2 (0, − 0,') +
0
2
These coefficients should be calculated and then R' may be
arranged in a series. When we have thus calculated R' we
dR' dR'
must find
da,
da
dR'
R' +
da
and substitute them in
dR'
a, ul + a'u' +
d a
and we shall have R expressed in a series of terms depending
on the time and the elements of the instantaneous orbits.
It is not our object to enter into the numerical calcula-
tion of the coefficients of the expansion of R: for this we refer
the reader to M. Pontécoulant's Theorie Analytique du
Système du Monde, Tom. I. p. 340, Mécanique Céleste,
Tom. III. and Mr. Lubbock's Papers in the Philos. Trans.
and Astron. Trans.
We proceed to demonstrate some Propositions relative to
the general nature of the terms.
PROP. To prove that the terms of R which depend on
the mean anomalies (n,t and n't) of the planets are of the
form P cos {(pn,−qn')t +Q} or P cos {(pn,+qn') t + Q},
where P is a function of the mean distances, eccentricities,
and inclinations of the orbits, and Q is a function of the
longitudes of the perihelia and nodes and of the epochs; and
p and q are positive integers.
368. We shall make use of the following elementary tri-
gonometrical formulæ :
THE FORM OF THE TERMS OF THE EXPANSION OF R. 341
I.
Cos a cos b =
1
cos (a - b) +
1
cos (a + b).
II.
Sin a sin b =
cos (a - b) -
½
c
1 cos (a + b).
III.
Sin a cos b =
sin (a + b) +
sin (a − b)
Now 0,0 = (n¸t + €,) − (n't + €)
+(2e, .....) sin (n, t + €, − w ) + ( — e² +...) sin 2 (n,t +e, −@,)+...
−(2e'+…..) sin (n't + e,' −w') + (½ e,'"*+...) sin 2 (n't+e' -@,') + ...
– tan² 1 i̟, sin 2 (n, t + €, − Q, ) +
-
e'
+ tan" i sin 2 (n't + e − 2) + ......
k
-
(n,t + e,) − (n't + €) + T suppose;
..cos l (0, 0) = cos {k (n,t + e,) − k (n't + e')} cos kT
− sin {k (n,t + €) − k (n' t + e'') } sin kT,
and cos k T = 1 − ½ k²T² + .
1
sin k T = kT
-
k3T3
+
1.2.3
Now by formulæ II. and I. the even powers of T, and .. cos kT,
will involve only simple cosines; and by formulæ II. and III.
the odd powers of T, and therefore sin kT, only simple sines.
Hence by formulæ I. and II. the expansion of cos k (0,−0,')
given by the above formula will contain only simple cosines.
In the same manner we might shew that sin (0,-) and
sin (0-2) will equal a series of simple sines (with no con-
stant term), and therefore by formula II. the squares or product
of these will contain only simple cosines.
We see then that when the complete development of R'
given in Art. 367. is worked out and arranged in a series, it will
consist only of simple cosines.
Again by Art. 367. we see that u and u' consist of a series
of terms involving the simple cosines of angles. Hence, by
formula I. each of the quantities u, u', u', uu', u'.... will
consist of a scries of simple cosines of angles formed by coin-
342
PLANETARY THEORY.
DYNAMICS.
bining the arguments* of terms of u and u' in endless variety
by addition and subtraction.
It follows, then, finally, that the series into which R is to
be developed (Art. 367.) will by formula I. consist only of terms
of the form
P cos {(pn, - qn') t + Q} and P cos {(pn, + qn') t + Q}
p and q being positive integers, P a function of the mean dist-
ances, eccentricities, and inclinations of the instantaneous orbits;
and Q a function of the longitudes of the perihelia and nodes
and the epochs.
369. We have already frequently remarked, that the
eccentricities and inclinations of the planets are so small, that
their higher powers will be of almost imperceptible magnitude.
It becomes important then to search for some means of deter-
mining the relative magnitude of P in reference to the argu-
ment (pn, — qn') t + Q for this will materially shorten the cal-
culation of R, by pointing out at once those terms of the
infinite series into which R is developed, which are of sufficient
importance to be retained.
In the following Article we shall prove a principle which
answers our purpose.
PROP. The lowest dimension of the quantities e,,
tan i,, tan i' in the coefficient of P cos {(pn, -qn,)t +Q} is
of the order p~q.
370. We have by Art. 367.
d R'
dR'
R = R' +
au +
da
da
a'u, +
(1)
A remarkable law prevails in the expansions of u
and u'. It is this (Art. 367). The number which multiplies
nt + € in the argument of any term in these expansions re-
presents the dimensions in e,, e,, tan i,, tan i, of the principal
part of the coefficient of that term.
In an expression a cos (pal +7), the angle pnt + q is called the argument
of the term a cos (pní + q).
THE FORM OF THE TERMS OF THE EXPANSION OF R. 343
2
Now the same holds good in any power of u or u'. Thus
in u² a term P cos (pnt + P') can arise only in the following
ways, partly from the multiplication of any two terms in u of
which the arguments are In,t+L and mn¸t+ M, where l+m=p;
and partly from such as have the arguments l'nt + L and
m'n,t+M, where l'm' p. In the former case the dimen-
sion of the principal part of the coefficient will be_1+ m =
in the latter it will be l'+m' and this is greater than p.
Hence the principal part of the coefficient of a term P cos
(pn,t + P') in u² will be of the dimension p.
=
The same is evidently true of u², u³, u'³,
P,
(2) In the product of any powers of u and u' as uªu'³,
the dimension is the sum of the multipliers of nt and n't.
For let us consider a term N cos {(In ± l'n') t + M }
Now this must evidently have arisen from the multiplication
of cos (Int+L) and cos ('n't+L') in ua and uß respec-
tively. The coefficient of this is of the order + l'.
l
(3) Let us next consider the law of the coefficients in
cos k (0, – 0,').
If we turn to Art. 368. and examine the expansions of
cos k T and sin k T, we shall find that the laws (1) and (2)
hold equally in them. But in cos k (0, – 0,′), since it is equal
to
cos {k (n,t + e,) − k (n' t + e'') } cos k T
− sin {k (n¸t + e,) − k (n't + e')} sin k T,
the dimension of the coefficient of any term calculated by
the laws (1), (2) will be higher or lower by 2k than it ought
to be according as the argument is formed by addition or
subtraction.
If, then, we turn to Art. 367. and examine the expression
given for R we see that the laws (1), (2) just proved hold for
R, if we leave out of consideration all the multipliers which
are of the form cos k (0,0). Bearing this in mind we shall
be able to prove our Proposition.
Any term Pcos {(pn,−qn')t+Q} in R has partly arisen
from the multiplication of cos {(kn, − kn) t + Q'} with
cos {[(p − k) n, − (q − k) n'] t + Q"} and partly from mul-
tiplication with cos {[(p + k) n, − (q + k) n']† + Q″"}, and
344
DYNAMICS. PLANETARY THEORY.
in no other way can it have been formed: k being any number
of the series 0, 1, 2, 3,......
First suppose k intermediate to p and q. Then the
first of these cosines becomes
cos {[(p − k) n, + (k − q) n'] t + Q"},
and the dimension of the principal coefficient of this and
therefore of cos {(pn, − qn') t + Q} is by law (2) equal to
(p ~ k) + (k ~ q) = p~q, since k is intermediate to p and q.
Second: suppose k is not greater than the smaller of p and
Then the dimension of the principal coefficient of
1.
་
cos {[(p-k) n,- (q − k) n'] t + Q"} is p+q-2k:
and therefore the dimension of the principal coefficient of
cos {(pn' - qn')t +Q} is the least value of which p+q-2k
is susceptible, and that is p~q.
Third: suppose k is not less than the greater of p and q.
Then the dimension of the principal coefficient of
cos {[(k - p) n,- (k − q) n']t - Q"} is 2k -p -q,
and therefore the dimension of the principal coefficient of
cos {(pn, −qn')t + Q} is the least value of 2k-p-q, and this
is pq, as before.
Hence the Proposition is true.
PROP. To prove that the principal coefficient of the
term Pcos {(pn, + qn')t + Q} in R is of the dimension
p+q in e,, e', tani,, tani.
371. This term arises from the multiplication of such
terms as P'cos {(kn, − kn') t + Q'} with
P'cos {[(p −k) n¸ + (q + k) n' ]t + Q"}
and P" cos {[(p + k) n, + (q − k) n'] t + Q"}
Hence, in both cases, the dimension will be p+q, since
(p − k) + (q + k) and (p + k) + (q − k) each equals p+q:
see law (2) of Art. 370. We have here supposed k is not greater
than p and q but if k be greater than p or q it will be easily
seen that the dimension will be greater than p + J..
Proposition is true.
Hence the
THE FORM OF THE TERMS OF THE EXPANSION OF R.
345
PROP. To determine the part of R which is independent
of the periodic terms.
372. We have
dR'
R = R' +
da,
au +
dR'
daja, u
ď² R' a² u²
d² R'
ď R' a2u²
+
+
a,a'uu' +
da²
10
da da
da
12
10
+
We shall neglect small quantities of the third order; hence
we need calculate the first differential coefficients of R' only
to the first order: and in the second differential coefficients
we may neglect all small quantities.
Let us turn to the expression for R' and that for u in
Art. 367, and it will be seen (after reduction) that the constant
part of R' is
m'a,e̟é̟
+
cos (w, - w')
{from the first term of R'}
12
αρ
е
- ½ m'C₁ - m'e̱e C, cos (ww)
1
{from the second term of R'}
{from the fifth term of R'}
12
+ } m' (a² tan³i, + a," tan³ ¿') D。
-m'a a' tani, tan i' D, cos (2,-){from the fifth term of R'
α
The constant part of
dR'
a u is
da
{from the first term of R'}
(e,² - 4 tan² ¿) {from the second term of R'}
m'ae,e!
+
cos (w, - w')
12
2a
d C。
Co
m'a
da
4
dC₁ m'a,e,e
da
10
cos (w, -w) {from the second term of R'}.
X x
346
PLANETARY THEORY.
DYNAMICS.
d R'
The constant part of
a'u' is
da,
cos (@
@')
{from the first term of R'}
(e'²
m'a,e,e,
12
a,
d C。 m'a,
da 4
d C₁ m'a'ee
da
2
-
tani) {from the second term of R'}
cos (-) {from the second term of R'}.
The constant part of
d² C₁ m'a e
Co
d² R' a² u²
is
da²
2 2
(from the second term of R').
da² 8
d² R'
The constant part of
a,a,uu' is
da,da
m'a
2a12
cos (ww) {from the first term of R'}
d² C₁ m'a,ae, e
C.
da,da
4
cos(w-w){from the second term of R'}.
The constant part of
d² R' a
12 2
U
is
da/2
α
2
d² C。 m'a'²e
12
(from the second term of R').
da 72
8
The part of R which is independent of periodic terms
equals the sum of these parts.
We shall call this sum F;
+
m'
.. F
Co
2
m'
12
8
m (a"D, + a, dc,) tan'i, + m² (a,"¹D, + a dc;)
m'
8
m'
4
Co
da
Co
da
a,atan i tan i D, cos (N, – N')
12 Do
/
tan² i
0
THE FORM OF THE TERMS OF THE EXPANSION OF R. 347
m'
4
m²
d Co
ď² Co
2
2
a
+ 1/2 a
e
da
da2
m'
4
d Co
d Co
12
12
a
+글​,
e
da
da2
d C.
Ci
+2α
(da) e, e
e e' cos (w, w').
d C₁
4 C₁+2a +a,a, da, da
da,
12
A
Now C₁+C₁cos (0¸ −0,') + ... = { a²+ a -2a, a' cos(0,−0,'} − };
d Co
14
+
da,
— { } Do +D₁cos (0,0)+...}. {a, -a,cos (0,–0,')}
a
-- (D.D.) ·
2
.. a² D₂+a,
12
Similarly, a Do+a,
+
= a¸a, D₁.
d Co
da.
d Co
= a¸ a¸D₁.
da,
Wherefore putting the coefficients of the last three terms.
of F equal to B, B', C, we have
F
11
m'
Co+
2
m'a, a
8
D₁ (tan² i, + tan³ ¿')
m'a, a
4
2
tan i tan i D, cos (N, – N')
12
1
- - w/)*
m' Be² – m' B'e," - m' Ce,e, cos (@,
-
-
in which we observe that C is symmetrical with respect to
a, and a'.
373. In Art. 366, we collected the formulæ for calculating
the elements of the instantaneous ellipse at any time. Since
the object of the present work is only to explain the theory
and not to enter into the numerical calculations of the per-
turbations, we shall proceed to demonstrate a few of the most
important and interesting results to which these equations
conduct us.
* We might shew that B = B' = 0, 0, D₁ ; but there is no occasion for this in
what follows.
348
PLANETARY THEORY.
DYNAMICS.
PROP. To shew that the effect of all the terms of R
(after the first) upon the elements of the planetary orbits is
periodical.
374. Any term P cos {(pn, ± qn') t + Q} will produce
a similar term in
ㄓ
​da, de,
d R
d R d R
and
but a term of the form
di,
d R
d R
d R
and
: since Q is
d Q
P sin {(pn, ± qn') t + Q} in
independent of a,, e,, i,; and
de, dw,
P is independent of e,, w,,,,
If then these be substituted in the equations of Art. 366. and
the integrations be effected, the elements a,, e,, w,, i,, Q, will
receive, in consequence, a term of the form
P
ㄓ
​COS
pn, qn, sin
de,
dt
{(pn, ± qn') t + Q};
since the formula for contains a term multiplied by t, the
element €, will receive, besides this, terms of the form (as may
be shewn by integrating by parts)
Pt
(pn, ±qn')
cos {(pn, ± qn') t + Q}
P
+
sin {(pn, ± qn') t + Q}·
(pn, ± qn,')²
360
the
pn, ± qn
£
It follows, then, that after a period of time =
perturbations of the elements, arising from the above term in R,
will have gone through their changes.
These variations of the elements are therefore termed
Periodic Variations.
It will be remarked that if pn, + qn, or pn, – qn, be
a very small quantity the integration described in the last
Article will increase the corresponding terms considerably:
and therefore it may happen that terms in R, of which the coeffi-
cients are so small as to appear of no consequence may rise to
INEQUALS. OF JUPITER & SATURN, EARTH & VENUS. 349
importance by receiving in the process of integration a small
divisor.
PROP. To find what terms in the development of R will
be much increased by the process of integration in determining
the elliptic elements.
375. By reference to the last Article we see that either
First, pn, + qn, must be a small quantity: hence, since p and
q are positive integers or zero, n, and n' must be small. By
reference to the first Table in Art. 391. we see that this is not
the case with any of the planets.
Or Secondly: pn,
qn must be small.
:
Hence p and q must be in the ratio nn, as nearly as
possible. Now the lowest dimension of the coefficient in terms
of small quantities is pq, Art. 370. If, then, we can find
two integers p and q nearly in the ratio n n, and having
a small difference, the corresponding term of R will rise into im-
portance by the integration. If we turn, now, to the first Table
in Art. 391, and by continued fractions find the convergents
which express the ratio of the values of n, for any two planets,
and choose those of them which have a small difference between
the numerator and denominator, we shall be able to detect the
most important of the terms of the development of R.
5 - 2
For Jupiter and Saturn n, n: 5:2 nearly, and
23: hence the dimension of the coefficient of a term
P cos {(2n, — 5n') t + Q} will be of the third order and will
be divided by the small quantities (2n,−5n') and (2n¸−5n')°.
For the Earth and Venus n,: n :: 8: 13 nearly and
13-8 = 5: hence the order of small quantities in the coeffi-
cient will be of the fifth degree and the argument
(13n, − 8n') t + Q.
376. These two examples present very remarkable in-
stances of the agreement of theory with observation.
The observations upon Jupiter and Saturn from the times
of the Chinese and Arabian Astronomers down to the present
day prove, that for ages the mean motions of these planets
have been affected by an inequality of long period. This
350
DYNAMICS. PLANETARY THEORY.
formed an apparent anomaly in the Planetary Theory till
Laplace pointed out the real cause of the inequality, and
rescued Newton's doctrine of Gravity from the reproach which
had long attached to it in consequence of its inability to assign
the cause of so remarkable a phenomenon. Laplace proved
that the inequality depends upon the near commensurability of
the mean motions of the planets (as explained in Art. 375),
and succeeded in calculating its period and amount.
Mr Airy has discovered a similar inequality in the motion
of the Earth and Venus. In the Phil. Trans. for 1832 he
shews that it amounts to no more than a few seconds at its
maximum, though its period is no less than 240 years. Mr Airy
had detected an error in the solar tables, and this induced him
to seek for the cause, which is so satisfactorily shewn to arise
from the near commensurability of the mean motions of the
Earth and Venus.
PROP. To explain the difference between Periodic Va-
riations and Secular Variations.
377. In the last Proposition we have supposed the ele-
ments which are involved in the right hand side of the equa-
tions to be constant, while they are in fact functions of t. The
only effect however which would result from this consideration
would be that the period of the variations would be slightly
altered.
But if we consider the effect of the first part of the ex-
pansion of R, which is independent of the periodic terms, and
which we call F, and suppose the elements involved in F
constant, it is evident, that by the integration of the equations
of Art. 366. the elements will receive additions which con-
tinually increase or decrease with the time, unless in any in-
stance the right hand side of the equation vanishes when F is
put for R. If, however, we make a nearer approximation,
and suppose that the elements in F are variable, and then
integrate the equations of Art. 366, the integrals may give
periodical values for the elements.
If this be the case in any
instance the variation is not called a Periodic Variation,
though in fact it is periodical, but a Secular Variation; since
SECULAR VARIATIONS OF THE ELEMENTS.
351
it arises from a cause quite different from that, which produces
the periodic variations. In short a periodic variation arises
from the fact of R involving r and 8 the co-ordinates of the
planet disturbed: but a secular variation arises from the fact
that the elements of the orbit vary. And since they vary very
slowly, the period in which they perform their secular vari-
ations is of immense duration *. Perhaps the following obser-
vations may throw light upon this subject.
The magnitude of the forces which disturb the elliptic
motion of the planets depends solely upon their relative posi-
tions, and not on their velocities and the directions of their
motion. When therefore, after a lapse of years, the planets
return to the same relative positions that they occupied at the
commencement of that period, the disturbing forces and the
perturbations in the places of the planets will have gone
through a series of changes, compensating in one part of this
period for the errors they have caused in some other part.
The inequalities produced during this interval of time are
termed Periodical Variations. But although the configuration
of the planetary system may become the same, yet, as was
before mentioned, the velocities and directions of the motion
of the planets will not necessarily become the same also; the
original and final orbits intersecting respectively in those points.
which the planets occupy at the beginning and end of the time,
which the periodic variations have taken to go through their
changes. The inequalities produced in this way are termed
Secular Variations in consequence of their very slow variation.
We proceed now to the examination of the Secular Vari-
ations.
PROP. To obtain the equations for calculating the
Secular Variations of the elliptic elements of a planet's orbit.
378. We must first find the differential coefficients of F
(the part of R independent of the periodic terms) with respect
to the elements: hence by Art. 372.
* The periodic variation of longest duration among those that are of sufficient
importance to be calculated has a period equal to 929 years. But some of the
secular variations have a period of 70000 and even more years.
352
DYNAMICS. PLANETARY THEORY.
dF
de,
dF
doo,
dF
de
0,
= m' Ce,e,' sin (w, - @'),
رة
=-2m' Be, - m'Ce, cos (w, w'),
d F
m'
dQ
4
dF
a tan i, tan i, D, sin (Q, – N'),
a a
a, a
4
m'
m D₁ tan i a a' tan i D, cos (2, − 2).
di,
1
Substituting these in the equations of Art. 366.
da,
dt
0,
de,
n,a,m' C'e
sin (w, -,'),
dt
и
do
n, a¸m' √1
e²
2
{2 Be, + Ce, cos (w, - w')},
dt
ме,
d tan i
na¸m'a,a' tani, D,
sin (2,-),
dt
4 μ
dQ, _ na, a'm'D
2
tan i
1
{− 1 +
cos (2, − 2,')}.
dt
4
tan i
u
We have retained the variable values of the elements on
the right hand of these equations: but should it be necessary,
we may use the constant values in approximating.
PROP. To prove the stability of the mean distances of
the planets from the Sun: and of their mean motions.
379. By Art. 378
da
dt
= 0;
α
a = const.
STABILITY OF THE ECCENTRICITIES.
353
This shews that the axis-major of any of the planets is
susceptible of no secular variation; and will suffer no per-
manent change the changes it undergoes in consequence of
the mutual attraction of the planets are wholly periodical.
The same is true of the mean motion
n
since it =
√√
a
μ
3
and does not alter.
μ
We are hereby assured of the impos-
sibility of any of the bodies of our system ever leaving it in
consequence of the disturbances it may experience from the
other bodies, and secures the general permanence of the whole
by keeping the mean distances and periodic times perpetually
fluctuating between certain limits (very restricted ones) which
they can never exceed, nor fall short of.
PROP. To prove the stability of the eccentricities of the
planetary orbits.
380.
By Art. 378 we have
de
na¸m' Ce
dt
μ
de
In the same way we should have
n'a'm Ce
sin (w, - w').
´ sin (w,' – ☎,).
dt
Ц
ก
m'
Multiply these by
e, and
n,a,
m
de,
e
Na
dt
+
m'
n a
n'a,e and add them
de,
e
= 0
idt
ጎ
m'
12
e
+
e,
= constant.
na
η α
If we had considered three planets we should have had
the following equations.
Y I
354
DYNAMICS. PLANETARY THEORY.
de, _ nam' Ce
dt
м
_
de_n'a'm Ce, sin (w;
dt
and
de"
n"a" m C'e
sin (w, -w') +
sin (w, −☎,) +
sin (@," - @) +
n¸a¸m"C'e,"
sin (-),
М
n'a'm"C"" e
sin (w/-w"),
M
n"a"m'C""e
sin (w,” – w').
dt
These equations give
M'
de
m"
+
dt
n,
a
m
de,
e
n a
d t
af
n a
n'
e
//
de,"
d t
= 0;
m
2
e +
a
nα,
m'
e
n'a
12
m'
+
112
e
n"a
= constant.
And the same formula would be true of any number of
planets.
Now observation shews that the eccentricities of the orbits
of the planets at present are very small indeed, with the ex-
ception of the Asteroids, the masses of which are very small.
Hence the above constant must be small. Since, then, all the
terms of the first side of the last equation are positive and their
sum always equals a small constant it follows, that the terms
are always small and the eccentricities are always small.
Hence the eccentricities of the orbits of the planets are
confined within very restricted limits: and therefore the forms
of the orbits are said to be stable.
381. The only quantities in the above equation, subject
to a change of sign in applying it to a system of bodies, are
the mean motions n,, n', n But observation shews that all
the planets revolve round the Sun in the same direction: and
consequently the terms are all positive.
PROP. To prove the stability of the inclinations of the
planets of the Solar System.
STABILITY OF THE INCLINATIONS.
355
382. By Art. 378 we have
d tan i,
dt
2
n, a'a' m' tan i 'D₁
sin (2, − N'),
in the same way
d tan i n'a am tan i, D,
sin (2′ – 2,),
d t
4
m
d tan i
m'
d tan i
tan i
+
tan i
= 0;
na
dt
n a
dt
m'
m
tan²i, +
tan i
= constant.
na
n'a,
The same equation would (as in the eccentricities) be true.
for any number of planets: and we see that the inclinations
must always be small. The certainty of this fact depends, as
before, upon the fact that the planets all revolve in the same
direction; Art. 381.
383. We a
We are thus led to the following remarkable conclu-
sion: The fact that the planets revolve about the Sun in the
same direction ensures the stability of the planetary system.
The converse of this would not necessarily be true, as we
shall see in Arts. 385, 387: the numerical relations of the
dimensions and positions of the orbits of the planets might be
such as to ensure stability although they revolved in opposite
directions. But the above is independent of particular nume-
rical relations.
384. We have given the two foregoing Propositions be-
cause of the simplicity of their demonstrations as well as the
beauty of the results. We shall, however, in the following
Articles obtain formula for calculating the magnitude of the
variations of the orbits in dimension and position.
PROP. To find the secular variation of the eccentricity
of the planetary orbits.
356
PLANETARY THEORY.
DYNAMICS.
385.
By Art. 378 we have for the planet m,
de,
n, a m' Ce
sin (w, w);
dt
dw,
na
n, a, m' √1-e;
2
{2 Be, + Ce, cos (w, −w,')}.
dt
ме,
And for the other planet m',
de' _ n'a¦'m Ce, sin (w; — ☎,),
μ
–
dt
απ'
n'a'm √1-e,"
12
{2 B'e + Ce, cos (w, - w)} ;
dt
ме
observing that C is the same for m and m', (Art. 372).
To integrate these equations assume
dr
dt
n, a m
m'
и
nam'
r' = e, sin w',
ሳ
e, sin w,,
S
e cos @,,
s' = e'
e
cos w/';
da,
de,
+
sin w
dt
= e, cos @, dt
{2 Be, cos w, + C'e, cos w
{2Bs + Cs'},
M
ds
dw
de
e sin @
+
cos w
dt
• dt
dt
nam'
{2Br + Cr'},
μ
dr' n'a'm
{2 B's' + Cs},
dt
μ
ds'
dt
n'a'm
{2B'r' + Cr}.
SECULAR VARIATION OF THE ECCENTRICITY.
357
These four are linear equations and their solutions are of the
form
r = D sin (gt + k) + E sin (ht + 1)
s = Dcos (gt + k) + Ecos (ht + 1)
r' = D′ sin (gt + k) + E' sin (ht + 1)
s' = D'cos (gt + k) + E' cos (ht + 1).
If we put these values in the differential equations we
arrive at the four following conditions connecting the eight
constants, four of which are consequently arbitrary and depend
upon the configuration of the planetary system.
nam'
Ꭰ g
{2BD + CD'},
m'
Eh
=
Ц
ѣ, а,
n m
ль
n'a'm
队
​{2 BE + CE'},
{2B'D' + CD},
D'g
E'h
{2 B'E' + CE}.
μ
n'a'm
By eliminating D' from the first and third of these,
we have
2n, a¸m' B
2n'am B
μ
M
n¸n'a,a' mm' C²
112
.. g
g=
n, a,m' B+n'a'm B' 1
±= √(n,a,m' B—n¦'a'm B')²+n¸n'a,a¦'mm' C².
#
μ
In a similar way we might shew that h has the same values.
n'
Now these values of g and h are possible when n, and n
have the same sign; that is, when the planets revolve in the
same direction about the Sun. But even if they do not
revolve in the same direction and n¸a¸m' B + n'a'm B′ be not
less than √n n'a a'm'm C, then g and h are still possible.
n¸n'a¸a'm'mC,
-
Now e2 20² + s² = D² + E² + 2 DE cos { ( g − h) t + k − l},
and a similar expression is true for e
358
PLANETARY THEORY.
DYNAMICS.
This shews that the eccentricity of m's orbit fluctuates
between the limits D + E and D E. Hence the form of
the orbit will be stable: the same is true of m''s orbit.
The values of D and E are very small in all the planets,
this is shewn by observation.*
The periods of the changes in the eccentricities of the orbits.
of the two planets are the same in each, being
360°
g - h
In
the case of Jupiter and Saturn this equals 70414 years! The
greatest and least eccentricities which Jupiter's orbit can attain
are 0.06036 and 0.02606, and those of Saturn 0.08409 and
0.01345; the maximum of each taking place at the time of
the minimum of the other, and vice versâ.
PROP. To find the secular variation of the longitude
of the perihelion.
386. By Art. 385, tan w
2°
S
D sin (gt + k) + E sin (ht + 1)
D cos (gt + k) + E cos (ht + 1)
در
The maxima and minima values of w or the greatest
deviations of the perihelion, from its mean place are found
by the equation
g D² + hE² + DE (g+ h) cos {(g − h) t + (k − 1)} = 0,
or cos {(g- h) t + (k − l) }
-
-
gD² + hE2
DE (g+ h)
which is obtained by equating to zero the differential coefficient
of tan w,.
* Sir Jolin Herschel finds that
D
– 0.01715, E = 0.04321 for Jupiter,
D'= 0.04877, E' = 0.03532 for Saturn,
g=21".9905, h=3".5851, k = 306° 34′ 40″, 7210° 16′40″,
being the number of years since the year A.D. 1700. See Article Physical
Astronomy in Encyclop. Metrop.
SECULAR VARIATION OF THE INCLINATION.
359
If this (disregarding the sign) be not greater than unity,
the perihelion will vibrate: but if, as is the case with Jupiter
and Saturn, this be greater than unity the longitude of the
perihelion has no maximum or minimum and therefore the
mean motion of the perihelion is continually in one direction.
PROP. To find the secular variation of the inclination.
387. By Art. 378. we have for m
d tan i
dt
n,a,m'a, a' tan i 'D₁
sin (2, − 2,')
4 μ
do
na'am'D₁
tan i
{− 1 +
cos (2,-,')},
tan i
dt
4 μ
and for the planet m'
d tan i
n'a'ma¸a'a¸tan i¸Ð¸
1
· sin (2,′ – N,)
dt
Ам
dQ, __ n'a¦² a¸m D,
tan i
1
{− 1 +
cos (2,-2)}.
tan i
dt
Аль
To integrate these, assume
p
tan i, sin,, q
Q,,q
=
tan i, cos,
p'
dp
dt
tani, sin, q' = tan i, cos Q';
= tan i, cos, dt
ΦΩ
d tan i
+ sin 2,
dt
2
na
n¸ a²a, m'D₁
(q' − q)
4μ
12
dq__ n‚¸ã‚¨“a'm' D、 (p − p'),
dt
dp' _ n'a, a 2 m D
dt
ηα
4μ
-
4 μ
(9 − q'),
dq__ n'a, a
a¸ am D₁
(p' − p).
dt
Аль
The integrals are of the form
p = G sin(at+y)+H sin(ẞt+d), q = G cos(at+y)+H cos(ßt+d),
p'=G'sin (at+y)+H'sin(ßt+d), q'=G'cos(at+y)+H'cos(ßt+d).
360
PLANETARY THEORY.
DYNAMICS.
Substituting in the differential equations we have
2
2
Ga=n, a² a'm' D₁
n, a² a'm' D₁
(G' –G), Hẞ=
=
(H'- H),
4 μ
αα
n'a,am D₁
G'a=
(G-G'), H'ß
n'a,
n'a a²m D₁
(H – H').
Αμ
4 μ
Eliminating G from the first and third
η α α
n'a,a,²m D₁
3
2
(a + ".º;a'm' D;) (a + "'¤¸‚ª"mD;) _ n‚n'a,a'mm'D;"
lat
na²
4μ
•
:: a² +
4
(n¸a¸m' + n'a'm) a¸a'D₁
4 μ
α
2
16μ²
a = 0.
We should arrive at the same equation for ß: hence
α
n¸a,m' + n'a'm
=
Аль
a,a'D₁ and ẞ = 0,
tan² i, = p² + q² = G² + H² + 2GH c
G² + H² + 2GH cos {at+y-d}.
Observation proves that G and H are very small for all the
planets (except the Asteroids). Hence the tangent of inclina-
tion fluctuates between the small limits G+ H and GH*.
The period of the changes in the inclination equals
360º
α
years. In the case of Jupiter and Saturn the number of years
is 50673! The maximum and minimum inclinations of Jupiter's
orbit to the ecliptic are 2° 2′ 30″ and 1° 17′ 10″: and those
of Saturn are 2° 32′ 40″ and 0° 47'. The maximum of each
takes place at the time of the minimum of the other, and vice
versâ.
* Sir John Herschel shews that when Jupiter and Saturn are the two planets,
G = -0.00681, H -0.02905 for Jupiter.
=
G' = 0.01537, H'= 0.02905 for Saturn.
α
25″.5756, y= 125° 15′ 40″, d=103° 38′ 40″,
being the number of years since A.D. 1700.
VARIATION OF LONGITUDE OF NODE.
361
PROP.
the node.
To find the secular variation of the longitude of
388. By Art. 378, we have
Ρ
G sin (at + y) + H sin d
tan
G cos (at + y) + H cos &
When attains a maximum or minimum value the differential
coefficient of tan Q, equals zero: hence
0 = a G² + GH a cos (at + - d)
G
... cos (at +7-8)
Η
If this (disregarding the sign) be not greater than unity, then
$60⁰
the node fluctuates, the period of its fluctuation being
α
years. But if this be greater than unity then there cannot be
any stationary positions; but the node continually moves in
one direction.
In the case of Saturn and Jupiter the node oscillates, the
extent of oscillation being about 13° 9′ 40″ in Jupiter's orbit
and 31° 56′ 20″ in Saturn's on either side of their mean po-
sitions the plane of the ecliptic being supposed immoveable.
389. The conclusions at which we have arrived in Arts.
379—383, with regard to the stability of the planetary system
are of especial interest. In consequence of the changes in the
elements we might have fancied that in the lapse of ages the
orbits would undergo such alterations in their dimensions as to
bring the planets into collision or hurry them into boundless
space. But we are assured that this can never be the case,
unless by the action of a resisting medium; since analysis
shews us that the orbits will continually fluctuate within very
small limits, never departing considerably from circles; and
the inclinations of the orbits will never change much.
390. Our calculations have not included the square of
the disturbing forces. But the same conclusions are found to
hold when the approximation is carried so far as analysts have
at present advanced: see the Mécanique Céleste, Liv. vi; Pon-
Zz
362
DYNAMICS. PLANETARY THEORY.
técoulant's Système du Monde, Tom. III; Plana's Planetary
Theory in the Memoirs of the Astronomical Society, Vol. 11.;
also a Memoir by Professor Hansen of Seeberg, the title of
this Memoir is Untersuchung üeber die gegenseitigen Störun-
gen des Jupiters und Saturns. In this method the true lon-
gitude is computed by means of the elements corresponding to
the invariable ellipse at the time of the epoch, taking a func-
tion of t, instead of t, which corrects for the perturbations.
See M. Pontécoulant's remarks on this in the Connaissance
des Tems for 1837. And lastly Mr Lubbock's papers in the
Transactions of the Royal Society and of the Astronomical
Society may be consulted.
PROP. To shew how the masses of the planets may be
discovered.
391. There are in general two methods of determining
the masses of the planets; either by observing the elongations
of a satellite, when the planet is accompanied by a satellite;
or by comparing the inequalities produced in their motion by
their mutual action. The secular variations are best adapted
to give the most exact results; but these are not yet known
with sufficient accuracy to allow of this use. We are therefore
obliged to recur to the periodic variations, and, by combining
a vast number of observations, gather from them the most
probable results. It is by these means that Astronomers have
obtained the following results.
Mass of Sun
1
1
Mercury
1909706
1
Venus
401839
1
Earth
356354
1
Mars.
2680337
MASSES OF THE PLANETS.
363
1
Mass of Jupiter.
1053.924
1
Saturn..
3512
1
Herschel ...
17918
We have taken these from M. Pontécoulant, Système du
Monde, Tom. III. p. 341. The following is the formula for
calculating the mass when the planet has a satellite.
Let 1, M, m be the masses of the Sun, the planet, and the
satellite: T',t the periodic times of the planet about the Sun,
and the satellite about the planet: A, a the mean distances of
the planet from the Sun, and the satellite from the planet.
Hence by Art. 269,
2π A
πα
M + m
a³ T2
T:
t
√1+ M
I+
√ M + m
1 + M
Á³ ť²
1
therefore (if we neglect m) M =
³ ť²
a³ Tc
In the case of Jupiter and his fourth satellite, we find by this
1
formula M
=
:
1048.69
this is more properly the mass of Ju-
piter with that of his fourth satellite.
The first value of the mass of Jupiter determined by
Laplace (Méc. Cél. Liv. v1. §. 21.) is
1
>
1067.09
and is founded
on the observed elongations of the satellites by Pound. These
elongations have been lately observed with much greater accu-
racy by Mr Airy at the Observatory of the University of
Cambridge, the result of his measures gives
1
1048.69
; Astro-
nomische Nachrichten, Vol. x. p. 304. Nicolai makes the
mass
1
1053.924
by observing the perturbations of Juno. Encke
364
DYNAMICS. PLANETARY THEORY.
1
makes it
1050.117
1
1054.4
by observing the motion of Vesta, and
by observations on the comet which bears his name. All these
concur in proving that the mass of Jupiter assumed by Laplace
is too small by about 4th part. The observations of Bouvard,
however, are at variance with this: he gives
of the Earth may be determined as follows.
1
1070.5
The mass
The attraction of the Earth on a body at its surface in the
parallel of which the square of the sine of the latitude is, is
very nearly the same as if the Earth were condensed into its
centre as we shall see in the Figure of the Earth in a sub-
sequent Chapter. Let sin, g= gravity in latitude 7,
b the mean radius of the Earth, 1 and E the masses of the Sun
and Earth, T the length of the year, a the mean radius of
the Earth's orbit: hence
E
T²g b²
مة
g
|}
and T = 2πa³; .. E
b2
4π² a²
Απ
b
α
sin Sun's parallax= sin 8".7.
1
The mass of the Moon nearly. Méc. Cél. Liv. vi.
74
§. 4.4.
But this is not yet very satisfactorily determined: we have
seen no value deduced from the observations mentioned in
Art. 348.
392. We extract the following Table from M. Ponté-
coulant's Système du Monde. These results are obtained by
the methods mentioned in Art. 270.
MASSES OF THE PLANETS.
365

Longitudes
of
Ascending Node.
Epoch is
Mean Motions
Longitudes
Longitudes
Mean Distance
Jan. 1.
1800.
in a Year of
Eccentricities.
of
of
Inclinations.
from Sun.
365 Days.
Epochs.
Perihelia.
Mercury 5323416″.79
0.38709812
0.2055149
110° 13′ 17″.9
74° 21′ 41″
7° 00′ 9″
45° 57′ 39″
Venus
2106641 .52
0.72333230
0.0068531
145 56 52 .1
128 43 6
3 23 29
74 52 39
Earth
1295977 .35
1.00000000
0.0168536
100 23 32 .6
99 29 53
0 00 00
0 00 00
Mars
689051 .12
1.52369352
0.0933061
232 49 50 .5
332 23 40
1 50 6
48 00 26
Jupiter
109256 .29
5.20116636
0.0481621
81 52 10 .3
11 7 36
1 18 52
98 25 45
Saturn
Herschel
43996 .72
15424 .54 19.18330500
9.53787090
0.0561505
123 5 29 .4
0.0166108
173 30 16 .6
89 8 20
167 30 21
2 29 38
46 26
111 56 7
72 59 21
366
PLANETARY THEORY.
DYNAMICS.
Table of Secular Inequalities of the Planets calculated
for the beginning of the Year 1801.

In the
Eccentricity.
In the Long. of
Perihelion.
In the Long.
of the Node.
In the Inclin.
of Orbit to
Ecliptic.
Mercury 0.000003867
9′ 43″.5
-
13' 29"
19″.8
Venus
0.000062711 4′ 28″
- 31′ 10″
4".5
Earth
0.000041200
11".9496
Mars
0.000090176
26".22
-
38′ 48″
1".5
Jupiter
Herschel
0.00015935 11' 4"
Saturn 0.000312402 31′ 17″
0.000025072 4'
- 26′ 17″
23"
- 37′ 54″
15' 5"
59′ 57″
3″.7
To obtain equations for calculating the effect of a resist-
ing medium upon a comet we must refer the reader to the
Mécanique Céleste, and also to Mr Airy's translation of the
dissertation on Encke's Comet in the Astronomische Nach-
richten.
Also for a very interesting paper on the orbits of re-
volving double stars the reader is referred to Vol. v. of the
Memoirs of the Astronomical Society, in which Sir John
F. W. Herschel has treated the subject in a very original
manner.
The following are Tables of the elements of the four small
planets Vesta, Juno, Pallas, and Ceres: and of the four known
periodical comets. The comet of Olbers has been observed
only once, at the time of its return to the perihelion in 1815:
the others have been observed in several successive revolutions.
It must be remarked that the elements of the small planets
given in the Table are not their mean values, but their values
at the specified epoch.
MASSES OF THE PLANETS.
367


Epoch 1831
July 23d. Oh.
Mean Time
Longitude
Mean
Mean
Longitude. Anomaly.
of
Perihelion.
Longitude
of
Inclination. Eccentricity.
Mean
Motion.
Mean
Distance.
Period.
Asc. Node.
at Berlin.
m
|
8447035 195 35" 26" 249 11m 375 103d 20m 285 707 578 0.0885601 9778.75540 2.361484 1325.5
13 02 10 0.2555592 813 .52533 2.669464 1593 .1
34 35 49 0.2419986|768 .54421 2.772631 | 1686 .3
10 36 56 0.0767379 769 .26059 2.770907 | 1684 .7
Vesta
Juno
Pallas
74 39 44 20 22 31
290 38 12 169 33 11
Ceres
54 17 13 170 52 34
121 05 01 172 38 30
307 03 26 159 22 02 147 41 23 80 53 50
Name of
the Comet.
Period.
Time of
Perihelion
Passage.
Longitude of
Perihelion
on the Orbit.
Longitude of
Asc. Node.
Mean
Inclination. Eccentricity.
Distance.
Halley's
Olbers's
76 years
74 years
Nov. 7, 1835
304 31
m
4.3s
55d 30m
170 44m 248
Ap. 26, 1815
149 02
Encke's
1204 days
Biela's
6.7 years
Jan. 10, 1829
Nov. 27, 1832
157 18 35
83 29
334 24 158
44 30
0.9675212
0.9313
17.98705
17.7
13 22 34
0.8446862
2.224346
109 56 45
248 12 24
13 13 13
0.751748
3.53683
事
​CHAPTER VII.
MOTION OF A PARTICLE ON CURVES AND SURFACES.
SIMPLE PENDULUM.
PROP. A material particle moves on a curve in a ver-
tical plane, and acted upon by gravity: required to determine
the motion.
393. Let A be the lowest point of the curve (fig. 94.)
Ax the axis of a drawn vertically upwards: P the position
of the body on the curve AP at the time t: AM=x, MP=y:
let R be the pressure of the curve against the body, this acts
R
in the normal line PG: M the mass of the body: then M
is the accelerating force resulting from the action of R
(Art. 225): g the force of gravity.
Now the forces acting vertically are g downwards and
R dy
upwards, the only horizontal force is
R
cos PGM or
M
M ds
R da
M ds
Hence, attending to the directions of the forces, we have
the following equations of motion :
ď² x
dt
& +
R dy
M ds
ď³y
R da
(1),
..(2).
d ť
M ds
dx
dy
Multiply these respectively by 2
2
and add, then
dt
dt
20
dx d²x dy day
+2
d x
2R /dx dy
dy dx
28
+
dt dť
dt dť
dt
M dt ds
dt ds
dx
2g
dt
MOTION AND PRESSURE ON A CURVE.
369
dx²
+
dy2
= const. - 2gx,
dt²
dt
ds2
or
=
dt2
const. - 2gx.
At the commencement of the motion let v = h;
0 = const.2gh;
ds2
dt
2g (h − x).
This expression shews that the velocity at any time is
independent of the form of the curve on which the body moves;
and depends solely on the vertical space through which it
passes.
Extracting the square root and inverting the two sides
of the equation
dt
1
1
ds
√
2 g √h-x
the negative sign being taken because s diminishes as t increases
(Note in page 208).
1
dx
ds
··. t
2g
S
Wh
-
v
dx
ds
We must determine
from the equation to the curve
d x
ds
dy2
by the formula
1 +
then by integration we shall
dx
dx²
know t in terms of x and therefore a in terms of t. In this
manner, then, we shall know the velocity and position of the
body at every assigned instant.
PROP. To find the pressure upon the curve.
394. The equations of motion being
ď²x
g+
dt
R dy
M ds
d³y
dť
R dx
M ds
3 A
370
DYNAMICS.
we multiply them respectively by
dy dx
dt dt
and subtract;
dy d³x
dx d²y
dy
R (dy dy
dx dx
g
+
+
dt dť
dt dť
dt
Mds dt
ds dt
dy
R ds
dy² dx²
g
+
+
1.
dt M dt
ds²
ds²
Now if ρ
be the radius of curvature of the curve on which
the body moves at the point (xy), then by the Differential
Calculus
dy d² x
dx dy
1
dt dť²
dt dť
P
d s3
dt³
t being a function of x and y, as is the case here:
R
M
dy v2
g + ;
ds ρ
v = velocity.
This expression shews that the pressure consists of two
parts, one the part of the forces which act upon the body
resolved along the normal, and the other the centrifugal force
arising from the motion. (Art. 254.)
PROP. A body moves on a cycloid, the axis of the
cycloid being vertical: required to find the time of an os-
cillation and to shew that it is independent of the extent
of the vibration.
395. We have shewn that
ds²
d t²
2 g (h − x) ;
dt
1
1
ds
2g √ h
X
the negative sign being taken because the arc decreases as
the time increases.
MOTION ON A CYCLOID.
371
Now the equation to the cycloid is
y = √ 2ax − x² + a vers
the lowest point being the origin;
X
1
a
dy
a X
a
✓
√2a-
x
2 ax
ევ2
dx √2ax
ds
dx
—
x²
1 +
dy²
dx2
2 a
X
α
1
8 √hx
dt
dt ds
Hence
√a
dx
ds dx
.. t = C -
√ vers-12
g
x02
h
- N
18120
g
when t = 0, x = h; .. 0 = C -
a
2x
t = √ = = - vers-12="}
g
π
h
ds
and, whenever the body stops, the velocity, or
= 0; and
dt
2x
therefore x =
h, and the values of vers
when a = h are
h
±π, ±3π, ±5π,...
and therefore the values of t are
2 П
√√
АП
g
8180
α
6 п
g
which shew that the body will oscillate backwards and for-
wards, the interval of time in which each oscillation is per-
Ve
formed being 2π N
مع
372
DYNAMICS.
4
This expression is independent of h and therefore points
out the remarkable fact that however large the arc of vibra-
tion be the time of oscillation is the same in all.
For this reason the cycloid is called a Tautochronous
Curve. *
PROP. A particle moves on a circular arc acted upon
by gravity: required the time of oscillating through a given
portion of the arc.
ds2
dt2
396. As before =2g
2 g (h - x) and the equation to the
-
circle from the lowest point is y² = 2ax − x²;
dy
dx
a
XC
ds
dy²
α
1 +
2 ax
x2
dx
dx²
√
2 αx
X2
* It may be interesting to ascertain whether there are any other tautochronous
curves when gravity is the force acting.
We have
dt
dx
1
1
ds
29
√h
dx
X
1
ds
1
]
X
1.3 x2
+
√2 g
d x
IN
+
+......+
hi
MW
2.4
5
ds
Now
d x
he
is independent of h: and consequently the integral of the general term
1
√2g
1. 3...(2n − 1)
2.4...2n
Xin
ds
2 n + 1 d x
h
2
1.3... (2n-1)
2.4...2n
xr
+
2 n + 1
h
2
must be of the form c.
(+1)
2n + 1
2
c being a constant, in order that when taken
between the limits = 0 and x = h the result may be independent of h: then
r
ds
A
fx dx
dx =
2 n + 1
ds
Ax
=
A
2
X
... s = A a²
2n + 1
V
2 A a constant ;
-
2
5² = A² x,
and this is the equation to the cycloid and therefore this is the only tautochronous
curve for gravity.
MOTION ON A CIRCULAR ARC.
373
dt
a
dx
1
√2 g √ (h − x) (2 ax − x²)
√(h
We are not able to integrate this function of x it is
reducible to one of the class called Elliptic Transcendents, the
properties of which Legendre has discussed in his Traité des
Fonctions Elliptiques: tables are given of the approximate
values of the integral for given values of *.
By means of series, however, the integral can be obtained
approximately.
dt
Va
dx
g
1
8 √hx - x²
1
8 √ hx-xº
X
1
2 a
2
X
1.3
ac
1.3... (2n-1)
X
+
+.
...
+
+...}.
2 a
2.4
2 a
2.4...2n
2 a
x" dx
2n-1
xn-1dx
xn
1
x²-¹ √ hx-x²
√hx-x²
Now
h
hx-x²
2η
√h x-x²
n
and between the limits a = h and x =
h and x = 0, we have
0 x" dx
2n - 1
h
h √hx - x²
2 ບ
h
S. Thes
0 x²-1dx
hx
0
x d x
h
vers-1
h
√hx - x²
2x
h
πλ
+- constant
2
* Let x = h sinº 0: then
=
when a h or t = 0;
2
dx
S:
2ax
√(h− x) (2 a x − x²)
2 sin cos de
cos² (2a - h sinº 0) sin² 0
2
d Ꮎ
√2a
h
1
sin² 0
2 a
which is an elliptic function of the first order.
374
DYNAMICS.
S
0 x² dx
1.3
0
203
h √h x
πh²,
x²
2.4
h
√hx - x²
and so on;
. T:
π
Į
1.3.5
2.4.6
Th³
g
2
h
2
1.3
h
2
1.3...(2n−1)
h
× {1+
+
+...+
2 a
2.4
2 a
2.4... 2n
+...}.
2 a
When the arc of vibration is very small, then
π
T
√a
а
2
g
and the time of an oscillation
= π
√
which coincides with
g
that in a cycloid, observing that the a in this case is four times
the a in that.
}]}
π
h
8 α
The next approximation gives a correction of the time
Va
h
g 8a
-; and the ratio this bears to the time of oscillation
(chord of angle of oscillation)³.
Thus if the body oscillate on each side of the vertical
through an angle of which the chord is, the time of oscillation
will be greater by a part than that calculated by the
formula 7
a
1 th
1600
397. Instead of supposing the body to move on a curve,
we may imagine it suspended by a string of invariable length,
or a thin wire considered of no weight. In this case the in-
strument is called a Pendulum, and is of great importance in
physical researches. For if
For if I be the length of a pendulum
oscillating in a second (or unit of time) then π-= 1,
g
and g
g = T
π² 1,
SIMPLE FENDULUM.
375
1
32
By this formula we may estimate the relative intensity of
the Earth's attraction at different stations on the surface, above,
or below it.
PROP. A seconds pendulum is carried to the top of a
mountain; required to find the height of the mountain by
observing the change in the time of oscillation.
398. Let be the radius of the Earth, considered
spherical; h the height of the mountain; 7 the length of the
pendulum: the force of gravity on bodies outside of the Earth
varies inversely as the square of the distance from the centre:
hence
is gravity at the top of the mountain.
2
Let n be
(r + h)²
the number of oscillations the pendulum makes in
in 24 × 60 × 60 seconds: then time of oscillation
a day, or
24 × 60 × 60
ጎ
.. 1=π
√
24×60×60
and
π
√√√1 (r+h)²
π (r+h) √√√!
g
n
gp2
ተ
h
24 × 60 × 60
1,
n
which gives the height of the mountain. For the sake of
example suppose the pendulum loses 5″ a day :
then n = 24 × 60 × 60 − 5,
-1
h
1
1
- 1
nearly;
"
24 × 12 × 60
24 x 12 x 60
4000
.. h =
mile nearly.
24 x 12 x 60
PROP.
To find the depth of a mine by observing the
change of oscillation in a seconds pendulum.
399. The gravity in the interior of the Earth varies
directly as the distance from the centre: if, then, h be the
g (r − h)
depth,
is gravity at the bottom of the mine:
376
DYNAMICS.
.. 1 = =
:π
П
1
:
180
"
24 × 60 × 60
N
T
1.1
r
g (r − h)
.'. 1
h
n
24 × 60 × 60
from which h can be found.
lose 5″ a day
h
2°
1
|}
1
سط
2
If, as before, the pendulum
2
1
1 - 1 –
nearly;
24 × 60 × 12
12 × 60 × 12
..hmile nearly.
400. The results deduced by the pendulum, as far as
we have at present explained its construction, would lead to
erroneous conclusions; since we have supposed the rod sup-
porting the bob, as the lower extremity is termed, to have no
weight. We must leave the correction of this to a future
part of the work, in which we shall shew that I must not be
taken equal to the length of the pendulum; but some other
expression which it is unnecessary to give here.
401. Owing to the remarkable property of the cycloid,
that its evolute is an equal cycloid, we can easily make the
bob of a flexible pendulum move in a cycloidal arc.
For let CA (fig. 95) be the pendulum when remaining at
rest: PAP' the cycloid in which the bob is to move, the length
of the axis being half that of the pendulum: CQ, CQ' the
evolutes of PAP'. Now move the bob to the right, and let
the upper portion of the pendulum bend round CQ and the
other portion remain straight, touching CQ in Q. Then since
CQ is the evolute of AP, the extremity of the pendulum will
be in the curve AP: and by this contrivance the bob will be
made to describe the cycloid PAP'.
This suggests the following means of correcting a common
pendulum which makes small oscillations. Let a small portion
of the upper extremity be flexible: (consisting of watch spring,
&c.) and let it be suspended between two cycloidal cheeks, as
in fig. 96. Then the small oscillations of the bob will be in
SIMPLE PENDULUM.
377
a cycloid, and in the expression for the time of oscillation the
h
correction depending on is avoided: see Art. 396.
2 a
402. The following Table contains the results of ex-
periments with a seconds pendulum on various parts of the
Earth. It is extracted from the Mécanique Céleste.

Places.
Latitudes.
Lengths of a
Seconds Pendulum.
Peru
0º.00
0.99669
Porto Bello
10.61
0.99689
Pondicherry
13.25
0.99710
Jamaica
20.00
0.99745
Petit-Goave
20.50
0.99728
Cape of Good Hope
37.69
0.99877
Toulouse
48.44
0.99950
Vienna
53.57
0.99987
Paris
54.26
1.00000
Gotha
56.63
1.00006
London
57.22
1.00018
Petersburgh
64.72
1.00074
Arensgberg
66.60
1.00101
Ponoi
74.22
1.00137
Lapland
74.53
1.00148
403. Mr Airy, in a Paper which was read before the
Philosophical Society of Cambridge in the year 1826, has re-
duced the usual theorems for the alteration in the time and
extent of vibration produced by the difference between cycloidal
and circular arcs, by the resistance of the air, by the friction
at the point of suspension, and by other disturbing causes, to
3 B
378
DYNAMICS.
a very general investigation which leads to results remarkable
for their simplicity. Since the principle of the pendulum is
of vast importance in physical researches we shall not scruple
to introduce large extracts from this valuable communication.
PROP. A pendulum is acted upon by a small disturbing
force: required the alteration in the time and extent of its
oscillations.
404. We shall suppose that the undisturbed pendulum
moves with its extremity in a cycloidal arc, since in this case
the calculation is not approximate.
Let s be the distance of the pendulum at the time t from
the lowest point of the cycloid, s being measured along the
arc described, the length of the pendulum. Then the re-
dx
ds
solved part of gravity along the tangent is g x being
measured vertically upwards: and s² = 21x is the equation to
the cycloid;
dx
g
S.
g
ds
Wherefore the equation of motion of the bob of the
pendulum is
ď s
d t2
مة
S
or if n²
d2 s
+ n³s = 0.
dt
مع
The solution of this equation is
s = a sin (nt + b),
where a and b are arbitrary constant quantities depending on
the length of the arc of vibration and the time of passing the
lowest point.
ds
The velocity at time t =
= na cos (nt + b).
dt
PERTURBATION OF A PENDULUM.
379
We shall now suppose that ƒ is a small disturbing accele-
rating force resolved along the tangent: the equation of
motion then is
d's
+ n²s = f.
dť
The solution of this equation we shall assume to be
s = a sin (nt + b)
(conformably to the principle of the variation of parameters)
a and b being considered unknown functions of t, which it is
our business now to determine.
Since there are two functions a and b we may assume any
relation between them that we please, since we have but one
quantity (s) to determine. Let this assumption be that the
velocity is still expressed by na cos (nt + b): the convenience
of this we shall soon discover.
Now sa sin (nt + b);
ds
da
db
= na cos (nt + b) +
sin (nt + b) + a cos (nt + b)
dt
dt
dt
da
d b
and ..
sin (nt + b) + a cos (nt + b)
0,
dt
dt
this is the assumed relation between a and b.
ds
Again since
= na cos (nt + b) ;
dt
d² s
da
db
n² a sin (nt+b)+n
cos (nt+b) − n a sin (nt+b)
d t2
dt
dt
d's
in this substitute for
its value;
dt2
da
db
... n
cos (nt + b) — na sin (nt + b)
=f,
dt
dt
this is the second equation between a and b.
380
DYNAMICS.
db
Eliminating successively
dt
da
and from these, we have
dt
da f
db
f
cos (nt + b),
sin (nt + b).
dt n
dt
na
If we could solve these equations we should have the complete
determination of the motion. In few cases is this practicable:
in all to which we shall have to apply the investigation an
approximation is sufficient.
Hence the va-
We suppose f to be a very small force.
riable parts of a and b are of the same order of magnitude as ƒ
and consequently may be neglected on the right-hand side of
the above equations if we agree to neglect the square and
higher powers of f
In order to find the alteration in the extent of vibra-
tion which takes place in one oscillation we must integrate
f
n
cos (nt + b) through the limits of t corresponding to one
= a to
oscillation: that is from a value of t which gives nt + b
the value of t which gives nt + b =πτα. Here a may be any
quantity in different cases we shall find it convenient to in-
tegrate between different limits.
ff
· increase of arc of semi-vibration == ƒ cos (nt + b) dt
between the above-mentioned limits.
n
To find the alteration in the time of oscillation, let T, T
be the values of t at two successive arrivals of the pendulum
at the lowest point; B, B' the values of b at these times. Then
nT' + B′ = (m + 1) . π ;
nT + B = m.π,
n (T′′ − T) + B' – B = т,
4
.. (T'.
π
T'- T
=
(B'- B).
n
n
Τι
db
T'
]
dt
dt
f sin (nt + b) dt
na
T
T
Now B'- B =
between the proper limits;
PERTURBATION OF A PENDULUM.
381
T'
1
.. the increase of time of oscillation =
n² a
T
f sin (nt + b) dt,
and the proportionate increase of time of oscillation
T'
1
f sin (nt + b) dt.
ппа
Τ
If the circumstances are such that we
must integrate
through two vibrations, then
1
proportionate increase of time of osc. =
fsin (nt+b) dt.
Ωπηα
These formulæ are convenient when ƒ can be expressed in
terms of t. If however ƒ be expressed in terms of s, as is the
case particularly in clock escapements, we must modify the
formulæ
da da dt
1
da
f
n² a
db
na cos (nt + b) dt
1
na cos (nt + b) dt
ds
dt ds
d b
and
ds
f
tan (nt + b)
n² a²
f
S
n² a³ √ a² – s²
.. increase of arc of semi-vibration
proportionate increase of the time of vib".
n² a
1
I
S
fds,
fsds
π N² a²
√ a²-s²
We shall subjoin a variety of examples.
1
Instead of vibrating in a cycloid let the pendulum
Ex. 1.
vibrate in a circle.
Here the force = g sin
g
En
68
esooja
- ga
67³
رکت
ترى
g's
g
nearly;
7
673
sin (nt + b);
382
DYNAMICS.
therefore proportionate increase in time of vibration
ga²
6πnt³ fsin¹ (nt + b) dt.
1
Now fsin' (nt+b) dt == [{³-
f {3-4 cos 2 (nt+b)+cos 4 (nt+b)}dt
||
11∞
2
1
{3t
sin 2 (nt + b) +
sin 4 (nt + b)} + C
N
4n
3 п
>
8 n
from nt + b = 0 to π ;
.. proportionate increase of time
The increase of arc of vib.
ga³
2
24 n² 13
ε a³
6n13
=
ga²
a²
g
since n²
을​.
16 n² 3 1672
on's fcos (nt + b) sin³ (nt + b) dt
sin¹ (nt + b) + C = 0 between the limits,
as we might easily have foreseen.
Ex. 2. Suppose the friction at the point of suspension
to be constant.
Here fc, since the friction retards the motion; and
the motion is considered from the lowest point. It will be
convenient to take the integrals during that time in which the
friction acts in the same direction: that is, from the beginning
of a vibration to its end, or from nt + b
П
to nt + b
2
C
.. increase of arc =
n
fcos (nt + b) dt
C
=
π
;
102
- sin (nt + b) + C = −
proportionate increase of time
C
2c
25
πna Ssin (nt + b) dt
ппа
PERTURBATION OF A PENDULUM.
383
C
πηρα
cos (nt + b) + C = 0,
π
between the limits nt + b and 7.
Ex. 3.
2017
Suppose the resistance of the air to produce a
force varying as the mth power of the velocity or = kvm, m
being any whole number.
The velocity in moving from the lowest point
f =
ds
= na cos (nt + b) ;
dt
- knm am cosm (nt + b);
therefore increase of arc
1
= − kn™-¹a″ fcos"+¹ (nt + b) dt from nt + b = −
||
11
- Κπηm-2 am
:
m (m − 2)
1
(m + 1) (m − 1)
1) ....
(m odd)
2
2 knm-
2
ат
m (m2)
2).
10
2
(m + 1) (m − 1)
(m even).
3
1019
T
to
arc =
When m = 2 (the law usually taken) the decrease of the
4 ka
3
The proportionate increase of time of oscillation
k
П
n"-1am-1 fcos" (nt + b) sin (nt + b) dt.
k nm-e
a
772 - 1
COS
m+1
(nt + b) + C
π (m + 1)
= 0 between nt + b
π
and
૭ | મે
7™
whether m be a positive integer or fraction.
1014
384
Ex. 4.
DYNAMICS.
Suppose the resistance of the air is expressed by
any function of the velocity.
Here f= (v) for the descent and (v) for the ascent,
and the increase of the arc of vibration
1
n³ a √p (v) sin (nt + b)
3
cos (nt + b)
1
d v =
η
n³ a
svp ( v ) d v
But it
ર
from v = 0 to v = 0
again.
v = 0 to
0 to v = na (that is, from s =
√ n² a² — v²
must be observed that from
a to s = 0) the radical must
be taken with a negative sign, because sin (nt + b) is then
negative. The increase of the arc is consequently
na
1
3
n³ a
0
vp (v) dv
√ n² a² - v²
1
+
୮
r⁰ vp (v) dv
n³ a
n a
na
√ n² a² – v²
v p (v) d v
2
n³ a
0
√ n³ a² - v²
,
and therefore decrease
The proportionate increase of time of vibration
1
ппа
[p(v)
1
Ø(v) sin (nt + b) dt
3
p (v) dv
1
π N³ a²
(v) = 0, from v = = 0 to v = 0.
Hence a resistance which is constant, or which depends on the
velocity, does not alter the time of vibration.
Ex. 5. Let the resistance be that produced by a current
of air moving in the plane of vibration with a velocity V
greater than the greatest velocity of the pendulum: and
varying as the square of their relative velocity.
k
Here (v) = (V - v) when the pendulum moves in
the direction of the current
$ (v) = k (V + v)2 when it moves in the opposite direction.
By the formula in the last Example, when the pendulum
moves in the direction of the current, the arc is increased by
PENDULUM ESCAPEMENTS.
385
and when it returns the arc is dimi-
2 V2
Καπ
4a²
k
+
2
N
n
3
2 V2
ναπ
4a2
nished by k
+
+
n²
η
3
The diminution in two vibrations
is unaffected.
2k Vаπ
ㅠ
​The time
n
Ex. 6. Let a force Fact through a very small space x at
the distance c from the lowest point.
The increase of the arc =
1
"
c + c
Fx
Fds=
nearly.
n² a
n² a
The proportionate increase of the time of vibration
1
éta
Fsds
π n² a²
a²
ઈમ્ફ
if the general value of the integral be (s), then the propor-
tionate increase of time = $ (c + x) − ¢ (c) = 4′ (c) x
Fa
n²ď² √ a²
π N
C
If, then, an impulse be given when the pendulum is at its
lowest point, c = 0 and the time of vibration is unaffected.
405. Since the preceding theory is applicable to every
case in which a pendulum is acted on by small forces, it can be
applied to determine the effect produced on the motion of the
pendulum of a clock, or the balance of a watch, by the ma-
chinery which serves to maintain that motion.
If a pendulum vibrate uninfluenced by any external forces
except that of gravity, the resistance of the air and the friction
of the point of suspension gradually reduce the extent of vi-
bration. But this diminution goes on very slowly. A pen-
dulum suspended on knife edges has been observed to vibrate
more than seven hours before its arc was reduced from two
degrees to th of a degree. In order to maintain vibrations of
the same or nearly the same length (which for clocks is indis-
pensable) a force must act on the pendulum: this force is
3 C
386
DYNAMICS.
generally given by the action of a tooth of the seconds wheel on
the inclined surfaces of small arms or pallets carried by the
pendulum: and the whole apparatus is called an escapement.
Now it appears from Examples 2, 3, 4 and 5 of the last
Article, that the friction and the resistance of the air do not.
affect the time of vibration. The maintaining force, therefore,
must be impressed in such a manner as not to alter the time
of vibration. The escapements of clocks in general use may
be divided into the three following classes: recoil escapements,
dead-beat escapements, and the escapements in which the action
of the wheels raises a small weight which by its descent acce-
lerates the pendulum: this last is Cumming's escapement.
A full discussion of these will be found in Mr Airy's com-
munication. He comes to the conclusion that the dead-beat ·
escapement is far superior to any other.
4.06. In this the wheel acts on the pallet for a small space
near the middle of the vibration, and during the remainder of
the vibration it has no effect except in producing a slight
friction. The impact also at the beat does not tend to acce-
lerate or retard the pendulum. Neglecting then the consider-
ation of the friction, we have a constant force F, which begins
to act when ∞ = − c and ceases when a = c. Hence by Ex. 6.
of last Article, proportionate increase of time
F
πη απ
F
2
ПП
یر
Ր
sds
√ a² - s²
12
c'² – c²
F
π n² a²
{√ a³ - c²
12
F
12
2 π n² a³
3
(c'+c) (c' — c) nearly;
πn²а² √a²-c² + √ a²-c²²
an extremely small quantity, since c and c' are very small when
compared with a, and c-c may be made almost as small as
we please, though it cannot be made absolutely zero; for the
wheel must be so adapted to the pallets, that when it is dis-
engaged from one it may strike the other, not on the acting
surface, but a little above it; that is, the instant of disen-
gagement from a pallet must follow the instant at which the
pendulum is in its middle position by a rather longer time than
that by which the instant of beginning to act preceded it.
Hence c' must be rather greater than c. But the difference
MOTION OF A PARTICLE ON A SURFACE.
387
•
may be made so small that the effect on the clock's rate shall
be almost insensible. This escapement, then, approaches very
nearly to absolute perfection: and in this respect theory and
practice are in exact agreement.
Mr Airy suggests a construction (Trans. Cam. Phil. Soc.
Vol. 111. p. 125.) for a clock escapement similar in its prin-
ciples to the best detached escapements of chronometers.
PROP. To prove that the velocity of a particle moving on
a smooth surface is independent of the path described, but
depends solely on the co-ordinates of position.
407. Let R be the normal pressure between the surface
and particle at the time t, M the mass of the particle; aßy
the angles which the direction of R makes with the axes: then,
X, Y, Z being the other forces acting on the particle, the
equations of motion are
d² x
d t
R
d'y
R
= X +
cos a,
= Y +
cos B,
M
dt2
M
ď² z
R
2+
cos Y.
dť
M
da
dy
dz
Multiply these by 2
2
2
and add; then
d t
dt
d t
d.v²
dx
dy
dz
2X
+ Į
+ Z
dt
dt
dt
dt
2 R
dx
dy
dz
+
cos a +
cos B +
COS
M dt
dt
dt
dx dy dz
But
ds ds ds
are the cosines of the angles which the
tangent line to the curve described makes with the axes; hence
dx
dy
cos a +
cos B +
ds
ds
dz
ds
cos Y
equals the cosine of the angle which this tangent makes with
the normal, and therefore equals zero;
388
DYNAMICS.
v2
…. v = 2 [(Xd + Ydy + Zdx),
and X, Y, Z being functions of x, y, z this expression when
integrated will be a function of x, y, z, the co-ordinates of
position, and does not depend on the path described.
PROP. A particle moves in a spherical bowl acted on by
gravity: required to determine the motion.
408. The equations of motion are (≈ being vertical)
ď x
R
d² y
R
d2 %
R
cos α,
cos B,
༡
g
cos y,
dt2
M
dt2
M
dt2
M
also x²+y²+x²=a² is the equation to the surface in this case,
cos a
=
ac
α
then (as in last Article)
Z
C cos B
B=2/, cos y =
α
α
dx² dy
dx
+
+
dt2
dť
dt2
C + 2gz.
d№2
+
+
dt2
df2
d t²
Let V and k be the initial values of the velocity and of ≈ them
dx² dy2
V² − 2g (k − ≈),
ď² y
ď² x
also x
d t
Y
0;
df
dy
d x
a
y
= const.
-
h
dt
dt
dx
dy
dz
likewise x
+ y
+ &
0 *
dt
dt
dt
By eliminating
dx dy
and
from these, we have
dt
dt
t
adz
√ (a² − x²) { V² – 2 g (k − x) } – h²
MOTION ON A SPHERE.
389
This is an elliptic function, Art. 396. If this could be inte-
grated, then ≈ (and consequently x and y) is known in terms
of t, and the motion is determined.
409. We may obtain approximate results by supposing
the oscillations to be very small.
In this case, let be the angle that the radius drawn to
the particle makes with the vertical, the angle which the
vertical plane in which is measured makes with the vertical
plane through the centre of the sphere and the point of pro-
jection; let the velocity of projection (V) = ß√gɑ, ß being
a small numerical quantity, the direction of V horizontal,
initial value of 0; then
α the
Ꮎ
-
k = a − 1 aa², ≈
z = a
- 1a0², h² = a³ga²ß²,
y
= x tan, x² + y² + +²
=
a²;
dt
dt
ав
√a
do
dz
d↓
d y d t
1
ᏧᎾ
dt do
x² + y²
(
dy
αβ
X
dt
g √ (a² − 0²) (0² – ß³)
Y
dx\ dt
dt dᎾ 0 √ (a² −0²) (0³ −ß³)
The first of these equations gives
2 t =
2d.02
-
√ (a² – ß³)² – {20³ − (a² + ß³)}~
81
20
α
(20°
Cos-1
COS
(a² + B²)
a² - B²
,
const. = 0;
0² = 1 / (a² + (²) + ½ (a² − ß³) cos 2
201
t:
a
this shews that the pendulum makes isochronous oscillations in
the moveable vertical plane: the extreme angles being a and ß,
and the time of oscillation being
π
✓
or half the time of
g
2
oscillation when the plane of motion is constant.
390
DYNAMICS.
d↓
Hence also
امة
dt
a
a² cos²
αβ
Et + B² sin²
g
t
са
a
.. a tan ↓ = ẞ tan
Y
امة
(L
2t,
from which the azimuth of the plane of oscillation is known at
any time.
By substitution we have
x2
ايف
y²
cos²
sin²
cos²
sin'
a² + B
=
(a* - ≈²)
+
a² в
༡
a²
B2
a²
Be
and substituting for 0 and their values in terms of t,
2
x²
yo
Gi
+
B2
a²,
which shews that the projection of the path on a horizontal
plane is an ellipse with its centre in the vertical radius of the
sphere.
COR.
If a = ß, then 0² = a², ↓ = √√§t, x² + y² = a²a,
and the pendulum describes a conical surface with a uniform
motion.
PROP. A particle moves on a curve surface, required to
find the pressure at any instant.
410. The equations of motion are
d² x
dt2
R
ď² y
R
ď² z
R
Y+
=
X +
cos B,
2+
cos Y.
cos a,
M
d t
M
dť
M
Multiply by cos a, cos ß, cos y respectively, and add, then
R ď²x
M
df
d²y
d t
d²
z
cos a + cos B +
cos Y
d t³
- {X cos a + Y cos B + Z cos y}.
To calculate the former part suppose that the co-ordinate
planes are so chosen, that, at the instant under consideration,
PRESSURE ON THE SURFACE.
391
the axis of x is the normal line at the point of contact of the
particle hence cos a = 0, cos B = 0, cos y = 1, and this part
becomes
d² z
d t² ·
Now ≈ is a function of x and y: x and y are functions of
t; hence
dz
dz dx
dz dy
+
dt
dx dt
dy dt
ď² z
d²≈ dx²
d² z dx dy
d²≈ dy
dz d²x dz d'y
+ 2
+
+
+
d t2 dx² dt²
dxdy dt
dt
dy² dť
dx dť dy dť
dz
dz
But
0,
O as the axes are chosen.
dx
dy
d² z
ds2
Hence
Jd² z dx²
d²
d'z dx dy
ď² z dy²
+2
+
d t²
d ť dv ds
dx dy ds ds
dy ds
(velocity)
radius of curvature
and the magnitude of this cannot depend upon the manner of
fixing the axis; therefore, in general,
v2
P
R
v2
(X cos a + Y cos ẞ + Z cos y)
M
P
centrifugal force - resolved part of the forces along the normal.
PROP. A particle moves in a groove in the form of a
curve of double curvature; required the pressure.
411. The equations of motion are the same as in the last
Article: aßy being the angles which the direction of the
pressure makes with the axes; this coincides with the radius of
absolute curvature.
Let
P
be the radius, and x,y,, the co-ordinates to the
centre of curvature, then
dx
ď y
2
x₁ = x + p²
Y₁ = Y + p²
ds²
d s²
22
+ p²
d²
ds
392
DYNAMICS.
x
d² x
ď² y
ď² z
1
.. cos a =
P
cos B = p
d s²
ds2
>
sy
cos y = p
ds2
p
R
M
P
[d² x d² x d²y d² y d² z d² z
{df ds²+ dt ds² + dt ds-{Xcosa+Y cosẞ+Zcosy},
the former part, by changing the independent variable to s (as
in Art. 255), becomes
ď² x
ds2
2
||
ds² ds I d s² ds? ds2
d²y
2
2
d2
+
+
d² t d [ d x ²
dy2
d2
+
ds2
+
ds2
dt2
dt³
ds2
d s³
1
ds2
1 ds2
2
(cos² a + cos² ß + cos² y)
ן
dt
p² dť²
R v2
M ρ
(X cos a + Y cos ß + Z cos y)
= centrifugal force resolved part of the forces along the
radius of absolute curvature.
CHAPTER VIII.
PROBLEMS ON THE MOTION OF BODIES CONSIDERED AS PARTICLES.
PROB. 1. A BODY is projected vertically upwards and the
time between its leaving a given point and returning to it is
given find the velocity of projection, and the whole time of
:
motion.
PROB. 2. Two bodies fall from two given points in space
in the same vertical down two straight lines drawn to any
point of a surface in the same time, find the form of the
surface.
PROB. 3. A semi-cycloid is placed with its axis vertical
and vertex downwards, and from different points in it a number
of bodies are let fall at the same instant, each moving down
the tangent at the point from which it sets out: prove that
they will reach the involute (passing through the vertex) all
at the same instant.
PROB. 4. From the top of a tower two bodies are pro-
jected with the same given velocity at different given angles
of elevation, and they strike the horizon at the same place:
find the height of the tower.
PROB. 5. A body acted upon by two central forces, each
varying inversely as the square of a distance, is projected from
a point between them towards one of the centres: required
the velocity of projection that the body may just arrive at the
neutral point of attraction and remain at rest there.
PROB. 6. A body, acted on by a force varying inversely
as the fifth power of the distance, is projected in any direction
with a velocity equal to that which would be acquired in falling
from an infinite distance: find the orbit.
a
PROB. 7. A body, projected in a given direction with
given velocity and attracted towards a given centre of force,
1
3 D
394
DYNAMICS.
has its velocity at every point: the velocity in a circle at the
same distance :: 1: √2; find the orbit described, the po-
sition of the apse, the magnitude of its axis, and the law of
force.
PROB. 8. Two bodies are connected by a string passing
through a hole in a horizontal plane; one of them is projected
in any direction in the horizontal plane, and the other descends
vertically by the action of gravity: find the motion of the
bodies, and the curve described on the plane.
PROB. 9. A body is projected in any direction from one
extremity of a right line, each particle of which attracts it by
a force proportional to the distance; prove that the body will
pass through the other extremity.
PROB. 10. A body projected from a given point in a plane
m
X®
m'
is attracted by forces in the direction of x, and in the
y³
direction of y: prove that if the velocity and direction of pro-
jection be rightly assumed, it will describe a circle round the
origin as centre, and find how the velocity varies in different
parts of the orbit.
PROB. 11,
A body, urged towards a plane by a force
varying as the perpendicular distance from it, is projected at
right angles to the plane from a given point in it with a given
velocity: find what force must act at the same time on the
body parallel to the plane, that it may move in a given para-
bola having its axis in the plane; and determine the circum-
stances of the motion.
PROB. 12. A body acted on by a force varying partly as
the inverse cube and partly as the inverse fifth power of the
distance is projected with the velocity which would be acquired
in falling from infinity, at an angle with the distance the
tangent of which = √2, the forces being equal at the point of
projection; determine the motion.
PROB. 13. A body is projected from a point near a centre
of force which varies inversely as the square of the distance,
in a direction perpendicular to the line joining the point of
projection with the centre of force, and so as to describe an
ellipse about that centre: shew that the point of projection
PROBLEMS.
395
will coincide with the nearer or further apse according as the
velocity of projection is greater or less than that with which
a circle might be described at the same distance.
PROB. 14. If a force vary inversely as the 7th power of
the distance, and a body be projected from an apse with a
velocity which is to the velocity in a circle at the same distance
:: 1:√3; find the polar equation to the curve described,
and transform it to rectangular co-ordinates.
PROB. 15. If a body be projected about a centre of force.
varying inversely as the square of the distance with a velocity
equal to n times the velocity in a circle at the same distance,
and in a direction making an angle ẞ with the distance; the
angle a between the axis major and this distance may be deter-
mined from the equation
tan (a – ẞ) = (1 ~ n²) tan ß.
PROB. 16. If a be the mean distance of a planet from
the Sun, and 7 the length of the line of nodes, then the time
of the planet's passage (supposed undisturbed) from node to
node through perihelion is
alp {tan-
l
7
2 a
1
2a-l
2 a
where p
=
π
the length of the year, and 1= mean distance of
the Earth from the Sun.
PROB. 17. If a body revolve in an ellipse round the
focus prove, that a progressive motion of the apse will be the
effect of any continual addition of force in the direction of the
radius vector during the progress of the body from the further
to the nearer apse, and point out the effect on the eccentricity.
PROB. 18. A body is acted on by two forces, one repul-
sive and varying as the distance from a given point, and the
other constant and acting in parallel lines: determine the
motion of the body.
PROB. 19. If a body can describe a given curve about
one centre with one law of force, about another centre with
another law of force and so for any number of centres, it is
possible to project the body with such a velocity that it may
describe the same curve under the action of all those forces.
396
DYNAMICS.
PROB. 20. A body describes a parabola about a centre
of force residing in a point in the circumference of a given
ellipse, the foci of which are in the circumference of the para-
bola, the force varying inversely as the square of the distance:
shew that the time of moving from one focus to the other is
the same, at whatever point in the circumference of the ellipse
the centre of force is placed.
PROB. 21. If P be a central force attracting a catenary,
and p be the perpendicular on the tangent at any point from
the centre of force; then, the force which would cause a body
to revolve in the curve formed by the catenary varies as
PP.
PROB. 22. A body P is projected with a given velocity
a√μ in a direction perpendicular to its distance SA from
a centre of force S, which itself moves uniformly with
velocity V in the direction AS produced; the force varies as
the distance: determine the equation to the orbit described,
and shew that the motions of P and S are parallel when the
co-ordinates of P measured from the original position of S
are a and (1) V.
PROB. 23. If two equal bodies, which attract each other
with forces varying inversely as the square of the distance,
are constrained to move in two straight lines at right angles
to each other; shew that they will arrive together at the point
of intersection of the lines, from whatever points their motions.
commence and having given their distance at the beginning
of the motion, find the time to the point of intersection.
:
PROB. 24. The times of oscillation of a pendulum are
observed at the Earth's surface, and at a given depth below
the surface; find from these data the radius of the Earth,
supposed spherical.
PROP. 25. If a pendulum oscillating in a small circular
arc be acted upon, in addition to the force of gravity, by a
small horizontal force (as the attraction of a mountain) in the
plane in which it oscillates; having given the number of
oscillations gained in a day, find the horizontal force.
PROB. 26. A body oscillates in a cycloid on an inclined
plane, and the friction on the plane = times the pressure:
shew that the friction will not affect the time of oscillation,
µ
PROBLEMS.
397
and that the body will stop after it has oscillated a number
α
of times
- tan a
27μ
, where a is the original distance
from the lowest point and a the inclination of the plane.
PROB. 27. A body acted on by gravity moves on the
convex surface of a cycloid, the vertex of which is its highest
point; the velocity at the highest point being √2gh, deter-
mine the point where it will leave the curve, and the latus
rectum of the parabola afterwards described.
PROB. 28. A body moving on the interior surface of a
vertical cylinder was projected with a given velocity, and
goes round precisely n times before it begins to descend:
find the direction of projection.
PROB. 29. A body acted on by a repulsive central force
varying as the distance, moves in a groove of the form of an
epicycloid, the pole of which is in the centre of force: prove
that the oscillations are isochronous.
PROB. 30. If a body move in an elliptic groove uni-
formly, round two centres of force situated in the foci; prove
that the forces at any point of the ellipse are equal, and
inversely proportional to the square of the corresponding
diameter.
PROB. 31. A body moves in a groove under the action
of two centres of force each varying inversely as the distance,
and of equal intensity at the same distance; the body is
projected from the mid-point between the centres: prove that
if the velocity be uniform the form of the groove is a
lemniscate.
PROB. 32. A body attracted to two centres of force
varying inversely as the square of the distance moves in a
hyperbolic groove, of which the foci are the centres of force:
required to find the pressure on the groove; and to shew that
if the particle begin to move from a point where it is equally
attracted by the two centres, the pressure on the groove is
zero during the whole motion.
CHAPTER IX.
PRELIMINARY ANALYSIS.
412. We now enter upon the calculation of the motion
of a rigid body.
In the following Chapters we shall repeatedly meet with
the expressions
Σ.mz,
Σ.mx,
Σ.my,
Σ.myz,
Σ.mxz,
Σ.my, Σ.mx²;
Σ.mxy,
Σ.mx²,
xyz being the co-ordinates to a particle m of a material
system, and Σ being a symbol which represents that the sum
of the quantities symmetrical with that before which it is
placed is to be taken throughout the system.
It becomes important, then, to enquire whether the axes
of co-ordinates may not be so chosen, as to simplify these
expressions.
PROP. The first three may be simplified.
413.
Let x, y, ≈ be the co-ordinates of the centre of
gravity of the system and let M be the mass of the system.
Then by p. 67, we have
:
Mã, Σ.my=Mỹ, Σ.mx= M≈.
Σ.mx = Mx
If it be allowable in any case to choose for one of the co-
ordinate planes a plane passing through the centre of gravity,
then, supposing this the plane of xy, we have
therefore .mz = 0.
0 and
If it be allowable to choose for the axis of x a line
passing through the centre of gravity, then y = 0,
these give .my = 0, Σ.mz = 0.
0, and
PRELIMINARY ANALYSIS.
399
Lastly, if it be allowable to choose the origin at the centre
of gravity, then ≈ = 0, y = 0, ≈ = 0; and therefore Σ.mx=0,
Σ.my = 0, Σ. mz = 0.
414. The second set of expressions, viz.: . may,
Σ.mxz, 2. myz may be made to vanish by properly
choosing the co-ordinates.
This simplification is so important that the axes which
possess this property are called the Principal Axes of the
system. They are likewise termed the Natural Axes of Ro-
tation for a reason hereafter to be assigned: see Art. 439.
Before proceeding to find these axes we must prove the
formulæ by which we pass from one system of axes to another.
PROP. To prove the formula for the transformation
of one system of rectangular co-ordinates to another, the
origin remaining the same.
415. Let Ax, Ay, A≈ be the original axes (fig. 97),
Ax,, Ay,, A≈, the new axes.
◊ = inclination of plane x,y, to plane wy.
the angular distance of the line of intersection of these
planes from the axis of a; i. e. the angle NAx.
= the angular distance of axis of a, from this line of in-
tersection; i. e. the angle NA∞¸·
xyz, x,y,, the co-ordinates to any point referred to the two
systems of axes respectively.
↑ = the distance of this point from the origin.
Then the cosines of the angles which
2°
* |
makes with the axes of
a
Y
X
xyz, x,y,z, are respectively
y,
Hence
J
グ
​グ
​Y
||
COS XX
2°
ૐ ।
Y
2
|}
2°
cos xx, +
cos yx,
+
Y
cos xy, +
*
cos yy, +
cos y
2°
p
&
a
Y
cos zx, +
cos zy, +
Jo
2°
→ |
COS ដ នី ) •
400
DYNAMICS. RIGID BODY.
Let us now suppose all the points where the six axes
meet a sphere of radius unity described about A to be joined
by arcs of great circles; then we shall have by the formula
for the cosine of the side of a spherical triangle in terms of
the other sides and opposite angle
cos Ꮖ Ꮖ .
cos o cos
+ sin
sin cos
cos xy,
sin
cos
+ cos
sin
cos Ꮎ
COS X Z
sin y sin
cos y x
cos sin
+ sin
cos y cos 0
Ꮎ
cos YY,
sin o sin
+ cos
cos y cos
Ꮎ
cos y z =
cos y sin 6
COS ZX
sin o sin
0
cos zy,
cos o sin
COS ≈
cos 0.
Hence by substitution
x = x, (cos & cos + sin o sin
- y, (sin cos
y = -
-x, (cos o sin - sin
cos 0)
- cos o sin cos 0), sin sin
y
cos cos 0)
+y, (sin & sin
+ cos
cos y cos 0) — ≈, cos sin ✪
z = x, sin & sin 0 + y, cos o sin 0 + ≈, cos 0.
416. In the same manner we should find
x = x (cos cos + sin o sin
cos 0)
-y (cos o sin
- sin
cos y cos 0) + ≈ sin sin 0
&
Y
x (sin cos - cos
sin
cos 0)
+y (sin & sin
+ cos
cos y cos 0) + ≈ cos & sin ✪
Z
y
x sin sin - y cos y sin 0 + ≈ cos 0.
PROP. To prove that in every body there is a system of
rectangular axes, and in general only one system, which will
satisfy the conditions 2.mx,y,=0, 2.mx,z,=0, 2.my,z,=0.
ALL BODIES HAVE THREE PRINCIPAL AXES.
401
417. Substitute in the equations .mx,y,=0, .mx,≈¸=0,
Z.my,x,=0 the values of x,y,, given in Art. 416, and putting
Σ.my,≈
Σ . m (y² + x²) = D, Σ. m (x² + ≈²) = E, Σ. m (x² + y²) = F,
Σ.myz G, Σ.mxx = H, Σ.mxy = K;
=
we have L sin 2
p
+
p
M cos 2 = 0...………….. (1),
N cos o
P sin = 0,
N sin + P cos & = 0,
where L, M, N, P are certain functions of 0, †, D, E, F, G,
H, and K; and are independent of ø.
The first of these equations gives when and ✈ are
from the second and third we have
replace P and N by their values,
known. By eliminating
P=0, N=0: or, if we
we have
sin 20 {D sin² - 2 K sin
√
+ 2 cos 20 {G cos
cos
+ E cos² ↓ − F}·
+ H sin }
Y} = 0
(2).
sin 0 { (D − E) sin
cos
– K (cos² ↓ – sin³ \)}
cos 0 {G sin †
–
H cos }
cos &} = 0
И
Let tanu, and .. sin
√1+u²
cos y = √1+w²
1
also let be eliminated from the above equations by the formula
(1 - tan³ 0) tan 20 2 tan ; and we have, after all reductions,
{(D−E) u − K (1−u²)} {(GD−GF+HK) u−HE÷HF+GK}
- (Gu - H)² (Hu+ G) = 0.
-
This equation, being a cubic, must give at least one real
value of u, and therefore of : and substituting this in one
of the equations (2) we shall have the value of and then
is known from equation (1).
We conclude, then, that we can always find a system of
co-ordinate axes which will satisfy our conditions. But, not
3 E
402
DYNAMICS. RIGID BODY.
only so, there is in general only one such system; for although
we might fancy that there could be three since the equation
in u is a cubic yet this will be found not to be the case when
it is remarked that this equation, which is to obtain the angle
between the axis of a and the intersection of the planes x,y,
and xy, ought likewise to give the angles which the axis of a
makes with the intersections of the two other planes xx, and
yx, with the plane xy. Hence all three roots of the cubic
will be possible and serve to determine the three angles
specified above.
Hence the Proposition is true.
COR. 1. The equation in u becomes identical whenever, in
any particular case, we have G = 0, H = 0, K = 0. In this
case every system of rectangular axes is a system of principal
axes; as is proved by these three equations: and for this
reason the equation in u gives no result.
COR. 2. Again, the equation in u is identical when G=0,
H 0. In this case also there is an infinite number of systems
of principal axes; but they must all have a common axis,
since F does not vanish.
It will be seen that in most cases the difficulty of calcu-
lating the position of the principal axes in a body is great.
But whenever we know one of them the other two are easily
determined, as we shall now shew.
PROP. To find the principle axes of a body when one
of them is known.
418. Let Ax, be the known principal axis, Ax,, Ay, the
others making an angle with the arbitrary axes Ax, Ay
drawn at right angles to Ax, (fig. 98).
Let x,y,,, xyz be the co-ordinates to a particle m referred
to these two systems of axes: then
x, = x cos √ + y sin, y, y cos
Hence Σ.mx,y,= 0 gives
x sin .
y.
(cos² - sin² ) Σ. mxy cos y sin Σ. m (x² - y³) = 0 ;
DETERMINATION OF PRINCIPAL AXES.
403
sin 2
2 sin cos
.. tan 2
cos 2
火
​cos
-
sin
2Σ.may
Σ. m (x² — y´)
Ex. 1.
One principal axis of a rectangular parallelogram
of uniform thickness is perpendicular to its plane through
the centre: required the other two.
Let 2a, 2b be the sides of the parallelogram: M its mass:
the sides parallel to the plane ry, and the centre of the origin:
then the mass of an element M
d x d y
4ab
and therefore
Σ.may
-[[Maydady=
ΜΙ
a
M
N
M
a
Σ.m (x²- y²):
(x²³ b − 1 b³)
4ab
2ab
- a
b
-a
M
ΜΙ
(a³b - b³ a)
(a² — b²) ;
3ab
3
4ab
(x² - y²) d x dy
xydx dy =
0.dx
0,
4ab
- a
tan 24 = 0; and .. 24 = 0 and 180º, or = 0 and 90º, and
the other two axes are parallel to the sides of the parallelogram.
COR. If the parallelogram be a square then ab and
tan 24
0
0
: which shews that in this case any pair of axes
x and y are principal axes.
X
Ex. 2. One principal axis of an elliptic board being
perpendicular to its plane through its centre; the other two
coincide with the axes of the ellipse.
419. The last three of the expressions in Art. 412, viz.
Σ.mx², Σ.my², 2. m², do not admit of much simplification.
The sum of the products of the mass of each particle of
the system and the square of its distance from any straight
line is called the Moment of Inertia of the System about that
line.
404
DYNAMICS. RIGID BODY.
We proceed to prove certain Propositions connected with
the Moment of Inertia.
PROP. The moment of inertia of a system about any
axis is equal to the moment of inertia about an axis through
the centre of gravity and parallel to the former, together
with the product of the mass of the system and the square
of the distance between the two axes.
420. Let the plane of the paper pass through the centre
of gravity G of the system and be perpendicular to the ori-
ginal axis and cut it in A (fig. 99): Ax, Ay the axes of x
and y, and P the projection of any particle of the system m
on the plane of the paper: x, y the co-ordinates of P from G;
the co-ordinates of G from A. Then the moment of
inertia
x y
= Σ. m AP² = Σ . m {(x + x)² + (y + y)³}
=
M (x² + y²) + 2x Σ. mx + 2y Σ. my + Σ . m (x² + y²)
M (x² + y²) + Σ.m (x² + y²): see Art. 413.
M. GA² + moment of inertia about an axis of which the
projection is G.
421. We shall now calculate the moment of inertia in
some particular cases.
Let k be such a quantity that the moment of inertia= M k².
Then it will be seen that k is the distance of the point at which
we may suppose the whole mass collected so as not to alter
the moment of inertia. This quantity k is called the Radius
of Gyration.
We shall always use the symbol k for this radius when the
axis passes through the centre of gravity, and k, (with a sub-
script accent) when it does not.
Ex. 1. A physical line about an axis through its centre
and perpendicular to its length.
2a = length; r = distance of any particle from the centre;
MOMENT OF INERTIA.
405
.. mass of a length dr
... moment of inertia, or M.k²
.. k, or radius of gyration,
-
M
dr
2 a
a
;
["M² dr = MC
-
-α
α
2 a
;
3
If the axis of rotation be at a distance c from the centre
of gravity and parallel to that used above, then
2
α
a³
k +c² by Art. 420.
3
Ex. 2. A circular body of uniform thickness and den-
sity about an axis through its centre and perpendicular to
its plane.
a = radius, ▲ BAP = 0, AP=r (fig. 25);
therefore element of the mass at P M
=
dr.rde
;
па?
a
2π
.. Mk²
=
-[" [" M = drão.
drde
J+3
=
Пач
2M
dr = Mª
0
a²
2
0 0
· ·. k² = ½ a².
For an axis parallel to the above at a distance c,
k² = 1 a² + c² by Art. 420.
Ex. 3. The same body about an axis through its centre
and in its plane.
Mass of element at P = M
drrde
πα
a
2π
23 sin20
Mk²
M
C2π
M
drde
па?
203
2πα
73 (1-cos 20) drde
0
0
M
a
23 dr
=
M
0
4
k² = 1 a².
406
RIGID BODY.
DYNAMICS.
1
About an axis parallel to the above at a distance c,
k² = 1 a² + c².
Ex. 4. A solid of revolution about any axis perpendicular
to the axis of the solid.
Let DA'E be the given axis cutting the axis of the solid
in A': let A' be the origin of co-ordinates (fig. 27): PM = y,
A'M=x: A'A = m, A'B = n, V the volume of the solid;
Μ
.. mass of elementary section PP' = M
mom. of iner. of this element about PP'
пуч
V
dx
May'da 2 (Ex. 3.),
M
DAE
=
V
π у² dx
4
'y²
4
+002
(Art. 420);
n M
πυ
V
MW? = ["
· Mk
y²
k? = [ " = = y² († + a) da : but V =
4
.. k² = √m² ( — y² + x²y²) dx ÷ fm y²dx.
Ex. 5. A sphere about a tangent.
2
-2a
.. k‚² = ƒª²ª (a²x² + ax³ − ‡ x¹) d x
8
0
24
2 a
n
=
πу³dx:
m
y² = 2 ax x²;
Co
10
÷ ƒª¹ª (2 a x − x²) dx
a5
15
28 a² ÷ 3 = } a².
2
5
— a².
(3 + 4 − 2 4) a³ ÷ ( 4 − 3 ) a³ =
-
2
Also k² = k² - a² =
PROP. To find the moment of inertia of a system
referred to any axis.
422. Let AC be the axis (fig. 100): P any particle m
of the system: PM perpendicular to AC: Ax, Ay, Az the
co-ordinate axes: x, y, z co-ordinates to P; a, B, the angles
AC makes with the axes;
γ
MOMENT OF INERTIA.
407
.. PM² = AP² sin² PAC =
= 2,2
202
AP² (1 - cos² PAC), AP = r
Y
²/
-+-+² (cosa + cos B + cos)
ጥ
2
x² + y² + x² − (x cos a + y cos ß + ≈ cos y)²
x² sin² a + y² sin² ß + x² sin²
Υ
2xy cos a cos ẞ - 2 x z cos a cos y - 2yz cos ẞ cos y.
Hence moment of inertia
sin² a Σ. ma² + sin ẞ E. my²+ sin² y Σ.mx²
-2 cosa cosẞ.mxy-2 cosa cosy.mxz-2 cosẞcosy Z.myz.
If the axes of co-ordinates be principal axes, then,
accenting the co-ordinates in accordance with the notation.
of Art. 417,
Mk² = sin² a Σ.mx² + sin² ßΣ.my, + sin² y Σ. mx².
Let A, B, C be the moments of inertia of the system
about the principal axes;
2
... A = Σ . m (y,² + ≈²), B = Σ . m (x² +≈²), C = Σ . m (x²+y,³),
then Σ.mx2 =
+ 1
2
( B + C - A), Σ.my,² = 1 (A + C − B),
Σ . m≈² = 1 (A + B − C) ;
.. Mk2 A (sin² ß, + sin y, - sin³ a)
Mk² = 1
½ B (sin² a, + sin² y, - sin² ß,) + ½ C (sin³ a, + sin² ß, – sin³ y,)
= A cos² a, + B cos² ß, + C cos² Y.
PROP. If A and C be the greatest and least principal
moments, then every other moment of inertia is intermediate
to these.
423.
2
For Mk = A - (A – B) cos² ß, ‒ (A – C) cos²
γι
and also = C + (A − C) cos² a, + (B − C) cos² ß,
since cosa, + cos² ß,
a, + cos² ß, + cos² Y = 1.
408
DYNAMICS. RIGID BODY.
The first is evidently less than A, and the second greater
than C.
PROP.
When two of the principal moments are equal
to each other, the moments about all axes lying in any
right cone described about the principal axis of unequal
moment are the same.
424. For let B = C: then
Mk2= A cos² a, + B (cos B, + cos y) = A cos² a, + B sin² a,,
==
and this is constant when a, remains the same although B,
and y, may vary.
PROP. If the three principal moments be equal to each
other, every other moment is equal to these.
?
425. For Mk = A (cos² a, + cos² ß, + cos² y) = A.
PROP. To find the points in a system with respect to
which the principal moments are equal to each other.
426. Let the centre of gravity be the origin, and the
principal axes the axes of co-ordinates:
ay,
x,y,, co-ordinates to any particle m,
the point which gives the principal moments equal: then
from this point the co-ordinates of m are
x-x',', y, -y'', ≈, -8';
by Art. 417. . m (x, − x') (y, − y) = 0,
Σ. m (x, − x') (≈, - x) = 0, and . m (y, − y) (≈, − ≈,) = 0.
·
Observing the origin and axes we have chosen, we see that
these conditions become,
Mx'y' = 0, Mx'x' = 0, My'x' = 0;
.. two of xyx must = 0.
Suppose y=0,x=0 and then remains indeterminate.
a
MOMENT OF INERTIA.
409
Hence by Art. 420,
moment about axis parallel to r, through (x,y,z) = A
y
Z
and these by hypothesis are all the same;
.. B = C, and a
A - B
10
M
Hence we derive the following corollaries.
=B+M x2
12
=C+Mx²,
12
1. If all the moments about the principal axes through
the centre of gravity be unequal, there is no point in the
system with respect to which the moments are equal.
2. If two of them be equal and the moment of the
unequal one be the greatest, there are two points equally
distant from the centre of gravity and on the axis of the
greatest moment corresponding to which the moments are all
equal.
X
3. When the principal moments are all equal, r = 0,
and there is no point but the centre of gravity with respect
to which the moments are all equal.
3 F
CHAPTER X.
MOTION OF A RIGID BODY ACTED ON BY FORCES OF FINITE INTENSITY.
427. IN considering the equilibrium of a rigid body
(Art. 27) we stated, that, in consequence of our ignorance of
the nature and laws of the forces by which the molecules are
held together, we are unable to deduce the conditions of
equilibrium of a body from those of a single particle. By
the aid, however, of the principle of the transmission of force
through a body (Art. 28) we deduced certain relations which
the impressed forces, that act upon the body when in equi-
librium must satisfy independently of the molecular forces.
It is evident that the system of molecular forces are them-
selves in equilibrium independently of the other forces which
act upon the body.
In considering the motion of a rigid body we fall upon
the same difficulty. We know nothing of the laws of the
molecular forces, and consequently cannot calculate the motion
of the body by calculating the motion of its molecules sepa-
rately. But
we may surmount this in the manner
overcame the difficulty just mentioned.
we
Let mX, my, mZ be the impressed moving forces which
act upon the particle m, not including the molecular forces
which act upon this particle. Let xyz be the co-ordinates
to m at the time t: then m
effective moving forces of m
ď² x
d t
,
day
d²
z
m
m
are the
dt
dt
(Art. 211).
Now by the first of the general principles enunciated
in Art. 226, the forces
d' z
m (x-da), m(x-dy), m (2-15)
t2
GENERAL EQUATIONS OF MOTION.
411
acting on m parallel to the axes of a, y, z respectively, and
similar forces acting on all the other particles ought, together
with the molecular forces by which the particles of the body
act upon each other, to satisfy the equations of equilibrium of
forces acting on a rigid body.
But the molecular forces are of themselves in equilibrium,
since the molecules retain their relative situations during
the motion.
z
Hence the forces m (X-), m (Y-y), m (2-1)
acting on m and similar forces acting on the other particles of
the body ought to satisfy the six equations of equilibrium of
forces acting on a rigid body, given in Art. 65. Wherefore
we have the six equations of motion
ďa
2. m (X
dt
Σ.»
y
2-
d²
=0, 2. m (Y - 1) = 0, 2. m (Z)-c
d²
==
(2
Y
-
y
1/2))}
-0.
0,
5. m {y (2-1) x (1
X-
df
dx
2. m = (X - ( ) - x ( 2 -
dt
df
ď z
( )} = 0,
df
dt
= 0,
Σ.m
{
(
d'y
dť
心
​ď²x
yr
0.
dť
By these six equations we shall be able to calculate the
motion of a rigid body acted on by any forces of finite in-
tensity. They lead immediately to two Principles, one of
which enables us to calculate the motion of translation of
the body in space; and the other the motion of rotation.
PROP. The motion of the centre of gravity of a body
moving free in space and acted on by any forces is the same
as if all the forces were applied at the centre of gravity
parallel to their former directions.
428. By the first three equations of Art. 427,
ď v
Σ.η
m X
= 0, Σ.m
dť²
Σ.m()
dy
Γ
0, .m2.
ď z
= 0.
dť
dt
412
RIGID BODY.
DYNAMICS.
Let xyz be the co-ordinates to the centre of gravity,
x'y' z'
m from the centre of gravity;
2
· · x = x + x', y = ÿ + y, z = z + z' ·
y
Now Σ.mx = 0, Σ.my = 0, E. mx 0 (Art. 413).
Hence, substituting for xyz, the above equations give, M
being the whole mass of the body,
d²x
ď x
Σ.mX ď²y Σ.my
dz
Σ.mΖ
dt²
M dt2 M
dt
2
M
and these are the equations we should obtain for the motion of
the centre of gravity supposing the forces all applied at
that point. Hence the Proposition is proved.
PROP. The motion of rotation of a body acted on by
any forces and moving freely is the same as if the centre
of gravity were fixed and the same forces acted.
429. The last three of the equations of Art. 427 are
Σ.myZ
2-
ď z
dť
ď y
Z
Y-
ი.
dt²
Σ.m
m { x (
ď² x
X-
2?
dt
に
​d² z
Z-
= 0,
dt
{
d'y
Σ.m/xY_
-yX
dt2
(*
ď x
= 0.
d t²
Now let x, y,
gravity, and let (as
be the co-ordinates to the centre of
before) x = x + x', y =ÿ+ y, z=2+z′.
Let these be put in the above equations, observing that
Σ.mx'=0, Σ.my' = 0, Σ.mx'=0 (Art. 413), and that there-
fore the differential coefficients of these with respect to t
vanish; also bearing in mind the equations of last Article we
have after all reductions,
MOTION OF ROTATION ABOUT THE CENTRE OF GRAVITY. 413
dz ૪
Σ.myZ
df
Y
(r
-
d² y'
dť
ď²
= 0.
2. m x ( X - 1 x ) = x ( 2 - 1 x )} = 0.
dť
- Z dť
d² x'
2. m (Y -
Σ.m r Y-
1)
-
(x-2)=0.
dť
But these are precisely the equations we should have
obtained on the supposition that the centre of gravity were
fixed, and that point taken as the origin of moments.
Hence
the Proposition is true.
430. From the first of the Principles demonstrated
in the last two Articles we gather, that all the calculations
we have made of the motion of a material particle will be
true also of the centre of gravity of a rigid body. It remains
then to ascertain the motion of the other parts of the body
relative to the centre of gravity: and this the latter Principle
enables us to accomplish, as we shall shew in the following
Chapters. We shall consider the motion of rotation of a
body first about any fixed axis, either passing through the
centre of gravity or not, and lastly about a fixed point.
CHAPTER XI.
MOTION OF A RIGID BODY ABOUT A FIXED AXIS: FINITE FORCES.
PROP. To calculate the angular accelerating force of a
rigid body moving about a fixed axis, and acted on by any
given forces.
431. Let the fixed axis be taken as the axis of ≈, and
let ay be the co-ordinates to the projection of a particle m on
the plane ay also let r be the distance of m from the axis of
rotation and the angle makes with the plane ≈: then
x = r cos 0, y = r sin 0.
v
Now by Art. 68, we are to take only the last of the equa-
tions of Art 427;
=
ď² O
dť
•. Σ. m { x
{
d³y
ď² x
Y
= 2. m (x Y-yX).
dt
dť
dx
do
dy
d Ꮎ
But
r sin e
= r cos e
d t
dt
d t
dt
d² y
d² x
d
dy
da
d
do
X
Y
dt
- y´dť²
X
У
dt
d t
dt
-y a t
dt
dt
dt
de
Hence Σ.mp²
dť
d Ꮎ
or, since
d t²
Σ . m (x Y − y X),
is the same for every particle,
d Ꮎ Σ . m (x Y − y X)
dť
Σ.mr²
moment of the forces about the axis
moment of inertia about the axis
By integrating this equation we shall know the angle through
which the body has revolved in a given time; and shall con-
COMPOUND PENDULUM.
415
11
sequently be able to determine the position of the body at any
instant.
PROP. A body moves about a fixed horizontal axis acted
on by gravity only: required to determine the time of a small
oscillation.
432. Let ABC (fig. 101.) be a section of the body made
by the plane of the paper passing through the centre of gravity
G and cutting the axis of rotation perpendicularly in C; P the
projection of any particle m on this plane; CX vertical; GH
perpendicular to CX; CG CP=r; PCX=0'; GCX=9.
h;
d Ꮎ
moment of forces
Then by Art. 431,
dť
moment of inertia
Σ. mgr sin '
Mgh sin (
Σ.mp
M (k² + h²)
gh
k² + h²
sin . Arts. 413, 420.
kc + h²
If we put
7, multiply by 2
h
d Ꮎ
dt
and integrate;
do
dt
2g
Z
g
cos + const. =
- § (a² – 9³), neglecting 9ª….... and
7
supposing a at first;
=
.. time of oscillation
80
a
d Ꮎ
= π
u
√ a² - (²
Hence the body will move as if collected in a material
k² + hⓇ
k² + h²
point at a distance
from the axis. Take CO
1.
h
in the line CG produced: then O is called the centre of oscil-
lation: and
k² + h²
h
is called the length of the isochronous
simple pendulum, the body itself being denominated, in con-
tradistinction, a compound pendulum. The point C is called
the centre of suspension.
PROP. The centres of oscillation and suspension are
reciprocal: that is, if the body be suspended on an axis
through O parallel to that through C, then C will be the
centre of oscillation.
416
DYNAMICS. RIGID BODY.
433. For let l' be the length of the simple pendulum in
this case; then
k² + OG² k2
で
​+l-h
OG
l-h
lh -- h²
+l-h (Art. 432.) = l.
l - h
From which the truth of the Proposition is evident.
PROP. To determine the length of the seconds pen-
dulum experimentally.
434. We have already shewn (Art. 396.) that if I be the
length of a simple pendulum, that is, a pendulum consisting of
a single particle suspended by a string without weight, t the
duration of each oscillation and g the force of gravity, then
t
= π
VI
50
The
But it is impossible to form a pendulum which may, with due
regard to accuracy, be considered a simple pendulum. It
becomes necessary, then, to measure the distance between the
centres of suspension and oscillation (see Art. 432).
practical difficulties in the way of determining the latter point
were considerable, and such as greatly to endanger the accu-
racy of the result, before Captain Kater removed the sources
of difficulty by using the property of the compound pendulum
proved in Art. 433, namely, that the centres of oscillation and
suspension are reciprocal. We proceed to explain this.
Let AB be the pendulum (fig. 102); C the point of sus-
pension; F a weight which may be shifted from one position
to another on the pendulum: O the centre of oscillation of
the pendulum including F.
The position of O is first found pretty accurately by
making the pendulum oscillate about C and O till the times.
of oscillation are nearly the same. Knife edges are then fixed
at C and O, and the weight F, which is placed near the middle
point between C and O, is shifted till it is found that the
KATER'S COMPOUND PENDULUM.
417
It
time of oscillation about C and O is exactly the same.
remains only to measure CO and observe the time of oscilla-
tion. For the details of the experiment we refer the reader
to the Philosophical Transactions for 1818. If t be the time
of oscillation in seconds and CO = l, then, since the length of
the simple pendulum varies as the square of the time of
oscillation, the length of the seconds pendulum
だ
​PROP. To calculate the effect produced on the pendulum
by shifting F.
435. Let l' be the length of the simple pendulum when
F is removed: M (1+n) and M the masses of the pendulum
with and without F, n being a small fraction: let be the
length of the simple pendulum when F is so situated that
the times of oscillation about C and O are the same: and let
L and L' be the lengths when the pendulum oscillates about
C and O, the weight F being then at a distance from C:
and let & be the value of a when Ll. Then, by
Art. 432,
1
+
square
of rad. of gyration about axis of suspension
dist. of centre of gravity from same axis
Ml' h + Mn({ l + d)²
l' h + n ( ½ l + d) *
Mh + Mn (l + d)
... l'
=
h + n ( l + d)
1 1 /
ጎ
1 + — ({ l² −
d³ ) .
h
d L
dx
Also L
l' h + n x²
h + n xx
n²x² + 2nhx – n l'h
(h+n)
2
;
n² (x − a) (x + B)
(h + na:)²
1
where a =
√ h² + n l' h − h}
Դ
h fnl nl r
Jn
n2 h
ľ
n l'"
neg. n°..
sh
2
Sh
3 G
418
DYNAMICS. RIGID BODY.
||
n de
2
2h
B is a positive quantity.
=
Let CD = DO 1: and take CP = a. Then if F be
below P (that is, a greater than a), the time of oscillation
about C will increase or decrease according as F is shifted.
from or towards C, since dL and do have the same sign: the
contrary will be the case when F is placed above P.
In like manner if the pendulum be suspended from 0,
we have a point Q, the distance of which from O equals
n 82
(h' being the distance of the centre of gravity from
2h'
O), such that when F is beyond Q from O the time of oscil-
lation about O is increased or diminished according as F is
moved further from O or nearer to it; and vice versâ.
Since DP
nde
2h
and DQ
n sz
2h*
and these are both less
than 8 (8 being by hypothesis a very small quantity), it follows
that F cannot be between P and Q when the times of oscilla-
tion about C and O are the same.
PROP. To shew that if the axes of suspension be equal
cylinders rolling on horizontal plates, instead of knife edges,
the length of the simple pendulum still equals the distance
of the axes.
436. Let AB be the pendulum (fig. 103); G its centre
of gravity, O its centre of oscillation, CDE the semi-cylin-
drical axis of suspension, C being the point of contact with the
horizontal plane of support when the pendulum hangs in its
position of rest: P the point of contact at the time t, when
the pendulum oscillates; CM = x, MG = y, the co-ordinates to
G, O the angle CG makes with the vertical, R the pressure at
P, F the friction on the plane of support, CG = h, M_the
mass of the pendulum, CO = l, k = rad. of gyration about G,
a = rad. of the axis at C.
Now by Art. 428 the motion of the centre of gravity is the
same as if all the forces were applied at that point;
KATER'S COMPOUND PENDULUM.
419
d² x
F
d²y
R
(1),
(2).
dt
M
d t
df
M
Also by Art. 429 the motion of rotation is the same as
if G were fixed; hence by Art. 431
d² 0 Fy - R (a + h) sin (
d t
Mk2
(3)
we have here three equations and five unknown quantities R,
F, x, y, 0: we must seek, then, two relations connecting
x, y, 0: these are
X = PM - PC = (a + h) sin ( − að ... (4)
A
y = (a + h) cos ◊ a.. (5).
By equations (1) (2) (3) we have
d Ꮎ
k2 + y
d t
ď² x
df
+ (a + h) sin 0 (g
ď²y
= 0;
dť
differentiating (4) and (5) we have
dx
dy
(a + h) cos
Ө-
a = = y,
· (a + h) sin 0.
de
d Ꮎ
Hence our last equation becomes
ᏧᎾ dᎾ dx d'x
dy day
d Ꮎ
12
+
+
+ (a + h) g sin
= 0;
dt dť
dt de
dt dť
dt
de dx²
dy³
+
dť dť²
+
df
C + 2 (a + h) g cos &
when 0 = a
= a, velocity
velocity = 0;
d 02
dx²
dy
... k2
+
+
dt2 dt
d t
d 02
dt²
= 2 (a + h) g (cos - cos a);
{k² + (a+h)² + a² −2 a (a+h) cos 0} = 2 (a+h) g (cos 0-cosa)
d Ꮎ
(a + h) g (a² −0²)
d ť²
k² + h²
neglecting powers of a and higher than the square.
420
DYNAMICS. RIGID BODY.
dt
dᎾ
k² + h²
1
(a + h) g √ a² − 0²
k² + h²
1
t =
COS
(a + h) g
Ꮎ
α
.
time of oscillation = π
const. = 0;
k² + h²
(a + h) g
Also if b be the radius of the axis at O, and if CO = m,
then
time of oscillation about O
= π
√
k² + (m − h)²
(b + m − h) g
and these times being equal, we have
k² + h²
k² + (m − h)²
-
l;
a + h
b + m - h
--
.. l (a + h) — h² = (k² =) bl + (m − h) l − (m − h)²;
−
(m − h)² – h²
-
m (m-2h)
... l
-
m 2 h + b
-
a
m
A
2h+b- a
If b = a, l = m; that is, the length of the simple pen-
dulum equals the distance between the axes, when the cylinders.
are of equal radii.
437. Mr Lubbock has calculated, in a Paper read before
the Royal Society in 1830, the errors in the length of the
simple pendulum corresponding to given deviations of the knife
edges. It is there shewn that a small deviation of one of the
knife edges in azimuth is quite insensible: but that this is not
the case for a small deviation in altitude: a deviation of one
degree increases by 3 the vibrations of a seconds pendulum
in 24 hours. A deviation from horizontality in the agate.
planes has a still greater influence for a deviation in hori-
zontality of 10' increases by about 6 the vibrations in 24 hours.
PROP.
When a body moves about a fixed axis, required
to find the pressure upon the axis at any instant.
រ
PRESSURE ON A FIXED AXIS.
421
438. We shall suppose that the axis is fixed at two
given points: let the axis of rotation be the axis of ≈, and
let a and a be the distances of the fixed points from the
origin: let P, P' be the pressures at these points, aßy and
a'B' the angles which their directions make with the axes
of xyz respectively: X, Y, Z the impressed accelerating forces
of the particle m, the co-ordinates of which are xyz at the
d² x ď² y ď² z
the effective accelerating
dť' de' dť
time t; and therefore
forces of m: but since the angular accelerating force about
the axis of rotation is calculated in Art. 432, we shall trans-
form these effective forces as follows. Let f be the effective
angular accelerating force, w the angular velocity of the body
at the time t, r the distance of m from the axis of rotation,
✪ the angle which r makes with the plane xx; then a=r cos 0,
= r sin 0; differentiating twice with respect to t and
does not vary with the time, and then re-
and y
observing that
placing x and y, we have
ď²x
yf- xw²,
dt
d²y
dť
= xf-yw².
Then the moving forces m (X+yƒ+xw³), m (Y−xf+yw³),
mZ acting parallel to the axes on the particle m, and similar
forces acting on all the other particles, together with the
pressures P, P' on the two fixed points of the axis ought to
be in equilibrium at the time t, according to the first Principle
of Art. 226. Hence by Art. 65,
P cos a + P' cos a' + Σ . m (X + yƒ + xw²)
ά
= 0
P cos ẞ+ P' cos B' + . m (Y - x ƒ + ywˆ) = 0
P cos y + P' cos y + Σ. m Z
= 0
- Pcosß. a - P'cos B'. a' + E. m {Zy − (Y − x ƒ + yw²) ≈ } = 0
Pcos a. a+ P'cos a'. a'
+ Σ. m {(X + yƒ + xw³)≈ − Zx} = 0
Σ. m {(Y_ xf+yw²) x − (X+yf+x w³) y} = 0.
These equations may generally be much simplified in
applying them to any particular case, as we shall sec in the
422
DYNAMICS.
RIGID BODY.
Chapter of Problems on this subject. The first, second,
fourth, and fifth equations determine the four quantities
Pcos a, Pcos ß, P' cos a', P' cos B'; from which the pressures
perpendicular to the axis may be obtained. The third equa-
tion is the only equation which contains Pcosy and P'cos y',
and it shews that these quantities are indeterminate but that
their sum must -Σ.m Z. Lastly, the sixth equation is
independent of the pressures, and, in short, determines the
motion as calculated in Art. 432: this is easily seen, since
the equation by reduction becomes
f. E. m (x² + y²) = Σ . m (Y x
Xy).
The following Proposition is an application of these equations.
PROP. The principal axes through the centre of
gravity are permanent axes, when the body is not acted on
by any forces.
439. An axis is said to be permanent when the body
permanently revolves about it when it is not fixed.
Let us suppose the body moves about a fixed principal
axis. Since no forces act upon the body it follows that
X, Y, Z each vanish, hence the equations of last Article
become (since the sixth gives ƒ = 0)
P cos a + P' cos a' +
Pcos B+ P' cos B' +
P cos
cos y + P'
"cos y'
w² Σ.mx = 0
w² Σ. my = 0
0
- Pa cosẞ- P'a' cos ẞ' - w². myz = 0
Pa cosa + P'a' cos a' + w² Σ.mxz = 0.
Since the axis of ≈ passes through the centre of gravity,
therefore .mx = 0, Σ.my = 0 (Art. 413): also if the other
two principal axes xy, make each an angle
of xy respectively at the time t, we have
with the axes
x = x¸ cos &+ y, sin o, and y = y, cos & − x, sin 8, ≈ = ;
.. Σ.mxz = cos pΣ. mr,x, + sin 2. my,≈, = 0 ;
THE PRINCIPAL AXES ARE PERMANENT AXES. 423
so also .myz = 0.
Hence the equations become
Pcos a+ P'cos a=0, P cos ẞ+P'cos B'=0, Pcosy + P'cos y' =0,
Pa cos ẞ+ P' a' cos B' = 0, Pa cos a + P'a' cos a' = 0,
these give P = 0 and P' = 0.
Hence there is no pressure
on the fixed axis, and therefore it would not move if the body
were to rotate about it when it is not fixed.
CHAPTER XII.
MOTION OF A RIGID BODY ABOUT A FIXED POINT: FINITE FORCES.
440. IN calculating the motion of a rigid body about
a fixed point it is found most convenient to transform the
equations of motion so as to contain angular co-ordinates and
angular velocities.
ω ω
Let the axes of co-ordinates be drawn through the fixed
point and suppose that w'w"w"" are three angular velocities.
such that if they were simultaneously impressed upon the body
about the axes xyz respectively at the expiration of the time t,
the motion of the body shall be what it actually is; then these
are called the angular velocities of the body about the axes at
that instant.
We shall always estimate those angular velocities positive
which make the body revolve from the axis of a to the axis
of y about; from y to ≈ about x; and from ≈ to r about
and those negative which act in the opposite directions.
z
Y:
When the axes of co-ordinates are principal axes we shall
use w₁w2w3 for w'w'w'
///
PROP. To find the linear velocities, parallel to the axes
of co-ordinates, of any particle of the body in terms of the
angular velocities about the axes.
:
441. Let xys be the co-ordinates to particle m at P
(fig. 104) draw PM perpendicular to the axis of x: PN per-
pendicular to plane xy: then at the time t the velocity of m
about the axis of x = w'PM: resolving this parallel to the axes
of y and ≈ and reckoning those linear velocities positive which
tend from the origin, and vice versa, we have
INSTANTANEOUS AXIS,
425
vel. of m arising from w' parallel to y=-wPM sin PMN = - w'≈
≈= 'PM cos PMN = w'y,
also velocity of m arising from w" parallel to x =
ર
w" z
w"x,
w Y
y =
w""x.
velocity of m arising from "" parallel to x
W
Adding together those velocities which are parallel to the
same axes, we have
ដ
velocity of m parallel to a = w"≈ - w″"y,
• Y = w x
w' 29
z = w' y - w" x.
If m be at rest at the instant of expiration of the time t
these expressions vanish; the third is a necessary consequence
of the other two.
W
w"
Hence x = ≈, Y
≈ are the equations to a straight
W
W
line through the fixed point which is at rest at the instant
under consideration.
This line is called the Axis of Instantaneous Rotation.
PROP. To find the position of the instantaneous axis
at any instant.
442. Let aẞy be the angles which this line makes with
the axes of xys at the proposed instant: then by fig. 104,
w'
cos α =
a
AM
AP
X
x² + y² + ~²
12
w²² + w
112
1712
+ w
"/
MN
cos B =
AP
Y
x² + y² + x²
√ w² + w
12
112
+ w
PN
W"
COS
Y
AP
x² + y² + x²
112
1119
@-tw
+ w
3 H
426
DYNAMICS. RIGID BODY.
By means of these we shall know the position at any instant
when w'w"w"" are known.
PROP. To find the angular velocity of the body about
the instantaneous axis.
W
443. Let
Let w be the required angular velocity: the
distance of the particle m from the origin: then the distance
of this particle from the instantaneous axis
= r sin ( < between r and inst. axis) = r
1- cos (same <)
√ x² + y² + x² - (x cos a + y cos ß + ≈ cos y)²;
y²+
-
.. the velocity of m=w√ x²+y²+z² − (x cos a+y cos ß+≈ cos y)².
But by Art. 441 the whole velocity
=
2
√(w"x-w""'y)² + (w""'x
w'"'y)² + (w''' x − w'≈)² + (w'y − w″x)².
Let us substitute for ww" w'
ω
then whole velocity
1112,
12
√ w " + w¹¹² + w
112
√w² + w
12
112
2
///
in terms of aẞy by Art. 442,
(≈cos B-ycosy)²+(x cosy−zcosa)²+(y cos a−x cos ß)"
+ω
///2
x²
+ y² + ≈² − ( x cos a + y cos ß + ≈ cos y)².
Hence by equating these expressions,
ω
Vo
12
W = √ w +ω
112 + w
+w
this is the angular velocity required.
444. Cor.
7/12
If a body revolve about an axis with an
angular velocity w, then the resolved part of this about another
axis inclined to the former at an angle a
=
12
112
11/2
(w' = √ w~t w + w cos a =) w cosa.
PROP. To find the inclinations of the instantaneous
axis to the principal axes.
α
445. Let a,ẞy, be the angles the instantaneous axis
makes with the principal axes, and www the angular velo-
cities about the principal axes.
z
VELOCITY ABOUT THE INSTANTANEOUS AXIS.
427
cos a cos x¸x + cosß cos x, y + cos
cos xx
... cos a,
///
w
w"
(
COS X X +
0,x
cos x y +
cos, (Arts. 443, 444.)
ω
ω
W
WI
ω
since by resolving the angular velocities
w'w"w"" about the axis of x, we have by Art. 444,
ωω
w"
w₁ = w' cos x¸x + w″ cos x, y + w
///
COS V Z.
W3
W2
Similarly cos B,
COS Y,
W
ω
2
112
1112
COR.
Also w+w
+ω
=
2
w² = w₁² + w;² +w3².
PROP. To obtain equations for calculating the angular
velocities about the principal axes at any instant.
446. Let A, Ay, A≈ be the axes of co-ordinates fixed
in space; A being the fixed point of the body;
Ax,, Ay,, Ax, the principal axes in the body.
Then the three equations of rotatory motion are by Art. 427.
=Σ.m {yZ-≈1}= L suppose
d²
d² y
Σ.my
df
dt's
Σ.m.
d² x
dt²
X
d²
d t²
Σ.m
{
ď y
ď x
W
- Y
dt
dt
Now by Art. +kl.
=
Σ . m {z X − x Z} = M .....
= Σ . m {x YyX} = N .......
dx
dy
dz
w" - w Y
= w
x - wz,
w'y – w″x.
dt
dt
d t
By differentiating these with respect to t
ď x
dz
dť
dt
#
dy
dw"
do"
+
-Y
d t
dt
dt
428
DYNAMICS. RIGID BODY.
SO
d2y
(w''² + w'¹¹²) x + w' w" y + w 'w" z +
W
(w²² + w ""'²) y + w "w' x + w″ w
w"
dw"
dt
મેર
dw""
dt
y,
dw"" do
≈ +
X
I
ર
d t²
dt
dt
d² z
do'
dw"
1/2
(w²² + w¹¹²) ≈ + w" w' x + w""w" y +
Y
x.
d t
dt
dt
ď² z
d² y
Hence Σ.m Y
=
(w'''² – w″²) Σ. myz
dt2
d ť²
dw
w""w
Σ.mxy
dt
(w" w
dw
ωω +
Σ.mxz
dt
dw
+ w
w" Σ.my² – w"w"" Σ.mx² +
Σ . m (y² + x²) = L ... (1).
dt
Now suppose the fixed axes Ax, Ay, Az were so chosen
that at the instant of expiration of the time t the principal
axes should coincide with them. Then at this instant
Σ.mxy=0, Σ.mxx=0, Σ.myz=0: also w'=w1, w" = w₂, w""=w3 ;
and likewise
d w
dt
dwi
dt
for the changes in the two angular
x
velocities w' and w₁ during a given small time after the axis
of x,
coincides with the axis of a will differ only by a quantity
which depends upon the angle passed through by the axis of
during that given small time: the difference between w' and w
will therefore be an infinitessimal of the second order and
therefore their differential coefficients will be equal.
equation (1) becomes at this instant
Hence
dwi
w₂ w3 Z. m (y² – 2,3) +
Σ . m (y + ≈ *) = L,
dt
the letters with subscript accents having reference to the prin-
cipal axes.
Now this equation is independent of the epoch from which
the time is measured: it is also independent of the angles which
CALCULATION OF THE POSITION OF THE BODY.
429
the principal axes make with the fixed axes in space. It follows,
then, that this equation will hold for every instant of the
time t; and is therefore generally true.
–
-
Now Σ.m (y²+) = A; and Σ. m (y,² − x²) = C − B ;
.. A
d wi
dt
+ (C − B) w2w3 = L
d w2
-
+ (A − C) w₂ w₂ = M
(2).
similarly B
dt
d w3
C
d t
+ (B − A) w₂ w₂ = N
By means of these three equations the three quantities
wwwę must be determined.
PROP. To determine the position of the body in space
when the angular velocities about the principal axes are
known.
447. We consider, as before, those angular velocities
positive which tend to turn the body from the axis of x, to
the axis y, about ≈,, from y, to z≈, about a, and from ≈, to x,
about y
Also by Art. 444 an angular velocity is resolved about
any new axis by multiplying it by the cosine of the angle
between the axes.
Now the position of the principal axes of the body at the
time t, is determined by the values of 0, p, y, these angles
being measured as explained in Art. 415: it follows, then, that
de do dy
w] w₂ wз must be functions of 0, 4, †, and dt dt dt
d Ꮎ
The resolved parts of
about the axes of x, y, z, are
dt
d Ꮎ
d Ꮎ
cos o,
sin, 0,
dt
dt
430
DYNAMICS. RIGID BODY.
аф
the resolved parts of
about these axes are
dt
аф
0,
0,
dt
d f
and the resolved parts of
about these axes are
dt
dy
αψ
d↓
COS ZX
cos zy,
cos ZX,,
לן
dt
dt
dt
dy
αψ
dy
or
sinosin 0,
cos sine,
cos (see Art. 415).
dt
dt
dt
Hence, adding those about the same axes,
Ꮷ Ꮎ
αψ
Οι
w₁ =
cos o
sin sin 0,
d t
dt
d Ꮎ
αψ
W2
sin
cos o sin 0,
dt
d t
аф
dy
W3
Wz =
cos 0.
dt
dt
In these we must substitute the values of w, W2 W3
obtained
by integrating the equations in Art. 446, and we shall find
0, p, 4, and so determine the position of the principal axes,
and consequently of the body, at any proposed instant.
448. COR. 1. By the above equations we obtain
do
dt
= w₁ cos & — w., sin p
Ф
d↓
sin
W1
sin
W₂ cos &
d t
аф
= W3
d t
sin
449.
cos
(w, sin &+w, cos P).
COR. 2. When is very small these become
ᏧᎾ
1
= w₁ cos o p - w…
w, sin q
аф
dt
MOTION ABOUT A POINT WHEN NO FORCES ACT.
431
αψ
W2
sin
cos &,
dt
Ꮎ
Ꮎ
W2
аф
WI
= W3
dt
1
sin
cos 0.
Ꮎ
Ꮎ
PROP. A body revolves about its centre of gravity
acted on by no forces but such as pass through that point:
required to integrate the equations of motion.
450. The equations (2) of Art. 446 become in this case
A
dwi
dt
+ (C − B) w½ w3 = 0,
d we
B
+ (A − C) w₂ w3 = 0,
dt
dw3
C
+ (B − A) w₁ w₂ = 0,
dt
Multiply these equations by w w w
the principal axes being drawn through the centre of gravity.
w₂ wз
respectively and
add; then
dwi
d we
dws
+ B w₂
Βω
+ C @ 3
= 0
dt
dt
add;
Aws dt
A w₁² + B w₂² + C wz² = constant = h².
Again multiply the equations by Aw₁, Bw, Cw, and
dwr
dw
dw
...
A² w1
+ B² w₂
+ C² w3
=
= 0;
W2
dt
dt
dt
2
2
.. A² w₁² + B² w₂² + C²² wz² constant k².
Α ως
2
Eliminating 3 from these two equations, we have
W3
2
2
A (A - C) w₁² + B (B − C) w₂² = k² – Ch² ;
and w32
1
B (B-C)
1
C (CB)
2
{k² - Ch² - A (A - C) w₂²},
{k² – B h² – A (A – B) wi²}.
432
RIGID BODY.
DYNAMICS.
Hence the first of the equations of motion gives
dur + |
dt
2
-
✓√(4−C') (4−B) k² − C h² \ [ k² − B h²
A (A – C)} \A (A – B)
ω
2
ω
the integral of this equation, which in the general case cannot
be found, will give w₁ in terms of t and then we and we will
be known.
w,
W2
W3
Knowing w₁w2ws the position of the body at any time is
determined by integrating the equations of Art. 447.
PROP.
When the body is acted on by no forces except
such as pass through the origin, there exists a plane to
which it may be referred, which plane is invariable in
position.
451. Let
abc be the cosines of the angles which x makes with a,y,,
a'b'c'
a"b"c"
Y
23
We shall now seek the values of the differential coefficients
of these with respect to t.
Let the planes yx, yx, cut in the line AK (fig. 105):
then this line is perpendicular to the plane x Ax,: let AI be
the instantaneous axis: describe a sphere about A of radius.
unity and cutting the axes of co-ordinates, AK, and AI in
the points marked in the figure.
Then
d. x A x
dt
= the angular velocity about AK
= wcos IAK, (Art. 444.)
w (cos a¸ cos Kx¸ + cos ẞ, cos Ky, + cos y, cos K≈)
= (cos ẞ, sin K≈, + cos y, cos K≈), ··· Kx, = 90°;
da
d cos xx
W2 sin xx,
sin K≈, – w3 sin xx, cos K≈,
dt
dt
sin
Kxz,
-
W2
We
xx, sin Kx,≈, sin
@3 xx, cos Kx z
sin xx¸ cos z¸x¸x + w3 sin xx, cos y¸x¸x
C
(၂၇ COS X X + W3 COS XY, = wer + wzb.
MOTION ABOUT A POINT WHEN NO FORCES ACT.
433
Similarly we should obtain
db
dt
d cos xy,
dt
W3 COS X X + W₁ COS X Z,
wzα + w₁ C,
dc
d cos x z
dt
dt
W₁ COS XY+W2 COS X X¸ = -w₁b+wa, and so on.
Now multiply the equations (2) of Art. 446 by a, b, c
respectively and add,
dw
la du
. Ala
dw3
+ cfo dury
{
dt
+ w; (bwz − cw₂) }
d wz
–
+ B/b
+ w₂ (cw₂ − αwz)
dt
C
dt
+ w3 (ɑ w₂ − b wz)
0,
dwi
da
d w z
dcl
or Ala
+ ω1
+ B√b
dt
dt
dt
+ W2
dt
+ W3
dt
dt
= constant = 1.
(aw₂ – bws)}
dby + c { c d ws
= 0;
.. Aaw₁ + Bbw z + Cc w z
Similarly, Aa' w₁ + Bb' w₂ + Cc' wz = l',
A a'"w₁ + Bb"w₂ + Cc″wz = l″.
Add the squares of these together; observing that since
the angle between any two axes of the same system of co-
ordinates equals a right angle, therefore
ab + a'b'+a"b"=0, ac + a'c'+a"c"= 0, bc + b'c' + b"c"= 0;
and we have
2
2
A² w₁² + B² w₂² + C² wz² = 1² + l'² + l'² = k² by Art. 450.
Hence if we draw a line AI' making angles with the fixed
axes of which the cosines arc
Aaw, + Bbw₂+ Сcw3
√ Aw
√ A² w₁² + B² w₂² + C w²²
Aa'w₁ + Bb'w₂ + Сc wz
2
2
√ A² w² + B² w₂² + C² w₂²²
A a" w₁ + Bb" w₂ + Cc" wz
ωι
2
A² w₁² + B² w₂² + C² w z²
に
​W2
this line, and therefore the plane perpendicular to it, will
remain invariable during the whole motion.
For this reason
I
434
RIGID BODY.
DYNAMICS.
In a future Chapter we
it is called the Invariable Plane.
shall speak more of this plane.
452. COR. 1.
cos. l'Ax = a cos l'Ax + a' cos I'Ay + a" cos I' A≈
A wi
k
Bw.
Cw3
also cos I'Ay,
cos l'Az,
,
k
k
453.
COR. 2. If the invariable plane be taken for the
plane of ay then
A wr
k
= COS ZX sin sin 0: Art. 415.
Bw2
= cos xy, = cos & sin 0,
k
Cw₂
k
= cos zx, = cos 0.
The equation in Art. 450 for finding w₁ can be integrated
when the principal moments are equal: and also when two
only of them are equal. We shall investigate these cases.
PROP. To find the motion of a rigid body about a fixed
point when its principal moments are equal to each other :
the forces all passing through the centre of gravity fixed.
454. In this case BCA: and the equations of
Art. 450 give
WI constant, w₂ = constant, w3 constant;
and therefore the instantaneous axis remains fixed in the body:
see Art 445.
of
Z
Since every axis is a principal axis (Art. 425); let the axis
coincide with the instantaneous axis.
... (1
=
0, w₁₂ = 0, w3 = constant = n suppose.
ως
Hence the equations of Art. 447 become
MOTION ABOUT A POINT WHEN NO FORCES ACT.
435
d Ꮎ
αψ
0 =
cos
sin o sin 0,
dt
dt
do
df
0:
sin
cos o sin 0,
dt
dt
do
d¥
N =
cos 0.
dt
d t
By the first and second equations, we have
do
df
0,
0 ;
.. and are constant.
dt
dt
Also by the third equation
аф
dt
= n;
$ = nt + const.: this
shews that the body revolves about a fixed axis: hence the
instantaneous axis is not only fixed in the body, but also in
space. The position of this axis and the magnitude of the
angular velocity depend upon the circumstances of projection.
PROP. When A B required to determine the motion.
455. The equations (2), of Art. 446 become
A
dwi
dt
dwz
+ (C − A) wyWz = 0,
+ (A − C) w] wz = 0,
A
dt
dw3
C
0;
dt
w3 = const. = n suppose.
... Wz
equations we have
Also by differentiating the first
d w z
A
ď or
+ n (C ~ A)
d t
dt
0,
and therefore by this and the second equation.
d'w
+
dt
C A\2
A
n' wi
W₁ = 0;
436
RIGID BODY.
DYNAMICS.
{nt + 1};
C
A
·· w₁ = e cos
A
A 1
dwz
... W2
=e
C - A n
d t
f
e sin {C-Ant +f},
where e and ƒ are constants to be determined by the circum-
stances of the motion at some given time.
The angular velocity (w) about the instantaneous axis
2
√w₁² + w₂² + w₂² (Art. 443.) = √√√n² + e², and is constant.
We shall now substitute the values of wz wą wz
in the equa-
tions of Art. 447: and put
C-A
A
n = m;
ᏧᎾ
df
cos &
sin & sin 0 = e cos (mt +ƒ)
dt
dt
do
dy
sin
cos o sin = e sin (mt +f)
dt
dt
аф
dy
cos
N.
dt
dt
Let us take the Invariable Plane for the plane of xy: then by
Art. 453.
A
Ae
sin & sin
ω
W₁ =
cos (mt +f),
k
k
A
A e
cos o sin
W2
W₂ =
sin (mt +f),
k
k
C
Cn
cos
W3
k
k
... tan p = cot (mt+f),
mt-f; and
π
.. Φ
2
Cn
Also, since cos
k
و
аф
m.
dt
Ø is constant.
MOTION ABOUT A POINT WHEN NO FORCES ACT. 437
And
dt
cos 0
dy-1 (-n)
m + n k
k
N
C Α
Hence the body revolves uniformly about the principal axis
Ax,; while the line of nodes (that is, the line of intersection
of the planes x y and x,y) revolves uniformly on the plane x y.
Also cos IAN
1 기
​= cos a, cos +cos ẞ, cos (π + ) + cos y, cos
π, see fig. 105,
W2
C
cos
sin & = 0 ;
W
U
and therefore the instantaneous axis of rotation is always per-
pendicular to the line of nodes.
n
W3
Again cos IAz,
and therefore the in-
W
√ n² + e²
stantaneous axis always makes the same angle with the axis
of %,·
z
We shall now obtain the arbitrary constants from the cir-
cumstances when t = 0.
Since any axis in the plane x,y, is a principal axis
(Art. 424), let the axis of x, be so chosen, that when t = 0 it
coincides with the line of intersection of the planes IA,
and x,y;
ωο
Let wo be the angular velocity about the instantaneous axis
when t = 0, and let z IA, = 8; therefore when t = 0,
w₁ = w。 cos Ix
cos Ix = w。 sin &,
w₂ = w₁ cos ly
W2
W3 =
0,
w₁ cos Iz
= w₁ cos d,
and consequently by the general values of w, we w3 we have
e cos fw, sin 8, e sin f= 0, n = w₁ cos &;
2
ƒ= 0, e = w。 sin d, n = w, cos d,
2
k = √ A² (w₂² + w) + Cw3 (Art. 450.) = √ sin" & + C² cos &.
2
w₂ √² d
438
DYNAMICS. RIGID BODY.
π
C - A
Hence o
W₁ cos d. t,
A
dy
wo
dt
A
cos
√ A² sin² 8 + C² cos² 8,
C cos d
√ A² sin² 8 + C² cos² d
tan
A
or
tan & C
456. These formulæ lead to the following geometrical
construction, fig. 106. Let the axes of co-ordinates ≈≈ and
the instantaneous axis cut a sphere of radius unity described
about 4 in the points ≈≈I respectively.
2
dy do sin &+ C² cos² &
√A
(C - A) cos d
C
sin d
(C-A) cos sin (8-0)
0
Then
=
d t dt
sin Iz
sin Iz
since AI is perpendicular to AN and
makes a constant angle with Ax, and is consequently always in
the plane ≈ Az,.
About, and ≈ describe two small circles on the sphere of
which the radii measured on the sphere are 8 and 8 -0: then
these circles touch in the point I, where the instantaneous axis
meets the sphere.
Suppose I' and I" are the points in these circles which
were in contact when
Then, since the
t
***
0.
angular velocities about A≈ and Az, are
d↓
аф
uniform and equal to
and
we have
dt
dt
dy
sin Iz
dt
arc I'I = angle I'≈ I sin I≈ = t
аф
= t
dt
sin Ix, = angle I'≈, I sin I≈, = arc I'I,
wherefore the motion of the body may be described by making
the circle I roll with its internal surface on the fixed circle ≈ I.
COR. If C be less than A, then I will be between ≈ and ≈
and the external circumference of the circle I would roll on
the circle Iz.
STABILITY OF THE EARTH'S ROTATION.
439
457. If we observe the apparent motion of the stars
night after night we remark, that they all seem to move in
parallel circles about the star named (on that account) the
Pole Star. This proves that the axis about which the Earth
revolves points towards the Pole Star, and never deviates from
that direction by an angle appreciable by ordinary observation.
Also geodetic measurements and other calculations for ascer-
taining the Figure of the Earth shew, that this axis of rotation
coincides (so far as the approximation is carried) with the
geometrical axis of the spheroidal form of the Earth's Surface.
Theory shews that there is a necessary connexion between these
two facts which are apparently independent of each other.
This we proceed to prove.
PROP. Suppose the Earth revolves about an axis nearly
coinciding with one of its principal axes at any given time :
required to find the motion, all external forces being neg-
lected.
458. Let the axis of ≈, be that near which the instanta-
neous axis lies at the given time t. Now the sine of the angle
which these two axes make with each other
=
2
2
w₁² + w ₂ ²
wi
2
2
w₁² + w₂ ~ + w z
2
2
(Art. 445), and this is small by hypothesis: hence w² + w² is
small, and w₁ and we are small: and the equations (2) of
Art. 446 give
(01
A
dwr
+ (C − B) w2w3 =
0,
dt
B
C
d w
dt
d w3
dt
+ (A − C) w] w3 = 0,
– -
+ (B − A) w₂ w₂ = 0,
1
then neglecting the product of w₁ and w, the last equation.
gives w3 = constant = n: and the others give
dw
A
–
+ (C − B) wḥn =
0,
dt
440
RIGID BODY.
DYNAMICS.
+ (A − C) w₂n = 0,
d w₂
Ᏼ
dt
& wi
dt
+
AB
... w₁ = e sin
{
AB
n² wr
w₁ = 0;
nt + f
+1}
(A − C) (B − C)

(A − C) (B – C)
e and f being constants which depend upon the circumstances
at any given time;
A dwr
.. W2 (B - C)n dt

A A-C
e
COS
BB-C
(AC) (BC)
AB
[nt+f}}
and since w+w2 is small at the given time, e is small: and
since e is constant it shews that w₁ and we are always small so
long as (AC) (BC) is positive.
If however (AC) (BC) be negative, then the trigo-
nometrical expressions for w, we must be replaced by exponen-
tials, and consequently they will not remain small.
From this we gather that if a body revolve at any time
about an axis coinciding nearly with the principal axis of
greatest or least moment, the axis of rotation will always.
nearly coincide with that principal axis. But if the axis be
that of mean moment the instantaneous axis of rotation will
deviate more and more from that principal axis till it ap-
proaches the principal axis of either greatest or least moment.
COR. If the instantaneous axis actually coincide with a
principal axis at first, then e = 0, and w₁ and w₂ each vanish.
Hence any principal axis is a permanent axis (see Art. 439).
If, however, the slightest cause tend to make the instan-
taneous axis of rotation deviate from the principal axis, the
rotatory motion may be said to be stable or unstable according
as the principal axis in question is not or is the mean principal
axis.
This points out an admirable adaptation in the laws of
nature that the motion of rotation which causes the heavenly
EQUATIONS OF ROTATORY MOTION OF THE EARTH. 441
bodies to bulge at their equators, in so doing, gives them such
a figure as to insure the stability of their rotation.
We shall now consider the action of the Sun and Moon on
the rotatory motion of the Earth.
PROP. To obtain equations for calculating the rotatory
motion of the Earth when acted on by the Sun and Moon.
459. The equations of motion referred to the principal
axes are by Art. 446,
+ (C − B) w₂ W3 = L,
+ (A − C) w1 w3 = M,
dwi
A
dt
dw2
B
dt
dw3
C
-
-
+ (B − A) w1 w2 =
N₁.
dt
To calculate L,M,N, let S be the mass of one of the
disturbing bodies: ays, the co-ordinates of the centre of S;
x'y' the co-ordinates to any particle m of the Earth's mass
referred to the principal axes: 2² = x² + y² + ≈
2
2
Then the difference of the attractions of S on the particle
(m), and the centre of the Earth (which we here suppose fixed,
see Art. 429.) resolved parallel to the axes y, and Z and esti-
mated positive in directions from the origin, are
2
S (y, — y')
{(x, − x',')² + (y, − y')² + (≈, − ≈,')²}
—
S (≈, − ≈,')
{(x, − x')² + (y, −y, )² + (≈, − ≈,')º } }
- ≈,
Sy,
Y suppose
203
S≈
-
·Z, suppose.
༡°
Hence L, .m (y, Z, Y), see Art. 446.
=SE.
p²
S
=
x,y) — y, x,
12
−2(x,x,'+y,y'+≈≈',')+(x,'²+y,"²+≈,'') } *
**
12
12
2(x¸x'+y¸y'+≈,x,')−(x,²+y,'²+x,
2.3
一卷
​= = = m (= x - y = ){ (1 - (")) = -1}
¡Σ.m(x,y)
༡॰
3 K
442
DYNAMICS. RIGID BODY.
11
3S
= = = = Σ.m(x,y,' −y,≈',')(x,x'+y,y'+≈,x,'),
215
neglecting the cubes of very small quantities,
3S
205
3S
205
12
2
Σ.m{(y,'² — x,'²)≈,y,−(y²−z,²)≈'y' +≈,x,y,' x' — y,x,x'x' }
12
12
x,y, Σ. m (y,"² - ≈2) by the property of principal axes,
3S
Z
.. L₁ = ~ ~, y, (C - B).
205
In the same manner we should find
3S
M, = ∞, ∞, (A — C'),
205
dwi
A
dt
d wz
3S
N
2.5
x¸y, (B – A).
Hence the equations of motion become
3S
20.5
+ (C − B) w₂ wz
25 Y, ≈, (C - B),
3S
B
+ (A − C) w1W3
x¸≈, (A − C),
-
dt
205
dw3
3S
C
+ (B − A) w1w2 =
dt
215
x¸y, (B − A).
In these equations the disturbing body is supposed to
be at a very great distance, as is the case with the Sun and
Moon; but it is remarkable that they are very nearly correct
even when the attracting body is very near the Earth, sup-
posing the Earth's figure to be spheroidal. For a demon-
stration of this we refer the reader to the Mécanique Céleste.
Liv. V. Chap. 1. §. 3.
It will be observed that we have taken account of only
one disturbing body S in these equations: but since the per-
turbations are small and the equations in www linear, we
LENGTH OF THE MEAN DAY IS INVARIABLE.
443
may calculate the effects of the disturbing bodies singly and
add them together, Art. 288.
PROP. To prove that the velocity of rotation of the
Earth, and consequently the length of the mean day, is not
altered by the action of the Sun and Moon, very small
quantities being neglected.
460. If we neglect the disturbing forces and suppose the
figure of the Earth to be one of revolution and not differing
much from a sphere, that is, BA, and each of these nearly
= C, the difference being of the order of the ellipticity of the
terrestrial spheroid; then in this case the equations of the last
Article give, for a first approximation, w, const. = n, w, and
w very small quantities. These values may be put in the
small terms of our equations in order to obtain a nearer
approximation.
3
If we multiply the three cquations of last Article by
wwwg and divide them by A, B, C respectively and add
them together, we have
z
2
d (w₁² + w₂² + wz²)
2
+ 2W1 W2 Wz
sc
-B A-C
B - A
+
d t
A
B
BA
6S (y,≈, C - B
X, Z
A - C
x₁y, B - A
w₁ +
W₂ +
2.3
зой A
B
zoz
C
С 003
C-BA
C
Now ww2 are each extremely small;
are
A
B
В - A
is
C
of the order of the ellipticity of the Earth; and
extremely small, since if we suppose the Earth a figure of
S
revolution this expression vanishes; also is very small,
2+3
because S varies as the cube of the radius of the body S.
Hence if we neglect extremely small quantities
2
2
2
√ wi² +w¸² + wz² = const.
= ԴՆ:
since the mean values of www are 0, 0, n.
444
DYNAMICS. RIGID BODY.
Hence the angular velocity of the Earth is constant, and
the length of the mean day is not affected by the action of
the Sun and Moon, when we neglect inappreciable quantities.
A full discussion of this important question will be found
in the Méc. Céleste, Liv. V. Chap. 1. §. 8, 9: and also in the
Mémoires de l'Académie Royale des Sciences de l'Institut
de France: Vol. VII. p. 199.
Astronomical observations shew in a remarkable degree
that the length of the mean day has been invariable for a
long period of time. We proceed to explain how this result
is obtained from observations.
PROP. To shew from observations made on eclipses that
the length of the mean day has been invariable for a great
length of time.
are the
461. Let us take for the unit of time the length of any
day at the present epoch: and suppose the day has been de-
creasing by a parts. Let n be the mean angular motion of
the Moon on the day which is taken for the unit of time;
then n is the number of degrees through which the Moon
moves on that day: and n (1 + a), n (1 + 2a),
angles described by the Moon during the days preceding
that day in order: and the angle described during t days
= nt + 1 na (t − 1)t, and if t be very large this angle
=nt + nat. Let n' be the mean motion of the Sun on the
day of which the length is the unit of time, then the angle
described by the Sun in the t days now elapsed=n't + 1 n'at³,
and the difference of longitude in the Sun and Moon being
λ now, was = λ + (n' − n) t + 1 (n' −n) at at the distance of
t days from the present time.
1
Let & be the error made in calculating the difference of
longitudes of the Sun and Moon at a distance of t days on
the supposition of the invariability of the length of the day:
then 1 (n' – n) ať² = d.
Now the values of 8 have been calculated in the Con-
naissance des Tems of 1800, from 27 eclipses observed by the
Chaldees, the Greeks, and the Arabs. The greatest value
of corresponds to an eclipse observed in the year B.C. 382:
LENGTH OF THE MEAN DAY IS INVARIABLE.
445
for this = 27′. 41". For the most ancient eclipse d= 2″ ;
this eclipse being observed by the Chaldees in the year
B.C. 720.
Let i be the number of centuries in t days: then
36525 i. By the mean of modern observations on the Sun
and Moon it is found that (n' – n) 36525 445268° for the
t
=
most ancient eclipse i = 25.56;
... d = (36525). (25.56)² a × 445268°.
Now if the day be shorter by a ten-millionth part than at
the epoch of the most ancient eclipse on record, then
(36525) (25.56) a = 0.0000001;
... d = ↓ (25.56). (0.0000001). (445268)°
34′,
a value which renders an eclipse impossible, since the sum
of the greatest semi-diameters of the Sun and Moon does not
exceed half a degree.
From this we learn that the length of the day has not
changed even by a hundred and fifteenth part of a second of
time during the last 2556 years. M. Poisson's Traité de
Mécanique, Seconde Edition, Tom. 11. p. 196—200.
462. By comparing the observed north-polar distances of
stars made at epochs distant from each other Bradley shewed
that the point in the heavens to which the Earth's axis of
rotation is directed is not stationary, although for periods of
time not very long this deviation, as we remarked in Art. 457,
is not perceptible. It becomes an interesting question, then,
to ascertain the cause of this perturbation. Since we neglected
the action of the Sun and Moon in the calculation of Art. 458,
we may readily conjecture that the action of these bodies is the
cause required. This we proceed to demonstrate.
PROP. To determine the position of the axis of rotation
of the Earth at any given time, the action of the Sun being
considered; and the figure of the Earth being taken to be
one of revolution.
446
DYNAMICS. RIGID BODY.
463. We shall refer the disturbing body S to the ecliptic.
Let the plane of the ecliptic be the plane of XY: the axis of
X being drawn through the first point of Aries, which is
moveable; the centre of the Earth the origin of co-ordinates;
x,y,, parallel to the principal axes,
Ꮎ
0 = the angle between the equator and ecliptic,
or the angle between the axes of ≈, and Z.
= the right ascension of the axis of x
לן
or the angle between the axes of x, and X.
7 = longitude of the Sun,
r distance of Sun from the Earth's centre;
then by Spherical Trigonometry,
XxXx
cos o cos 1 + sin o sin l cos 0,
J'
Y
r
ド
​Z
2?
2°
sino cos l + cos o sin l cos 0,
sin 7 sin 0;
Y, Z,
p2
sin 27 sin 0 sin - sin² 7 sin 20 cos 0,
&
X,Z
202
sin 27 sin cos sin l sin 20 sin o,
2
substituting these in the equations of motion of Art. 459, and
putting
P =
3S
27.3
sin 21 sin 0, and P' =
3S
27.3
sin² / sin 20,
we obtain, since B = 4,
A
SOLAR PRECESSION AND NUTATION.
447
dwr
C- A
+
W2 W3
dt
A
C- A
A
ф-р
(P sin - P' cos P)
d wz
C - A
C- A
WlWz =
W1W3
(P cos + P' sin ()
(1).
dt
A
A
dw3
= 0.
dt
The third equation gives w;= constant
= nt + small terms (see Art. 447).
Ф=
=
n; and therefore
Let the time be measured from the epoch when the Sun
was in Aries: then n't. Since B = A any axis in the
plane x,y, is a principal axis: let the axis of x, be so chosen,
that when t = 0 it passed through Aries: then = nt neg-
lecting small terms;
.. P =
3 S
27.3
sin ( sin 2n′t; P'
3S
4203
sin 20 (1 – cos 2n't).
We shall neglect the variations of the inclination (0) of the
equator and ecliptic in calculating small terms.
Since the equations (1) are linear we may take one term
only of P and P' in the calculation: let k sin it and k' cos it
be corresponding terms: then i admits of two values 0 and 2n′.
Considering these terms we have
dw C-A
C-A
+
n w₂ =
{(k-k') cos (n−i) t−(k+k') cos (n+i)t},
dt
A
2 A
d w₂
C-A
ηωι
dt
A
C-A
2 A
{− (k—k') sin (n−i)t +(k+k') sin (n+i)t}.
To solve these differentiate the first with respect to t and
d we
by the second;
eliminate
dt
F² wi
C- A
2
+
dť
A
C-A
C-A
{(k-k')
n−n+i) sin (n−i) t
-n+i) sin
Α
N
กา
– i) sin
sin (n + i) t}.
A
−
n² w₁ =
2 A
(k + k')
C A
448
RIGID BODY.
DYNAMICS.
The integral of this is of the form
W₁ =
C₁ cos
C- A
A
nt + C₂) + M sin (n − i) t + N sin (n + i) t,
where C₁ and C₂ are constants independent of the disturbing
forces; and M and N are to be found by putting this value for
w, in the differential equation, we find that
2 M
=
2 N
(k − k') (C - A) {Cn − (2n − i) A}
(C − A)² n² – A² (n − i)²
(k + k') (C − A) {Cn − (2n + i) A}
(C ~ A)² n² – A² (n + i)²
i
2n'
Now the only values of are 0 and
2
since there
N
365
i
n
are 365 days in a year): hence by neglecting - in the small
n
terms, we have
k-k' C-A
M
N
>
2n
C
k + k' C - A
2n C
Now when there are no disturbing forces w₁ = 0, and conse-
quently C₁ = 0;
... w₁ = M sin (ni) t + N sin (n + i) t,
(1
W2 =
M cos (n − i) t + N cos (n + i) t.
Returning to the axes fixed in space and choosing the plane of
the ecliptic for the plane of ay we must put the values of w₁
and we in the equations (Art. 448.)
W2
ᏧᎾ
w₁ cos — w₂ sin o,
dt
sin e dy
dt
аф
dt
= n
w₁ sin – w₂ cos P,
8-
cot 0 (w, sin + w₂ cos P),
·P
SOLAR PRECESSION AND NUTATION.
449
in which is the obliquity of the ecliptic; the right ascen-
sion of a fixed terrestrial meridian; the longitude of Aries
measured in a retrograde direction: fig. 97.
Since vanishes when there are no disturbing forces;
dy
dt
.. φ
nt for a first approximation;
d Ꮎ
dt
w₁ cos nt wą sin nt
C - A
= (N − M) sin it
.k sin it,
n C
dy
sin
01
o₁ sin nt
w₂ cos nt,
dt
C- A
nc
k' cos it,
(M + N) cos it
and by replacing k sin it and k' cos it by P and P', of which
they have been the representatives,
de
C- A
P =
dt
n C
273
3 S C - A
n C
sin sin 2n't.
dy
C- A
P'
dt
n C sin o
2 703
3 S C - A
n C
Ꮎ
cos & (1 − cos 2n't),
ا
=
n', the mean
дов
integrating these equations and putting A
motion of the Sun,
0 = I +
3 n' C - A
sin I cos 2n't,
An
C
2n
3n'² C - A
C
3n' C-A
cos I.t-
cos I sin 2n't,
4 n
C
I being the mean value of 0: and the axis of a being so chosen
that t = 0 when Aries was in that axis.
3 L
450
DYNAMICS. RIGID BODY.
We should obtain analogous expressions for the perturba-
tion of the Earth's axis by the Moon.
464. The first of these expressions shews that the obli-
quity of the ecliptic fluctuates; but preserves its mean value
equal to the value it would have if there were no disturbing
forces.
The second shews that the first point of Aries, or the
vernal equinox, has on the whole a retrograde motion on the
ecliptic, though at the same time it is subject to a small oscil-
latory motion.
The steady retrograde motion is called the Precession
of the Equinoxes; the solar precession (i. e. the precession
3n'² C – A
caused by the Sun) equals
2n C
cos I in a unit of time.
This precessional motion causes the pole of the Earth to de-
scribe a small circle about the pole of the ecliptic.
The oscillating motion of the pole, arising partly from the
change of the. obliquity and partly from the periodical term
in, is called the Nutation of the Earth's Axis.
465. It will be seen that the Precession and Nutation of
the Earth's axis arise from the attraction of the Sun and Moon
upon the protuberant parts of the Earth, i. e. upon the portion
by which it exceeds a sphere touching it internally. For if
the form of the Earth were spherical, then CA and the
variable terms in and would vanish.
We proceed to calculate the effect of the Moon upon the
position of the Earth's axis.
PROP. To find the motion of the Earth's axis with
respect to the plane of the Moon's orbit caused by the action
of the Moon.
466. Let and be the same quantities as ✪ and √
in Art. 463. with this difference that the plane of the Moon's
orbit is used instead of the ecliptic; i the inclination of the
Moon's orbit to the ecliptic; this does never much exceed 5º,
and is therefore so small that we may neglect its square: we
shall also consider i constant, since its variations are very small,
as is shewn by observation: I' the inclination of the equator
INCLIN. OF EARTH'S AXIS TO LUNAR ORBIT CONSTANT. 451
to the Moon's orbit: M the mass of the Moon: a the radius
of the Moon's orbit.
Now for
S
in Art. 463. we must put
M
a³
९.३
Let n" be the mean motion of the Moon about the Earth:
then n" = √√√ M + E
M
Μη
Mn'2
#12
N
>
a³
a
M + E
1 + ν
where E is the mass of the Earth, and the ratio of this
mass to that of the Moon.
S
Hence in Art. 463. must be replaced by
2.3
n''2
1 + ν
;
do'
3n'
112
C – A
sin I' sin 2n't
dt
2 (1 + v)
nC
3n"2
2 (1 + v)
C- A
n C
d↓
dt
cos I'(1- cos 2n't).
The periodical quantities sin 2n't and cos2n't go through
their changes in half a month; in consequence of the short-
ness of their period and the smallness of their coefficients they
never accumulate so much as to produce a sensible effect: and
are therefore omitted. Hence the inclination of the Earth's
axis to the Moon's orbit suffers no sensible change from the
Moon's attraction; but the line of intersection of the equator
and the plane of the Moon's orbit does change its position,
which is determined by the equation
3n"
C - A
2 (1+r) nC
cos I'. (n't const.)
+
In order to calculate the Lunar Precession and Nutation,
we must refer the angle to the ecliptic. Since the oscilla-
tions of the plane of the Moon's orbit are insensible no
Lunar Nutation can arise from them; but the Moon's line of
Nodes continually regresses (Art. 342.) performing a revolu-
tion in 18 years and 7 months and this is the cause of Lunar
Nutation. We proceed to calculate this and Lunar Precession.
452
DYNAMICS. RIGID BODY.
PROP.
To calculate Lunar Precession.
467.
Let K, K', P be respectively the poles of the ecliptic,
Moon's orbit, and the Earth's equator (fig. 107).
df
Now P revolves about K' with an angular velocity dt
dý
dt
hence the linear vel. of P about K'
sin e', rad. of sphere = 1
dý
sin 'cos KPK': and
dt
the resolved part of this about K
therefore
P revolves about K with an angular velocity
dysine
dt sine
cos KPK'
and...
Y...
dt
dy' sine' cos Y PK';
dy
dy sin o'
3n"2 C-A cose'sin'
dt
dt sin 0
3n"2
cos KPK'
cos KPK'
2(1+v) nC
sin
2 (1+v)
C - A
n C
cos 0' cos i
cos e cos '
sine
sin
3n"2
11
(cos e cos i + sin Ø sin i cos )
C- A 1
2 (1+v) nC sin20
× (cos i — cos² cos i cos e sin sin i cos )
where is the longitude of the Moon's node measured in a
retrograde direction
3n"2
1/2
n' C- A
2(1+v) nC
(cos cosi+sinesini cos N) (cos i-cote sinicos)
3n'2 C-A
2(1+v) nC
(cos@cos³i-
cos 20
2 sin
sin 2icos-cosesini cos)
3n"2 C-A
2(1+v) nC
cos 2 I sin 2 i
2 sin I
COS
cos I (cos² i - sin² i)
2π t
+ 12 - 1/1 cos I sin² i cos
T
t
(+220)}
T
LUNAR PRECESSION AND NUTATION.
453
in which the periodic time of the Moon's nodes: and
σ =
is the longitude of the ascending node when t = 0;
3n"2
C- A
↓
2 (1+v)
nC
{co
cos I (cos² i – 1 sin² i) t
Т cos 2 I sin 2 i
2πt
П
sin
+
4π
sin I
T
T
4πt
cos I sin² i sin
+ 200
8π
T
20)}
+ const.
Hence the Lunar Precession
3n"
2 (1 + v)
C- A
n C
cos I (cosi - sin² i) n't.
The second term of is periodical as well as the third;
but the third is so small, in consequence of its coefficient
sini, that it may be neglected. They are both parts of
Lunar Nutation.
PROP. To find the effect of the Moon on the obliquity
of the ecliptic.
468. We have from the last Proposition
ᏧᎾ
dy
3n"
112
sin 'cos YPK'
dt
dt
2 (1+v)
C- A
nC
cos' sin e' cos YPK'
3n"2
C- A
2 (1+v)
nC
(cos e cos i + sin 0 sin i cos ) sin i sin (2,
3n2
||
2 (1+v)
C- A
n C
π
cos I sin 2i sin
+ Ωρ
т
t
3n""
C- A
T
.. 0 = I
2 (1+v)
nC
4π
+ sin I sin' i sin (+22)};
T
t
cos I sin 2i cos 1 (2=² + (20)
T
T
4π t
+
sin I sin² i cos
+ 200
Επ
T
454
RIGID BODY.
DYNAMICS.
the variable terms in this are periodical, and the last is so
small as to be insensible. These terms and the periodical
terms in the value of make up the whole of Lunar
Nutation.
469. Let x and Y be the parts of Lunar Nutation which
have been determined in Arts. 467, 468;
8 πn (1+v) C
37n" (C-A) cos 2 Isin 2i)
112
This is the equation to
in the ratio cos 21: cos I.
2
+y
2
8 πn (1+v) C
3 тn"² (C-A) cos I sin 2i
2
=1.
an ellipse of which the axes are
This explains the construction
mentioned in works on Plane Astronomy. Woodhouse's Plane
Astronomy, p. 857. Maddy's Plane Astronomy, 2nd Edition.
The whole Precession, both Solar and Lunar, equals
3 C- A
2n
C
112
n''s
cos I {n'" + (cosi - -sini)} t,
1 + ν
(see Mécanique Céleste, Liv. v. Chap. 1. §. 14.) and the Nuta-
tion is given by the equations of Arts. 467, 468.
470. Annual Precession.
C – A зn'
112
3
n
sin² i
I
cos 1+
180º
C n
12
n
1 + v
//
N
N 36526
I = 23° 28′ 18″, i = 5º 8′ 50″,
i:
8'
365.26,
n'
n'
2732
log 10 sin i = 2.9528656
=
log 10 (sin² i) = 2.0818225 log 10.0120732;
... log 10 (1-sin" i) = log 10.9879268 = 1.9947248
log 10
n
112
n'
12
2.2522428
... log 10
In"²
78'2 (1 − 3 sin² i
sin')} = 2.2469676 = log 10 176.5906 ;
SOLAR AND LUNAR PRECESSION.
455
log 10
Jзn'
[sn'
n
1/2
N'2
(1 − 3 sin² ¿) = 176.5906
× 180 × 60 × 60 × cos I
n
}
= 2.3856065
3.9030900
ī.9625076 – 2.5626021
6.2512041
2.5626021
3.6886020 = log 10 4882.05;
C- A
176.5906
.. Annual Precession
1 -
+
900)
4882".05.
C
1 + v
471. We have supposed in these calculations that the
Earth is wholly solid. Laplace shews, however, that the
variation of the motion of the terrestrial nucleus, covered by
a fluid, are the same as if the sea formed a solid mass with it:
Mécanique Céleste, Liv. v. §. 10—12.
1
CHAPTER XIII.
MOTION OF A RIGID BODY ACTED ON BY IMPULSIVE FORCES.
472. IN the preceding Chapters we have obtained dif-
ferential equations for calculating the motion of a body acted
on by any forces of finite intensity. Since these differential
equations are of the second order their first integrals will be of
the first order, and will therefore be functions of the velocities
and co-ordinates of position of the various parts of the body.
The values of the arbitrary constants introduced by the process
of integration are determined by knowing the velocity and
position of the parts of the body at any given instant of the
motion: the instant generally chosen is the epoch from which
the time is measured. In calculating the motion of a heavenly
body the values of the arbitrary constants are found by obser-
vations made on the position and velocity of the body at any
given time; since we are altogether unacquainted with the
initial circumstances of the motion. But we may wish to
calculate the motion of a rigid body when the original cir-
cumstances of projection are known. Now instantaneous
velocities are generated by forces of the nature we have termed
impulsive (Art. 201). It becomes necessary, then, to calcu-
late the motion of a body which results from the action of
impulsive forces.
By finite forces we mean such as require a finite time.
to produce motion, or a change of motion, in a body: such
as the moving force produced by the attraction of the Earth.
They are measured, when uniform, by the momentum gene-
IMPULSIVE FORCES.
457
rated in a unit of time; when variable, by the momentum
that would be generated if they were to remain uniform after
the epoch at which they are to be estimated.
But by
Impulsive Forces we mean such as act only for an in-
definitely short time, and yet produce a finite velocity in a
body; such as the force of explosion of a cannon; the force
of impact of one body against another. These forces are
measured by the momentum generated in the body on which
they act.
In Art. 226. we have enunciated a Principle by means
of which we can make the calculation of the motion depend
upon the conditions of equilibrium of forces acting on a
rigid body.
PROP. To obtain the equations of motion of a rigid
body acted on by impulsive forces.
473. Let V be the velocity impressed on a particle m of
the system, the co-ordinates of the particle being wys: then
dx dy
dz
the effective velocities parallel to the axes
dt
are
dt' dt
of co-ordinates, that is, the actual velocities which the particle
acquires in consequence of the action of the impulsive forces:
a, b, c the angles which the direction of the velocity V makes
with the axes.
Then according to the Principle explained in Art 226. the
impulsive forces
กาง
い
​V cos a
da),
M
い
​V cos b
dy
dt
du), m (v
dz
I cos c
dt
acting on m parallel to the axes of co-ordinates, together with
similar forces acting on the other particles of the system ought
to be in equilibrium.
Hence we have from Art. 65. the equations
E.mV cos a
(v
dx
0.
Σ.»
(Icos b
dy
= 0,
dt
dt
い
​dz
Σ.ml cos c
dt
B M
458
DYNAMICS. RIGID BODY.
2. m {y (V cos c-dz) - x (V cos b-
COS C
dt
m fx (1
da
Σ.mx V cos a
a V cos c
-
dt
2.
Σ. m {x (V cos b - dy) -y (V cos a
dt
Y
dt
dy)}
0,
dz
0,
dt
dx
= 0.
dt
By means of these six equations we shall be able to
calculate the motion of a rigid body acted on by any im-
pulsive forces.
They lead immediately to two fundamental principles,
analogous to those of Arts. 428, 429. for finite forces.
PROP. The centre of gravity of the body will move in
the same manner as if the forces which act upon the various
particles of the body were all transferred to that point, their
directions being parallel to their former directions.
give
474. For the first three of the equations of last Article
dx
Σ.m
Σ.m V cos a, Σ.m
dy
dt
Σ.m V cos b,
dt
dz
Σ.m
Σ.m V cos c.
dt
Now let xyz be the co-ordinates of the centre of gravity
at the instant the impulsive forces act: then
x Σ.m Σ.mx, y E. m =
Σ.my,
| મેર
≈ Σ.m=
2.ms.
Differentiate these with respect to t, and we have by the
equations of motion above
dx
dy
Σ.m = Σ.m Vcos a,
Σ.m
.m V cos b,
dt
d t
dz
Σ . m = Σ . m V cos c.
dt
IMPULSIVE FORCES.
459
But these are the equations we should have obtained
by supposing the forces transferred to the centre of gravity,
their directions being preserved.
Hence the Proposition, as enunciated, is true.
PROP
The motion of rotation of the body will be the
same as if the centre of gravity were fixed.
475. Let x,y,, be the co-ordinates of m measured from
the centre of gravity parallel to the original axes: then
x = x + x, y = ÿ + Y₁, ≈ = ≈ + ≈, ;
Y Y
and by substituting these in the last three equations of
motion in Art. 473. the first of these becomes
{
dz
dz
dy
dy
Σ . m { (y + y)
+
dt d t
(~~ + ~~)
+
dt dt
Σ. m {(y+y,) V cos c − (~ + ≈,) V cos b}.
But Σ.ma, =
0, Σ.my, = 0, Σ. m2 = 0, by Art. 413;
... Σ.m
(y.
dz
dy
.m (y, V cos c
≈, V cos b),
Y
d t
d t
and the other equations by a similar process become
Σ
5. m (3
dx,
di
=
心
​Σ. m (≈, l'cos a − x, l'′ cos c),
≈
• dt
Σ.mx
m (x,
dy,
dt
-
dt
dx
- Y,
2. m (x, V cos b − y¸Ïˇˇcos a).
dt
But these are exactly the equations we should arrive
at by supposing the centre of gravity fixed.
Hence the Proposition, as enunciated, is true.
476. The Principles proved in the last two Propositions
reduce the calculation of the motion of a rigid body moving
freely and acted on by impulsive forces to the calculation of
the motion of a single particle, and of a rigid body moving
about a fixed point. We shall now determine more con-
460
DYNAMICS. RIGID BODY.
venient equations for calculating the rotatory motion of a
body about its centre of gravity when acted on by impulsive
forces.
PROP. To calculate the rotatory motion of a body
moving about a fixed axis and acted on by impulsive forces.
477. The equation for determining the rotatory motion
is (Art. 473.)
n { x (
Σ.mxV cos b -
dy
dt
(
dx
− y |V cos a
=
0,
dt
the axis of being the axis of revolution; let
X
of any particle m from this axis:
with the plane ≈≈;
be the distance
the angle which makes
.. x = r cos 0, y
y = r sin 0;
dx
d Ꮎ
do
dy
d Ꮎ
d Ꮎ
" sin e
Y
= 7' cos 0
V
dt
dt
dt' dt
dt
dt
.. Σ.m
(2
dy
da
de
d Ꮎ
-y
Σ . m (x² + y²)
Σ.mr²;
d t
dt
dt
dt
do
dt
Σ.mp.2
Σ.m V (x cos b − y cos a)
moment of the impressed forces about the axis
moment of inertia about the axis
PROP. A body in which one point is fixed is acted on
by impulsive forces: required to determine the motion.
478. Let the fixed point be the origin: and let rr2r3 be
the distances of the particle m from the axes of x, y, ≈ and
let 0,020 be the angles which ₁₂ make respectively with
717273
3
the planes xy, ZY, ZX,
=
Y "'₁ cos 0₁,
= 1, COS 02,
λ =
≈ = 7'₁ sin 0₁,
2 = 12 sin 0,
Өз
13 cos 03,
COS
≈ = 7'3 sin 03 ;
AXIS OF SPONTANEOUS ROTATION.
461
dz
y
dt
3
dy
2
2°
dt
do₁
dt
dx
d z
dᎾ,
2
Z
20
12
dt
dt
dt
dy
dx
d Ꮎ.
2
X
У
2°.
73
dt
dt
dt
Hence the last three equations of Art. 473. become
de,
Σ.mr² = Σ. m V (y cos c
≈ cos b),
dt
dᎾ
2
Σ.mr₂ = Σ. m V (≈ cos a
x cos c),
dt
dox
dt
Σ . m r² = Σ . m V (x cos b − y cos a),
ᏧᎾ .
dt
do
dt
Οι
dex
dt
sum of moments of the impressed forces about axis of a
moment of inertia about axis of a
sum of moments of the impressed forces about axis of y
moment of inertia about axis of
Y
sum of moments of the impressed forces about axis of ≈
moment of inertia about axis of ≈
When these are integrated we shall know the position of the
body at the time t with respect to the fixed point.
We shall apply the principles proved in the last Articles to
the solution of a few questions.
When a body at rest is acted on by any forces there is a
line about which it begins to revolve. This line is called the
Axis of Spontaneous Rotation.
PROP. To find the position of the axis of spontaneous
rotation of a body when it is acted on by an impulsive force.
479.
Let P be the momentum which measures the im-
pulsive force: M the mass of the body.
462
RIGID BODY.
DYNAMICS.
Then by Art. 474. the centre of gravity moves with the
P
velocity in a direction parallel to that of the impulse.
M
Let the line of the impulse be taken for the axis of a:
and the plane through this and the centre of gravity for the
plane of xy: h and the distances of the centre of gravity
from the line of impulse and from the projection on the plane
ry of any particle m: the angle which r makes with the axis
of x.
Hence by Arts. 475. and 477.
ᏧᎾ
moment of P
Ph
dt
moment of inertia
Mkt
k being the radius of gyration about the centre of gravity.
Let wy be the co-ordinates of the projection of m: then
by compounding the velocities of translation and rotation, we
have (fig. 108).
P
də
P
Phr sin ✪
vel. of m parallel to x =
r sin
M
dt
M
Mk2
dᎾ
Phr cos 0
Y
?' cos
dt
Mk²
To find the point in the plane ay which is at rest at the be-
gining of the motion we must equate these two velocities to
zero;
k²
... k2 hr sin = 0,
.. 0 = 90º, and r
0
=
k2
h
=
» cos 0 = 0;
GO in the figure,
and therefore the axis of spontaneous rotation is at right
angles to the direction of the impulse; and also cuts at right
angles the perpendicular from the centre of gravity upon the
k2
direction of the impulse at a distance the centre of gravity
h
lying between the axis of spontaneous rotation and the impulse.
The point O coincides with the centre of oscillation, if H
be the projection of the axis of suspension: see Art. 432.
CENTRE OF PERCUSSION.
463
PROF.
A body revolves about a fixed axis and impinges
upon a fixed point, so that the direction of the impulse is
perpendicular to the plane passing through the axis and the
centre of gravity: required to find the position of the fixed
point so that the pressure on the fixed axis at the instant of
impact may be wholly in the plane perpendicular to the direc-
tion of the impulse. The fixed point so found is called the
Centre of Percussion.
z
480. Let the fixed axis be the axis of ≈ and the plane
passing through the centre of gravity at the instant of the
impulse the plane yz. P the momentum which measures the
impulse on the fixed point: y,, the co-ordinates to the point
in which the direction of the impulse cuts the plane yz: r the
distance of any particle m from the axis of rotation: the
angle makes with the plane a at the instant of impact:
then arcos 0, y = r sin 0: and the velocities of (m) parallel
to x and y at the instant of impact are
dy
dt
N
do
dt
:
dx
dt
(dar = )
d Ꮎ
-Y
and
d t
and therefore the momenta parallel to the
axes are at that instant my
d Ꮎ
dt
ᏧᎾ
aud mx
dt
fixed at two points (since
reduced to two) of which
We shall suppose the axis to be
if fixed at more they can always be
the distances from the origin are a and a'; let R cos a,
R cos B, R cos y, R' cos a', R' cos B', R' cos y be the mo-
menta which measure the impulsive pressures parallel to the
axes of and y on these points at the instant of impact:
distance of the centre of gravity from the axis of ≈.
the
The forces, then, which act upon the body at the instant
of impact are
de
R cos a, R' cos a', - my
on each particle m, and P parallel to v,
dt
də
R cos ß, R' cos B',
M X
on each particle m,
·Y,
d t
R cos Y,
R'
cos y', ..
X.
464
DYNAMICS.
RIGID BODY.
But the body is reduced to rest, by hypothesis; and conse-
quently by the six equations of Art. 65. we have
ᏧᎾ
R cos a + R' cos a'
Σ.my + P = 0,
dt
do
R cos B+ R' cos B' +
Σ.mx=
0,
R
R cos y + R' cos y' = 0,
dt
d Ꮎ
-
R cos B. a
R' cos B'. a'
Σ.mxz = 0,
dt
d Ꮎ
R cos a. a + R' cos a'. a'
Σ.myz + P≈₁ = 0,
Pz,
dt
do
Σ . m (x² + y²) − Py, = 0,
dt
in these Σ.mx = 0, and Σ.my = Mh, as the axes have been
chosen. From these equations the pressures may be found.
Now for the centre of Percussion we must have R cos a =
and R' cos a' = 0: hence
= 0
d Ꮎ
do
Mh
+ P = 0,
Σ.myx + P≈, = 0,
dt
dt
do
Σ . m (x² + y²) − Py, = 0 ;
dt
therefore P = Mh
dᎾ .
dt
is known, since the motion previous to
ᏧᎾ
the impact, and consequently , may be calculated by the
dt
principles of Chapter XI: and the co-ordinates to the centre
of percussion are
Y
Σ. m (x² + y²) k² + h²
Mh
h
22
Σ.my z
Mh
If the body be symmetrical about a plane through the centre
of gravity and perpendicular to the axis of, then, if I be
CENTRE OF PERCUSSION.
465
the distance of the centre of gravity from the plane xy, and
if x = + we have
≈
Σ.myx = xΣ.my + Σ. myx' = zΣ.my = Mõh.
In this case y,
k² + h²
h
X = and the centre of per-
cussion will then coincide with the centre of oscillation: see
Art. 432.
In the Chapter of Problems we shall give some examples
of the impact of bodies.
SN
CHAPTER XIV.
THE MOTION OF A FLEXIBLE BODY.
481. IN the present Chapter we shall calculate and ex-
plain some of the simpler cases of the motion of vibrating
strings: for more information on this subject and on the
motion of elastic springs we refer the reader to M. Poisson's
Traité de Mécanique, Tom. II. Seconde Edition; to the
Journal de l'Ecole Polytechnique, Cahier xvIII, p. 442; and
lastly to M. Poisson's Memoir on the equilibrium and motion
of elastic bodies in the Mémoires de l'Académie des Sciences,
Tom. VIII.
PROP.
To determine equations for calculating the motion
of a perfectly flexible cord, very slightly extensible, of the
same thickness and density throughout, fixed at its two
extremities, and very little disturbed from its position of
rest.
482. Let A and B be the fixed extremities of the cord
(fig. 109), we shall suppose that the cord is straight when in
equilibrium: Let P be the position of a particle of the cord in
motion at the time t which is at Q when the cord is at rest:
AQ = x, AM = x + U,
= x+u, MN = y, NP =≈, AB=l: M the
mass of the cord, T the tension at the point P: the resolved
parts of T parallel to the axes are
d (x + u)
dy
dz
T
T
T ; where ds = PP';
ds
ds
ds
the excesses of the corresponding tensions at P' over those.
at P are therefore, by Taylor's Theorem,
VIBRATING CORD.
467
T
dx
d (pd (x+u)) dx
dx
d (rdy)
d
T dx,
ds
dx
(Tas)
dz
ds) dx:
these are the impressed forces acting on PP': the mass of
ds
PP' M hence the effective forces acting on PP' are
ī
M
ds d'u
l d f
M
ds d²y
l dť
ds dⓇ z
M
l
1 de:
dť
hence by the first Principle of Art. 226 we have
u)]
d {p d ( x + x)}.
T
dx
ds
d
T
M ď² u
l dť
dx ds
d (1 dy)
M dy
I dť
T
dz
M ď²
2 (142) - 1 F
(ra
dx
ds
dť
Let IV be the tension of the cord when at rest; then it
is found by experiment that the change in extension (as
ds – dx,) of a piece of cord de varies as the change in tension
T-W: suppose that T - IV
ds²
Now (1.
dx²
+
du
dx
+
=
Q
ds dx
dx
dy
dz²
+
dx²
dx²
(1 + du) ²,
dx
و
neglecting small quantities of the second order. Hence
and our equations of motion become
du
T - W = Q
dx
ď u
dt
d² u
d² y
ď y
ď z
d&
b2
a²
a²
,
dx²
dt
da
df
dx2
23
if we neglect small quantities of the second order, and put
Ql = Mb², and WI Ma²: hence a and b are in the ratio
WV : Q.
The variables u, y, are separated in these equations;
from which we conclude that the vibrations of the cord parallel
to the axes of x, y, are independent of each other, and
468
FLEXIBLE BODY.
DYNAMICS.
co-exist without any interference. The transversal vibrations
are the same in the directions of y and x. We shall calculate
the motion parallel to y.
PROP. To integrate the equations of motion, and to
interpret the integrals.
483. For the transversal vibrations,
ď y
d² y
a²
d t²
dx2
To integrate this add to each side a
d²y
dxdt
.
d [dy
dt dt
dyl
d (dy
dy
+ a
a
dx
d v
d v
dy
or
a
if we put
+ a
dt
dx
dt
da dt
dy
dx
dx]'
+ a
=v:
d v
d v
d v
.. dv
dt +
dx =
d (x + at),
dt
d x
dx
dy
dy
.. v, or
+ a
'(x + at);
dt
da
representing any arbitrary function of x + at.
In like manner by subtracting a
d² y
from each side we
dxd t
have
dy
dt
dy
α
=
'(x — at)
dx
representing another arbitrary function.
dy
Hence
dt
p' (v + at) + 1 f'(x − at),
dy
1
1
g'(x + at)
d x
2a
a
2 / 4' (x - at);
VIBRATING CORD.
469
dy
dy
:. dy
=
dt +
dx
dt
dx
·
1
1
— 4' (x + at) d ( x + at) — — — y' (x − at) d (x − at) ;
2 a
2 a
.. y = F(x + at) + f (x − at),
F and ƒ being arbitrary functions depending on o' and '; but
we shall cease to use and y'.
We proceed to explain how to determine the values of
these functions. We are supposed to know the initial cir-
cumstances of the motion, namely the values of y and
dy
dt
for
all values of a between a O and xl when t=0: hence
F(x) +f(x) is known for all values of a between 0 and 7: and
also
dF(x) df(x)
is known and consequently F(x) and
dx
dx
f(x) are known between these limits; but the values of these
quantities for values of a greater than 7 and less than 0 are
not given, nor are they necessarily known since the functions
F(x) and f(x) may be discontinuous; that is, the original
form of the curve need not be such as can be expressed in
analysis, but may be a series of pieces of curve so long as they
have the same tangent at their points of junction.
The condition that the extremities of the cord are always
stationary enables us to determine the values of F(x+at) and
f(xat) for all values of a and t. For by this condition
= 0 and
O when x = 0 and 7 whatever t be: hence by
dt
putting at = v we have
У
dy
F (v) + ƒ(− v) = 0 ... (1), _F (l + v) + ƒ (l − v) = 0 ... (2)
for all positive values of v.
Put + v for v in (2), then by (2) and (1)
F (2l + v) = F (v)
(3).
The initial circumstances make known F (v) and ƒ (v)
from v = 0 to v = 7: then (2) gives F (v) from v = l_ to v = 2,
470
FLEXIBLE BODY.
DYNAMICS.
0 to v = ∞ :
and then (3) gives F (v) from v = 21 to 4l, then to 67 and so
on to hence F(x + at) is known for all positive values
of t. Again, by (1) ƒ(− v) is known from v
but also ƒ(v) is known from v = 0 to v by the initial
circumstances, hence f(x − at) is known for every point of
the cord during the whole motion. Hence the value of y,
and therefore the form of the string, is known at every instant.
There is nothing to make known F (v) for negative values of v
or ƒ (v) for values of v between 7 and ∞.
In (1) put v + 2l for v, then
ƒ (− 2l − v) = − F (21 + v) = − F(v) by (3)
Hence y
=
= f(v) by (1) ………………….. (4).
F' (x + at) + f (x − at)
F (x + at + 21) + f (x at 21)
→
−
F (x + at + 2nl) + f (x − at − 2nl) by (3) (4)
-
n a positive integer: hence the cord repeatedly assumes the
same form relatively to the plane a≈, performing a vibration
21
in the time
; substituting for a its value,
α
time of vibration
MI
= 2
W
the same is true of the motion parallel to ≈: and also parallel
to a the oscillations take place in the time 2
Μι
Q
PROP. A portion only of the cord is set in motion at
first, as a piano-forte wire by the sudden blow of the ham-
mer of a key: required to determine the motion.
484. To simplify the calculation we shall at first suppose
that one end of the string is at an indefinitely great distance.
Let the original displacement extend over a small space 2a;
and let the origin of a be at the mid-point of this space: h the
VIBRATING CORD.
471
な
​distance of the nearest extremity.
Then when t = 0 we have
dy
y = 0 and
0 from x =
∞ to x = a and from x = a
dt
to x =
h;
... F(v) = 0 and ƒ (v) = 0
(1),
from v =
∞ to v =
ɑ,
and from v = a to v =
= h.
Because of the fixed extremity
F' (h + v) + f (h − v) = 0,
for positive values of v.
Hence when v is greater than h,
F (v) = − ƒ (2h − v)
(2).
Since xat is always negative for negative values of a, it
follows by (1) that beyond the limits of disturbance, that is when
isa, we have f(x-at) = 0: also for negative values of a
x
we have F (x + at) = 0 unless t lie between
α Ꮗ
α
and
a x
α
Hence the initial disturbance is propagated to the left and each
particle of the cord begins to move after a time
for a time
2 a
and then returns to rest.
α
α- x
,
vibrates
a
In the same way the motion is propagated to the right:
but in consequence of the fixed extremity this will require
a little further examination.
Let us consider the motion of a particle at a distance ~;
y = F(x + at) + f (xat) is its general displacement.
When to this particle is at rest, for y = 0 by (1): and
it remains so till t
for, then y = f (a), and the particle
2
α
α
x + a
moves till t
and after this is at rest again for a time.
a
Ever after this a -at is negative and < -a, and therefore
f(x − at) = 0 by (1). But when t becomes
472
FLEXIBLE BODY.
DYNAMICS.
Xx
α
2h-2x
2h-x
a
+
or
α
a
a
-
y = F(2h − a) = f (a) by (2),
x + a 2h-2x
and as t increases and becomes
a
+
a
2 h − x + a
;
a
y = F(2h + a) = f(-a) by (2).
G
2h -2x
Hence at a time
a
after the particle began to move, it
again begins to move: and ceases to move at the same time
after it ceased before. Likewise the displacements of the
particle are exactly the same that they were before, but on the
opposite side of the line of rest.
2h
x + a
When t is >
x + at is > 2 h + a;
a
.. F(x + at) = -ƒ {2h - (x + at)} = 0 always.
2α
Hence the particle oscillates for a period
commencing
at the time
α
α
α
: it then rests: and after a time
2 (h − x)
a
-
it oscillates for the same period in a manner precisely similar
to the former; except on the opposite side of the line of rest:
after this the particle remains permanently at rest. This is true
2 (h − x)
whatever be: and it is to be remarked that
a
-
the
time the bent portion of the cord, or the pulse, would take to
move from the particle to the fixed point and back again.
Hence, since this second motion arises solely in consequence
of the fixity of one extremity of the string, it follows that when
the right hand pulse reaches the fixed point it is reflected, but
to the opposite side of the string.
Hence the original disturbance divides itself into two
pulses, one moving continually to the left; the other to the
right till it reaches the fixed point, after which it moves back,
towards the left, on the other side of the line and with the
same velocity as before.
VIBRATING CORD.
473
Let the string be of definite length. Then the pulse after
reflexion will be reflected again at the other extremity and move
on the upper part of the string to the right.
Let C, D (fig. 110) be the fixed extremities of the string
and A the origin of disturbance. The initial disturbance
divides into two a and b: b is reflected at D and moves along
to b'at the same time that a moves to a' having been reflected
at C: a and b meet at B and confirm each other forming
a disturbance exactly similar to the original disturbance.
Evidently B is such a point that
AD + DB = AC+ CB, or AD CB.
Now the interval between two maximum disturbances at
A = time of describing AC + CD + DA
velocity of the pulses = a.
21
=
Hence the
α
3 0
..
CHAPTER XV.
GENERAL DYNAMICAL PRINCIPLES.
485. In the present Chapter we shall prove some general
principles of the motion of a material system, which are con-
sequences of the laws of motion.
PROP. When a material system is in motion under the
action of forces, whether finite in intensity or impulsive, but
none of them being extraneous to the system, then the centre
of gravity moves uniformly in a straight line or remains at
rest.
486. The internal forces which act upon the particles of
the system may be of the nature we have denominated im-
pulsive, or they may be of finite intensity, or they may be
some impulsive and some finite. But impulsive forces are such
as act only for an indefinitely short time; and consequently
during the period of their action the finite forces can produce
no effect.
We may therefore consider the action of these two
systems of forces on the system separately.
I. Suppose the forces are of finite intensity.
Let XYZ be the internal accelerating forces acting ou
a particle m of the system parallel to the axes, not including
the molecular action: xyz the co-ordinates to m. Then the
first three equations of Art. 427, give
Σ.m (x-6x)
dx
=
0,
Σ.m Z
d²
dt
A)
0.
2. m (Y - 1) =0, 2.m(2-)=0
d
Now since, by hypothesis, the forces acting on the system
are all internal, or, in other words, arise from the mutual action
MOTION OF THE CENTRE OF GRAVITY.
475
-
of the particles of the system, it follows that if m' X', m' Y',
m'Z' be forces acting on the particle m' there must be in the
system forces equal to m'X', m'Y', - m'Z' acting on
some other particle: hence these forces will disappear in the
expressions Σ.mX, E.mY, Z. mZ; and this will be the case
with all the forces;
.. Σ.mX=0,
and consequently we obtain
Σ.mY=0, Σ.mZ = 0,
E. mY = 0,
d² x
d'y
ď z
Σ.m
0, Σ.m
=
0,
Σ.m
0.
dt
dť
d t2
Let x y
y be the co-ordinates to the centre of gravity of
the system at the time t; x,y,, co-ordinates to m measured
from the centre of gravity: then x = x + x
≈ = ≈ +≈,; and since Σ.mx,
the last three equations give
درنا
У
y = y + Y,,
0 and Σ.mx, = 0,
0, Σ.my,
ď x
d²y
d²
0,
0,
: 0;
d t2
dt2
dt2
dx
dy
dz
0
V₁ cos a,
V₁cos B,
V₁cos y,
dt
d t
dt
= 0 and
V₁ being the velocity of the centre of gravity when t
aßy the angles its direction makes with the axes;
0
α

da²
dy²
dz
d≈2
.. vel. of centre of gravity at time t
+
+
Vo,
dt
d t
dt
and consequently is constant.
If V₁ = 0 then the centre of gravity remains at rest.
Again, by integration we have
2
a = V₁t cosa, y = Vt cos B,
Vt cos y,
the origin of co-ordinates being chosen at the point where the
centre of gravity lies when = 0;
cos a
cos B
cos Y
cos Y
X
Y
476
SYSTEM OF BODIES.
DYNAMICS.
These equations shew that the centre of gravity, if it be not at
rest, moves in a straight line.
II. Suppose the forces are impulsive.
Let V be the velocity that any particle m has when the
impulsive forces begin to act; a, b, c the angles its direction
makes with the axes: V' the velocity which measures the in-
ternal impulsive force acting on m: a'b'c' the angles of its
direction.
Then the first three equations of Art. 473, give
dx
Σ.mV cosa + V' cos a'
0,
dt
dy
-
0,
dt
dz
0.
dt
(v
Σ.mV cos b + V' cos b'
い
​Σ.m (V cose + V' cos c'
Now since the impulsive forces are all internal we shall have,
as may be shewn in the same manner as before, that
Σ.mv' os a' = 0, Σ.m V′ cos b′ = 0, Σ.m V' cos c' = 0.
Hence the equations become
Σ.m (V cos a
-da)
0
dt
=
Σ.m (V
0, Σ.m Veos b
V COS C
-
dz
dt
0.
dy
<
0,
dt
Let Vo be the velocity of the centre of gravity at the com-
mencement of the action of the impulsive forces: aßy the
angles its direction makes with the axes: then (Art. 413.)
Σ. mV cos a = V₁ cos aΣ.m, Σ.m V cos b =
0
2. mV cos c = V, cos yΣ.m.
0
βΣ.n
Vo cos BΣ.m,
0
CONSERVATION OF AREAS.
477
dx
dx
Also Σ.m
Σ.m,
dt
dt
2. m dy dy z.m
Σ.m,
d t
dt
d z
dz
Σ.m
Σ.m;
dt
dt
d x
dt
dy
dz
V₁ cos a,
=
V₁ cos B,
0
Vo cos y,
ولا
0
dt
dt
and
consequently the velocity of the centre of gravity is V,
therefore is not altered by the action of the impulsive forces.
Also, if the centre of gravity be in motion, the indefinitely
small space it describes during the action of the impulsive
forces is a continuation of the straight line in which it was
moving before their action.
COS a
For
Y
COS Y
cos B
COS Y
Z; the origin of co-ordinates
being taken at the point where the centre of gravity is situated
when the forces begin to act.
Wherefore the Proposition, as enunciated, is true.
This
Principle is called The Principle of the Conservation of the
Motion of the Centre of Gravity.
PROP. When a material system is in motion under the
action of forces, whether of finite intensity or impulsive, but
none of which are extraneous to the system; then the sum
of the products of each particle multiplied by the projection on
any plane of the area swept out by the radius vector of this
particle measured from any fixed point varies as the time
This is called the Principle of the Conservation
of motion.
of Areas.
487. I. Suppose the forces are of finite intensity.
Using the same rotation as in the last Proposition the
last three equations of Art. 427 are
Σ.η
m{y
d² & d²y
Σ . m (y Z − ≈ Y),
dt
dť
478
SYSTEM OF BODIES.
DYNAMICS.
Σ.m
{
ર
dt2
a
d2
Z
Σ. m (≈ X − x Z),
dt2
Σ. m / x
{
d² y
ď² x
y
Σ. m (x Y − y X).
dt2
dt2
We shall now shew that the second sides of these equations
vanish when all the forces are internal.
Let m'X', m' Y', m' Z' and m"X", m"Y", m"Z" be the
resolved parts parallel to the axes of the mutual actions
P' and P" of two particles m' and m"; then P' - P" and also
m'X' = - m" X", m'Y' - m"Y", m' Z' = -m" Z".
If the particles be in contact let x'y'' be their co-ordi-
nates; then these mutual actions will enter the expression
Σ. m (y Z - ≈ Y) in the form
m' (y' Z' − x' Y') + m" (y' Z″ – x'Y")
but this vanishes and in the same manner it may be shewn
that the mutual actions of particles in contact will disappear
from all three of the equations of motion.
Again; suppose the particles are not in contact, their
co-ordinates being x'y'z' and x"y"z": let r be their distance;
x'′ — x"
then
y' - y" ≈'
are the cosines of the angles
p
r
2°
which makes with the axes; and we have
—
m' X' = p' x' — 'x'', m' Y' = P
I'
y' — 'y'"', m' Z' = P'
2.
2°
and consequently the mutual actions of m' and m" enter the
expression . m (y Z-Y) under the form
Ρ
γ
P"
{y' (≈' — x″) — x' (y' − y″)} + · {y″ (x' — z″) — ≈″ (y' − y″)}
2
and this vanishes since P' - P": and consequently the mutual
actions in this case also disappear from the equations of
motion.
CONSERVATION OF AREAS.
479
Since then all the forces are supposed to be internal the
equations of motion become
ɛ.m(
ď² z
d²y
Σ.my
જ
0, Σ.m
d t²
dt2
(:
d² x
d
ď z
Z
X
0,
Σ.m
(
d²y
x
-y
dt2
d² x
dť
d t²
=
dt2
0,
and therefore by integration
Σ.my
(
dz
dy
dx
dz
Z
h, E.m
Z
ac
= h',
dt
dt
dt
dt
=h",
2. m (xdy - y da)
h, h', h" being constants.
У
dt
Let А½ be the areas swept out by the projections
of the radius vector of the particle m on the co-ordinate
planes respectively perpendicular to the axes of xyz during
the time t then by the last three equations
Σ.m
d Ac
d t
d A y
Ay
=
h, Σ.m
=h', Σ.m =h";
d Az
dt
dt
.. Σ.mÄ = ht, Σ.mA₁ = h't, Σ.m A₂ = h't,
since the areas are measured from the epoch when t 0.
Wherefore the Principle, as enunciated, is true for the
three co-ordinate planes arbitrarily chosen; and consequently
true for any plane.
II. Suppose the forces are impulsive.
Using the same notation as in Art. 486, the last three
equations of motion of Art. 473, are
Σ.m (ydz - zdy)
dt
dt
= Σ. m {V(y cos e − ≈ cos b) + V' (y cos c′ ≈ cos b')},
480
DYNAMICS. SYSTEM OF BODIES.
(
dx
dz
X
dt
dt
Σ.mx
-
x cos c) + V' (≈ cos a' — x cos c')},
= Σ.m {V (≈ cos a
-
Σ.m х
·Y
dt
dt
dy dx
= Σ. m {V (x cos b − y cos a) + V' (x cos b'
- y cos a
cos a')}.
As in the previous part of this Proposition, we might
prove that the mutual forces will disappear from these
equations. Hence
d z
dy
Σ.my
22
Σ.m V (y cos c − ≈ cos b) = h,
dt
dt
Σ.m
m ( 2
dx
dz
Z
ર
N
= Σ. mV (≈ cos a
--
x cos c) = h',
dt
dt
2. m (x dy - y de )
dx
dt
dt
ะ
Σ.m V (x cos by cos a) = h",
and therefore the principle of the Conservation of Areas is
true whatever the internal forces be.
488. COR. This Principle is also true when extraneous
forces act on the system, provided their directions all pass
through the same point and the areas are estimated about
that point.
For, this point being the origin of co-ordinates, suppose
that P' is one of the extraneous forces acting at the point
(x'y'x') at the distance r' from the origin: then P'. P', and
a
༡༩
y'
P' are the resolved parts of P': and this force consequently
vanishes from the expressions E. m (y Z-xY) and Σ.m V'
(y cos c'
cos b') and from all the analogous expressions.
-
PROP.
To prove that when a material system is acted
on by forces, whether of finite intensity or impulsive, but
INVARIABLE PLANE.
481
none of which are extraneous to the system, there is a plane
invariable in position during the motion, with respect to
which the motion may be estimated: and to find its po-
sition.
489. We have seen in Art. 487. that
Σ.m A₂ = ht, Σ. mA, = h't, Σ. m A₂ = h″t,
hhh" being constant quantities which depend upon the con-
figuration of the system and which may be found by observing
the motions of the bodies of the system and then calculating
Σ.md., Σ.mA, Σ.mA..
Hence a plane drawn at the time t perpendicular to the
straight line which makes with the axes the angles of which
the cosines are
Σ.m Ar
Σ.m Ay
(E. m A¿)² + (Σ . m A„)²+(E.mA.)²' √(Σ.m A¿)² + (Σ· mA₁)² + (Σ.m A.)²
and
Σ. 4.
2
(2.m A¸)² + (Σ.m Aŋ)² + (Σ.m A₂)²
remains invariable in position during the motion.
For this reason it is termed the Invariable Plane.
PROP. To prove that the Invariable Plane is that with
respect to which the sum of the moments of the momenta of
the different particles of the system is a maximum.
490. Let Aur be the angles which the Invariable Plane
makes with the co-ordinate planes respectively perpendicular to
the axes of xyz.
yz
The momentum of the particle m parallel to the plane
dive
is m
√dyⓇ
+ : the perpendicular from the origin upon
dť d t²
the tangent to the projection of the curve in which m is moving
dz dy
Y
dt
dt
on the plane yz =
Hence the moment of the
dy²
dv2
+
dt
dt
3 P
482
SYSTEM OF BODIES.
DYNAMICS.
dz
dy
momentum of m about the axis of x = m Y
and
dt
dt
dz dy
the sum of the moments
Σ.my
Z
=
dt
dt
h and the
sums of the moments about the axes of y and ≈ are h′ and h".
Hence the sum of the moments about the line perpendicular
to the plane which makes the angles λuv with the co-ordinate
planes equals
h cos λ + h' cos µ + h" cos v ;
and when this is a maximum
dv
dv
h sin λ+h'" sin v
= 0 and h' sin u + h' sin v
=
0,
d x
αμ
but cos² + cos² μ + cos²
cos² v = 1;
h
sin v dv
cos λ
h
sin v dv
cos μ
and
;
h"
sin λ dλ
COS V
h"
sinu du
COS V
h² + h¹² + h″²
12
112
1
... cos v =
h'2
cos² v
h"
12
√ h² + h'½ + h'¹²
and in like manner
COS
cos μ =
h'
12
√ h² + h²² + h'¹2
and cos
h
12
√ h² + h²² + h's
But these determine also the position of the Invariable
Plane (see Art. 489.) Hence the Invariable Plane possesses
the property that the sum of the moments is greater on that
than on any other plane. Also the maximum sum of moments
is always the same, since it equals
s v = √ h² + h² + h″².
h cosλ + h' cos μ + h' cos v =
491. If the position of this plane be calculated upon the
supposition that the heavenly bodies are intense particles with-
out rotatory motion it is found that h, h', h" are constant even
in carrying the approximation to the squares and products of
the masses, whatever changes the secular variations may induce.
in the course of ages: hence it follows that the invariable plane
retains its position notwithstanding the secular variations in the
INVARIABLE PLANE.
483
elliptical elements of the planetary system. The determination
of the position of the invariable plane requires a knowledge
of the masses of all the bodies in the system and of the elements
of their orbits. Now we know the masses of the planets only
approximately; but of the masses of the comets we are in total
ignorance. But from the agreement of theory and observation,
mentioned above, we learn, that, hitherto at least, the action
of the comets on the planetary system is insensible. Laplace
has shewn that the comet of 1770 passed through the system of
Jupiter and his satellites without producing the smallest effect,
though its own motion was much perturbed.
If the position of the ecliptic in the beginning of 1750 be
taken as the fixed plane of ay and the longitudes measured
from the line of the equinoxes, it is found that at the epoch
1750 the longitude of the ascending node of the invariable plane
was 102°57′30″, and its inclination on the ecliptic was 1°35′31″:
and if these be calculated for 1950 they are 102°57′15″ and
1°35′31″; these differ but very little from the former, and
therefore shew that the motion of the ecliptic in space is
exceedingly slow.
492. It is important to remark that the terms in the
equations of motion of Art. 487. which depend upon the mutual
action of the parts of the system would disappear even when
the intensity of the forces varies with the time, independently
of the distance: i. e. when the expression for the force is an
explicit function of the time. For in this case the invariability
of the principal moment and of the direction of its axis is
preserved.
493. This shews that the loss of heat sustained by the
particles of the system by radiation though it diminishes the
intensity of their mutual action, yet has no effect on the
position of the invariable plane or on the principal moment.
So that if we leave out of consideration the action of the Sun
Moon and planets on the Earth, and suppose that our planet
were at one time in a gaseous state, and become solid by re-
frigeration without losing any position of its ponderable matter,
we may feel assured that the principal moment of the system
has not altered in magnitude nor has its axis changed its
position during the change of condition of the globe.
484
SYSTEM OF BODIES.
DYNAMICS.
If M be the whole mass: k the radius of gyration about
the axis of principal moments through the centre of gravity at
the time t, w the angular velocity about this axis; then
Mk² w =
principal moment = constant.
This shews that if the Earth radiate its heat into space so as
to diminish its radius by contracting its dimensions, then, since
k varies as the radius, w will be increasing and the length of
the day shortening.
Now it has been proved in Art. 461, by calculations of
eclipses, that within the last 2556 years the length of the day
is not become shorter by even a ten millionth part: and there-
fore, since w varies inversely as k³, or the length of the day
varies as the square of the mean radius, the mass remaining
the same, the mean radius of the Earth has not changed.
within the last five and twenty centuries by even a twenty-
millionth part.
494. The appearance of fossil remains of tropical plants
and animals in these higher latitudes has induced geologists
to adopt the hypothesis that the temperature of the Earth
was in ages gone by far higher than at present. The
results of the last Article shew that no objection can be
urged against this theory, at least upon mechanical prin-
ciples. If this hypothesis be true we learn that the radia-
tion goes on now very slowly, whatever its rapidity may
have been at more ancient epochs.
495. The Principle of the Conservation of Areas, or
rather the Principle of the Conservation of the Principal
Moments which springs from it, proves that earthquakes,
volcanic explosions, the action of winds upon the surface of
the Earth, the friction and pressure of the Ocean upon the
solid nucleus of the terrestrial spheroid, produce no variation
in the principal moment on the direction of its axis: since
these forces all arise from the mutual action of the parts.
of the system. And since the displacements produced by
these causes in any portions composing the Earth's mass are
too inconsiderable sensibly to alter the value of k, it follows
that their effect on the angular velocity (w) and upon the
length of the day will be inappreciable.
PRINCIPLE OF VIS VIVA.
485
PROP. When a material system is in motion under the
action of forces not impulsive, and none of which are
extraneous to the system; then the change of the Vis Viva
of the system during a given time depends only on the
co-ordinates of the particles of the system at the beginning
and end of the given time, and not at all on the curves
which the particles describe.
496. This is called the Principle of Vis Viva.
Let XYZ be the impressed accelerating forces which act
upon the particle m resolved parallel to the axes of co-
ordinates: xyz the co-ordinates to m at the time t: then
the forces
Ꮓ
m (x-da), m (y-dy), m (z-d).
acting on m, and similar forces acting on the other particles
of the system will satisfy the conditions of equilibrium;
(Art. 226.) Wherefore by the Principle of Virtual Velocities
(Art. 72.) we have
x
Sx
Ꮓ
d²
. m { ( x − de ) 3.x + (Y - 1) by + (z – dz) az} = 0,
Σ.m
d t
d
dt
Sx, dy, dx, being any small spaces geometrically described
by m parallel to the axes, in a manner consistent with the
connexion of the parts of the system one with another at
the time t.
Now the spaces actually described by the particle m
during the instant after the time t parallel to the axes
are consistent with the connexion of the parts of the system
one with another: hence we may take
Sx st, бу=
dx
dt
dz
dy
dt, dz
St,
dt
dt
and the above equation becomes
Σ.m.
Ida
dx d²x
dy day
dz d² z]
+
+
Σ. Χ
m
dt dť
dt dť
dt dť
(
dx
dy
dz
+ Y
+ 2
d t
dt
dt
486
SYSTEM OF BODIES.
DYNAMICS.
(dx²
dy²
d22
.. Σ.m
+
dt dť²
+
d t²
dx
dy
dz
dt
- 2 2. mf (x + y + z) dt + C.
Σ.m
But by the Differential Calculus
d.x2 dy² dx2 ds2
dt
dt
+
+
(velocity)² = 12,
d t2
dt2
d t2
dt
dx
dy
dz
+ Y
+ 2
dt + C.
dt
dt
dt
::. Σ. mv² = 2 Σ .m
Σm f ( x
Now let P be the mutual pressure of two particles
m and m' in contact at the point xyz: aẞy the angles
its direction makes with the axes.
Then the expression mfx
dx
dy
dz
+ Y
+ Z
dt,
dt
dt
dt
for the particle m becomes
dx
dy
dz
cos a
+ cos B
dt.
+ cos y
dt
dt
d t
and for the particle m' it becomes
-SP (
dx
dy
d
Cos a
+ cos B
β
+ cos Y
dt,
dt
dt
dt
and the sum of these 0, and therefore P will not appear
in our final equation.
=
Again, let xyz, x'y'z' be the co-ordinates of two par-
ticles m and m' not in contact; their distance; P their
mutual action, supposed to be a function of r: then the
cosines of the angles which the direction of P makes with
y-y'
༧-༧
the axes are
:
and the expression
2°
2°
m f(x
dx
dy
dz
+ Y
+ 2
dt becomes, for the particle m
d t
dt
dt
PRINCIPLE OF VIS VIVA.
487
SP
x − x dx
+
Ju
dt
y - y' dy
+
グ
​dt
*
ર
z dz
ጥ dt
dt
+
dt
↑
and for the particle m'
-SP
z dz
x - x' dx'
day-'y dy ==' da dt,
Hence P will appear in 2 E.m
under the form
dt
dx dy
+ Y
dz
+2
dt
dt
dt
+
ተ dt
グ
​√(x
dt
P
2 ) = {(x-a') d (x=x')
-
d (y-y')
+(y-y')
+ (x−2') d ( x = x ) } dt.
(z-z')
dt
dt
dt
}
•
7º
or 2 SP dr; since = (x - x)² + (y − y′)² + (≈ − x')².
Wherefore we have finally
Σ.mv² = 22. [Pdr + C,
and since P is a function of r the second side of this
equation, when taken between limits, will be a function
solely of the initial and final co-ordinates of the particles
of the system.
Hence the Principle is true.
497. COR. 1. The expression for the vis viva of a
system acted on by any forces (not impulsive) is given by
the equation
pdy
Σ. mv² = 2 Σ.m
22. m. f(x
dx
+ Y
dt
d t
dz
+ Z dt.
dt
498. COR. 2. Any force which acts upon a fixed point
of the system will not appear in the equation of vis
viva, since the velocities of the point are nothing. In this
way the mutual pressures of any parts of the system against
immoveable obstacles will not appear. Neither will the force
of friction which acts upon a body rolling (not partly rolling
and partly sliding) upon a fixed obstacle appear in the vis
viva; since the point of contact is for an instant at rest.
1
488
SYSTEM OF BODIES.
DYNAMICS.
499. COR. 3.
COR. 3. If forces act upon none of the particles
of the system except such as remain invariably connected
during the motion, then the vis viva remains the same
throughout the motion. For in this case
fore .mv² = C.
dr
dt
0; and there-
This is called the Principle of the Conservation of Vis
Viva.
PROP. The vis viva of a material system in motion is
equal to the vis viva arising from the motion of translation
of the centre of gravity in space added to the vis viva arising
from the motion about the centre of gravity.
500. Let xyz be co-ordinates to m at time t,
xyz be co-ordinates to centre of gravity of the system,
and let x = x + x,, Y = y + Y,, Z = ≈ + ≈,;
2
da² dy dx² dx2 dy² de 2
+
+
dt dť dť
.. v² = (vel. of m)
+
+
d t
dt
d t²
x
+2
jda dx dy dy
dz dz
dx
2
dy2
dz²
2
+
+
+
+
dt dt
dt dt
d t d t
dt2
d t²
dt2
dx
dy,
dz
and observing that E.m
0, Σ.m
= 0, Σ.m
0
dt
dt
dt
we have
Σ. mv² = √² Σ . m + Σ. mv²,
being the velocity of the centre of gravity of the system
and v, the velocity of m relative to the centre of gravity.
501. By Art. 496. we have
2 Σ. m Pdr
d (E. mv)
(Σ.m
dt
dt
therefore whenever during the motion the particles of the
system assume such a relative position that the vis viva is
a maximum or minimum 2. m Pdr =0, and therefore (Art. 81.)
the system is at that instant in a position in which the forces
are in equilibrium.
VIS VIVA; IMPULSIVE FORCES.
489
When the vis viva is a maximum the position which the
system assumes would be a position of stable equilibrium, if
all velocity be destroyed: and when the vis viva is a minimum
the position would be one of unstable equilibrium. This
readily follows from considerations analogous to those men-
tioned in Art. 79. Also since a function passes through its
maximum and minimum values alternately as the variable
increases continuously, the system when in motion will pass
through the positions of stable and unstable equilibrium alter-
nately.
PROP. When a material system in motion is acted on
by impulsive forces, none of which are supposed external
to the system, vis viva is lost or gained according as the
impulse is of the nature of collision or explosion. When
the system is perfectly elastic the vis viva is the same before
and after the impulse.
502. Let V cosa, Vcos b, Vcos c be the velocities of
any particle of the system m parallel to the axes at the com-
mencement of the impulse: P the resultant of the internal
forces acting on m, and aßy the angles its direction makes
with the axes. Then the forces
dy
m V cos a + P cos a m
dx
dt'
m V cos b + P cos ß - m
dt
d z
dt
m V cos c + P cos y — m
acting on m parallel to the axes and similar forces acting on
all the other particles of the system will satisfy the conditions.
of equilibrium (Art. 226).
Hence by the Principle of Virtual Velocities (Art. 72.)
{(cosa -
P
m
dv
Cos a
Sxx
dt
P
dz
V cos c +
COS
y
M
dt
d) 8 x ) = 0,
Σ.m
V cos a +
P
+ (Vcos
V cos b +
dy
cos B - du
Sy +
M
dt
3 Q
490
DYNAMICS. SYSTEM OF BODIES.
Sx, y, z being any small spaces geometrically described by
m parallel to the axes in a manner consistent with the con-
nexion of the parts of the system one with another at the
time t.
We shall first observe, that P will disappear from the
equation above. For if P be the action between two bodies
of the system which touch each other in the point wyz, then
Sx, Sy, Sz will be the virtual velocities of the point of contact
with respect to P acting on one, and - da, -dy, - 8x those
with respect to the other body; and consequently in the above.
expression when a term of the form Pcos ada occurs we find
also P cos ada; and therefore P disappears, and the equa-
tion becomes
Σ.m
V cos a
dx
dt
Sx +
(v
V cos b
―
бу
dy) Sy
dt
z
+
V cos c
dt
da) 8=} -
ર
= 0... (1).
In applying this equation to calculate the motion of a
system suddenly acted on by impulsive forces we must make
a few important remarks. When a body yields or expands
the centres of its particles approach or recede from each other;
but, during the action of the impulsive forces, the spaces
through which they yield or recede are so extremely small,
that we wholly neglect them; but this is not the case with
their velocities, for although the change of distance of the
centres of the particles during the impulse is indefinitely small,
yet this change divided by the time elapsed during the impulse
will give a difference of velocities which is not necessary in-
sensible. In consequence of this, when two bodies come into
collision the particles in contact do not move with the same
velocity at the first instant of the contact, but after all com-
pression ceases and the restitution of figure has not begun to
take place, at this instant and at this instant alone, do the
particles in contact move with the same velocity. Again,
when two bodies are acted upon by impulsive forces of the
nature of internal explosion, the particles in contact move with
the same velocity at the first instant of the action of the forces,
VIS VIVA; IMPULSIVE FORCES.
491
but at every other instant of the action they move with
different velocities.
Sz
Now da, dy, dz may be any small spaces provided they
be consistent with the connexion of the parts of the system
one with another at the time of the impulse; this connexion
remains the same during the impulse, because all small spaces
described in that time are insensible. Wherefore we must not
give to these quantities such arbitrary values as will imply,
that the particles in contact at the point (xy) separate, or
penetrate each other, or (in other words) move with opposite
or unequal velocities.
If, then, the impulsive forces be of the nature of collision
and xyz be co-ordinates of the point of contact, the initial
velocities of the particles in contact will not be the same, but
after the collision ceases they will have the same effective
dx dy dz
Hence, in this case we may put
dt
dt' dt
velocities
dx
dx = dadt, dy
dt
dy St. Sz
dt
dz
St.
dt
since these virtual velocities are consistent with the connexion
of the parts of the system one with another, and they imply
that the particles in contact remain in contact when the prin-
ciple of virtual velocities is applied to the system in its
imaginary state of equilibrium.
If the impulsive forces be of the nature of internal ex-
plosion, then it will easily be seen, after what has been said,
that we may put
Sx
dx = cosast, Sy
dz
V cos adt, dy = V cos bdt, dx = V cos côt,
but we must not put the other values for da, dy, dz.
I. Suppose the impulse is of the nature of collision, the
bodies being inelastic.
Then substituting for da, dy, d≈ in equation (1), and
putting v for the resulting velocity of m
Σ . m v² = Σ . m V
dz
COS
E. mv (da
d
dy
cos a +
cos b +
dt
d t
dt
de cos of:
492
SYSTEM OF BODIES.
DYNAMICS.
-Σ.m
m{(1
2
dx
.. Σ.mv² = Σ. m V²
V cos a
sa
dt
-da) + (Vcos b-dy) + (V cose -
dx \ 2
dt
dt
2
and, since the last term of this is essentially negative, we see
that vis viva is lost during the collision.
II. Suppose the impulse is of the nature of internal
explosion.
By substituting in (1) the values of dx, dy, dz above
specified we have
Σ . m v ² = Σ . m v (dx
dt
dz
dy
cos a +
cos b +
de cos e
COS
dt
dt
}
.. Σ.mv² = Σ.m V²
2
+ Σ.m V cos a
-
da)² + (v
+ V cos b
dt
dy)
2
dz
+
V cos c
dt
t
and consequently vis viva is gained during the separation.
2
III. Suppose that the impulse is in part of the nature
of collision, and in part of the nature of explosion.
In this case we must combine the cases already mentioned.
When, for instance, the bodies are perfectly elastic the im-
pulsive forces which act during the collision are the same
exactly as those which act during the separation of the bodies
it follows, by examining the above expressions, that the vis
viva lost during the collision is exactly regained during the
separation, and that the state of the system is consequently
unaffected by the whole impulse.
503. The degradation of rocks and the consequent action
of collision which is incessantly taking place in large portions
of matter on the surface of the Earth, the unceasing action of
waves on the sea shore and the collision of the waters of the
ocean upon the solid nucleus of the Earth, and other like
causes are continually causing a loss of vis viva in the Earth's
mass, and if allowed to act without any compensating pheno-
EFFECT OF EARTHQUAKES &C. ON LENGTH OF THE DAY.
493
mena would in the course of time produce a sensible effect in
the length of the day but on the other hand the explosions of
volcanoes are compensating causes. Also the downward motion
of rivers, the descent of vapour and cloud in the form of rain,
the descent of boulders and avalanches, and various other
causes, all tend to remove large portions of matter nearer to
the Earth's centre and would in the course of time produce
a sensible increase in the length of the day, since we have seen
(Art. 499.) that the vis viva of the Earth is constant, if we
neglect the attraction of the Sun, Moon, and planets and con-
sider only the action of finite forces. But the ascent of vapour
by evaporation, and the effect of earthquakes and volcanoes in
removing masses of matter to a greater distance from the centre
have an opposite effect. On the whole all these causes balance
each other, since observations have shewn that the length of
the day has been invariable for many ages, Art. 461.
PROP. To prove that the variation of E. m/vds taken
between given limits equals zero, where v is the velocity and
ds is the element of the space described in the short time
dt by the particle m of a material system in free motion:
if any particle move on a surface it is supposed to continue
on the surface in taking the variation.
504. This is called the Principle of Least Action; be-
cause, in general, Σ.mfvds is a minimum.
Let & be the symbol of variation in the Calculus of
Variations: then
d (E. mfvds) = Σ. m f8 (vds) = Σ . m f(v d. ds + ds dr)
= Σ. mf (vd. ds + ½ dtd. v²).
Suppose the particle m rests on a curve surface, and that
R is the normal pressure, aßy the angles of its direction;
X, Y, Z the accelerating forces acting on m, then (as in
Art. 407)
ďa
dt²
R
X +
cos a
ďy
dť
R
ľ +
cos ß,
dz
R
Z÷
cos Y.
df
772
M
494
SYSTEM OF BODIES.
DYNAMICS.
Let L = 0 be the equation to the surface; then
d L
d L
d L
cos a
a =
V
cos B = V
s3
cos Y
V
dx
dy
d z
1
d L2
d L2
d L2
where
+
+
V2
dx²
dy
dz
Hence v² = 2(Xdx + Ydy + Zdz) = p(x, y, z) + const.
1 d. v² = X 8x + Ydy + Zdz
ď² x
ď²y
d² z
R
Fx
dy
d² z
Sx +
Sy +
V&L=
Sx +
Sy +
Sz,
d t²
d t²
2
dt2
dt2
dt2
dť
m
for if the particle do not rest on a surface R 0, and if it do
still SL
=
0, because we suppose the motion to be such that
particles on curve surfaces remain on the surfaces.
છે
Again, ds²= dx² + dy² + dz²,
... dsd. ds = dxd. dx + dys.dy+dzd.dz;
dx
dy
dz
:. vd.ds
8.dx +
=
S.dy + d.dz.
dt
dt
dt
dx
=
x +
d t
dy dz
Sy +
бу
S+ const.
dt
Hence (vd.ds+ ½ dtd. v²):
and at the limits dx
=
last positions are given,
dt
δια
0, dy = 0, dx = 0 because the first and
[ (vd. ds + ½ dtd. v²) = 0,
S (Σ.mfvds) = 0,
and .mfvds is a maximum or minimum.
It is evidently
a minimum, because a path of an indefinite length can always
be found for any particle of the system.
COR. 1. Since ds = vdt we learn that . mfvidt is a
minimum, or the quantity of vis viva generated or expended
during any given time is a minimum.
THE PRINCIPLE OF VARYING ACTION.
495
COR. 2. If the system consist of only one particle moving
on a surface and no forces but the normal pressure act, then
fvds is a minimum: but v is a constant (Art. 407), therefore
fds is a minimum, or the particle will describe the shortest
curve line that can be drawn on the surface between its posi-
tions at the beginning and end of the time t.
505. If we compare the principle of least action with
the principles of the conservation of the motion of the centre
of gravity, of the conservation of areas, and of vis viva we see
that this principle only serves to determine the equations of
motion, and is therefore comparatively useless since these are
found by much simpler means; but the other principles, which
develop important properties, have the advantage of furnishing
three general integrals of the equations of motion, which are
in most problems the only integrals that can be found.
PROP. To shew that the calculation of the motion of a
material system may be made to depend upon the integration
of a single function.
506. We shall shew this by proving a new dynamical
principle discovered by Professor Hamilton and published in
the Philosophical Transactions, 1834.
We have seen, Art. 497, that the Principle of Virtual
Velocities leads us to the dynamical equation
Σ.m
Jdx²
+
dy2
+
de
dx
=
- 2 Σ.m
X
d t² dt² dt
Now it is easily shewn that
dt
+ pdy
dz
+2
dt
dt.
dt
Σ.m X
dx
dt
dy
dz
+ Y
+ 2
d t
dt
is a perfect differential coefficient with respect to t for all the
forces which exist in nature; viz. forces tending to the centres
of the particles of the material universe, whether fixed or
moveable. Let therefore the second side 2 (U+ H), H
being independent of t: and let 2T be the vis viva of the
system at the time t; T, Ho the values of T and H when
t = 0 ;
.. T
=
0
U + H, and T, = U。 + H.
496
SYSTEM OF BODIES.
DYNAMICS.
Now if the initial circumstances of the motion be varied,
then I will vary, and so also will T and U: let & be the
symbol of these variations;
.. & T = SU + SH
S
dydy dzdz
б
dt dt dt dt dt dt
or Σ. m [ d x d x
+ б +
xy Sy +
ď² z
бу
б
+SH;
dt2
dt2
dz
б
-Σ.m 8x +
d²x
dt
and therefore 2 Σ. m
J d x
d
dt
dx dy dy dz
+ б +
dt dt dt dt dt
dz
dy 3y+ d² 8=} + 81
d [ d x S x +
- Σ.m dx + dy dy +
dt dt
dt
dt
Sx SH.
Now let the accumulation of the vis viva from the com-
mencement to the termination of the time t be V;
·. V =
=
Sz
t
Z. m l d
[ d x²
dy²
dve
+
+
dt2
dt.
dť
0
Σ.m
l t²
Then V is a function of the initial and final co-ordinates
of the material particles, and
SV
SV
SV
SV
SV
SV
SV = Σ.
8x +
бу
Sy +
Sz
S& +
δα + 8b+
(бос
dy
8%
δα
бъ
бе
=2 /=
Σ.m
SV
SV
+ Σ
ба
Sa+
бе
ба
Sb
Sc
- Σ.m
dt
[ d x 8 x +
dy
dz
бу
dt
Sy +
dt
Sz
S≈ + t & H + H₁,
dx dy dy dzdz
8 +
dt dt dt dt dt dt
31 db + 3x de
+
dt
H, being a function of the initial co-ordinates a, b, c.
THE PRINCIPLE OF VARYING ACTION.
497
But when t = 0, SV = 0, hence
(da 8a + dy 8y + 15 3 = }
dt
dx
dz
SV = Σ.m
Sx
x
dt
dt
-Σ.m
dt
dt
{da da + db db + de de} + 18 H.
dc
dt
From this equation we obtain the following groups
equations; ₁₁₁ being co-ordinates to m₁.………….
1
SV
= m1
dx
;
SV
dx2
1
= m2
;
δαι
dt
Sx 2
dt
SV
dy
SV
dy2
(4).
M2
Mi
бу
dt
буг
dt
SV
dzi
SV
dz 2
= m2
= m1
;
M2
dt
ဝ
dt
Second group,
SV
da
SV
d a 2
M²
M1
δα,
dt
δα
dt
S V
db₁
S V
db₂
(B).
M2
M1
ახა
dt
боз
dt
SV
dc
SV
dc2
m2
;
m1
dt Sc₂
dt
Sc₁
Lastly,
SV
t ....
SH
(C).
of
The problem is therefore reduced to finding the function V,
which Professor Hamilton denominates the characteristic func-
tion of the motion of a system. When V is calculated, then,
by eliminating H from the equations (A) (C), we shall have
the 3n integrals of the first order of the equations of motion by
simply differentiating V. And by eliminating H from the
equations (B) (C) we have the 3n final integrals by simple
differentiation.
3 R
498
DYNAMICS. SYSTEM OF BODIES.
It may be observed that V must satisfy the two following
partial differential equations
ΙΣ
1
for
SV2 SV 2
SV21
m \ S x 2
Sx² + Sy²
+
+
S22
း
U + H,
SV ²
SV² SV21
and
+
m 1 Sa²
Sb 2
0
U。 + H.
These equations furnish the principal means of discovering the
form of the function V and are of essential importance in
Professor Hamilton's Theory.
The equation
SV = Σ.m
Jd i
x
dy
dz
δυ
Sx +
бу
Sy +
Sz
dt
dt
dt
-Σ.m
Ida Sa
db
dc
Sa+
8b+
Scr + t . SH
dt
dt
dt
is denominated the law of varying action.
507. "It has been shewn by Lagrange and others, in
treating of the motion of a system, that the variation V
vanishes when the extreme co-ordinates and constant H are
given (Art. 504): and they appear to have deduced from this
result only the principle which is called the law of least action :
namely, that if the particles of a system be imagined to move
from a given set of initial to a given set of final positions, not
as they do, nor even as they could move consistently with the
general dynamical laws, or differential equations of motion, but
so as not to violate any supposed geometrical connexions, nor
that one dynamical relation between velocities and configuration
which constitutes the law of vis viva: and if, moreover, this
geometrically imaginable, but dynamically impossible motion,
be made to differ infinitely little from the actual manner of
motion of the system, between the given extreme positions,
then the varied value of the definite integral called action, or
the accumulated vis viva of the system in the motion thus.
imagined, will differ infinitely less from the actual value of
that integral.
THE LAWS OF SMALL OSCILLATIONS.
499
But when this principle of least action, or," as Professor
Hamilton proposes to call it, "of stationary action, is applied
to the determination of the actual motion of a system, it serves
only to form, by the rules of the Calculus of Variations, the
differential equations of motion of the second order, which can
always be otherwise found."
In this, then, appears the excellence of this new principle
called the law of varying action, that we pass from an actual
motion to another motion dynamically possible, by varying the
extreme positions of the system and (in general) the quantity H:
but more especially that it serves to express, by means of a
single function, not the mere differential equations of motion,
but their intermediate and their final integrals.
We hope that the slight sketch we have given of this
new principle will tempt our readers to consult the original
Memoirs in the Transactions of the Royal Society of London
for the years 1834, 1835, from which this notice has been
gathered.
PROP. To prove the general laws of the very small
oscillations of a vibrating system of particles.
508. If the oscillations of the particles be extremely
small we may always reduce the equations of motion to linear
equations and obtain approximately the co-ordinates in terms
of the time. Very many and various phenomena depend upon
the principles of small oscillations.
Let i be the number of particles, and m the number of
equations L = 0, L'= 0, …….. connecting their co-ordinates: let
3imn, then these equations determine m of the variable
co-ordinates in terms of the other n, or, more generally, all the
co-ordinates may be determined by means of these equations in
functions of n independent variables. Let a, ẞ.... be the
initial values of these variables, and a +u, B+v.... their
values at the time t; in which we suppose that u, v . are
very small during the whole motion: hence the co-ordinates
x, y, z, x', …….. can be expanded in very converging series of
24, V....
Let x = p + au + b v + ½ c u² + ¦ e v² +ƒ uv + ......
1
y = P₁ + a, u + b₁v + { cu² + { c, v² +ƒ¡uv +
500
SYSTEM OF BODIES.
DYNAMICS.
≈ = P₂ + a2u + b¿v + ½ c₂u² + ½ ev² + f₂uv +
2
½
x' = p' + a'u + b'v + c'u² +
e' v² +ƒ'uv + ......
Also since the forces X, Y, Z, X', ..... are supposed to be
functions of the co-ordinates, these may be expanded in con-
verging series: let
X = P + Au + Bv +
Y = P₁ + A₁u + B₁v + ..
Z = P₂+ A₂u + B₂v + ...., &c.
2
2
P, A, B ... being functions of p, a, b, c ...
Now by Art. 496, we have
x
d² z
Σ.m
бас
dt
m { (JE - X) 3x + (dy − r) 8y + (1 - Z) 8=)
dt2
8x } = 0,
- 0.
dt2
and Sx = (a+cu + fv + ……..) du + (b + ev + fu + ....) dv +.
+....
If we substitute these and put the coefficients of the n arbi-
trary quantities du, dv ...... equal to zero, we have
d²y
d t²
Σ.m
り
​{(da-x) (a+cu+fo+...) +
·Y) (a₁+c₁u+ƒiv+…..)
d² z
+
dt2
Z) (α₂+ c₂u + ƒ₂v + ...
0.
X, Y, Z
this
It remains to substitute for X, Y, Z ...
substitution being made we shall neglect the squares and pro-
ducts of u, v and of their second differential coefficients
with respect to t: we shall thus have n linear equations of the
form
d² u
d² v
D
+ E
+
+ Fu + Gv +
d t²
dt2
(1),
THE LAWS OF SMALL OSCILLATIONS.
501
We may
where D, E, F, G, Q... are given functions of the constants
which enter the formulæ for x, y, z X, Y, Z ...
suppose Q = 0, since we can always add to u, v….. such con-
stant values as to strike Q out: this amounts to supposing that
a, ß, y ... are the values of the n independent variables which
correspond to a state of equilibrium of the system; since
when u
u = 0, v = 0.
... the accelerating forces vanish.
RN
We may satisfy the equations (1), putting Q = 0, by u =
sin (t√p-r), v = RN' sin (t√p-r)
r) ...... R and r being
arbitrary constants of which the second may be considered
positive and less than T, and p, N, N'.
N, N', .... are constants to be
determined. By putting these values in (1), we have n equations,
(DN + EN' + ....) p = FN + GN' + .....
(2).
In eliminating from these n-1 of the quantities N, N'
N'....
the nth equation will be of the nth degree in p and will be free
from all the quantities N, N'.... in consequence of the form of
equations (2). And the values of n 1 of N, N', .... viz.
N'.... suppose, obtained from (2) will be rational fractions of
the nth degree with respect to p, having a common denominator,
and being each multiplied by N, which remains indeterminate;
we may therefore choose N equal to the common denominator,
and N, N'………. will be expressed in
will be expressed in terms of symmetrical
functions of of the nth degree.
ρ
In consequence of the linear form of equations (1) they
are satisfied not only by the values of u, v corresponding
to each of the n values of p, but also by taking for u, v ....
the sums of these particular values, in which we may change
the values of R and r as p changes.
P
If then, we call ppi P2 P3
the values of and use
corresponding subscript figures for the other letters, we have
the following general solutions of equations (1),
- ^')) + ....)
u = RN sin (t√√p-r) + R, N, sin (t√ R-
R¸N₁ r₁)
+....
v = RN'sin (t√√p − r) + R₁N'¸ sin († √ p₁ − r) + ..
ρι
•
(3),
502
SYSTEM OF BODIES.
DYNAMICS.
R, R₁ …….. ~', ~'1
being the 2n arbitrary constants in these
complete integrals. The constants must be determined in
terms of the initial values of u, v and their differential
coefficients: they are small because the original displacements
are small. If the values ppi P2
be all real, then the
motions of the particles will be periodical and will always
be very small. If, however, one or more of ppi P2
imaginary, we must replace the circular functions by expo-
nentials, and therefore as the time increases u, v…….. will increase
indefinitely and the above formula will cease to be true.
the first case the state of equilibrium of the system is stable;
in the second unstable.
be
In
509. Suppose, for example, that all of R, R₁ ………. except
the first vanish: then
x = p + (a N + b N' + ....) R sin (t √p-r),
y = p₁+ (a₁ N + b₁ N' + ....) R sin (t√√√ p−r),
(t√p−r),
≈ = P² + (a₂ N + b₂ N' + ....) R sin (t √ p −1),
x' = p' + (a' N + b' N' + ....) R sin (t√√p − r),
.... (4).
Hence the particles all perform their oscillations in the
2 п
same period, viz. : and all the particles return to their
places of equilibrium at the same instant.
510. A system of material particles, in which the relations
connecting the co-ordinates are of such a number as to leave
n of them independent variables, will, when slightly disturbed
from the position of rest, assume a number (n) of oscillatory
motions, each analogous to that described in the last Article,
corresponding to the n values ppipe And in virtue of
equations (3) and the corresponding values of x, y, ≈………. all
the oscillations, or only some of them may exist at the same
time in the system: and conversely, whatever be the initial
derangement we may always resolve the motion of each particle
parallel to each co-ordinate axis into n or less than n simple
CO-EXISTENCE AND SUPERPOSITION OF SMALL MOTIONS. 503
oscillations analogous to that represented by equations (4), the
when these are commensur-
periods being
2 п
2 π
何
​P1
able the whole system will return to the same state in a period
equal to the least common multiple of these periods: this is the
case in vibrating cords, and vibrating surfaces.
The prin-
ciple proved in this Article is called the Principle of the
Co-existence of Small Vibrations.
when
511. Suppose that U, V ………. are values of u, v
the system is in vibration under the action of one set of forces,
du
d v
the initial values of u, v
dt
dt
being u。, v。....
du dv
u, v
dt' dt
U19 VI
И, v
Again suppose that U', V'.... are the values of
when the system is under the action of a second set
the initial values of
of forces and u, v
ví
ur, vi
and so on then, if the initial values
du
dv
of u, v
be u + u +
dt' dt
....., vo + v₂ +
vó
U₂ + U₁ +
are
u = U + U' +
., v₁ + v₁ + .... the general values of u, บ
This principle, the truth of which arises from equations (1)
being linear, is called the Principle of the Superposition of
Small Motions: see Art. 288.
v = V + V' +
CHAPTER XVI.
PROBLEMS ON THE MOTION OF RIGID BODIES, AND ANY MATERIAL SYSTEM.
512. WE shall commence this Chapter with some observ-
ations upon the best methods of solving dynamical problems,
and the application of the general principles proved in the
last Chapter in facilitating their solution.
To determine the motion of a rigid body in space we
have six differential equations of the second order: these
contain the three co-ordinates to the centre of gravity and
the three angles of position of the principal axes of the body;
see Arts. 428, 446 and 447. These are the only relations
that can exist among the mechanical quantities (Art. 144).
If all the forces and other quantities involved in these
equations be known, then we have sufficient equations for
solving the problem, and determining the position of the
body at every instant.
If, however, the equations involve unknown forces, or un-
known geometrical quantities (as angular and linear measures),
or both, then there must exist as many more equations as
there are of these unknown quantities; and, moreover, these
relations must be among the geometrical quantities, since
the six equations of motion, as we have mentioned, are the
only mechanical relations that can exist.
Suppose that from the nature of the problem we have,
involved in the six mechanical equations, one unknown force,
and n unknown geometrical quantities besides those necessarily
contained in the six equations: then we must have n + 1
additional equations among the n + 6 geometrical quantities:
ON THE SOLUTION OF PROBLEMS.
505
when we have obtained these we have enough equations for
the solution of the problem. To determine then, these n + 6
geometrical quantities, and therefore to determine the position
of the body, we have already n + 1 equations free from un-
known mechanical quantities, and must therefore obtain five
more such equations; these are found by eliminating the un-
known force from the six equations of motion. In the same
way we should proceed if there were two, three, or more
unknown forces. The equations which we obtain among the
unknown geometrical quantities must be integrated, that we
may have these quantities in terms of the time.
The same remarks will apply when the system is acted on
by impulsive forces.
Now the principles of the conservation of motion of the
centre of gravity, and the conservation of areas, and the
principle of vis viva demonstrated in the last Chapter are the
first integrals of the equations of motion under peculiar
suppositions as to the nature of the forces which act upon
the system. If, then, in any proposed problem one or more of
these principles apply we may write them down as the integrals
of our equations, and so diminish the labour of elimination and
integration. If the integrals involved in these principles can-
not be obtained in consequence of their involving unknown
forces, the principles, though they may be true in these cases,
will nevertheless not answer our purpose.
To find the unknown forces we must obtain their values
from the equations of motion in terms of the geometrical
quantities and their differential coefficients; and since these
are supposed to be found the forces will be known also; see
Problem 16.
If we find, after all the equations are written down, that
there are more unknown quantities than equations, then the
general solution of the problem is indeterminate; though it
does not necessarily follow that all the unknown quantities
are indeterminate (as in Art. 438). If we find more equations
than unknown quantities it follows, that the general solution
of the problem is impossible unless certain relations among the
known quantities are fulfilled, the number of these relations
being equal to the number by which the equations exceed
3 S
506
SYSTEM OF BODIES.
DYNAMICS.
the unknown quantities. Nevertheless, as in the last, some of
the unknown quantities may be independent of these conditions.
We shall illustrate the remarks which we have made upon
the solution of problems by referring to Art. 436.
Here we
have a case of motion in parallel planes, and therefore only
three equations of motion: but these contain the unknown
forces F and R beside the three necessary geometrical measures.
of position x, y, 0: hence two more equations must exist,
and these among x, y, 0; these are equations (4) (5) in that
Article; and we require only one more relation connecting
x, y, 0; this we have by elimination from the equations of
motion. But since the point of application of the forces
F and R has no velocity, (for the body at each instant is re-
volving about that point as an instantaneous centre of rotation),
F and R will not appear in the equation of vis viva of Art. 497.
Hence this equation gives the integral we require; and we
have (by Art. 500),
Jd dx² dy
M
+
dt
d t
d 02
+ Mk2 = 2 Σ.mfgdy',
dt
y' being the vertical ordinate of m,
= 2gΣ. m (y + constant) = 2 Mg (y+ constant), (Art. 413).
This is the equation obtained in Art 436, by elimination.
We shall now give some Problems; we shall solve a few,
or give hints to guide to their solution.
PROB. 1. A sphere rolls down an inclined plane; re-
quired to determine the motion: (fig. 111.)
Since the motion of the centre of gravity is evidently
parallel to the fixed inclined plane we shall measure its dist-
ance (*) from the point C which it occupies at the commence-
ment of the motion, E the point which was then in contact at
B with the plane, EOD = 0, P the pressure of the plane,
▲
F the friction acting upwards, a the radius of the sphere. Then
for the motion of the centre of gravity (Art. 428.) and the
motion of rotation about the centre of gravity (Art. 429, 431),
d2%
F
d Ꮎ Fa
g
(1),
(2);
dt2
M
dt
MK2
PROBLEMS.
507
three unknown quantities; we want another equation, this is
að ...... (3).
Since the object is to determine the position of the body at
a given time we must obtain an equation between ≈ and ✪ in
addition to (3); this is obtained by eliminating F from (1)
and (2): we thus have
ď z
dť
ď² z
ટર
df
સ્વ
80
k2 d20
a dť
a² g
50
dz
k² d² z
a² dť
by (3);
a²gt
constant = 0,
a² + k² ³
,
dt
a² + k²
a²g ť
ag
t2
0 =
a² + k² 2
a² + k² 2
We might have used the principle of vis viva to obtain the
second equation between ≈ and 0, since F does not occur in
the equation of vis viva, because the velocity of its point of
application equals zero: but the elimination was so simple that
we preferred that method.
COR. If the body partly roll and partly slide, then Fis
constant, and must be determined by experiment. Hence
equation (3) does not hold, and, in short, (1) (2) are sufficient
for determining the motion in this case.
PROB. 2. Suppose the inclined plane, or wedge, on which
the cylinder rests is capable of moving on a smooth horizontal
plane to determine the motion of the sphere and wedge:
(fig. 111.)
The quantities as before, except that x and y are the
horizontal and vertical co-ordinates of O measured from A in
the horizontal plane: a' the horizontal co-ordinate to the
point K of the wedge, M and M' the masses of the sphere and
wedge. Then for the sphere we have the three equations (Art.
428, 429, 431.)
ď² x
F cos a
P sin a
ďy
F sin a + P cos a
(1),
...
(2),
dt2
M
dť
M
ď◊ Fa
(3).
dt
Mk²
508
SYSTEM OF BODIES.
DYNAMICS.
d² x'
P sin a - F cos a
For the wedge
(4).
dt2
M'
Here are six unknown quantities, there must therefore be two
relations connecting x, y, 0, x': these are
x' — x — a sin a = al cos a
(5),
y=h-a0 sin a
….. (6),
h the initial value of y.
We must obtain two relations connecting x, y, 0, x′ from
(1) (2) (3) (4). But since there are no forces acting exter-
nally to the system of the sphere and wedge parallel to the
horizon, there is a conservation of the horizontal motion of the
centre of gravity (Art. 486): hence
dx
dx'
M + M' = constant = 0,
dt
dt
in our case, since there is supposed to be no initial velocity;
.. Mx + M' x'
= constant = 0 ………….. (7),
if we properly choose the origin A.
Again, the principle of vis viva gives us an integral; for
although the point of application of P and F does move in this
case, yet the velocity of this point will have exactly opposite
signs relatively to P and F acting on the sphere, and P and F
acting on the wedge, and therefore P and F will not occur in
the equation of vis viva: in short, they are internal forces;
Ida
[dx²
dt dť
The equations (5) (6) (7) (8) will determine the position.
.. M
(dx²
dt
dy²
dᎾ
+
+k²
do²r
+M'
dx'2
dt2
· 2Σ.mf-gdy'
=
2g M (h − y) ....
(8).
dx'
By (5) (7) (6)
dt
Ma cos a do
M+M' dt
M dx dy
M' dt dt
d Ꮎ
a sin a
;
dt
d02
.. by (8)
{a²
a² + k²
d t2
M
M+ M'
a² cos³ a} = 2ag 0 sin a ;
:. 0
=
{ a² + k² -
M
M+ M'
α
a² cos² a}-¹.agt,
this coincides with the result of Prob. 1. if we put M'
= ∞.
PROBLEMS.
509
The equation to the path of O is, by (5) (6) (7)
y = h +
M + M'
M'
x tan a;
therefore the path of the centre of the sphere is a straight line.
PROB. 3. A groove in the form of a cycloid with its
vertex downwards and base horizontal is cut in a solid ver-
tical board: determine the motion of a ball moving along it
while the board itself is capable of moving freely along a
smooth horizontal plane, and the curve which the ball describes
in space.
Let x, y be the horizontal and vertical co-ordinates to the
ball at time t: x' the co-ordinate to the vertex of the cycloid
supposed to be in the horizontal plane: s the distance of the
ball from the vertex measured along the groove, R the mutual
pressure of the ball and groove, M and m the masses of the
board and ball: then the equations of the problem are
ď² x
R dy
ď²y
Rd (x − x')
(1),
- g +
(2),
dt2
m ds
df
M
ds
d²x'
R dy
(3),
dt²
M ds
Y
x - x' = a vers
1
√2ay - y³
·(4).
α
The principle of conservation of the horizontal motion of the
centre of gravity and the principle of vis viva both apply:
they give
Mx' + mx = 0
0....
(5),
by choosing the origin under the initial position of the centre
of gravity, and also
dx'2
(dx²
M
dy)
+ m
+
dt2
d t²
= 2mg (hy) ...... (6),
dt
h the initial value of y.
510
SYSTEM OF BODIES.
DYNAMICS.
M + m
M
-
∞ = a vers
By (4) (6) the equation to the path of the ball in space is
1
Y
-
a
+ √2ay − y²
(7).
dx
M
2 a
-
By (7) (5)
y dy
M dx'
;
dt
M + m
y
dt
m dt
dy²
M
.. by (6) m
2a-y
+1
dť² | M + m
Y
}
2mg (h − y),
from which the motion must be calculated.
PROB. 4. Two equal balls are fixed to the end of a
rod without weight; the rod is connected at its middle
point with a fixed vertical axis, so as to allow the rod to
move in a vertical plane passing through the axis, and to
revolve with the axis in a horizontal direction: required the
motion of the balls.
Let M be the mass of each ball, 2a the length of the
rod, the angle the rod makes with the vertical axis at
the time t, the angle the plane in which is measured
makes with the initial position of that plane. We shall not
write down the equations of motion in this case, but resort
immediately to the principles of the conservation of areas
(which applies since the resultant of the weights of the balls
always passes through the fixed point,) and the conservation
of vis viva. The principle of areas gives for a horizontal
plane
аф
аф
2 Ma²
= const.
= a.
dt
dt
The conservation of vis viva gives
do
d t
2 M {a² dope + a² 24}
dt
ግ
const.
d Ꮎ
B.
dt
a and ẞ being the initial angular velocities: then
Ꮎ
= at,
0 = 0, + ßt: and if xyz be co-ordinates to one of the balls
from the centre of the rod, the vertical axis being the axis
of %, we have
x=a sin (0+ẞt) cosat, y = a sin (0, + ẞt) sinat,≈=a cos (0,+ßt).
PROBLEMS.
511
PROB. 5. A rod acts by one extremity with a uniform
force in the direction of its length on the fly-wheel-crank of
a steam engine, the other extremity moving in a straight
line passing through the centre of the fly-wheel, and a uni-
form resistance is to be overcome by the fly-wheel. Find
the velocity of the wheel at any time: and find the relation
between the forces when they are so adjusted that after half
a revolution the velocity may be unaltered.
α
=
PROB. 6. A uniform lever ACB, of which the arms
AC and BC are at right angles to each other, rests in equi-
librium when AC is inclined at aº to the vertical: shew that if
AC be raised to a horizontal position (C being fixed) it will
fall through an angle 0, such that cos 0 cot (45° + a).
PROB. 7.
Given the radii and masses of the wheels in
Atwood's Machine (Art. 216.) and the constant friction on
the fixed axles of the wheels A and B (fig. 75); shew that
the accelerating force of P and Q when in motion is much
less effected by the friction at A and B, than if the wheel
C turned about a fixed axle.
PROB. 8. A horizontal wheel moves freely about a ver-
tical axis through its centre; a string of definite length is
wrapt round its circumference, and passing through a ring
has fixed to it a weight which falls by gravity; determine
the whole motion.
PROB. 9. A hemisphere rests on a horizontal plane with
a string fastened to its edge, which, passing over a pully,
supports a weight: when the string is cut find the motion
of the hemisphere.
PROB. 10. A beam is drawn from a horizontal to a
vertical position about one extremity, which is fixed, by
means of a string which is attached to the other extremity
of the beam and after passing over a pully placed above.
the fixed extremity at a height equal to the length of the
beam is attached to a falling body; determine the motion.
PROB. 11. A beam is projected perpendicular upwards,
and has a rotatory motion round its centre of gravity in a
vertical plane; it is observed at a given altitude to be in
one of its horizontal positions, and to be then ascending
with a given velocity; after this it performs a given number
512
SYSTEM OF BODIES.
DYNAMICS.
of revolutions and strikes the ground at a given angle:
find the angular velocity.
PROB. 12. An inflexible straight rod is set in motion
round a vertical axis passing through one extremity, about
which it is capable of revolving freely in an horizontal plane:
determine the motion of a ring sliding freely along it: and
prove that the whole vis viva of the system is constant.
PROB. 13. A body is placed on a smooth wedge which
rests upon a smooth horizontal plane, and the wedge is
acted on by a horizontal and constant force ƒ in a vertical
plane perpendicular to the inclined plane of the wedge:
determine the motion: and find ƒ when the body is at rest
on the plane.
PROB. 14. A semi-cylinder rests with its plane surface
on the ground, on which it is capable of moving freely;
shew that a body sliding down its curved surface will de-
scribe an ellipse; and determine the time of descent.
PROB. 15. Determine the motion of two heavy particles
connected by an inflexible rod without weight, one of which
moves on a surface of revolution and the other is constrained
to move in the axis of the surface, this axis being vertical.
Find the velocity of the particle on the surface when the
other continues stationary.
PROB. 16. A cylinder rolls down a fixed quadrant; find
where the cylinder will leave the quadrant.
The pressure must be calculated; the body leaves at
the instant that this is zero.
PROB. 17. A sphere revolves round an axis touching its
surface, find the length of the simple isochronous pendulum.
PROB. 18. A sector of a circle revolves round an axis
perpendicular to its plane, and passing through the centre
of the circle; find the angle of the sector when the length
of the isochronous simple pendulum equals one half the
length of the arc.
PROB. 19. For what axes of suspension is the time of
a small oscillation of a solid body an absolute minimum?
Take the case of an ellipsoid.
PROB. 20. A rough vertical cylinder, capable of revolving
about a concentric but smooth and smaller cylinder as an axis,
PROBLEMS.
513
rests upon a rough horizontal plane, on every point of which
the pressure is the same: determine the force applied by
a string wrapt round the cylinder which will just make it
If the force be greater than this determine the
move.
motion.
PROB. 21. A cylinder is made to rotate about it axis, and
is then suddenly placed in contact with a rough horizontal
plane with its axis parallel to the plane; the force of friction
is of finite intensity and is not sufficiently great to prevent the
line of contact of the cylinder from sliding on the plane at the
beginning of the motion: required to determine the motion,
and to shew how long the cylinder will continue to combine.
a sliding motion with its rolling motion.
Let o be the angular velocity communicated to the cylinder
before the contact: the friction does not affect this velocity at
the first instant of the contact because the force of friction by
hypothesis is of finite intensity: the angle described in the
time t by that radius of the cylinder that was in contact with
the plane at first: a the radius: a the distance of the axis of
the cylinder at time t from its initial position: F the friction.
Then F is constant and has its greatest value so long
as the
cylinder slides as well as rolls; in which case the equations of
motion are
d Ꮎ
d t
Fa
Mk²
….. (1),
ď² x F
d t² M
(2),
but when the sliding motion ceases, if F' be the friction, which
is then not necessarily constant, we must put F' for F in (1)
and (2), and add the equation
I.
x = ɑ
a0..
(3).
So long as the sliding motion continues we have,
then,
de
Fat
dx
Ft
= w
dt
MK2
dt
M
Fat
Fť
8 = wt-
2 Mk2
M
3 T
514
SYSTEM OF BODIES.
DYNAMICS.
The sliding motion ceases when the motion of translation
and the motion of rotation give exactly equal and opposite
motions to the point of contact, or when
dt
dx ᏧᎾ
dt
and .. t =
M k² a w
Fa² + k²
de
kw
and
,
dt
a² + k²
II. After the sliding motion ceases, the equations of
motion are
d Ꮎ
d t²
F'a
Mk2
ď² x
F'
(1),
(2),
X = a
= að ….. (3).
dt2
M
k2 d20
d² x
d Ꮎ
By (1) (2)
+
0,
... by (3)
0 ;
a dť
d t²
dt2
ᏧᎾ
k² w
.. F'= 0 and
= constant =
dt
a² + k²
From this we learn that the friction has gradually reduced the
angular motion of the body till the velocity of the point of
contact is zero, and after that the body proceeds to move.
uniformly and to rotate uniformly and no friction is called
into play.
PROB. 22. A rough body lies upon a rough board, and this
lies upon a smooth horizontal plane, the friction between the
body and board is of finite intensity (as in the last Problem):
the board is projected with a given velocity, determine the
motion of the body and board.
PROB. 23. A sphere is fastened by an inflexible rod to
a horizontal axis fixed at two points: when the sphere revolves
about the axis required the pressure on the two fixed points.
If we
use the notation of Art. 438, and put y = 90º,
y= 90° and therefore ẞ = 90° - a, then, the axis of rotation
being the axis of ≈ and the plane in which the centre of the
sphere moves the plane of xy and the axis of a drawn vertically
downwards, the moving forces m (g+yf + xw²), m (y w² − x ƒ),
O acting on m parallel to the axes of x, y, z and similar forces
acting on all the other particles of the system, together with
PROBLEMS.
515
the pressures of the fixed points ought to be in equilibrium at
the time t.
Hence
P cos a + P' cos a' + Σ. m (g + yƒ + xw³) = 0,
P sin a + P' sin a' + Σ. m (y w² - xƒ) = 0,
z
- P sin a. a - P' sin a'. a-Z.m (gz + yzf + x % w³) = 0 ;
P cos a.
a + P' cos a' . a' + Σ . m (y z w² – xxƒ) = 0,
Σ.m (yzw² xzf)
Σ.m (xy w² — x²ƒ—gy—y³ƒ—xyw²³) = 0.
Let xyo be the co-ordinates to the centre of gravity;
then, since every axis through the centre of a sphere is a prin-
cipal axis, we have
Σ. m (y ~ÿ) x = 0, Σ.m (x − x ) ≈ = 0, Σ. m (y−ÿ) (x − x ) = 0 :
.. Σ.my z = ÿ Σ.mz = 0, Σ.mxz = 0, Σ.myx =
y
= Mxy.
Hence the equations become
P cos a + P' cos a' + M (g + fy + w³ x ) = 0,
P sin a + P' sin a' + M (w²y -ƒx) = 0;
Pa sina+P'a' sin a=0, Pa cos a+ P'a' cosa =0, Mfk²+Mgy=0.
From which P, P', a, a' may be found.
PROB. 24. If a body revolve round an axis by the action
of a constant force in a direction always perpendicular to the
plane passing through the axis and the centre of gravity of the
body, determine how the point of application of this force must
vary with the time, so that there may be no pressure on the
axis, except in the plane to which the direction of the force
is perpendicular.
PROB. 25. A hemisphere oscillates about a horizontal axis,
which coincides with a diameter of the base; shew that if the
base be at first vertical, the ratio of the greatest pressure on
the axis to the weight of the hemisphere = 109 ÷ 64.
516
SYSTEM OF BODIES.
DYNAMICS.
PROB. 26. A sphere, when acted on separately by three
forces, revolves round three diameters inclined at the same
angle to each other and with the same angular velocity, deter-
mine the angular velocity and the new axis of rotation when
the three forces are applied at the same instant.
PROB. 27. A sphere attracted to a given centre of force
varying as the distance is projected with a given velocity along
a plane passing through that centre, friction being such as to
destroy all sliding: prove that the path will be an ellipse, and
find the velocity that the ellipse may be a circle.
PROB. 28. A cone of given form, and supported at G
its centre of gravity, has a motion communicated to it round
an axis through G perpendicular to the line joining G with a
point in the circumference of the base, and in a plane passing
through this point and the axis of the cone: determine the
position of the invariable plane; and explain the motion of
the cone's vertex.
PROB. 29. Explain how the rotation of a hoop preserves
it from falling.
PROB. 30. A solid of revolution moveable about its centre
of gravity G, which is fixed and is the origin, and having its
axis inclined to the axis of at an angle o, has an angular
motion impressed upon it about a line between these two axes,
and inclined to the former at an angle 0, such that k² tan
k'tan 0, where k and k' are the radii of gyration about its
axis and a line perpendicular to the axis through G: prove
that the axis of the solid will constantly preserve the same
inclination to the axis of ≈, and will revolve uniformly about
it; and the solid will at the same time revolve uniformly
about its own axis, which is in motion.
PROB. 31. If the Moon moved in the ecliptic, shew that
the force of the Earth to produce rotation about her axis
3 μ sin 20 A-B
C
perpendicular to that plane would nearly
u
22.3
u, r being the Earth's mass and distance from the Moon,
A, B, C the principal moments of inertia of the lunar sphe-
roid, and ✪ the angular distance, at the Moon's centre, of the
Earth from one of the principal axes which are in the ecliptic.
PROBLEMS.
517
We shall now give some problems in which the action of
impulsive forces is considered.
PROB. 32. Two inelastic balls impinge upon each other,
their motions being in the same straight line: required their
velocity after impact.
Let M, M' be the masses of the balls: V, V' their velo-
cities at the instant the contact commences; v, v' their velo-
cities after the impulse ceases: P the momentum which
measures their mutual pressure during the collision. Then
by Art. 474. for the motion of the centre of gravity of M
MV - P- Mv
= 0......
for the ball M'
.(1),
M'V' + P - M'v′ = 0.
(2).
Here we have two equations with three unknown quan-
tities P, v, v'. The third equation is the condition that
the particles in contact move with the same velocity the in-
stant the compression ceases.
Hence v-v' = 0………
Eliminating P from (1) (2)
(S).
Mv + M'v'
MV + M'V';
MV + M'V'
..
. by (3), v = v′
M + M'
and the balls will therefore each move with a velocity
MV + M'V'
M+ M'
and remain in contact.
Also P M (V − v)
:
=
MM'
M + M'
(V – V') by (1).
PROB. 33. Suppose the two balls are imperfectly elastic.
In this case the Problem divides itself into two parts: first
the motion during compression, and secondly the motion during
the restitution of the figure of the bodies. Now bodies differ
in their elasticity owing to their physical constitution; but
the law to which we are led by experiment is this, that for
518
SYSTEM OF BODIES.
DYNAMICS.
the same material the momentum gained by the restitution
bears a constant ratio to the momentum lost by the collision:
this ratio we write e, and is called the elasticity of the material
of which the bodies are made (Art. 220).
By the previous Problem the bodies are moving with a
MV + M'V'
at the instant the restitution of figure
M + M'
velocity
commences: and the momentum which measures their mutual
action = P =
MM'
M+ M'
(VV'):
V'): therefore the mutual pressure
during restitution = Pe=
MM'e
M+ M'
(V − V'):
Let И, u' be the velocities after the restitution ceases:
then
M
MV+M'V'
M+ M'
MV + M'V'
P'-Mu= 0, M'
· + P' − M' u' = 0 ;
M+ M'
MV + M'V'
M' (V – V')
•*. U =
e
M+ M'
M + M'
MV + M'V'
ú
+ e
M + M'
M (V – V')
M + M'
COR.
e = 1.
If the bodies are considered perfectly elastic, then
2 M'
u =
V
(V − V'), u' = V' +
M + M'
2 M
(V – V').
M + M'
PROB. 34. A smooth but imperfectly elastic ball moves
in a horizontal plane and impinges on a hard vertical plane
obliquely, its direction making an angle a with the normal to
the plane: find the velocity and direction of the motion after
impact.
t
Let V be the velocity before impact; v the velocity and 0
the direction of motion at the instant compression ceases; u the
velocity and the direction of motion after impact: P the
mutual pressure during compression, Pe the pressure during
restitution, M the mass of the ball: then when compression
ceases
PROBLEMS.
519
MV cos a–Mv cos 0=0...(1), MV sin a -P-M v sin 0=0…….(2),
but we have three unknown quantities P, v, 0: a third equation
is given by the condition that the plane is immoveable, and
hence the velocity of the body perpendicular to the plane is
zero at the instant compression ceases, therefore
v sin ◊ = 0 ……….
when restitution of figure ceases,
Mv cos - Mu cos
... .
(3),
(4), Pe – Mu sin p = 0 ... (5).
By (1) (3) (4) u cosp=V cosa, by (2) (3) (5), u sine V sin a;
.. tan -e tan a, and u= V
Vcos
´cos² a+e² sin²a, = V cosa÷cos &,
which determine the direction and the velocity after impact.
PROB. 35. Two imperfectly elastic and smooth balls im-
pinge upon each other, the motion of their centre taking place
in the same plane: required their velocities after impact.
Since the balls are perfectly smooth there will be no
rotatory motion produced by the impulse. We must first
consider the motion till the compression ceases. Let P be
the mutual pressure acting in the common normal at the
points in contact; V, V' the velocities at the commencement
of the contact: aa the angles their directions make with the
line passing through their centres when the contact takes place;
v, v' the velocities of the balls at the instant the compression
ceases: 0, 0' the angles their directions make with the axes.
Therefore
M V cos a-P-M v cos 0 =0... (1), M V sin a − M v sin 0 =0... (2),
M'V' cos a+P—M'v' cos 0′ =0 ….. (3),
M'V' sin a'—M'v' sin ('=0 ... (4) ;
and since the points in contact move with the same velocities
in the direction of the normal at the instant the compression
ceases, then
v cos ✪ – v' cos (′ = 0 ……….
(5).
520
SYSTEM OF BODIES.
DYNAMICS.
Again, during the restitution of figure the mutual pressure
Pe: and if u and u' be the velocities after the restitution of
figure is complete, and p and o' the angles of the directions
of motion, the equations of motion are
M v cos ◊ –Pe-M u cos p=0...(6), M v sin ◊ – M u sin=0... (7);
M'v'cos 0'+Pe-M'u' cos p=0... (8), M'v'sin '-M'u' sin d'=0... (9).
In these nine equations are involved nine unknown quantities
P, v, v', 0, 0', u, u', p, p': we have to determine u, u', p, p'.
By (1) (2) (5) (M + M') v cos 0 = MV cos a + M'V' cos a' ;
eliminating e by (1) (6) (M + M′) u cos o
= (M + M') (1 + e) v cos ◊ − ( M + M') V cos a
=
M'V' cos a'
sa),
M'V cos a + e (MV cos a + M'V' cos a
by (2) (7) u sin = V sin a;
&
from which u and may easily be determined: in the same
way u' and ' may be determined.
PROB. 36. A rough ball A is placed on a rough horizontal
table, and another rough ball B lying on the table is struck
in a direction not passing through the centre of gravity, but
so as to cause B to strike 4: find the motion after impact, the
bodies being inelastic.
PROB. 37. Supposing, in the last Problem, that the
friction of the Table is so slight as not altogether to prevent
sliding, find the conditions that B may move through its
original place of rest.
The four following Problems are intended to illustrate the
action of springs in removing the shock arising from the sudden
collision of bodies.
PROB. 38. A ball A moves along a smooth horizontal
plane with a velocity V, and sets in motion another ball B,
equal to A and originally at rest, by impinging upon a spring
CD (fig. 112), which is fastened to B at the point D: the
inertia of the spring is neglected, and we suppose the force
of the spring to vary as the space through which it is com-
pressed required to determine the motion of the balls.
:
PROBLEMS.
521
Let O be the place of A, the centre of the first ball, at the
time of first contact with the spring: OA=x, CD=≈ (=b when
the spring is not compressed), OB = x': then the force exerted
by the spring on the balls at the time t varies as b-≈; let
it = c² (b − ≈). The equations of motion are
ď²x
ď² x
c² (b − ≈) ………….. (1),
ď² x
dť
= C
c² (b − z) ………….. (2) ;
dt2
also x'
x = 2 a + ≈
2a
.... (3), a the radius of the balls,
three equations and three unknown quantities x, x', z.
Differentiating (3) and subtracting (1) (2) we have
d² &
dt
ર
2 c² (b − ≈) ;
d z
dz
constant – 2c² (b − ≈)² = V² − 2 c² (b − ≈)º,
—
d t²
since the point C of the spring (having no inertia) instantly
acquires the velocity (V) of the body A at the first contact;
V
= b - sin (c√2 t + C') = b
V
sin c√2 t ... (4).
∞ √ 2
This shews that the greatest compression of the spring is
equal to
C
V
and that the time of compression
π
20 √2'
after an equal duration of time the spring is restored to its
original form, since ≈ equals b when c√2t=π.
C
By (1) (4)
ď²x
dť
Vc
sin c√√√2t;
√2
dx
V
V
= const. +
cos c√
C
c√2t
(1 + cos c
os c√2t);
dt
20
V
1
.. X
(t +
2
C
sin c√√√2t);
3 U
522
SYSTEM OF BODIES.
DYNAMICS.
V
1
·. by (3) x' = 2a + b +
(t
sin c√2t);
dx'
V
(1'- cos c√2t).
dt
2
From these equations we readily gather the following results.
dx
The ball stops when
0, or t
dt
C
π
; but at this
c √ 2
instant (as we have shewn) the spring has returned to its na-
tural form, consequently the contact between A and the spring
at this instant ceases, and A remains permanently at rest: the
space through which A has moved during the action of the
The velocity of B is zero when the spring
spring
π V
20√2
begins to act and is V when its action ceases, and with this
velocity B henceforth moves uniformly along the plane. Hence
A gradually imparts all its velocity to B: and the duration
of time which this communication of velocity occupies is
П
If the elastic force of the spring be of very great intensity, as
is the case with the forces put into play by the impact of hard
balls of ivory, c is very great, and the duration of collision is
exceedingly short.
PROB. 39. Suppose that A and B (in the last Problem)
are of the same size, but of different masses M and M', and
that they move with the velocities V and V' before they come
in contact required to determine the motion.
PROB. 40. Suppose, in the last Problem, that the force
exerted by the spring during the restitution of its figure is less
than the force exerted during the compression in the ratio
e 1, but that a complete restitution of figure takes place:
required to determine the motion.
PROB. 41. A heavy carriage (represented in fig. 113.)
rests upon a spring B, and is also held in its place by two
springs pressing at C and C': the carriage moves uniformly
along a horizontal plane with a velocity V, and its four wheels
(two only of which are seen in the figure) which are all of the
PROBLEMS.
523
very
same size suddenly impinge at the same instant on four
small and equal pointed obstacles, and move over them; the
force exerted by each spring is supposed to vary as the extent
of displacement of its point of contact with the carriage, and
the springs are supposed to be bent into such a form that for
all small displacements of the body of the carriage the re-
sultant of their pressures always passes through the centre of
gravity of the body and so prevent rotatory motion required
the motion of the centre of gravity.
We shall merely give the results with a few of the steps
of the calculation.
Let the dotted lines in the figure represent the state of
things at the instant of the impact: and the dark lines the
state of things at a time t after the impact: a the radius of
each wheel. In consequence of the elasticity of the springs
(which is supposed perfect) the body of the carriage is not
rigidly connected with the wheels and axle-tree, and therefore
the body can produce no instantaneous effect upon the
velocity when the impulse takes place. By
By the impact
of the wheels on the obstacles the parts of the springs
which are connected with the axes of the wheels and the
axle-tree have their motion suddenly changed, this causes the
springs to assume new forms and in that way the forces
are brought into action which gradually change the motion
of the body.
Let x'y' be the horizontal and vertical spaces described
by the point B of the axle-tree in the time t; a and y the
spaces described by the centre of gravity of the body: let c
and e² be constants which depend upon the elasticity of the
springs at C and C' and that at B; we neglect the downward
effect of the spring B on the axle-tree but consider only the
dead weight to act at B: let O be the angle which the spoke
of each wheel, which passes through the obstacle, makes with
the vertical at the time t; 0 Ꮎ = a when t = 0: M' = mass of
each wheel the angular velocity of each wheel after impact.
The equations of motion are, for each wheel,
d Ꮎ
dt2
(M' + 1 M) ga sin ( sin
M'k²
N2
, suppose
(1),
524
SYSTEM OF BODIES.
DYNAMICS.
for the motion of G
ď² x
dt2
d² y
· c² (x − x') ………….. (2),
− g + e² (y' − y) ………….. (3),
d t2
and x', y', 0 are connected by
the equations
x'= a (sin a − sin ◊) ……………. (4), y' = a (cos ·
cos a) .... (5).
These equations are sufficient to solve the problem: but they
cannot be integrated unless a and (and therefore the ob-
stacles) be supposed small: we shall neglect powers higher
than the second. After reducing the equations their integrals
will be found to be of the forms
x = A sin (ct + B) + aa + hɛ n + kεn,
2t
23
2t
y = C sin (et + D) — 1 — mε™" - pen,
− mɛ
A, B, C, D being arbitrary constants to be determined by the
initial circumstances, and h, k, l, m, p being written for
known quantities. After determining these it will be found
that when t
= 0,
=
dy
dx
= 0 and therefore the original rectilinear
path of G is a tangent at the first point to the curve described
by G; also the values of the constants will shew that the
velocity is V at first, and gradually decreases: hence there
is no jerk in the body of the carriage.
We might in the same way obtain the circumstances after
the wheels again come to the horizontal plane.
PROB. 42. A rectangular parallelopiped slides down a
smooth inclined plane and meets a fixed obstacle determine
the impulse and the subsequent motion.
PROB. 43. A beam is projected in any manner along
a smooth horizontal plane and impinges upon a fixed obstacle:
determine the impulse.
PROB. 44. A beam is fixed at one extremity, what ver-
tical force applied instantaneously at the other will throw it
exactly vertical?
PROB. 45. A rectangular parallelopiped revolves about
one of its edges, which rests in a horizontal groove, and
PROBLEMS.
525
impinges on a fixed line parallel to the groove and in the
same horizontal plane with it: find the angle through which
the parallelopiped must fall so that it may be just on the
point of revolving about the fixed line as a new axis, all
sliding being prevented by friction.
PROB. 46. A beam is placed with one end against a
smooth vertical wall and the other on a smooth horizontal
plane so as to move in a vertical plane when left to the
action of gravity; the horizontal plane does not extend to
the wall, but is terminated by a straight edge parallel to
the wall: find the distance of this edge from the wall that
the beam may just be prevented from revolving about the
edge and finally falling beneath the horizontal plane.
PROB. 47. At what point must a given uniform circular
body be struck by a force perpendicular to its plane, that
in the first instant of the body's motion one extremity of a
given diameter may remain at rest?
PROB. 48. A beam falls from a vertical position by
revolving about one extremity which rests on a rough hori-
zontal plane, and impinges on a vertical post: determine
the magnitude and direction of the impulse on the post and
on the horizontal plane at the immoveable extremity of
the beam.
PROB. 49.
In the last Problem determine the initial
circumstances that the beam may just fall over the post.
PROB. 50. If a rough ball be projected against a rough
beam on a smooth horizontal plane determine the centre of
spontaneous rotation.
PROB. 51. An elastic beam falls upon a horizontal fixed
line: determine the motion.
PROB. 52. A beam, moveable about a fixed horizontal
axis at a given altitude above a horizontal plane, falls through
a given angle: determine the point at which a given sphere
should be opposed to its impact, that it may be projected
to the greatest possible distance on the horizontal plane, the
beam being in its vertical position at the instant of impact.
PROB. 53. A hoop rolling down an inclined plane suddenly
comes in contact with a horizontal plane; find the change in
angular velocity.
526
SYSTEM OF BODIES.
DYNAMICS.
PROB. 54. In lowering a bale of goods from the higher
story of a warehouse by means of a given crane, the whole
weight of the bale is allowed to wind off the rope freely from
the axle, and when the bale is half way down, the handle
of the crane suddenly flies off; determine the motion.
PROB. 55. Explain the use of fly-wheels in machinery,
and if a fly-wheel of given dimensions and weight move with
a given angular velocity what force applied perpendicularly
at a given point of one of the spokes of the wheel will in-
stantaneously destroy the motion.
PROB. 56. A perfectly flexible chain has one end fixed
to a peg, which is at the extremity and highest point of a
quadrant of a circle of which the plane is vertical, and all the
chain is collected at that point; it will just cover the quadrant,
and being suffered to descend freely, it is required to find the
stress upon the peg at the end of the motion.
PROB. 57. A groove is cut in a horizontal table in the
form of a regular hexagon and an inelastic ball is projected
with a given velocity along one of its sides, find the velocity
with which it will successively describe each of the other sides
of the figure.
PROB. 58. A perfectly elastic solid of revolution, turning
about its axis at a given rate, impinges on a hard smooth
plane: if before impact the centre of gravity move perpen-
dicular to the plane with a velocity V, determine the motion
of rotation after impact, and prove that the centre of gravity
will move in the same direction with a velocity
p² - k²
p² + k²
V, where
p is the perpendicular from the centre of gravity on the normal
at the point of impact, and k is the radius of gyration round an
axis through the centre of gravity perpendicular to the axis
of the solid.
PROB. 59. A solid sphere is placed in a hollow sphere,
which rests on a smooth horizontal plane; determine the small
oscillations, when they are slightly disturbed from the state
of rest.
PROB. 60. Prove, by means of the principle of least action,
that the orbit a body describes about a centre of force varying
inversely as the square of the distance is a conic section.
PROBLEMS.
527
PROB. 61. Prove the laws of reflexion and refraction
of light by the principle of least action, on the supposition
that light consists of luminous particles moving uniformly
in the same homogeneous medium, but with different velocities
in different media.
PROB. 62. A.bullet is fired into a thick board hanging
from a fixed horizontal axis about which it is capable of
revolving; the board has a sheet of iron on its back to
prevent the bullet from passing through a ribbon is fastened
to the bottom of the board and runs through a ring touching
the bottom of the board in its position of rest: shew how to
compare the velocities of bullets by observing the lengths of
ribbon drawn out by the motion of the board.
This is Robins' Ballistic Pendulum.
PROB. 63. The weights suspended from a wheel and axle
are in motion, the wheel and axle move about a fixed axis very
nearly fitting into the cylindrical aperture concentric with the
axle, so as to suffer only one point to be in contact: determine
the position of this point when friction is considered and when
it is neglected.
HYDROSTATICS.
CHAPTER. I.
DIFINITIONS AND PRINCIPLES.
513. HYDROSTATICS is the science which treats of the
equilibrium of fluid bodies.
By a a fluid body we mean an assemblage of material
molecules which yield without resistance to the slightest
effort which we can make to separate them. No fluids with
which we meet in nature are exactly of this character; but
they approach more or less to perfect fluidity, as it is
termed. The adherence that exists among the molecules of
most fluids, known by the term viscosity, prevents the se-
paration of its parts by the slightest forces; but, in this
work we shall suppose the fluids to be perfect, for, if we
except certain fluids the viscosity of which is considerable
and of which we do not treat, the laws of equilibrium which
we deduce are true, without sensible error.
514. Fluids are divided into liquids and aeriform
fluids: they are also said to be incomprehensible and elastic.
In truth, all fluids are more or less elastic: but some, as
water, are compressible, and that but slightly, only when
subjected to enormous pressure. Aeriform fluids may be
divided into vapours, and permanent fluids, such as air and
gasses. A given space will not contain above a determi-
nate quantity of vapour under a given temperature: so that
HYDROSTATICS.
529
if the vapour attain the limit of temperature, and we di-
minish ever so little the space or the temperature, a portion
of the vapour becomes liquified. Experiment proves that
the maximum quantity of vapour is the same, when the
temperature is the same, in a space void of air, and in a
space filled with air more or less dilated or compressed.
On the other hand, air and the gasses do not under any
circumstances become liquids; though it is the opinion of
some that this would be the case if sufficient force of com-
pression could be exerted, or if the temperature could be
reduced to a sufficient degree of cold.
515. In order that a fluid mass may be in equilibrium
the forces acting upon each molecule must satisfy the
equations of equilibrium of a particle deduced in Art. 65.
But we are ignorant of the forms of the particles of fluids
and of the forces by which they influence each other and
consequently we fall upon the same difficulty as in the
Article cited, and must therefore seek for some principle to
serve us the office which that of the transmission of force
did when we considered the equilibrium of a rigid body.
The following Principle is proved by experiment: that
any pressure communicated to a fluid mass in equilibrium
is equally transmitted through the whole fluid.
This is perfectly independent of the form of the mo-
lecules: we may therefore suppose the fluid to consist of
an indefinite number of small parallelopipeds formed by
planes drawn very near to each other and parallel to the
co-ordinate planes.
The following experiment will illustrate the principle of
the uniform distribution of pressure: we take the case of
an incompressible fluid. Let fig. 114. represent a closed box
full of water; A, B two vertical pistons of equal transverse
section neatly fitted into the upper face of the box, and
made to move as freely as possible. It is found by experi-
ment that if a weight be placed on A an equal weight must
be placed on B to preserve the equilibrium, shewing that
the pressure of the weight A is propagated through the
fluid to the under surface of B, and equably too, since it
requires an equal weight on B to balance this pressure.
3 X
530
HYDROSTATICS.
Again if a piston equal to A or B be fitted at C, it is
found that to preserve equilibrium, a pressure must be ex-
erted at C: and when the equilibrium exists if additional
pressure act at C, or a weight be placed on A or B, an
equal force must act on the other two pistons to preserve
equilibrium which shews that the pressure upon any portion
of the fluid is transmitted through the fluid, and acts equally
upon every equal area upon which it presses.
516. When a fluid is placed in a vessel and is acted
on only by gravity and the pressure of the sides of the
vessel, it transmits this pressure throughout its mass in the
same way as if it were deprived of its weight: but it exerts
on the sides of the vessel a pressure due to its weight and
variable from one point to another. The same is the case
when other forces, beside gravity, act upon the fluid. Since
the only direction in which any small portion of the surface
of the vessel can sustain pressure is that of the normal to the
surface, it follows, that the pressure of a fluid on any small
element of a surface, containing the fluid or immersed in
it, is perpendicular to the surface. The magnitude of the
pressure at any point in the fluid is, as yet, an unknown
quantity: it depends upon the position of the point and
upon the forces which act upon the fluid; and therefore, in
the general case, varies as the position of the point varies,
the forces being given. We measure the pressure at any
point in terms of the force exerted on a plane of a unit
of area and acted on at every point by a pressure equal to
the pressure to be measured. Thus let p be the pressure
upon a unit of area acted on uniformly by a pressure equal
to that at the point (xy), then pw is the pressure sus-
tained by an indefinitely small portion (w) of this surface.
The coefficient p is a function of the three co-ordinates (xyz)
and is termed the pressure referred to a unit of surface.
517. The pressure of the same elastic fluid against the
sides of a containing vessel is proportional to the density
of the fluid, when the temperature is constant: p = kp,
where ρ is the density at the point (xyz) and k a constant
when the temperature is constant: when this is not the case
k is a function of the temperature.
PRESSURE IN THE INTERIOR OF A FLUID.
531
PROP. To find the pressure at any point in the interior
of a fluid mass in equilibrium.
518. Let the fluid be referred to three rectangular co-
ordinate planes and let xyz be the co-ordinates to the angular
point nearest the origin of a small parallelopiped of which the
sides are dx, dy, d≈ drawn parallel to the co-ordinates. Let p
be the density of the fluid at the point (xyz), X, Y, Z the
accelerating forces acting upon the fluid, and p the pressure.
at the same point estimated as explained in Art. 516. Now the
parallelopiped is held in equilibrium by the forces acting on its
particles and the pressure of the surrounding fluid on its sides.
Let us suppose that X, Y, Z and p are the same for each
particle of the parallelopiped. Now the pressure at a point
dp
(x, y, z + dx) is p + Poz, and therefore if we suppose the
dz
pressure to act uniformly over the faces of the parallelopiped
parallel to the plane of xy, then the forces acting parallel to
dz da dy acting from the origin
the axis of x are a pressure
ρ X
dp
dz
and pZdx dy dx acting towards the origin: and since the pa-
rallelopiped is in equilibrium the sum of the forces parallel
to each axis must vanish.
dp
S≈ = pZox, also
d z
.. Sp, or
dP Sx +
dx
Hence
dpsy=pYdy, and
dy
dP Sy + dz
d P S x,
dy
dp
Sx = pXdx;
dx
p(Xô + Yông+Z8x.)
Now let us diminish the parallelopiped indefinitely, in
which case the supposition we have made respecting the uni-
formity of the density of the parallelopiped and of the action
of the forces will be true; and
dp = p (Xd® + Ydy + Zdã).
x
From this equation we shall obtain the conditions of equi-
librium of a fluid mass.
PROP.
To find the conditions of equilibrium of a mass
of fluid acted on by any forces.
532
HYDROSTATICS.
519. The first member of the equation of last Article
is a perfect differential, and therefore when the equilibrium
is possible the second member must be so too: hence the
forces must satisfy the condition that p (Xda + Ydy + Zdz)
shall be a perfect differential.
If this condition be fulfilled then equilibrium will subsist
in the interior of the fluid provided the surface be of a proper
form for since at the surface p = 0 it is easily seen that
XYZ must satisfy the additional condition that for all points
at the surface Xdx + Ydy + Zdz=0; or, in other words, this
must be the differential equation to the surface.
This latter condition amounts to the same as saying that
the resultant of the forces acting on any particle of fluid at
the surface must be in the direction of the normal. For the
cosines of the angles which the normal at a point (xyx) of a
surface makes with the axes are
dz
dz
1
dx²
- V
- V
V; where
1 +
dx
dy'
V
dx2
+
dx² dy²
or in this case,
X
Y
Ꮓ
√ x² + Y² + Z²
√ X² + Y² + Z²
√ X² + Y² + Z2
Hence our
But these are the cosines of the angles which the direction of
the resultant of X, Y, Z makes with the axes.
remark is correct.
520. When p(Xdx + Ydy + Zdz) is a perfect differ-
ential, then
d.pXd.py d.pZ_d.pX d.pY_d.pZ
J
dy
"
dx
dx
dz
dz
Performing the differentiations we have
Jd X d Y
-
d p
X
dx dy
d p
Y
p
dy
dx
Jdz
dZ
d X
dp
X
Ꮓ
dp
dx
dz
d&
dx
2
dy
CONDITIONS OF EQUILIBRIUM.
533
dY dz
P
dz
dy
}
d p
dp
Ꮓ
Y
dy
dz
Multiply these respectively by Z, Y, X and add, then
dX dY
dZ dX
Ꮓ
+ Y
dy
dx
dx dz
dY
dx) + x (dr dz)
dz dy
= 0.
This is independent of the density and furnishes a partial
criterion before we make use of the others.
PROP. To prove that for central forces tending to fixed
centres Xdx + Ydy + Zdz is a perfect differential.
521. Let a, b, c be the co-ordinates to any one of the
centres of force, P the corresponding force: then the resolved
parts of its action on the particle (xyz) are
p?
20 a
P
Y b
P
C
P
グ
​J'
where 7² = (x − a)² + (y − b)² + (≈ − c)³.
Hence, by resolving all the forces in this manner and
adding together those parallel to the same axes,
20
a
X = Σ. P
,
Y = Σ.py − b¸
C
Z=Σ.P
r
2°
P
P
2
.. Xd +Ydy+Z dã =
Σ . — { (x − a) dx + (y − b) dy + (≈ − c) d≈} = Σ. Pdr.
But P is a function of r, and therefore Pdr and the
similar expressions for the other forces are perfect differentials.
Hence Xdx + Ydy + Zd≈ is a perfect differential.
All the forces in nature are central forces. It follows
then that equilibrium will always be possible provided the
surface of the fluid be of the proper form.
·
PROP. To prove that the particles of a mass of fluid
when in equilibrium are so arranged, that the same sur-
faces are surfaces of equal pressure, of equal density, and
of equal temperature.
534
HYDROSTATICS.
522. By Art. 518. we have
dp = p(Xdx Ydy + Zdx), = pdp suppose
+
where is a certain function of xyz, (Art. 521.)
But dp is a perfect differential, therefore pd is so also:
hence p is a function of p, and then p is a function of
and therefore of p; consequently any two of p, p, & can be
expressed in terms of the third: hence all values of xyz
which make one of them constant make all the others con-
stant also.
If the fluid be elastic the density varies directly as the
pressure, as was first shewn by the experiments of Mariotte.
Let p kp; then it is found that if the temperature vary,
k is a function of the temperature.
Now dp = pdp, ..
dp аф
p k
و
and
Гаф
p = II e√dd
Пе
р
where II is an arbitary constant: this expression shews,
that if there be equilibrium, k must be a function of p,
and therefore p and and the temperature must be func-
tions of and consequently any three of these quantities
can be expressed in terms of the fourth, and therefore when
one is constant the others are also. Hence the truth of
the Proposition is manifest.
COR. It is evident, then, that the atmosphere can never
be in equilibrium; for the sun heats unequally masses of
air which are equally pressed by the superincumbent air;
consequently the layers of equal pressure, density, and tem-
perature do not coincide; a condition necessary when there
is equilibrium.
523. If, then, we integrate the equation Xdx + Ydy
+ Zdx = 0 and give to the arbitrary constant introduced by
the integration as many particular values as we please, the
determinate equations which we thus obtain belong to sur-
faces each of which has the equation Xdx+ Ydy + Zdz = 0
for its differential equation, and, in consequence, possesses
the property that it is equally pressed on every part, and
cuts at right angles in every point the direction of the
resultant of the forces X, Y, Z. These internal surfaces
CONDITIONS OF EQUILIBRIUM.
535
are called surfaces de niveau. If we make the constant
vary by indefinitely small degrees we divide the mass of
the fluid into an indefinitely great number of thin shells;
these are called couches de niveau.
The value of the constant which corresponds to the
external surface is determined from knowing the volume of
the fluid.
CHAPTER II.
FIGURE OF THE EARTH.
524. We have already remarked that the heavenly bodies
are nearly spherical in their figure. It is found, however, by
measuring degrees of latitude in places near the poles and
equator, that the figure of the Earth approaches more nearly
to a spheroid than a sphere, Also experiments made with
pendulums lead to the same conclusion. The polar radius
is found to be about 14 miles shorter than the equatorial radius.
Now the altitude of the highest mountains is not greater than
five miles above the level of the ocean. It follows from this
that the form of the sea cannot be spherical, but must partake
more or less of the spheroidal form of the land. It becomes,
then, a matter of especial interest to ascertain whether the
ocean covering the solid nucleus of the Earth and the nucleus
itself would according to the theory of gravitation assume a
spheroidal form. The calculation is one of great difficulty,
and indeed would be impracticable did we not know that the
figure does not differ greatly from a sphere. As a first ap-
proximation we shall enquire whether a homogeneous fluid
mass revolving about a fixed axis with a uniform angular
velocity will assume a spheroidal figure. It is nevertheless
à priori highly improbable that the density of the Earth
should be homogeneous since the weight of the superincum-
bent mass must be sufficient of itself to produce a great
condensation of the matter in the interior of the Earth.
PROP. A homogeneous mass of fluid in the form of a
spheroid is revolving with a uniform angular velocity about
its axis required to determine whether the equilibrium of
the fluid is possible.
MASS OF THE EARTH SUPPOSED HOMOGENEOUS.
537
525. Let a and c be the axes of the spheroid, e being
that about which it revolves: also let c²= a² (1 − e²). Now
the forces which act upon the particle (xy) are the cen-
trifugal force and the attractions of the spheroid parallel to
the axes, and these latter are given in Art. 158. as follows:
1
Σπρ {√ 1 – e² sin¯¹ e − e (1 − e³)} x,
e3
Σπρ
e³
Απρ
e³
{√1-e² sin'e - e (1 − e²) } y,
-
{e - √// 1 − e² sin¯' e} z.
Let these be represented by Ax, Ay, Cz.
Let w be
the angular velocity of rotation, then w²²+ y² is the centri-
fugal force of the particle (xy), and the resolved parts of it
parallel to the axes are w²x, w²y, 0.
Hence X (A− w²) x, Y = (A - w²) y, Z = Cz.
=
Now these satisfy the criterions given in Art. 520; hence,
so far, the equilibrium is possible. Then
2p
P
1 dp = Xd® + Ydy + Zd»
P
(A − w²) (x dx + ydy) + Czdz;
+ const. = (4 – 2) (x + y) + Cổ,
and at the surface p = 0;
A - w²
and ..
(x² + y²) + x² = const.
C
hence the surface is a spheroid and therefore the equilibrium
is possible and the eccentricity is given by the equation
A - w²
1 - e²,
C
a
2
3 Y
538
FIGURE OF THE EARTH.
HYDROSTATICS.
2
w2
1 - e²
- e²
or
sin-¹e - 3
1
e³
e²
-3-(1-e)
(1 − e³)³ sin−¹ e,
or
πρ
2
πρ
Now
+
e³
3 (1 − e³) (3 − 2e²) √T – e²
e
e³
centrifugal force at equator
gravity at equator
sin¹e = 0.
w² a
πρα-ωα
and observation proves that this equals
1
289
Απρ
3 w2
w2
1
290;
435
2 пр
Then by expanding in powers of e and neglecting powers
higher than the second (since we know that the figure of the
Earth is not far from spherical) we have
de
sin-¹e =
1
= e +
1.3
{ 1 + 1 e² +
· e¹ + ....} de
2.4
e³
1.3
3 e³
100
+
+
3
2.45
1.1
/ 1 - e² = 1 - 1 / e²
e² +
2.4
1
435
2
に
​1.1
1.3 e
-2) (1 - 1 e².
e²) 1 +
+
+3
2.4
3
2.4 5
3
2
ß e)
e²;
15
( e²
(1
= (-2) α1 - 3 0 - 0) 2 + 3 ==
... e² = 116·
If e be the ellipticity, then
·
e²
α
C
e
€
= 1 -
- e²
1
232
•
α
2
1
This result is so much greater than that obtained by
measuring the arcs of a meridian, which gives e, and
which agrees very nearly with results arrived at by other and
306
MASS OF THE EARTH SUPPOSED HOMOGENEOUS.
539
independent processes, that we are led to the conclusion that
the mass of the Earth is not homogeneous.
526. Another value of e, nearly 1, also satisfies the
equation but this evidently does not give the figure of any of
the heavenly bodies, for none of them are very elliptical.
There are no other possible values of e (except - 1 which
of course we reject): this may be shewn by putting the first
side of the equation for calculating e equal to y, and tracing
the curve considering e the variable abscissa: then those
values of e which make the curve cut the axis of abscissas are
the values we are seeking. It is found that the curve cuts the
axis in only two points on the positive side of the origin.
PROF. To shew that there is a limiting angular velocity
beyond which the equilibrium is impossible.
we alter
527. The equation of Art. 525. shews that as
the value of w the value of e will also vary. To find the
dw
de
greatest value of w we must put = 0: this gives after some
long numerical calculations.
time of rotation = 0.10090 day, and € =
e
17197
27197*
528. Since there are two forms of equilibrium it might
perhaps be imagined, that the form of the equilibrium of a
homogeneous fluid mass is unstable; and that when a fluid
mass is set in rotation it will be indifferent which form it will
ultimately assume. Supposing that there is a cohesive force
among the particles, which is the case with all known fluids,
the mass would after revolving for a greater or less time attain
a rotatory motion comprised within the limits of equilibrium
and maintain itself in that state, which of the two states being
apparently doubtful. But Laplace has shewn (Méc. Céles.
Liv. 111. §. 21), that for a given primitive impulse there is but
one form of equilibrium. In fact, it will easily be seen, that
for a given value of the angular velocity w the vis viva of two
masses so different in their form as to have e small and e 1
nearly must be very different.
529. Since the ellipticity of the Earth deduced on the
supposition of its being a homogeneous mass is greater than
540
FIGURE OF THE EARTH.
HYDROSTATICS.
that given by geodetic measurements and experiments made
with pendulums, we are constrained to reject the hypothesis of
the Earth's homogeneity, and shall proceed to calculate its
ellipticity on the supposition of its being heterogeneous. But
before proceeding to this we shall investigate the following
Proposition, since we shall find it of use hereafter.
PROP. To calculate the ellipticity of a mass of fluid re-
volving about a fixed axis and attracted by a force wholly
residing in the centre of the fluid and varying inversely as
the square of the distance.
530. Let M be the mass of the fluid, the other quantities
as before;
X
Mx
203
+ w² x, Y
My
203
+ w³y, Z
Mz
203
Then the equation Xdx + Ydy + Zdz = 0 becomes
M
2.3
( d +ydy+d%) - ( d +ydy) = 0;
M
1°
M
dr
w²
2
شه
2
X
d (x² + y²) = 0,
+ (x² + y²) = constant =
C ;
let the ratio of the centrifugal force at the equator to the
gravity at the equator be represented by am, where a is a very
small numerical quantity, and am = 285 then
1
w² a
M
a m
am
.. w²
;
M
a² 1 + am
- w² a
a²
1
1
C
1
am
or
グ
​x² + y² + ~²
M
3
(x² + y²),
2a³ 1 + am
when a=0 and y=0 then
=0:
when ≈ = 0 then a² + y² = a²,
MASS SUPPOSED HETEROGENEOUS.
541
1
C
1
1 +
C
M
a
01
2 + 2am
C
€ = 1 -
a
2 + 3am'
a
2 + 3am
Now am
1
1
289 >
€
581
am
nam)
C
2 + 2am
M
am
We shall find this result of use in a following part of this
Chapter. The value of e is far too small on this hypothesis,
as the value of e was too large on the hypothesis of every par-
ticle attracting. We shall, therefore, pass on to consider the
form of a heterogeneous mass.
PROP. To find the equation of equilibrium of a heteroge-
neous mass of fluid consisting of strata each nearly spherical,
and revolving about a fixed axis passing through the centre
of gravity with a uniform angular velocity.
531. The general equation of equilibrium of fluids de-
duced in Art. 519, gives
dp
= f(Xdx + Ydy + Zd≈),
p being the density of the mass at the point (xy≈); p the
pressure at that point; and X, Y, Z the sums of the resolved
parts parallel to the axes of the forces acting on the particle
at (xyz). These forces are the attraction of the mass itself,
and the centrifugal force.
V
Let the sum of the quotients formed by dividing each
particle of the body by its distance from the attracted point:
d V
then
dx'
d V
dy
d V
are the attractions parallel to the
dz
axes of the body on the particle at (xy≈).
Let w be the angular velocity of the mass, the distance
of (xy) from the origin of co-ordinates, μ the sine of the
latitude of this particle. Then the resolved parts parallel to
the axes of a and y of the centrifugal force of this particle
are wa, wy: the axis of revolution being the axis of ≈.
542
FIGURE OF THE EARTH.
HYDROSTATICS.
1
It is usual to express the angular velocity in terms of the
ratio of the centrifugal force at the equator to the equatorial
gravity, a fraction which observation shews to be 2: we shall
call it am, a being used throughout the calculation as a very
small numerical quantity, of which the square may be neg-
lected.
Then am
w2a3
Mass
289
, a being the equatorial radius: to calcu-
late the mass in this fraction we suppose its strata spherical,
because of the smallness of the numerator. Hence mass
= 4π fª p'a²da', a being the radius of any stratum. Let
3f" p'a'² da' = p(a);
a
3 w² a³
4π
... am =
.. w2
φ (a)
am
4πφ (a)
$(a)'
00
3
a³
d V
4. π
Hence, then, X =
+
am
dx
3
d V
4.π
Y
+
a m
dy
3
(a)
a³
Ф(а)
3
a³
X
Y
d V
Z
;
dz
2π ФС a
· [dp = V + 2 = am (2) (1 - µ³) P.
3
a❞
If we suppose the mass partly solid this equation deter-
mines the figure of the strata of the fluid part. We shall,
however, consider the whole fluid, or else suppose that the
solid and fluid parts follow the same law of density in passing
from the circumference to the centre.
f
dp
p
Now p is a function of p': and is a function of p' and
+
is therefore constant for the outer surface and for every level
surface, (or surface de niveau, Art. 523) or stratum.
MASS SUPPOSED HETEROGENEOUS.
543
Hence the general equation to the strata is
2 П
(a)
constant =
am
(1 − µ³) gºº
3
a³
= V +
4 π
9
(a)
2 п
(a)
am
a³
3
am (1 − µ³) 2º²
3
a³
this arrangement being made because each of the last terms, as
they now stand, satisfies the partial differential equation of
Laplace's coefficients; they are of the orders 0 and 2 respect-
ively. (See Art. 170).
Now by Art. 186, we have
a
Απ
d
a
V
√ =
12
a² + a
da'
31
GK
Y₁ + ... +
2'
a'i + 3
(2 i + 1) pi
Y!
Y' + ...) } da
a
d
a'r
202
+4π
+a
Y₁ + ... +
da 3
(2i + 1) a'i-2
Y;'
Y +
...) da.
a
In this put = a (1 + a Y¸ + a Y₁ + ... + a Y¿ + ...) and
ƒ“³ p'a² da' = p(a) as before.
0
0
1
as before. Then substitute this value of
V in the equation to the strata, and equate* terms of the
order i.
The constant parts when equated give
dp
4π φ (α)
3 a
+= ["p'a'do'
4. π
p'a'da' +
ama²
p(a)
9
a³
3
*It is proper to equate separately the terms of the order i on each side of the
equation because, as we have shewn, in Art. 181, the functions Y₁, Y... ; ... are
all determinate, consequently we have upon substituting for in the equation
of the text, performing the integrations, and transposing all terms to one side, a
series of Laplace's coefficients equal to zero; suppose Qo + Q₁+ Q₂+ + Qit... is
the series. If then we multiply by any arbitrary coefficient of Laplace of the ith order,
viz: Z, and integrate between proper limits we have whatever Z; be,
1
£, £ (Q + Q₁ + + Qi + ….. ) Z; d µd w = 0 ;
™
dµdw=0
Q; Z; dpdw=0 by Art. 180.
Hence, since Z, is arbitrary, Q;=0.
544
HYDROSTATICS. FIGURE OF THE EARTH.
and the terms of the order i give, after dividing by -4πa,
p(a)
3 a
1
3
a
d
Y₁
i
(2i + 1) ai+1
ρ
da' (a'i + ³ Y') da'
0
except when i
a² а
Y
2141 [Pad (144) da = 0,
2i+1
a'i-
2
0,
2, in which case the second side is
m a² (a)
(1 − µ³).
6
a³
By this equation Y is to be calculated, and then the form
of the stratum of which the mean radius is a is known by the
equation r = a (1 + a Y₁ + ... + a Y ¿ + ...).
gives
1
532. COR. At the surface a = a and the above equation
a d
(a'i + 3 Y) da'
da'
2i+1
3
a' (a) Y, when i is not = 2,
13 a² 4 (a) Y, − 5 ma² µ (a) (} − µ³), when i 2.
2
This equation we shall hereafter find of great use.
PROP.
of Y¿.
To simplify the equation for the calculation
i
533. Since Y, is a function of a, suppose it expanded
in a series of the form
W
i
Y¿ = a® W¿+ aº W{ +
W; as + as!
Wi
}
-Wh; suppose,
then h; is a function of a and equals aª +
W, ... are independent of a.
Hence, since Y is the same
W
as +
and W;
W;
i
MASS SUPPOSED HETEROGENEOUS.
545
function as Y; when a' is put for a, we have Y = W;h{ : h{
being the value of h; when a' is put for a.
i
Let these values of Y, and Y be put in the equation
deduced in the last Proposition;
α
a²
i
... (@) h
(p(a)
W
1
3 a (2i + 1) a² + 1
hi
hi
Ρ
da' la'i-
d
3
(a' ¹ + ³ h;') da'
da
0, when i does not = 2
m a²(a)
(} −µ³), when i = 2.
6 a³
-26 + 1 [Pdz (14-2) da') = 0,
Ja
Divide both sides by a and differentiate with respect to a,
observing the remark in the note*; then
´dh; p(a) i + 1 h¿p(a)
W
i
da 3a+1
1
d
3
ai + 2
+
3
q2i+2
P
da'
(a'¹ + ³h;') da'} =
= 0.
ď² hi
;
da²
Multiply this by a2i+2, differentiate with respect to a, and
divide by the coefficient of
then
ď
dh;
6 pa² dh;
W
+
¿
da2
(a) da
(i
βρα
6 pa³ hi
(i + 1) -
0,
p(a)) a²
this is true for all values of i in the series 1, 2, 3,
By assuming any law of the density we can calculate
p(a), or 3 fop'a da'. We must then solve the above equation,
* In order to explain how to differentiate a definite integral with respect to a
quantity involved in the limit, suppose that
d
ff(x)dx= F(x)+C;
t.
ƒ,, f(x)dx=F (t) − F(t');
t
-- Se 8(x)dx=(0) = (1)
åt Sq, f(x) dx=dF (t)
dt
d
and =
dt I t
dt
=f(t),
F(t')
f(x)dz-d() = -5(c).
dť
=-f(t).
3 Z
546
FIGURE OF THE EARTH.
HYDROSTATICS.
i
and knowing h; we shall know the law of variation of Y; in
terms of a, and therefore the law of variation of the different
strata of the fluid mass. Before this is attempted we can shew
that the expression for the radius, viz. a {1 + a Y₁ + a Y₂+ ...}
admits of great simplification.
2
i
1
2
=
PROP. To prove that Y; O excepting the case of i = 2.
534. The equation in the last Article shews that W0
(and . Y = 0) unless the part multiplying W, vanish.
We assume that the density of the mass decreases from
the centre to the surface. For it is evident that if a stratum
were placed over one less dense, the particles of the upper
would penetrate into the lower in the same manner that a
body sinks in a fluid of less specific gravity than its own.
d p
Hence is negative. Let p = ẞ- B'a'"...... ß, B'......
da'
B-
being constants: and n... positive indices;
a
·· 4 (a) = 3
S
p'a²d a'
=
Ba³
3 ß'
an+3
0
n+ 3
ρα
1
$ (a)
Also hi
as +
dh;
ď hi
= sas−1 +
da
s( s − 1) as −2+
da
n
B'
an +
β
n + 3 ß
(see Art. 533);
Substitute these expressions in the equation of Art. 533, and
neglect higher powers of a; then
0 = W; {s (s - 1) as-2 + 68a-2 - [i (i + 1) − 6] a³ −2}
near the centre.
Hence, unless W;
W₁ = 0, we must have
8 (8 − 1) + 68 = i (i + 1) − 6, ... s² + 58 +24
-
25
¿ ² + i + 1}{ ?
4
5
S · − / ± (i + 1 )
= ¿ - 2 ori - 3,
i
SIMPLIFICATION OF THE RADIUS OF THE STRATA.
547
i
the last value is inadmissible, because it would make h;, and
consequently Y; (since we suppose W; not to vanish), infinite
at the centre; i. e. the radius of the central strata indefinitely
great, which is contrary to the original hypothesis of the strata
being nearly spherical.
When i = 1,
1, i - 2 1, and therefore the same objection
applies to the value si-2 when i = 1.
=
Hence W₁ must
equal 0: and therefore Y₁ = W₁h₁ = 0.
For other values of i, s = i − 2 :
1
dhi
•. h; = ai−² +
ai-c
and
= (i − 2) a¹³ +
i-3
da
dh₂
therefore when i is greater than 2, is positive near the
da
centre, and therefore h; increases from the centre to the sur-
face for strata near the centre.
If h in increasing attain a maximum value for a value
6 pa³ | h;
p(a)] a
a positive quantity when i
dh;
of a less than a, then
0, and the equation of Art. 533
da
ď h;
gives
(i + 1)
da²
is greater than 2.
For
a
0
dp
da'
βρα
$(a)
is always less than 6
$ (a) = 3 [ " "p'a”da' = pa³ – S″
and since is negative, ..
a
13
dp
de da':
da'
but i (i + 1) is greater than 6 when i is greater than 2 and
consequently the above expression is positive when i is greater
than 2.
Since this would indicate a minimum, and not a
maximum, it follows that h; can obtain neither a maximum
nor a minimum value, and that it therefore increases continu-
ally as we pass from the central strata towards the surface.
Let h, be the value of h; at the surface: then
than unity.
h;
is less
h;
548
FIGURE OF THE EARTH.
HYDROSTATICS.
Now by Art. 533, putting a = a we have, when i is greater
than 2,
Wi
{
f
0
a d
p
da
h'
i
h) da
li + 3
h;
a
12
da' - (2i + 1) a' f "p'a" da'} = 0,
0
ρα
or, if we integrate by parts and multiply by 3,
+3
do'
da'
W,{−p (2i−2) a'+*+(24+1)2' ["aºdg_da-s fashidp a}=0.
13
p
α
0
d p
hí
h;
0
h;
Now is negative, and is less than unity; hence
da'
i
the multiplier of W; is negative and not zero when i is
greater than 2 and consequently W; 0, i greater than 2;
=
.'. Y₂ = 0, Y₁ = 0,……………. and we have shewn that I₁ = 0.
3
4
2
Hence r = a {1 + a Y₂ is the equation to the stratum
of which the mean radius is a.
:
PROP. To prove that the strata are all spheroidal and
to find the law of their ellipticity.
535. We have seen that r = a { 1 + a Y₂}
2
Now a Y₂ = a W2h2. To calculate W₂ we have from the
equation for the stratum at the surface (Art. 532.) by putting
Y₂ = W₂h₂ and Y = Wh
2
5a²
2
a
d
5m
a²
6
w, { "a" (p (a) h₂ = ƒ ƒ —, (a'h,') da' } = 5T 2° op (a) (} − 1)
W 2
3
- St h₂)
da'
Dividing by the coefficient of W, we have
2
a Y₂ = a W₂h₂ = ε (} − µ³),
ε being a small quantity given by the equation
am
h2
2 h₂
દ
3
d
1
5a² (a) h₂
(a¹³h₂') da'
da
FORM AND ELLIPTICITY OF THE STRATA.
549
a { 1 + ε ( } − µ³) }, μ = sin (latitude) = sin l,
Hence r = a
६
- -
− a (1 − & e) { 1 + ε cos² l}
This is the equation to a spheroid from the centre, of which ɛ
is the ellipticity: and the axis minor coincides with the axis of
revolution of the whole mass. But varies as h; and we
have proved that h₂ increases from the centre to the surface.
Hence we arrive at the following result.
2
If we assume that the density increases from the surface
to the centre, then the strata are concentric spheroids the
axes of which coincide with that of the revolution and the
ellipticities increase from the centre to the surface.
PROP. To obtain equations for calculating the value of
the ellipticity when the law of density is known.
536. Let e be the ellipticity of the external stratum;
.*. & =
દ
h₂
h₂
e by the value of & in the last Article:
and if we put (a) for
S
a d
P (a) da' we have by the
da'
value of ɛ given in the last Article
3
(a)
a m
€
5a²
(a)
2
am.
3
√(a)
.'. €
2
5а² ф(а)
h₂
in the differential equation of
6
Also by putting h₂
Art. 533, we have the following equation for calculating &;
+
6pa² de
da² (a) da
-
f.
ρα) σε
0.
p(a)] a²
This may be simplified thus: multiply by (a),
d
da
(a) de}
d
+
da
da
{3pa²e}
a²
60(a) ɛ
+3a² dp
e
da
550
FIGURE OF THE EARTH.
HYDROSTATICS.
d2
6
da { $(a) (} = 1 4 (a) € + 3a²e d
dp
"
da²
da
from this equation & is to be calculated when the law of density
is known.
PROP. To calculate the ellipticity of the surface in the
two extreme cases of the mass being homogeneous, and of the
density at the centre being infinitely greater than at any
other point.
530.
537. This Proposition will verify the results of Arts. 525,
3 (a)
am = 5a² (a)
Now by Art. 536, e - 1 a
α
I. When p is constant: (a) = 3 fª p′aª² da' = Da³: D the
density at the surface;
d
also (a)
15
=
(a'³¿') = Da³ e ;
da'
0
II.
.*. € —
€, ... € € = 5 am.
// am = 응 ​6.
When the density at the centre is infinitely greater
than that at any other point
φ (a) = 3√ p'a da' = element at centre = 3p'a da',
a being indefinitely small.
Also (a) = p
d
da'
5
(a'³ ε') da' = 5 p'a'¹é'da' + p'a' de';
12
a
.'. €
11 a
dε
am =
a²
2
ર
+!
+ á
da'
= 0;
· € = 1 am.
1
In the case of the Earth observations shew that am = 289
and therefore the ellipticity of the Earth in these two hypo-
theses is and The ellipticity of the Earth deduced from
the measurements of degrees of the meridian is intermediate to
1
231
1
578 ·
CLAIRAUT'S THEOREM.
551
these: and therefore we may conclude that the Earth is not
homogeneous, and also that every particle of the Earth's mass
attracts and not the central parts only (see Art. 260).
We shall presently seek for a law of density which is likely
to be an approximation to the truth. Before this, however, we
shall calculate the value of gravity and the length of a seconds
pendulum at the equator, and the length of a degree of lati-
tude, in order to compare the results of theory and observation.
PROP.
To calculate the value of gravity at any place
on the Earth's surface, and the length of the seconds pen-
dulum: and to shew that their changes as we pass from one
place to another vary as the change in the square of the sine
of the latitude. Clairaut's Theorem.
538.
Let g be the action of gravity at a place of which
the latitude is l, or sin-¹
By Art. 531 we have
м.
f(xda
4π
dx+Ydy+Zdx)=V+
(a)
2 п
φ (a)
am.
am
9
a³
3
(-u²) r².
3
a³ 3
Now this expression is the sum of all the forces which act upon
any particle multiplied by the elements of their respective di-
rections. Hence the resolved part of the resultant attraction
in the direction of is obtained by differentiating this with
respect to r and changing the sign; let the result be g: the
resolved part perpendicular to this will be of the order
α, let
it be ag';
therefore gravity = √g" + a'g'² = g, neglecting a²
d V
8 π
Now g
a m
dr
9
(a)
4π
φ (a)
am
( − µ³) r
3
a³
4.π
702
Sπ
9
am
a
a
12
p' { α²² +
φ (a)
a³
ቀ
a³
3 a
d
5r² da'
4π
3
(a'³ Y₂') } da'
am Þ (a)
(1 − µ³) r.
a³
552
FIGURE OF THE EARTH.
HYDROSTATICS.
But r = a {1+aY₂} at the surface. Also by Art. 532, we
have
a
2
[* p' d /, (a'³ Y₁') da' = { } a² Y, − ‡ ma² († − µ²)} † (a) ;
da'
• g
11
0
4π φ (a)
3
૨૭
{1 − — am + a Y₂ –
5 am
2
· (3 - μ²) }
=
4π φ (a)
3 a2
4π φ (a)
3 a²
3 am + (ε − 3 am) ( − u²)} by Art. 535.
(1 + e − 3 am) { 1 + (½ am − e) sin² 1}
G {1 + (½ am − e) sinº 1},
G being the equatorial gravity.
Hence, the ratio of the excess of polar over equatorial
gravity to equatorial gravity (that is ame, by the for-
mula just proved) added to the ellipticity equals & × the ratio
of the centrifugal force at the equator to the equatorial gra-
vity, whatever be the law of the density. This is called
CLAIRAUT'S THEOREM after its discoverer.
539. Letland L be the lengths of the seconds pendulum
at the places at which g and G are the values of gravity;
g
.. l = L
L { 1 + (½ am − e) sinº 7}.
G
PROP.
To calculate the length of a degree of latitude at
any place on the Earth's surface, in terms of the length of a
degree at the equator.
540. Let C and c be the lengths of a degree correspond-
ing to the values of gravity G and g.
dr²
+
dl2
Now rad. of curv. =
sin l
==
µ, r = a (1 + a Y₂),
dr²
d²r
202 +- 2
dl2
dl2
LENGTH OF A DEGREE OF LATITUDE.
553
dr
dr du
= aα
1
di
d u d l
d² r
d
dr
2
1
dľ²
αμ
dl
d
dY
d Y2
αμ
dY₂
2
аам + aa (1 − µ³)
dμ
d Y 2
-
ď² Y 2
du²
= aa
1 – µ²)
+аам
du
du
du
.. rad. of curv. a
ift
+ a
d
1 + a Y ¿ + a
2
αμ
((1 − μ²)
dY₂
+ αμ
-
du
d Y₂l
du
2
dY,
α
2
- a fi - says + aux
=
5a 2
2
d² Y
1- μ² dw² f
by the equation of Laplace's coefficients (Art. 170.)
= a {
dY 2
શે
1 5a Y₂+ aμ
2
du
because Y₂ is independent of w (see Art. 535.)
2
Now the
length of the degree varies as the length of the radius of
curvature;
dY,
· = d'{1
c = c'f1 - saY, + anxi), c' being a constant.
·.
2
But a Y₂ = € (
2
6 (3
αμ
− µ³) ;
. . c = c′ { 1 − 3 € († − µ²)}
... c'
= c' (1 − e) (1 + 3e sin² l);
-
.. c C (1+3e sin² 1).
-
These formulæ for the length of the seconds pendulum and the
length of a degree of latitude enable us to test the truth of our
calculations by comparing observed results with those given by
these formulæ.
2
541. It will be observed that throughout this calculation
we have neglected a²....
α
It is very desirable, then, to apply
some independent test of the truth of our results.
The two
conclusions to which we are come are, that the figure of the
Earth is an oblate spheroid, and that the law of variation in
:
4 A
554
HYDROSTATICS. FIGURE OF THE EARTH.
gravity in passing from one place to another on the surface of
the Earth is represented by Clairaut's Theorem. Mr Airy has
estimated the probable magnitude of the errors committed in
these calculations, by considering the two following extreme
laws of density: 1st, the case of homogeneity: 2nd, the case in
which the matter at the centre only attracts; or, which amounts
to the same, the density of the matter at the centre is infinitely
greater than that of the rest of the mass. The deviations
from a spheroidal figure, in these cases, are found to be in-
sensible. The same is found to be the case with Clairaut's
Theorem. It follows, then, that the error in omitting small
quantities in our calculations is inappreciable. (Ency. Met.
Figure of the Earth, Arts. 69–72.) As far, then, as these
two tests are concerned in determining the law of density in the
Earth's mass we should be at a loss in attempting to decide.
what the actual law of nature is. But the magnitudes of the
ellipticity being and in these extreme laws of density, we
learn, that since the first is greater and the second less than
that given by geodetic and other measures, we must suppose
that the density increases towards the centre; but that it does
not become infinitely great.
1
231
1
577
542. We are now able to calculate the principal moments
of inertia of the Earth, which were required in Art. 470, in
calculating the Precession of the Equinoxes and the Nutation
of the Earth's Axis.
PROP. To calculate the principal moments of inertia of
the Earth, supposing it to consist of layers nearly spherical
and of different densities.
543. Let A, B, C be the principal moments of inertia
about the axes of x, y, z, respectively: x,y,≈, the co-ordi-
nates to an element of the mass. Then the mass of this
element p'r² dr dudo (Art. 183.); also
2
y² + ≈,² = r² { 1 − (1 −µ³) cos² w} = r² { + [− (1 − µ³) cos² w]},
//
-
2
2
2
x ² + ≈ 2² = r² { } } + { } -(1-µ³) sin²w]}, x² + y² = r² { ÷ +(}−µ³)},
[
Z
we have arranged these in this manner because they are then
each of the form U+ U. Hence
PRINCIPAL MOMENTS OF INERTIA.
555
2π r
2
A = ƒˆ‚ ƒ³˜˜˜ p' r³ { ÷ + ‡ − (1 − µ²) cos² w} dµ dw dr.
But
S
4
'p'r¹ dr = t
0
d
= + [ ' §' ~ /
0
0
ρ
da'
d
a d.75
ρ da'
"'d
5
da'
{a'³ + 5 a'³ a (Y' + Y₁' + Y½' + ...) } da'
5
15
ρ a'³ + 5 a'³ a Y₂'} da' by Art. 534,
da
2
= } σ (a) + ½ a a³ † (a) Y₂ — § am a² ¤ (a) (} − µ³)
a
d. a
a'5
putting p
ρ
da
2
da' = σ (a). (Art. 532).
Also a Y₂ = e (u) by Art. 535;
r
·· ₤" p'r¹ dr = + σ (a) + §a²
ལ་
(a) (e – † am) (} — µ²)
— σ (a) + ↓ (a) ( − µ²) by Art. 536;
2π 2
15
... A = S_₁₂ {}, σ (a)
+ † (a) (} − µ³) [} − (1 − µ²) cos²w]} du dw by Art. 180.
8π
8 п
σ (a)
† (a).
15
45
In the same manner
8π
π
B = -
o (a) — 8 — y (a).
15
•2 T r
45
Also we obtain C = ↓₁₁ ƒ ²™ ƒ p² r¹ { } +
ƒ ƒ³¹¤ p' (}
( - μ²) }
8 п
16π
σ (a) +
y (a).
15
45
C-A
Hence
(a)
C
σ (a)
556
FIGURE OF THE EARTH.
HYDROSTATICS.
544.
We proceed to obtain an approximate law of the
density of the strata, and then to prepare the formulæ
deduced in the foregoing Articles for numerical calculation.
PROP. To obtain an approximate law of the density
of the strata.
545. By Art. 531, we have for calculating the pressure
on the stratum of which the mean radius is a, neglecting a,
୮
a
a
1 dp
p' da'
da
4π φ(α)
a
+4π
p'a'da'.
3
a
a
Laplace has integrated this equation upon the supposition
that the change in pressure in descending through the strata
varies as the change in the square of the density (Mémoires
de l'Institut, Tom. 111. p. 496). This law of compression
differs from that of fluids, in which the change in pressure
varies as the change in density. The law used by Laplace
is à priori more probably true than the law of the compression
of fluids, since tenacious and semi-fluid bodies must require
a greater compressing force to produce a given compression
and density under given circumstances than a fluid body does, in
consequence of the greater cohesive force of the particles of the
semi-fluid body. See also some remarks by Professor Challis
The ap-
on this subject in the Phil. Mag. Vol. xxXVIII.
proximate truth, however, of this law is shewn by the accuracy
of the results to which it leads us. Putting, then, dp = ½kd.p
k being a constant, we have
a
1 dp
da'
da' = k
འ
a
a
dp, da' = k (p − D),
da'
D being the density at the surface;
a
12
.. ka (p − D) = 4𠃪 р'a² da' + 4π a
12
because (a) = 3" p'a da'.
p
a
So p'a'da',
12
APPROXIMATE LAW OF DENSITY.
557
Differentiate with respect to a, having regard to the note in
page 545;
d. pa
• k − k D = 4 π р а² + 4π [ª p'a'da' — 4 π pа² = 4π↓ª p'a'da';
da
a
a
ρ
a
d². pa
da2
4π
-q2pa, putting
q²;
k
A
.. pa = A sin (qa + B),
P
sin (qa + B).
a
A sin B
When a = 0, ρ
0
.. B=0, otherwise p would
be infinite at the centre, which cannot be;
ρ
A
sin qa,
a
ά
A and q being unknown constants.
A
The formula
P
α
-
sin qa appears à priori to be well adapted
to represent the law of the density since it gives a density
increasing from the surface to the centre.
PROP. To calculate the ellipticity of the strata on the
approximate law of density deduced in the last Article.
A
a
546. We must put p
sin qa in the equation
d²
{$ (a) ε } = √2 $ (a) & + 3a² €
dp
(see Art. 536).
da²
a
12
a
6
a²
da
Now (a) = 3√ p'a² da' = 34" a' sin qa' da
= A
α
a
- 3 4 - = cos
Չ
9
also de=1{2 cos qa -
dp
da
A
a
and our equation becomes
ď³ { $ (a) ε }
da²
ga +
1
a²
sin
1
q
ε
sin qu},
ga},
qa
q²
ga} = — — (a),
6
3 a²
+ q° $ (a) & = −₂ $ (a) ε.
$
558
FIGURE OF THE EARTH.
HYDROSTATICS.
The integral of this equation is
C
$ (a) ε =
& c{(1-2)
3
3
sin (qa + C') +
cos (qa +
a²
qa
C and C' being arbitrary constants. In our case C' = 0, other-
wise the ellipticity at the centre would be infinite, as is easily
shewn by expanding & in powers of a.
Hence, if we substitute for (a), we have
3
3
1
tan qa +
Cq²
q2a2
qa
દ
3 A
tan qa
qa
3
3
1
દ
tan qa - qa
q²a²
2
tan qa +
α
qa
and .
...
(1).
6
tan qa - qa
3
3
tan qa +
q² a²
qa
This gives the law of decrease of the ellipticity in the strata
in passing from the surface to the centre.
To determine & and e we must combine with the above
the following equation
€
1
2
3
(a)
am =
(see Art. 536).
5a²
(a)
Now y (a) = √
a
d
a
sin qa'
d
P
(a'³ ε') da' = A
da'
α'
da'
—¿ (a'³ é') da'
= A {a' e sin qa + fa³e' (sin qa' - qa' cos qa') da'}, by parts.
α
The integral contained in this.
3
*{(«^-°a)sinv«+°"cosqc}d«
tan qa
qa
13
3
3
q2a2
tan qa +
qa
1
tan qa - qa
—-= {3 (2 a² q² — 5) sin qa− (q³a³—15qa)cosqa}.
3 q
3
q² a
2
tan
tan qa+
qa
ELLIPTICITY OF THE STRATA.
559
€
3 A
Also (a)
{sin qaqa cos qa}.
92
3
(a)
Hence the equation e 11/1/14
1 am =
becomes
5a²
(a)
am =

(q*a*-3 q²a²) tan² qa+3 q³a³tan qa+(tan qa−qa) {(6 q²a²–15) tan qa—(9³a³–159a)}
5 q² a² (tan qa—qa) { (1 –,313)
tan qa+
(2 - qa²) tan³ qa qa tan qa
3
qa
5 am
q³a²
€
to
2
(tan qa – qa) {
3
3
1
tan qa +
q² a
qa
–
This equation determines e: and consequently is also
known by equation (1).
547. To facilitate the calculation of e let
qa
tan qa
= 1 − ≈ ;
1
'. € =
5 am
2
--
3z
q² a²
q*a*
3-z
Before we reduce this to numbers we shall calculate the
principal moments of inertia deduced in Art. 543, in terms of
the approximate law of density.
PROP. To calculate o(a) and ↓(a) and
approximate law of density.
A
C- A
with the
C
13
Now (a) = ["p' (a) da' = 5.4 ["a" singa'da
548. 5 =
-
ρ
d
da'
ეპ
3a²
6a
{-
q
q
q³
5 A
0
6
qa
qa}
= 54 { - cos ga+ sin ga + cos gasinga
{(3 qa² -6) sin qa (qa³ - 6 qa) cos qa}.
560
HYDROSTATICS.
FIGURE OF THE EARTH.
Also (a)
5 a²
- (a) (e − 1 am)
=
3
5 Aa
q²
(e − 1 am) (sin qa — qa cos qa);
(a)
or
o (a)
2
qa
q² a²/ tanqa
qa
1
C - A
tan qa
,
C
(ε - 1 am)
6
6
3
q2a2
(ε − 1 am) ≈
qa
x = 1
2+
(1
6
tan qa
ર
q² a
2
We now proceed to reduce, these quantities to numbers.
PROP. The law of density being represented by
A
p == sin qa; it is required to find the value of q in the
a
case of the Earth.
549. The experiments of Cavendish and the observations
of Maskelyne shew that the mean density of the Earth is
about five and a half times that of water. Now since the
superficial stratum of the Earth consists partly of the Ocean
we must not include this among the strata of the Earth when
we speak of their law of variation, because the density of water
is far less than that of rock or earth. We shall consequently
suppose the superficial stratum to be of the density of granite
or thereabouts; i. e. nearly in the ratio of 5 to 2 to that of
We shall therefore suppose the mean density is to
that of the superficial stratum as 11 to 5.
water.
Let & be the mean density: D the density of the outer
stratum: then
1
a
a
a
12
√ √ª 4 π á²² da' = ƒˆ±π p
= √² 4π p'a²² da'
a
— = A
a²d - 4 ƒ a' sin qa' . da' = 1 { —
9
cos qa+singa};
NUMERICAL VALUE OF THE ELLIPTICITY.
561
d
D
"
3
qa
11
by hypothesis;
q² a²
tan qa
5
15qa
... tan qa
15-11q2a
5π
qa
A very small value of q will satisfy this equation; but
a small value would give a very slow change of density in
the strata of the Earth. After repeated trials we find that
6
2.618 nearly satisfies this equation for then
:
π
15 qa
log10
I.8130736
=
log10 tan 33° 2′
log 10 tan
nearly
6
11 q2a2 - 15
5π
... qa
6
2.618 very nearly satisfies the equation.
If we had taken d = 2.4225 D, then qa = 2.618 would have
satisfied the equation as far as four decimal places.
We shall take, then, qaπ = 2.618.
The remarkable agreement found to subsist between the
calculations made with this value of qa and by other means
is the most satisfactory proof of its correctness.
PROP. To calculate numerically the ellipticity of the
Earth with the value of qa equal to 2.618.
3z
5 am
q² a²
550. By Art. 547, € =
3−z
-
q² a²
12
Ja
where = 1
tan qa
1
s
hence = 5.5345: also am =
289
by observation;
1
.'. € = .00325401 =
307.313 ↑
measures, which give e =
= 306 •
this value of € accords remarkably with the result of geodetic
1
4 B
562
FIGURE OF THE EARTH.
HYDROSTATICS.
PROP. To calculate numerically the value of
C- A
C
and the Annual Precession of the Equinoxes with the ap-
proximate law of density.
C- A
(a)
(ε - 1 am) x
551. By Art. 548,
C
σ (a)
6
Also the Annual Precession of the Equinoxes
92 a2
2 + (1 - 2²²) =
= .00313593.
C - A
C
1 +
176.5906
1 + v
4882".05 (Art. 470.)
= (.00313598) (3.3545) (4882".05); v = 74
= log₁0¹
1
10
(3.4963664)
"/
.5256278 = log-'(1.7105962)" = 51″.3566,
3.6886020
the observed precession is 50".1. The exceedingly remark-
able agreement of the calculated values of the precession and
ellipticity of the Earth with their observed values afford a
convincing proof of the correctness of the principles involved
in the calculation,
552. Before we quit this subject we shall give a few im-
portant Propositions which tend to throw additional light upon
the determination of the figure of the Earth.
The permanent state of equilibrium of the heavenly bodies.
makes known to us some of the properties of their radii. If
the planets did not revolve about one of their three principal
axes, or very near to one of them, there would be produced
in the position of the axes of rotation, some variations which
would become sensible, particularly in the Earth (see Art. 458).
Now by the most accurate observations, no such variations
are perceived. Therefore we must infer, that a long period
of time has elapsed since all parts of the heavenly bodies,
and particularly the fluid particles on their surfaces, have
been arranged in such a manner as to render their state of
equilibrium permanent, consequently also their axes of rotation :
NUMERICAL VALUE OF PRECESSION.
563
for it is very natural to suppose, that after a great number of
oscillations, the bodies must assume the forms corresponding
to the state of equilibrium, on account of the resistances
suffered by the particles of the fluid. We shall now examine
into the conditions, arising from this supposition, in the ex-
pression of the radii of the heavenly bodies.
2
PROP. To prove that in consequence of the perma-
nence of the rotatory motion of the Earth Y₂ = K (} − µ³)
+ K' (1 − μ²) cos 2w, K and K' being constants which depend
upon the internal structure of the Earth.
553. Let the Earth be referred to its principal axes:
and let be the radius of any particle: the angle which
r makes with the axis of ≈,, w the angle between the planes
of x,x, and rz, also let cos 0 = µ ;
is
.. x,
=
r sine cosw = ? √ 1
2
μ
u² cosw,
'}'
= 1'μ.
y₁ = r sin✪ sin w = √1 - μ² sinw,
Z z = 1 cos
u
Let p be the density of the stratum of which the radius
then the element of the mass (m) = pdr rôė r sine dw
=-pr² dr dμ dw, and by the properties of the principal axes,
viz. Σ.mz¸y, = 0, Σ. mx, z 0, .mx,y, = 0, we have
2
So ♪ - ₁ fo²™ pr² µ √ 1 − μ² sin w dr du dw = 0
1
pr¹µ√1 - µ² cos w dr dµ dw
√ L₁ f² pr²
1
(1 - µ³) cosa sino drdµdw = 0
Now r = a {1 + a Y₁ + a Y₂ +
1
2
•
0... (1),
+ a Ÿ¿ + ..
and ..
S
dr
a
d. 2.5
prt dr
pr
da
Ja5
0
а
P
d
da 5
+ aa”
αα
Un + a U₁ + a V₂
da
(Y₁ + Y₂ + + Y; +
...
+ + a U; + ... šuppose.
р
da,
da
da
...
564
HYDROSTATICS. FIGURE OF THE EARTH.
Then by substituting this series in equations (1), bearing in
mind the property of Laplace's Coefficients proved in Art.
180, we have
S₁₁
S₁S
~2π
10
2π
2 M
U₂ μ √ I
2
PR
u² sine du dw = 0,
U₂p √1
μ² cosw dµ dw = 0,
2
-
and ƒ¹,ƒ³″ U₂ (1 − µ³) sino cose du dw = 0.
0
2
Now since U₂ is a function of
2
1
М
2
2
2
and
2
1 — u² cos w,
Mg
1
sin w of the second order, and satisfies the Equa-
tion of Laplace's Coefficients, it is of the form H(-u²)
+H'μ1-u sino + H" 1-μ² cosw+H"" (1-µ³) sin 2w
+ Hiv (1 − µ³) cos 2 w. Then by putting this for U, in the
last three equations we find
H'
=
ль
: 0, H″ = 0, H"
-
0,
··· U₂ = H (} − µ³) + Hiv (1 − µ³) cos 2 w.
..
But U was written for
So
a
p
μ
d
2
(a³
da
2
Y₂) da, and by Art. 532 this
—
= § a² 4 (a) { Y, − 1 m (} − µ²) },
K'
.. Y₂ = K (} − µ³) + K′ (1 − µ³) cos 2 w,
2
K and K' being constants which depend upon the internal
structure of the Earth.
Thus far, then, the condition that the Earth rotates
about a principal axis, determines the form of the function
Y₂ in the radius of the surface, r = a (1 + a Y₁ + a Y₂ +
+aY¿+...); shewing that three of the terms in the most
general form of Y vanish.
2
2
554. Laplace has obtained some remarkable formulæ for
the value of gravity, the length of a seconds pendulum, and
the length of a degree of latitude which are independent of any
supposition of the internal structure of the Earth, except that
it consists of nearly spherical strata. They
They are deduced as in
Arts. 538, 539, 540; but a {1+ a Y₁ + a Y₂+aYs + ......} is
put for instead of a {1+aY2}. The results are
I'
r
1
3
GENERAL FORMULE.
565
g = G {1 + a Y₂+2a Y3 +
+ (i − 1) a Y ¿ + ......}
i
- 1898 7
10
(a)
3
πα
( − µ²),
a²
1 = L {1 + a Y₂+ 2a Yg + ...... + (i − 1) a Y; +
Y3
2
- — am ( - μ²)},
·
c = C { 1 − 5a Y½ – 11 a Y3 ………….. — (¿² + ¿ − 1) Y¿ − ...
+аль
dY, dY 3
αμ
2
du
dY;
T
+
+
+
+
dμ
d p
.)
α
de Y₂
d2 Y 3
d² Y
2
+
+
+
2
Sedan
dw²
dw²
dw°
...)
555.
If we compare the expressions for the radius of the
Earth with that for the length of the pendulum and the length
of a degree of the meridian we shall perceive that the term
a Y; in the expression for the radius is multiplied by ¿ - 1
in the length of the pendulum, and by i²+ i − 1 in the degree of
the meridian. It follows, then, that however small i-1 may be
this term will be more sensible in the lengths of the pendulum
than in the horizontal parallax of the Moon, which is propor-
tional to the radius of the Earth: and it will be still more
sensible in the measures of the degrees than in the lengths of
the pendulum.
1
2
These three expressions are very important, inasmuch as
they are independent of the internal structure of the Earth;
that is, they are independent of the figure and density of the
strata, since the functions Y, Y... all refer to the surface. It
follows, then, that if we can determine the functions Y, Y₂ ...
by the measures of degrees and parallaxes, we shall obtain
directly the length of the pendulum. We may by this means
ascertain whether the law of universal gravitation agrees with
the figure of the Earth and with the observed variations of
gravity at its surface. These remarkable relations connecting
the degrees of the meridian and the length of the pendulum
also serve to verify any hypothesis, assumed to represent the
measures of the degrees of the meridian. Laplace (Méc. Céles
566
FIGURE OF THE EARTH.
HYDROSTATICS.
III.
Liv. 111. §. 33.) makes an application of these formulæ to an
hypothesis of Bouguer with respect to the lengths of degrees:
and his calculation shews that it must be rejected. The by-
pothesis is that the variation of a degree of the meridian is
proportional to the fourth power of the sine of the latitude.
556. Laplace shews that the greatest minimum probable
error in calculating the length of a degree from observations.
made at seven places is 97.2 toises, the mean length of the
degrees being about 51307.4 toises (Liv. 111. §. 41). The ratio
the error bears to this 0.00189.
It is also shewn (Liv. III. §. 42.) that the greatest pro-
bable error in the calculation of the length of a seconds pen-
dulum is 0.00018, the mean of the lengths of the pendulum at
the fifteen places at which the observations were made being
0.99922: the ratio the error bears to this = 0.00018. Now this
is more than ten times less than the error of the measures of
the degrees, and remarkably confirms the theory in Art. 555;
viz. that the terms of the expression of the terrestrial radius,
which cause the Earth to vary from an elliptical figure, are
much less sensible in the lengths of the pendulum than in the
lengths of the degrees of the meridian. Later observations,
as Mr Bowditch observes in his Commentary on this part of
the Mécanique Céleste, do not confirm this result. The dis-
crepancies among the observations of the length of the pen-
dulum are greater than in those of the best observations of the
measured arcs of the meridian. Various causes have been
assigned for these differences in the observations of the pen-
dulum: as the local attractions of neighbouring bodies; the
peculiar action of the substance of the stratum of the Earth
over which the pendulum is placed; and finally magnetic
action. With respect to the first Bouguer has found by ob-
servation that the attraction of the mountain Chimboraço pro-
duced a deviation of 7".5 in the plumb line: and Dr Maskelyne
observed the attraction of the mountain Schehallien to be 5".8.
CHAPTER III.
FORM OF EQUILIBRIUM OF THE OCEAN UNDER THE MOON'S ATTRACTION,
AND THE FORM OF THE ATMOSPHERE.
THE following Proposition we shall find of use when we
come to the Chapter on the Tides.
PROP. Supposing that the Earth is a sphere surrounded
by a sea of small depth: required to determine whether the
form of the sea attracted by the Moon will be spheroidal,
the Earth and Moon being both supposed held at rest.
557. If the spheroid be the form of equilibrium it must
evidently be prolate, the axis of the spheroid passing through
the Moon. Let c' be the distance between the Earth and
Moon, a the mean radius of the Earth, a' the radius of the
solid nucleus of the Earth, and the distances of any par-
ticle (xy) from the Earth's centre and the Moon, E and M
the masses of the Earth and Moon,
P
and p' the mean density
of the Earth and the density of the sea, e the ellipticity of
the spheroidal figure of the ocean caused by the Moon.
Then the excess of attraction of the nucleus above what
it would be supposing it of the density of the sea, gives the
following resolved forces on a particle (xy) parallel to the
axes
13
4 π (ρ − p) a¹³ x 4 π (p-p')a¹³ y
3
7.3
3
2.3
4 π ≈
+ = (p=p') a'³
3
The attraction of a whole fluid spheroid gives
203
Απρί
2 €
Απρί
1 +
X,
1 +
3
5
3
2 €\
5
Απρ
y,
3
(1-0)
8.
568
HYDROSTATICS.
The difference of the attraction of the Moon on the par-
ticle and centre of the Earth gives
Mx
2013
و
My
M (≈ - c')
M
'3"
13
Jo
C
Adding the respective forces together and substituting in
the equation
we have
Xd » + Ydy + Zdx=0 (Art 519),
3
4 π (p −ρ') a³ x dx + ydy + zdz
3
203
Απρ
2 €
+
1 +
( d + ydy) +
Απρ'
3
5
3
ATP (1-1) xdz
5
M
Mdz
+
2013
c') dx} +
= 0.
C2
x dx + ydy + (≈ − c') de
But y² = x² + y² +² and p¹² = x² + y² + (≈ − c')².
Hence substituting and integrating,
4π
_ += (p − p) d'² 1 + 2 = p² (1 + 2/2²) (a² + y')
3
a'
13
Σπρ 26
3
5
πρ
4€
M Mz
+
1
~2
+
= constant.
3
5
c/2
Now if b be the semi-axis minor of the spheroid, then
¿³ (1 + €) = a³, and ba (1 − e), and the equation to the
spheroid is
x² + y² + (1 − 2 €) ≈² = b² = a² (1 − † €) ;
1
1
=
=
(x² + y² + ≈²) − ³
-
1
2 €
+2€
1
J
a
3
a²
a
(1+
€
3
a²
28
77 = (x² + y² + x² − 2xc′ + c'²)−4
x² y² x² -−1}
y² -
20'2
12
= (1 - 2/3 + 2 + 3 + ~ ) = (1 + 2 - ² + 1 - 2 ~ ^).
1
/
C
C
12
C
7
C
FORM OF THE OCEAN.
569
Substituting these in the above equation to the surface
of the ocean, we have
5
(x² + y²)
πρ
3
(1 + 2 €)
+
ૐ
+ *² {2mp' (1 - 4) + +
πρό
3
5
4 π −
+
M
2013
13
Ꮇ
(p − p) a "e_M}
3 a³
= constant.
13
Then in order that this may coincide with the equation
x² + y² + (1 − 2 €) = constant, we must have
x²
(1-26) (2 mg (1 + 2) + 1M)
Σπρ
3
2€)
—
4€
5
πρ
3
+
5
4 π (p − p') a'³ M
3 a
€
13
с
'. €
8 πρ
15
+
3 M
2 c'3
20
4 π (p − p') a
3
3 a³
13
+
M
'3
с
11
8 пр
+
15
3 M
2c'3
4 π (p − p') α
3a²
(neglecting &² as before).
Now E = mass of the Earth
4πρα
pa³
3
πρ'
+
15
4.
Απρ
+ (a³ — a´³)
3
a's
4π (p-p') d'"_
3a²
−
€
4π(ρ - ρ)α" Απρίας
3
4π −
2 3E+ 4x (p = p') a
5a³
13
3
5a³
2 E
ΩΠ
1 +
5a³
€ =
1 +
15 M
4 E
Σπα
13
α
E (p-p')
−
13
2 = (p = p²) a" };
E
4 C
570
HYDROSTATICS.
α
M
1
E
74
= the parallax of the Moon =
4π
3
1
nearly,
60
nearly, E ρα nearly;
13
2πα
E
(p − p) = 21 (1
(1.
nearly, since a = a' nearly,
p = 5p' nearly.
These values shew that e will be very small: and that
the spheroidal figure may be taken as the form of equilibrium.
We must remark that the spheroidal figure of the nucleus has
been neglected in this calculation; also the centrifugal force
of the particles arising not only from the rotation of the Earth
about the centre of gravity of the Earth and Moon; but also
that arising from the rotation about its own axis, which is far
more important.
558. It is deserving of observation, however, that the
greater the mean depth of the ocean is (i. e. the less a' is) the
greater is e This shews that the depth of the sea affects the
Tides. Also e is greater the greater the density of the sea is
in comparison with the mean density of the Earth.
=
P
the value of e is more than double what it is when
If p
If p' be greater than
p = p (the value in nature). If p
ρ
وم
but
a little less than we find e =
5 3p
1, when the above numerical
values are substituted. In this case our solution is not even
an approximation: but it shews that if the density of the sea
were somewhat greater than the mean density of the Earth
the figure of the Ocean would be very prolate and consequently
the Tides much greater than they are in nature.
PROP. To determine the form of equilibrium of the
atmosphere of the heavenly bodies.
559. Let be the co-latitude of any particle of the
atmosphere, r its distance from the centre of gravity of the
spheroid. Then if w be the angular velocity the centrifugal
force of the particle is w'r sin : and therefore
FORM OF THE ATMOSPHERE.
571
dp
P
dp
р
= dV + w² (xdx + ydy),
=
√ +
p being a function of p.
w2
2 sin² 0,
2
Then, if we neglect the ellipticity of the spheroid,
const.
w2
E
+
2°
2
r² sin² 0, E = mass of the Earth.
Let R, R' be the polar and equatorial radii of the atmo-
sphere: hence
E
E w?
const. =
const.
+ R'2;
R
R' 2
E E
w² r² sin² 0 ;
2
R
7°
R' - R
and
R'3
2 E
R
The greatest possible value of R' is that which extends
to the point where the centrifugal force equals gravity: that
is, where w² R'
=
E
R'
>
or w² R'3
=
E, .. R' = 3 R.
This is the greatest possible ratio of R' to R: for let
w² R'³ = (1 − ×) E, in which
cannot be negative, then
R'
3 Z
R
2
It appears also that the equatorial is the greatest radius
of the atmosphere: for by differentiating the equation between
r and 0
dr
w² sin cos 0
2
d Ꮎ E – w² 73 sin²0'
but the centrifugal force resolved in the direction of the radius
is w²r sin² 0, and this must not be greater than
E
and there-
༡? ?
,
572
HYDROSTATICS.
dr
fore is always positive, or increases from the pole to
d Ꮎ
the equator.
560. There is but one form of equilibrium.
For the
equation to the surface of the atmosphere may be written thus:
203
2 E
Rw² sin² 0
2 E
:)' +
0,
w² sin² 0
and, since the last term is positive, one value of r must be
negative, and therefore does not serve our purpose:
: let ri
and r be the other roots, then since the second term does
not appear the third root is − (~1 + r½).
E
cannot both be less than
w² sin2 0
Now r₁ and r
for then the third
3
E
root would (disregarding its sign) be less than 2 w² sin20
and their product would be less than
2 E
w² sin² 0
: but this is
the product of the roots. Hence only one positive value of
r is less than
equilibrium.
3
E
2
w² sin² 0
or there is only one form of
Laplace draws the following conclusions with respect to
the Zodiacal Light from these results (Liv. 111. §. 47). The
Sun's atmosphere can extend no farther than to the orbit of
a planet, of which the periodic revolution is performed in the
same time as the Sun's rotatory motion about its axis; or in
twenty-five days and a half. Therefore it does not extend
so far as the orbits of Mercury and Venus: and we know
that the Zodiacal Light extends much beyond them. The
ratio of the polar to the equatorial diameter of the solar
atmosphere cannot be less than and the Zodiacal Light
2
appears under the form of a very flat lens, the apex of which
is in the plane of the solar equator. Therefore the fluid
which reflects to us the Zodiacal Light is not the atmosphere
of the Sun and since it surrounds the Sun, it must revolve
about it according to the same laws as the planets: perhaps
this is the reason why its resistance to their motions is so
insensible.
HYDRODYNAMICS.
CHAPTER I.
EQUATIONS OF MOTION.
561. THE equations of the equilibrium of fluids which we
have found in Art. 519, are deduced from the characteristic
property of fluids, both incompressible and elastic, viz. the
equable transmission in every direction of pressures applied at
their surface. This property arises from the fact, that the
molecules of the fluid when compressed or dilated rapidly
assume the same relative positions that they previously had.
The time that the particles occupy in passing into this state
has no influence on the laws of equilibrium, since these are
observed only after the fluid has attained its state of rest.
But this time, small as it may be, must influence the laws of
motion of fluids, so that the principle of the equality of pres-
sure in every direction is true in Hydrostatics, but is not
always applicable in Hydrodynamics; this is Poisson's view
of the subject, Traité de Mécanique.
Laplace remarked an analogous difference in the state of
rest and motion of fluids relative to Mariotte's law. This law,
which teaches that the density of an elastic fluid varies as the
pressure, requires that the temperature of the fluid should
become the same after the change in volume that it was before.
It is ascertained that heat is given out or absorbed when a vo-
lume of air is suddenly compressed or dilated; and in this
way the elasticity of the air is greatly modified by the nature
of the motion. This circumstance introduces into the equa-
574
HYDRODYNAMICS.
tions of motion terms which cannot be deduced from the
equations of equilibrium. In the present work we shall
suppose, as is ordinarily made the supposition, that the equality
of pressure holds equally in the state of rest and motion of
fluids: when we adopt this hypothesis the equations of equi-
librium conduct immediately to those of the motion of fluids.
PROP. To determine the equations of motion of a mass
of fluid, the molecules of which are acted on by given forces.
562. Let xyz be the co-ordinates to any molecule at the
time t, and X, Y, Z the sums of the resolved parts of the acce-
lerating forces which act upon this molecule parallel to the
axes of co-ordinates respectively. Now in accordance with the
Principle enunciated in Art. 226, the acceleration of the mole-
cules would cease if that at the point (xyz) were acted on by
the accelerating forces
X
ď² x
dt2
dy
Y
Ꮓ
―
dt'
,
,
d t2
and all the other molecules acted on simultaneously by similar
forces.
Hence if p be the pressure at the point (xyz) at the time
t referred to a unit of surface and p be the density, we have by
the equation of equilibrium of a fluid mass
Sp
= (x - de) 8x + (y - dp)
Y
бу +
P
(z
d² z
Ꮓ
Sz
dt²
where the differentials dx, dy, d≈ do not refer to the motion,
but are arbitrary; and may therefore be taken equal to the
differentials of the spaces described by the molecule parallel to
the axes.
Hence
1 dp
dx
P
dt
ď²x 1 dp
p dy
dy
1 dp
d²
z
Y
=
2
dt
dz
dt
P
Now let u, v, w be the velocities of the molecule (xyz)
parallel to ~, y, respectively at the time t. Then each of
these will be a function of the time and the position of the
molecule;
EQUATIONS OF MOTION.
575
1
ď² x
du
du
du
du
du
+
u +
V
v +
W,
dt2
dt
dt
dx
dy
dz
ď y
d v
d v
d v
+
W +
d v
d v
v +
W,
dt2
dt
dt
dx
dy
dz
d² z
dw
dt2
dt
(dw)
dw
dw
d w
+
u +
v +
w.
dt
dx
dy
dz
Hence the three equations involving p become
1 dp
X-
P
dx
(du)
du
du
du
V
w,
dx
dy
dz
1 dp
d v
d v
d v
Y
2
V
W,
p dy
d.x
dy
dz
1 dp
Z -
ρ
dz
- (dwo)
dw
dw
dw
น
27
พ.
da dy dz
These are three equations connecting the five unknown quan-
tities u, v, w, p, p which we wish to determine in terms of
x, y, z, and t.
563. Two more equations will be furnished by the
following consideration. Suppose we consider the motion of
the molecules which at the instant t form an indefinitely small
parallelopiped with its sides parallel to the co-ordinate planes.
The various molecules will change their situation, and we can
determine the volume of the figure which they form after a
short time but since the number of molecules remains the
same the volume which these molecules occupy when multi-
plied by the density of the fluid must remain the same during
the motion.
Let dr, dy, d≈ be the sides of the parallelopiped at the
time t: m the molecule nearest the origin; n the molecule
immediately over this. Then at time +&t the co-ordinates
of m are
x+ust, y + vôt, ≈ + W wdt,
and the co-ordinates of n change from x, y, ≈ + d≈ to
576
HYDRODYNAMICS.
x+u'st, y + v′dt,
x + dx + w'&t,
u', v', w' being the values of u, v, w at n;
du
dz
•. u' = u + Sz, v′ = v +
d v
d z
dw
Sz, w' = w + Sz.
dz
Hence the co-ordinates of n at the time t + St are
du
d v
x+ubt+ бъбва
Szst, y + vdt +
ozst, and
dz
dz
dw
z + d z + w & t +
Sz & t;
dz
and the distance between m and n at that time is

du²
2
dx² St²+
dx2
d v2
dx2
dw
dw
dx² Sť² + (8x +
Sx St)² = Sx +
SzSt,
dz
dz
neglecting small quantities of the third order.
Now let us consider the two angular points of the parallelo-
piped which lie in the same diagonal plane with m and n: we
shall call them m' and n': we shall obtain their distance at the
time t + dt from that of m and n by putting a + dx,
for x and y then the distance between m' and n' is
y + Sy
dw
8x+
+
ď² w
[ d w
Sx +
dz d z d x
ď² w
d x d y
s y sz st
·S t = 8x +
Sz St.
dz
Hence the distance between m' and n' is the same as that
between m and n, when we neglect small quantities of the third
order. In the same manner it may be shewn that all the edges
of the parallelopiped which are parallel at time t are equal to
each other at time t + St and therefore still form a parallelo-
piped, the sides being
dw
d v
du
8% +
SxSt, Sy+
бъбна
Syst,
8x +
Sx St.
dz
dy
dx
Also the density at the time t + &t is
dp
St,
{de + deu + dev + dp w} &t.
dp dp
p+
dt
dx
dy dz
EQUATIONS OF MOTION.
577
and therefore the mass of the parallelopiped, which at time t
is pdadysz, becomes at time t + St,
(p +
dp st+
dpust +
d p v dt +
de wdt)
dt
dx
dy
dz
d v
du
dx
× (1+ St) (1 +
dy St) (1 +
du
St) Sx Sysz.
dz
Equating these expressions, dividing by Sady dxst, and
then taking the limit, we have
[du dv
d w
0 =
+
+
+
+
dp do u +
dp
dp
v +
dy
р
dx dy
dz dt dx
dz
W.
This equation is called the Equation of the continuity of the
fluid; since it expresses analytically the relation between the
velocity of the molecules and the density of the fluid, which
are necessarily dependent on each other, if the fluid be sup-
posed to be continuous in its constitution.
564. If the fluid be incompressible then the variation of
p equals zero, and the above equation gives two
du
d v
d w
+
+
0,
dx
dy
dz
dp
and +
dp
dp
u +
dt dx
dy
dp
v t w = 0;
dz
these complete the five equations for computing u, v, w, p,
in terms of x, y, z, t when the fluid is incompressible.
p
When the fluid is homogeneous and incompressible then
is constant throughout the fluid and given in value and
therefore the last equation becomes identical.
P
565. If the fluid be compressible the fourth and fifth
equations are
Jdu
d v
0 =
P
+
+
dx
dy
dw dp dp
+
+
dz dt dx
dp
dp
2+
dy
W
w;
dz
and 0 = F(p, p),
the function F depending upon the nature of the fluid.
+ D
578
HYDRODYNAMICS.
The equations admit of great simplification in the case of
an incompressible homogeneous fluid mass, when udx+vdy
+wds is a perfect differential.
PROP.
To find the pressure at any point of a homoge-
neous and incompressible fluid mass in motion.
dz
566. Assuming udx + vdy + w dx = dp a perfect differ-
ential, we have
аф
v =
аф
dy
W =
dz
аф
dx
Hence the equation of the continuity of the fluid becomes
ď² O
dx dy²
ď²
ΦΦ
0.
+
d&e
By the integration of this equation
is to be found. We shall
proceed to eliminate u, v, w from the equations involving p.
ď² & dy +
ď & d≈
dt dz
du
d v
dw
Φ
dx +
dy +
dz =
dx +
dt
dt
dt
dt dx
dt dy
аф
аф
аф
d-
d
d
dt
dt
dt
аф
dx +
dy +
dx = d
dx
dy
dz
dt
du
d v
dw
аф
dx +
dy+
d z =
d
in the same way
dx
dx
dx
dx
du
d
v
dw
do
dx +
dy +
dz
= d
dy
dy
dy
dy
du
d v
dw
dx +
dy +
dz
dz
dz
dz
Now let the three equations in p be multiplied respectively
by dx, dy, dz and added together;
dz
d dp
dp
= Xd + Ydy+Zdã
P
EQUATIONS OF MOTION.
579
_ααφ_ αφ, αφ
аф аф аф
d-
d
dt dx dx
аф, аф аф, аф
d
d
dz
αφ, αφ
dy dy
dz
= Xd + Ydy + Z đã - d
аф [dø²
1/2 d
2
dp² dp²
+
dt
dx² dy² + dx²]
567. COR. When the excursions of the molecules are
small we may neglect the squares of the velocities and the
equation becomes
dp
P
аф
= Xdæ + Ydy + Zdượp
Id+Ydy+Z
dt
PROP.. To prove that if udx + vdy+wdz be a perfect
differential at any instant it is so during the whole time of
the motion..
568. For at the time t+&t the value of udx+vdy+wdz
v
dw
{dda + do dy + 10 dx} &
becomes
udx+vdy +wdz +
Jdu
dx
dt
= dp + d⋅
аф
dt
dt
dt
St = dp + (Xdx + Ydy + Zdz) St
Stjdp²
[dø² dø² døp²
2
+
d.x²
dy²
d2
dx²);
t
dp St
P
wherefore if udx + vdy+wdz be a perfect differential do
at the time t it will be so also at the time t + St and will
consequently be so throughout the motion.
PROP. To determine equations for calculating the motion
of an elastic fluid, the excursions of the molecules being
supposed small, no extraneous forces acting.
569. We shall suppose uda + vdy + w dz = dp, then
the equation of continuity is
P
Jdu
|dx
d v
dw
dp
+
+
+
0,
dy
13
dt
580
HYDRODYNAMICS.
neglecting small quantities of the second order;
dloge p
dΦ
d² Q
d² p
+
+
+
= 0.
dt
dx²
dy² dz
We shall suppose that p = a²p, which is the law of nature
if the motion be so slow as not to absorb or develop heat.
Also since the excursions are small we have, as in Art. 567,
dp
P
аф
d
;
dt
...
a²
d loge f
dt
d2p
dt
аф
a²
[d² &
Φ
d φ
dφ
d* φι
+
+
dt
dx²
From this equation
dy² + dxf
must be determined and then p
and P calculated by the equations given above.
570. For the Theories of Sound and Light which depend
upon the principles in this Chapter we must refer the reader
to two Articles on those subjects in the Encyclopædia Metro-
politana, by Sir John W. F. Herschel, to Mr Airy's Tracts,
and also to Mr Webster's Theory of Fluids. We shall
proceed with applications of the principles more peculiarly
adapted to the nature of the present work, viz. to the Tides
and the Stability of the Ocean.
CHAPTER II.
1
TIDES AND STABILITY OF THE OCEAN.
571. THERE remains yet another phenomenon, which
is evidently connected with the mechanism of the Solar
System, the Tides of our Ocean. In the calmest weather
the vast body of waters that wash our coasts advances on
our shores, inundating all the flat sands, rising to a con-
siderable height, and then as gradually retiring to their
former level; and all this without any visible cause to impel
the waters to our shores or again to draw them off. Twice
every day is this repeated. In many places this motion of
the waters is tremendous, the sea advancing, even in the
calmest weather, with a high surge, rolling along the flats
with resistless violence, and rising to the height of many
fathoms.
572. In searching for the cause of this remarkable phe-
nomenon philosophers readily conceived, that, since the Sun
and Moon each cross the meridian twice in twenty-four
hours, these bodies may by their attraction influence the
waters of the ocean. Accordingly various theories have been
adopted for the calculation of the tides upon this hypothesis
of lunar and solar attraction, of which the most noted have
been those of D. Bernoulli and Laplace. If the hypothesis of
universal gravitation be adopted, there can be but one correct
theory based upon it for calculating the oscillations of the
ocean; but in consequence of the difficulties of the analysis,
which have hitherto been insurmountable, other hypotheses
must be resorted to in addition to that of gravitation in
order to obtain an approximate solution of the problem.
582
HYDRODYNAMICS.
The irregularity of the depth of the ocean, the manner in
which it is spread over the earth, the position and declivity
of the shores, and their connexions with the adjoining coasts,
the currents, and the resistances which the waters suffer,
cannot possibly be subjected to an accurate calculation,
though these causes modify the oscillations of the great fluid
mass. All we can do is to analyze the general phenomena
which must result from the attractions of the Sun and Moon,
and to deduce from the observations such data as are in-
dispensable for completing in each port the theory of the
ebb and flow of the tides. These data are the arbitrary
quantities, depending on the extent of the surface of the sea,
its depth, and the local circumstances of the port. In the
absence of these data we must resort to the best expedients
that can be found.
Bernoulli, in his theory, assumed that the attraction of the
Moon causes the ocean to assume at every instant the form it
would have if the Earth and Moon were stationary. It is
found, by calculating the tides upon this hypothesis, supposing
the pole of the prolate spheroid (which is the form of equi-
librium nearly, see Art. 557.) to lag behind the Moon, that
this hypothesis gives results according very well with obser-
vations in some of the more ordinary phenomena of the
tides. This theory is termed by Mr Whewell the Equili-
brium Theory. Laplace, however, has taken a different
course. He calculates the attractive forces of the Sun and
Moon upon the ocean, and finds them to contain constant
terms and periodical terms. He states, that in consequence
of the resistance and friction of the waters they would soon
have assumed a form of equilibrium under the forces which
are represented by the constant terms: and then, assuming
this as
a general dynamical principle, that the state of a
system of bodies in which the primitive conditions of the
motion have disappeared by the resistances it suffers is pe-
riodical when the forces themselves are periodical, he obtains
an expression for the height of the tide the same as that
obtained from the Equilibrium Theory of Bernoulli. But
there are so many assumptions in this, that we may, as far
as we know a priori, as readily adopt the equilibrium theory
TIDES.
583
as Laplace's we must test their accuracy by comparing their
results with observations. With this laborious task many
calculators and observers are at this time employed under
the superintendence of Mr Whewell and Mr Lubbock: and
we must look to the general empiric laws to be deduced
from the enormous mass of observations that is in the pro-
gress of accumulation to guide us in adopting such proper
hypotheses as shall bring the subject under the dominion of
analysis without materially vitiating the rigour of the approxi-
mation. A most interesting Essay towards a first approxi-
mation to a map of cotidal lines by Mr Whewell will be
found in the Philosophical Transactions for 1833. In the
present state of our knowledge of the tides we are constrained
to confess that the laws we possess are only empiric. All
we shall attempt in this work will be to obtain the formulæ
for the calculation of the tides upon the equilibrium theory
of Bernoulli.
PROP. To calculate the height of the tide at any place
at a given time upon Bernoulli's hypothesis.
*
573. Let c, c (1+e) be the semi-axes of the prolate
spheroid into the form of which the Moon attracts the Ocean
a the radius of the sphere the volume of which equals that of
the Earth h the elevation of the pole of the spheroid above
the mean level of the sea: r the distance from the centre of
any point of the surface of the ocean, and the angle r makes
with the axis of the spheroid: n the angular velocity of rota-
tion of the Earth about its axis: w, w, the right ascensions.
of the point on the surface and of the Moon: 0, 0, their
north polar distances. Hence we have
c³ (1 + €) = a³, h=c(1+e) a;
−
.. c = a (1 − e), h = ÷ α €; ac = &h, c = ah.
-ㅎ​),
{
'.
Also r = c(1 + e cos³ Ø)
= a
a - 1
1 h + 3 h cos² ( ;
584
HYDRODYNAMICS.
therefore height of tide at the place of which the terrestrial
co-latitude and longitude are ✪ and w
λ
= r α 1 h (3 cos² ( − 1).
Let the difference between the right ascensions of the
Moon and the pole of the tidal spheroid (see Art. 572), then
nt+w-w-λ is the hour angle of the pole, t being reckoned
from noon;
−
.. cos & = cos 0 cos 0, + sin 0 sin 0, cos (nt + w - w, − λ) ;
therefore height of tide =
e
3h {[cos cos 0, + sin 0 sin e, cos (nt + w - w, - λ)]² - } } .
Then considering 0,, the north polar distance, nearly 90º,
we shall neglect cos 0,. Also we shall put
1 + cos 2 (nt + ww, λ) for 2 cos (nt +w-w, -λ);
therefore height of tide =
2
→
3h {sin 0 sin²0,- + sin³0 sin²0, cos 2 (nt + ww, -λ)},
and since during a day 0, remains nearly constant, we have
change in height = h sin² 0 sine, cos 2 (nt + ww, -λ).
Supposing accented letters to apply to the Sun in the same
way that the unaccented letters apply to the Moon, we have
the whole variation in the height of the tide arising from
the combined action of these two luminaries
3h sin20 sin²0, cos 2 (nt + ww, -λ)
+ 3 h′ sin² 0 sin² 0, cos 2 (nt + w − w,' — X').
4
PROP. To find the time of High Tide at a given place.
574.
The height of the tide at a given place (by the
last Article)
-
= 3 sin² 0 {h sin² 0, cos 2 (nt + w - w, − λ)
+ h' sin² 0, cos 2 (nt + w − w, − X')},
−
TIDES.
585
and when the tide is full the differential coefficient of this
vanishes. We shall suppose the angular velocities of the Sun
d (w, +\)__d (w' + λ')
and Moon to be the same or
... 0 = h sin² 0, sin 2 (nt + w
dt
w, - λ)
dt
+ h' sinº 0'sin 2 (nt + w — w,' -λ'),
or 0=h sin² 0, sin 2 (nt + w −
w₁ - x)
—
+h' sin² 0, sin 2 (nt + w - w, - λ + w, — w,' + x − x');
.. tan2(nt+w-w,− λ) =
=
h' sinº 0' sin 2 (w, — w, + λ' − λ)
2
h sin 0,+h'sin' 'cos 2 (w,' — w,+λ' - λ) '
It will be seen by referring to Art. 295. that the force of
S
the Sun on the Ocean varies as and that of the Moon as
3
M
2+3
h' S 7.3
SII³
hence
h
M 3
Mr
M II'
13
•
II, II' being the parallax of the Moon and Sun;
S II \ 3
sin³0'sin 2 (w,' -w,+λ'−λ)
Mn
... tan2 (nt+w-w-λ)
SII\3
sin20,+
2
sin²0,cos 2 (w/-w,+λ'—λ)
Μ MII
This expression shews that the time of the Moon's meri-
dian passage precedes the High Tide by an interval which is
not the same for all ages of the Moon.
The mean of all these intervals is λ and nt + w
ω, - λ
is the excess of any interval above the mean: and X' A is the
time of the Moon's meridian passage when A (the mean) is the
interval of time between that event and high tide.
The value of the interval, at any port, when the Moon is
full or new is called the Establishment of the Port.
-
Let 0, 0, 90° or the Sun and Moon be supposed in the
equator. Then the above formula leads to the following table
4 E
586
HYDRODYNAMICS.
See Com-
as the result of calculation compared with good observations of
the time of high water made at London Docks.
panion to the British Almanac, 1830.

Time of Moon's
Meridian Passage
Time by which the Moon's
Meridian Passage precedes
the time of High Water.
w, - w,.
Observed.
Calculated.
h.
202
m.
h.
2
m.
0
10
1
1 47
1 47
10
2
1 32
1 32
Co
3
1 18
1
17
4.
1 5
1 4
5
0 55
0 55
6
0 52
0
54
7
1 43
1 6
8
1 32
1 32
9
1 59
1 58
10
2 9
2
10
11
N
2
10
2 9
The mean of the observed results gives λ = 1 hour, 32
minutes;
-
and .. X'λ = 2 hours, by the Table.
PROP. To calculate the tide at a Port at which the
tidal wave arrives by two distinct routs.
575. We shall consider the action of the Moon only.
Let n
-
dw,
dt
αλ
dt
=m: T the time of transmission
up
the
TIDES.
587
11
first channel, then at the time t the tide at the port produced
by the tidal wave up the first channel
DM
2.3
cos 2 (nt + w - w, — X — m T)
-
D depending upon the height of this wave when it was at the
mouth of the channel. T'the time the tidal wave takes to
move to the second mouth and up the second channel: then
the tide at the time t at the port arising from this second wave
EM
203
-
cos 2 (nt + w - w, − λ — m T′)
E depending upon the height of the tidal wave when it reached
the mouth of the second channel.
M
༡•3
Hence the height at the port
D cos 2 (nt+w-w,−λ −m T) + E cos 2 (nt+w-w,−λ-mT′)
MF
703
cos 2 (nt+w-w, − λ — G),
where F² = D² + E² + 2 DE cos 2 m (T′ − T),
and sin 2 G
D
F
E
sin 2 m T +
sin 2 m T'.
F
Hence F and G depend upon m and therefore on the
rapidity of the Moon.
If F = 0, that is, if T = T' and D - E or if it be high
water at one mouth when it is low water at the other, and
if the tides require the same time to reach the port after
the great tidal wave has reached the first mouth, then there
will be a complete interference at the port, or no tide at all, if
we consider the height of the two poles of the spheroid above
the mean level of the ocean to be the same. Now this is not
strictly the case, the height of the pole furthest from the Moon
is less (as might be shewn by a nearer approximation in
Art. 557) than the other. Hence there will only be one ebb
and one flow in twenty-four hours, and that very small. This
*
588
HYDRODYNAMICS.
singular fact has been observed at Batsham, a port of the
kingdom of Tonquin, 20° 50′ north latitude. The two waves
seem to come by two channels which run, one from the China
seas between the continent and the island Luconia, the other
from the Indian sea between the continent and the island of
Borneo. (Principia, Tom. 111. Prop. 24).
PROP. To determine equations for calculating the mo-
tion of an incompressible fluid mass surrounding a body
nearly spherical, the body having a uniform rotatory motion
about a fixed axis, and the fluid being supposed to be de-
ranged but very little from the state of equilibrium by very
small forces.
Z
576. We shall refer the fluid mass to polar co-ordinates.
Let the axis of ≈ be the axis of rotation, n the angular velocity:
γ the distance of any molecule at the commencement of t from
the centre of gravity of the body which the fluid covers; this
centre of gravity we shall suppose at rest (see Art. 428); 0, the
angle between, and ≈, w, the angle between the planes rx and
xx at the commencement of t. Suppose that at the end of the
در
rz
time tr 0,, w, r,+
are become + ar, 0, + að, nt + w, + aw,
a being a very small fraction of which the square and higher
powers may be neglected;
.. ∞ = (r, + ar) sin (0, + a0) cos (nt + w, + aw),
y = (r,+ ar) sin (0, + a0) sin (nt + w + aw),
≈ = (1, + ar) cos (0, + a0.)
We shall substitute the expressions in the equation of
Art. 562, neglecting a,......
2
dx
α
dt
{sin 0, cos (nt + w,)
d r
d Ꮎ
+ r, cos 0, cos (nt + w)
dt
dt
d w l
ďx
dt
— r, sin 0, sin (nt + w)
-afsino,
- nr, sine, sin (nt + w),
dt
d Ꮎ
d²r
= asin 0, cos (nt + w,)
+ r, cos 0, cos (nt + w)
dt2
d t²
TRANSFORMATION OF EQUATIONS.
589
dr
I w
r, sin 0, sin (nt + w)
.
- n sin 0, sin (nt + w)
d t²
dt
dw
d Ꮎ
- nr, cos 0, sin (nt + w)
- nr,
sin 0, cos (nt + w,)
dt
dt
– n²r, sin 0, cos (nt + w,).
ď² y
From this we can obtain
by putting w,+ nt + aw
1
2019
π
for
dt2
ď z
w, + nt + aw,
and
by putting w, + nt + aw = 0
and
dt2
π
0, + a0 +
ď² y
.*.
dt²
2
for 0, + a0;
{sin
+r, sin 0, cos (nt + w)
+nr¸cos 0¸cos (nt + w,)
— n²r, sin 0, sin (nt + w),
ď² r
d Ꮎ
=asin 0, sin (nt + w)
+r, cose, sin (nt + w)
dt2
df
d2
ď² w
dr
+ n sin 0, cos (nt +w,)
dť
dt
dᎾ
dw]
nr, sinθ sin (nt +w,)
dt
dt
d2
d² z
α
dt2
{cos e
d²r
d20
dw
Ꮎ . -r,
›, sin 0,
nr, cos
dt2
10. at
n²r, cos 0.
dt2
Hence making the substitutions and putting Xda + Ydy + Zdz
- SV for the attractions, we have
Sp d²x
S V
Sa+
dť
d'y
бу
dť
ર
·Sy +
df
P
= ar 280
ar³80,
+ ar²dw, {sin² 0,
I d²
d Ꮎ
dw
J
2n sin e, cos 0,
dt
dt
dt
I w
+ 2n sin 0, cos 0,
do
2n
dr
+
sin² 0
dt
༡་
dt
dw
n²
+ aor far
2nr, sin² 0,
♪ {(r, + ar) sin (0, + a0)}².
dt2
dt
g
590
HYDRODYNAMICS.
At the external surface of the fluid we have dp = 0: more-
over, in the state of equilibrium we have
n²
2
8. {(r, + ar) sin (0, + a0) }² + (§ V),
since r, 0, w are constant when there is equilibrium: (8V) is
the value of SV corresponding to this state, it is therefore
the force of gravity multiplied by the element of its direc-
tion. Let g be the force of gravity, ay the small elevation
of a particle of the fluid above the surface of equilibrium,
which we shall consider as the true level of the sea. The
variation (V) will increase by this elevation in the state of
motion, by the quantity - agdy, since gravity acts nearly in
the direction of y towards the origin of that line: then
SV = (SV) – agdy + ad V′,
where ad V' is a variation depending upon the new forces which
in the state of motion act upon the particle.
No
2
Likewise 8. { (r, + ar) sin (0, + a0) }² will be increased
2
by the quantity an² dyr, sin20, by means of the elevation of
the particle above the level of the sea: this quantity may,
however, be neglected in comparison with - agdy, because
the ratio of centrifugal force to gravity at the equator,
n² r
g
which equals. is a very small fraction, being nearly
289
Lastly, the radius, is very nearly constant at the surface
of the sea, since it differs but slightly from a spherical
surface; we may therefore neglect dr. Then the equation
becomes, at the surface of the sea,
2
ጥ
80
J d² 0
d Ꮎ
+r Sw, sin² 0,
ď w
df
-
dw
2n sine, cose, at
do
2. ԴՆ
dr
+ sin Ꮎ
J'
dt
+ în sine, cose, at
= gdy + dV',
the variations dy and SV being in reference to do, and dw,
TRANSFORMATION OF EQUATIONS.
591
577. We shall now consider the equation relative to
the continuity of the fluid.
Volume of the element at the commencement of t,
δω
= dr, r,de, r, sine, dw, or r2 sine, dr, de, dw,.
And the volume of the same element after a time t,
(r,+ ar)² sin (0, + a0) d (r, + ar) d (0, + a0) d (w,+aw+nt).
б
But since the density of the sea is supposed to remain the
same, these volumes must be the same,
·´`· (~, + ar)² sin (0, + a0) d (r, + ar) d (0, + a0) d (w,+aw+nt)
= r² sine, dr, de, Sw,.
We have then, by
equating, expanding, and neglecting the
squares and higher powers of a,
0 = r² sin
[
dr
d Ꮎ
+
dr
do,
dwi
d w
+ + r² 0 cos 0, +2r, r sin 0,,
dw
dw
d.r²r
or 0
dr,
fe cose,
sinė,
d Ꮎ
+
+
dw,
Ꮎ
des
Let
Y
and
w,.
be the mean depth of the sea corresponding to 0,
Now since the oscillations are small we may assume
that all the particles which are on any one radius, will
remain on the same radius when 0,, w, change to 0, +að
and w, + aw+nt: i. e. the relative change of situation of the
particles will be chiefly in the direction of the radius vector.
Hence the integral of the above equation is
O cose, dw
d Ꮎ
0 = r² r − (r ² r') + r² y
+
+
sin 0,
doj
Ꮎ
2
r r
dw,
where () is the value of 12 at the bottom of the sea:
and equals r²(") nearly; since the change in the radius of
the Earth between the bottom and surface of the sea is so
small. The mean depth even of the Pacific Ocean is only
about of the radius of the Earth.
1 th
1000
Wherefore,
0 = r
(~) + Y
Je cose,
dw
d Ꮎ ]
do
+
+
sinə,
da, des
Ꮎ
Now the depth of the sea corresponding to 0, + a✪ and
nt + w + aw = y + a {r − (r)}.
592
HYDRODYNAMICS.
Also the depth = y +
dy
dy
α a0 +
aw + ay,
ão,
dw,
where ay is the elevation of the particle above the mean
surface.
x − (1)
=
dro+
dy
Ө+
w + Y ;
ᏧᎾ .
dw,
d.yo
d. Yo
Yo cose
y=
də,
dw,
sin 0,
d.yesine, 1
d Ꮎ .
d. yw.
γω
sine,
dw
Let cose,
=
d. 701-u³,
d. yw
Y
du,
dw,
By means of this and equation of last Article, we have to
determine the oscillations of the ocean.
PROP. The depth of the sea being supposed uniform,
the Earth to have no rotatory motion, and its figure to be
a sphere; required to prove the stability of the Ocean.
578. By the last Article we have, since y is constant,
d w
Y = Y
d. O√1 - µ,
αμ
M
-Y
dw
d.
Jd²
ď w
2
d
d t
dⓇ y
γ
γ
dw,
d t
dt
άμ
But if we put cose, u, in the equation of Art. 576. and
=
d V
d V'
dy
dy
observe that dy
80, +
Sw, and 8 V' =
80, +
de
dw,
Sw,"
dᎾ
dw,
and equate the
Ꮷ Ꮎ
dy
dv' dy
coefficients of 80, and also of ow, in that
equation, we obtain (putting n = 0,)
2
g
+
g
•
dt2
ᏧᎾ .
dᎾ .
du.
dv'
2
du,
d² w
dy
2
r² (1 μ²)
g
dt2
+
dw,
d V
dw,
STABILITY OF THE OCEAN.
593
ď² y
Let these be substituted in the value of
then
و
dť
d²y
gy
d
dy
d
γ
d vn
μ,2)
d to
r² dμ,
2
άμ
dµ,)
r² du,
du
d² y
d² V'
EY
dw²
y dw
2
+
(1).
2
2
a
—
2
1
We have already shewn (see Art. 179.) that y can be ex-
panded in a series of Laplace's Coefficients of the form
a { Y₂ + Y₁ + Y₂ + ...... + Y ¿ + ......}
1
2
i
The part of a V' relative to the spherical stratum of fluid
of which the general radius is r, or (a + ay), is by Art. 182
70
α
4 π pa² a {Y + Y₁ + } Y₂ + ... ... +
1
2
1
2 i + 1
Y₂+......}
Likewise the part of a V' which depends upon the action
of the Sun and Moon can be expanded in a series of Laplace's
Coefficients,
0
α aVo + a U₁ +
+ a U; +
1
i
This being premised, let us substitute the values of
r₁, y, V' in the equation (1): the comparison of the similar
functions U, and Y, will give
ין
i
¿Y, i (i + 1) #7 { (2i + 1) - * *pa)} r, − Zi (i + 1) U, .
Y;
+
d t
2 i + 1
gy
a²
= .
4 πρα
g
Let p' be the mean density of the Earth, then
cra
g
11
4 πρα
then putting ³
3
Απα
3
g
ررة
+
a²
i (i + 1) gy
2 i + 1
4 F
594
HYDRODYNAMICS,
and the integral of this is
the above equation becomes
d2 Y;
¿
2
de + X² Y₁ = Zi (i + 1) U,,
Y₁ = M¿ sin λ; t + N¿ cos λ¿ t
λε
a³
¿ (i + 1) Y sin λ¡t ſU¿¡ cosλ¿ tdt
i (i + 1) Y
i
i
cos λ¿t ſU¿ sin λ¿ tdt.
λ;
α
M; and N, being two of Laplace's Coefficients.
i
Y¡
When i = 0, the differential equation in Y; becomes
Y
a
d² Y
dt
0
0,
... Y₂ = M¸t + No ;
0
Yo + Y₁ + Y₂ +
1
2
= M₁t+N+ M₁ sin λ, t + N, cosλ, t
0
1
1
+ M½ sin λ2 t + N₂ cos λ½ t
+
2
2
-2
+ M¿ sin λ; t + N¿ cos λ¿
+
2
t
If the quantity M, be finite, then y will increase without
any limit, and the oscillations of the sea will not be stable.
But it is easy to shew that M. and N. vanish, because the
mass of the fluid is constant.
This condition gives f₁₂ydµdw = 0;
-2π
.. {
J, JT { Yo + Y + Y +
。 + Y₁+ Y₂+ ......} dµdw = 0,
0
1
2
=
ƒ ƒ₁₂
but by Art. 180. " Ydudw=0 when i is > 0;
1
2π
i
.. also, Y dudw = 0, or 4 π Y = 0,
.. Y₁ = 0,
0
0
.. Mot+ No = 0.
STABILITY OF THE OCEAN.
595
The stability of the sea depends then on the signs of
Mr M............
λι,
for if one of these be negative the value of y will have an
exponential term in its expansion, and all the terms will not
be periodic functions of t. The condition that this should not
3 p
happen is, that 2i+1- shall not be negative for any
positive integral value of i,
P
must not be greater than any value of
2 i + 1
3
ρ
must not be greater than 1,
P
or the density of the sea must not be greater than the mean
density of the Earth; otherwise the equilibrium of the waters
of the Ocean would not be stable.
579. Laplace extends this investigation to the case where
the rotatory motion of the Earth is taken into consideration and
the depth of the sea is not uniform and arrives at the same
result as before (Liv. IV. §. 3, also §. 13). He likewise shews
in §. 14, of the second Chapter that the converse Proposition
is true in many cases.
The part of the oscillations which depends on the primitive
state of the sea must have quickly disappeared by the resist-
ances of different kinds which the waters of the Ocean suffer
in their motions: so that if it were not for the action of
the Sun and Moon the sea would long since have subsided
into a permanent state of equilibrium. It seems pretty evident
that the same would be the case if the ellipticity of the
Earth and the rotatory motion were taken into account, the
only permanent effect of the rotatory motion being to modify
the action of the Sun and Moon and so to alter the period of
the oscillations but not their nature. For these reasons we
may consider the solution in Art. 578, as applicable to the case
of nature.
Now the experiments made by Maskelyne on the attraction
of the mountain Schehallien in Scotland and by Cavendish
596
HYDRODYNAMICS.
on the attraction of leaden balls shew, that the mean
density of the Earth is somewhere about five times that of
the sea.
We are hereby assured, then, that, provided no
geological convulsion change the form of the Earth's surface,
no inundating catastrophe can overwhelm us, so long as matter
obeys the laws which at present regulate its motion: a striking
illustration of the words "hitherto shalt thou come, but no
further." Job xxxviii. 11.
CHAPTER III.
THE MOTION OF BODIES IN A RESISTING MEDIUM.
580. It is found by experiment that when bodies move
in a fluid, whether incompressible or aeriform, they meet with
a resistance which tends continually to diminish their velocity.
In consequence of the great difficulty of making accurate ex-
periments on the resistance of media and also because of the
extreme complication of the analysis which prohibits our
making any extensive use of the facts which are brought to
light, the laws of the resistance of fluids have not yet been
very satisfactorily ascertained.
The general approximate law seems to be that the resist-
ance on a plane surface moving with its plane at right angles
to the line of motion is proportional to the extent of surface,
the density of the resisting medium, and the square of the
velocity taken conjointly. Some recent experiments upon the
motion of boats on canals seem to indicate, that beyond a cer-
tain degree of velocity this law is not even an approximation
to the truth, but the simple velocity better suits the experi-
ments than the square of the velocity.
In a work of the nature of the present we should not
have thought of entering upon this subject were it not in-
timately concerned with celestial phenomena. It has been
computed that Encke's comet has since its appearance in
1786 been moving round the Sun with an increasing mean
motion. Encke attributes this to the resistance of a medium
pervading space. We shall therefore proceed to calculate the
effect that such a medium must have upon the motion of the
planets, and then explain the process of calculating the per-
turbation produced by this cause in the motion of comets.
598
HYDRODYNAMICS.
PROP. Supposing that a resisting medium pervades the
planetary spaces, and that its resistance varies as the square
of the velocity, required to find the effect on the motion of
the planets, supposed to describe elliptic orbits nearly circular
about the Sun.
581. Let x and y be the co-ordinates to a planet and r its
distance measured from the Sun: S the mass of the Sun and
V
d s²
the resistance of the medium; V we shall consider con-
dť
stant, because the orbit is nearly circular and therefore the
planet nearly at the same distance from the Sun, and also we
shall suppose V very small. The equations of motion are
ď²x
Sx
ds² dx
ď³y
Sy
ds² dy
V
- V
d t2
203
dt2 ds
dt2
203
dt ds
dx
dy
multiply by 2
and 2
add, and integrate,
dt
dt
ds2
2S
C+
2 V
dt2
Sv
1°
Tv ds 3
df³
dt.
But in the instantaneous ellipse (Arts. 350, 352), the first
differential coefficients are the same as in the actual orbit,
because the contact of this ellipse with the orbit is of the first
order; hence
2S
S ds²
グ
​(L
d t²
C +
in ellipse (2 a = axis-major)
2S
1*
2 f
Sv
ds3
dt,
dt3
1 C 2
ds3
dt;
a
S
d t³
da
2 a² V ds3
2 a² VS
dt
S
dt
PLANETS IN A RESISTING MEDIUM.
599
d2y
£² x
dy
Again, x
Vx
Y
dt2
d t²
dy
but in the ellipse x
S
dt
y
dx ds
dt dt dt'
dx
Y √ S'a (1 − e²) ;
dt
α
1
d√a (1 − e²)
ds
V
a (1 − e²)
dt
dt
d.a (1 − e²)
dt
ds
· = − 2a (1 −
2a (1 − e²) V
dt
de
1 - e²
2 V
dt
e
Also, since
performing the differentiation, and substituting,
α
ds
2 V
1 − e²
e
α
S
(1) (1) (-1)*
a (1 − e²)
dt
= 1 + e cos (0 −w),
ጥ
d w
dt
1 - e²
re sin (0 – ₪)
da
2 a
1
de
+
dt
r sin (0 – ₪)
cot (0 - w
e
dt
2 V
ds
sin (0 – ₪)
e
dt'
after all reductions.
582. Since the orbit is nearly circular we shall neglect
e³, e³………: hence v = a {1
e cos (0 – w)};
dt
22
a
(Art. 242.)
| –
- 2 e cos (0-)};
do
h
S
.. nt = 0 - To 2e sin (0 – ☎),
t being reckoned from the perihelion passage;
da
Hence
dt
@= nt + 2e sin nt.
2 V √ Sa {1 + 3 e cos (0 – w)}
2 V na² {1 + 3 e cos nt};
... a = const. - 2 Va² {nt + 3e sin nt}.
N
600
HYDRODYNAMICS.
Hence the effect of the medium is to diminish the mean
distance by 4π Va² during each successive revolution, and
2
therefore to increase the mean motion
by 6π Vna.
n, which
✓
α
de
2 V na
Also
{1-e-1-ecos (0)} {1+ e cos (0 –w)}
dt
e
Vna { 2 cos (0 – w) + e[3 + cos 2 (0 – π)]}
Vna {2 cos nt + e (1 + 3 cos 2 n t) } ;
...e const. Va {2 sin nt + e (nt + sin 2nt)}.
=
½
Hence the eccentricity is diminished during each revolu-
tion by 2π Vae.
dw
2 V na
1 + e² + 2 e cos (A
Lastly,
sin (0-w
= w)}}
dt
1 - e²
e
- nA₁ sinnt. n A₂ sin 2 nt +
A1, A2...being functions of V, a, and e;
.. ☎ = const. + A₁ cos nt + ¦ A₂ cos 2nt +...
hence during a revolution is unaffected, or the axis major
remains stationary.
583. The variations of the elements of the planetary
orbits calculated on the principles of Chapter VI. of DYNAMICS
on the Planetary Theory give the position of the planets very
accurately: hence the variations arising from a resisting me-
dium, if one exist, are insensible in the motion of the planets:
hence V is extremely small. But V varies directly as the
density of the medium and inversely as the mass of the body
acted on, consequently although the medium produces no
effect on the motion of the planets, yet its influence on the
comets may be sensible, since their mass is extremely small,
Art. 491.
584. In order to apply the variations in Art. 581, to
find the change in a comet's orbit, we must use the method
PLANETS IN A RESISTING MEDIUM.
601
of quadratures; that is, we must calculate the differential co-
efficients for values of the elements at certain instants not far
apart, and then multiply these results by the respective inter-
vals of time between these instants and add them all together:
see Mr Airy's Translation of Encke's Dissertation on his
comet in the Astronomische Nachrichten, Nos. 210, 211.
The result deduced in Art. 582, with respect to the po-
sition of the perihelion is true for all orbits, as there shewn,
1 G
CONCLUSION.
SUMMARY OF ARGUMENTS IN FAVOUR OF UNIVERSAL GRAVITATION.
585.
THE perpetual recurrence of similar phenomena
under similar circumstances suggests the idea, that the mate-
rial world is regulated in its movements by laws, that the
changes and vicissitudes we witness, whether in the heavens
or in the planet we inhabit, are not the results of mere chance
and caprice, but spring from the secret influence and oper-
ation of certain principles and properties with which matter
is endowed. The ordinary observer cannot fail to trace in
many instances the connexion between one set of phenomena
and another; as for example the relation between the length
of the day and the interval of time which elapses between the
times of high tide; the connexion which this bears with the
position of the Sun and Moon in the heavens: then again the
analogy which exists between the notion that the Sun and
Moon have an affinity for the waters of the Ocean, and the
fact that terrestrial bodies are drawn towards the Earth when
left to themselves. In this way our conception of order and
regularity in the changes of the natural world is strengthened,
and by carrying on our researches we begin to discover that
many effects, which seemed to be independent of each other
and linked by no natural connexion, are collateral results of
one and the same principle.
Our object in the present Chapter is to gather the ar-
guments which convince us of the truth of the Theory of
Universal Gravitation, arranging them in order and present-
ing them under one view. By the labours of philosophers
extended over a long period of time the celestial phenomena
UNIVERSAL GRAVITATION.
603
have been traced to the action of a few simple laws. These
laws we have pointed out in the course of this work.
586. All our knowledge of external objects is the result
of experience; by experience we accumulate facts, and by
the comparison and classification of facts we are led through
a process of induction to the discovery of the general laws
from the operation of which these facts spring merely as
limited and individual results.
Experience teaches us that bodies, when left to themselves
and when unresisted by external objects, fall downwards.
This constant tendency downwards in preference to any other
direction suggests the first idea of an affinity which one por-
tion of matter has for other portions. The greatness of the
size of the Earth when compared with that of any body upon
which we can perform experiments sufficiently accounts for
the fact, that bodies do not appear to influence each other.
Experiments shew, however, that when two bodies are placed at
rest and near each other on the surface of a fluid (under which
circumstances the least possible resistance is offered to their mo-
tion) they will begin to move and finally come in contact. Also
Cavendish's experiments with leaden balls prove the same.
Let us examine the consequences to which this conception
of an attractive property of matter leads us.
587. The examination of numerous experiments led us
to conceive that the following laws regulate the motion of
bodies; That a body in motion will continue in motion and
move uniformly in a straight line when not acted on by ex-
ternal forces; and That when a force acts upon a body in
motion the change in motion is the same as if the force acted
on the body originally at rest (Arts. 195, 208).
Now the bodies of the Solar System do not move in
straight lines. The laws just enunciated shew, then, that
these bodies are acted on by external forces; and, since we
have seen that a principle of attraction does prevail in matter
on the Earth's surface, we are fully justified in adopting as an
hypothesis, to stand or fall by the comparison of calculated
results with the observed phenomena, that this principle of
attraction prevails throughout the Universe.
We shall now
enquire into the nature of this force of attraction.
604
CONCLUSION.
588. It is by a combination of the first and second laws
of motion that we calculate by the use of mathematical symbols
the relations connecting the path described by a body moving
in space and the forces which act upon the body. In order,
then, to discover the nature of the forces we must examine the
nature of the orbit described by the body. Kepler has fur-
nished us with the necessary facts respecting the configuration
of the Solar System; he deduced them from observations made
upon the motion of the planets. These, as we shall shew,
incontrovertibly strengthen the testimony in favour of the uni-
versal gravitation of matter, and moreover point out the law
of variation of the attraction. The following are the laws
which Kepler discovered: they are very nearly verified in the
Solar System.
Each planet describes an ellipse about the Sun, the Sun.
being in one of the foci.
The areas described by the radius-vector of each planet
about the Sun vary as the times of describing them.
The squares of the periodic times of the planets about the
Sun bear to each other the same relation as the cubes of their
mean distances.
These laws have been shewn to hold for the satellites re-
volving about their primaries if small inequalities be neglected.
The same gravitating principle therefore that retains (as we
shall see) the planets in their orbits about the Sun, binds
the satellites to their primaries.
589. The Sun's magnitude is very enormous in compa-
rison with that of the other heavenly bodies; also the mutual
distances of the heavenly bodies are never very small in com-
parison with their distance from the Sun: this is shewn by
astronomical observations which are independent of all theory.
For this reason we may neglect, at least for a first approxima-
tion, the mutual action of the heavenly bodies in comparison
with the action of the Sun upon them. Also the diameters of
the Sun and planets bear a very small ratio to the distances of
the planets from the Sun : therefore we shall not be very far
from the truth if we consider the Sun and planets as intense
particles, condensed into their centres. If we adopt this sup-
position, Kepler's second law proves, that the forces acting on
UNIVERSAL GRAVITATION.
605
the planets pass through the Sun's centre (Art. 257): and con-
sequently confirms the notion of a principle of attraction, and
shews that the attraction of the planets, as we have supposed,
is far more feeble than the attraction of the Sun. Kepler's
first law proves that the attraction between the Sun and planets
varies in intensity inversely as the square of the distance of the
bodies. The third law shews that, not only must the force on
each planet pass through the Sun's centre, and the law of at-
traction be the same; but the intensity of the attractive force
on each planet must be the same at the same distance for
every planet.
590. Thus far then the laws which Kepler discovered to
prevail in the Solar System give great weight to the evidence
in favour of the universality of the principle of attraction;
and moreover they point out the law of variation of the at-
traction when the distance varies, and shew that it is the in-
verse square of the distance. But we have hitherto considered
the heavenly bodies to be merely intense particles, whereas the
diameter of the Earth is nearly 8000 miles, and that of the
Sun between 800,000 and 900,000 miles. And, moreover, we
are convinced that it is not the centres alone of the heavenly
bodies that attract, since Dr Maskelyne ascertained by ob-
servations on the stars made near the mountain Shehallien, in
Scotland, that the direction of the plumb line was affected by
the attraction of the mountain, and consequently the Earth's
attraction is not accurately directed towards its centre. The
same is likewise proved by measuring degrees of latitude near
the pole and equator. It becomes necessary, then, to enquire
more minutely into the legitimacy of the hypothesis we have
adopted in Art. 589.
591. The force of the objection, that all the particles of
a body attract and not the centres only, will be considerably
weakened by referring to Art. 165, in which we proved that a
mass of matter composed of particles attracting according to
a certain law will have nearly the same attraction for a distant
body as if we considered the particles to be condensed into
their common centre of gravity. But upon a further examina-
tion we find that the objection is, as far as the accuracy of
our results in this stage of the question is concerned, quite
606
CONCLUSION.
removed. For in the Article following that last cited we shew,
that the law of attraction of the constituent particles and the
resultant law of attraction are always the same. But the re-
sultant law is that of the inverse square of the distance, as
we have proved by Kepler's laws: this, then, is the law of
attraction of the constituent particles. If we turn now to
Art. 153, we shall find that the inverse square of the distance
is one of those three laws of attraction which give accurately
the same resultant for a spherical shell as if we conceive it
condensed into its centre.
When we bear in mind, then, that astronomical obser-
vations have proved that the figures of the Sun and planets
and their satellites do not differ much from spheres, the
objection which arises from effecting the calculations of the
motion of the heavenly bodies on the hypothesis of their being
intense particles is entirely removed.
592. We come, then, to this conclusion, that, if we
neglect the minute errors which accurate observations on the
heavenly bodies detected in Kepler's laws, if we neglect the
errors arising from the deviation of the figures of these bodies
from spheres, and the probable variation of density in their
interior, of which we have made no account, then the only
simple hypothesis which will account for the phenomena is,
that all particles of the universe attract each other with a force
which varies inversely as the square of the distance and di-
rectly as the mass of the attracting particle.
It remains to be seen whether this law, simple as it is,
will upon effecting the calculations give correct numerical
results for the test of a theory consists in a comparison of
the exact results to which it leads with the observed phe-
nomena; and not solely in its power of explaining the nature
of the phenomena.
593. Now the calculations of the position of the planets
made upon this hypothesis of their gravitating towards the
Sun with a force directly as their mass and inversely as the
square of their distance from the Sun are found to agree
very well with the observed positions, if the calculations extend
over only a few years. After the lapse however of a con-
siderable interval of time, as a century, minute crrors
are
UNIVERSAL GRAVITATION.
607
detected in the calculations, and they are then found not
to agree exactly with the observed positions of the planets.
But this disagreement is in fact precisely what we should
have anticipated, since if the principle of gravitation be
universal the planets would attract each other and conse-
quently disturb the elliptic motion and the equable description
of areas; likewise the deviation of the figures of the planets
from perfect sphericity and their heterogeneous structure give
rise to additional errors. An idea of the extreme small-
ness of the perturbations may be learned from the fact, that
if we trace on paper an ellipse ten feet in diameter to represent
the orbit in which the Earth is moving at any instant
about the Sun, and if we trace by its side the path actually
described in its revolution round the Sun, the difference be-
tween the original ellipse and the curve actually described
is so excessively minute, that the nicest examination with micro-
scopes, continued along the outlines of the two curves, would
hardly detect any perceptible interval between them: Herschel's
Astronomy.
594. Our next enquiry should therefore be whether the
magnitudes of the minute deviations from elliptic motion
accord with the calculations effected on the hypothesis of the
universality of the gravitation of matter. And here we enter
upon an investigation so complicated and depending upon such
a variety of disturbing causes all in simultaneous operation,
that it is desirable to seek for a compendious method of treating
the subject.
The course we shall pursue is exactly the reverse of that
we have hitherto followed: for we shall now assume the truth
of the Law of Universal Gravitation and calculate by means
of mathematical reasoning the phenomena which would result
from the operation of this Law in combination with the Laws
of Motion. In adopting this method we entirely disentangle
ourselves from the multitude of difficulties which were sur-
rounding us; we have not now to consider the influence which
this or that observed fact may have upon our calculations ;
we no longer have to modify our original notions upon each
discovery: we commence entirely anew, and assuming the
Laws of Motion and of Universal Gravitation we investigate by
608
CONCLUSION.
means of the rigorous and infallible engine of mathematical
calculation, the phenomena which would naturally arise from
the action of these laws. The complete accordance which is
found to exist, in all the instances submitted to this test,
between the calculated and the observed phenomena of the
Solar System is the surest proof of the truth of the assumed
laws.
We were unable to proceed in this way from the begin-
ning since mathematical reasoning is incapable of application.
unless we have laws to reason upon: and in consequence of
the number and complexity of the observed facts, it was
impossible, a priori, even to conjecture, that the law of at-
traction was that of the inverse square of the distance. But
having been led by a process of induction step by step to
this great principle, we descend through a process of deduc-
tion to examine its consequences, and so in the end incon-
trovertibly to establish its truth. So strong is the evidence
in its favour, and so firm the basis on which it rests, that
in many cases we attribute slight discrepancies to errors in
the observations rather than in the assumed law.
595. Notwithstanding what we have said of the advan-
tage of considering the subject in this point of view, it
must not be denied that great difficulties beset our path in
effecting the mathematical calculation. Were it not indeed
for the peculiar configuration of our system, the enormous pre-
ponderance of the mass of the Sun over that of the planets,
the grouping of the heavenly bodies into clusters consisting
of a primary and its satellites, the mass of the central body
greatly exceeding that of its attendants, were it not for these
peculiar arrangements the calculation would baffle the powers
of analysis.
It must not be imagined, however, that these difficulties
are such as to vitiate the results to which they lead us :
they add to the labour of the calculation, but do not
subtract from their certainty; since we may by successive
approximation arrive within an inappreciable distance of the
exact results.
596. The motion of a body moving freely in space
consists of two parts, one a motion of translation from one
UNIVERSAL GRAVITATION.
609
situation to another, and the other a motion of rotation: and
these may be considered independently of each other; for
the motion of the centre of gravity would be precisely the
same if the whole mass were concentrated in that point,
and all the forces which act upon the body were transferred
to the same point: and the motion of rotation would be the
same if the centre of gravity were held at rest. These
principles we have proved in Arts. 428, 429, 474, 475 and
they enable us to divide the investigation of the motion of
the planets and other heavenly bodies into two branches,
which we shall consider separately.
597. The forces which act upon a heavenly body are
the attractions of the Sun, planets, satellites, and comets
upon every particle of its mass, and the resistance of the
medium in which the body moves.
But in consequence of the enormous magnitude of the
Sun in comparison of that of the other bodies, and in con-
sequence also of the tenuity of the medium which pervades
the planetary spaces, the great preponderating force is the
attraction of the Sun: and since the figure of the Sun differs
but little from a sphere, and since the same is likewise true
of the other bodies of the solar system, it follows, that the
resultant of all the forces which act upon the various par-
ticles of the body, the motion of which we are considering,
when transferred to the centre of gravity of the body will
differ very slightly from a force varying as the sum of the
masses of the Sun and body divided by the square of the
distance of their centres: the small additional forces arc
called disturbing forces.
598. Now if the disturbing forces did not exist, the
centres of the bodies of the Solar System would each move in
a conic section: and all the orbits would have one common
focus in the Sun's centre: (Art. 252.) But in consequence
of the disturbing forces slight deviations from these paths
will arise which it will be necessary to calculate.
Lagrange has put the subject in a most lucid point of
view. Let us imagine the disturbing forces to cease acting,
the centre of the body would ever after move in a conic
section, the magnitude and position of which would depend
4 H
610
CONCLUSION.
upon the state of things at the instant the disturbing forces
ceased to act.
Now this curve and the path actually described have a
common tangent, and the velocity at the instant under con-
sideration is the same as that in the conic section calculated
according to the principles of elliptic motion. For these rea-
sons we may suppose that for an instant the body is moving
in the conic section, the elements of its position and magni-
tude depending upon the time. And in short the entire
motion of the centre of the body may be represented by sup-
posing it to move in a conic section, of which the elements
are subject to continual variation. The paths of all the
heavenly bodies (with the exception perhaps of some of the
comets) are nearly elliptical, and the ellipses in which they are
moving at any instant are called the instantaneous ellipses.
We thus reduce the calculation of the motion of the
heavenly bodies to that of the elements of the instantaneous
ellipse in which the body is moving at the instant we wish
to calculate the position of the body. Nothing remains to
be done after this, but to substitute the values of these
elements in the expressions of the radius vector, latitude, and
longitude. See DYNAMICS, Chap. vi.
599. When a number of small disturbing forces act upon
a body and alter its motion the aggregate effect is very nearly
equal to the algebraical sum of the separate effects which they
would produce if they acted independently of each other. For
the real effect differs from this sum merely by the effect which
the disturbing forces produce in modifying each others action,
and must consequently be of the second order of magnitude,
and therefore so trifling as to be inappreciable except in ex-
treme cases. (Art. 288.) This principle we find of vast im-
portance since it greatly facilitates the calculations.
600. In order to determine the elliptic elements of
a heavenly body corresponding to any instant, we investigate
their value in terms of a small arbitrary disturbing force
expressed in general symbols, as will be seen by turning to
Chapter vi, of DYNAMICS.
The various disturbing forces are then calculated and
substituted singly in the formula which give the variations
1
UNIVERSAL GRAVITATION.
611
of the elements: and the variations being added together with
their correct signs the whole variation is known: the value of
each element being then determined we are prepared to apply
the formulæ of Arts. 278, 280, already mentioned.
Having thus explained the analytical machinery, so to
speak, which we make use of in order to calculate the motions
of the heavenly bodies, we must shew how the results of the
calculation bear comparison with the observed phenomena.
601. The planets are subject to perturbations of two
kinds; both depending upon their reciprocal action; a full
explanation of these will be found in Art. 377. They are
termed periodic variations and secular variations. Some of
these remarkably prove the correctness of the principles which
have guided the calculations, such as the great inequality of
Jupiter and Saturn, which at one time was the great stumbling
block in the way of receiving Newton's theory of gravitation,
but which Laplace so entirely removed by shewing that the
defect lay in the analysis and not in the principle of gravita-
tion: again the inequality of the Earth and Venus discovered
by Mr Airy as clearly demonstrates the truth of the law of
gravitation, especially when we consider the extreme minute-
ness of this error and the complex character of the analysis.
(Art. 376).
There is a remarkable perturbation in the motion of the
Moon arising from the secular inequality of the eccentricity of
the Earth's orbit: it is called the Secular Acceleration of the
Moon's mean motion. It had been observed by Halley, upon
comparing together the records of the most ancient lunar
eclipses of Chaldean Astronomers with those of modern times,
that the periodic time of the Moon about the Earth is now
sensibly shorter than it was at that distant epoch. This result
was confirmed by a further comparison of both sets of obser-
vations with those of the Arabian Astronomers of the eighth
and ninth centuries; and it was proved that the mean motion
is increasing by about 11" per century, a quantity small in
itself, but becoming of importance by the accumulation of
ages. This had long been a stumbling block to mathema-
ticians, and so difficult did it appear to render an exact
account of it, that the theory of gravity was declared to be
612
CONCLUSION.
inadequate satisfactorily to remove the difficulty by explain-
ing the cause of the phenomena. It was in this dilemma
that the penetrating sagacity of Laplace was once more
called into action to rescue Physical Astronomy from its
reproach. If the solar ellipse were invariable the alternate
dilatation and contraction of the lunar orbit would, in the
course of a great many revolutions of the Sun, at length
bring about an exact compensation in the distance and peri-
odic time of the Moon.
But the solar ellipse is not invariable, its eccentricity has
been decreasing since the earliest ages, and will continue to do
so till the orbit becomes a circle, after which epoch the orbit.
will again dilute and increase in eccentricity. It was from
this variation of the eccentricity of the solar orbit that Laplace
shewed that the variation in the Moon's mean motion arose.
This phenomenon is a very striking instance of the propagation
of a periodic inequality from one part of a system to another.
The masses of the planets are too small and their distances
from the Earth are too great for their difference of action
on the Earth and Moon ever to become sensible. Yet their
effect on the Earth's orbit is propagated (as we have seen)
through the Sun to the Moon's orbit; and, what is very
remarkable, the transmitted effect thus indirectly produced
on the angle described by the Moon round the Earth is more
sensible to observation than that directly produced by them on
the angle described by the Earth round the Sun.
602. But without adding more suffice it to say, that the
calculations of the inequalities of the planets and of Jupiter's
satellites is arrived at such a degree of precision, as to agree
exactly with the observations, omitting only unavoidable in-
strumental errors.
603. Again in the Lunar Theory, we have many proofs
of the truth of Newton's law (Arts. 341–346). The Variation,
the Evection, and Annual Equation, the motion of the perigee
and node all agree in their numerical results with observation :
numerous other inequalities, a few of which are mentioned in
Art. 346, give additional evidence in support of our theory.
604. Likewise the calculation of the motion of comets
very remarkably agrees with observation. The time of the
UNIVERSAL GRAVITATION.
613
re-appearance of Halley's Comet in the year 1835, after an
absence of 76 years, was predicted correctly within nine days of
its actual appearance! A most astonishing fact, when we con-
sider that the light of the comet is diminishing, and that con-
sequently we could not expect to have the time of re-appearance
very accurately calculated.
605. In short, when we consider the simplicity of the law
to which we are led, the variety and different characters of the
tests we use, the labyrinth of calculations through which we
have to wind our way, and the exact character of the results
upon which any reliance can be placed, we are irresistably
constrained to admit, that no theory has ever been based upon
a firmer foundation than that of Universal Gravitation. In
many instances, if the law departed in the slightest degree from
that of the inverse square, inequalities, which are now calcu-
lated numerically in the theory and agree with observation,
would not give results near the truth: this is the case with the
motion of the Moon's perigee.
606. But we have hitherto gathered our evidence solely
from the motion of the heavenly bodies considered as intense
particles: when, however, we descend deeper into the conse-
quences of the law of gravity and enquire into the minute
errors caused by the attraction of the various particles which
form the masses of the heavenly bodies, we obtain an accession
of sound arguments in favour of Universal Gravitation.
607. By measuring degrees of latitude as near as possible
to the pole and equator it is found that they increase in length
as we pass from the equator to the pole; this shews that the
vertical lines (or normal lines) to the Earth's surface are less
and less inclined to each other as we proceed towards the pole
from the equator. We learn from this that the form of the
Earth is not spherical, but flattened at the poles: and when
the calculations and observations are combined, the geodetic
measures shew that the Earth is very nearly spheroidal, having
an ellipticity This is a result not of theory but observa-
tion and trigonometrical calculations.
1
306*
608. With this fact to guide us we entered upon an
enquiry whether this form were given to the Earth in a fluid
or semi-fluid state, since we easily see a priori that the rota-
614
CONCLUSION.
tory motion would, in that case, cause the parts near the
equator to bulge. We proved in Art. 525, that this would
not give the proper numerical value of the ellipticity if we
suppose that the mass of the Earth were homogeneous, an hy-
pothesis in itself highly improbable, since the pressure of the
upper strata must produce a condensation of the lower: also
it is contrary to the results of Maskelyne's and Cavendish's
results respecting the mean density of the Earth. We here
have then a negative argument in favour of gravitation. We
therefore proceeded (Art. 531.) to investigate the figure of the
Earth upon the hypothesis that it consists of strata differing
but little from spherical shells (an hypothesis extremely pro-
bable, since the ellipticity of the surface is by observations),
and increasing in density towards the centre according to an
unknown law. The result we arrived at was, that the form
of all the shells is spheroidal, decreasing in ellipticity towards
the centre (Art. 535). An equation was obtained for calcu-
lating the ellipticity of the surface, when the law of density
was discovered: this law we obtained upon hydrostatic prin-
ciples in terms of arbitrary constants (Art. 545), and having
determined the values of one of these constants by means of
the facts given us by Maskelyne and Cavendish respecting the
mean density of the Earth, we reduced to numbers the for-
mula for the ellipticity and obtained a result, according most
remarkably with that given by geodetic measurements.
1
306
609. As a further test we reduced to numbers the for-
mula for the precession of the equinoxes, which had been
previously calculated in Arts. 463—470, and obtained a result
according with remarkable exactness with the observed pre-
cession (Art. 551).
610. Laplace calculated the errors caused in the latitude.
and longitude of the Moon by the bulging portion of the
Earth at the equator, and obtained results very closely agree-
ing with those we have given above (see Art. 346).
611. Also pendulum experiments give, by the formula
of Art. 539, an ellipticity according very well with the other
values.
612. It will be readily granted, then, that we have an
abundance of evidence (and more might be given) to justify
UNIVERSAL GRAVITATION.
615
the conviction, that not only the heavenly bodies attract each
other with forces varying as the attracting mass and inversely
as the square of the distance, but that the individual particles
of which they are composed attract according to the same law.
613. The theory of the Tides is at present in so imper-
fect a state, that we must not look for evidence in that quarter.
Nevertheless some of the observations collected by Mr Whewell
and Mr Lubbock seem to indicate that the force of attraction
of the Moon in raising the waters varies inversely as the cube
of the distance, the theoretical law according to the theory of
gravitation (Art. 574).
614. We may well draw this epitome to a conclusion in
the words of Sir John Herschel. "There is one feature in
physical astronomy which renders it remarkable among the
sciences, and has been the chief, if not the only, source of the
perfection it has attained. It is this, that the fundamental law
embracing all the minutiae of the phenomena so far as we yet
know them, presents itself at once, on the consideration of
broad features and general facts, deduced by observations of
even a rude and imperfect kind, in such a form as to require
no modification, extension, or addition when applied to minute
detail. In other sciences, when an induction of a moderate
extent has led us to the knowledge of a law which we conceive.
to be general, the further progress of our enquiries frequently
obliges us either to limit its extent or modify its expression.
...In physical astronomy, however,...our first conclusion is
our last. The law on which all its phenomena depend flows.
naturally and easily from the simplest among them, as pre-
sented by the rudest observation; and, in point of fact, such
has really been the order of investigation in this science. The
rude supposition of the uniform revolution of the Moon in
a circle about the Earth as a centre led Newton at once to
the true law of gravity, as extending from the Earth to its
companion. The uniform circular motions of the planets
about the Sun, in times following the progression assigned
by observation in Kepler's rule, confirmed the law, and ex-
tended its influence to the boundaries of our system. Every
thing more refined than this, the elliptic motions of the planets
and satellites-their mutual perturbations, the slow changes
616
CONCLUSION.
of their orbits and motion denominated secular variations, the
deviation of their figures from the spherical form, the oscil-
latory motions of their axes which produce nutation and the
precession of the equinoxes, the theory of the tides both of
the ocean and the atmosphere have all in succession been so
many trials for life and death in which this law has been, as
it were, pitted against nature; trials, of which the event no
human foresight could predict, and where it was impossible
even to conjecture what modifications it might be found to
need." Enc. Met. Physical Astronomy, p. 647.

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