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THE ROTATION OF THE EARTH,
LOIN DON
P. b. I N T E D BY SPOTTI S W O O D E AN ID CO.
NEW-STREET SQUARE
THE
EARTH AND ITS MECHANISM:
BEING AN ACCOUNT OF THE WARIO US
PR00FS OF THE ROTATION OF THE EARTH.
WITH A DESCRIPTION OF THE INSTRUMENTS
TJSED IN THE EXPERIMENTAL DEMONSTRATIONS : TO WHICH IS ADT). ED THE THEORY
OF FOUCAULT’S PEN DUI,UM AND GY ROSCOPE.
BY HENRY WORMS, F.R.A.S., F.G.S.
sºme"
‘E pur si muove.’
WITH NUMEROUS WOODCUTS AND DIAGRAMS
LON ID ON :
LONGMAN, GREEN, LONG MAN, ROBERTS, & GREEN.
1862
‘That spinning sleeps
On her soft axle as she paces even,
And bears us swiftly with the smooth air along.’
MTITON.
TO
PROFESSOR WHEATSTONE, LL.D., F.R.S., &c.
WITH FEELINGS OF
THE DEEPEST RESPECT, AND IN
ADMIRATION OF THAT FERTILE GENIUS
WHICH HAS PRODUCED SO MANY USEFUL INVENTIONS IN SCIENCE
AND ART, THIS TREATISE IS INSCRIBED
BY HIS OBEDIENT SERVANT
THE AUTHOR.
1961.21
PREFACE.
THE present work consists of two parts, the one
popular and adapted to the comprehension of the
general reader; the other analytical and designed
for the perusal of the mathematical student. The
table of contents will sufficiently exhibit the plan
of the whole as a complete collection of the various
demonstrations of the Earth's rotation and the
theorems connected therewith.
The principal object, however, has been to pro-
mulgate a knowledge of the more recent discoveries
and experiments relating to this branch of science.
In the first instance, therefore, particular atten-
tion is drawn to Arago's application of Wheatstone's
revolving mirror for the determination of the finite
velocity of light, which not only affords evidence
of the motion of translation as thought hitherto,
but also of that of rotation.
Then a minute account is given of the latest ex-
periments on falling bodies made by Professor Reich
in the mines at Freiberg in addition to those of
viii PREFACE.
Benzenberg and Guglielmini. They deserve special
notice because their results are trustworthy, and the
idea of undertaking them for ascertaining the ro-
tation of the Earth originated with the illustrious
Newton.
The Author has further endeavoured to give a
clear and distinct idea of the phenomena exhibited
by Foucault's pendulum and gyroscope, inasmuch
as they furnish the most remarkable and direct
proofs of the Earth's rotation.
In order to elucidate the principle on which the
apparent change of the plane of Oscillation depends
(without having recourse to abstruse reasoning and
calculation), the Author has made use of a number
of auxiliary instruments originally invented for this
purpose. A full description of these instruments in
connection with that of the main apparatus is here
offered to the public, accompanied throughout with
corresponding graphical illustrations.
Finally, new modes for the suspension of the
pendulum are proposed as improvements, modifi-
cations of the gyroscope with practical directions
for using them are pointed out, and the conical
motion of the Earth's axis itself, causing the pre-
cession of the equinoxes, is explained by Maxwell's
dynamical top and Burr's apparatus.
With regard to the theoretical portion of the
work containing the solution of intricate problems
PREFACE. ix
of analytical mechanics, and intended for the ma-
thematical student, it is necessary to state that in
the theory of the motion of the pendulum taken
from ‘Hansen's Theorie der Pendelbewegung, the
Author has allowed himself some omissions and
additions, for which he is responsible; at the same
time he ventures to express a hope that the Reader
will be lenient in his criticism of the imperfections
and shortcomings of this and other parts of the work,
and take into consideration the extreme difficulty
and abstruseness of the several subjects he has
endeavoured to treat.
GoLDSMITH BUILDING, TEMPLE:
October 1862,
CONTENTS.
PART I.
INTRODUCTORY CHAPTER
DID THE ANCIENTS KNOW THE COPERNICAN SYSTEM 7
The admission of Copernicus. – Nicetas. – Philolaus, Plato and Eudoxus,
their want of inductive reasoning. — Aristotle's ‘De Coelo;' the Pytha-
goreans place the Sun in the centre, because fire is nobler than earth. –
Aristarchus; the planetary theory unknown to him, his correct notions
are mere surmises.—Hipparchus disregards them. — Ptolemy.—Seneca's
“Naturales Quæstiones; ” his scanty knowledge. — The most important-
point of the Copernican system completely unknown to the ancients.
—The errors of Bailly and Dr. Gregory . * tº e . Page 1
CHAPTER I.
THE DISCOVERY OF THE EARTH's MOTION.
The beneficial effects of the discovery. — The complexity of previous
systems.— Copernicus dedicates his work to the Pope. — The fanatic
outcry against his doctrine. — A letter of Galileo to Kepler. — The per-
secution of Galileo, the form of his abjuration; his monument a century
after. — The objections raised against the earth’s motion. — The ob-
jections of Riccioli, of Tycho and others, stated and refuted. — The
objections on religious grounds are shown to be erroneous . tº 11
CHAPTER II.
THE PROBABILITIES OF THE EARTH's MOTION.
The earth is a globe floating in space. — An oblique impulse produces
two motions, viz. translation and rotation. — Geometrical proof of it.
— The probability of the earth having received an oblique impulse. — It
is increased by analogy with other planets. – A rotation of the celestial
sphere is improbable. — The velocity and uniformity of the apparent
motion of the stars. — The want of a physical axis. - The law of central
forces.—They cannot be seated in mathematical points & º 24
xii CONTENTS.
CHAPTER III.
PROOF OF THE EARTH's ROTATION BY THE TRANSMISSION OF LIGHT.
The apparent position of the stars. — The phenomena are incompatible
with the fixity of the earth. — The finite velocity of light proved by
experiments. – Professor Wheatstone's revolving mirror. — Arago's
suggestions,—Foucault's combination of mirrors. — The angular velocity
of the reflected light double that of the revolving mirror. — Geometrical
proof of it. — Foucault's results . † & g . . Page 33
CHAPTER IV.
THE EARTH's ROTATION DEMONSTRATED BY THE VARYING FORCE OF
GRAVITY IN DIFFERENT LATITUDES.
Richer's observations in Cayenne.—Borda's and Biot's pendulum.—Kater's
invariable pendulum. — Tables of the lengths of the seconds' pen-
dulums. – The theorem of centrifugal forces proved geometrically. —
Diminution of gravity. — Clairaut's theorem. — Near coincidence of
theory and observation g * º wº e tº * º 43
CHAPTER V.
THE DESCENT OF BODIES FROM GREAT HEIGHTS.
Newton's suggestion. — Dr. Hooke's experiment. — Guglielmini’s expe-
riments. – His errors. — Benzenberg's experiments in Hamburg's St.
Michael's tower and in the coalpit at Schlebusch. — Tables of results. –
Analytical investigation. — Comparison of theory with observation. —
The results of 106 experiments made by Professor Reich in the mines
of Freiberg. * * º g © * © g g e 61
CHAPTER VI.
ON THE APPARENT CHANGE OF THE PLANE OF OSCILLATION OF THE
FREE PENDULUM.
Definition of a pendulum. — Dr. Hooke's experiment in 1774. — The Acade-
micians del Cimento first noticed the apparent change of the plane of
oscillation, — Foucault first showed that this change depended on the
CONTENTS. xiii
earth's rotation. — His mode of liberation and suspension. — Expe-
rimental proofs of the parallelism of the plane of oscillation. — The
composition and resolution of rotatory motion. — The apparent angular
velocity of the motion of the pendulum in the azimuth is proportional
to the sine of latitude of the plane of observation.—Theoretical proofs
of it.—Experimental proofs of it by means of Professor Wheatstone's
apparatus . tº ge e e e * º t . Page 84
CHAPTER VII.
EXPERIMIENT MADE WITH THE FREE PENDULUM TO PROVE THE
ROTATION OF THE EARTH.
Description of the experiments made by General Dufour. — Mr. Morren's
experiments. – Mr. Bravais' observations on levogyric and dextrogyric
oscillations. – Experiment made in Rio de Janeiro. — M. Plana con-
siders Foucault's mode of suspension defective.—Hansen's suspension. —
Dettmann's suspension. —Experiments made by the author in the Hall
of King's College, London . g e * * sº e ... 103
CHAPTER VIII.
{ }
DESCRIPTION OF THE GYROSCOPE.
Description of the gyroscope. — Its application to show the laws of
rotation. — Fessel's gyroscope. — The phenomena of rotatory motion
exhibited by it and their explanation. — Professor Wheatstone's magnetic
gyroscope. — Professor Magnus' apparatus with two discs. – Foucault's
gyroscope to show the rotation of the earth. — The fixity of the plane
of rotation. — The force of orientation. — The results of Mr. Quet's
mathematical investigation. — The objections raised against Foucault's
mode of suspension. —Bohnenberger's original apparatus. – Its ap-
plication for showing the rotation of the earth . * * . 116
CHAPTER IX.
THE PRECESSION OF THE EQUINOXES A PROOF OF THE EARTH's
ROTATION.
Definition. — Historical account. — Timochares and Hipparchus. – The
spinning top affords an illustration of astronomical precession. —Bro-
fessor Maxwell's dynamical top, its description and application. — The
mutation of the earth's axis discovered by Dr. Bradley. — Definition of
it. — Mr. Burr's apparatus for the imitation of the two motions of pre-
cession and mutation . . . . º tº e . . 135
xiv CONTENTS.
PART II.
CHAPTER I.
THEORY OF THE MOTION OF FALLING BODIES BY LA PLACE AND
GAUSS, WITH REGARD TO THE ROTATION OF THE EARTH PAGE 149
CHAPTER II.
HANSEN'S THEORY OF THE MOTION OF THE PENDULUM WITH REGARD
TO THE ROTATION OF THE EARTH tº * e tº . 164
CHAPTER III.
THEORY OF THE MOTION OF THE GYROSCOPE WITH REGARD TO THE
ROTATION OF THE EARTH e e g g tº . 24.7
CHAPTER IV.
POINSOT's NEW THEORY OF ROTATION * © sº . 275
NoTE—Page 167.
TO DERIVE EXPRESSIONS FOR THE MOTION OF FALLING AND PRO -
JECTED BODIES FROM THE DIFFERENTIAL EQUATIONS FOR THE
MOTION OF THE SIMPLE PENDULUM WITH REGARD TO THE ROTA-
TION OF THE EARTH e tº º g * º . 291
DART I.
–ºe-
INTRODUCTORY CHAPTER,
DID THE ANCIENTS KNOW THE COPERNICAN SYSTEM 2
TTEMPTS have been made at various periods to show
that Copernicus borrowed his ideas from the Pythago-
reans, and he has therefore been called by Bailly the restorer
of the true system of the world. The object which we have
in view in this introduction is to prove that the ancients
did not possess that amount of knowledge which would
have enabled them to explain the motion of the celestial
bodies in the manner which is now universally recognised.
Before citing all the passages relating to the subject which
are to be met with in the classics, we will quote those
which are admitted by Copernicus, in his preface to the
* Revolutionibus Orbium Celestium,’ to have led him in
some degree to his discoveries. He says, “Ac reperi qui-
dem apud Ciceronem primum : Nicetam sensisse terram
moveri; postea et apud Plutarchum inveni quosdam alios
in ea fuisse opiniome. Inde igitur occasionem nactus,
coepi ut ego de terrae mobilitate cogitare.’ The two
passages to which he alludes are the following. Cicero
having spoken (Quaest. Acad. 2) of men who believe in the
existence of antipodes, continues thus: “Theophrastus
B
2 DID THE ANCIENTS KNOW
narrates that Nicetas of Syracuse was of opinion that the
sun, moon, and stars were at rest, and the earth alone
moved, turning about its axis, by which the same pheno-
mena were produced, as if the contrary were the case.'
From this it is evident that either Cicero misinterpreted
Theophrastes, or the latter did not understand Nicetas, or,
what is still more probable, Nicetas himself had but a very
imperfect knowledge of the subject, for it is impossible to
explain the apparent movements of the sun, the moon, and
other planets by the rotation of the earth alone. It is
difficult to determine whether Copernicus meant this pas-
sage of Cicero, or whether he did not rather allude to the
one in Lib. I. Tuscul., which briefly states, “Primus Nicetas
Syracusanus terram moveri,’ etc. These five words might
have led us to believe that Nicetas had a correct idea of
the universe, were we not aware of the passage above cited.
We may here aptly apply the old rule of the oracle, “He
who knows but little of a subject should be sparing of his
words or hold his tongue. He who veils his meaning by
the obscurity of his speech defies contradiction.”
The passage in Plutarch referred to by Copernicus is in
the third Book, ‘De Placitis Philosophorum,’ caps. 11 and
13. “The followers of Thales placed the earth in the centre
of the universe: Xenophanes was the first who believed
it to be in infinite space. The doctrine of Philolaus the
Pythagorean was, that fire was the centre and focus of the
universe; while the other Pythagoreans maintained that the
earth was at rest, he asserted that it revolved in an oblique
circle about a central fire like the sun and moon.’ “Ceteri
terram manere dixerunt. Philolaus vero Pythagoraeus,
circum ignem verti obliquo in circulo similiter soli et lunae.’
* Heraclides of Pontus, and Ecphantes the Pythagorean,
THE COPERNICAN SYSTEM P 3
believed that the earth had no translation in space, but re-
volved like a wheel upon its axis from west to east.” We
gather from these quotations that mention is made of a
motion of the earth, but not of the true one, and the fol-
lowing fact proves that no notice of it was contained in
the writings of Philolaus. Plato put this question to the
astronomers, “How is the retrograde motion of the planets
to be explained P’ Eudoxus solved the problem by means
of concentric spheres. The great Plato undoubtedly saw
that this explanation was very complicated and far-fetched,
and as he was in possession of Philolaus' manuscripts,
having purchased them of his heirs for a hundred Mines,” it
is very unlikely that he neglected to read them, especially
as he held the author in such high esteem that he travelled
to Italy on purpose to see him, which was in those days no
small undertaking. We learn, therefore, from Eudoxus that
the Pythagoreans did not know the Copernican system,
otherwise he would never have introduced so complicated a
combination of concentric circles in order to explain the
celestial phenomena. In fact, none of the notions of the
Greeks were founded on a true knowledge of astronomy,
or upon a comparison of facts; everything was speculation
without regard to inductive reasoning, and they conse-
quently arrived at contradictory results. Some considered
the earth to be a cube, others a mountain, raising its lofty
and rounded summit from the midst of the universe.
Besides the passages quoted by Copernicus, and which
we have just criticised, there are others in the writings of
antiquity in which mention is made of a motion of the
earth. Aristotle, in his second Book, ‘De Coelo,” says,
* Most people place the earth in the middle, but some
* A ‘Mine” was worth about £4 1s. 3d.
B 2
4 DID THE ANCIENTS KNOW
who live in Italy and are called Pythagoreans assert that
fire is the centre, and the earth is only a star, which, by
its revolution, produces day and night.” At first sight this
passage might lead us to infer that they were aware of
one of the earth's motions, but examining closely their
mode of investigation, we find that this isolated truth was
not the result of observation, but an hypothesis based on
illusory premises; for they reasoned in the following way —
the noblest body should have the noblest place. Fire, in
their estimation, ranked higher than earth, therefore to
the former they assigned the ‘place d’honneur.” “Ignem
hanc habere regionem dixerunt, quem Jovis carcerem sive
custodiam nominant.”
In the “Arenario of Archimedes, it is stated that Ari-
starchus of Samos was of opinion that the stars moved and
the sun was fixed, that the earth revolved in a circle round
the latter, and that the sphere which carried the stars
round rotated also about it. Aristarchus lived in the year
280 B.C., and the superstitious Athenians believed that his
doctrines were contrary to religion, because they deprived
the earth of its central position. Cleanthus, a disciple of
Zeno, indicted him for despising the glory of the goddess
Vesta. This prosecution by the Athenian inquisition is
assigned by many as the reason that the imperfect know-
ledge of the true system was kept so secret as now to be
lost altogether.
Athenian power, however, did not extend far, and the
philosophers in other parts of Greece, in Asia Minor, and
in Italy, where civilization was quite as high, could pro-
pagate their opinions without fear of inquisitorial perse-
cution. This much we may admit on the testimony of
Archimedes, that the opinions of Aristarchus excelled
THE COPERNICAN SYSTEM P 5
in clearness and correctness those of all the other Greeks;
but they still did not possess any great astronomical im-
portance, as they were not employed to explain the motion
of the planets in their direct or retrograde motion. The
whole planetary theory was unknown to him, and he, more
Pythagorean than astronomer, never conceived that so
complicated a mechanism was capable of so simple an ex-
planation; and had he even imagined it, he never could
have established it, owing to the total want of observations
at that period. But the strongest proof that the true
system was not known, is that no allusion is made to it by
the greatest astronomers of that time. If the assertions
made by the Greeks about the motion of the earth and the
place of the sun were more than mere philosophical sur-
mises, then Hipparchus would have availed himself of
them for his own observations. In order to avoid the in-
troduction of epicycles, he refrained from considering the
earth as the centre of the system, but did not go so far as
to assume its annual and diurnal motion. He calculated
the first astronomical tables which served as a model for
future ages, but he did not base them upon the Copernican
hypothesis, though he might have done so without reveal-
ing the truth to the uninitiated, and neither Greek nor
Roman priests could have discovered those points which
they considered heretical. He despaired of explaining the
stationary position or retrograde motion of the planets, and
contented himself with collecting the results of observations
previously made and comparing them with his own, in order
to bequeath them to posterity. His successor Ptolemy is
known as the champion of the principle of the earth’s im-
mobility, and we shall hereafter notice a few of his argu-
ments. Having thus hastily glanced over the opinions of
6 DID THE ANCIENTS KNOW
the Greeks, we will turn our attention to those of the
Romans. We find that Seneca, their greatest philosopher,
speaks Several times in his ‘Quaest. Nat.’ of a motion of the
earth. In the second chapter he says: ‘It seems that the
comets have something in common with the planets;
therefore, if all the stars are worlds, the comets must be
considered as such. But if the stars are mere elementary
fire, which can last for six months without being extin-
guished by the rapid movement, then comets may also
consist of this thin and subtile material without being
endangered by the revolution of the firmament. This
inquiry will also serve to show us whether the earth is at
rest and the firmament moves round it, or the latter is at
rest and the former turns. There were men who main-
tained that the heavens did not move but we did. This sub-
ject is worthy of investigation, in order to learn whether
our habitation is at rest or endowed with rapid motion,
whether God causes us to move, or merely the celestial
bodies around us.”
In the 25th and 26th caps. of the same book, we find:
— ‘We have so little knowledge of nature, that we must
not feel surprised at not understanding the laws of the
comets which only appear occasionally. There are even
now nations which do not know why the moon has phases
and is sometimes eclipsed, nor is it long since we found
out the cause of these things. The time will come when
that which is now hidden will be brought to light by the
industry and intelligence of future generations. A life-
time is not sufficient to analyse the laws which regulate
these phenomena, and are not the few years of our exist-
ence improperly divided between our studies and our
passions? A time will come when our descendants will
THE COPERNICAN SYSTEM P 7
look back with astonishment and wonder at our obtuseness
in understanding things which were so clear and obvious.
Of the five stars which attract our attention from their
continual change of position, we are at a loss to know
which rises in the morning and which in the evening,
whether they are in the direct course, and why they be-
come retrograde. Some wise men say: you are mistaken
in maintaining that a star is stationary or goes back, for
celestial bodies must go forward in a fixed direction.
Their path will always be the same in eternity, the
movements of the universe are immutable; were they
to cease, the world would sink into ruins.”.
Seneca continues, Cap. 27:
‘What is the reason that some of the stars appear to go
backwards? It is because the sun moves towards them
with such rapidity that their motion is relatively slower,
and then their circular orbits are in such position to each
other as to deceive those who look upon them; on the
same principle that a ship in full sail sometimes appears
to be at rest.’
We gather from these passages that the want was felt of
establishing a simpler system of astronomy, and that a
slight conception existed of the manner in which it might
be accomplished; but Seneca, whose genius excelled that
of all his contemporaries, had but an obscure idea of the
earth's true motion, and was even unable to apply the
scanty knowledge he possessed to solve the great problem
of the mechanism of the universe.
Before we decide the question which is the object of this
chapter, it is necessary to define the three essentially dif-
ferent points of the Copernican system.
1. The diurnal motion from west to east.
8 DID THE ANCIENTS KNOW
2. The annual motion around the sun. y
3. The motion of the planets in their direct and retro-
grade courses.
With regard to the first, we have seen that Heraclides
and Ecphantes were aware that by it the phenomena of
day and night could be explained. Aristarchus of Samos
had a notion of the second, and most of the Pythagoreans
placed the sun, or fire (two expressions which they often
confounded), in the centre of the universe, for the absurd
reason we have stated above.
But the third, the most important of all, was completely
unknown to the ancients, they did not possess the requisite
astronomical observations. How trifling a knowledge of
astronomy the Greeks had, and how utterly at a loss they
were to invent a system, we gather from Plutarch’s ‘De
Placitis Philosophorum,” in which propositions are laid
down which prove nothing but the infantine state of science
at that period.
Anaximander says: ‘The figure of the earth is that of a
column, and the sun and moon are great vessels filled with
fire; at the top of them is an opening through which the
fire escapes, but should the aperture be closed an eclipse
takes place.’ Anaximenes believed the heavens to be
solid, and to consist of a kind of fine earthenware, and the
stars to be like golden nails driven through it.
Anaxagoras considered that the solstices were caused
by the air at the poles being so thick that the sun could not
pass through. He also thought that the sun and moon
were rocks cast off by centrifugal force, and ignited in the
upper regions of fire.
Aristotle was of opinion that the spots on the moon
were the reflected images of the seas and lands of the
THE COPERNICAN SYSTEM P 9
earth. Theophrastes believed that the celestial hemi-
spheres were knitted together by means of the Milky Way,
but so carelessly that spaces were left through which the
fiery heavens could be seen. Xenophanes’ doctrine was
that the sun consisted of particles of fire extracted from
the humid vapours of the earth, which were extinguished
every night and renewed every morning, and that the
stars were fiery clouds collected in the higher regions. This
simple notion is expressed by Lucretius, when he asks:
Unde mare, ingenui fontes aeternaque longe
Flumina suppeditant 2 Unde aether sidera pascit?
Summing up impartially the opinions enunciated by
the ‘Savants’ of that age, we think ourselves justified in
coming to the conclusion that they did not know the true
system, notwithstanding the laudable efforts made by
Bailly to prove the contrary; assigning the knowledge the
Greeks had to instruction they received from a highly
civilized but, alas, extinct race called the Atlantides. This
last part of his hypothesis renders refutation unnecessary.
The following example will serve to illustrate the erro-
neous ideas conceived by some interpreters of the ancients
prejudiced in their favour. Pythagoras had, amongst
others, the very remarkable doctrine that the sun and stars,
in their motions, emitted sounds, which varied in intensity
according to their distances from the earth. Dr. Gregory
draws from this a proof that the Pythagoreans were ac-
quainted with the Newtonian law of the decrease of
gravity. He observes that these philosophers spoke alle-
gorically when they asserted that Apollo touched the seven-
stringed lyre, which he supposes to represent the sun
and the seven planets, and to indicate that the former
retained the latter by attractive forces in harmonic
10 THE COPERNICAN SYSTEM.
proportion, and because the tones obtained from chords of
equal thickness are inversely proportioned to the squares
of their lengths, he infers that the harmonic proportion
alluded to is the inverse duplicate of the distances of the
planets from the Sun. For our own part, we are not dis-
posed to believe that the doctrines of Pythagoras, or of any
of his disciples, lay as deep as Dr. Gregory imagined.
CHAPTER I.
THE DISCOVERY of THE EARTH's MOTION.
VERYONE, however limited his thirst for knowledge, }
must be eager to learn whether this orb, which it mº
pleased the Creator to give us as a temporary home, is en-
dowed with motion or not ; and those who have already
decided this question in their minds, must feel gratified
either to find sympathy with their creed, or additional
proofs in support of their opinions. Moreover, the pro-
pagation of the doctrine of the earth’s motion presents one
of the most conspicuous and peculiar passages in the his-
tory of the human intellect ; for it is the first instance
where the thinking world freed itself from deep-rooted
illusions in which the greatest philosophers were spell-
bound — it is the first grand example of proud scholars
renouncing long-established doctrines, and accepting —
nay, teaching — precisely the contrary of what all, laity
and priesthood, had believed and preached through so many
centuries. The system of Copernicus has not only laid a
new foundation for astronomy, but it has emboldened in-
vestigators to doubt any dogmas in natural science, it
having been proved that man lived 5,000 years in blind
and obstinate error, and given to the world the practical
lesson first followed by our great countrymen, Bacon and
Newton, only to take that for granted which is the result
12 THE DISCOVERY OF
of proper deduction from a sufficient number of facts and
observations.
The system of the celestial motions that prevailed before
Copernicus was both complicated and erroneous. Pytha-
goras may be said to have been the first who entertained
the idea that the earth moved and that the sun was fixed,
but unfortunately the arguments brought forward in sup-
port of this doctrine by his school of philosophy were not
sufficiently strong to withstand the attacks made upon
them by Aristotle, who did not maintain that the stars and
planets moved individually round the earth, but that they
were all fastened to the concave surface of a crystal sphere,
which rotated, carrying them with it in its course. Wild
and fanciful as this theory must appear to us, it neverthe-
less was universally adopted and developed by successive
generations of astronomers, up to the time of Purbach, in
the middle of the fifteenth century.
At this period the earth was supposed to be immovably
fixed in the centre of the universe; immediately surround-
ing it were the atmospheres of air and fire, and beyond
these the sun, moon, and planets were supposed to travel
round the earth, each fixed to a separate orb, or heaven,
of Solid but transparent matter. The stars were considered
fixed in an outer orb, beyond which were two crystallime
spheres; and on the outside of all the ‘primum mobile,”
which sphere was supposed to revolve round the earth in
twenty-four hours, and by its friction, exercised on the in-
terior orbs, to carry them round with a similar motion:
hence the diversity of day and night. But, besides this
principal and general motion, each orb had one of its own,
which was intended to account for the apparent changes
of position of the planets. This supposition, however,
THE EARTH's MOTION. 13
proving insufficient to account for all the irregularities,
other hypotheses were introduced :—
Firstly. That to each planet belonged several concentric
spheres.
Secondly. That the centres of these revolving spheres
were placed on the circumference of a secondary revolving
sphere, the centre of the latter being the earth.
Thus originated the names of Eccentrics and Epicycles.
The whole art of astronomers was now directed towards
inventing and combining different eccentric and epicyclical
motions, so as to represent the phenomena of the heavens.
The name of Aristotle lent its powerful assistance to
enable the false system to prevail against the true One,
which was partially conceived by some philosophers before
his time, as is gathered from his own writings.
Accordingly the heavens rapidly became, under this
system, to use the words of Milton —
With centric and eccentric scribbled o'er,
Cycle and epicycle, orb in orb.
The bigotry of the middle ages accepted and protected
willingly these heathen notions, rendering it most dam-
gerous to confute them. Copernicus was well aware of
this, and prudently delayed the publication of his immortal
work, ‘De Revolutionibus Orbium Coelestium,’ which ap-
peared in 1543, the year of his death. His wish seems to
have been to refute as gently as possible the opinions of
the followers of Aristotle, who exercised an intellectual
monopoly which he justly held to be absurd. It seems
never to have occurred to him that an outcry might be
raised against his system on religious grounds, for if so he
would not have dedicated his book to Pope Paul III. or
14 & THE DISCOVERY OF
mentioned in the preface that he was induced to publish
it by the persuasion of his friends the Cardinal of Capua
and the Bishop of Culm. Apparently the prelates did not
think that a case of impiety could be made out of an astro-
nomical theory. But the Aristotelians were determined to
punish as heresy that which they could not overturn by
scientific reasons, and the clamour of irreligion which they
set up deterred many astronomers from avowing openly
his doctrines, though they were fully convinced of their
Sublime and simple truth.
In a letter from Galileo to Kepler, written from Padua
in 1597, we find the following passage:—
* Congratulating you on your elegant discoveries, I
shall only add a promise to peruse your book dispas-
sionately; this I shall do the more willingly, because
many years ago I became a convert to the opinions of
Copernicus. I have arranged many arguments and con-
futations of the opposite opinions, which, however, I have
not yet dared to publish, fearing the fate of our master
Copernicus, who, although he has earned immortal fame
among a few, yet by an infinite number (for so only can
the number of fools be measured) is exploded and derided.
If there were many such as you, I would venture to pub-
lish my speculations; but, since that is not so, I shall take
time to consider it.”
This time for consideration lasted till about 1612, when
the all-powerful love of truth vanquished his fears; and
having observed some solar spots, he printed that dis-
covery, and ventured in the same pamphlet to assert the
justness of the Copernican system, adding some new argu-
ments to confirm it. For this he was cited before the
Inquisition, condemned to prison, but released after some
THE EARTH'S MOTION. 15
months, on the promise that he would renounce his here-
tical opinions. However, in 1632, he published his
“Dialogues of the Two Systems of the World, Ptolemaic
and Copernican.” He was again cited before the Inqui-
sition, and sent to prison by that abominable court at
Rome. In June the same year the Congregation con-
vened and pronounced sentence against him and his books,
obliging him to abjure what they termed his errors.
The following is the form of abjuration taken from
Delambres’ ‘Histoire de l’Astronomie : ” —
“I Galileo Galilei, son of the late Vincent Galileo of
Florence, seventy years of age, being of sound mind and
on my knees before you, most eminent and reverend
Cardinals of the Universal Christian Republic, Inquisitors
General of Heretic Malice, having before my eyes the
Saints, and the Holy Testament in my hand, do swear that
I always have believed, that I do now believe, and, with
the grace of God, always shall believe, everything that is
taught by the Holy Roman Catholic Church; but because
this holy institution justly ordered me to abandon entirely
the false opinion that the sun is the centre of the world
and immovable, that the earth is not the centre and
that it moves, and because I could neither support it nor
defend it, nor teach it in any manner whatsoever, either by
word of mouth or in writing, and after the said doctrine
had been declared contrary to Holy Writ, I wrote and
caused to be printed a book in which I treat of the said
condemned doctrine, and give powerful reasons in favour
of the said doctrine without bringing it to any result,
it is for this that I have been strongly suspected of heresy,
for asserting and believing that the sun was the centre
of the world and immovable, and that the earth was not
16 THE DISCOVERY OF
the centre and did move. It is for this reason, that
wishing sincerely and from my heart to obliterate from
the minds of your Eminences and all good Catholics the
strong suspicion justly conceived against me, I do abjure,
curse, and detest the errors and heresies aforesaid, and
generally any other error or sect whatsoever contrary to
the Holy Church aforesaid, and I do swear that in future
neither by word of mouth or in writing will I state or
affirm anything which could raise similar suspicions
against me; and if I knew ºf any heretic or any one sus-
pected of heresy, I would denounce him to the Holy Office,
or to the Inquisitor, or to the priest of the place in which
I might be. I swear, moreover, and promise that I will
strictly fulfil and observe all penances which are or may
be enjoined me by the Holy Office; if I should act con-
trary to any of these my words, promises, protestations,
and oaths, which God forbid, I will submit to all pains and
penalties which by the holy canons and other constitu-
tions, general as well as special, have been instituted and
used against such delinquents. So help me God and the
Holy Apostles.
* At the Convent de Minerve, on the 22nd of June, 1633.
‘(Signed) GALILEO GALILEI.’
Thus did a few contemptible bigots exercise their tyran-
nical authority over this venerable man (he was then
seventy years of age), making him abjure doctrines
which he knew to be founded on sound reason. Happily,
however, for science, this infamous court has long been
disarmed of its terrors, the Copernican system has pre-
vailed, and the philosopher can now investigate the works
of Nature, and benefit mankind by the expression of his
THE EARTH'S MOTION. 17
Opinions, unfettered alike by the cruel intolerance of
Jesuitical tribunals like this, and the calumniating fana-
ticism of a bigoted priesthood. Since the time when the
Reformation dispelled the clouds of Superstition, science
has advanced without requiring martyrs like Galileo to
maintain that the earth rotates.
In the year 1737, a century after this monstrous sen-
tence had been carried into effect, a more civilized gene-
ration rendered that justice to Galileo, dead, which their
forefathers had refused him living, by erecting in the
most conspicuous part of the church of Santa Croce a
monument to perpetuate the memory of that great and
persecuted philosopher. The erroneous notions of the an-
cients, so obstinately supported by the absurd superstitions
of the middle ages, have vanished before the light of sound
philosophy; but not without a struggle, for many direct
arguments were brought to bear against the Copernican
system.
One of the most plausible objections to the earth's mo-
tion seems to be that raised on the ground that we are
unconscious of that motion. We can, however, easily de-
monstrate, by observation and experiment, that we may
actually be moving without being aware of it, by attribut-
ing the change of the relative position of surrounding
objects to a motion in themselves which they do not
possess. For instance, a person travelling along a rough
road is only reminded of the motion of the carriage by
the roughness of the road, whereas in a vessel on a smooth
sea he is led to believe, not that he is progressing, but
that fixed objects on land are receding from him —
Provehimur portu terræque urbesque recedunt.
C
18 THE DISCOVERY OF
Even motion produced by our own exertions, such as in
walking, is sometimes delusive. Strange as this may ap-
pear, the following experiment will suffice to illustrate it.
Let a person walk close to railings, nearly his own height,
so that their outline may be about on a level with his eye,
then, as he moves along, he will see them continually
dancing up and down, and the impression on his senses is
that the railings possess in themselves a vertical motion.
A little reflection will teach him that, by his mode of step-
ping, the eye is at different elevations at different periods,
causing the apparent variation of level in the outline of
the railings. It is easy to multiply such experiments, the
results of which will be similar. In fact, we may come to
the conclusion, that we are insensible of motion unless
made aware of it by mechanical obstructions or muscular
exertions.
It is obvious that when we treat of the question of the
earth's motion, no evidence can be deduced from our own
personal movements. Nor is it possible to show that
mechanical impediments exist in the perfectly smooth
frictionless path of our planet. Therefore we are led to
|believe that, whatever be the motion of the earth, we must
be unconscious of it. Many other objections have been
raised against the Copernican hypothesis, which at one
time possessed weight and importance in proportion to the
celebrity of their propounders. Riccioli collected seventy-
seven arguments against the earth’s motion, of which he
himself refutes forty-nine as favourable to the Copernican
system, and of all the remaining only those deserve any
consideration which arefounded on the argument of Ptolemy
(‘Almagest, I. 7), viz., that the birds in the air would see
the earth moving under them, and therefore never leave
THE EARTH's MOTION. 19
its surface, for fear of losing their nest and young; or, as
Buchanan expresses it:-
Ipsæ etiam volucres tranantes aera leni
Remigio alarum ; celeri vertigine terrae
Abreptas gemerent sylvas, nidosque tenella
Cum Sobole, et cara forsan cum conjuge; nec se
Auderet zephiro solus committere turtur.—Sphaera, i.
Tycho Brahe’s principal objection is similar, though
brought forward in another form. In a letter to Rothmann,
he asked him, ‘how it was possible that a ball dropped
from the summit of a tower should always fall close to the
foot of it, for if the earth completed an entire revolution
in twenty-four hours, the tower must have advanced to-
wards the east and moved a considerable distance from the
ball in proportion to the time of descent of the latter.” Thus
assuming the rate of the earth’s motion at the equator to
be eighteen miles per minute, as it is nearly, and the height
of a tower to be 400 feet, which a body would take about
five seconds to fall through, then the falling body would
reach the ground 1% miles westward from the foot of the
tower, which is certainly contrary to all observation.
The fallacy of this reasoning is obvious, and it is aston-
ishing that men of science should have fallen into an error
like this, which could only have arisen from ignorance of
the first principles of mechanics. It is a known fact, that
when a stone drops from the mast of a vessel in motion,
it falls at the foot of it, just as if the ship were at rest —
and why? Because the motion of the ship was communi-
cated to the stone before falling; and thus (supposing it to
be possible) though birds might fly even higher than the
atmosphere, they have received before that the impulse of
the earth’s motion, its direction and velocity, and must
therefore continue to rotate with it.
C 2
20 THE DISCOVERY OF
The actual motion of a ball, while falling from any
height, is compounded of two motions, one in the direction
of the earth's rotation, the other in that of gravity; the
path described, therefore, is a curved line. But as the
change in the relative position only of the falling body to
the earth is produced by the action of gravity, this result
is the only one that can be observed; and the apparent
motion is the perpendicular.
Another objection was raised, which at the present time
appears childish, yet it is worthy of mention, inasmuch as
it hindered many from becoming converts to the true
opinions of Copernicus. The diurnal motion of the earth
was imagined to be an absolute impossibility, for, reasoned
the astronomers of former ages, ‘if it were not so, after a
lapse of twelve hours we should find ourselves head down-
wards’—a contingency which they contemplated with feel-
ings of the deepest awe. We may, however, excuse their
limited notions, for the voyages of discovery had not then
informed them of the existence of antipodes, and that
there is no difficulty in conceiving that we can rest in the
same position as the New Zealanders. In the letter of
Tycho Brahe, before alluded to, we find another objection
to the earth's rotation founded upon the parallelism of its
axis. ‘How can it be imagined, he writes, ‘ that one
and the same body should have two motions so different,
one which translates its centre of gravity, the other which
changes the position of its axis?’ The answer is, that
the parallelism of the axis does not require a specific
motion, as he supposed, for it is a fixed position.
It is sufficient that the axis has been once directed to a
certain point of the heavens in order to remain in that
direction, though the earth has an annual motion. There
THE EARTH'S MOTION. 21
exists no physical or mathematical reason whence we
might conclude that the axis of diurnal motion must be
directed perpendicularly to the annual orbit, for between
these two movements there is neither connection nor
dependence. As soon as the particles composing a globe
are projected simultaneously in the same path, they will
acquire velocities and directions equal and parallel to
each other, and hence do not undergo any change in their
relative positions. Let us apply this reasoning to our
earth; moving on a fixed axis, and hurled into space, all
particles of it received the same impulse, and preserved
their original rotation. This may be illustrated by a top
spinning on a table; whatever motion is imparted to the
latter, the top will continue to revolve on the same axis.
So a ball projected from a cannon always preserves its
axis of revolution. Having disposed of the objections
which were raised against the Copernican doctrine on
mechanical grounds, let us consider those which were
founded on the testimony of Holy Writ, and which
the fanaticism of the middle ages rendered perilous to
refute.
The principal text for controversy was taken from the
Book of Joshua, x. 13: “And the sun stood still, and
the moon stayed, until the people had avenged themselves
upon their enemies. Is not this written in the book of
Jasher ? So the sun stood still in the midst of heaven,
and hasted not to go down about a whole day.” Reading
this verse, together with the one preceding it, in which it
is stated that Joshua said before Israel, “Sun, stand thou
still,’ &c., we find that he could not have expressed
himself differently without making himself unintelligible
to those he addressed.
22 THE DISCOVERY OF
Astronomers of the present day say “the sun rises,”
* the sun sets,’ without being reproached with misunder-
standing the true state of nature. Joshua, conversing
with his nation, could not speak otherwise than in their
own language. It is absurd to maintain that a general—
for such was Joshua — could give lessons of astronomy to
his army, at the very moment when he was to show them
the power and glory of the Supreme Being manifested in
their favour; for if he had commanded the earth to stand
still, he would have been obliged to explain the reason of
his strange phraseology, so contrary to all usage. There-
fore, if Joshua as a prophet knew that which in his time,
and especially in his country, was unknown, he could not
have expressed himself otherwise than he did.
Some other passages occur, such as in Isaiah xxxviii. 8,
Ecclesiastes i. 5, which may appear contrary to the
true hypothesis of the earth’s motion, if taken in their
literal sense and not in that in which they should be
understood, namely, of ordinary parlance, or of poetical
metaphor.
The number of texts which touch generally on the
subjects of astronomy and natural philosophy is very large,
but for the most part they are not scientific, but popular,
suited to the common apprehension of mankind; for in-
stance, David says, Psalm xviii., ‘Tellus fundata super
maria.” Solomon asserts, “Terra in aeternum stat;’ lastly,
in Kings xxiii. 7, we find it stated that a bath was con-
structed in a circular form, ten cubits in diameter, but
measuring only thirty cubits in circumference, whereas
mathematically, the latter must have been more than
thirty-one cubits.
The deduction which we must draw from all these quo-
THE EARTH'S MOTION. 23
tations is, that they will not bear scientific interpretation,
as they were not meant to convey any scientific instruction;
as when we read that the sun moved, it is obvious to us
that the inspired writers did not aim at deciding a question
Of philosophy, nor did they intend to establish or ordain
any opinion about it. These passages are either inferior
in importance, or are not matters of faith, but are merely
remarks accompanying an historical narration. After due
and impartial consideration of the most celebrated ob-
jections brought during ages to bear against the theory of
the earth's rotation, we must arrive at the conclusion that
none of them, either moral or physical, are sufficiently
cogent to impress our minds with the faintest idea that
they are founded on reason or tenable in their conse-
quences.
24 THE PROBABILITIES OF
CHAPTER II.
THE PROBABILITIES OF THE EARTH'S MOTION.
HE earth being a globular mass floating freely in space,
without pedestal or material suspension to keep it
in its place, it follows that if once this globe received an
impulse, it would move for ever in a straight line, since
there is no friction and no resisting medium.
Now, if we grant the possibility of a power existing
capable of communicating this impulse, we can easily
imagine that this globe should be in motion.
But we may carry our speculation on the probabilities
of this case much further.
FIG. I.

THE EARTH's MOTION. 25
It is a principle of mechanics, that if motion be com-
municated to a mass in any other direction than through
its centre of gravity, it will acquire two motions, one of
translation, of which all the particles partake equally, the
other of rotation, which affects only those which lie with-
out the axis. It is a remarkable fact that these two mo–
tions are independent of each other, and if the different
impulses of a central force were to act on a globe possessing
these two motions, the effect would be only to change its
rectilinear path of translation into a curvilinear orbit
without disturbing the axis and velocity of rotation.
The mathematical investigations of these mechani-
cal truths may be effected in the following elementary
Iſha,010.6.1 : —
Let E B L be a great circle of a homogeneous sphere,
placed ‘in vacuo, and suppose the latter to receive an
impulse represented by the line A B in magnitude and
direction. It is plain that if A B were perpendicular to
the surface of the globe, and consequently passing through
its centre, the action of the force would make the globe
move uniformly in that perpendicular; but in the case
under consideration we assume the force to act obliquely,
whence it follows that the force A B must be resolved into
two, the one, Bo, perpendicular to the surface of the
sphere, the other, B H, parallel, i. e. tangential to it,
so that the straight lines Bo, B H become sides of a
parallelogram H B O I, the diagonal of which is B I = A B.
Now Bo expresses the component of the force A B which
impels the globe uniformly in the direction B C B, within
the plane of the great circle B E L, termed the equator,
whilst B H expresses the component of the force A B
which gives to the point B at the surface of the globe
26. THE PROBABILITIES OF
a tendency to recede from the centre C along the tangent
B H. Hence there is a force pushing the point B uniformly
~towards H, but at the same time this point B will, on
account of the reaction of cohesion, return the same dis-
tance towards the centre C as the component B H made
it depart from it. The result necessarily is that the point
B must move in a circle uniformly about the point C, or,
which is the same, about the axis passing through C. The
oblique force therefore imparts to the globe two uniform
motions, one which causes it to turn on its axis perpendi-
cularly to the plane of the equator, termed ‘rotation,”
the other which causes it to advance in space in the direc-
tion of the plane of the equator, called ‘translation.’
If the globe receive another impulse represented by the
line D E in the plane of the same equator, this new force
must also be resolved into two, the one, E P, pushing the
globe in the direction E G with a velocity expressed by
EP, the other, EF, tending to make the globe turn uni-
formly on the axis of that equator. Now making CP, -
EP and C B, = op, then the two uniform forces of trans-
lation which act upon the globe are expressed by CB, and
C P. Therefore the latter will move through the diagonal
CM, which is in the plane of the equator, turning at the
same time on the axis of it with a velocity expressed by
E F + B H, since these forces together tend to the same
effect and hold the same position relatively to the globe's
surface. It is also obvious that the different degrees of
obliquity of the impulse will produce different ratios in
the velocity of rotation to that of translation; but any cem-
tral force acting on a globe which is already endowed
with the two motions of rotation and translation, will only
affect the latter, as may be shown in the following way.
THE EARTH'S MOTION. 27
If a globe, the section of which is A P B Q, while re-
volving about the axis P Q, be impelled by a force in any
direction cs, passing through its centre, and, conse-
quently, perpendicular to its surface, it is evident that no
part of this force can act tangentially to it, but its whole
effect on the two hemispheres, A P B and A Q B, on which it
acts equally, is only to direct them towards the point S
FIG. 2.

s
B
or away from it, according as the force is one of attraction
or repulsion. Therefore this force must necessarily ac-
celerate or retard the motion of translation, according as it
acts with it in the same or opposite direction. Should, how-
ever, the line of action of the force at S include an angle
with the rectilinear path of translation, then the direction
of the latter will continually change, as is obvious from
the theory of the composition of motions, and the globe
in this case will describe a curvilinear orbit without mo-
difying the motion of rotation.
Hence we can understand how the annual motion of
our earth would not interfere with the diurnal motion,
assuming that there be one. Let us now proceed to weigh
28 THE PROBABILITIES OF
the probabilities of the truth of this assumption. For the
sake of illustration, let us imagine the force of gravity to
be destroyed, and let us then suppose that a person holding
a ball should open his hand and set it free. The result
would be that it would commence spinning about one of its
diameters, at the same time advancing uniformly through
space. If the ball had received no impulse when the hand
was opened, there being no force of gravity to attract it, it
would have remained stationary in space. But we know
from experience that it is impossible to open the hand
without communicating some impulse, however slight, to
the body grasped, and the chances against the direction
of this impulse passing through the centre are infinite.
We may therefore conclude, with a degree of probability
almost amounting to certainty, that the result will be two
motions — translation and rotation. Let us now apply
this illustration to our earth. No one can deny that the
Almighty hand which created it could have left it qui-
escent in space; but that it moves is by far the more ra-
tional hypothesis. For after the Creator had covered the
face of this earth with animal and vegetable life, the
alternations of day and night and the variations of the
seasons became absolutely necessary for the preservation
of that life, and since all these changes could be effected
by a compound motion of rotation and translation result-
ing from an original impulse, and as all objections to .
such motion were shown in the preceding chapter to be
groundless, it is on the whole more probable that it
moves than that it is at rest. This probability is fur-
ther increased by analogy. We have ocular proof of the
rotation of other celestial bodies, by observing through
telescopes the movement of spots on their surface. The
THE EARTH's MOTION. 29
moon turns upon its axis in about 27# days, the Sun in
25+ days, Mercury, Venus, and Mars in nearly a day,
Jupiter in about 94% hours, and finally, Saturn in about
104 hours.” Moreover, all the revolutions of the planets
about their axes take place in the same direction, from
west to east, and our earth must rotate in a similar direc-
tion in order to produce the phenomena of day and night.
Having shown the last hypothesis to be highly probable,
let us now consider the pros and cons of the other.
Either the globe revolves upon an axis every twenty-
four hours, or the whole universe, including the sun, moon,
planets, comets, and fixed stars, accomplishes a revolution
around the earth in the same time: no third opinion upon
the subject can possibly be held. In the latter case, it is evi-
dent, from the distance of the celestial bodies, that their
diurnal revolution round our planet must involve a rate
of motion utterly inconceivable. The sun must travel at
the rate of 400,000 miles a minute, the nearer stars with
a velocity of a thousand million miles in a second, and the
more distant with a rapidity which no words can express.
It is, indeed, absurd to suppose this, when the end
which was to be gained required only our little globe to
revolve upon itself.
Is it probable that the planets, endowed with their own
motion, the comets, which seem to have no resemblance
with other celestial bodies, the millions of fixed stars dis-
persed throughout the firmament; all these, I say, which,
have no common relations with each other, which are
placed at distances our imagination has difficulty to
* To the rapid rotation of these planets is attributable the great flat-
tening at their poles, which may be distinctly seen through the telescope,
amounting, in Jupiter, to # of its radius, and in Saturn to #.
30 THE PROBABILITIES of
conceive —- is it probable that these should unite in one
single mass, and perform a daily revolution round the
atom which in the infinity of space represents our earth P
No | The regularity and sameness in the motion of num-
berless bodies so different from each other in all other
respects, indicate that these motions are not real; and the
more we reflect upon it, the stronger becomes our con-
viction that the earth rotates.
This uniformity of motion, which appears to us so in-
comprehensible, was explained by the ancients by the as-
sumption that the stars were fixed to hollow spheres made
of a crystalline substance, and moving together one within
the other; but since we have observed that the planets
vary their distances from the earth, that at one period they
advance towards it, while at another they recede from it,
and that the comets sometimes come close to us and then
disappear altogether, we may conclude that the hypothesis
of a solid heaven is an absurdity. If, however, the heavens
consist of millions of detached bodies, then the uniformity
of their movements can be attributed to no physical cause,
and the laws of motion which nature observes in all other
things must in this case be checked by direct spiritual in-
fluence. We cannot hold with Riccioli, who, in his “ Al-
magest Nov.” (II. 248), states that the celestial bodies are
carried round daily by intelligences called angels, spe-
cially employed in that behalf. Rejecting such mystical
powers as incompatible with the dignity of true science,
the following mechanical considerations present themselves.
If the diurnal movement which we see be attributable to
a motion of the stars, them all of them, except those lying
in the equinoctial, will rotate in circles, the centres of
which are situated without the earth—in fact, in an ima-
THE EARTH'S MOTION. 31
ginary line termed the “Celestial Axis;" and the same may
be said with regard to the planets, since their orbits inter-
sect the equinoctial in two points only. But whenever a
body describes a circle with a uniform velocity, it is at-
tracted towards the centre of its orbit by a constant force,
the magnitude of which depends jointly upon its rate of
motion and the radius of the circle it describes. The theo-
rem relating to central forces was demonstrated by Huy-
ghens, and is as follows, viz., that the centrifugal force
which is equal and opposite to the centripetal, is directly
proportional to the square of the velocity, and inversely
proportional to the radius, or, in other words, it is in-
versely proportional to the square of the periodic time,
and directly proportional to the radius. Since the periodic
times of apparent rotation are the same for all the stars,
namely, twenty-four hours, it follows that the magnitude of
the force towards which each star is attracted will merely
depend on the radius of the circle of declination which the
latter describes. The central forces must therefore con-
tinually decrease, the nearer the stars upon which they
act are to the poles of the heavens, and the locus of all
these different forces extending from both sides of our
earth to infinity is nothing but an imaginary line without
physical dependence. It is against our notions of me-
chanics to suppose that a multitude of moving forces should
have their seat and origin in as many mathematical points,
and yet, if we admit the immobility of the earth, we must
accept this hypothesis. From these and the preceding
considerations, it follows that an actual diurnal rotation of
the firmament is highly improbable, and this improbability,
greatin itself, is infinitely strengthened by the probability
of the other hypothesis, that the earth rotates. But
32 THE EARTH'S MOTION.
conclusive as may be this balance of probabilities, the
question admits of more rigid investigation, for there exist
direct, positive, and even experimental demonstrations of
the earth's rotation about its axis, the elucidation of which
will form the subject of the following chapters.
33
CHAPTER III.
PROOF OF THE EARTH's ROTATION BY THE TRANSMISSION OF
LIGHT.
E have stated in the last chapter that the apparent
phenomena of the heavens must be the same whether
the latter complete an entire revolution from east to west
in twenty-four hours, the earth remaining fixed, or whether
the earth itself performs an entire revolution in the same
time. This proposition is true, if the transmission of light
be instantaneous, however great the distance through
which it has to pass. But if the velocity of light be not
infinite, that is to say, if the rays take some perceptible
time to travel from the stars to us, then this proposition
does not hold good, as Arago has proved by the following
ingenious method of reasoning. Let us suppose our earth
to be fixed, and a star to move round it from east to west.
The star is always the centre of diverging rays, but the
position of this centre relatively to the horizon and the
meridian of a given place will vary continually. The rays
emanating from the star move in straight lines, and the
star itself will therefore become visible on the horizon by
those rays of light which were emitted when it actually
was in the horizon; in the same manner it will appear in
the meridian, by rays lying exactly within the plane of the
D
34 PROOF OF THE EARTH's ROTATION
meridian, and this will occur at the moment of its real
culmination. We may deduce from this, that if the earth
does not rotate, but the heavens do, a star. can only be seen
in the horizon or meridian by means of those rays which
it emitted when actually in the prolongation of either of
those planes. Let us now suppose the velocity of light to
be such, that for a given star it requires six hours to reach
us, then its culmination would be apparent to us six hours
after its actual transit; therefore, if a star were at such a
distance that its light required twelve hours to reach us, it
would appear to rise when in reality it was setting; and if
its distance were still greater, it would be seen in the east
long after it had passed below the horizon. Following up
this reasoning, and assuming the distance of two stars from
our earth to be different and proportional to their real po-
sitions, then, though to us they may appear to be side by
side, yet in reality they may be separated by distances
scarcely expressible, even by taking the diameter of our
orbit as the unit of measure.
Before examining whether these results derived from
the double assumption of the fixity of the earth and the
successive propagation of light are reconcileable with the
facts observed, let us consider the case in which the earth
moves and the firmament is fixed. The centres whence the
light proceeds in straight lines are fixed in space, therefore
any of these luminous points will appear to rise when the
horizon in its motion of rotation from west to east coincides
with any of the descending rays; and similarly a star will
pass the meridian when the prolongation of this plane
coincides with the invariable position towards which all the
rays, which render it visible to us, converge. With regard
to the rise of a star and its culmination, it matters not
BY THE TRANSMISSION OF LIGHT. 35
whether the luminous particles, by which these phenomena
are produced, quitted the star days, years, or even centuries
before they were observed, since they travel in straight
lines terminating in fixed points of the firmament in which
the stars are situated. Hence the velocity of light does
not, in the latter case, exercise any influence on their ap-
parent positions. When a star appears to pass the meri-
dian, it really does so, and when two stars seem to be near
each other, the lines drawn to their centres from the earth
are really in close proximity.
The deductions arising from the hypothesis of the fixity
of the earth must have appeared strange even to those who
made them, but in scientific as in other researches, the
singularity of a proposition is no proof of its fallacy. Let
us now consider if we cannot trace some facts in connection
with the motion of the stars, which are irreconcileable
with the deduction that their apparent position must de-
pend upon their distance from the earth; for instance, the
apparent time of culmination of Mars, when in ‘conjunction,”
is equal to the time of its real transit plus the time which
light takes to pass from it to the earth, and the same is
true when the planet is in ‘opposition.” But the distance
of Mars from the earth at the time of conjunction is less
than that at the time of its opposition by double the dis-
tance of the sun from the earth. Therefore, comparing
the observed with the real transit, we should have an in-
equality between the ‘opposition' and the “conjunction'
amounting to about 16.5 minutes of time, that is, double
the time required for light to travel from the sun to the
earth. It is further evident that from the same reason
the apparent motion of the planet between its conjunction
and opposition must take place from east to west, but the
D 2
36 PROOF OF THE EARTH'S ROTATION
existence of such inequalities is in no way indicated by
observation; similar perturbations must take place with
Jupiter and Saturn, and, with regard to the double stars,
it may easily be shown that the hypothesis of the earth’s
immobility would lead to results still more inadmissible.
When the principal star and its satellite are both at the
same distance from the earth, they will appear very close
together (which they in reality are), but if the satellite in
the course of its revolution round the principal star takes
a position by which its distance from the earth is increased
by a quantity equal to the diameter of the terrestrial orbit,
then, for seeing it almost in contact with the centre of its
motion, it ought to appear distant in right ascension by a
quantity which, expressed in time, would amount to more
than eight minutes, a result so discordant with all observed
facts, that we may consider it as a mathematical demon-
stration of the falsity of the hypothesis from which we
started, namely, the immobility of the earth. -
An objection might be raised to this mode of reasoning,
namely, that the determination of the velocity of light
presupposes a knowledge of the true system of the world,
and it would have been a very important one in the days
of Römer, when the measure of that velocity was solely
founded upon the observations of the eclipses of Jupiter's
satellites. But now that recent researches have shown us
how to ascertain the velocity of light on our own globe,
this new method, first suggested by Arago, seems to rest on
a firm basis. &
This great advocate of the undulatory theory consi-
dered the retardation of light in dense media a crucial proof
between the rival hypothesis of Newton and Huyghens.
He, therefore, as early as 1838, recommended the appli-
BY THE TRANSMISSION OF LIGHT. 37
cation of Mr. Wheatstone’s admirable invention of the
revolving mirror as a means of measuring extremely
Small intervals of time, in order to compare the velocity
of light in air with that in a corresponding length of
Water.
Professor Wheatstone contrived and used the apparatus
alluded to in the year 1834, for measuring the velocity of
electrical conduction. The principle upon which it was
constructed may be understood from the following de-
Scription. Let a copper wire, half a mile long, be so con-
voluted that the middle and the two ends may be brought
near together, the whole being perfectly insulated. Let
the wire be interrupted at these three places, which are
so arranged as to lie in one line, and then be placed into
connection with an electric machine or battery. The
contact will effect three sparks, which, taking place close
to each other, can easily be seen at Once reflected in a
small plane mirror. Let now the latter be put in very
rapid rotation round a horizontal axis, so placed that the
sparks, if possible, may be reflected together to the eye of
the observer. In Mr. Wheatstone's apparatus the velocity
reached 800 times in a second, consequently the mirror
described one degree in a gººd = ###555th part of a
second. But for one degree of rotation of the mirror the
reflected image will describe an arc of two degrees. If
then the three sparks occur at the same instant of time,
they will be seen in one line, but if either spark occur
later than the others by only rºwth of a second, the
mirror will have revolved so much in this interval as to
displace the image of that spark relatively to the others,
by the very palpable angular amount of two degrees.
In the copper wire half a mile long the end-sparks took
38 PROOF OF THE EARTH's ROTATION
place simultaneously, while the middle spark occurred later
by about one-millionth part of a second, giving a velocity
of transmission somewhat greater than that of light.
It will be understood, from the account of this method,
that the retardation of light is shown by the displace-
ment of the image of an object seen through water re-
latively to the image of the same object seen through
air. Suppose the mirror to be turned from left to right,
then, if light moves faster in water, as Newton thought,
the water-image will be to the right, but if slower, accord-
ing to Huyghen’s theory, it will be to the left. The amount
of angular deviation to be appreciated may be increased
with the same speed of rotation, by applying two or more
rotating mirrors. Regard, of course, must be taken to
the loss of light by reflection and refraction. According
to the experiments of Bouguer, light passing through ten
feet of water preserves only #ths of its original intensity.
When, therefore, Arago proposed the application of
Mr. Wheatstone's revolving apparatus for a purpose
different from that for which the inventor used it, he
endeavoured first to prove the possibility of making
sensible the slight differences of angular deviation, with-
out surpassing the limits of the velocity of rotation and
those of the length of the tube of water. His calcula-
tions on the subject are contained in a memoir presented
by him to the Académie des Sciences, on December 3rd,
1838. The results of these show that one mirror, re-
volving a thousand times per second, requires a tube of
water ninety-two feet long, in order to cause a deviation
of one minute; but this length of the tube may be easily
reduced to its half, third, or fourth, &c., by having
recourse respectively to two, three, or four mirrors, &c.,
which must be done in all those cases where the image
BY THE TRANSMISSION OF LIGHT. 39
appears too faint. The practicability of the plan was
therefore sufficiently proved by its projector; but per-
haps, owing to his failing sight, he never realised his own
idea. It was adopted by M. Foucault, who succeeded, in
1850, in demonstrating the retardation of light in a tube
of water only 64 feet long, and with a velocity of rotation
not exceeding 200 turns. The latter, however, was increased
for determining the absolute rate of its transmission.
It will not be out of place here to explain M. Foucault's
combination of fixed and revolving mirrors, by means of
which he was enabled to ascertain the prodigious rapidity
with which light travels. The construction of the essential
portions of M. Foucault's apparatus will be understood
by the annexed figure (3). The camera obscura, C, has a
FIG, 3.
TOUCAULT's ARRANGEMENT FOR MEASURING THE VELOCITY OF LIGHT.
Small square opening, behind which is suspended a fine
platinum wire P; a pencil of solar rays, reflected from a
mirror placed outside, passes through the Square opening
into the camera, and, meeting the platinum wire, is thrown

40 PROOF OF THE EARTH's ROTATION
on to an achromatic lens L, placed at a distance from the
wire less than double its focal distance; then the image of
the wire, more or less magnified, will tend to form itself
on the axis of the lens, but the pencil of rays, after
having passed through the glass, meets a plane mirror M,
revolving with great velocity, from which it is reflected,
and forms an image of the wire in space which moves with an
angular velocity equal to double that of the mirror. This
is obvious from the following geometrical consideration.
Let M M (fig. 4) be the revolving mirror, P a fixed object
forming its image in I, P I being perpendicular to M M, them
FIG. 4.

M?
I'
the mirror having come into the position M'M', the image will
be formed at I', P1’ being perpendicular to M'M'. Now Z. MoM'
= Z. I PI', because the sides of the one are perpendicular
to the sides of the other; hence, by Euclid, Lib. 3, Prop. 20.,
the are I I' is double the arc MM', that is, the image travels
with an angular velocity double that of the mirror. (Q.E.D.)
The image is again reflected in space (see fig. 3) by M",
a fixed concave mirror, of which the centre of curvature
lies in the axis of rotation of the revolving mirror, and
BY THE TRANSMISSION OF LIGHT. 41
coincides with its centre. The pencil of rays reflected
from the mirror M' turns back on itself, is again re-
flected from M, passes the lens a second time, and forms
an image of the wire upon the wire itself as long as the
mirror M turns slowly. In order to observe this image
without obscuring the pencil of rays which passes on the
camera at C, a plane mirror M" is placed between the pla–
timum and the lens at such inclination, that the reflected
rays fall upon a powerful eye-piece L’. Now, if the mirror
M is at rest, or is only turned slowly, the returning ray
M. M' will meet the mirror M in the same place as during
the first reflection; it will therefore take its original direc-
tion, meeting the mirror M" in a, whence it is partially
reflected, and forms in c, at a distance a c = a P, the image
which the eye observes at L’. The image appears at every
revolution of M, and when its velocity of rotation is uni-
form, remains fixed in space. For velocities which do not
surpass thirty revolutions per second, the successive phe-
nomena are distinct, but above thirty the impressions on
the retina are continuous, and give the body the appear-
ance of being absolutely at rest. Finally, if the mirror
turns with sufficient rapidity, its position is sensibly
changed during the time the light required to travel from
MI to M and back again; the return ray, after its reflection
from the mirror M, takes the direction M b, and forms its
image in d, hence the image has undergone a total de-
viation c d. Strictly speaking, a deviation ought to occur
as soon as the mirror turns, however slowly, but it will
only be appreciable when it has attained a certain mag-
nitude, which requires either the rotation to be rapid, or
the distance M M' to be sufficiently large. In Foucault's
experiment, the distance between the two mirrors was
42 PROOF OF THE EARTH's ROTATION.
13 feet, and by communicating to the mirror Mavelocity of
between 600 to 800 revolutions per second, the deviations
obtained amounted to about 0.01 inches. Denoting the
velocity of light by V, the deviation by 8, and the number
of revolutions per second by m, and assuming M M = l',
M L = l and L P = r, Foucault arrived at the formula
_ 8 ºr mºr!”
~ (IIT): ;
by means of which he obtained results respecting the ve-
locity of light closely approaching those derived from
observations of the eclipses of Jupiter's satellites. The suc-
cessive propagation of light being thus established without
astronomical observation, forms therefore a legitimate
premise in our demonstration of the earth's rotation.
We are inclined to think that D’Alembert was the first
who drew attention to the fact that if the velocity of light
be finite, and the earth be at rest, the stars could not be
seen in their real positions, though Montucla asserts that
Empedocles was the author of this opinion; however, be
this as it may, these remarks would have remained incon-
clusive and sterile, had not Arago singled out phenomena
in the celestial motions as necessary consequences of the
hypothesis of the earth's fixity totally at variance with
observed facts. Bradley’s discovery and ingenious ex-
planation of the aberration of the stars afforded the first
striking evidence of the translatory motion of the earth,
and nothing can be more interesting, or mark more highly
the progressive character of science, than the fact that the
same hypothesis, namely, the finite velocity of light, which
led Bradley to his result, should years afterwards serve to
confirm, with equal strength, its rotatory motion.
43
CHAPTER IV.
THE EARTH's ROTATION DEMONSTRATED BY THE WARYING
FoRCE OF GRAVITY IN DIFFERENT LATITUDEs.
HE first direct proof of the earth's diurnal motion is
connected with the investigation of its figure. Picard,
in his measurement of the earth, published in 1671, speaks
of a conjecture proposed to the Academy, ‘ that assuming
the rotation of the earth to be true, heavy bodies should
descend with less force at the equator than at the poles;"
and observes, for the same reason, “there should be a
difference in the length of the pendulum vibrating seconds
in different latitudes. In the same year Richer was sent
by the Academy of Paris to Cayenne, on a scientific
mission; and one of the Several objects of his voyage
was to ascertain the length of the pendulum vibrating
seconds. He returned in 1672, and mentions the obser-
vations of the pendulum as the most important he made.
The same measure, marked at Cayenne on a rod of iron,
according to the length of the seconds' pendulum for that
place, having been brought back and compared with that
marked at Paris : the difference was found to be 14 lines,
that at Cayenne being the shorter. The arcs of vibration of
the pendulum with which the experiment was made were
very small; they continued for fifty-two minutes, and were
compared with an excellent clock. Moreover, the clock
which Richer took to Cayenne, having been adjusted to
44 THE EARTH's ROTATION DEMONSTRATED
beat seconds at Paris, retarded there two minutes a day;
so that no doubt remained of the diminution of gravity
at the equator.
Newton, whose Sagacity at once recognised the cause
of this phenomena, did not hesitate to declare it as the
experimental demonstration of the earth's rotation, so
long sought for by men of science.
It is evident that all places on or near the equator
must revolve quicker than those situate near the poles;
and hence the centrifugal force being increased by the
revolution in greater circles, must proportionately neu-
tralise the centripetal force; i. e. the gravity towards
the centre of the earth, upon which the oscillations of
the pendulum depend.
Richer’s experiment has since been repeated by other
scientific men with the greatest care, and has always
yielded the same results. It must not be supposed that
to swing a pendulum, count its vibrations, and measure
its length is an easy task. Experience shows that a more
difficult practical problem can hardly be proposed, and
that persons therefore who have overcome these difficulties
rank high as promoters of Science; such are especially
Lacaille, Borda and Biot in France, and Foster, Kater,
Sabine, and Bayly in England.
The accuracy with which these observations must be
made, and the extreme delicacy of the instruments re-
quired, will be seen by the following description of Biot's
pendulum. It is constructed on Borda’s principle, which
consists in making it as near as possible resemble the
simple pendulum. The bob is spherical, and made of
platinum, suspended by a metallic wire, the lower end
of which is fastened by means of a screw into the neck
BY THE WARYING FORCE OF GRAVITY. 45
of the spherical segment. The upper end of the wire
is attached to another screw, which forms part of the
prismatic bar of suspension. The edge of the latter rests
on an agate plane, to which the Oscillations are per-
pendicular. This plane of suspension is placed upon a
plate of iron, and can be exactly adjusted by a small
spirit level and parallel screws, the plate itself being sup-
ported by standards. The prismatic suspension is adjusted
so that the duration of oscillation corresponds nearly with
that of a clock-pendulum; this is effected by turning a
small nut, screwed on the top of the knife-edge. When
the times of oscillation coincide approximately, the wire,
with the bob attached, is suspended from the rod imme-
diately under the prism by means of the screw. The
length of the wire must be so regulated that the oscil-
lations of the entire system differ very slightly from those
of the clock, and consequently from those of the prismatic
suspension.
As the centre of gravity of the knife-edge is extremely
near to the plane of suspension, its mass can, under the
conditions by which the apparatus acts, exercise no ap-
preciable influence upon the length of the pendulum. At a
distance of twenty-five feet from the instrument, a telescope
is placed, the eye-glass of which is intersected in its focus
by a vertical wire, which must be made to coincide with
the thread of the pendulum when in a state of rest, at
the same time a small piece of paper is fixed upon the
bob of the clock-pendulum to serve as an indicator.
These preparations being completed, the clock-work is
set in motion, and the simple pendulum also. The
telescope enables us to judge the difference between the
oscillations of the clock-pendulum and those of the
46 THE EARTH's ROTATION DEMONSTRATED
simple one; it also increases the apparent velocity of the
thread of the pendulum relatively to that of the indicator
of the bob, and finally, by its means, we can ascertain
exactly the times of coincidence of the two threads.
But it is evident that the pendulum must either lose or
gain upon the clock two oscillations during an interval of
two consecutive coincidences, and hence the entire number
of oscillations lost or gained by the pendulum, during a
mean solar day, can be computed by simple proportion.
Since the time of the oscillations of the pendulum in-
creases with the magnitude of the circular arcs described,
for only the vibrations in cycloidal arcs are isochronous,
it becomes necessary to observe the amplitude at the
moment of each coincidence. For this purpose a small
circular scale, divided into minutes, is fixed to the bob
of the clock, by means of which the observer can easily
ascertain the magnitude of the arcs described by the
pendulum.
In order to measure the length of the pendulum from
the plane of suspension down to the globe of platinum,
the following apparatus is employed. A metallic plane,
resting upon supports, is placed at the commencement of the
experiment just below the pendulum ; this can be raised
or lowered by means of a screw, of which the number of
turns and the fractional part of each turn are indicated by
a divided circle, which moves in front of a vertical rule
similarly divided. A few trials will in general be sufficient
to obtain an exact contact between the metallic plane and
the platinum ball, great care being taken that the latter
be not raised; for if it be, the effect of the tension of the
ball upon the thread would be destroyed, and it could
only be restored to its original state of rest after a
BY THE WARYING FORCE OF GRAVITY. 47
considerable time. To ascertain when the contact of the
ball and the plane is complete, the eye must be so placed
as to observe any ray of light between the ball and the
plane; the moment such ray disappears, the contact is
perfect. In order to measure the distance of this plane
from that of suspension, a sheet-rule is employed, at one
end of which a knife-edge suspension is adapted, at the
other a metallic index, which can be drawn up and down
by means of a screw; both the index and rule are graduated
equally. The pendulum is now removed, the rule sus-
pended in its place, and the index lowered until it comes
in perfect contact with the plane underneath; then, read-
ing off the number of degrees the index has moved, the
distance required will be known ; subtracting from this
result the radius of the platinum ball, the length of the
pendulum is obtained. English observers generally prefer
an invariable pendulum, in the form of a flat bar, pro-
vided with a weight. A pendulum of this description,
after being compared with an astronomical clock at
Greenwich, or any other observatory, is taken to other
stations, and the rate of its vibrations determined; it is
then brought back to the first place, and again compared
with the chronometer. This method requires the re-
duction of the length of the compound pendulum used
to that of the corresponding simple pendulum —a process
by no means easy, considering the great accuracy requisite.
Captain Kater's ingenuity overcame this difficulty, by
applying Huyghens' beautiful theorem of the convertibility
of the centres of oscillation and suspension to practical
use. The pendulum he employed is represented in the
annexed diagram (fig. 5).
The bar cq is furnished with two axles, by either of
48 THE EARTH's ROTATION DEMONSTRATED
which it may be suspended, one passing through c and
the other through o. Besides the principal weight, d, it
is provided with a small sliding-weight, f,
FIG. 5. which can be moved along the stem c d, and
-- this weight is to be moved till the number of
| Q oscillations in a given time (as twenty-four
[L]e ||* hours), is the same whether the pendulum is
[E]f suspended from c or from o. If f be placed
- in such a position that, by moving it from o
as to f, the number of oscillations about c, in
twenty-four hours, will be increased, and by

Fº the same change the number of oscillations
Je about o, in the same time, will be still more
9 , increased. The adjustment is thus made —
– Let the weight be at f, and let the number
{} of oscillations in ten minutes about c be
606, and about o be 601; now let the weight
be moved to f, and let the oscillations in ten minutes be
607 about c and 609 about o (because the latter are more
affected than the former); then the proper position of the
slider is somewhere between the first and last points. Let
it be placed at f, and let the oscillations in this case be
606; and 606; then the proper position is somewhat
changed, observing always that if the number of vibra-
tions about c be greater the slider must move towards c ;
and, if the contrary, it must move towards o. By this
means, continually halving the distance last moved, we
may make the oscillations about c and o approach within
any required degree of exactness; the distance between c
and o being then measured will give the length of a pen-
dulum which makes a known number of oscillations in
ten minutes.
BY THE WARYING FORCE OF GRAVITY. 49
With a pendulum of this description, Captain Kater
made his important experiments. In pursuance of the
resolution of the House of Commons of March 15, 1816,
the Royal Society of London requested this active astro-
nomer to ascertain the length of the seconds' pendulum
at Unst, Portsoy, Leith Fort, Clifton, Arbury Hill, and
Shanklin Farm. He arrived at Leith on June 28, and
reached Unst on July 9. After finishing his observations
in the Orkneys, he arrived at Portsoy on August 1; at
Edinburgh, August 28; at Clifton, September 28; at
Arbury Hill, October 15; and at Shanklin Farm, in the
Isle of Wight, about May 10, 1819, where he completed
the series of laborious experiments which he had under-
taken. The following are the results of these operations,
as given by Captain Kater himself in the ‘ Philosophical
Transactions' for 1819, Part III. : —
“It now remains to give in one view the results of the
operations that have been detailed. These are comprised
in the following table. It would have been desirable to
have expressed the length of the pendulum vibrating
seconds, in parts of the scale which forms the basis of the
Trigonometrical Survey of Great Britain, the Commis-
sioners of Weights and Measures having agreed to re-
commend that the “ Standard used in the Trigonometrical
Survey of Great Britain should be considered as affording
the most authentic determination of the linear measure of
the United Kingdom.” But, as experiments are yet
wanting to enable me to do this with sufficient accuracy, .
I have given the length of the pendulum in parts of Sir
George Shuckburgh's standard scale, the correction for the
difference between which and the national standard of
linear measure may be readily applied hereafter.
E
50 THE EARTH'S ROTATION DEMONSTRATED
* The length of the pendulum vibrating seconds in
the latitude of London is stated in the “Philosophical
Transactions” for 1818, to be 39°13860 inches. But I
have here to notice a very important omission, which I am
obliged to Mr. Troughton for having pointed out in the
first number of the “Edinburgh Philosophical Journal.”
It may be seen that in computing the specific gravity of
the pendulum I have neglected to include the deal ends.
Anxious to supply this omission in the most unexception-
able manner, I thought it best to take the specific gravity
of the whole pendulum, and for this purpose requested
Mr. Barton, comptroller of his Majesty's Mint, to allow me
the use of the fine balance lately constructed under his
directions, a request with which he most obligingly com-
plied, and favoured me with his assistance, and with every
requisite for making the experiment.
* A deal trough was prepared, seven feet long, nine
inches wide, and the same depth. The pendulum was
slung horizontally from the scale pan, by a fine iron wire.
The weight of the whole was carefully determined in air,
and found to be 66904 grains. The trough, which had
been previously placed beneath the pendulum, was then
filled with distilled water, and the weight of water dis-
placed was found to be 9066 grains. The small portion
of iron wire which was immersed in the water was care-
fully noted; the weight of the wire by which the pen-
dulum was suspended was 56 grains, and the weight of
water equal in bulk to that part of the wire which was
immersed was 2.5 grains. The temperature of the water
was 68°, and that of the atmosphere 62°; the barometer
29.9 inches. Hence we have the weight of the pendulum
66858-8 grains in vacuo, at the temperature of 62°; the
weight of an equal bulk of water at the same temperature
BY THE WARYING FORCE OF GRAVITY. 51
9068°4 grains; and the resulting specific gravity of the
pendulum 7.3727. Employing this specific gravity in
computing the allowance for the mean buoyancy of the
atmosphere, we obtain '00624 for this correction instead
of .00545, the former erroneous conclusion. Besides this,
the allowance + -00031 for the height of the pendulum
above the level of the sea should, according to Dr.
Young's investigation, have been multiplied by Hº,
making + 00021 of an inch. These corrections being
applied, we have 39-13929 inches of Sir G. Shuckburgh's
standard scale, for the length of the pendulum vibrating
seconds in the latitude of London.
* Wishing to compare with this the result which would
have been obtained by means of the weights and specific
gravities of the different parts of the pendulum, I care-
fully measured the deal ends, and found them to contain
3.956 cubic inches. The weight of the knife edges was
370 grains, and their specific gravity 7-84. With these
data, and taking the specific gravity of deal at 0:49, the
specific gravity of the whole pendulum will be found to
vary from the more accurate determination above given a
quantity which would have occasioned a difference in the
length of the seconds' pendulum of only g-gºing of an inch.”

Vibrations *... of ...,
Place of Observation. Latitude. ..". . *::::::: 'H.
e burgh's scale.
O I A / Inches.
TJnst . 60 45 28°01 || 86096-90 39°17146
IPortsoy e 57 40 58-65 86086-05 39-16159
Leith Fort . 55 58 40-80 || 8607.9°40 39°15554
Clifton © 53 27 43-12 || 86068-90 39-14600
Arbury Hill 52 12 55-32 || 86065-05 39°14250
London º º 51 31 8-40 || 86061-52 39°13929
Shanklin Farm . 50 37 23-94 | 86058'07 39°13614
E 2
52
THE EARTH's ROTATION DEMONSTRATED

Diminution of Gravity
Names of the Stations. from Pole to Equator. Compression.
Unst and Portsoy . •0053639 sº
Leith Fort •0054840 sº
Clifton •0056340 sº
Arbury Hill •0054282 sº
‘London •0055510 #
Dunnose . •0055262 in
Portsoy and Leith Fort . •0056920 sº
Clifton •0058.194 sº
Arbury Hill •0054620 i;
London •0056382 sº
Dunnose . •0055920 *
Leith Fort and Clifton •0059033 sº
Arbury Hill •0053615 sº
London . •00561.86 sº
Dunnose . •0055614 sº
Clifton and Arbury Hill . -0042956 #:
London •0052590 sº
Dunnose . •0052616 ñ
Arbury Hill and London. •0069767 sº
*. Dunnose . •0060212 sº
London and Dunnose •005.2837 1
BY THE WARYING FORCE OF GRAVITY. 53
By comparing the observations of each station suc-
cessively with those at all the others, Captain Kater
obtained the results in the opposite table, containing the
diminution of gravity and the resulting compression.
From the number of experiments on the length of the
seconds' pendulum at various latitudes made for the English
Government by Kater and Sabine, and for the French by
Biot, Arago, Duperrey, and Freycinet, the following simple
equation has been deduced. If L denote the length of a
seconds' pendulum at any latitude A, and 39,017 inches be
the length of a seconds' pendulum at the equator, then
L=39,017-F0,200 sin”).
The numerical values of this formula have been de-
rived from most accurate observations, the results of which
are exhibited in the following table. According to Mr.
Airy all the different lengths for the seconds' pendulum,
there noted and arranged in the order of latitude, are,
with one exception, erroneous to the amount of the error
in the correction for the density of the air. The mag-
nitude of this error is nearly the same for all, and is
about 0,0018, except the length given by Bessel, which
is accurate, owing to this astronomer's method of deter-
mining the atmospheric correction. It consists in ob-
serving the length of the seconds' pendulum in air, first
with a bob of brass, and, secondly, with a bob of ivory,
The correction, being greater for the latter than for the
former in the proportion of the specific gravities, is then
calculated with great accuracy from the difference of the
observed lengths: —
54 THE EARTH'S ROTATION DEMONSTRATED
TABLE of THE LENGTH OF THE SECONDs' PENDULUM.
Place. Latitude. tº in Observer.
1. Spitzbergen * 79° 50' N. 39.21469 Sabine.
2. Melville Island . 74 47 39-20700 Sabine.
3. Greenland º 74 32 39-20335 Sabine.
4. Port Bowen . 73 14 39°20419 Foster.
5. Hammerfest . e 70 40 39-19475 Sabine.
6. Drontheim & 63 26 39-17456 Sabine.
7. Unst . e º 60 45 39-17162 Biot.
8 Stockholm . o 59 21 39°16541 Svanberg.
9. Portsoy. e • 57 41 39°16159 Eater.
10. Leith Fort . e 55 59 39°15546 Rater.
11. Königsberg . e 54 42 39°15072 Bessel.
12. Clifton . º e 53 28 39°14600 JKater.
13. Arbury Hill . ſº 52 13 39°14250 Rater.
14. London. º e 51 31 39°13929 Eater.
15. Greenwich Ö1 29 39°13983 Sabine.
16. Dunkirk e 51 2 39-13773 Biot.
17. Isle of Wight ſe 50 37 39° 13614 JKater.
18. Paris . & * 48 50 39' 12864 Borda.
19. Clermont e 45 47 39°11615 Biot.
20. Milan o e 45 28 39°11603 Biot.
21. Padua . p º 45 24 39°11896 Biot.
22. Fiume . e 45 19 39°11788 Biot.
23. Bordeaux 44 50 39°11296 Biot.
24. Figeac . e 44 37 39°11215 Biot.
25. Toulon . Jº 43 7 39°10952 Duperrey.
26. Barcelona 41 23 39 10432 Biot.
27. New York 40 43 39°10120 Sabine.
28. Formentera 38 40 39°095.10 Biot.
29. Lipari . . . 38 29 39°09828 || Biot.
30. California e 21 30 39-03829 Hall.
31. Sandwich Islands . 20 52 39°04690 Freycinet.
32. Jamaica e g 17 56 39°03503 Sabine.
33. Marian Islands 13 28 39-03379 Freycinet.
34. Madras , 13 4 39°02630 Goldingham.
35. Trinidad 10 39 39°01888 Sabine.
36. Sierra Leone . 8 30 39°01997 Sabine.
37. Galapagas () 32 39-01717 Eſall.
38. St. Thomas e 0 25 39°02074 Sabine.
39. Pulo Gaunsah Lout 0 2 39°02126 Goldingham.
40. Rawak . tº e 0 2 39°01433 Freycinet.
41. Maranham 2 32 39 01213 Sabine.
42. Ascension 7 55 39°02363 Sabine.
43. Bahia 12 59 39°02433 Sabine.
44. Isle of France 20 10 39°04684 Freycinet.
45. Rio de Janeiro 22 55 39°04:350 Freycinet.
46. Paramatta . 33 49 39-07452 Rumker.
47. Port Jackson. e 33 52 39°07919 Duperrey.
48. Cape of Good Hope 33 55 39°07800 Freycinet.
49. Isles Malouines 51 35 39°13781 Freycinet.

BY THE WARYING FORCE OF GRAVITY. 55
Since at the pole A = 90°, and therefore sin” X = 1, we
have by substitution in the formula derived from these
observations
the length of the polar pendulum = 39,217
while the length of the equatorial pendulum = 39,017
hence the difference between the length of the seconds'
pendulum at the pole and the equator amounts to 0-2
inches, which is the Tºgth part of the whole length of the
seconds' pendulum at the equator; but the length of the
Seconds' pendulum is a true representation of the intensity
of gravity, the one being proportional to the other, since
g=T*L, and therefore the decrease of the force of gravity
for the quadrant is equal to about T+3th part of its value
at the equator. Let us compare this result with the
theoretical determination of the centrifugal force of
rotating bodies.
A knowledge of the first elements of geometry and me-
chanics will enable the reader to understand the annexed
proof of the theorem that ‘ the centrifugal force equal and
opposite to the centripetal is directly proportional to the
square of the velocity, and inversely proportional to the
radius of the circle described,” or, expressed analytically,
Q)?
f=+.
Let oo' D (fig. 6) be a circle described by a body in uniform
motion, and let oo' be an indefinitely small arc passed over
in the unit of time, and therefore representing the velocity
of the body. Let oo', which may be considered straight,
be resolved into two components, the tangential OP and the
central oo, which form the sides of the rectangle OPO'Q,
the diagonal of which is oo'. Joining o' with the extremity
56 THE EARTH'S ROTATION DEMONSTRATED
FIG. 6.
Q ---------------- >0.
of the diameter D, we have the right-angled triangle oo' D,
whence, by Euclid. Prop. VIII. Lib. 6:
oq : oo' - oo' : od,
but oo = #f, oo' = v, and op = 2 r
... ºf : v = v : 2 r
f= g (Q. E. D.)
Or, if the periodic time required to describe an entire re-
volution be expressed by t, then, since the circumference
of the circle = 2 rar, we have
Q) = 27° 7'
t
Substituting this value of v in the preceding formula,
we have
7"
f=47%;

BY THE WARYING FORCE OF GRAVITY. 57.
i.e. the centrifugal force is directly proportional to the
radius of the circle described, and inversely proportional
to the square of the periodic time. Applying this theorem
to the rotatory motion of the earth, we shall find its centri-
fugal force at the equator by taking the equatorial radius
r= 20,923,596 feet, and the number of seconds in a sideral
year t = 86,164", and, substituting these values in the last
formula, we obtain, as the final result,
. f = 0,11126.
Let G denote the attraction of the earth at the equator
(not influenced by rotation), and g the actual gravity at
the same place, then
G = g + f;
|but it has been found by observations made in the island
of Rawak, that
g = 32,088 feet
•. G = 32,088 + 0,11126 = 32,19926,
G. 32,19926
f*TO.III26
or, expressed in whole numbers, the force of the earth's
attraction at the equator is 289 times the centrifugal force
at the same place.
At first sight, there seems to be a discrepancy, since the
decrease of gravity (from the poles to the equator) exceeds
the centrifugal force (at the same place) by
1 I I
*
I95 T 255 F 595
whence = 289,4,
very nearly; a small quantity in itself, but far too large as
compared with the others in question not to be accounted
for. A little consideration will convince us that this
58 THE EARTH's ROTATION DEMONSTRATED
difference, instead of proving fatal to our hypothesis of
the earth’s rotation, tends to confirm it.
The rotation of the earth gives rise to the centrifugal
force, the centrifugal force produces an ellipticity in the
form of the earth itself, and this ellipticity modifies its
power of attraction on bodies placed at its surface. Thus
the same cause, viz., the rotation of the earth, exer-
cises a direct and an indirect influence. The latter, viz.,
the oblate figure of the earth, has been deduced and cal-
culated by Newton, Huyghens, and Clairaut, from the
theory of gravitation and rotatory motion, and also by
Laplace from the theory of the moon, and from precession
and nutation; but it is not necessary for our purpose to
enter into their abstruse calculations, or to examine the
comparative merits of their analytical results, it is suffi-
cient for us to state that the spheroidal form of the earth
has been proved by the actual measurement of arcs of the
meridian in different latitudes. We may cite as examples
of the most important geodetical operations the French
arc of 12° 22' 12", extending from Dunkirk to the Bale-
aric Isles, and the English arc from Dunnose to Burleigh
Moor, amounting to 3°57'13", and both the English and
French measurements, when compared with those of Bou-
guer in South America, Lambton in India, and Svanberg
in Lapland, give a general ellipticity of about gº, which
is probably as near the truth as local inequalities will ad-
mit. But it is evident that, the earth being elliptical, the
attraction exerted by it on bodies placed near the equator
must be less than that exerted on bodies near the poles,
which are flattened, and therefore nearer to the centre.
This diminution of attraction, which is owing to the oblate-
BY THE WARYING FORCE OF GRAVITY. 59
ness of the earth, has been investigated by many eminent
geometers, and the results show that independent of the
centrifugal force, gravitation, considered in the direction
of the pole to the equator, ought to decrease by almost
exactly the ###th part, which, together with gºth part
due to centrifugal force, makes up the Tºgth part, the
whole quantity actually found.”
Such a near coincidence of theory and observation ren-
ders our hypothesis the more incontestable, as there is no
other to account for these differences of terrestrial gravity,
even roughly estimated. Before concluding this chapter,
we may briefly allude to another geographical phenomenon,
which may be explained by the earth's rotation, namely,
the trade winds. The difference of temperature of the
air over the torrid and frigid zones could only generate
southerly or northerly currents, but, in consequence of the
rotation, every particle of air transferred from the pole to
the equator must slacken its rotatory motion, and therefore
acquire a relative direction towards the west, and thus as-
sume the character of permanent north-easterly or south-
easterly winds. But as they approach the equator from
both sides simultaneously, their easterly tendency ought
to diminish, and, upon meeting, be destroyed; so that the
* According to Clairaut's theorem, we have the ellipticity, as measured by
the quotient of the difference of the axes by the smaller, equal to # times
the ratio of the centrifugal force at the equator to the force of gravity
diminished by the ratio of the whole increase of gravity to the equatorial
gravity. By this means we obtain the value of
tº a e - 5 1 1
the ellipticity = 3 * 359 TI35
while its value found from the geodetic measures is - 0-00335
= 0.00352,
Difference . . = 0-00017
60 THE EARTH's ROTATION DEMONSTRATED.
equatorial region should be comparatively calm and free
from easterly winds. All these consequences are agreeable
to observed facts, and may therefore be taken as circum-
stantial evidence that the cause alleged, namely, the
rotation of the earth, is the true one.
61
CHAPTER V.
THE DESCENT OF BODIES FROM GREAT HEIGHTS,
DIRECT and most convincing demonstration of the
earth’s diurnal motion is furnished by those experi-
ments which establish the fact that balls truly turned and
carefully dropped from a great height do not fall exactly
in the perpendicular, but deviate towards the east. Newton,
in 1679, was the first who proposed to prove the rotation
in this practical manner. The deviation of which Newton
speaks is not, as Lalande (tom. 1, § 1082) asserts, to be
accounted for by the curvature of the globe, but by the
principles of mechanics. For if the earth rotates, the
velocity of rotation must be greater at the summit of a
tower than at its base, and a body dropped from the
former will move faster in an easterly direction throughout
the whole of its descent than the base does, and therefore
reach the ground a little in advance of the foot of the
perpendicular. Dr. Hooke, who was secretary of the Royal
Society at the time Newton suggested this method, and
who was renowned for his many mechanical contrivances,
undertook to perform the necessary experiments. On
January 14, 1680, he reported the results to the Society,
stating that the height from which he dropped the balls
was twenty-seven feet, and that they fell to the south-east.
It was resolved that these experiments should be repeated
62 THE DESCENT OF BODIES
before some of the members, but we find no further
mention of them in the Transactions, and we are therefore
inclined to conclude that they were not attended with
success, owing to the smallness of the height, which could
scarcely have caused a deviation of # line. The idea,
which originated with a philosopher whom all Europe
revered, remained barren for a period of 112 years, when
Guglielmini, a geometer of Bologna, revived it, and com-
menced making experiments in the tower degli Asinelli,
which is 300 feet high. After many unsuccessful trials
and various alterations in his apparatus, he gave up
working in the day because he discovered that the traffic
caused the tower to vibrate; and selected nights when the
air was calm. The liberation of the balls was effected by
burning the threads by which they were suspended, and
this was never done until they had entirely ceased to
oscillate, as observed by means of a microscope.
The balls fell upon a cake of wax spread upon the
ground, and the holes made by them nearly coincided, the
greatest distance between two centres scarcely amounting
to half a line. The persevering philosopher finally dropped
a plumb-line in August 1791, but was obliged to wait till
February 1792 before it came to a state of perfect rest.
He then found that the mean deviation of the sixteen balls
he had dropped was 7.4 lines east and 5'27 lines south.
These results were received by the scientific world with
the greatest interest, when Lalande announced that
Laplace's theory gave only a deviation of five lines to the
east and none to the south. This striking difference of
theory and observation was at once explained by a trifling
circumstance, overlooked in the otherwise most accurate
experiments of Guglielmini, but the explanation over-
FROM GREAT HEIGHTS. 63
threw at the same time all their force of demonstration.
The difference was occasioned, first, by a local disturbance
of the air, owing to the numerous perforations in the walls
of the tower, and secondly, by a slight curvature of its
structure. The balls were dropped in the summer, but
the vertical was only ascertained six months afterwards, in
the winter, and there is no doubt that the difference of
temperature was sufficient to shift the point of suspension.
Although these experiments did not achieve the result for
which they were undertaken, yet they were the precursors of
others which furnished most satisfactorily the required
proofs. Dr. Benzenberg commenced in 1802 a series of
experiments in St. Michael's tower in Hamburgh, which is
402 feet” in height and most suitable for the purpose, for
an uninterrupted fall of 340 feet may be obtained, exceeding
that in the Asinelli by ninety-nine feet and that in St. Paul’s
by eighty-five feet. But as only 240 feet of the tower, from
the base upwards, were completely closed, Benzenberg chose
a height of 235 feet for his investigations, in order that even
strong winds might not influence the result. The balls
he used were 1-6 inches in diameter, and made of equal
parts of lead and tin with the addition of a little zinc, an
alloy which is easily turned and polished; they were tested
by being placed upon quicksilver, and those which betrayed
eccentricity in floating were rejected. His mode of sus-
pension is represented in fig. 7. The suspending thread
was cut, for burning might have caused a slight draught
and an oscillation of the ball before being detached; and
though he could not obtain permission to work at night,
yet the vibrations of the tower were not perceptible, and
* Throughout this chapter, unless otherwise stated, the lengths are given
in the French measure, viz. 1 foot = 12.78933 English inches.
64 THE DESCENT OF BODHES
G. 7,
iii.;;}}#} }
EI
ſº º
|
*
...]iſ:{i}; :::
ºilii
º |
#,
|Hüll l
mň
the weather proving very favourable, his experiments may
be looked upon as accurate. The last thirty-one balls he
dropped fell as follows: —
| -
|
i
|
|
| :=== *
t -
; : til t
|
- - |
I,
ſ
it.
- - ---
t -
l
21 towards the east
8 22 , West
2 neutral
16 towards the south
11 35 , north
4 neutral



FROM GREAT HEIGHTS.
65
The special results of these thirty-one experiments will
be seen in the following tables: —

North South East West
July 23, 1802 5 - * 5
55 3 y *sº 5 8-5 cº-º-º:
33 93 *mº &=º * tºm-º.
35 32 g 1 sºmeº gmemº *===
92 92 tºº--> 3 21-5 sº
August 14 sº *º *º- 6
39 39 * = ºr &ºm=} 16 sºmsmsº
93 99 * , tºm-º. 2-8 1-8 *
93 2? 6-8 * = a 5-2 *-*
August 16, *-i- 1-7 *=º 10
35 5 y 2.2 tº º 12 *º-ſº
37 93 *º- 0-8 * 1.5
33 55 § * 2 ſº * 1.
September 17 * * 4 **sº * 8
99 95 e tº 8 e- 7 *-ºsº
. 33 33 tºm- 9 1-5 -º-
October 14 g 5-7 &ºmºmº 0-5 ºmº
53 39 g * 17 12 tºmsº
October 15 ge *º 10-5 17.3 *
93. 33 g i. *º 2 7 *
99 35 { } * 8 0-8 *===
33 33 e dº ** 2 17°5 | —
33 yy wº 0-2 *-ºº: 14-7 *º
53 5 3. * — | 167 0-2 gºmº
October 23 © * ºms-º - 14-8
32 32 * 5 8 *
93 33 * 5-6 * 4-2
October 26 1. *Eº 6 *sº
33 92 - 5 13°5 —
33 27 10°5 - 1-8 *==º
33 33 * 0.5 1-7 &===º
Sum 46-4 92.6 174'5 50-5
South = 92.6 East = 174°5
North = 46-4 West = 50-5
46-2
tºº & Asºº
124°0
These numbers divided by 31 give respectively a mean
deviation of four lines to the east and 1% lines to the south.
It will be seen that the first sixteen trials gave a mean de-
F
66 THE DESCENT OF BODIES
viation of 0:53 north and 2-6 east, while the last fifteen gave
a deviation of 3:53 south and 5'46 east. This difference is
explained by the fact that the apparatus was reversed, that
is to say, the nippers which first opened towards the south
afterwards opened towards the north, and at the same
time the position of the suspending hook was changed
from the west to the east. As this reversion seems ne-
cessary to avoid constant errors, we may accept the given
deviations as fair results; but how do these agree with
theory P. We have already remarked, that the theory of
Laplace does not give any southerly deviation, nor do the
fundamental formulae of Gauss allow any. Our object
now, therefore, will be to investigate the question on
physico-mathematical principles. In the preceding chap-
ter we have seen that if the radius of the earth be denoted
by r, and its periodic time of rotation by r seconds, we
have for centrifugal force at the equator
f = 4r ().
At any other point of the earth's surface the effect of
the centrifugal force must be resolved into two components,
the one acting in the prolongation of the radius, the other
perpendicular to it.
Let the circle Po P' represent any meridian of the earth,
o any place on it, and let its latitude o E be denoted by L.
Draw o M perpendicular to the axis of rotation PP' and
produce it till o N=0 M', then by the last formula the whole
effect of rotation at O is expressed by
4ON (...) 2.
T
Joining o with the centre C producing Co, drawing a
FROM GREAT HEIGHTS. 67
FIG. 8.
P.
tangent at o and completing the parallelogram O A N B,
we have
40A (...) 2.
T
4OB (...) 2.
T
the respective expressions of the two components of the
centrifugal force at o, but
O A = r cos” L and O B = r sin L cos Lº
hence by substitution, the component opposite, to the direc-
tion of gravity, and employed to diminish it, is expressed
by 40' cos” 1. (...)
and the other perpendicular to it in the plane of the
meridian is expressed by
º T © T
47° cos L sin L (...) * = 27° Sin 2L ().
T T
If, therefore, from a point vertically above any place, the

F 2
68 THE DESCENT OF BODIES
latitude of which is L., and at which the actual gravity is
g, a plumb-line be dropped, it will hang in a direction
different from the vertical, and p, its angle of deviation
from the perpendicular, will be determined by the equation
27" sin 2 L ( i).
T
p = 9.
Let the height from which the plumb-line is dropped be
denoted by h, then 8, its southerly deviation, will be ex-
pressed in linear units by
2hr sin 2 L (...)
T
9
Now, in order to find the southerly deviation, 8', of the
point at which a ball dropped from the height h strikes
the earth, let us assume that the falling body remains in
the plane of a great circle perpendicular to the meridian
(which must be the case if the resistance of the air is not
taken into account), then this great circle will touch the
small one of rotation of the place in question. If t
denote the time of descent of the ball, then the angle of
rotation for that time is
©
8 =
s = 2+!.
T
Let P be the complement of the point o, and let the
point where the ball strikes the earth be expressed by P’;
then
tan P’ = tan P 5
COS 8
therefore
tan P tan” P
tan P-r)=( — t r) : (1
( ) COS 8 8.I) + COS 8.
_tan P (1 — cos g) 2 tan P sin” # s
cos g + tan” P cos g + tan” P
FROM GREAT HEIGHTS. 69
Now as s in these experiments only amounts to about 36",
we may take cos s = 1; thus we have
2 tan P sin” # 8
Secº P
tan (P’ — P) = = sin 2 P sin” # 8;
but sin 2 P = sin 2 L,
hence tan (P’ – P) = sin 2L sin” # e.
No appreciable error can arise by taking the arc (P’ — P)
for tan (P’ — P), whence we obtain the southerly deviation
expressed in linear measure,
8' = r sin 2 L sin” # s.
In order to eliminate s, we take the equation s = 27t ,
T
from which we derive
sin” # s = ºr” . ;
therefore, by substitution,
8' = ri” sin 2 L (...) 2,
and if the ball fall in vacuo T
h.
# g
t” =
#
****(·).
9
which is identical with the expression for 8, as determined
above.
From this it will be obvious that if there is no resist-
ance of the air, there is no southerly deviation, but as it
cannot be neglected,
whence we have 8' =
h
t” > *- :
# g’
therefore 8' > 8,
70. THE DESCENT OF BODIES
whence it would appear that a deviation exists. We must
emember, however, that the expression for 8 was found
by assuming that the ball never quits a certain plane of
a great circle: now this cannot be true in a resisting
medium, for the conical motion of the atmosphere will
rge the ball towards the north, just sufficiently to com-
pensate for the apparent difference 8–8. This may be
proved in the following manner. If in the formula
8' = ri” sin 2 L (...)
T
2r sin 2 L (...)
TZ
9
we substitute tam p =
we have 8 = } gi” tan p.
Now as , gº expresses the height through which the ball
would fall in vacuo during t seconds, it follows that if
there be no resisting medium the ball must pass in the
direction of the plumb-line; but owing to the resistance
of the air, the space of descent is not # 9t” but h, and
hence,
2hr sim 2 L (...)
g T / .
8' = h tan p = 3.
p 9
we have therefore again the result
8 = 8,
i. e., no southerly deviation can exist, even if the ex-
periment be made in air.
With regard to the easterly deviation, we know that
a ball dropped from a height h has a velocity caused by
the rotation of the earth
2T (ºr + h) cos I,
T
V -
FROM GREAT HEIGHTS. 71
in the direction of the tangent to the circle of latitude of
the place of observation, with such a velocity that the
ball would describe a space
Wi.
_ 27tt (ºr + h) cos L
sº 5
T
while the point of the earth vertically below the place of
observation passes through an arc the length of which is
expressed by
2T troos L
3.
T
and as are and tangent have in this case no appreciable
difference, we obtain for the easterly deviation
27 th cos L
T ©
This is true if the directions of gravity towards the falling
body remain parallel, and, though their angles of incli-
nation are small, they must not be neglected.
FIG. 9.
C
Suppose the body during its descent to have arrived

72 THE DESCENT OF BODIES
at H, and let M H = a, let w denote its velocity in the
direction M! H, and v that in the direction H C, then
div = galt,
therefore dw = =º • di.
taking H C = M C = r.
da: da: .
But w = dź or dt = 70’
whence by substitution
_ ga; da;
QUT"
ga, da;
— »
7°
dw =
or walw =
which by integration gives
2
w” = — º + constant;
and since for a = 0 the original tangential velocity is
w = ** (r + h) cos L, we have by correction
T
2
w? = y2 – 9°
7"
gaº
Or QU = W 1 – 4:3),
Taking again the equation
2
di = º we have by substitution
da: ga”
*-VII: = d. (1 + 3 +. & © tº gº ..)
ºv2
Neglecting the other terms of the series, we obtain by
integration,
3
gº- # 900
wi = a + #4,
FROM GREAT HEIGHTS. 73
no correction being necessary for a = 0 when t = 0. It
is also evident that the second term of the right side of
the equation is very small compared with aſ, and we may
therefore without sensible error substitute in it w8 tº for acº,
and write
gvtº
vt = a + +
7°
3
or a = vi – ºf
7"
Now if A denote the excess of descent in vacuo above
that in air, we have
which is true for the 0-000001 part of a line;
therefore
Ø E *7 t (r. 4 h) COS L - #t (h+ A) cos L.;
T T
and because the plane 0 advances towards the east in t
seconds by
* troos L5
T
we get for the easterly deviation of the fallen ball the
expression
27 th cos 1 – #t (h+A) cos I.
T 3T
= 4T t (h – 3 A) cos I.
37-
Applying this formula to Benzenberg's experiments, we
must take L = 53° 33', the latitude of Hamburg, and
74 THE DESCENT OF BODIES
h = 235 feet, from which the other quantities t and A
may be calculated, t = 4". 1''", A = 8-787, while r and at
are known as constants, r = 86164"1, and T = 3-14 . .
The calculation is rendered very simple by the use of
logarithms, as will be seen by the following: —
Calculation of Eaſperiments made in St. Michael's Tower,
Hamburg, ammo 1803, from a height of 235 feet, and
wnder 53° 33' latitude, by Dr. Benzenberg.
47 t (h – # A) cos I
FORMUL.A. a.
3T.
h = 235
t = 4// 1” == 4”-017
# g = 157° 108
h + A = 243'-787 = # gt?
A = 87-787
h — # A = 230/.607
L = 53° 33' T = 86164”-1
log. 144 = 1'158362 . . (The factor 144 is inserted
to reduce the feet into lines.)
log. 4T = 1.099212
log. t = 0-603902
log. (h — ; A) = 2,362872
log. cos L = 9-773875 – 10
5°998223
— log. 3T = 5:412447
O-5857.76
No. of which = 3-852 lines, result of theory.
3.997 25 35 observation.
Difference 0-145
This result shows how, as regards the easterly deviation,
FROM GREAT HEIGHTS. 75
theory and observation agree, but there still remains a
southerly deviation of 1% lines which is not accounted for
by theory. Benzenberg, not quite satisfied with the result
of his experiments, determined to repeat them, and chose
for this purpose an abandoned coal-pit at Schlebusch, in
the county of Mark, Westphalia, situated in a latitude
51° 25', and about 264 feet deep, of which 262 feet were
available for the free descent of the balls. With these
data, and taking; = 15-105 and t = 4". 164, we get by
the formula
_ 47r t (h. — ; A) cos L
3T.
an easterly deviation of 4:64 lines. The following is the
calculation: – &
log. 4T = 1.099212
log. t [4”-164] = 0.619510
log. h. — A [262 – 5:36] = 2.406926
log. 144 = 2-158363
log. cos L [51° 25'] = 9-794942 – 10
6-07895.3
log. 3r = 5:412441
0-666512
the number of which = 4:64 lines
Having now examined the theory, we will turn our
attention to Benzenberg's new experiments, twenty-
nine in number. They were commenced in the autumn
of 1804, all precautionary arrangements having been
made, such as covering the entrance of the pit and block-
ing up its lower outlets.
76 THE DESCENT OF BODIES

S
O
Ul
t
h
North East West
October 7, 1804
33 35
5 y 95
October 8 m,
i
:
-
8
12
;
13
20
5
11'5
i
12
1
:
#
13-5
º-
O
C
to
b
e
r
9
1
1
i
1
3º
;
1
l
tº-10
8-5
10
5-5
189°5
4'
5
95 35 s
Sum . * te . | 12
4
4
10
North = 124 East = 189°5
South = 103 West = 42
-*-
Difference 21 Difference 147-5
*-*. *-*-
*m-
These numbers equally distributed among the twenty-
nine trials, give as a mean result for a single ball,
Northerly deviation = 0-7 lines
Easterly deviation = 5-1 ,
We see from this that no deviation took place towards
the south and the small amount, viz. Fºr of a line nor-
FROM GREAT HEIGHTS. 77
therly deviation, is obviously due to the errors which can
never be avoided in practice. On the other hand, the
deviation towards the east given by observation ap-
proaches so closely to the theoretical result 4,6 calculated
upon the hypothesis of the earth's rotation, that the latter
is confirmed, and the more so as no other reason can be
assigned for the aggregate deviation of all the balls towards
the east exceeding half the sum of the eastern and western
deviations. An additional and very strong confirmation
has since been furnished by the favourable result of a long
series of similar experiments made by Professor Reich in
the year 1831 in the mines of Freiberg. The depth he
could command for the free descent of the balls was 488
feet, nearly double that of Schlebusch, the number of
trials he made amounted to 106, and the mean result gave
only a slight southerly deviation, viz. less than half a line,
and an easterly deviation of 10,32, which differs in defect
by 1,85 from the results of theory.
Calculation of Eaſperiments made in the Mimes of
Freiberg, from a height of 488,06 feet, and wnder
50° 53° 23' latitude, by Professor Reich.
— 1
FORMUL.A. a = 47ſ t (h — ; A) cos L.
3r
h = 488-06
# = 6'-0098
h. -- A = 545.317
A = 57°257
h — # A = 459°432
L = 50° 53' 23”. r = 861647.1
78 THE DESCENT OF BODIES
log. 144 = 2-158362
log. 4T = 1.0992.12
log. t = 0.778216
log. (h — ; A) = 2-662217
log. cos Z = 9-799902 — 10
6-497909
— log. 3T = 5:412447
1.085462
the number of which is - 12:17 lines.
The following table shows the results of 106 experi-
ments, expressed in millimètres: —
FIRST SERIES of 18 ExPDRIMENTS.


No. South North Fast West
August 23 l - 61, 8 38, 7 -
,, 25. 7 - 102, 8 179, 0 -*.
, , - . 13 | 84,15 *- *** 77,05
, , – º 19 85, 0 - * 12,85
,, 27 3 — | 129, 3 || 11,55 -
2 3 - 9 3,85 - 16,65 -
,, 29 3 || 41,55 - *mº 80,15
, - 9 — 129,65 60,95 -*
Sept. 1 . 3 - 2, 6 56, 6 -
53 I 9 - 16,55 146, 6 -
2 3 6 3 - 71, 7 96,05 -*
35 — . 9 13, 2 - 4, 6 -
52 — , 15 - 1, 7 * 57,95
33 - . 21 — 34, 9 -* 7,15
33 7. 6 - 147,55 47,19 -
53 — . 12 193, 8 - 94, 4 -
, , - . 18 — | 130, 9 * 19, 9
421,55 837,75 820, 14 255,05
FROM GREAT HEIGHTS. 79
SEcoRD SERIES OF 18 ExPERIMENTs.
No. South North East West
August 24 . º 2 6, 3 -*. 49, 3 -*
,, 25 . tº 8 - 56,95 37, 7 -
3, - º 14 19,65 -* 60, 3 -*
33 - 20 -- 51, 9 52, 7 -*.
,, 27 . 4 -* 97,55 a- 65,35
33 - 10 65,15 -*- - 105,85
, 29 . º e 4 - 6, 2 *- 58, 6
23 - - e g 10 65, 5 -*- *-* 61,85
$ept. 1 . º tº 4 62,85 -*- 72,65 --
35 1 . º o 10 27, 7 - 46,10 -**
25 1 ſº se e 16 - 22, 9 •- 38, 7
3 y 6 . e 4 47, 4 *s- --> 6, 3
29 * e 10 62, 1 - 62, 5 -
, , – te 16 26, 5 -*- 58,15 *
.., 7 º 1 -* 97, 2 41,09 -
, - e 7 3, 0 * 127,19 -
, - e 13 117,95 - 48,65 -
, - 19 104, 0 * - 163, 3
608,10 332,70 656,33 499,95
THIRD SERIES OF 18 ExPERIMENTs.
No. South North East West
August 24 3 - 5,05 - 37,15
,, 25 9 - 123, 1 - 18,75
, , – 15 79,85 * - 9,52
, , – 21 123, 0 --- - 4, 5
,, 27. 5 15, 6 -º-º-º-º: -- 14, 4
, - 11 11,85 - - 41,15
, 29. e 5 97,55 -º- 66, 8 -
3, T • º 11 26,45 -- 41, 6 --
Sept. 1 . º 5 - 31,05 ... — 45,45
33 1 . e 11 13, 8 *- 92, 9 -**
32 1 . tº 17 19, 6 -*- 85, 8 -mº
59 6. 5 14, 9 - 45, 4 --
, , – º 11 -* 11,85 69,15 -
, , – & 17 - 4, 8 61, 4 -ms-
, 7 º 2 66, 9 - - 141,81
93 -º- 8 -> 8, 9 68,24 -
,. — 14 -* 114,65 32, 8 -
, - 20 - 31, 9 3, 8 -º-
469,50 332,30 567,49 312,73



80 THE DESCENT OF BODIES
FourTH SERIES OF 18 ExPERIMENTs.
No. South North East West
August 24. º 4 62, 8 *- - 60,75
,, 25. tº º 10 - 12,45 - 36,95
3, - . º {º} 16 18, 5 gºmº- 63, 7 -
3, - . de º 22 - 109,95 19,55 -
,, 27. e 6 184,35 * 75, 8 -
, , - © 12 11, 7 -- 90, 0 -
, 29. g ſº 6 - 79, 8 29, 3 -
, , - . © º 12 110,55 amº- 79,05 -
Sept. 1 . gº © 6 - 0,75 3,75 -
32 1 . tº º 12 33,15 ** 60,15 -
33 1. g º 18 18, 9 as-mº 37,35 *
33 6. º º 6 - 99, 4 57,85 -
, , – º © 12 35,75 * 42, 5 -
33 - & º 18 - 2, 9 6, 0 -
33 7 {º º 3 º- 39, 6 - 45,31
3 y - e e i9 25, 5 e-º 99,64 -
33 - 15 34, 2 *-*- - 30,95
33 - 21 - 151, 2 27, O -
535,30 496,05 691,64 173,96
FIFTH SERIEs of 17 ExPERIMENTs.
No. South North East West
August 24. 5 6,65 *- 48, 1 -
,, 25. & 11 21, 3 *-* 69,45 -
, , – º 17 - 26, 5 - 6, 6
,, 27. • o 1 - 11,65 - 9,75
55 - º - 7 141,1 - 88, 1 -
, 29. te 1 - 86, 0 17,55 -
9 y — , º 7 33, 0 *- 24,95 -
Sept. 1 . º 1 44,55 * *-*. 43,15
33 1 . 7 70,05 *- 22, 8 -
33 1 . 13 1,15 *- 33, 9 -
35 6. 1 - 86,85 - 23, 9
3 5 – , 7 - 20,45 30,15 -
39 -* 13 - 38, 3 -* 34, 4
, , – 19 - 27,85 12, 8 -
5 y 7 4 - 27, 2 10,79 -
, , – 10 - 130, 8 57.94 -
, , —- 16 8, 1 *- 15, 0 -
325,95 455,60 431,53 117,80


FROM GREAT HEIGEITS.
SIXTH SERIEs of 17 ExPERIMENTs.

No. South North East West
August 24. 6 83, 6 *º º 40,75
,, 25. 12 11, 6 gºss tºmº 5,55
2, - 18 95, 2 *=º 74,95 gººmsº
,, 27. 2 sº 7, 3 , 5 tº sºm
, , – 8 88, 8 smºº 119,15 ºmº
,, 29. 2 tºº 128, 7 49, 0 &=º
, - . 8 39,45 *=º 30, 9 gº
Sept. 1 . 2 * 30,65 90, 9 gºmº
35 1 . 8 72,55 * 49,75 * =
35 1 . 14 «-º 26,65 94,35 tºº
,, 6. 2 &ºmº 34, 9 46,55 sº
, , – . 8 18, 9 &=º 95, 0 tº-º-º-
, , - 14 Essºms 20,25 32,95 *sº
, , – 20 25, 2 *==º 55, 0 **
55 7 5 * = 104,75 tºº 134,66
3 y tºº 11 117, 9 t=== 29, 6 *º
, - 17 * 21,95 57,85 *-º
553,20 375,15 832,45 180,96
FINAL RESULTs.
1. Series 421,55 837,75 820,14 255,05
2. , 608,10 332,70 656,33 499,95
3. , 469,50 332,30 567,49 312,73
4. , 535,30 469,05 691,.64 173,96
5. , 325,90 455,60 431,53 117,80
6. , 553,20 375,15 832,45 180,96

From which we derive—
1st. For the northerly and Southerly deviation.
South — North
Millimëtres
*: 8°7.7% = — 23,122.
18
608,10 Is 332,70 = + 15,300.
G.
82 THE DESCENT OF BODIES
South – North
Milbimètres
496,50 is * = + 7,622.
535,30 Fº = + 3,686.
325,90 H 499.99 – – 7,629.
553,20 H ** = + 10,473.
2nd. For the easterly and westerly deviation.
East — West
Millimëtres
820,14 is ** = + 31,393.
656,33 is 498,92 = + 8,687.
567,49 is *73 = + 14,153.
691,14 is 173,96 = + 28,760.
431,53 H 11789 – 4: 18,454.
832,45 H 189.9% = + 38,323.
Mean Result.
North–South IEast–West
37,081 – 30,751 = 6,330 + 139,770
These results divided by six, the number of series, give
the following mean deviations : —
Towards the south Towards the east
º F: 1,055 millim. lsº – 23,295 millim.;
.*
FROM GREAT HEIGHTS. 83
or, since one millim. = 0,4433 Paris lines, we have by
reduction,
Southerly deviation = 0,46 Paris lines
Easterly deviation = 10,32 , 25
We cannot conclude this chapter without expressing a
wish that some scientific man, emulated by the truly prac-
tical spirit so often met with in Englishmen, may repeat
these delicate and arduous experiments. This country
offers singular local advantages for the prosecution of such
researches. We have deep mines and high buildings, and we
only require an enterprising and persevering philosopher to
corroborate by his testimony the results of those inquiries
which, though emanating originally here, have as yet only
found successful votaries abroad.
84 THE CHANGE OF THE PLANE OF
CHAPTER VI.
ON THE CHANGE OF THE PLANE OF OSCIILATION OF THE
PENDULUM.
HE pendulum, which has become so useful in science
and in art (ever since the day when Galileo reflected on
the swinging of a lamp)—the pendulum, which is the
most perfect measure of time, which serves to determine
the figure of the earth, which is so intimately connected
with the progress of modern astronomy; this simple and
beautiful instrument is again chosen in this investigation
as the best auxiliary in physical researches. It is a well-
known fact, that if any weight, attached to the end of a
string, be suspended from a fixed point it will remain in
equilibrum, when in a vertical position, but if drawn into
any other and then liberated, it will descend by the action
of gravity with accelerated motion, and by reason of its
inertia continue to move, and, therefore, ascend with re-
tarded motion; moreover, the Swinging down and up
would last for ever if the point of suspension had no fric-
tion, and the air offered no resistance. It is further self-
evident, that if the pendulum be not acted upon by any
other force than gravity —if it commenced with no initial
velocity, and no extraneous impulse interfere, it must
preserve its original plane of vibration. Should, on the
other hand, the pendulum be exposed to a jerk, however
slight, it will describe the surface of a cone with elliptical
base. This elliptical motion was once made use of by Dr.
OSCILLATION OF THE PENDULUM. 85
Hooke to establish a representation of the laws of nature in
the Solar system. It was one evening, in the year 1774,
when he read before the Royal Society his paper on the
laws of attraction, that, for sake of illustration, he sus-
pended a ball from the ceiling, and, while the pendulum
was vibrating, he communicated to it an impulse, laterally
with respect to the plane of vibration; immediately the
ball was found to describe an ellipse about a centre which
was in the vertical line passing through the point of
Suspension.
He succeeded in some degree in explaining by this
method the principle upon which the elliptical motion of
the earth round the sun depends, but his experiment is
imperfect, inasmuch as he assumes the sun to be in the
centre instead of the focus of the terrestrial orbit; besides,
it affords no demonstration of the earth’s motion, for, in
order to show its rotation by means of the pendulum, all
elliptical swing must be carefully avoided. The ocular
evidence of the earth's motion is founded upon the prin-
ciple of mechanics, that the simple pendulum cannot, by
its own agency, change its original plane of Oscillation;
if, therefore, any change take place it must be attributed
to some external cause.
The Academicians del Cimento, at Florence, who made
many experiments on the pendulum, established the fact
that the plane of vibration varies. We may here quote
two passages which occur in the ‘Saggi di Naturali
Experienze,” edizione del 1841, p. 20, and in the “Notizie
degli Aggrandimenti delle Scienze fisiche in Toscana,’ vol. ii.
p. 669, which show the antiquity of the discovery.
The first is as follows: —
“The simple pendulum with a single thread suspension
86 THE CHANGE OF THE PLANE OF
being at liberty to move round deviates farther and far
from its original position until it comes to rest; it does
not move in a vertical arc but follows an oval spiral.’
The other paragraph in the notes published by Targioni
runs thus: —
* November 28, 1661.
“If the pointed bob of a simple pendulum with a single
thread suspension be made to touch during its swing a
plane strewed with sand, it will be observed that its
vibrations cease to be in a vertical arc and follow a spiral
curve, thus tracing in the sand an oval spiral the branches
which all intersect in the centre.”
Notwithstanding that these quotations prove that the
academicians of years gone by were aware of the deviation
of the pendulum from its vertical plane, yet we have no
evidence to show that they applied this discovery to the
demonstration of the earth’s rotation. It is to Mr.
Foucault that the merit is due of having first shown the
relation and dependence of these two motions. Mr. Fou-
cault communicated a description of his experiments, on
February 3, 1851, to the Academy of Sciences at
Paris; his apparatus consists of a steel wire fastened at
the upper extremity in a metal plate by means of a screw
(see fig. 10) and carrying at the other end a heavy ball of
brass, to which is attached a pointed index. In order to
avoid any initial velocity, the pendulum is drawn up out
of the vertical and held in that position by means of a
thread passed round the bob and made fast to a fixed
object; this being done, and the ball in a perfect state of
rest, the thread is burnt through, the pendulum commences
its swing in the vertical plane of its point of suspension;
OSCILLATION OF THE PENDULUM. 87
it will soon, however, be observed to leave this and deviate
more and more towards the west. We shall proceed here-
after to determine theoretically the amount of this devia-
tion and compare it with the results of observation, but it.
will now suffice to remark that the motion of the pendulum
FIG. 10.
illllllllllllllllliºn.
l. Thiſ Tºllſ l
sº-
| º
- |
| | º
|||||||ſiº, ill|| ||
FoucAULT's SUSPENSION
in the azimuth is only apparent, that its plane of oscilla-
tion remains fixed, and that it is the earth which travels
under it. Objection may be taken to this proof, inasmuch
as it may be suggested that the point of suspension being

88 THE CHANGE OF THE PLANE OF
fixed to the earth, participates in its motion, but the tor-
sion of the wire thus occasioned is so small as to exercise
no sensible influence on the whole. It is necessary, in
order to have a clear conception of the fixity of the plane
of suspension, to be thoroughly acquainted with the follow-
ing two principles of mechanics : —
1st. If the pendulum commencing its swing in a vertical
plane rotate at the same time round its fixed point of sus-
pension then the oscillations will continue in the same
plane.
2nd. If the centre of suspension rotate also, then the
oscillations will take place in planes parallel to the first
plane of motion.
In order to illustrate this tendency of the pendulum
to move in the same plane or in one parallel to its
original motion, I have used the following apparatus (see
fig. 11). R R, represents a brass ring fitted in a wooden
frame WW, and capable of being turned round by means
of the handle H, the whole supported by a tripod stand;
D is a disc of metal fastened to the wooden frame by a
strip of brass, concentric with the ring but of smaller
radius, so that ample space is left for the rotatory motion
of the pendulum PP, bent at the upper part, which termi-
nates in a small cone of hardened steel fitting into a cor-
responding cup drilled in the centre of the disc; Q Q
represents a similar pendulum of smaller dimensions
capable of free movement at the extremity E of a rod E E,
bent at right angles and fastened to the brass ring.
If now the heavy bob of the pendulum, PP, be raised
and carefully dropped, it will be observed to swing in a
vertical plane; let its direction be marked by comparison
with the position of a fixed object, and let a lateral
OSCILLATION OF THE PENDULUM. 89
FIG. 11.
E
Q \b i E
“ull|| º
Cº) DDS
º,
º
º,
'.
| | | |
º
y
Wº v.
º,
º
|
º
. .
'''
W.
\
|
º
º
".
|
".
º
------
-—-
_--—-
__––-
-ms-
*===
ams-
•==s*
*-*
•ºmsºmº- - P-
- --—-
- T = ---sºmºmº-T" Af-
-- ſ
†ºsmºs-smºs- - ------"
- c. ſ/-----
- ass=-“ --- A **
*
sº-ºmºs º-º-º-º- smsºmºmºmº-mm-º-º-mºsºm"- -** ====s ---
*-** A.----ms-sm, ------------ =". ****
*==-, , , * -ºs.
smºsºm-mºmºsºm- -E" *-
*-*-*=-sº .* * * -
------ms-------sºmeº _z= EFT * A ===
**-*m-ºsmº. *:- - - sºm-ºsmºs---
J.-s-s-s-s-s- A E -—-A-T
--—e---------" " º -- --- ... ------s-s-sº-" ºsmºs-ºs-ºs-
--- -- -- ... * *-m-sº-sº-º-º-º- 2 --- *
a '•c-—- - -
*===== sº-sºm-mm-ºs-Tºi º
*-*-mºm-smº. ; : 3 ---
*E*m-. -
*-
tº-º-º-º-º-º:
-- **-
(§ - -º-º-º-º:
APPARATUS FOR SHOWING PARALLELISM OF PLANE OF OSCILLATION.













90 THE CHANGE OF THE PLANE OF
movement be repeatedly communicated to it so that it
may rotate through arcs of different degrees about the
disc D ; then the following curious phenomenon will
manifest itself, viz. the plane of oscillation remains un-
changed. If we take into consideration that the motion
of the earth can exercise no appreciable influence on the
instrument in the short duration of the pendulum's
motion, we may take this experiment as furnishing a
convincing proof of our first proposition. In order to
investigate the truth of the other proposition, we set the
Smaller pendulum, Q Q, Swinging in a plane, for instance,
in one parallel to the rod, SR, and turn the brass ring by
means of the handle H steadily round, it will then be
apparent that while the rod rotates the pendulum will
continually shift about its point of suspension so as to
swing parallel to its original plane of oscillation.
Having now demonstrated in a practical manner these
two mechanical principles, we may proceed to investigate
the effects upon the free pendulum which should result
from the earth's rotation. Let us first imagine that the
place for experiment is exactly over the North Pole. Let
the pendulum be swung so as to coincide with the plane
of the meridian of London; them, as the earth rotates, the
point of suspension will remain fixed; but the motion of
the mass originally impressed parallel to a given plane will
continue in that plane, and consequently the latter will
coincide in the course of twelve hours with every meridian
in succession, and the apparent rotation of the pendulum
from east to west will be entirely completed in twenty-four
hours. In any other latitude between 90° and 0, a con-
tinuous and regular apparent change of motion must
occur; for, though the point of suspension rotates, the
OSCILLATION OF THE PENDULUM. 91
planes of oscillation remain parallel to each other (at least,
their intersections with the horizon); but a meridian line
of the globe does not preserve its parallelism during
diurnal motion, except at the equator; therefore, at all
points of the earth's surface, except at the equator, we
must observe this phenomenon, that the plane of motion
varies uniformly, apparently remaining to the westward,
while the earth rotates eastward quicker, however, in
high than in low latitudes.
To find the angular velocity of the pendulum's apparent
motion mathematically for any place, it is necessary to
resolve the rotation of the earth round its axis into two
directions, the horizontal and the vertical of the place of
observation; one of which, namely, that round the vertical,
is alone effective. -
The method of resolution and composition of rotatory
motion is analogous to that of rectilinear, founded on
the theorem known in mechanics by the name of the
parallelogram of forces, as may be proved in the following
way: —
Suppose a body rotate about an axis C A with an angular
FIG. 12.
C Q. 1A
velocity p proportional to the length CA, and at the same
time about another axis C B with an angular velocity q

92 THE CHANGE OF THE PLANE OF
proportional to the line C B, then the resultant motion will
be a rotation about the diagonal CD of the parallelogram
C B D A, having an angular velocity m. proportional to the
line CD, so that
p : q : n = CA : CB : DC = sin & : sin a ; sin (a +/8)
With the centre C and a radius taken as unity, let a
circle be described in the plane of the axes C B and CA
intersecting the lines CA, CD, C B respectively in a d b,
then the point d will, in consequence of its rotation about
the axis CA, describe during the instant of time T a small
arc perpendicular to the plane through the axis with a
radius equal to sin a, so that its elevation above the plane
becomes, after the lapse of T,
Tp sin a
but, in consequence of the simultaneous rotation of the
point d about the axis CD, it must undergo at the same
time a depression expressed by
To sin &
but because
\ p : q = CA : CB = sin & : sin a .
therefore
Ta sin 6 = Tp sin a...(1)
The same is true for any other point of the line CD,
hence the diagonal CD is the resultant axis of rotation.
Considering the point b we have for its elevation, in con-
sequence of its original rotation about CA,
Tp sin (a +8),
but in consequence of the resultant rotation we have for
the same expression
Tn sin 8
OSCILLATION OF THE PENDULUM. 93
therefore :
p sin (a +8) = n sin &
which gives by substitution in equation (1)
p : q = n = sin & : sin a sin (a + 8)
This theorem may now be applied to determine the
angular velocity with which the plane of oscillation of a
freely-suspended pendulum moves in the azimuth from east
to west. Let L be any place on the earth, and let C A repre-
sent the amount of rotation of the earth about its axis PP'2
FIG, 13. y

, -T-
Pſ
for a short interval of time, then drawing through L the
vertical c L and the line H H', perpendicular to it through
the centre of the globe, we have by completing the rectangle
A B CD, replaced the whole rotation CA by its two compo-
nent parts C B and CD. But with regard to the rotation
CD about the axis HH, the pendulum at I will undergo the
94 THE CHANGE OF THE PLANE OF
same conditions as if placed at the equator, whence it
follows that the direction of the plane of oscillation and
the velocity with which it moves are not influenced by the
component rotation CD about the axis H. H. It must
therefore be admitted that the velocity will be the same
as if a rotation about C B alone took place. Hence we
conclude that the plane of oscillation of the pendulum at
L must appear to turn about the vertical of that place with
a velocity of rotation equal to that which the earth would
have if it moved with the speed expressed by the com-
ponent C B instead of that represented by the resultant O C.
In other words, the velocity of the plane of oscillation is
to that of the earth's rotation as CB to CA, but this ratio
* = cos Poi, - sin Lc E.
The velocity of rotation is therefore proportional to the
sin : of latitude of the place at which the experiment is
made.
The following demonstration is simpler than the prece-
ding, and perhaps appears more evident to those who are
not familiar with the theorems of mechanics : —
Let A (fig. 14) be the place of observation, and q denote
its latitude, draw AZ and the tangent A P which intersects
the earth's axis in P, then the point A describes daily the pa-
rallel circle AA’ a aſ while the vertical Aztraces the surface
of a right cone. The pendulum swings in a vertical plane,
and its intersection with the horizon indicates every mo-
ment the direction of the plane of oscillation. If the earth
were at rest the angle between this line of intersection and
the meridian would remain constant; it is now our object
to find the changes of this angle, caused by rotation. To an
observer standing at P the horizon at A will appear to revolve
OSCILLATION OF THE PENDULUM. 95
about a cone the apex of which is at P, while the vertical
A Z through which the pendulum passes in each oscillation
must always remain perpendicular to the revolving horizon.
The point. A retaining throughout the same distance from
P will appear to a person standing at P as if it described
FIG, 14.
L-- *
an arc A a with a radius PA. The length of the arc A. Q.
must be the same as that described on the parallel circle,
but the corresponding angle at the centre P of this arc is
smaller than that at G, the centre of the parallel circle.
Therefore, while the point travels in twenty-four hours

96 THE CHANGE OF THE PLANE OF
round the whole circumference A A'a aſ its apparent revolu-
tion round P will not be fully accomplished. The radii
P A and P A represent in every position the meridian of
the place. Let A B (fig. 15) be the direction of the plane
IFIG. 15.

P/ p
E B/
of oscillation in A, then we have for the direction of the
same plane at the point A’ the section A'B' parallel to AB,
the angle P A B has passed to B', A' P, i. e. the meridian has
apparently moved from the plane of oscillation by the
OSCILLATION OF THE PENDULUM. 97
angle APA’ = a, Let p denote the angle of rotation be-
tween the positions G A and G. Aſ and R the radius of the
earth, then
A G = R cos q, A P = R cot ºp
AA’ in the circle about G = p R cos p
/ *E= }
A. A 22 92 ,, P = a R cot ‘p
whence a = p sin ºp.
To this theoretical proof of the theorem relating to the
change of the plane of oscillation we may add an experi-
mental one due to the ingenuity of Professor Wheatstone.
In the apparatus we use to show the principle of parallelism
of the planes of oscillation of a pendulum suspended freely,
we can only make the pendulum swing vertically with re-
gard to the ring the plane of which produced is identical
with the horizon, and, so long as we make gravity the moving
force in our experiment, we cannot effect vibrations in
planes which have any other inclination. Professor Wheat-
stone hit upon the expedient of substituting for the force
of gravity that of elasticity. For this purpose he had a
wooden semicircular arch A A' constructed (see fig. 16),
graduated in 180°, so supported at the centre c as to be
movable round its vertical axis. This was intended to
represent one of the earth's meridians, which, in the course
of twenty-four hours, would complete a whole revolution;
D D' would therefore be a section of the equator. He then,
in accordance with his adopted principle, applied a spiral
spring C C, one extremity of which he fixed on the dia-
meter, so that its prolongation passed through the centre,
the other was attached to a small clasp capable of sliding
along the arch; by this means the chord CC' could be
brought to any position relatively to the equator D D, the
H
98 THE CHANGE OF THE PLANE OF
arc c D always representing the latitude of the place c’.
If the chord be made to vibrate in a certain plane, and the
arch is at the same time turned about its axis, the following
remarkable phenomenon will become manifest, viz. that
the planes of vibration will always remain parallel to the
first, in whatever oblique position the chord may have
been placed; but if the chord be placed at right angles,
FIG. 16.
r:EEs-
e= * * *-āº, º T. #
WHEATSTONE's APPARATUS.
then the vibrations will always take place in one and the
same plane, whence it follows that in this case the chord
and the arch complete their revolutions in the same time,
illustrating the motion of the free pendulum at the poles.
In all other cases the revolution of the arch will be com-
pleted before that of the chord, and the ratio of velocity
with which this takes place depends on the magnitude of
the angle which the chord makes with the horizontal, or,
in other words, on the degree of latitude. Let us suppose

OSCILLATION OF THE PENDULUM. 99
that the chord is placed at an angle of 30°, then it will be
observed to make only half a revolution during the time
that the arch completed the whole, and
as sin 30° = },
the theorem is proved for this angle. A similar result
would be obtained from any other angle, the frac-
tional part of a chord’s rotation during a complete revo-
lution of the arch being expressed by the sine of the
angle at which the chord is placed; but as the sines are
in general irrational, we must be satisfied with an approxi-
mate verification.
Another apparatus for the same purpose was invented
by Mr. Sylvestre. He wished to exhibit mechanically the
simultaneous motions of the earth and different horizons
as they really occur in nature. Let P KN E (fig. 17) be a
FIG, 17.
P
circular ring representing a meridian of the earth. Z.
is a point taken upon the circumference at any latitude,
N P is the axis of the earth, o/, the vertical of the place

H 2
100 THE CHANGE OF THE PLANE OF
of observation, and y its angle of latitude. Let O R be
the radius of a circular plane, the centre of which always
coincides with that of the earth, and the axis of which is
the line o / which passes through the line of observation.
This circle, movable about its axis, and with a diameter
which remains invariable after having been once chosen,
is capable of taking different inclimations towards the plane
of the equator according as the latitude changes. Let S R
be the path upon the meridian of another circular plane
the axis of which lies in that of the earth. This plane
can only be raised or lowered along its axis, so that it will
always be in contact with the circle of latitude, whatever
the inclination of the latter may be. It is also obvious
that the two planes o R and S R must always include an
angle equal to the complement of latitude. According
to the position of these two planes, which are dependent
for motion on each other, the angular velocities of their
circumferences in contact are in the inverse ratio of their
radii. We have, therefore,
ang. vel. O R : ang. vel. S R = S R : O R
but S R = O R sin ºy
... ang. vel. O R = ang. vel, S R sin y.
Hence, if we choose a certain place L, and then turn the
ring about its axis, the latter will by its motion cause the
vertical of the place to describe a cone, and the circle of
latitude will also turn about its axis, so that when the ring
has completed an entire revolution, the horizon of the
place chosen will only have turned a fractional part, ex-
pressed by the sine of its latitude.
A similar and more practical instrument is shown in
fig. 18; it was invented by a schoolmaster of Haverford-
OSCILLATION OF THE PENDULUM. | 0 |
west, and improved by Professor Wheatstone. On turning
the stand round the ball revolves, but while it completes
an entire revolution the arrow on the top of the axis will
only describe a fractional part depending on the angle of
inclination of the axis to the horizon. The graduated arc
serves to set the axis at any angle required, and the screw
fixes it in this position during the experiment.
FIG. 18.
• *-*
=5
5
*::=
--
It now remains for us to compare with the results of
observation the theory that the angular velocity of the
plane of oscillation is equal to that of the earth's rotation

102 OSCILLATION OF THE PENDULUM.
multiplied into the sine of latitude. We may here pre-
pare the reader for certain discrepancies which must occur
in practice, and which are due to the following causes: the
resistance of the air, the torsion of the suspending wire,
and the lateral impulse. The disturbing influences of
these forces we intend investigating theoretically in a
subsequent chapter. We now proceed to state the results of
actual observation.
103
CHAPTER VII.
EXPERIMENTS MADE WITH THE FREE PENDULUM TO PROWE
THE ROTATION OF THE EARTH.
FTER the well-known original experiments with the
pendulum made by Foucault, those of General
Dufour, who was assisted by Messrs. Wartmann and
Marignac, are the next in importance. The pendulum
which they used was 20 mêtres in length, the bob, which
was of lead, weighed 12 kilogrammes (about 26% lbs.) and
it was capable of swinging rather more than three hours, but
it could only be observed during two hours and a half, for
after that time the oscillations were too small. Four ex-
periments were made in the meridian and four in the line
perpendicular to it, commencing alternately from either
extremity without any appreciable difference. Each ex-
periment was continued till the pendulum had attained a
deviation of 25°; the time was measured by a second-beat
chronometer. The exact moment when the pointer fixed
to the bob passed over one of the diameters, traced on the
disc underneath, could be accurately ascertained so long
as the amplitude of the oscillation was more than 0' 70.
The following is the mean result of four observations on
each line : —
A deviation of 25° in the meridian was
accomplished in . e º e . 2,376 hours
A deviation of 25° in the meridian was
accomplished in . º g wº . 2,110 ,
Difference . º . 0,266 ,
*mºmºmºrº
104 EXPERIMENTS MADE WITH THE FREE PENDULUM
The difference is, therefore, about a # of an hour. This
proves that the velocity of deviation is not the same in the
meridian as in the perpendicular, and the formula a -psin ºp
(p being the angular velocity of the earth, and p the
latitude of the place) is not rigorously true, for by it we
obtain for a deviation of 25° at Paris 2,306 hours, a
number which lies between the former two, but which
is nearer to the first than to the second. This difference
of time is so great that it cannot be attributed to an error
in the experiment, but is owing to an acceleration in the
line perpendicular to the meridian.
Mr. Morren repeated these experiments at Rennes and
obtained similar results. The pendulum was 19-7 mêtres in
length, the bob weighed about 66%lbs., the amplitude of
the first swing amounted to little more than three mêtres.
He also found that the deviation was greater when the
oscillations were started in the perpendicular plane and
less when in the meridian plane. He was well aware that
these differences could not be due to currents of air or to
the attractive force of the neighbouring walls, for the great
mass of the pendulum rendered fallacious any such surmise,
but he still could not arrive at the real disturbing power,
which General Dufour conjectured to be the centrifugal
force. In order to decide this point other experiments
were necessary. On the 10th of February 1851, Mr.
Bravais communicated to the French Academy the results
of the experiments he made in the Great Hall of the Obser-
vatory with a pendulum ten mêtres in length. -
He employed two distinct methods—that of direct ob-
servations, and that of coincidences. By the direct method
he observed the duration of from 900 to 1,200 levogyric
oscillations of the pendulum, i. e. from west to east over
the south, and then the deviation of a similar number of
TO PROVE THE ROTATION OF THE EARTH. 105
dextrogyric (from east to west) oscillations of the same
pendulum; the distance from the point of suspension to
the centre of the bob varied from 10187 to 101.97 milli-
mêtres. The following table exhibits the observed
deviations : — *

L ge Dextrogyri *
§s *. Difference
May 5 6,39887 6,39823 0,00064
, 11 6,39925 6,39849 0,00076
,, 16 (A) 6,39815 6,39751 0,00064
,, 18 (A) 6,39959 6,39863 0,000.96
,, 16 (B) 6,40106 6,40032 0,00074
,, 18 (B) 6,401.16 6,40044 0,00072
Mean difference . . . . . . 0,00074
The difference furnished by theory is equal to 0,000716,
which agrees satisfactorily with the above result.
The other method offers still more accurate results.
Two pendulums of different lengths, but similar in other
respects, are placed in the plane of the meridian at a
distance of 7 decimètres from each other. The space
between the point of suspension and the centre of the
bob was 10216 millimètres for the longer pendulum,
and 101.15 millimetres for the shorter. The levogyric
and dextrogyric oscillations were communicated to the two
pendulums respectively at the same time, and the moment
of passing the meridian was observed by means of a
telescope, the optical axis of which intersected the two
threads when at rest. The number of oscillations between
two successive coincidences of the two pendulums was
noted down; let them be mºnſ, then n’= n + 1 where m/
refers to the short pendulum. The series of observations
106 EXPERIMENTS MADE WITH THE FREE PENDULUM
being finished, the pendulums were stopped and afterwards
set in motion with reversed directions. The coincidence
was again observed, and the two new numbers determined,
N and N' = N + 1 denoting by A the latitude of the place,
r the durations of a sidereal day, t the duration of the
conical oscillation of the long pendulum uninfluenced by
the effect of terrestrial rotation, we must obtain, if the
theory be exact,
sin A 1 { 1 1 }
T T #Tº l n + 7 NTN'
The experiment of the 25th of May gives
q) = 207-86 N = 217.82
70’ = 208-86 N’ = 21.8-82
The experiment of the 10th of June gives
m = 206-31 N = 215-96
70’ = 207-31 N’ = 216-96
After making the necessary corrections, we derive from
these the difference of duration between the levogyric
and dextrogyric oscillations:
Difference found May 25th. . . O-000725
59 ,, June 10th. . ... 0-000710
Whence it follows, that for a pendulum of 10 mêtres which
turns from west to east, the angular velocity is retarded
at Paris by 11.4 seconds of space for each second of time,
and that it is increased by the same quantity when the
rotation takes place from east to west.
This inequality in the duration of dextrogyric and
levogyric oscillations of the free pendulum was confirmed
by experiments made in Rio de Janeiro (in 22°54' south
TO PROVE THE ROTATION OF THE EARTH. 107
latitude) during the months of September and October
1851 by Mr. D’Oliveira. The following is an account of
his experiments : —
Mass of the pendulwm employed.
A hollow sphere 24lbs. in weight furnished with a
pointer.
*
Suspension of the pendulum. &
An untwisted hempen thread fastened to the ceiling,
and having at its other extremity a piece of iron fitting
into the hollow sphere, so adjusted that the extremity of
the pointer corresponded with the point of suspension,
the length of the pendulum being 4,365 mêtres.
Arcs described by the oscillations of the pendulwm.
The first experiments were made by imparting to the
pendulum a swing of 5° 14' 44", but in the later ones the
arcs were 7° 51' 41’’. The arcs corresponded to the length
of the oscillations measured by their tangents, which
amounted in the first to 4 decimétres, and in the second
to 6 decimétres.
Trajectory described by the pendulum.
The trajectory formed almost always in a double os-
cillation a very elongated ellipse, the minor axis being
hardly large enough to show the deviation of the pendulum
by the tracings in fine sand made by its pointer. The
square frame containing the sand had the cardinal points
marked on it, and so placed under the pendulum that the
intersection of two diagonals corresponded with the ex-
tremity of the pointer.
The pendulum was put in motion first in the direction
108 EXPERIMENTS MADE WITH THE FREE PENDULUM
of the meridian, and started alternately from the north
and south sides, and afterwards the experiments were
repeated in the direction of the parallel circle. Similar
experiments were also made in several directions inter-
mediate to these, two south-west and south-east. The
results obtained from those experiments, which were
considered as accurately carried out, are the following : —
1st. The motions of the pendulum describing ellipses
in the meridian or parallel circle always took place in op-
posite directions, that is, when the ellipse in the meridian
was traced from right to left (like the rotation of the earth
looking towards the south pole) the pendulum moved
from left to right in an ellipse described in the direction
of the Parallel. A reduction in the thickness of the
thread of suspension lessened the length of the minor
axis.
2nd. Lifting the pendulum three decimétres from the
vertical, in order to make it describe the greatest arc, and
allowing it to swing during 30° first in the direction of
the meridian, and afterwards in the direction of the
parallel circle, the deviation was found in the first case to
be 5° 13' towards the east, and in the other case 5° 12'
towards the south.
3rd. Measuring the lengths of the major axis of the
ellipses by means of the tracings on the sand, they were
found to be #metres and #metres respectively with
regard to the meridian and Parallel.
4th. Moving the pendulum in various directions, a line
was found along which the trajectory had no tendency to
move either to the right or left, and at the same time the
pendulum continued to move without deviation, so that
TO PROVE THE ROTATION OF THE EARTH, 109
this line marked out the position of an invariable plane
of oscillation. This plane made with the Parallel an
angle of 11° 18' 40", or very nearly half the latitude of the
place of observation.
5th. The motion of the pendulum made in two directions
midway between the meridian and Parallel, showed that
the deviation of the south-west portion was much slower
than in the meridian, while that in the south-east was
Quicker; but all these deviations were in the same direc-
tion, namely, in that of the rotation of the earth.
Before proceeding to give an account of our own
observations, we will give the opinion of a great authority
on an experiment made in Turin. Plana, in the “Memor.
dell Acc. di Torino, xiii. 55, 56, says that the time of an
entire revolution of the plane of oscillation was found to
be 34 hours 49 minutes instead of 33 hours 48 minutes,
according to theory. To clear up this difference the ex-
periment was repeated, but with a worse result, for this
time the whole revolution occupied 40 hours. Plana
justly attributed this to the defects of Foucault's mode of
suspension, which it will be remembered consists in
squeezing one end of the thread between the metallic
plates screwed to the ceiling (see fig. 9). The disadvan-
tages of this mode of suspension are the bending of the
thread at its upper extremity, and its twisting so as to
cause a motion of the pendulum round its axis; the former
causes the amplitude of the Swing to diminish, the latter
tends to change the azimuth of the plane of oscillation.
The disturbances produced by the torsion may be calculated
provided the thread is homogenous throughout, and its
section a perfect circle. It is impossible to preserve the
cylindrical form of the thread at its upper extremity owing
110 EXPERIMENTS MADE WITH THE FREE PENDULUM
to the compression of the two plates: hence changes will
occur which vary with the different azimuths, and are
consequently beyond calculation. It was therefore evident
that another mode of suspension was required which
should give these researches a higher degree of accuracy.
Hansen suggested the following method:—Instead of a fine
thread, a thick wire should be employed, so that torsion
would be impossible; the bob is so fastened that its centre
of gravity lies in the prolongation of the axis of the wire,
the upper end is curved, and teminates in a spherical seg-
ment of hardened steel, which, placed on a horizontal plate
of the same material, forms the suspension (see fig. 19).
HANSEN's SUSPENSION.
In order to avoid all sliding motion and to effect a rolling
one, during the swing of the pendulum, the segment and
plate are ground but not polished, by which friction is

TO PROVE THE ROTATION OF THE EARTH. 111
greatly diminished. Though the theory of this arrange-
ment has been very ingeniously developed by the learned
inventor, yet we are not aware that as yet any practical
trial of it has taken place. We ourselves, in conducting
Some experiments, made use of a suspension similar to that
proposed by Hansen, with one essential difference, viz. a
cone was substituted for the spherical segment working
freely in a corresponding socket (see fig. 20). Another mo-
Fig. 20.
DETTMANN’s SUSPENSION.
dification was made in the mode of liberating the pendulum;
in all previous experiments the bob was drawn up to the
required angle and maintained in that position by a thread,
which was either cut or burnt after perfect rest was ob-
tained. Now, it will be easily understood that by this
method the particles of the thread cannot be severed
simultaneously, so that there is a slight jerk before com-

112 EXPERIMENTS MADE WITH THE FREE PENDULUM
plete liberation. To obviate this, I fixed on the graduated
board underneath the pendulum, at the proper distance,
two small perpendicular posts pierced at the top by two
horizontal screws, the points of which were accurately
ground and capable of coming together; one end of the
loop destined to hold the pendulum was then compressed
between them, the pendulum was then raised till the
pointer could be looped in; after a short interval, to allow
the instrument to become quiescent, a turn of one of these
fine-threaded screws set the pendulum free instantaneously,
while the loop fell to the ground (see fig. 21).
The following is a short account of the apparatus
used, and the experiments I made, in the Hall of King's
College, London, in the year 1859 : —
The bob was a truly-turned ball of brass, weighing
40lbs., the suspending medium was a thick steel wire, the
length of the pendulum was 17 feet 9 inches. The
amplitude of the first oscillation, as measured by its
tangent, amounted to 6°42'22", and during the time of the
experiment, which was about half an hour, the arcs were
not much diminished. As I had to demonstrate to a large
number of spectators I encountered considerable difficulty
in rendering the small deviations of the plane of oscilla-
tion visible to all. I accomplished this in the three
following ways: —
1st. I placed on the first two lines of division (count-
ing from the meridian in which the pendulum commenced
its swing) two small wooden cones, so that the diameters
of their bases coincided with the line that marked the
degree; their weight was regulated so that when the plane
of oscillation deviated sufficiently to pass through the
points of their azimuth, they were struck by the pointer
TO PROVE THE ROTATION OF THE EARTH,
113
FIG. 21.
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S--i >=
* º
--~~-
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THE PENDULUM WITH IMPROVED SUSPENSION.
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114 EXPERIMENTS MADE WITH THE FREE PENDULUM
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TO PROVE THE ROTATION OF THE EARTH. 115
and knocked down. By this means everyone present was
enabled to observe the time taken for each degree of
deviation.
2nd. At the fourth degree from the meridian line I
placed a small cannon loaded with gun cotton, across the
touch-hole of which was a fine platinum wire in connec-
tion with one of the poles of a galvanic battery, the
metallic suspension being connected with the other pole.
The object of this arrangement was that, when the deviation
amounted to four degrees, the pointer should come in
contact with the platinum wire, thus completing the circuit
and firing the cannon. This effect took place after the
proper time furnished by theory. &
3rd. The third method will be best understood by
reference to the figure. (Fig. 22.) All lights in the apart-
ment having been lowered, the lamp A was lighted. B
represents a rectangular box open at the top and back.
The front G being of ground glass and slightly slanted,
D D' are divisions corresponding to the degrees of deviation.
A well-defined shadow of the bob and pointer was by this
arrangement projected on the glass, and the gradually
increasing deviation could throughout its progress be ob-
served with the greatest ease by every person in the room,
and the moments of passage through the points of division
noted down. The results on the whole were most satis-
factory, owing mainly to the accurate suspension and
mode of liberation, for, as far as ocular observation went,
no elliptical swing took place.
Although the number of pendulum experiments which
we have described in this chapter is but small, yet we
venture to believe that they afford sufficient evidence of
the earth's rotation.
I 2
116 THE GYROSCOPE.
CHAPTER VIII.
THE GyRoscoPE. -
N the autumn of 1852 Mr. Foucault presented the Aca-
demy of Sciences of Paris with another experimental
proof of the earth's rotation, which was founded upon the
fixity of the plane of rotation of a body suspended freely
and revolving about one of its principal axes. The instru-
ment he used he called the “Gyroscope; many modifications
have since been made, and indeed, many years before Fou-
cault's invention, Bohnenberger, a German astronomer,
constructed a similar apparatus, which he employed to show
the precession of the equinoxes, and which, by the sugges-
tion of Laplace, was introduced in the Ecole Polytech-
nique for the purpose of explaining this phenomenon. In
order fully to understand Foucault's application of the
gyroscope, it is necessary to have a clear conception of the
laws of rotation, which form a very difficult part of me-
chanical science, and as the modified form now so com-
monly used demonstrates them practically, we will first
give a description of it.
It consists of a heavy solid disk D (fig. 23), turned with
extreme accuracy on an axis A A', which terminates in points
PP, the centre of gravity of the disk and axis being exactly at
These points are inserted into suitable sockets EE, on oppo-
site sides of a light but strong brass ring R R. At the ex-
tremities of the diameter of this ring, perpendicular to the
THE GYROSCOPE. 117
THE GYROSCOPE,

118 THE GYROSCOPE.
axis, it is inserted in the same manner, into another
ring or semicircle SS, which by means of a vertical axis v
secures the perfect mobility of the inner ring, and thus the
central heavy disk can turn in any direction, or remain at
rest in any position in which it may be placed. The disk is
made to rotate by pulling a string wound round the axis.
The following are the experiments which can be made
with this instrument : —
First experiment—When the wheel is in rapid rota-
tion, detach the inner ring from the outer, lifting it
carefully together with the revolving solid which it sup-
ports; then, holding the ring firmly in one or both hands,
attempt to change the position of the body. It will be
found that when the solid is rapidly revolving, the diffi-
culty of displacement is very great, and the displacement
which is effected is perpendicular to the direction of the
force you apply. This is a phenomenon in itself remarkable,
and the explanation of it is the key to most of the other
experiments with the instrument, and generally to the pe-
culiarities of rotation, which it is intended to illustrate.
This may be expressed in the form of a theorem as follows.
When a disk is thrown into very rapid rotation round an
axis, and another power tends to turn it round another
axis perpendicular to the first, then the result will be a
rotation round a third, perpendicular to the plane con-
taining the other two; i.e. the power applied may be said
to act ‘ tangentially.”
Second eacperiment—Suspend the apparatus which you
had removed for the last experiment by a string attached
at the outer part of the ring at the extremity of the axis
of the revolving solid. The ring being then placed in a
THE GYROSCOPE. II 9
horizontal position, the string held in the hand, and the
apparatus left to pursue its own course, it will be seen,
contrary to expectation, that the apparatus instead of falling
straight down, as it would do if not rotating, retains its
horizontal position, and at the same time marches round
the string with a slow uniform movement. The string
remains during this motion in a vertical position. The
external forces impressed on the solid in this case are
the weight of the wheel acting downwards at its centre
of gravity, and the force of the string, which is equal to
the weight of the wheel, acting upwards at the point of sus-
pension. This pair of forces would evidently produce a
rotation about a horizontal line in the equator of the wheel,
and we see that their effect is, like the forces applied in
the first experiment, to make the axis of the wheel deviate
towards this line.
Third ea periment — Let the apparatus be replaced in
its stand. Having set the wheel rotating, hang the
smaller weight furnished with the instrument at the point
where the string was applied in the preceding experiment.
Then, if the fixed vertical circle be set free, this weight with
the whole apparatus will be carried round with a slow
uniform rotation in a horizontal plane. The explanation
of this will evidently be the same as that of the last
experiment.
Fourth eaſperiment—While things are in the condition
just described, hang the heavier weight at the other end
of the axis; the motion will then be reversed.
Fifth eaſperiment — While the wheel is rapidly rotating,
and the vertical semicircle perfectly free, hang the weight
120 THE GYROSCOPE.
as in the third experiment, and while the apparatus is
moving round suddenly fix the vertical semicircle. The
moment this is done the precessional motion ceases, and
the weight falls as it would have done from the action of
gravity, if there had been no rotation. It appears, then,
from this experiment, that the supporting of the weight
contrary to the natural effort of gravity is only effected by
means of the rotation of the ring, when the accompanying
precessional motion is not prevented.
Siaºth eaſperiment — In the last experiment it ap-
peared that when the precessional motion is prevented,
the force applied to the axis has the same effect whether
the wheel be in rotation or not. To illustrate this further,
remove the weight, and set the wheel in rotation. It
will then be found that the wheel may be turned com-
pletely over, as if it were not in rotation.
If the gyroscope were made to revolve with great rapidity,
its axis would, if perfectly unrestrained, retain a constant
absolute direction, and therefore perform an apparent revo-
lution about the direction of the earth's axis, just as the
star to which it points. If, however, the axis be restrained,
so that it can only move in a given plane, it will oscil-
late about that line in the plane, which is nearest to
the earth’s axis. These oscillations are isochronous,
and will be more frequent the greater the velocity of
rotation.
This principle may be thus enunciated in the words of
Foucault: —
When a body turns upon a principal axis, and a force
or system of forces tends to produce another rotation not
parallel to the former, then the resulting effect is a dis-
THE GYROSCOPE. 121
placement of the axis of original rotation, which directs
itself in the way most favourable to the parallelism of the
two rotations. The new axis of rotation may be either
fixed or movable, and dependent on the position of the
body; when it is fixed, the turning body tends to a definite
position, and when movable with the axis of the body,
the latter changes its direction continually, without ever
arriving at a state of equilibrium. In order to elucidate
these remarkable facts of rotary motion, we cannot do
better than introduce to the reader the gyroscope, invented
by Mr. Fessel of Cologne (see fig. 24), which is perhaps
FIG. 24.
FESSEL's GYROSCOPE MODIFIED BY WHEATSTONE.
the best adapted for the exhibition and explanation of them,
especially in the modified form given by Professor Wheat-
stone. A beam is capable of moving round a horizontal
axis, which is itself movable about a vertical one, so that
it may be turned in any direction about a fixed point. At
one end of the beam a horizontal ring is fixed, which carries
a heavy disk, the axis of which is in a line with the beam.
At the opposite extremity a shifting weight is attached, by

122 THE GYROSCOPE.
means of which the equilibrium may be established or
disturbed at will. If the instrument be put in equilibrium,
and the disk be made to revolve rapidly by means of a
string wound round its axis, then the beam has no tendency
to displace itself in any direction; but on moving the weight
towards its centre, and causing the disk to preponderate,
then, if the latter rotate-from right to left, the former will
turn about its vertical axis in the same direction, and vice
versâ. If the weight be allowed to preponderate, contrary
effects will take place. The velocity of rotation round the
vertical axis increases in proportion to the amount of dis-
turbance of the equilibrium, but notwithstanding the in-
creased or diminished action of gravity on the disk, its
axis of rotation always preserves that inclination to the
vertical axis at which it was originally placed. When the
equilibrium is disturbed while the disk is at rest, gravity
acts so as to turn the beam round a horizontal axis, that is,
about an axis perpendicular to the plane which contains
the axes of the two original rotations.
A similar composition of forces takes place by impressing
on the beam in equilibrium a motion round the vertical
axis. If the disk rotates at the same time from right to left,
then the end of the beam conveying the latter will ascend,
provided that the pressure tending to produce motion round
the vertical axis acts in the same direction, but if it act in
the opposite, it will cause the disk to descend; hence, as in
the preceding case, the beam is constrained to move about
a horizontal axis perpendicular to the plane which contains
the other two. The beam ascends and descends in a like
manner whenever the motion round the vertical axis is
accelerated or retarded. If the disk rotate from right to
left with its own weight predominating, the rotation round
THE GYROSCOPE. 123
the vertical is from left to right. Accelerating it causes
the disk to descend, and retarding, to ascend.
Let the ring carrying the disk be removed, and the
tightening screw unfastened, then the axis of the rotating
disk is free to move in a plane; and if the outer is con-
strained to move about the line in this plane, the movable
axis will place itself so as to coincide with that line; but if
the fixed axis be in a different plane, the movable will
assume that position which approaches most to the former.
This phenomena may be thus generally explained.
Suppose the apparatus be in the position indicated in
the figure, let the axis D be held horizontal, the weight at
F being removed, and let the disk be made to rotate so
that its front A has an ascending motion. This rotation
communicates to all parts of the disk tangential velocities,
which may be resolved into their horizontal and vertical
components. Let the ring now be liberated, so that it can
be acted upon by the force of gravitation. The next effect
will be that the axis of rotation sinks down slightly, and
the disk in consequence takes an oblique position; the
horizontal components of the former motion are not
changed by it with respect to the position of the disk, but
the vertical are: they leave its plane, and are in front when
ascending towards the left, and behind when descending
towards the right. A lateral force is consequently exerted
upon the disk, which draws its front towards the left, and
its back towards the right. Both actions supporting each
other effect a rotation of the whole apparatus round the
vertical axis E, from left to right. The striking phenomenon
of the pressure on the axis of a freely rotating body acting
in a direction perpendicular to the plane of rotation is
therefore easily explained, but this effect does not terminate
124 THE GYROSCOPE.
the process of actions which modify the motion, for as soon
as the disk begins to turn about the vertical axis E, the
horizontal components leave the plane of rotation, and
exercise a force upon the disk which draws its lower part
to the right, and its upper to the left. The resultant of
these two directions is an ascending motion of the disk about
the horizontal axis D ; that is, contrary to the direction im-
§§
Jīlī ||||||.
wRCATSTONE'S MAGNETIC GYROSCOPE.
parted to the system by gravity, yet equal to it in power.
Hence, by putting weights at F, the horizontal motion may
be stopped, or changed into one from left to right. Other
forces may be substituted for weights to act upon the ring
of the gyroscope, which tend to produce similar effects. This
has been practically carried out by Professor Wheatstone in



THE GYPOSCOPE. 125
his magnetic gyroscope. (See fig. 25.) In this instrument
a portion of the ring consists of magnetised iron, immediately
underneath which, at a slight distance, is a small horse-shoe
magnet, movable on a vertical axis; by this means its
poles can be reversed, while those of the upper magnet
remain fixed. By this arrangement a repelling or at-
tracting force can be applied as a substitute for putting on
or removing weights.
Another apparatus worthy of mention was invented by
Professor Magnus of Berlin. It consists of two disks (see
jig. 26) A B and C D, which can be put in rapid rotation,
with a nearly equal amount of velocity, by means of the
handle E F ; these are supported by arms connected by the
rod 'm n, which can turn about the horizontal axis q r.
The apparatus rests on the vertical axis v w, round which
the whole moves easily. The screw 2 serves to arrest the
motion about the horizontal, and the piece of metal t w to
arrest, accelerate, or retard that about the vertical. The
following phenomena exhibit the results of combined
rotation : —
1. If both disks rotate in the same direction, and no
weights or equal ones are suspended at m m, then the
axis m 'm remains in its original position.
2. If the weights suspended on both sides of m n are of
different magnitudes, then the whole apparatus will turn
about the vertical axis in one or other direction, according
to the side on which the weight predominates, whereby
the position of the axis m n is unappreciably altered, even
though the overweight be very great.
3. If the motion round the vertical axis be accelerated
by means of the small rod t w, then the loaded disk will rise,
and vice versä. - -
126 THE GYROSCOPE.
FIG. 26.
MAGNUs’ APPARATUS.

THE GYROSCOPE. 127
4. If the screw 2 be tightened while the apparatus is in
motion, then the rotation about the vertical axis ceases
suddenly, and again commences on its being loosened.
5. If the disks rotate with equal velocities in opposite
directions, then the side which is the heavier loaded will
sink.
6. If the disks are made to rotate with opposite directions
and equal velocities, after having been placed at unequal
distances from the centre, the whole apparatus will be found
still extremely sensitive. Whence we conclude that the
distance at which a revolving body is placed from the
vertical axis has little influence on the motion of the in-
strument. The explanation of these phenomena rests on
the same principle as that of Fessel's gyroscope.
The instruments we have described, although they
illustrate the general phenomena of rotation, are not
capable of rendering apparent the diurnal motion of the
earth : for this purpose, not only greater accuracy, but
additional arrangement in the contruction, are required.
The principal parts of M. Foucault's gyroscope are, as in
the other instruments, the heavy disk and ring; but the steel
axis of the former is provided with a toothed wheel, and
from the ring two knife edges project in a direction per-
pendicular to it. By this means the apparatus, with its ring
in a horizontal position, can be placed in connection with a
combination of multiplying wheels, in order to impart to it a
very rapid rotation. This having been effected, it is placed
with its knife edges upon a vertical ring, which is sus-
pended by a thread without torsion, and rests on a pointed
pivot. (See fig. 27.) The small screws about the ring, one
pair of which is movablein a horizontal, the other in avertical
direction, serve to adjust the apparatus, so that the centre
128
THE GYROSCOPE.
| |
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isiº 23.
i_
ſº
:
... =
#| ||
FoucAULT's GYROSCOPE FOR SHOWING THE ROTATION OF THE EARTH.


THE GYROSCOPE. 129
of gravity may lie in the prolongation of the suspending
thread. If this is well carried out, the force of gravity can
have no influence either on the rotation of the disk, or
upon the entire system. The plane of rotation of the disk
remains fixed, therefore, in the original position, not par-
ticipating in the diurnal motion of the globe. The relative
displacement arising from it may be observed by a micro-
scope placed near the apparatus. Foucault described the
phenomenon in the ‘Journal des Débats’ of September 22,
1852, in the following words : —
‘The axis of a whirling body being fixed in absolute
space, appears when examined under the microscope to
retrograde from east to west, and travels in the same con-
tinuous manner upon the glass of the instrument as the
image of the celestial bodies in the focus of the astronomi-
cal telescope. I have furthermore established by experi-
ment a singular property of rotating bodies, which I found
à priori by pure reasoning — I mean the force of orienta-
tion, which tends to place the axis of the body parallel to
that of the earth, and at the same time to place the two
rotations in the same direction. This force of orientation
shows itself always when the axis of the whirling body is
constrained in a fixed plane, but can move freely within
it. The principle which has guided me in my researches
on the subject may be enunciated as follows. When a
body turns upon a principal axis, and a force or a system
of forces tends to produce another rotation not parallel to
the former, then the resulting effect is a displacement of
the axis of the original motion, which places itself in the
most favourable direction for the parallelism of the two
rotations. The new axis of rotation may be either fixed
or movable, and dependent on the position of the body.
IX
130 e THE GYROSCOPE.
When fixed, the turning body tends to a definite position,
which is in accordance with the motion of the earth; when
movable with the axis of the body, the latter changes its
direction continually, without ever arriving at a state of
equilibrium.
These experiments of M. Foucault to render the rotation
of the earth visible have been made the object of mathe-
matical investigation by Mr. Quet, but as we purpose giving
amother analysis of it in the second part, we will content
ourselves with stating here the principal results this emi-
ment philosopher arrived at. He supposes the centre of
gravity of the turning body to be fixed, and its axis movable
only in a certain plane, which is also fixed, termed the
directing plane.
The following are the various modes in which the axis
influenced by the diurnal motion of the earth oscillates.
1. When the directing plane is horizontal the axis of
the body can only be in equilibrium when in the meridian;
this equilibrium is stable if the rotation of the body pro-
jected upon the terrestrial equator lies in the same direc-
tion as that of the earth, otherwise it is unstable.
2. When the directing plane lies in the meridian the
axis of the body can only be in equilibrium when in a
position parallel to the terrestrial axis.
3. The axis of the rotating solid will assume the same
position as in the preceding case when the plane itself lies
in a position parallel to the terrestrial axis.
4. When the directing plane is perpendicular to the earth's
axis then that of the body is in indifferent equilibrium.
5. In general, whatever be the direction of the plane, we
obtain by projecting the earth's axis upon it the direction
of equilibrium of the axis of the rotating body.
THE GYROSCOPE. 131
6. When the axis of the body lies without its line of
equilibrium, it oscillates in a directing plane about its
stable position. The oscillations, whether great or small,
obey the same laws as those of the pendulum.
7. If the axis of the body is made to oscillate alter-
nately in the meridian and in the horizontal plane, we
find for the same velocity of rotation that the vibrations
in the former are quicker than in the latter.
8. The duration of the oscillation in the meridian may
serve to determine the time of the earth’s rotation.
9. Comparing the squares of the numbers of the oscil-
lations which take place respectively in the horizontal
plane and in the meridian, during equal intervals, and
with the same velocity of rotation, their ratio will give the
cosine of the latitude of the place of observation. Hence,
supposing an instrument to be accurately constructed on
the principles laid down by the theory, an observer could
determine by it, without seeing the heavens, the direction
in which the stars seem to revolve, and the duration of
the revolution, which would afford a most convincing proof
of the earth's diurnal motion.
We cannot conclude this chapter without laying before
the reader an objection to the primary hypothesis made
by Mr. Foucault, with regard to his gyroscope, viz. that
its only connection with the earth is through its centre of
gravity. This cannot be exactly true, for neglecting even
the torsion of the thread by which the ring is suspended, the
friction of the pivots is sufficient to communicate the earth's
motion to the gyrating body. This state of things does
not exist in Bohnenberger's original apparatus, which we
will now describe, and explain its application.
The principal part of this instrument, see fig. 28, is a
K 2
132 THE GYROSCOPE.
massive globe * capable of turning in all directions about
its centre of gravity, which rests in a ring, which itself is
suspended within two other rings. If the ball be set to
FIG. 28.
BOHNENBERGER's GYROSCOPE.
rotate about a horizontal axis, then the diurnal motion of
the earth, or in our latitudes, the component part of it,
will tend to move it about the vertical. The axis will,
instead of a horizontal motion, receive one in a plane
perpendicular to the horizon, thus furnishing us with a
plane invariable in the azimuth, by means of which we
can easily see whether the earth turns or not. Strictly
speaking, this plane is only indicated by the extremities
of the axis, but the apparatus offers a more substantial
one, in the shape of the middle ring, which, like the
sphere, cannot move in the azimuth ; its stability is most
* The substitution of a spheroid in the place of a globe is not advisable,
as it leads to the erroneous opinion that the phenomena observed depend
on the flatness of the figure.

THE GYROSCOPE. 133
striking, for we may turn the instrument in whatever way
we please, it will always return to its original position.
Therefore, inasmuch as the instrument is not constrained
by the rotation of its support, it affords a means of
verifying that of the earth; it must, however, be borne in
mind that our assumption of a sole tendency to move round
the vertical is only true at the poles; at any other lati-
tude, the component horizontal of the terrestrial rotation
interferes, which complicates the phenomena. We will
now examine this force, and show how, by elimination, we
may always produce the same results as at the poles.
Let ZPH represent a meridian, A the latitude of the place
z, m, the angular velocity of the earth on its axis, CP, then
by Euler's theorem we have m sin X and m cos X for the
angular velocities about cz, and the horizontal ch. Sup-
posing Bohnenberger's apparatus to be at Z, its rotation
about the vertical is obvious, since the latter passes through
the centre. The rotation about the horizontal, C H., may
be resolved into a translation in a straight line, and into a
rotation; during the translation the apparatus remains
parallel to itself, turning at the same time with an angular
velocity m, cos A. The motion of translation having no
influence in this case, a “couple’’ is generated, the axis of
which lies in the meridian and the forces in the first
vertical; the axis will, therefore, move in the azimuth.
Such motion, however slow, would render doubtful the
observations made to prove the rotation of the earth, but
this difficulty may be obviated in the following way. The
two components of terrestrial rotation, acting separately,
impart to the apparatus the angular velocities n sin A and
m cos A; therefore acting together they give it the velocity
m, about an axis parallel to that of the earth. Let us now
134 THE GYROSCOPE.
place the general axis of the instrument in this direction,
then the vertical component of terrestrial rotation is re-
placed by the total rotation, and no force exists which
tends to give it a motion in the azimuth. The earth's
rotation communicates to the little globe a very slow
motion in the meridian in which it was placed, whereas
the apparent motion in right ascension is much quicker,
being equal to that of the earth, so that the middle ring
will appear to make a complete revolution in twenty-four
hours as at the poles.
We have thus pointed out the arrangements by which
Bohnenberger's apparatus may in all latitudes be used to
prove the rotation of the earth, under conditions as simple
as at the poles; it is free from the objection raised against
Foucault's gyroscope, viz. being connected with the earth
by other points than its centre of gravity. We propose
to show theoretically in a subsequent chapter, that by
Foucault's mode of suspension the same effect is produced
as by supporting the revolving body solely on its centre of
gravity.
135
CHAPTER IX.
THE PRECESSION OF THE EQUINOXES A PROOF OF THE
EARTH's ROTATION.
HE ecliptic, i. e. the apparent path described by the sun
during a year, cuts the equator in two opposite points,
dividing it, and being divided by it in two equal parts. It
has been observed from the earliest periods of antiquity,
that when the sun was in either of these points of inter-
Section, his diurnal revolution coincided with the equator,
and therefore the days and nights were equal; hence the
equator was called the equinoctial line, its intersections
with the ecliptic the equinoctial points, one of them the
vernal, the other the autumnal equinox.
It was evidently an important problem in practical
astronomy to determine the exact moment of the sun's
occupying these stations; for it was natural to compute
the course of the year from that moment. Accordingly,
this has been the leading problem in the astronomy of all
nations. It is susceptible of considerable precision,
without any apparatus of instruments. It is only neces-
sary to observe the sun's declination on the noon of two
or three days before and after the equinoctial day. On
two consecutive days of this number, his declination must
have changed from north to south, or from south to north.
If his declination on one day was observed to be 21’ north,
136 THE PRECESSION OF THE EQUINOXES
and on the next 5' south, it follows that his declination
was nothing, or that he was in the equinoctial point about
twenty-three minutes after seven in the morning of the
second day. Knowing the precise moment, and knowing
the rate of the sun's motion in the ecliptic, it is easy to
ascertain the precise point of the ecliptic in which the
equator intersected it.
By a series of such observations made at Alexandria
between the years 161 and 127 before Christ, Hipparchus,
the great father of astronomy, found that the point of the
autumnal equinox was about six degrees to the eastward
of the star called spica virginis. Eager to determine
everything by multiplied observations, he ransacked all
the Chaldean, Egyptian, and other records, to which his
travels could procure him access, for observations of the
same kind; but he does not mention his having found
any. He found, however, some observations of Aristillus
and Timochares, made about 153 years before. From
these it appeared evident that the point of the autumnal
equinox was then about eight degrees east of the same
star. He discusses these observations with great Sagacity
and vigour; and on their authority he asserts that the
equinoctial points are not fixed in the heavens, but move
to the westward about a degree in seventy-five years, or
somewhat less.
This motion is called the precession of the equinoaces,
because by it the time and place of the Sun's equinoctial
station precedes the usual calculations: it is fully con-
firmed by all subsequent observations. In 1750 the
autumnal equinox was observed to be 20° 21' westward
of spica virginis. Supposing the motion to have been
uniform during the previous period, it follows that the
A PROOF OF THE EARTH's ROTATION. 137
annual precession is about 503" ; that is, if the celestial
equator cuts the ecliptic in a particular point on any day
of this year, it will on the same day of the following year
cut it in a point 50'3” to the west of it, and the sun will
come to the equinox 20' 23" before he has completed his
round of the heavens. Thus the equinoctial or tropical
year, or true year of seasons, is so much shorter than the
revolution of the sun or sidereal year. -
It is this discovery that has chiefly immortalised the
name of Hipparchus, though it must be acknowledged that
all his astronomical researches have not been conducted
with the same sagacity and intelligence. It was natural,
therefore, for him to value himself highly on the discovery.
It must be acknowledged to be one of the most singular
that has been made, that the revolution of the whole
heavens should not be stable, but its axis continually
changing. For it must be observed that since the equator
changes its position, and the equator is only an imaginary
circle equidistant from the two poles or extremities of the
axis, these poles and this axis must equally change their
positions. The equinoctial points make a complete revo-
lution in about 25,745 years, the equator being all the while
inclined to the ecliptic in nearly the same angle. There-
fore the poles of this diurnal revolution must describe a
circle round the poles of the ecliptic at the distance of
about 23% degrees in 25,745 years; and in the time of
Timochares the north pole of the heavens must have
been thirty degrees eastward of where it now is.
Hipparchus has been accused of plagiarism and insin-
cerity in this matter. It is now very certain that the
precession of the equinoxes was known to the astronomers
of India many ages before the time of Hipparchus. It
138 THE PRECESSION OF THE EQUINOXES
appears also that the Chaldeans had a pretty accurate
knowledge of the year of seasons. From their saros we
deduce their measure of this year to be 365 days, 5 hours,
49 minutes, and 11 seconds, exceeding the truth only by
twenty-six seconds, and much more exact than the year of
Hipparchus. They had also a sidereal year of 365 days,
6 hours, 11 minutes. Now what could occasion an atten-
tion to two years, if they did not suppose the equinoxes
movable 2 The Egyptians also had a knowledge of
something equivalent to this; for they had discovered that
the dog-star was no longer the faithful forewarner of the
overflowing of the Nile, and they combined him with the
Fomalhaset in their mystical calendar. This knowledge
is also involved in the precepts of the Chinese astronomy,
of much older date than the time of Hipparchus.
But all these acknowledged facts are not sufficient for
depriving Hipparchus of the honour of the discovery, or
fixing on him the charge of plagiarism. This motion was
a thing unknown to the astronomers of the Alexandrian
school, and it was pointed out to them by Hipparchus in
the way in which he ascertained every other position in
astronomy, namely, as the mathematical result of actual
observations, and not as a thing deducible from any
opinions on other subjects related to it. We see him on
all other occasions eager to confirm his own observations,
and his deductions from them, by everything he could
pick up from other astronomers; and he even adduced
the above-mentioned practice of the Egyptians in cor-
roboration of his doctrine. It is more than probable then
that he did not know anything more. Had he known the
Indian precession of 54” annually, he had no temptation
whatever to withhold him from using it in preference to
A PROOF OF THE EARTH's ROTATION. 139
one which he acknowledges to be inaccurate, because
deduced from the very short period of 150 years, and
from the observations of Timochares in which he had no
great confidence. *
This motion of the heavens was long a matter of
discussion as a phenomenon for which no physical
reason could be assigned, but the Copernican system
rendered it easy of explanation. A simple experiment
with the ordinary spinning top will serve to show
that a motion may exist about an axis which has itself a
conical motion without, at the same time affecting its
inclination. No more perfect illustration of astronomical
precession can be derived than that presented by a properly
balanced top which exhibits at the same time the other
phenomena of rotation. In the experimental portion of
our subject we have noticed Bohnenberger's rotatory
apparatus, which is the model of the gyroscope by which
Foucault made apparent the effects of the earth's rotation.
We may here mention Troughton’s nautical top, and
Elliot's, which is similar to it, though an original inven-
tion; but we will confine ourselves to a description of
Professor Maxwell’s dynamical top, which differs from
Elliot's in having more adjustments, and in being designed
to exhibit far more complicated phenomena: there is also
an essential addition in the coloured disc for showing
the motion of the axis. The conditions which must be
observed in the construction of such instruments are the
following. The body must be balanced on its centre of
gravity, and capable of having its principal axes and mo-
ments of inertia altered in form and position; the axle of
the instrument must be capable of being made the greatest,
least, or mean principal axis, and the position of the
140 THE PRECESSION OF THE EQUINOXES
invariable axis of rotation must admit of being made
visible at any time.
FIG. 29.
MAXWELL’s DYNAMICAL TOP.
There must be three adjustments to regulate the posi-
tions of the centre of gravity, three for the magnitudes of

A PROOF OF THE EARTH's ROTATION. 141
the moments of inertia, and three for the directions of
the principal axes in all nicer independent adjustments
which may be distributed ad libitwm among the screws of
the instrument.
The form of the body of the apparatus which Professor
Maxwell has found most suitable is that of a bell, see fig.
29. C is a hollow cone of brass, R is a heavy ring cast
in the same piece, six screws with heavy heads, a y z and
a: y’ 2' work horizontally in the ring, and three similar
screws, l m. m., work vertically through the ring at equal
intervals. As is the axle of the instrument, S S is a brass
screw working the upper part of the cone c, and capable
of being firmly clamped by means of the nut c. B is a
cylindrical brass bob which may be screwed up or down
the axle and fixed in the required place by the nut b. The
lower extremity of the axle is a fine steel point finished
without emery, and afterwards hardened; it runs in a little
agate cup set in the top of the pillar P. The upper end
of the axle has also a steel point by which it may be
kept steady while spinning; a coloured disc is attached
to it when the instrument is in use.
Dimensions and weights of the parts of the dynamical top.
I. Body of the top.
Mean diameter of ring 4 inches.
Section of ring # inch square.
The conical portion rises from the upper and
inner edge of the ring a height of 1% inch lb. oz.
The whole body of the top weighs . e . 1 7
Each screw is one inch long, and weighs one
ounce, together . <> º e g e 9
142 THE PRECESSION OF THE EQUINOXES
II. Axle, &c. lb. oz.
Length of axle 5 inches, of which # inch at the
bottom occupied by the steel point, 3% inches
are brass with a screw turned on it, and the
remaining inch is of steel with a sharp point
at the top. The whole weighs . © º
The bob B has a diameter of 1-4” and thickness of
I
#
0.4. It weighs . * ſº gº © ſº 2#
The nuts b and c for clamping the bob and the
body of the top on the axle weigh 1
Weight of whole top 2 5}
The disc of card is divided into seven quadrants, co-
loured with vermilion, chrome yellow, emerald green and
ultra-marine. They combine into a greyish tint when the
revolution is about the axle, and burst into brilliant colours
when the axis is disturbed. It is useful to have some
concentric circles, drawn with ink, over the colours, and
about 12 radii drawn in strong pencil lines. It is easy to
distinguish the ink from the pencil lines, as they cross
the invariable axis by their want of lustre. In this way
the path of the invariable axis may be identified with
great accuracy and compared with theory.
The first thing to be done with the instrument is to
make the steel point at the end of the axle coincide with
the centre of gravity. The adjustment for it being made,
the rotation of the top has no tendency to shake the steel
point in the agate cup.
The next thing to be done, is to make one of the prin-
cipal axes of the central ellipsoid coincide with the axle of
the top. We effect this by spinning the top gently about
A PROOF OF THE EARTH's ROTATION. 143
its axle, steadying the upper part with the finger at first.
If the axle is already a principal axis, the top will con-
tinue to revolve about its axle, when the finger is removed.
If it is not, the top begins to spin about some other axis,
alternately widening its circles and contracting them.
The path which the invariable axis describes upon the
disc may be a circle, an ellipse, a hyperbola, or a straight
line according to the arrangement of the instrument.
In the case in which the invariable axis coincides at
first with the axle of the top, and returns to it after sepa-
rating from it, its true path is a circle or an ellipse, having
the axle in its circumference. The true principal axis is
at the centre of the closest curve. It must be made to
coincide with the axle, by adjusting the vertical screws
l, m, n.
Let us suppose that the principal axis is that of the
greatest moment of inertia, and that we have made it
coincide with the axle of the instrument. Let us also
suppose that the moments of inertia about the other axes
are equal, and very little less than that about the axle.
Let the top be spun about the axle, and then receive
a disturbance which causes it to spin about some other
axis. The instantaneous axis will not remain at rest, either
in space or in the body. In space it will describe a right
cone, completing a revolution, in somewhat less than the
time of revolution of the top. In the body it will describe
another cone of larger angle, in a period which increases
inversely with the difference of axes of the body. The
invariable axis will be fixed in space, and describe a come
in the body.
Another small and slow movement of the earth's axis,
which is dependent on the variable position of the lunar
144 THE PRECESSION OF THE EQUINOXES
orbit, afford the most cogent proofs that our globe is
endowed with rotation, otherwise we could only account
for the phenomena it produces by the absurd hypothesis
that the whole celestial sphere is kept in a constant state
of tremor by the influence of the moon. The small but
important inequality we speak of, is called “Nutation,” a
name given to it by its discoverer, Dr. Bradley, in a
memoir communicated to the Royal Society in 1748. It
is a kind of tremulous or nodding motion of the earth's
axis, whereby its inclination to the plane of the ecliptic
varies backwards and forwards to the amount of some
seconds. By this motion, if subsisting alone, the pole
would describe among the stars, in the period of about
nineteen years, a small ellipse, with axes respectively
18° 5 and 13” 75, but in conjunction with the motion of
precession, the path pursued will be a gently undulated
ring.
The two motions of precession and mutation are conse-
quences of the earth's rotation, and ellipticity, combined
with the unequal attraction of the sun and moon on its
polar and equatorial regions. In order to exhibit these
motions jointly, and to illustrate the causes which pro-
duce them, an ingenious instrument was invented by
Mr. Burr, of which the following is a description. See
fig. 30.
It consists of a small globe representing the earth as
seen from the north pole, from which a small axle of steel
projects, while the equator is widened into a broad
metallic ring; the globe is supported at a point within it,
to which an axle descends from the foot of the apparatus,
which at the same time carries an horizontal ring, repre-
Senting the ecliptic. The whole is so balanced, that in
A PROOF OF THE EARTH's ROTATION. 145
a state of rest, the equator and ecliptic coincide. If
a small weight be attached to the ring of the equator,
representing the attraction of the sum and moon, the axis
FIG. 30.
BURR's APPARATUS FOR THE IMITATION OF PRECESSION.
will be inclined, and the latter being put in quick rota-
tion, it will be observed that the projecting axle is describ-
ing a small motion round the pole of the ecliptic in the
surface of a cone, and in a direction contrary to that of
the earth's rotation, in consequence of which, a shift-
ing of the equinoctial points (at which the equator and
ecliptic intersect), is produced in a retrograde direc-
tion. The phenomenon of the precession of the equi-
noxes is hereby clearly imitated; and more, if the rotation
of the projecting axle be closely and attentively watched,
it will be observed to undergo a number of oscillations in
its path round the pole of the ecliptic — Oscillations which
exactly correspond with the nutation of the earth.
L

146 EQUINOXES EVIDENCE OF EARTH's ROTATION.
These motions are easily explained by the first principles
of mechanics. A small weight on the plane of the equator
distributes the mass of the apparatus unequally with regard
to the axis of rotation, in consequence of which centrifugal
forces are produced, the resultant of which does not pass
through the point of suspension, viz. the centre of the
globe.
On the other hand, the globe being hollowed out at the
bottom in order to fit in the lower axle, it has more mass
in the upper than in the lower part, the centre of
gravity and the geometrical centre of the globe do not
coincide, and thence again a resultant of centrifugal forces
not passing through the centre is the natural consequence.
The least inclination of the axle, without any overweight
at the equator, is sufficient to produce the phenomenon of
precession, while the small weight attached to it acts as a
second disturbing force, thus producing the phenomenon of
mutation as observed in the minute oscillations of the axle
or nodding motion in its circular path round the pole of
the ecliptic.
We cannot do better than conclude this, the first part
of our work, with the words of Sir John Herschel: “ These
movements of ‘precession’ and “nutation' are common to
all the celestial bodies both fixed and erratic, and this
circumstance makes it impossible to attribute them to any
other cause than a real motion of the earth's axis such as
we have described.”
PART II.
PART II.
CHAPTER I.
THEORY OF THE MOTION OF FALLING BODIES, BY LA PLACE
AND GAUSS.
N the following analysis, La Place gives the true ex-
pression for the deviation of falling bodies from the
vertical, taking into consideration the resistance of the air,
and showing that whatever that resistance may be, and
whatever may be the form of the earth, no deviation takes
place towards the equator.
Let a y z be the three rectangular co-ordinates of the
body, the origin of co-ordinates being the centre of the
earth, and the axis of a; its axis of rotation. Let r be the
radius drawn from the centre to the summit of the tower
from which the body falls; 6 the angle which r makes
with the axis of rotation, and a the angle which the plane
passing through r, and through the axis of the earth,
makes with the plane passing through the same axis, and
also through one of the principal axes of the earth,
situated in the plane of its equator; and let m, t be the
angular motion of rotation of the earth. Taking XYZ,
the co-ordinates of the summit of the tower, we have
X = r cos 0
Y = r sin 6 cos (n t-i- w)
Z = r sin 6 cos (n t-i- w)
150 THEORY OF THE MOTION OF FALLING BODIES,
m t + o being the angle which the plane passing through
r and through the axis of the earth makes with the plane
of a; and y. Let us now suppose that, relatively to the
falling body, r changes itself into s—a s, 6 into 6 + a w,
and o into a + a v, then,
a = (r — as) cos (0 + au)
3) = (r — as) sin (6 + au) cos (n t-i- w -- a v)
2 = (r — as) sin (6 + au) sin (n t-i- w -- a v)
Let V be the sum of all the particles composing the
terrestrial spheroid, divided by their distances from the
body attracted; the forces which act on this body, by the
attraction of these particles, are parallel to the axes of a y 2
d V d V G. V.
(#) (#) and #
(see No. 21, Vol. II. of the “Mécanique Céleste”). In
order to consider the resistance of the air, we may repre-
sent it by the expression
ºp (as a #)
For, the speed of the body relatively to the air, considered
at rest, is considerably greater in the direction of r than
in that perpendicular to r, we shall see that the expression
for this relative speed is approximately
a de
d t
If now, for simplicity, we make
* = 1
then the relative speed of the body in the direction of 6 is
a *and in the direction o it is a “sin 0.
d t d t
d
(as “àº) d s
The resistance of the air is therefore –I, a ſi in the
g * Ol
direction of r, t
BY LA PLACE AND GAUSS. 151
tººs © (as Ol #)
d t du . º º
d's a 7, in the direction of 6
d t
Ø ( S d's
tº- CIS C. "T
d t e tº o º
ds Cl # sin 6 in the direction of o.
* aſ:
d's
p (as a #)
a de
d t
Let K represent the factor , then we have,
by the principle of virtual velocities,
dala: , s dal dolz GV & V
0=3 a “** a dº/ - * - * -
* +89 #4-82% **(#) 8 y (#)
—s, (d. VY_ d's du ;.2 a. d."
32 dź Kºra;+K80s; + Kö a sin 0a;
the differential 8 relating to the co-ordinates r, 6, and o,
of which a y z are functions. Substituting for a y z their
preceding values, and neglecting the terms of a”, we have
mºst dols do ..., 2 d's
0= 3 r * dº 2 an ri Sin? 0 a K+.
+ r? 3 6 (a #–2an; sin 6 cos a+ak;
º d d v . du d's sin 6 d v .
2 tº- sº *-*.
+ r? 3 w in 9 (a d t? sin 9+2an ºf cos 6 2am # 7" +a Kłsin 6)
—ºv-º (r—as): in-Q+aw) (1)
By the nature of the equilibrium of the stratum of air in
which the body is, we have
O=3 V-H ‘. 3 (r—as): sin? (0+aw) (2)
(vide “Mécanique Celeste,” Vol. I. p. 98),
provided that the value of 8 r be considered in relation to
152 THEORY OF THE MOTION OF FALLING BoDIES,
the level surface of the stratum of air. Take on this surface
T= a + y; y being a function of 6, of o, and of a, a being con-
stant for the same stratum, then by equation (2), we obtain
= d Qſ (dy d y d Q d Q
0=+. ((#):0+(# **)+ #) 20+(#)*,
Q being supposed = to V + ; (r—as) Sin 2 (0+aw)
Subtracting this equation from equation (1) we have
-* dols dº sº., 2 d's
O= 3 r (—a dº? 2 anr; SIIl 0–ak;)
dalw d v . du
+ r? 36 (a dº? –2an #sino cost-Hak #)
+ r? 3 to sin 6 (a #in 942an; cost-2an ##" +a Kºsin
t 7.
–% (ºr-(%) 36– (#) so).
If now we take the coefficients of the three variations
87, 86, and 80 respectively equal to 0, and observe that
º represents the gravity which we designate g (vide “La
Mécanique Céleste,” Vol. II. p. 104), we shall obtain,
taking the radius r as unity (which may be done without
appreciable error), the three following equations:—
_. dds dº sº.2 d's
0=a+2an; in 0+ a K #, g
dolu dv . dw d r
0=a;-2an; in 9 cost-Haki-g (#
dal v sin 0+2an # cos 0–2 a n *ēsin 9
0=a is d :
** sin a 9 dy
+ akº in 0- iº (%)
If we take the second decimal, or the Tatººn of the
mean day as the unity of time, m is the small angle de-
scribed in one second by the rotation of the earth. This
angle is exceedingly small, and as a w and a v are very
0)
BY LA PLACE AND GAUSS. 153
minute quantities in relation to a s, we may neglect in the
first of these three equations the term
dº sº, 2
2 an iſ sin 6
in the second
2an?” sin 6 cos 6
d t
and in the third
2an; co- 6
which reduces them to the following:—
dal's d's
0=a; + ak dº —g
dolu
Oſt 2
dolv
*º e d's . dv . g /d y
0=a; sin 9–2an; sin 6+ a K 7, sin 0– sin Ó (#
0= a
dw dy
Kºº – a ſ 2
+ an ar-g (#
K being the function of a s and of a #, the first of these
equations gives a s in functions of the time t. The second
of these equations can be satisfied by making
O. M. E. CLS (#)
d 6
because g and # may be supposed constant through the
duration of the movement, from the smallness of the
height from which the body falls, as compared with the
radius of the earth. This is the only manner of satisfying
du
d's dt
and iſ are nought at the commencement of the movement.
the second equation in the present analysis, for w, ..., s,
If now we imagine a plumb-line of the length a s sus-
pended from the point whence the body falls, it will deviate
towards the south from the radius r, by the quantity
as (*
#)
154 THEORY OF THE MOTION OF FALLING BODIES,
and consequently by the quantity a w; the body in falling
is always on the parallels of the points of the vertical,
which are at the same height as itself; it does not diverge
to the south of this line.
In order to integrate the third equation we will make
:... a- as dy /
a v sin 6== sin 6 #4-aw
and we have
__ da v.' dºw' d's .
The body deviates to the east from the radius r by the
quantity:
& d *
6= C. S ..)
D. Q) SII]. Ü - in 6 \do + d v
but the plumb line deviates to east from the same radius
by the quantity
as / dy
sin 6 *)
a v' is therefore the deviation of the body to the east of
the vertical.
Let us now suppose the resistance of the air to be pro-
portional to the square of the velocity, so that
K=7ma :
77.00. dt
m being a coefficient depending on the form of the body,
but which we may here, without appreciable error, consider
constant.
Then we have
dals ds?
0=a #; +*m; g
In order to integrate this equation we will make
1
= + 1 . S'
as =glog.s
BY LA PLACE AND GAUSS. 155
then
0–88; — mgs'
d;2
which by integration gives
s'=Acºmº --Bc-tºna
c being the number the hyperbolic logarithm of which is
unity, and A and B being two arbitrary quantities. In
order to determine them, we must remark that a s should
be nought when t = 0, which makes 8' = 1; therefore,
A-H B=1.
Besides aft must be nought with t, consequently with
ds which gives
d :
A–B–0
therefore A= B-#, consequently
1 - 7\
— as-it log. [ ]-c”9-4-4-6- ?/mg
Cl8 7??, Og (, C + #C /
and reducing into a series,
2 2 2
s=9 –”9. *** gºtº &
2 TTET "THE
In order to determine a v', we must consider that
ds 1 dºs'
* Jºn g7.j .
and that therefore the differential equation in a v' becomes
0=asºta ds' do' _2? ds'
d :2 dt dt m dt
C.
from which we obtain by integration,
C being an arbitrary constant, in order to determine which,
156 THEORY OF THE MOTION OF FALLING BODIES,
we must remember that t being nought # = 0, and that
then s'=1 which makes
c=–?”
777,
... dividing by S' and substituting the value of C
Ol dv' 2n 1Y 2n 1 Ol
dt 7, \* Ts’ $72, cºma-i-o-tºmy
integrating so that a v' be nought with t we have
..'mg ºwmg
2 m, 4n. c” c. *
av'- tº t— *= tang" W —
77? 7 * - fºſi
n v'mg C 2 - 2 7mg
+ C
and reducing into a series
ng tº sin 6 _mgt” 61 202:4— & )
aw = --—a (l *†-Fai, mºſt-we
We must consider in these expressions of a 8 and a v',
that t, expressing a number of units of time, g is double
the space caused by gravity in the first unit of time, m, t is
the angle of rotation of the earth during t unities, and 'm g
is a number depending on the resistance offered by the air
to the movement of the body. In order to obtain the time
of falling and the deviation towards the east in functions
of the height from which the body falls, let us represent
the height by h; then, by the preceding, we have
whence
1 *-*- e-ºs-
t= — — 1. Vmh Wmh– )
Vmſ, * (s + 1 + C 1
and
ſ 2 m. *-*. — \” cºmh. 1
- =ſlog 4. 7mh Wmh — —2 tan ( - ))
Clº) #(ºg #(e. + 1 + C 1) ang-UºTI
The height h being given, the observation of the time t
BY LA PLACE AND GAUSS. 157
will give the value of m, and we shall derive a v', or the
deviation of the body towards the east of the vertical.
The coincidence of this theoretical result with the practical
experiments, renders the rotary motion of the earth mani-
fest; m, could also be determined by the figure and density
of the body, and by experiments already made on the
resistance of the air.
In vacuo, or which is the same thing, when m is exces-
sively small, we have
a 0 vº.
9
6 is very nearly the complement of the latitude of the
place, and for Paris we may suppose 6=41°9'46", n is
the angle of rotation of the earth during a unit of time.
If now we take for this unit the Targºwn part of a day, we
shall obtain
, 2n h :
Cl ()' = — Sl
m = 1.233.9.9
because the duration of the earth’s rotation is 0.997:27 of
a day, and for Paris,
# g=3,66107 mêtres.
Supposing, therefore,
h–54 mêtres,
we have
a v/=5,7337 millimètres.
The following are the fundamental formulae for the
motion of a falling body with regard to the rotation of the
earth, as developed by Gauss. The position of a point
may be determined in two ways: —
1st. By its three perpendicular distances, X, Y, Z, from
three fixed planes perpendicular to each other. Their
common point of intersection C is placed in the earth's
axis, the plane of Z is parallel to the equator, the plane of
158 THEORY OF THE MOTION OF FALLING BODIES,
Y in that meridian which passes through the original
position of the body, and the plane of X in the meridian
perpendicular to the former. The co-ordinates of Z are
positive towards the north, those of X towards the primi-
tive place of the body, and those of Y towards that side
to which the first place is carried by rotation.
2ndly. By the three perpendicular distances a, y, z,
from three movable planes, which relatively to the earth
are at rest, but rotate with it; their common point of in-
tersection is best taken in the primitive place of the body.
The plane of z is perpendicular on the direction of
gravity, the plane of y lies in the meridian, and the third
plane is perpendicular to the two former. The poles of
these three planes are respectively the zenith, the east
point, and the South point, towards which the co-ordinates
2, y, aº, are taken positive.
Let for the point C, a = a, y=0, z = -c, and the comple-
ment of the latitude of the place of observation = b, and
let the angle of the earth's rotation during the time t be
denoted by 6, then we have the following equations: —
3/=Y sin 64-Y cos 0
z=X cos p cos 0 +Y cos q sin 6-i-Z sin p-c
X=(a!—a) sin ºp cos 0–9 sin 6-1-(2-Hc) cos p cos 0 }
(2)
a;=X sing cos 0--Y sing sin 6–Z cos (p + a }
(1)
Y=(a — a) sin ºp sin 0+y cos 0--(z+c) cosº sin 6
Z= —(a!—a) cos p + (2 +c) sin (p
The co-ordinates X, Y, Z, may be regarded either as
functions of t, or as those of the four variable quantities
6, a , y, z, and with regard to the latter they have four
partial differentials. Therefore
d x. d X d X d Y dZ
dx = | * ** - I — º-mºm sº-ºm- *E=-
(# dt (# ),04 (º)ast (;)avi (; dz
BY LA PLACE AND GAUSS. 1.59
The velocity of the body may be resolved into three, the
directions of which are perpendicular upon the planes of
X, Y, Z, and are therefore expressed by
d #) (dy d2
Jº Jº V ºf ) #)
The velocities of the particle of air in which the centre
of gravity of the body is at any moment, referred to the
Same axes, are
d x. 6. Y dZ
#) d6, (# d6, (#) d/6
hence the relative velocities of the body in these three
directions,
i-(; da: (#):# (; d2
㺠Jºã t \dy Jā; t \d, Jºdi
— sº da: in a 39 — d2
= Sin ºp cos 0 + – sin 0# cos p cos 6 dt
=(#):#+(#)}+(#);
"TVä.Jält (dy) iſ + \ſ, Jºãº
tº . A da; dy . A da
= Sin ºp sing; +cos 0 Jº +cos p sin 6 dź.
_/dZN da: , /dz \dy dZ\dz
&=(# 7-F #)}+ dz / dt
= u- da: ſº d2
tº cos p º +sin ºp dº
The total relative velocity is therefore
*==ºmºsºmsº ă 2 / 7.2.2
u-º-A/# +º] dº
772. " dº? " dº?
The square of this quantity is proportional to the resist-
ance of the air, which is therefore expressed by Mw, and
its three components in the directions above stated are
Muć, Mum, Muć,
160 THEORY OF THE MOTION OF FALLING BODIES,
Considering the earth as a spheroid of revolution, the
direction of gravity will pass through the earth's axis. Let
the point where these two limes intersect be above C by
the quantity q, i.e. its co-ordinate Z=q, taking the force
of gravity=p, and X*=Y”--(Z-q)*=r”. We have, by
the principles of Dynamics,
— dºx p?& , Mr. Y
_ d?Y p Y
d°Z p(Z—q) *
0= * *
#-F 7" + Muſ
From the values X, Y, Z, in (2) putting for the constant
#=n, we obtain the following equations:—
d”X as d’a ..., a d”y d22 •
dº. T sing cos 0 2:2 sin 6 # +cos ? cos 0;
-> * da: dy e dz
2n sin ºp sin dt 2n cos 6 dt 2n cos q sin 6 dt m2).
d2Y —sº º d2a: d?y º d22
# =sinº sin 9 Jºž + cos 6 # +cosº sin 9 772 - (4)
º da: :... a dy d2
+2n sin ºp cos d : 2 m sin 0; +2n cosºp cos 0 dt m2Y
d?Z da;2
# =-cos ? Hä
. . d22
+ sin (p Jºž -
Multiplying the three equations (3) respectively by
sin ºf cos 6, sin ºf sin 6,-cos p, and adding the products;
multiplying again these equations by – sin 6, cos 0, 0,
and thirdly by cos ºf cos 0, cos p sin 6, sin b, and adding
each time the products, we obtain after substituting the
G2X d2Y d2Z
values of É, m, , as also those of dº?’ dº?” d? from (4)
BY LAPLACE AND GAUSS. 161
and those X, Y, Z, from (2) the following three equa-
tions:—
d2a: tº dw cº-º-º: £ —nº P7–n? )
d? —2 n sin ºp Jº + (, c)(# *)+cºs(*. m?Z
da:
* = 0
+ Mu dt
d?y . . daº d2 P – º) M *4 =0
d;2 +2 n sin * † +2n cos ?; +y (; 70, * if
d/22 dy p © 7, q
# –2 m cosº; +(240) (#– w)+in *(# 7,22 )
d”
+ Mu # =0.
If, therefore, the body is at rest with regard to the earth,
or if da;= dy= d2=0, then it is acted upon perpendicu-
larly to the planes of ac, y, z, by the following forces:—
(a — a) (? —nº) + COS *(*-*z) |
y(; – ) | . . . (5)
(2+c) (. —nº) +sin *(*! —ºz) J
But a body already in motion is affected another way;
for besides the resistance of the air, which urges it in three
directions, like forces measured by Mu #. Mu % Mu, #,
there are three other forces acting in the same directions, viz.
tº-º i., a dy e da: dº dy
27, sinº #. 2 m sin º ż +2n cosp Øg' 2 n cos p gtº
and these serve in falling bodies to show the rotation of
the earth. As in all experiments of this kind, the space
described by the body is comparatively very small, we may
be allowed to assume the force of gravity acting upon it as
a constant = g, and its directions always parallel, i. e.
M
162 THEORY OF THE MOTION OF FALLING BODIES,
perpendicular to the plane of 2. On this assumption, we
may substitute for the three quantities (5) respectively
0, 0, g; by this means the fundamental formulae are trans-
formed into the following:—
d2a: da
..., , dy
d”y . . daº d2 dy —
dź +2 n sin q. dź +2n cos p 7? +Mu; =0
d22 dź
dy
#. —2n cos ?:, +g +M u : = 0.
dt d t
Neglecting the resistance of the air, i. e. taking M-0, we
obtain by integration,--
sin ºp
X=A-D cos p, t-i-E cos (2nt + F)+} sin ºp cos p g tº
—R– * si cos p g t
gy=B ; in (2nt-F)+ 2 m.
CO
2=C + D sin p, t + E # cos (2 m t + F)—# sin ºg t”.
Granting that the initial velocity of the falling body was
nought, then the values of the arbitrary quantities are
easily determined, viz. –
__g cos p sin (p tº- __ 9 cos” (p
A=– ºr, B=0, C-- *|†,
D=0, E=4;t, F= 0
... by substitution, —
– ºne ( 2 – l l cos2nt
(CE 2 m. 7, t ãº, t 2 m. )
— goos? (, sin **)
3/ 2 m. ( 2 n.
= –1 a #2 *( ---- sºn)
==–19944; (nº-g; 4*.**
BY LAPLACE AND GAUSS. 163
This integration is only true so long as the body does
not move far from its original place; but if the distance be
appreciable, we resolve the circular functions into series,
and thus we obtain —
=} cos p grº tº
2=} gt?--# cos”q gn°tº
Fi:. . . . . . . . . )
. . . . . . . . . k . (6)
As the time of falling lasts only a few seconds, the
quantity mit cannot exceed a few minutes of space, there-
fore a, and the second term of Small z become inappreciable;
whence
3/= —#2 cos pºnt
Taking into consideration the resistance of the air, we
must be satisfied with the approximate value of the in-
tegrals; and developing the expressions of a, y, z, in series
according to the ascending powers of m, M, we get for the
highest term of a, as before, cos (pg mºtº, which may be
neglected, and for y and 2 we get, omitting the second and
higher powers of n, and M, the following values: —
3/=# cos p gn tº—ºr cosp Mg2ntº
2= —# g tº +1]; Mg?tº
Assuming the actual descent —2=f, and that in vacuo
#g t”=f-i-8, we have
3/=# cos pn t(f-Hö)—cos p n tº
=# cospºnt (f-#6)
a formula which we obtained in Chapter W., Part I., by
a more elementary method.
164
CHAP. II.
THEORY OF THE MOTION OF THE FREE PENDULUM WITH
REGARD TO THE EARTH's MOTION.
7ſHE simple pendulwm. Let a, y, z be three rect-
angular axes, the former two parallel to the equator
of the earth, the third to its axis of rotation. Denoting
by ac, y, z the co-ordinates of the heavy particle, by aſ,
gy', 2' those of the point of suspension, and by l the con-
stant length of the pendulum, we have
(a —a')”--(y—y')”--(2–2/)*=l”.
We may consider the material point as in free motion,
if we introduce some indeterminate force p acting upon
it in a direction towards the point of suspension. The
components of this force are, —
–4 (2–2): -; (v-y);– (e–) . . (1)
I,et V express the attraction of the earth, then the com-
ponents of gravity are, —
#): (); (#)
and if m denote the angular velocity of rotation, and
a", y'', 2'' the co-ordinates of the centre of the earth,
we have for the components of the centrifugal force, —
m*(a —a"); m”(y—y”); 0
THEORY OF THE MOTION OF THE FREE PENDULUM. 165
hence, by subtraction, the components of attractive force
acting upon the pendulum are, —
(#)-we-º"); ()—wo-y), (). (2)
The expressions (1) and (2) for the components of the
accelerating forces acting upon the pendulum give, by
addition, —
d?a, G. V. / A/
# = (#) —nº–,")– #(2–2)
d°y G. V. —m°(al-a’’) – “ (al-
iſ = (#) m*(y—y”) fo Aſ)
d22. 6, V – “ (2–
d F = d. # (2–2)
Let a, y, 2 be the co-ordinates of the pendulous mass
referred to the point of suspension, so that,
(a —a')=X; (y—y')=y; (2–2)=Z
then we obtain by substitution,--
*_(&V)—ºx_nº, wo— d.” tº
#=(#)-nºx-nº-w")-;-#x
#=(#)-wy—wº-y)—#–#y . (3)
dź = \dy d #2 T 7
d’Z_/dV _d 2* *
2, #2 * \d, 2 T dź TT
Denoting by aſ the distance of the point of suspension
from the centre of the earth, and by $ the angle between
a' and the plane of the equator, we have —
a’—a’’= a' cos p' cos nt
gy’ —y’’= — aſ cos p' sin mt . . . (4)
z –2 = a sin º'
166 THEORY OF THE MOTION OF THE FREE PENDULUM.
In differentiating the latter equations, we may consider
a", y”, 2" as constants, because the annual motion of the
earth can exercise no appreciable influence upon the mo-
tion of the pendulum; we therefore have —
# =-wº—"); (#)=-woº'-y); #-0
whence the equations (3) are transformed into
# = }; -º-; Y . . . . (A)
2
# = (#) – 4
Changing the directions of the axes of X, Y, Z into those
of £, m, f, the first coinciding with the plumb-line through
the point of suspension, the other directed to west, and
the third to south, we have, denoting the elevation of the
pole by p, —
X=# sing cosmit-- m sin m t—& cos ºf cosm t
Y= —# sin ºp sin m t--m cosm, t + & cos @ sin mt . (B)
Z= —É cos (b —& sin ºp
where
£*-i- m” + £2=l”.
Differentiating the equations (B) twice, and substi-
tuting the results into (A), we obtain, –
2
# sin ºp contº, sin m t— º; cos p cosmit
–2n; in ºn at #2n}} count = #)—fx
d :
+ 2 m, d : cos psin mt
THEORY OF THE MOTION OF THE FREE PENDULUM. 167
d%. e Cl2 d? g *
—iſ, sin & SIIl nt : cos m.t-H #cosº. sin ºn t
* d # 2. d m ... (d. WY_4 y
2 m, J; sin ºp cos nt–2n+sin nt :=(#. l,
d&
2%, ’3
+270, dt cos p cos mt -
— "#cosa d’é. _ (dWY_4.
#cos ? #sin # =(...) #Z.
Multiplying respectively by N
sin ºp cosmit sin mt —cos p cosmit
—sin ºp sin mt cos nt cos p sin mt
— cosp O —sin (p
then adding and considering that these factors are equal
to the partial differential co-efficients
#) (#) (;
(# dºm d :
dy dy dy
(#) (; (#
(# #) d2
dź dºm d :
we obtain
d’é — —9, dºns; dV). A ~
dź T 2n}sin #4 (...) ; :
* —9, dº sin 4–9, d'é dVN Aw
# =2n , sinº 2n; coºt (; in (C)
d”& dºm dWY_Aw
# =2n + cost + (#) #: -
The assumption u = 0 would lead us here to the equa-
tions of the motion of a falling or projected body, but in
order to find those of the motion of the pendulum, we
must eliminate this indeterminate quantity, and con-
sidering that
* M: 4
168 THEORY OF THE MOTION OF THE FREE PENDULUM.
#4.7% +:#=0
di
we have,
d’m — mol” g \
*- 2ncosº; –2nsing:# + £ (...)
--> (;
"Vää
£d’é-Édº dm — e dºm C/V
—H-– 2ncosº, 2n sing tº ++(# D
—# (; (D)
dº
mdº-ºdºn dé e dé G|W
—ia-- 2ncosłł 27, sin&#. + m (#.
—t (; |
The function V consists of two parts, the one is the sum
of all the particles of the earth divided by their respective
distances from the attracted pendulum, the other accruing
from the centrifugal force is # m” (a.”--y”), the origin of
co-ordinates being placed in the centre of the earth. On
the same supposition let w, v, w denote the co-ordinates
of a particle, the density of which is q, then
q dw dw dºw
v=/=#Hiſtºry)
If the earth be considered a homogeneous ellipsoid, with
semi-major axis = a, ellipticity= x, and mass=M, denoting
by r the distance of the attracted point from the centre,
we would have, neglecting higher power than 2°–
v=x, (1 –22 + &”)+ *($2–2)
q,3
2 4
–Mº (; – ), * *.
q” 5 2 5 35
M 0.422 12 MGA24 3 I 2
; : * * * : * : *(***).
THEORY OF THE MOTION OF THE FREE PENDULUM. I.69
sº
But if the ellipsoid is not homogeneous, and the law of
its density not known, we must find the values of the co-
efficients in the last expression by observations. Since, for
the surface of the earth, the resulting gravity
g=9780194-0050754 sin”?
we may safely neglect 2", and thus obtain
O . P 2 P22 1
=;+#-F#-F#" (º-Fy’).
V
2 y.º.
Taking
ac=r cos S cos S'; y=r cos S sin 9'; z=r sin 9
then the components of gravity will be expressed by
d V ( 0 W ) Gl, V )
#). poly)? (rºy
and neglecting again &”, we have
C. V. O P 9 P sin 29
g=-(+)=; +8 = −3 =
but also on the surface of the ellipsoid
r= a (1 — a sin”.9)
and considering that here q may be written for 9 we
obtain
_O +*—nºa: 20, 9 P + 70, w) sin” p.
—mºr cos”.9
Tº Tai 0.2 2 a.4
m°o, 1 1
— = — = t—- whence
Nowº-gº anº-ào "
9–978183; ; –0-010725; nºw-0038813;
Cº, OW,
and thus the coefficients in the expression for V are
determined.
| 70 THEORY OF THE MOTION OF THE FREE PENDULUM.
In order to find the relations between ac, y, z, and É, m, ,
we must put in equations (B)
X=2–aſ cos $'; Y=y; Z= z—a' d'; and n=0
which gives
a;=a' cos p'+* sin ºp-É cos p; y=m
z=aſ sin p' —# cos ºp-8 sin (p
therefore, by Substitution and differentiation,
d V\ / O L3 P_15 Pz”) , s: /
(#)=-(++*-***) win (4–4)
+m” aſ cos p' sin 44 ºf a' sin ºp' cos p
7.3
_º # co-º-º: & sin ºf cos q + n’É sin”q,
—m”: sin ºp cos (p
(l. V O 3 P 15 P 22
(#)=-(++*-**::) nºn”
(l, W O 3 P 15 P 22
(...) * (++ *-*::) a' cos (p-ºp')
2 / / 3 P / º / g
—mºa' cos p' cos $4: a sin ºp' sin ºp
2
-(? +3 P_15 P2
+++ -, *) {
3 P . . 3 P . .
-º-; sing cost–º tsinº
—m”: sin ºf cos (p + m”& cos” (p.
In the development of these expressions we may omit
the squares and higher powers of É, m, , since these quan-
THEORY OF THE MOTION OF THE FREE PENDULUM. 171
tities are very small compared with a'. The values for
a', 'ſ 2, give then,
*= aſ”--2 a'É sin (ºp-q')–2 aft cos (q)-(b)
2*=a^*sin ºp' –2 aft sin ºp' cos ºp-2 aftsin b’sin (p
therefore by substitution, and taking cos (ºp-ºff’) equal to
unity, we have
G. V. O . / P .
dif =–73. sin (ºp-p') +3 A sin b cos ?
+ n’a'sin ºp cos p – # —3 #: sin (p-b')
P 21 P -- . P . .
75° T2 ; : sinº. 12; sin #cos ?
+ m”: sin”,p-nºsing cos (p
P 15 P .
#, n- + + n sinº-nºn
—6 § --
d WY_ _ 0 ,
C. VN O P 9 P . /
(#. =275 -F3 g – 3 ºr sin”q—mºa' cosºp
O O . . / P
+ 2 º t—3; #sin (?-?)+12+.
—is, tsinº #12; # in boost
—m”# sin ºp cos q + nº cosºp
But referring the value of V to the point of suspension
we must write
O P 3 P . I
W = 7 t aſ; - 2 aſ; sin”q/+ Tº m*a*cos°4'
and the components of gravity are then expressed by
G. V. O P 9 P . 2.1/
#) =– a –3 gºt 3 ºr sin”q'--nºaſcosºp
C. V P ... 1, s: A
(#) = — 3 a sin $' cos p' –mºa' sin ºp' cos p
(zº) = 0
172 THEORY OF THE MOTION OF THE FREE PENDULUM.
Denoting by g’ the resultant of gravity at this point we
have
C/V
g' cos(p-p')= –(º); g' sin (q –$)= -(#y
and therefore, with omission of the squares of ellipticity
and centrifugal force,
O P 9 P
9–4+3,. - a
sin” p-m” aſ cos” (p
o=9 si "Y-3 * sin cos ºp-m” aſsin ºp cos ºf
- aſ sin(q – q')— cy. q,
whence by substitution,
Tº
*WY__0 e-6 P 21 P e.g., P . .
(; *- ; : 6; #4 ##sin $43;&sing cost
- + m” # sin” p-40° à sing cos p
CIV O P 15 P e
(#)=-#7–8; 4%; n in ºwn
GV O P P . .
(#)= g4. 2%, 4-12; ; – 18 # sin”?
+8. # sin ºp cos (p + m” & cos” p-4 m” # sin ºp cos (p.
For brevity’s sake let us assume
T=(3 #4 w) cos” (p; A=(3 #. *- Anº) sin ºf cos p
O P 51 P . ©
II=3344-15 75T 2 aſ; sin”q —n” sin ºp
then the latter equations give by subtraction,
C/V G|V
# (...)—w (#. § m—Amé.
GV Cl V
t(...)-f(#)=-of-(n+1); tı A&—tº
m %)– º (º)= g m+II m {-|- A £ m
THEORY OF THE MOTION OF THE FREE PENDULUM. 173
By means of these values the formulae (D) are trans-
formed into
d &
& d” m—m d” # = — 2ncos (p. ##–2nsing.8%; Fren-Ant i
diº
*#*- –27, cosp. #–2nsinº.º-g:
—(II+T) ##-F.A (?–?”)
ºn d’é-- & dºn_ –2ncos (p. #–2nsinº.4% § + g m
Gli2 c. *
+ II m {-|- A &m
Multiplying the second of these equations by d #, the
third by (-dºm), adding the products and substituting
& d #4-m d m- – £d £, we get after integration,
d :” + d m” + d ?
d #2
where k denotes the arbitrary constant. This formula,
which is the only complete integral obtainable from (E),
expresses the relative velocity of the simple pendulum in
term of the co-ordinates of its material point.
The Compound Pendulum.—The differential equations
of the motion of the compound pendulum, which can turn
freely about its point of suspension, are obtained by mul-
tiplying the preceding equations by (d m), and then
integrating with respect to this differential, and for the
whole extent of the mass of the pendulum.
Let us refer all the particles of the vibrating body to the
three rectangular axes w, v, w, which intersect at the
point of suspension, and are at the same time principal
axes of the body, the centre of gravity of which is assumed
to lie within the axis of w. Denoting the angle between the
(E)
ad
= k + 2 g &-H II t”—Té” + 2 A # & . . (F)
174 THEORY OF THE MOTION OF THE FREE PENDULUM.
axes w and &by 6; further, the angle between the planes w &
and § { by Jº; finally, that between the planes wº, and w w
by X, we have, with introduction of auxiliary co-ordinates,
£=#' cos Jr.—m sin ºp; É'- £" cos 6+ £” sin 6
#"=w cos x–v sin X
m=# sin ºr +7' cos / ; ) = m”
m” =w sin x + v cos x
&= {"; º' = — £' sin 6 + &” cos 6
£’’=w
whence, by elimination,
#= c. w --8 v-H 'y w; m = aſ w--8'v + Y w
{= a'w-H g”v + y” QM), (G)
where
a= — sin X sin ºr +cos x cos / cos 6
£8= — cos x sin ºr – sin X cos / cos 6
y=cos ºr sin 6
a' = sin X cos ºr +cos x sin ºr cos 6
Á' = cos x cos \le-sin x sin ºr cos 6
Y=sin ºr sin 6
a" = — cos x sin 6
Á’’=sin x sin 6
y’=cos 6
and the equations of condition between these quantities are
a”-- £8°-H y?= a” + 8”-- y”= a”-- 8'24- Y”= 1
aa' + 83' +y|Y=aa" + 88" +yy" = a'a' +8/8” + y^y/-0
Or,
a*-i- a”-- a”=8°-H 8”-- 6”=y” + y^2 + y^2 = 1
aft-Ha'8' +a"8’’=ay--a'y'--a"y"–8) +8//+ 6%)” =0
THEORY OF THE MOTION OF THE FREE PENDULUM. 175
from which we may derive
aft'— a'8=y” ; aſ 'S- ag’’=y' 3 a'8"—a'8' = 'y
aſy —ay’=/3" 5 aſy” —a"y=8' 3 a’y'— aſy” = 3
Áy’ —8'y=a"; 6"y—Sy’’= &’; 8')"—S”y' = a
Denoting the elements of rotation about the axes w, v, w,
by p di, q dt, r dt, respectively, we have,
'r dt = d x+cos 6 d ºr
q dt=sin X sin 6 dºle--cos x dé
p dt= — cos x sin 6 dole--sin X d6
and the differentiation of the preceding values of a, &, &c.,
gives
da =(8” —yg) di ; d. 3 = (yp – ar) dt
dy = (a g— 8p) dt
da’ = (8'r — y'g) di ; d6' = (y'p — a'r) dt
dy' = (a/q— & p) dt
da’’=(8” r—'y'g) di; d8’’=(y"p —a"r) dt
dy” ==(a"q—8"p) dt
whence, differentiating the three equations of (G) we
obtain,
dé– a (wq—vr) dt-H 8 (wr-wp) dt ~
+ y (vp-wq) dt
dm=a' (wg—vr) di-H 8" (wr—wp) dt
+y’ (vp-wg) di ... . (H)
dg=a” (wg—vºr) dt-F8" (wr-wp) dt
+ y” (vp-wg) dt J
Since w, v, w are principal axes of the pendulum, we
have,
Juv d'm =0, ſww dm =0, ſww dm)=0
176 THEORY OF THE MOTION OF THE FREE PENDULUM.
and if A, B, C denote the momentums of inertia in
respect to the same axes, we have
A=ſ(v*-i-w”) dim, B=ſ(wº-Hw”) dim, C=ſ(w” +v°) dm
and the foregoing formulae become, by means of the equa-
tions of condition,
ſ(£dm—m dé) dim-(Aa"p +B8"q+Cy” r) dt
ſ(£d£–#d £) dm-(Aa' p +B8' q + Cy’ r) dt . . (I)
/(ndt—td m) dm-(Aa p-B3 q+C, r) di
and by Squaring the equations (H), and then adding,
ſ(d£”--dn’--dº”) dim-(Ap” +Bq*--Crº) di” . . (K)
Differentiating the three equations (H) in respect to the
time, we obtain
ſ(£d” m—mdº) dm-(Aa” dp-HB8" dº-FCy” dr) dt
+[(C–B) a” qr--(A–C) 3" pr
+(B–A) y” pg]dt”
ſ (ºd” #—#d” () dºm-(Aa' dp--B8 da-H Cy dr) di
+[(C–B) a' qr--(A–C) & pr
+(B—A) y' pºl dt”
J (nd” &–gdºm) dºm-(Aa dp--B8 da+Cy dr) dt
+[(C–B) a qr-H (A–C) 3 pr
+(B–A) y pg] dt”
whence we derive
C dr dt-F(B-A) pg dt°-ſ[y (mdº-gdºm)
+y (&l’É–£dº)+y" (ºdºn–nd'É)] dm . . (L)
and this equation is transformed by means of the formulae
(D) into
THEORY OF THE MOTION OF THE FREE PENDULUM. 177
C #4 (B-A)n+ſſºn-º-ºncº(ºr/#+7 #):
—2n sin *(, # +y}+" #):
+II (Ym—y'É) {+T(y"m—y'É) {
+A [(Yº-Y"m) {-|-(ym—y'É) #| | dºm.
The equation (H) gives
ydä-- y’dn + y^dć–(vp- wº) dt
from which we obtain by means of (G)
ſ(yd£4-ydn +y'dg) # dm-- C dy–3 (B-A) (24+3p) di
f(ydé+/dn +y'dº)&dm––4 C dy”–3 (B-A) (2”q+8"p) dt.
The formulae (G) give further in conjunction with the
equations of conditions
ſym—Yé-6"w—o.”v ; y’m —'y' = C.?) —£8w
whence
/()m —'y' £) & dim- 2' 6" (B-A)
ſ(Y'm—y'ſ) #dm− – 28 (B-A)
ſ(ym—y' à) £dm––#y' C++ (28" +2"8) (B-A)
f(y'{-Y'm) & dim = #y C++ (28’’-- a”8) (B-A).
Since the axis of w passes through the centre of gravity,
we have denoting the distance of the latter from the point
of suspension by A., L
ſudm–0; ſo dim–0; ſw dm-Am.
and therefore,
ſ(ym—y'É)dm–0.
N
178 THEORY OF THE MOTION OF THE FREE PENDULUM.
Substituting these integrals, we obtain
dr º * dir sº a dy"
C; +(B A) pg=Cn cos p #Cn sin # dt
+(B–A) m, cos p (2g+8p)
+(B—A) m sin ºp (2”q+8"p)
+(B–A) II2"|3”—(B—A) Tag
+(B–A) A (28’’--2'3);
but in all well-constructed pendulums we will have two
momentums of inertia equal to each other. Assuming,
therefore, as is generally the case, that
A= B,
we have, by integration,
r= n/+ym cos (p + y”m sin (p
=m/+m cos p cos ) sin 6-1-n sin ºf cos 0 (A1)
where m/ is the arbitrary constant, and expresses the
angular velocity which was originally communicated to the
pendulum about the axis that passes through its centre of
gravity and point of suspension.
The equations (F) and (K) give the following expression:
Apº Bº-cº-º-º/tam +II): dºm,
—Tſä” dm--2 Aſºdm.
But
Jā’dm– —2*A–8°B—y” C++ (A+B+C)
ſº dim– 2"A–8” B—y” C++ (A+B+C)
ſég dim--aa"A-38"B—ºy"C
ſ&dm-Mnvy",
THEORY OF THE MOTION OF THE FREE PENDULUM. 179
therefore, denoting by k the sum of all constants
Ap°-i-Bq”--Crº– c +2 gx my”—II (2” A+8°B+y^C)
+T (2*A+8°B+y^C)
–2A (22"A+88"B+ yy” C).
Substituting the value of r" from (AI), taking B-A,
2 2
and observing that p”--q*= #4% sin” 6, while
d Gli2
'y=cos ºr sin 6; and y”=cos 0, the last equation becomes
2 2 cºv. 2
A d6 #. SIIl *=s+2 (gMºn–Cnn/ sin ºp) cos 6
—2 Cnnſ cos ſp cos ºr sin 6
+ [II (A—C)—Cn” sin” (b) cos” 6
+2 [A (A–C)—Cn” sing cos (b) cosº), sin 6 cos 6
— [T(A–C)+Cnºcos” plcos”, sin”6. . . (Bi).
This integral, in conjunction with (A1), shows that the
conservation of vis viva takes place.
We further have by means of the first equations of (E)
and (I),
d (A&"p-HB8"q+ Cy^r)=–2n cos ºbſºd; dm
–2n sin ºbſºd; dim
+dt Tſän dim—dt Aſmºdm.
But
ſ fit dm-–ada"A–3d6"B-ydy"C
+} (xda”--8d,3”--ydy") (A+B+C)
ſłd £dm– —o." do.” A—S'd&" B —y’d Y'C
ſén dm → —22'A-88'B—yy'C
ſm: dn = — 2'2'' A-8'8"B–"Y Y'C'
N 2
180 THEORY OF THE MOTION OF THE FREE PENDULUM.
Therefore, by substitution and integration:
A&"p-HB8"q+Cy’r=c--n sing (2” A+3*B+y^*C)
+2ncosºp ſada"A+8d8"B+ydy"C)
—n cos @ (A+B+C)/(xda”--8d.6’ 4-ydy")
–T/(22'A+88'B+y/C) dt
+Aſ(2'2"A+8'8"Bx YYC) di.
Taking, again, A= B, substituting the value of r from
(A1), and observing that
do.” =(8” r—'y"q) dt; d. 3" =(Y"p—2"r)dt; etc.
c."p + 8"g = } sin”0; y' = sin ºr sin 6; etc.
we obtain finally,
A º sin”6= c-An sin ºp cos” 0–Cn' cos 6
+ 2 An cos p ſcos ºr sin” 6d.6
+ Cnn' cos p'ſsin ºr sin 6dt
—[A(A–C)—Cn” sin ºp cos plſ sin ºp cos 6 sin 6 dt
+[T(A–C)+Cn” cos” (b)ſ sin ºr cos / sin” 0dt.. (C)
This integral shows that the conservation of areas does
not take place.
The further integration of (Bl) and (Cl) can only be
effected by a series of approximations; the first of which
will suffice to show the laws of the motion of the free
THEORY OF THE MOTION OF THE FREE PENDULUM. 181
pendulum. For this purpose we take the following terms
from the equation (B1) and (Cl),
. . a d alº - d. 6” es
A (into *}: + #) = k +2(g»m–Cnn' sin ºp) cos 6
A sin 26 *} =c—An sin ºp cos”6–Cn' cos 6.
Assuming for shortness
plan sin ºp; p = n/ . g’=g *
we have
. . a d \lº” . dé” /
A (sinº: + dº = k + 2 A (g'—up/) cos 0 . . (DI)
A sin” 6 º = c-A a cos”6–Ap' cos 0 . (E.)
Denoting by s the angle which the pendulum made with
the plumb line at the moment of liberation, and by v the
velocity of the lateral impulse communicated in a direction
perpendicular to the primitive plane of oscillation, then
We have at the commencement of motion,
s. d 6 a.d. l
6=#; 77 T 0; *}=w
We find, therefore, by substitution the values of the
Constants, 1.
k= A v" sin” g–2 A (g’—pp/) cos s
c=A v sin” s-- Ap, cos’s + Apl' cos a
and the equations (DI) and (E.) are transformed into
. ..., d alr” d 62 / sº
sin 26 º + i + 2 (g’—upſ)(cos s—cos 6)=v° sin” s
sin?6 º = p (cos” s—cos” 6)+A' (cos s—cos 6)+ v sin” s
where v and s are the arbitrary constants which we will
182 THEORY of THE MOTION OF THE FREE PENDULUM.
eliminate by introducing two others k and k determined
by the equations.
io9 a dº d6% f
2 — tº 11’ -º -
sin'6% +; + 2 (g'—ap/) (1 — cos 0)=k
(F)
Sin?6 º = p sin” 8 + a' (1—cos 6)+ic' 1
whence, by comparing these with the preceding equations,
we have,
k = 2 (g'—pp/)(1 — coss)+ v” sin” s } . . . (GI)
k'= —p sin” s —p/(1-cos s) + v sin” a
These equations show that k and k are small quantities
of the second order, if s is a small quantity of the first
order as will be assumed hereafter agreeable to practice.
Taking the value of dy” from the second equation (FI),
diº
and substituting it into the first, we obtain,_
2
O = Sin? 6 º: +2g' sin” 6 (1–cos 6) + u” sin" 9–r" sin” 6
+p.”(1—cos 6)”--2 k' uſ (1—cos 6)+k”
where for shortness * >
ic' ^ = K. – 2 k'º.
Let now
a = 1 — cos 6
and the preceding equation becomes —
Claº te -
– #44°-2 Da' +E a”–2 Faſº.--a” a . . (H,)
where D= k"—k'u'
E=4g' -- k” +44%-i-p'º
F =g'+2p*.
Since we assume the initial value of 6, that is s, to be a
THEORY OF THE MOTION OF THE FREE PENDULUM. 183
small quantity, we may be satisfied with an approximate
integration by means of a converging series. For this
purpose we differentiate a second time, thus —
_ d’a:
T º
where E and F are finite quantities, but D a small quantity
of the second order, and the coefficient of acº at least of the
fourth order; as we will see hereafter, We assume, there-
fore, the integral to have the following form,
O —D+E ac—3 Faż-H 2p*a*
a;= a + b cos 2 (a t-i-m) + c cos 4 (ot--m)
+e cos 6 (ot-Fm) +; etc.
where a and b are small quantities of the second Order,
c of the fourth, e of the sixth order, and so on.
Differentiating and substituting this value of a; in the
last differential equation, we have for the determination of
the coefficients the following equations,—
O= —D+E a-3 F (a^+} b”)
0= —4 o” +E–3 F (2 a +c)
O= — 16 o” c +E c–3 F (2 ac-1-4b*)
O= — 36 o” e-H E e—3 F be.
These equations leave b and m arbitrary, but since
da, = 0, when t=0 we have
dt
m=0
and since at the same time a = 1–cos s, we have
1 — cos s = a +b+c+ e-H; etc.
As it is sufficient for our purpose only to consider the
small quantities of the second order, we take
ac= a + b cos 2 ot
184 THEORY OF THE MOTION OF THE FREE PENDULUM.
where the coefficients are to be determined by the equations
D D F
a +b=#s”; a = E3 4 o” = E – 6 F a = E–6 TE'
The equations (GI) become, not considering smaller
quantities than those of the second order,
ic=(v*-i-g’—pp') s”
k'=# (2 v–2p–p') s”.
Substituting the values of D and E we obtain, ne-
glecting s”,
_ 2 vº-H2 g’–2 plu'-(2 p +p')(2 v-2A-p') s”
O, 4 g’ + 4 p.” + p.” 2
which, combined with
a + b =# s”
gives
q_5= (2 v-**-*)' s = },
4 g’ +4 a”--A’” - 2 2
where
g’ = 2 v —20—pſ
aſ 49' +4p.” + p.”
But from a - a + b cos 2 ot we derive, considering that
a = 1 — cos 0–2 sin” # 6, and interchanging sine and arc,+
ac-, 6°–(a +b) cos” ot--(a-b) sin” ot
8.
that is, g
6 *= g” cos” o 4-g" sin” a t . . (Ag)
Whence we see that while e expresses the greatest, sº
expresses the least deviation of the pendulum from the
plumb line.
Substituting the values of D, E, and F in the equation
for o”, we get, —
THEORY OF THE MOTION OF THE FREE PENDULUM. 185
1 1 3 (g'+ +2 u”) (k”—k'u')
*= * + + uſ?-- it cº — B
*=ſ+tº+3*4; "-ºº-Hºa (B2)
and by neglecting the quantities smaller than those of the
Second order,
2 v-2 u — uſ
*= **-*- . . . . . (C2)
whence
k'=o es' . . . . . . (D.)
Putting this value of k' in the second equation (FI),
and neglecting again quantities of higher than the second
order, we have
tº- 1 ..., o a s”
dº= (a+gu') d t + 62 d t
and substituting for 6* its value from equation (Ag),
1 os aſ dt
d alr= + uſ) d t
Aft (A+. uſ) + g” cos” a t + s^* sin” ot
But o a sº dt =::/ adt,
ſº cos” a t + 5° sin” o tº s/ cos” ot (1+ #tan” ot)
8
ſ d tan ot g’
F — 2 = —l ( ~ t
8 1 + s' tan” ot tan (: tan ot)
g?
Therefore,
/
*=a+(44; 2) t+ tan- (f-tan ot) . (E.)
8
.The arbitrary constant c, denotes the azimuth of the
plane in which the pendulum commenced to swing.
The equation shows that the plane of oscillation changes
and moves in consequence of the rotation of the earth
from east through south to west except u' be negative, or
greater than 2 p. Let us find the curve of intersection of
186 THEORY OF THE MOTION OF THE FREE PENDULUM.
the moving pendulum, with a horizontal plane in the dis-
tance = 1 from the point of suspension. Taking in this
plane the axis of a positive towards south, and the axis of
gy positive towards west, we obtain for the co-ordinates of
the pendulum the following expressions:—
a;=6 cos ºr 9–0 sin ºr
but the preceding integral gives
tan (Jr.— M)={tan o t . . . . (Fº)
if we assume
1
M = 2 + (94; p/) t
hence
s” cos” (Jr.—M)
s” cos” (ºr-M)+ 5° sin” (!-M)
s” sin” (Jº-M)
s” cos” (J.-M)+ 5° sin” (!-M)
and substituting these values in equation (A2) we obtain
6° cos” (JP-M) , 6° sin” (Jº —M)
2 + /2 = 1.
8 8
This is the equation to an ellipse, the co-ordinates of
&
cos” cot–
sin” ot-
which are
a'=6 cos () —M); y’=6 sin (ºp-M)
and its principal axes are 2 s and 2 s'. Comparing these
co-ordinates with those of a; and y above, we find
a;=a' cos M–y' sin M
Ay=a' sin M+y' cos M
whence we infer that the axes of this ellipse move from
east to west, with an angular velocity expressed by
l
(P +;p}.
THEORY OF THE MOTION OF THE FREE PENDULUM. 187
The pendulum, therefore, moves on the surface of a
right cone, with elliptical basis, and the axes of this basis
change continually with the stated velocity their position
with regard to the meridian of the place of observation.
The magnitude of the minor axis 2 g/ depends chiefly
on the velocity v of the lateral impulse which we assumed
the pendulum might receive at the commencement of its
motion.
If
2 v–2 p. — A' = 2 o
then the base of the cone on which the pendulum moves
is circular.
The change of the plane of oscillation of the pendulum
in the azimuth amounts during each mean second of time,
in consequence of the earth’s rotation, to —
p=(15”-041) sin ºp
We must now examine the amount of the velocity of
this change in consequence of the rotation of the pendu-
lum on its own axis.
We must first remark that since p is always positive on
the northern hemisphere, and uſ either positive or mega-
tive, according as the rotation of the pendulum on its own
axis takes place from east to west or from west to east, the
two changes in azimuth will either conspire or oppose
each other.
Let us denote by T the duration of oscillation between
two consecutive greatest elongations of the pendulum,
then the change in azimuth after each oscillation is ex-
pressed by
I
(a + 2 p/) T.
188 THEORY OF THE MOTION OF THE FREE PENDULUM.
º
Let us assume, as is generally the case, that the ...bob
consists of a homogeneous ball, the diameter of which is
Small compared with the length of the suspending wire,
while the mass of the latter is very small compared with
that of the former; then
A=X” m,
and
C =} p” 770,
where p denotes the radius of the ball of the pendulum,
and substituting this value in the expression for aſ, we
have,
2
Let the number of rotations of the pendulum on its
axis during each oscillation be denoted by w, then
mſ T = 2 ºu Tr.
and sufficiently approximate
T=7r A. N
9.
whence
P.'- 4 (?!, PA/4
5 X? X.
therefore,
1 ..., 2 Tr
- uſ T = (**) (#): = 72° (...) º
3 * #) (k) u-72° (;)"w
Taking M-40 p, as nearly was the case in Foucault's
experiments, then
#2 T= 162// 'w,
Whence we see that a moderate velocity of rotation of
THEORY OF THE MOTION OF THE FREE PENDULUM. 189
the pendulum on its own axis, caused by a lateral impulse,
may effect a change in the azimuth equal to, or even
greater than that in consequence of the earth's rotation,
and either in the same or contrary direction.
This change in the azimuth does not depend on the ab-
solute length of the pendulum, but on the ratio of the
latter to the radius of the bob, which for this reason ought
to consist of a matter of great specific gravity. We also
infer from the last formula how necessary it is to avoid a
rotation of the pendulum on its own axis.
Let us suppose that this rotation is scarcely appreciable
and that at the same time sa=8' very nearly, i. e., that
the pendulum is moving on the surface of a come with
nearly circular base.
Taking in this case
y=s—s'
then y is, compared with s and sº, a very small quantity,
the square and higher powers of which may be ne–
glected.
The equation (F2) gives therefore, by making uſ=0
tan (Jr.— «–pt)=tan ot–% tan ot. . . (G2)
from which we derive, neglecting the square of y,
* = c + (4 + oyt– # sin 20t.
8.
Since the value of A is always positive on the northern,
and negative on the Southern hemisphere of the Earth, it
follows that the time of an entire revolution of , is dif-
ferent according as o is positive or negative, that is,
according as the pendulum moves from east through south
190 THEORY OF THE MOTION OF THE FREE PENDULUM.
to west, or from west through south to east. On the
northern hemisphere of the Earth, ſº will pass through the
whole circumference in shorter time, if the motion of the
pendulum takes place in the first direction; the contrary
will be the case on the southern hemisphere.
The equation (Ga) gives Jr.— 2–0 for t=0; hence
taking Jr.—2 = +267t, according as o is positive or negative,
we obtain for the determination of the time during which
tº passes i times the whole circumference, the following
equation—
+ 2ir=(•+2)-3. sin 2 ot.
Taking approximately
t= +267t
a) +/w
we have
20t= +2it+2it *
(2)
therefore
sin 20t= F2 it *
(º)
and by substituting this value in the first equation
+ 2 iT=(o-Fu)t-E****
8 (2)
whence we derive, neglecting the square of pu,
1 — A7 º
t=+2it++=+*(1–4–4).
o +/w T to o 2 so
The difference of the times of revolution of , arising
from the difference of the direction in which the pendulum
moves, is therefore
#( y 4 T ( y º
1 + -i- ) = — tº ſ 1 + -i- e
.# ("tâ)=1374.7 +%) A sin ºp
THEORY OF THE MOTION OF THE FREE PENDULUM. 191
We infer therefore that this difference is not depending
on the primitive amplitude of oscillation, if the motion of
the pendulum takes place on the surface of a cone with
circular base, where y=0, and furthermore that this dif-
ference is but very little changed if the base differs little
from a circle, because the quantity; is in this case very
24 8
small.
Expressed in seconds of time, we have, since
=5,9705–10
9.
and taking A = 10 metres, p=51°, and y=0, the following
numerical value for the difference of the times of a revo-
lution of lº //
0",00073.
This is, indeed, a small fraction of time, yet it may be
ascertained by observation in well-conducted experiments,
as Mr. Bravais has shown. (See p. 105.)
In order to find expressions for ſrdt and x, we take the
equation (A1), neglecting the square and higher powers of
6; thus we get
T=m/+ a--n cos (p. 6 cos \p.
But the equation (FA) gives
6 sin () —M)=g' sin ot
6 cos () —M)=s cos ot
whence we obtain
6 cos le=}(s-Hs') cos (ot-HM)+}(s— sº) cos (ot—M).
Substituting this value of 6 cos Jr, multiplying by dt and
integrating, we have
192 THEORY OF THE MOTION OF THE FREE PENDULUM.
- /rat-6+(n+ºf+ (***** (ot--M)
+ (5–3)” cost sin (ot–M)
2 (o
where we have neglected in the denominator the quantity
(a4-#1), and where 8 denotes the arbitrary constant. But
ſrdt=x+y
hence, by means of equation (E2),
/
x=8–2+ ("-ºf-tan (; tano)+ Grººt sin (ot--M
8 . (º)
- +(5–4)” cost
200
sin(ot–M).
In this equation the effect of the Earth's rotation on the
change of the plane of oscillation in the azimuth is elimin-
ated; from the same expression we derive the values
which X assumes at the greatest amplitudes, viz:—
x=8–2+(n/–#!/) iT+ it +
s'm cos d . &
** sin [x-H (u +}/)iT
the coefficient of the last term of this expression is very
small, never exceeding one second, moreover it is = 0, if
g’ =0, and may therefore be neglected, hence
X=8–2+ i r +(n/–$//) iT
º C \ .
= |\º * * | 1 — #) T
Á–2 + i T-H m ( 3A)*
Assuming that the pendulum received no rotation on
its own axis at the commencement of its motion, we have
the coefficient of T equal to zero, and therefore
X = 3–2 + i. 7t
THEORY OF THE MOTION OF THE FREE PENDULUM. 193
that is, at any two consecutive greatest amplitudes, the
values of X are respectively 8–2 and 180°--8–2; hence
the following theorem.
If we at the commencement of the motion of the pen-
dulum draw through its axis and the vertical line a plane
which remains fixed relatively to the pendulum, then this
plane will, notwithstanding the motion of the plane of
oscillation in the azimuth, always pass through the ver-
tical line at every greatest amplitude, except some ex-
traneous force communicated to the pendulum at the
commencement of its motion a rotation on its own axis.
Further approacimation to the integration of the equa-
tions for the motion of the compound pendulum.
In order to effect the following approximations to the
integration of our differential equations, we will make use
of the method for the variation of arbitrary constants.
The equation (F) to be integrated may be written:
dº dºl’
* - = — sin” 6
d #2 Ol 2 2(g’—ppſ) (1-cos 6)+ k
sin?6 # = p sin”0+A'(1—cos 6)+k'
where, abstracting from the correction for k which is not
wanted in this case,
=II cos”0+2 A cos ) sin 6 cos 0–T cos” ºr sin”6
–2% m m'cos p cos / sin 6
d 6
'd k e
– 1. 26 .
aſ Tā (#) + 2 m cos p cosºlº sin”;
O
194 THEORY OF THE MOTION OF THE FREE PENDULUM.
By differentiation we obtain
30 & 9 — 3 sing a d iſ d’ſ d 6 d J2
dº º żT 2 sin 9;; #-2 sin ºcos ?:, ; ;
te º d 6 d k
—2 sin 6 (a' – uſuſ) " " + “ ”
(9-4) ###.
:... a dº y - d 6 dip - d 6
* 6 º' Y = – 2 s - -
SII] aſ tº 2 sin ecosº, +2 a sing cosº.
+, sin 94° 18′.
d”), d t d t
Eliminating #; from the first by means of the second
equation, we have
d 6 dº? 6 º d 6 d ? . . d. 6
— — F 6 * : * Y--2 (a' — w w/ 8 * *
###-2 sin ºcos ?:, ; ;-2 (g'-º') sinº,
d 6 dip a d8 d iſ 13 k – 2 & W 3 º'
tº-º * 6 * : * : — f ~3 - - -— " 4-, -e
4 * sin 9 cos 0; ; ; ;-2 º' sin 93%; ++, − 2 +, +,
But,
—cos 6 ic'
I
i- /
dºlº-pº-Hp. sin; GT tsin’ 6
and for very small arcs of oscillation
g’= o”—p.”—#1,”; c' – oss'
hence by substitution
2 sin” # 9 (1 + cos” # 9)
# =-& sin 6+ a” sin 6 (1 — cos 0)—pal 2 cos” # 0
d : d iſ did
, sin # 6 a 2.2 cos 0 , d t d t d ?
-** ****, ****#; Td, 6 -ād-
d : d :
dº iſ d0 (, , , , cos 6 , sin # 9 ) –– dº.
d #2 d t Sinº. 6 2 cos’ 0) sin” 0 d t
The integrals with which these expressions must be
compared, are the equation (Ag) and (E2) after intro-
ducing in them the arbitrary constant m, viz.
6°= s” cos”(ot-H m)+ 5° sin” (ot-Fm)
o, +(A ++/w')t-Htan' (# tan (ot-ºn)
THEORY OF THE MOTION OF THE FREE PENDULUM. 195
where s, g,' m and c, denote the four arbitrary constants.
If these constants are considered as not variable, then
these equations express the approximate integrals of the
preceding differential equations of the second order, and
they become the complete integrals if the four constants
are taken variable and their variations suitably deter-
mined. Differentiating the integrals and taking that
part of the differential which arises from the variation of
the arbitrary constants equal to zero, we obtain
# = –0(s”—s”) sin(o) t + m) cos(a t + m)
d oss'
#=e ++1/+ TāT
0=** a cos” (oti n)+% sin” (a t + m)
dt d : ".
–# (s”—s”) sin(ot-i-m) cos (ot-Fm)
O= 62 #–4; 'sin (ot-i-m) cos (a t-i-m)
de' . tº d m
+++ - sin (a t + m) cos (a t + m)+#sº
Differentiating a second time and considering that
2 2
# #=;(e–ºysinº (otºn) cos’(otºn)
2-2-/2 2ſ c2 – 272
= o”6– º: _º º s' Moos' (o t+m)—sin”(o t+m)]
we obtain
**=-& o," sº sº 2 de we sin (a t + m) cos (a t + m)
d; tº * 68 d : 6
+ 2 de' we sin (a t + n) cos (a t + m)
d : 6
d (º) e”— /2 te
–% ( 6 G 2[cos' (a t + n)—sin” (w t + m)]
O 2
196 THEORY OF THE MOTION OF THE FREE PENDULUM.
#- _2 : 24; -#: *: [e” cos” (a t + m)—e" sin” (a t + m)]
# º: [e” cos” (w t + m)—eſ” sin” (w t + m)]
+ 2}} e; (e” – e”) sin (a t + m) cos(a t + m).
Comparing these values of d”6 and d”), with those found
before, and taking
2,-2,-/2
D=a^0°– * # —o"6 sin 6+ a”6 sin 6 (1–cos 6)
— w” {} sin°46(1 + cos”36) , sin #6
Al 2 cos” # 6 poss 6 2 cosº. 6
de 6 dºlº dº'
+ coºg”g/? 6 cos 6 dt dt dt
Siné 6 2 Jó T d6
dź dt
2 oss' d 6 d 6 cos 6 sin 46 62 g/c/
E = * * * * – 62 “‘ſ 2 ogg’ ‘’’’ ‘‘ — uſ **2 ( ) -- * *
##-”;(*º-ºs.º.)+;i,j,
we obtain the following equations
f
D––2}, . sin (a t + m) cos (a t + w)+2}: we'sin (wt + m) cos (w t + m)
–% w (e” – e’”) [cos" (a t + m)—sin” (a t + m)]
*E=-3: we' [e” cos” (a t + m)— s” sin” (w t + m)]
+% we [e” cos” (a t + m)—eſ” sin” (a t + m)]
+ 2 # * * (e”—e”) sin (a t + m) cos (w t + m).
These equations in conjunction with the two others, viz.,
f
#: e cos” (w tº tº e' sin” (at-w)-; (e”—e”) sin (a t + n) cos (a t + m)=0
62 #-# e' sin (a t + m) cos(a t + m) + # e sin (a t + n) cos (w t + m) +}} e e' = 0
are sufficient to determine the differentials of the con-
stants which are considered to become variable.
THEORY OF THE MOTION OF THE FREE PENDULUM. 197
We derive from them the following equations,
# = -ºin (ot-Fm) cos (ot-Fm) + #in' (a t + m)
#= # sin (ot-i-m) cos (ot-Fm)+ #sor (a t + m)
#= tº- airº–Aſſ” cos” (ot--m)—s” sin” (ot--m)]
zºº, sin (ot-H m) cos (ot-H m)
d x D ga'
di Tºº (; ;) [cos” (ot-H m)—sin” (a t + m)]
E (sº +s”)
o6° (g”— s”)
We see that the quantity 6% occurs in the denominators of
all the terms. If now the motion of the pendulum is
such that s’’= 0, then likewise 0–0 at every oscillation
when the pendulum coincides with the plumbline, con-
sequently the values of the variations of e, sº, m and a would
become infinitely great unless this factor 6 is expunged
by the substitution of the values of D and E containing
the same factor in the numerator. Since we know by
observation that the motion of the pendulum is continuous,
we may conclude that all the terms of D and E must be
so composed, that after substituting the values of the
latter in the preceding equations, the factor 6% is generated
in every numerator, by which means the same factor is
eliminated from the denominators. It will be seen here-
after that this is really the case.
Furthermore, the factor s”—a" in the denominator of
the expressions for dºm and d a shows that the variations of
these two constants become infinitely great if s = s/. In
this case the arbitrary constants s, s', m and 4...Betºne
sin (a t + m) cos (ot-Fm).
198 THEORY OF THE MOTION OF THE FREE PENDULUM.
inapplicable, and we must, therefore, make use of others.
Let us denote the new constants by 8, h, l, c.', taking
8 = }(s--s); h-# (s—s') sin 2 m
2' = x + m ; l =# (s—s') cos 2 m
whence
d6=}(ds--ds"); dh–#(d s—ds') sin 2m+ d m (s—s")cos 2 m
do.'-do. + dºm; dl= }(d s—d 8') cos2m-dim(s—s') sin 2m.
Dividing these equations by dt, and substituting the values
of the differentials of s, s', o, and m, we obtain
d 8 D g E te
Žiž T ## (, in 2 oth cos 2 & orgia (6+! cos 2 at-hsin 2 wo
#--siºn (a cos 2 - 12 al-ki in 2 strº ***)
+g; (5 in 2 at-sh–Alcos 2 at-rein 2 wo
d 7 D 2 -- 2 c.;
-gira (a cos 2 stral-hi in 2 ot-ºooºoo
d aſ T) g;#0 in 2 * *** cos 2 - )
# =-gira (8+! cos 2 w ł–h sin 2 & t)—
where the divisor (s—s') has disappeared. It remains to
be shown that 6 and ſº may be expressed in functions of
the new constants. For shortness we put
£3'=# (s—s')= x/hº-Flº
then the equation
6*=s* cos” (a t-i-m)+ 5° sin” (a t-i-m)
gives by substitution the following
6*=8°4-8°4-28 (l cos 2 ot—h sin 2 ot)
by: means of which, after substituting for 8 its value, the
THEORY OF THE MOTION OF THE FREE PENDULUM. 199
Quantity 6 will be expressed in functions of 8, h, and l.
From the equation (F.) we derive
sin (Jr.—M–o t—m) = — in (a t-i- m) cos (ºr + M)
but the same equation gives
w/8+8'. cos (a t-i-m)
w/8+l cos 2 ot—h sim 2 ot
cos () —M)=
whence by substitution
! sin 2 w t + h cos 2 ot
A/8 + 8'. A/8 + / cos 2 a t-h. Sin 2 w &
iſ =(a + u + 4 º') t + a -sin
by means of which, since 3’- whº-Flº, the quantity ºr is
expressed in functions of 8, h, l and 2'.
In order to effect the elimination of D and E, we must
successively substitute the several parts of D and E. Let
us first take that part of D and E which is not depending
on the differentials of k and k", retaining in each expression
only the first not-vanishing term; then because
cos 6 1
we have
I 1 I 1 1
D = 64 [ _t o” # *- : *)–6. ' (#1, * ')
6 (; o'+ . . a P. 6°o g a i P'-Fi, oss
d 6 / 1 2
E=63 tº ſet uſ -- *).
*# (#24 is os
In these expressions we may consider as vanishing u and
Aſ with respect to o, and likewise the terms of the sixth
order in comparison to those of the fourth order; thus we
obtain
D = . o” 64; E=0.
l
6
:
;
.
*
:
sº
200 THEORY OF THE MOTION OF THE FREE PENDULUM.
Substituting these values and that of 6% into those of the
differentials of e, s', m and 2, we get
de 1 - º
37--à" [e cos” (a t + n) sin (a t + m) + e e” sin” (a t + n) cos (a t + n)]
d
de' l 2 -/ 8 - /3 cºr,8
-- 㺠[. e' cos” (a t + n) sin (a t + n) + e” sin" (a t + n) cos (a t + n)]
d :
d m_ 1 ., t_2 w (e"+ e”) Gº)
Hit-iā" (e **)-###"wº (w t+h)-is (e” + s”) cos 4 (w t + m)
d a 1 , , w (sº + e”) (º)
dº." # ****i; º; “to 2(****)+; e e' cos 4 (a t + m)
Integrating and denoting the increment of the constants
by a prefixed 8, we have
*=# e° COS4 (****)-; e e” sin" (a t + m)
* --. e” e' cos" (w tº ºriº sin" (a t + m)
1 2 1 e” + e” . l *
6 m = T 16 ° (sº + s”) *Tāºri. sin 2 (****)-is (e” + e”) sin 4 (a t + m)
2 /2
*a-; *****::::::: * sin 2 (a t + m) +; “sin 4 (a t + m)
where m-0 on the right-hand side of the equations. The
terms which are affected with the time, t, require particular
notice, but the other terms are so small that they modify
the motion of the pendulum very little, though they hold
good for every case, even if s = s/, as will be seen by sub-
stituting them in the equation to the base of the conical
surface on which the pendulum moves, the factor (3*— s”)
being destroyed by this substitution.
The term in the value of 8m which is multiplied by t
shows that, instead of the arc ot of the first approxima-
tion, we must substitute [*-*. o (sº + s^*)]t, where o-
v'g' + p.”--haſ”, whence we obtain for the time of oscillation
T ( s? -- eſ?
T vſ Hº Hº \' " IG /*
• * T
* *
4,
THEORY OF THE MOTION OF THE FREE PENDULUM. 201
The term in the value of 8 & which is multiplied by t,
shows that the motion of the plane of oscillation, or, which
amounts to the same, the motion of the great axis of the
elliptical base of the conical surface on which the pendulum
moves, is expressed by
1 ..., , 1 ')
- . . . . . " " - t
(*#####" is
instead of by (a + #!/)t as found in the first approximation.
The additive correction which this expression has received
by the more approximate integration of our differential
equations may under circumstances become very consi-
derable.
Taking the equation (C.) viz—
, 2v-24- p'8
2 o
referring it to the initial values of s and s', and assuming
the initial lateral impulse communicated to the pendulum,
v=0, we would have
8
/
s' = — 2 p., + p. 8
2 o
and therefore the correction for the motion of the plane of
Oscillation is
I 1 ') 2
- - - t.
#(ºp).
This value is even for the greatest initial amplitudes ge-
nerally allowed more than a thousand times smaller than
(u +}uſ)t, and consequently of little effect. But if the
condition v- 0 is not completely fulfilled, the correction
may become considerable. It will be always possible to
ascertain the initial lateral velocity by means of the value
of s” which can be observed as well as that of s, and these
202 THEORY OF THE MOTION OF THE FREE PENDULUM.
two quantities in conjunction with the length of the pen-
dulum will furnish the value of the correction.
Let us illustrate by a numerical example how great the
latter may become in certain cases.
Suppose the length of the pendulum
X = 10”
then since very nearly
(a) - A/º ; g=9”, 8
o:= 1.
we may assume
Let, as has been the case in several experiments of this kind,
I I
=t: s =—
==#3 s =ion
then we obtain
1 /
gos º-2", 4
but for the middle of Europe we have nearly
Al = 1 1”
so that the effect of the correction amounts almost to 3 of
the value of the principal term, and much smaller values
of a than the one here assumed may exert an appreciable
influence upon the motion of the plane of oscillation.
Hence it follows that in all experiments with the free
pendulum it is necessary to observe not alone the greatest
but also the least deviation of the pendulum from the
plumb line in order that we may be able to compare the
results.of observation with those of theory.
The corrections here found exert on the difference of the
times of revolution of ſº, if s = e, only an inappreciable in-
THEORY OF THE MOTION OF THE FREE PENDULUM. 203
fluence. For by means of them we get, since s and sº are
always positive, and taking pº'-0, the following expression
for Jr.
2
*=a+bt+o (1–3).
whence we obtain for the difference of the times of revo-
lution of ºr which takes place according as the pendulum
moves from left to right, or from right to left the ex-
pression
4 Trp, #)
—f ſ 1 + =
o” ( + 6
so that, if for instance s- the numerical value given in
the first approximation is only increased by its 486th part.
We now proceed to investigate those parts of D and E
which depend on the differentials of k and c'. Having
only regard to these parts, and taking sin 6=6, we obtain
adº gº dº
_º_" dºlº, p_2&
= 15-aa-, *-at-
* 277 dº
But k is a function of the variables "le and 6, and therefore
#=(#) dºt (dº) dº
di T \dº) dº (#. d t
further sufficiently exact
d x 1 (die 3 d6
ºr 3 #)+2 nº co-º-º:
whence by substitution
D-30G.)–2 noos & cosº. *}}
d k C. ^
= 1 / … º. . 62 * *-
E #(#)+2" co, * * *;
204 THEORY OF THE MOTION OF THE FREE PENDULUM.
by means of which the differential equations of s, s', m and
& are transformed into the following:
de
d :
3 e!
d :
dºn
d t
d a
a. *
sin (a t + n) T/d k\ , s: , /. –6 (3 &
*#”[(#) via (etº-e (#) cº (stºl
2 ºn c & d d 6 , ...
+**** cos / sin (, trº) [º co, (• tº "sin(•+)]
cos (a t + m) T(d. ic d k\ , ...:
*#”[(#) or (~~~)+" (#) sin (etrºl
2 d * 8
3 * cos ?*** cos / cos (•+)[*#2 sin(•+)-; co- (•++º]
1 d k e
=-----→ – * f
+H: [(#)“sin (•+) cos(***)
-* (#) (*(*)-ºir (etro)]
2 m cos p cos \, [e; (e cos” (a t + m)—e” sin” (***)
a (e”—e”)
d 6 - 2, ...:
+ 2 iſ a ‘‘ sin (a t + m) cos (****)]
1 d k\ z_2 e'?) si t *
T2 … Jº (º-gº) [(º (e” + e”) sin (w t + m) cos (a t + m)
– 6 (#) e e' (or (a t + m)—sin” (w **))]
2 ºn cos q) cos ), [9% e e' (cos' (a t + m)— sin” (w tº)
* a (,e” gº e”)
+}} (e” + e”) sin (a t + m) cos (a tººl
and
We will first consider the terms arising from ( #) and
d k
dé,
Ic = II cos”6+2A cos / sin 6 cos 0–T cosº, sing 6
whence by differentiation and taking sin 9=6
). Neglecting . as small compared to A, we have
(#)-- 2A 6 sin iſ +2I 6 sin iſ . 6 cos )
d ic
6 (#)=–2 m ºr 2 A co, y—4 A 6°. 6 cos y–2 T 6 cos iſ . 9 cos ).
THEORY OF THE MOTION OF THE FREE PENDULUM. 205
In these formulas the term affected with II is much
greater than the others, and requires therefore the first
consideration. We substitute therefore in the differential
equation of 8, 8", m and o.
dic d k
* : } = 0; lº. } = —2 2
#) 6 (# 3) II 6
making on the right hand side of these equation m=0;
thus we get
d s II a
ºf F2 as sin 2 ot
/
#=-º: sin 2 ot
d m_II, II (sº + s^*) cos 2 ot
dt T2 o' 2 oGs”—s”)
d x IT es'
*-* *mº sºm-ºsmºsºm-mºsºm-ºmms 2 ot
dt To(g”—s”) COS Z (z)
Integrating these equations so that the variations of the
elements (arbitrary constants) become zero for t=0, and
denoting them by a prefixed 8, then
TT a
8 g- 4 o
a (1–cos 2 ot)
8s'= — # (1—cos 2 ot)
Cº)
II II (sº + s^)
= − t + →-----4- sin 2
&n=#####, in ºt
II a aſ &
The extreme smallness of the recurring terms in these
equations shows that the curvature of the earth, con-
sidered as a sphere, has no appreciable influence on the
change of the plane of oscillation in the azimuth, as might
206 THEORY OF THE MOTION OF THE FREE PENDULUM.
have been seen à priori. The expression for 8 m shows
that we must substitute in the value for the time of
Oscillation (e + #) for (o); thus we get
(º)
T= 7 (1– 11
o, \*T2 …?)"
Taking again the approximate value
? =
(º)
%
the factor for the correction of the time of oscillation
arising from the spherical figure of the earth, is
TIX.
1-#.
Now considering only the greatest term in the value of
II, we have (see page 169)
II=3%=29,845. .
Cº, Oſ,
where a denotes the radius of the terrestrial equator to
be expressed in metres; therefore
log Oſ, E 6,80366
which gives for the factor in question
1––*—
4254200
a very small correction but of greater effect on longer than
on shorter pendulums, and always diminishing the time
of oscillation.
The term in the differential of c, which we have just
considered, is the greatest, the others arising from the
ellipticity and centrifugal force of the earth; but as they
THEORY OF THE MOTION OF THE FREE PENDULUM. 207
give occasion to other forms, it will be useful to inves-
tigate them.
Let us first take the two parts
(#) = —2A6 sin ºr ; 6(#)=2A 6 cos \r.
Before we substitute these values in the differential
equations of 5, s', m and 2, we must find the expressions for
6 sin ºr and 6 cos ).
From the equation (F.) we have, after introducing m,
tan (*—M)=: tan (o t+ m),
therefore
6 sin (ºp-M)=s' sin (a t + m)
6 cos(\!--M)=s cos(a t + m)
where as before,
M= 2 +(u, ++M/) t
whence
6 sin ºp-8 sin Micos (o t+m)+s' cos M sin (o t+m)
6 cosº-s cos M cos (ot--m)—s' sin M sin (ot--m).
With regard to these values, the substitution of the partial
values of (# and 6(#) give the following differ-
ential equations:
de A º
ºf T Tz, cos M sin (a t + m)
/ *
**=-“sin Moos (a t + m)
d : @
d m A e A sº dº *
d a A s e ſº A sº
— = *-m-m-m-m- #
d t w (e.”—e”) sin M sin (a t + n)+++,(z=aj" M cos (a t + m)
whence we derive, neglecting the small factor (a + 4/),
208 THEORY OF THE MOTION OF THE FREE PENDULUM.
and taking on the right hand side m = 0, the following
integrals of the variations:
8 g =*, cos M cos ot
(º)
&=-à sin M sin ot
(º)
A s g A g’ &
an=-zº ºf sin ot + oſ(#Egº) sin M cos ot
& /
8.---º-º: sin M cos o t + As cos M sin ot
o°(sº + s^*) o”(s”—s”)
These terms are extremely small, they exert, however, on
the motion of the pendulum a very remarkable though
imperceptible effect, which consists in soliciting the pen-
dulum to perform its oscillations, not about the plumb line,
but about an other, deviating from it towards north. In
order to show this, we will introduce again the co-
ordinates
a;=0 cos / ; y = 6 sin J.
By means of the preceding values of 6 cos ) and 6 sin ºr,
we obtain
8 ac-8 scos M cos ot—8 s' sin M sin ot
–8 m (scos M sin ot--s’ sin M cos ot)
–8 o' (s sin M cos o t+s' cos M sin ot)
8 y=8 s sin M cos o t+8 = cos M sin ot
–8 m (g sin M sin ot—s' cos M coso t)
+8 oz (8 cos M cos o t—s' sin M sin ot)
and substituting the values of 8 s, 8 s', 8 m and 8 ×, we
obtain
A
THEORY OF THE MOTION OF THE FREE PENDULUM. 209
whence we see that all the abscissas are changed by the
A.
quantity ... or, since A is negative, are shifted towards
north, while the ordinates undergo no variation; so that if
the origin of these co-ordinates is transferred towards
north by the quantity -** the motion of the pendulum
about the new origin becomes symmetrical. The trigo-
nometrical tangent of the angle which the plumbline
makes with the straight line passing through this new
origin, and the point of suspension, is expressed by the
quantity -*. Taking as before
()
o?=?,
9
and substituting the value of A (page 172), we have
4n−3°
A a” . .
-:- —7– A sin #cos ?
9
which by means of the numerical values (page 169) gives
for the maximum of the corresponding angle the very
Small quantity
0",0002}\.
In order to substitute the partial values
d ic * dºc
— ) = 2 T º - Už | – || – - º
#) 6 sinº. 6 cos ºf 6(#) 2T6 cosº), .600s),
we multiply the second members of the last differential
equations of s, s', m and cº, by the expression for 6 cos \,,
VIZ. —
6 cos / -g cos M cos (ot--m)—s'sin Msin (ot-1 m)
and write—T instead of A. Thus we obtain, making at
P
210 THEORY OF THE MOTION OF THE FREE PENDULUM.
the same time m = 0, the following differential equa-
tions: —
d T tº t tº
* = – “. *in 2 M t; e cos? M sin 2 & 4. 'sin Moos M cos 2 at
(8) Cº)
d : 4 w
de' g º º º
— = *-s in 2M-5 & sinº M sin 2 & 4 + · sin Moos M cos 2 at
d : 4 w 2 w 2 w
d m T , T (sº + eſ?) T
+ = — 4-----4-cos 21M + — (e” cos” ’2 cºrn 2
d : 4. "Tº OS * g. (…I.T.) (e” cos” M + e” sin” M) cos 2 a t
d a Te e/ Te e' I" . º
— F - — C 2M – –– * =
d : 2 w (e”—e”) OS 2 w (e”—e”) cos 2 w t 2 a. sin M cos M sin 2 w t
Since the terms affected with the factor sin 2 ot, or
cos 2 ot, can never become considerable, they will be
omitted in integrating, likewise the indeterminate quan-
tity Aſ will be neglected, so that
M= 2 + A t- c. -- n t sin ºp
and the arbitrary constants to be added to the integrals of
the preceding equations will be so determined that the in-
crements 8s, 8s", 8m and 8x vanish for t=0. Thus we obtain
5 *—[cos 24-cos 2 (a+1 ). º
6 = - – &-
8 wºn sin ºp M.
5 *- : * – ſcos 2 a-cos 2 (a+, t)]
8 w n sin ºp ; (1)
T T (e” + e”) - º º
67) = —- t - sin 2 a. – sin 2 (a + u :
m= I. "sºº" (a + u t)]
Tee' e º
5 a = - sin 2 a -sin 2 (a + u ü)].
Q. 4 w n. Hº-Jº" (a + u t)]
Here we have terms which depend on the azimuth in
which the pendulum commences to swing, and which
therefore show that the primitive azimuth of the plane of
oscillation exerts an influence upon the motion of the
latter.
Since the quantity sin ºp occurs in the denominators of
these expressions, the latter will be of no use if the place
THEORY OF THE MOTION OF THE FREE PENDULUM. 211
of observation is at the equator. In this case p =0, and
the integrals therefore assume another form.
With regard to places near the equator, where at re-
mains a small arc, even in experiments of long duration,
we are able to expand sin 2 (2 + ut) and cos (2 + a t) into
highly converging series, by means of which the integrals
(1) are transformed into the following:
*** e' cos 2 a + &c.
se-–t-º- e' sin 2 a-t”
4 w
8 – t t e sin 2 a.4 °F tººlte cos 2 a 4 &c.
4 w (º)
5 # T Ž T (e” + eſ”) 2a–?” sin p (e" + e”) ; 2 & > (2.)
- - ——— COS O. - —4. Sln C.
7) 4 Cº) + 4. Cº) (e”—e”) 4. Cº) (e”—e”) C. --
f - f
8 a re- Te e COS 2 a 4- 42 F * sin Pe = sin 2 a + &c.
T“3. (FIF) 2 a (sº-J)
J
All the terms except those multiplied by the first power of
t vanish if p=0, and the latter therefore express the
values of 8 s, 8 g’, 8 m and 8 cº, when the point of suspension
of the pendulum is over the equator.
It is not necessary to discuss any further the integrals
just found; they are of no importance, since there exist
other terms much greater, and yet scarcely exerting a per-
ceptible influence. We will now proceed to determine
these greater terms.
In the several preceding integrations we have considered
the quantities s, s', m, and c, as constants on the right-hand
side of the equations, which is not strictly true, since they,
in consequence of the method for the variation of arbitrary
constants, have become variable. But on the assumption
of these quantities being invariable, only very small terms
have been added to their values, resulting from the first
P 2
212 THEORY OF THE MOTION OF THE FREE PENDULUM.
approximation, and their variation therefore may be found
by a method similar to that by which we determine the
square of the disturbing force in the theory of planetary
perturbations.
The greatest terms which we can obtain in this case
result from a combination of the integrals (1) with the
first terms of the values of the differentials of m and a,
viz.:
—4-go (s”-- s”)
=
— 1 A
=#4 o 88
##
.
Considering here s and s' as variable quantities which take
the increments 8 g and 8 g/, we have
#=– o (; 8.4, 8,0
*= o(; 8,48,
# =#, o ( s+ 8 8 g/).
Substituting in these expressions the values of 8 g and 8 g/
from the first two equations of (1), we obtain
dºn_
dº T
d x_ T
* † = — (a"—a’” *
dt Iºninº" s”) [cos 2 2-cos 2 (2 + p t)].
By integration therefore determining the arbitrary con-
stants to be added so that the integrals vanish for t=0,
we have
6 m = 0
2 — 2/2 2– e'?
t T (e.”—e !cos 2a+ F (º- q, [sin 2 a-sin 2 (a + a t)].
5 a = --- ---
192 m sin ºp 384 77° Sin”
THEORY OF THE MOTION OF THE FREE PENDULUM. 213
*
We see that our more exact determination exerts no in-
fluence upon the value of m ; but the effect on the value of
c. is far greater than that indicated by the equations (1)
and (2), for we have now in the denominator of the second
term the square of the small quantity m, and moreover the
expression is freed from the general factor g'.
If the place of observation is near the equator, we have,
by expanding sin 2 (2 + A t) in an infinite series,
sa=&#(*-*) sin 2 a + tº *** (e”—e”) cos 2 & + &c.
therefore at the equator
8 & = t2 T (s”—s") sin 2 a.
192
In order to illustrate the magnitude of the effect of this
term, we will take a numerical example.
Let s=# and gº inappreciable, then by means of the
value of T we have
8 &=t” (3,1383–10) sin 2 &
where the number within the brackets denotes the logarithm
of the coefficient expressed in seconds. Supposing the
pendulum-experiment lasted as long as 6 hours, then
t=6 x 3600
whence
8 & = 64!" sin 2 &.
Hence the greatest change of the motion of the plane of
oscillation at the equator generated by the terms just
developed during the time of 6 hours amounts to only
214 THEORY OF THE MOTION OF THE FREE PENDULUM.
64", and takes place when the initial azimuth of the plane
of oscillation is either 45° or 225°, or 135° or 315°. In
the first two cases this change is positive, in the other two
it is negative. This change will be zero if the pendulum
commences to swing either in the meridian or in the direc-
tion perpendicular to it.
For shortness, let
# T
T
- 2 – e’2\ . * — eſ?
*Iºninº." s”); S (s )
R T384 nº sinº
then the maximum of 8 a takes place for any latitude ‘p
and duration t, if
S (1-cos 2 p. t)
tan 2 & =
all Z, C& R—S sin 2 p., t
and the maximum itself is therefore expressed by
v R”—R S sin 2 p. t + 4 Sº sin” pot
whence we see that if t is not greater than 6 h., the maxi-
mum maximorum takes place for p=0, id est, at the
equator, so that for any other latitude
8 & Z. 64".
Considering the length of duration of the experiment we
have here assumed, this deviation is so small that perhaps
the most accurate observation will be scarcely able to
ascertain it.
We now substitute the remaining third term of the
value of 6 (#) into the differential equation (a) and
integrate; thus we obtain
THEORY OF THE MOTION OF THE FREE PENDULUM. 215
2
6 e = — #. cos Moos & ot—ºt
2 o” cos M cos w ł.
Cº)
/ * *º / sº gº
A e e sin M sin 3 ot—ºft sin M sin a t
6 ay” 2 o'
f /
A ee cos Moos 3 & 4 ºt: cos M cos w ł
a * 2 o”
/2 /2
** sin M sin 3 & #4 ºf
aſ” 2 a.”
f
6 m = *; cos M sin at #: sin M cos w t +
Cº) Cº)
+
5 e =
sin M sin w ś
Ae (e” + s”)
6 w” (e”—e”)
A e (e.” + e” e 2 .../2
#so M sin **ś
_A e (sº + e”)
2 w” (e”—e”)
6 a = A e” ef A e” eſ
— — — cos M sin 3 w t——:-- cos M sin a t
6 w” (e”—e”) ****-āzū-zº, Cº)
A e e” § A e e”
-–tº– sin McOS 3 w t + =::::=
6 w” (e”—e”) 2 w" (e”— e”)
cos M sin 3 w ł
sin Micos 3 ot
sin M cos w t
sin M cos w t.
The influence of these expressions is imperceptible.
It remains, finally, to consider the second lines of the
differential equations (a), but the terms arising from it
are likewise extremely small, and it may therefore suffice
to state the results.
By means of the equations
6*= s” cos” a t-H a” sin” ot
o, 6)
6:= –0(s”—s”) sin ot cos ot
t;
*}}=oss. He 62
where ºf is neglected in the last equation, we find by sub-
stitution and reduction
9% a cos o t + d 6, sin ot-6 o s' cos o t + 6 p. 8 cos ot
d t
{} º s' sino t — d 6. a cos o t = 6 o a sin ot-i-6 pas' sin ot
d t
216 THEORY OF THE MOTION OF THE FREE PENDULUM.
6 º (sº cos” ot — sº sin” ot)+2 d 6 a s' sin ot cos ot
dt
=0o (se’—2 s a sin” ot)+ 6 p. (s” cos” ot—s” sin” ot)
dº ..., 2 : … 2 d 6/2 . .”.
6 i == (cos” ot—sin” ot) + clºſ (sº + s^*) sin ot cos ot
= –0 o (s” sin” ot—g’” cos” ot)+6 u ss' cos 2 ot
whence by integration and neglecting the factor u, as small
compared to o, we have
se-–?” cos (p
3 w
5 e' = 2 *: q) [e”
[e e' cos M cos” a t + e” sin M sing a t)
cos M cos” a t + e e' sin M sin” w t|
6 m = n cos (p e” eſcos M sin a t—e e” sin M cos a t
w (e"—e") L +; e” e cos M sin 3 w ł +; e e” sin M cos 3 ot
sa_ _n cos ? Tº e (e–3 e”) cos M sin ot-3 e! (3 e”—e”) sin M cos ot
w (e”—e”) L-# e (e” + e”) cos M sin 3 w t—# e' (e” + e”) sin M cos 3 ot
Summing up the results of all these developements, we
observe that of all the expressions found in the second
approximation, only one term is of appreciable value,
viz. – º
8 & =# os st.
We have shown by a numerical example that the latter
may become considerable under circumstances which can
not always be avoided.
Its effect, however, is modified by the circumstance that
the resistance of the air is continually shortening the arcs
of the greatest oscillations of the pendulum, and it is
therefore necessary to consider the influence of this re-
sistance.
Effect of the resistance of the air on the motion of the
free pendulum.
The resistance which a medium opposes to a body
THEORY OF THE MOTION OF THE FREE PENDULUM. 217
moving in it, is a force acting in the direction of the
tangent to the curve which that body describes, and
vanishing if the relative velocity of the body and the
medium is zero ; it is therefore a function of the relative
velocity. Moreover this force is also a function of the
figure of the body, its density and the density of the
medium.
Taking here into account only the resistance of the
atmosphere, which moves uniformly with the earth, we
may assume
d s”= d £2–H dºm” + d ?
where º expresses the velocity of which the resistance
of the air is a function. Denoting by W a function of
d's
of
of the air, we have for the components of the resisting
force acting parallel to the axes #, m, & the expressions
, of the figure of the body, of its density and of that
d : d m d &
—W “º ; –W “” ; –W “º
d t d t d t
and these are the quantities which must be added to the
second members of the equations (C) if the effect of the
resisting medium is taken into consideration.
For the same reason we must add the quantities
w(ºft-tº), w (#-tº), w (sº-n}}
to the right-hand sides of the equations (D) and (E); but
the equation (F) would by this means become an incom-
plete integral.
In order, therefore, to find the additions to be made to
218 THEORY OF THE MOTION OF THE FREE PENDULUM.
the integrals (AI) (BI) and (Cl) we take the equation (L),
viz—
C dr d t-(B–A) p q dº=ſſy (n dº C-C dº n)+Y (; dºt-É dº ()
+ y” (£ d” m—n d’ê)] dºm
and in a similar way as we have derived this equation, or
by merely interchanging letters, we obtain the following
tWO —
B d g d t +(A–C) p r dº=ſſB (n dºg-gdºm)+8 (£d” – d’ ()
+ 8" (š d” m—m d” $)] d m
A d p di + (C–A) q r dºº-ſa (nd” (-(dºm)+a (; d. 3-5 dº ()
+ a” (; dºm—m d’ê)] dºm.
Considering only those terms which arise from the re-
sistance of the air, and making at the same time A= B, the
preceding three equations become
C dr. . . . . . =/W [(Y" m—y () dº +() (–)" （) dºm
+ (Yê-y m) d $1 dºm
A da-(A–C) prºt–ſ W [(8" m—8 () dº 4-(8 (–8"t) dºm
+ (8' 8–8 m) d £1 dºm
A dip—(A-C) q r dt =/W [(a" m—a () di + (a C-a";) dºm
+ (a' — a m) d [] d m.
From the equations (G) we easily obtain
'Y'm—y' = 2 v —8 w; y – y'É= o'v–8'w;
y'É—y m = 2"v–8"w;
Á"m—8% = y wicz w; 8 &–8''{= Yw—2'w;
Á'É–8 m = y”w—o.”w ;
2"m—o.' HS w—y v; c, č–2"º-B'w —y'v;
2'É–2 m = 8"w—y’v.
Hence by means of the equations (H)
THEORY OF THE MOTION OF THE FREE PENDULUM. 219
o;; • * e .--r/w (***) d mºrpſ w w w a m +q_ſw w w dºm {
A%+(A-C)pr—-a ſweet wº) an ºr ſw v w dºm-p/W u v dºm
A#-(A–C) ar--pſw (*4 w”) d m+gſ Wu v d m4. r.ſ Wu w dºm.
Since a well constructed pendulum must be symmetrical
about the axis of w, we have on this supposition
fW w w dºm-0; ſ Wv w dºm—0; ſw w w dºm-0
fW (w” +w”) dºm-ſw (v*-Hw”) dºm.
Assuming
fW (u2+v°) dim-C k'; f W (wº--w”) dim- A k
we have by substitution
dºr
dº = —k/r
dq_1 A–C = 'ºmº
d p_A –C
II --R- q r= —k p.
We may here remark that though the effect of the resist-
ance on the angular velocity r, can under these conditions
not become zero, yet k' is much smaller than k, the more so
the smoother the surface of the pendulum.
Let us first take the hypothesis that the resistance of
the air is proportional to the first power of the velocity,
then k' and k are both constant.
The integral of the equation dra —k'r dt is therefore
r=nº"
where e denotes the base of the hyperbolic logarithms and
mſ is the arbitrary constant having the same meaning as
in the integral (A).
220 THEORY OF THE MOTION OF THE FREE PENDULUM.
The equation shows that the resistance of the air coun-
teracts the rotation of the pendulum about its axis; but
since k’ is a very small quantity, the exponential function
can only exert a perceptible effect after the lapse of a long
time, and we may therefore neglect it for the short duration
of a pendulum experiment. Consequently, we take r=m/
and the expression (A,) remains unaltered.
But the case is different with the other two differential
equations, viz—
*q-LA-C, ,-
#+**pra-ka
dp_A–C
#—tº ar=-kp
because k is much greater than k'. Multiplying the first
of these two equations by q dt the other by p dit, then ad-
ding and integrating, we have
p°+ q2 = —2 ſk (p” + q’) dt
and we must therefore add to the right-hand side of the
equation (B,) the quantity
–2 Aſk (p”--q*) dt.
Multiplying the last three differential equations respectively
by CY'dt, A 3’dt, and A &"dt, then adding and integrat-
ing, we obtain, by means of the equations derived from (G);
A (a” d p + 8" d g –r dºy") + C (y" d r + r dy")
= — A k (a" p + 8"q) d t—C #' y” r d #
but the same equations derived from (G) give
A (p do.” + q d'é" + r dy”)=0
therefore adding this equation to the preceding, and inte-
grating,
A (a"p 4.3"g)+C Y' r=–A/h (2) p4 B" g) d t—C ſk. Y' r dt.
THEORY OF THE MOTION OF THE FREE PENDULUM. 221
Neglectingk as small compared tok, it follows that the quan-
tity to be added to the right-hand side of the integral (C) is
—Aſk (c."p +8°g) dit.
Consequently, the quantities k and c' determined by the
equations (G) must on account of the resistance of the air
receive the following terms:
k= —2 ſk (p” +q*) dt
k'+ —ſk (c."p +8°g) dit
and for the same reason the values of D and E in our
second approximation receive the additions
––1. " Tra-la”-* (,”, Laº
D= *#[p +g #(ºp.43%)]
E = —k (2"p + 8"q).
Eliminating the known values of p, q, &c., and neglecting
the higher powers of 6, we obtain
d 6
D= —k 6 #=ko (s”—s”) sin (ot--m) cos (ot-H m)
and substituting these values into the first expressions for
the differentials of 8, 8', m and 2, we have
dé- s' l. 1,” ; v.2 - \
d? T k[i+. (2442)] sin” (a t-i- m)
ds__ 8 l. 11/ 2
d : T k[*4; (2442)] cos” (ot--m)
dm = —k (****) sin (a t + m) cos (a t + m)
d (s”—s”) | (b.)
_2 k s = (A +% u’) si o
o (s”—s”) sin (a t + m) cos (ot-H m)
d x_ s sº tº
# =2k (s”—s”) sin (a t + m) cos (ot-H m)
+k(*-Fs")(P+} uſ)
o (s”—s”)
sin (ot-m) cos(ot-m))
222 THEORY OF THE MOTION OF THE FREE PENDULUM.
But s'=*T —# aſ 8.
(º)
Substituting this value in the second of the preceding
equations, we have
#= - U º: cos” (ot-i-m)
whence the following theorem.
Whatever be the law of the resistance of the air, the least
arcs of oscillations of the free pendulum undergo no change
in consequence of this resistance, unless the pendulum
received a lateral impulse at the commencement of its
motion.
If we take each of the quantities a pu, aſ equal to zero,
the four preceding differential equations become integrable.
The first and third of them are transformed into the
following:
ds__ 3-2
JºT k a sin” (o t+m)
#= —k sin (ot-H m) cos (ot--m).
In order to integrate them we assume
p=s sin m; q = 8 cosm
therefore
dp d g sº, a , dºm . d g de —ºn . .
dº - d ? Slm m +ji a coS m ; # =4; cosm clºſ 8 SIn m
or substituting the preceding values of d g and dºm
# = -k s sin (ot-i-m) coso t ; #=-k sin (a t + m) sin ot
THEORY OF THE MOTION OF THE FREE PENDULUM. 223
and eliminating a sin m and a cosm, we obtain
# = —#k p—# kp cos 2 ot–4k q sin 2 ot
# = —# k g + , k g cos 2 ot—# k p sin 2 ot.
In order to simplify the integration of these equations
we must transform them into others in which the co-
efficients of the variables are constant quantities. This
is effected by introducing two new variables C and D,
which are connected with p and q, by the following equa-
tions : —
C= —e”p sin a t-i-e”g cos ot (2)
D=e^*p cos o t+e”g sin ot
where e as before denotes the base of the hyperbolic
logarithms. Differentiating these equations and sub-
stituting the preceding values of dip and d q, then we
obtain
d t
o, D A k 1) C © * > © e e o © (8)
ji=-4K D+o
The differentiation of the first of these equations gives
d”C – 1 1. dC 2, d. D
# =# k; -o #,
but multiplying the two equations (8) by # k and o re-
spectively, we get by subtraction
d. C d. D
# kº-o;;−(3*-*)0
224 THEORY OF THE MOTION OF THE FREE PENDULUM.
Therefore
d2C gº
Jºž = —(ó” rºm- +k”) C.
Let us assume, since a 7 k, that
O*=o°–3k”
then we have by integration
C=X cos Qt--X' sin Qt . . . . (y)
where X and A' denote the two arbitrary constants.
Substituting the first differential of this integral, viz.-
#= —Q X sin () t-i-QX' cos Qt
and also the integral itself into the first equation of (8),
then we obtain the integral of the second equation of (8)
D =(;x +}x) cos () t-H (;x-#x) sin Qt. (8)
Denoting the initial value of s by [s], to distinguish it
from the variable quantity s, then for t=0 we have s—[s]
and ma 0, therefore for the same moment of time p = 0,
q=[s], whereby the equations (2) give C=[s], D=0; con-
sequently by means of the equations (y) and (8)
k ().
– : -º- N + + \'= 0
X. [*]; g. +.
whence it follows that
k
X/= — “– ſal.
in ſºl
The values of C and D are therefore transformed into
C=[s] cos Qt ; D = —#[s] sin () t.
THEORY OF THE MOTION OF THE FREE PENDULUM. 225
Now from the two equations (c.) we find
p= —C e-* sin ot-HD e-” cos ot
q= C e-* cos o t+D e-” sin ot
therefore
p=ssin n=-Hil(1+...) e-"sin(•+0)t
—#[s] (1 -:) e-"sin (o-Q)t
q = 8 coS m = 3II(1 +#) e-* cos(a)--Q) t
l – “’Y a-ºkt gºsº tº
+}[s] (1 & ), ) e-º" cos(a)-(2) t
Since Q”= @*—#k”, but k many times smaller than o, we
may in the integral assume ot-Q, and therefore
s sin m= —[s] e -* sin 2 ot
s cosm– [s] e -º cos 2 ot
whence, by adding the squares and by division,
s=[s]e-º; n=27–2 ot. . . . (s)
Now, if g = 0,
6= 8 cos (ot--m)
therefore, by the substitution of the equations (s),
6=[s]e-*cos ot.
This expresses the following known theorem.
On the hypothesis that the resistance of the air is pro-
portional to the velocity, the arcs of oscillation decrease in
a geometrical proportion, while the time of oscillation re-
mains the same.
Rigorously speaking, a very small decrease of the time
of oscillation takes place, since Q = w/o”—#k”< 0, but the
difference is not perceptible.
Q
226 THEORY OF THE MOTION OF THE FREE PENDULUM.
The developements which hold good for the case g = 0
may also be applied when g is very small, for in this case
the variation of sº, arising from the resistance of the air,
is a small quantity of the second order. The expression
for do, in the equations (b) becomes on this supposition
Z l ºf
# -: + sin 2 (oti n)+****) sin 2 (ot-i-m)
and substituting in it the equations (s),
l. 1."
d4– —ºgº sin 2 ot– 8 (w-H p") sin 2 ot.
olt [s] 2 o
Since the second term of this equation gives after integrat-
ing much smaller terms than the first, we will omit it.
Now
º 8oe #" 2 lºe #k' .
ſºng ot. di- -i. Tacos?ot-Fi.Hi. sin 2 ot
of which again the second term is of a higher order than
the first, and may therefore be neglected. Determining
the arbitrary constant so that for t=0, also 8 & = 0, then
k; s’
2 o [s] (e” cos 2 ot–1)
8&=
and hence at the moments of the greatest arcs of oscil-
lation
k g’
2 o [s]
Cº. tº:
(e” – 1).
We have seen in the second approximation that the term
which has a sensible effect on the motion of the plane of
oscillation in the azimuth, is
do.
*::=++ o 8 s'
d t 2 4
THEORY OF THE MOTION OF THE FREE PENDULUM. 227
substituting herein for s its value [s]e-*, and integrating,
there results,
_o|s|s' —#k
&a=&# (1—e-º).
This term is the greatest in the expression for the disturb-
ance of the motion of the plane of oscillation in the
azimuth.
We now proceed to investigate the effect of the resist-
ance of the air on the motion of the pendulum if the latter
moves on a conical surface of nearly circular base.
Substituting the additional values of D and E, viz.:
D=k o (s”—s”) sin (ot--m) cos(a t + m)
E= —k (u-- aſ) 6*—ko sa’
in the equations for the differentials of S, h, l, and 2', or,
what amounts to the same, substituting in the equations
(b) for 8, s', m, and 2, and their differentials, the quantities
AE, h, l, and 2', and their differentials, then we obtain
#= —# k (8–l cos 2 ot--hsin ot)
A
—k *# (8+l cos 2 o t—h sin 2 ot),
(º)
#= -# (8” sin 2 ot--& h--!” sin 2 ot-Fh l cos 2 ot)
—k # (8°sin2 ot—8h–lºsin2 ot—h lcos 2 ot)
# =; (8” cos 2 ot—8 l-H hl sin 2 ot--hº cos 2 ot)
+k *:::: (8°cos2 ot--8!—hlsin2 ot—h” cos2 ot)
#= -i. (l sin 2 ot--h cos 2 ot)
+k *:::: (lºsin 2 ot--h cos 2 ot).
(…)
Q 2
228 THEORY OF THE MOTION OF THE FREE PENDULUM.
In these equations the terms multiplied by (a ++//) are
much smaller than the others, they may therefore be
neglected. Further, h and l are small quantities, on the
supposition that the base of the conical surface is nearly a
circle; therefore and because after integration the terms
multiplied by h”, h l, and lº have no small divisors, they
will likewise be neglected. Consequently we have
4––4kg++kleosot–4khsin 20t
d t
#=-kg in 2 ot—ºkh
#=#3co. 2 ot-ski.
These equations are integrable, on the hypothesis that
the resistance of the air is proportional to the velocity.
Let
B= a e -3°, h- aſ e -**, ! = Q,” e -3°.”
where e denotes the base of the hyperbolic logarithms, and
a, aſ, aſ" are functions of t. Substituting the differentials
of these equations in the preceding, we have
# = ka"cos 2 ot—ºka in 2 ot
/
# = —# k a sin 2 ot
//
# =# k a cos 2 ot.
In order to obtain herefrom linear equations with con-
stant coefficients, we assume
j= — aſ sin 2 ot-i- a” cos 2 ot
g= aſ cos 2 ot-i- aſ 'sin 2 ot.
THEORY OF THE MOTION OF THE FREE PENDULUM. 229
Therefore by differentiation, and with regard to the pre-
ceding equations, $.
#=ºf n
d
#=3%a–2 og; & ſº tº g e (c. 2)
d g_
#=2 of
The combination of the first and third of these equations
gives by integration
a=#943 . . . . . . (88)
where A is the arbitrary constant, consequently the second
equation is transformed into
d 2
#=#Ex- (*-**)g.
The third equation, again differentiated, gives
dºg o ..., d f
##=2 o ż
and this is transformed by the preceding into
dºg
d :2
the integral of which is
= —4 (o”—its k”) g-H ko A
g=Xſ cos 2 (2 t +Xſ' sin 2 O t-- #.
where A/ and A’ are the arbitrary constants, and
Q= Wo-ſºº.
From the third equation of (c. 2) we derive
_ I dºg
f=g., +
230 THEORY OF THE MOTION OF THE FREE PENDULUM.
whence by differentiation and substitution
f= —x. sin 2 Qt-i-X/7 g cos 2 Qt . . (88)
In order to determine the arbitrary constants, we denote
by [8] the initial value of 8, and by [8] the initial value
of 8', that is, of # (s—s'). Since the initial value of m is
zero, the initial value of h will likewise be zero, and [9]
will be the initial value of l. The same quantities are
respectively the initial values of a, aſ, aſ", whence it follows
that [6] is the initial value off, and zero the initial value
of g. Consequently the integrals give
X=[8]; M = —ºft#: x=2...].
Substituting these values, we obtain
k
a=1;94 [8]
g=—[8] #. cos 2 Qt 4-[8] ; sin 2 ot-[3]
f=[3]; in 2 ot-Igloosa at
ko
4 Q2
and by means of our preceding assumptions
a/= —f sin 2 ot--g cos 2 ot
a’’=fcos 2 ot-# g sin 2 ot
S=a e-º-; h- aſ e-º-; l-a” e-#";
therefore taking again Q= 0, that is, neglecting k”, we
obtain
8–Igle-º-'811. e-" in 2 ot
h= —[3] * e-º-HI8] #. e -*** cos 2 ot
l=[8]e-º'-[3] ... e-"sin 2 ot.
(a)
THEORY OF THE MOTION OF THE FREE PENDULUM. 231
Since & = w/h.” +!”, and tan 2 w=} we have, neglecting
k”, tº - - -
3-[3]e-**[3] ... e-"sin 2 ot
_ [S] k B] k
tan 27– –4; zºº. cos 2 ot.
At the moments of greatest amplitudes we have always
cos 20–1, hence the last formula shows that the resistance
of the air does not affect the time of oscillation.
If the pendulum commences its motion on a conical
surface with circular base, then [8]=0, and the preced-
ing equations give
8=[3]e-º
&=[6]+, "sing ot.
(º)
The arcs of oscillation, therefore, decrease in a geometrical
proportion, just as in the case when g = 0. Since & =0 at
the points of 0, -90°, &c., we see that at these four points
of a revolution the amplitude of the arcs of oscillation re-
mains the same, so that the circular form of the base is
preserved during the decrease of the arcs of oscillation, the
deviation at the other points of revolution being imper-
ceptible on account of the smallness of the factor k, which
enters the expression for 8”.
If the pendulum commences its motion on a conical
surface with elliptical base, or [3] is not equal to zero,
then the equation for 8 and 8 give by division
[8]+ [5]; sin 2 ot
[8]+ [&#. sin 2 ot
* =
8
232 THEORY OF THE MOTION OF THE FREE PENDULUM.
therefore at the four points above mentioned
B = [8]
GTIST
whence it follows that
where [s] and [sº] denote the initial semi-axes of the
ellipse.
This equation proves the similarity of all the ellipses
which the pendulum describes in consequence of the de-
crease of the arcs of oscillation caused by the resistance of
the air.
Substituting the last values of £8, h and l in the equation
for d c.', we obtain, neglecting again k”,
da' k[8]
dt 2[&]
This equation shows that the time of rotation of the pen-
dulum is not sensibly changed by the resistance of the
air.
In order to justify our neglecting k”, we will illustrate
by a numerical example the smallness of the coefficient k
sin 2 ot.
in comparison with o.
We have stated at the commencement of Chap. 7. Part I.
the experiments made by General Dufour and Messrs.
Wartmann and Marignac. The pendulum which they
used was 20 metres in length, the amplitude of the first
arc of oscillation amounted to 3*, 25, which was reduced,
at the end of their experiments, to about 0”, 70. The
duration of any of their experiments was upon the average
2", 243. The least arcs of oscillation aſ were extremely
THEORY OF THE MOTION OF THE FREE PENDULUM. 233
Small, and we have therefore here a case where s'= 0,
whence the formula 3=[8]e-º' is transformed into
g=[s]e-º
where [s] denotes the initial value of s. Let [si] express
the final value of s, and T the number of seconds elapsed
between [s] and [si]; then we have
[s]=1",625
[s]=0*, 350
T=2*,243 x 3600
and applying the above formula to this time and the two
end-values of s, we get
k=2}og[s]-log ſºil
MT
where M denotes the modulus of common logarithms.
The result of the numerical computation is
k=0,0003803.
But we have very nearly
= . /g.
•=A/ X.
and we may assume in this case the value of the acceler-
ating force of gravity
g=9”,807
while the length of the pendulum is
X = 20 m
whence it follows that
o:= 0,7002145.
We see therefore that k, and the more so k”, is an ex-
tremely small quantity in comparison to o, and need only
be retained where it enters an exponential function.
234 THEORY OF THE MOTION OF THE FREE PENDULUM.
We will now examine the effect of the resisting medium
on the comical motion of the free pendulum, on the other
hypothesis that the resistance of the air is proportional to
the square of the velocity.
The square of the angular velocity of the pendulum is,
by equation (BI)
d 6*, and Jº
#4 ºr
and we have therefore

tº- d 6* , and Jº
k=PA/º; +6%.
where p is a constant depending on the figure of the pen-
dulum, and to be determined a posteriori, like k in our .
last example.
From the equations (p. 195)
9%=-o (*-*) sin (oti n) cos (oti n)
62 #=o sº-(e +} p’) 0°
we derive
d 62 diº c)\ }) ... •
62 (#4 62 #) = o” (s”—s”)” sin” (ot-Fm) cos” (ot--m)
+ o” s” s”--2 (a+} p") oss'6°-1-(u ++ p^)” 64
=6° o” [s” sin” (ot--m)+s” cos” (ot--m)]
+2 (a +} p") os & 6”--(a--##!)*6.
Omitting the terms multiplied by (u-F# Aſ), as they are
much Smaller than the others, we obtain
k=p ov (ot-H m)+s” cos” (a t-i-m).
The square root in this expression can always be expanded
THEORY OF THE MOTION OF THE FREE PENDULUM. 235
in a converging series, ascending in the cosines of multiples
of (ot-i-m), thus
k=pos [a”--2a” cos 2 (ot--m)+2a" cos4 (ot--m)+&c.]
For the determination of the coefficients aº, aº, aº, &c.,
we obtain, taking
g? º Es s/2
2
C
a 2
the following expressions
wº-l ſºv 1 — c” cos” a . daº
7T el/o
a"—". ſºv- cos” as . cos 2 a da;
T O
&c.
which depend on the elliptic integrals the modulus of which
is c, and they can therefore be calculated.
From the differential equations (b) we obtain by making
herein p + . Aſ = 0, and transforming the products and
powers of the circular functions into linear dimensions
#= —# k g + , k g cos 2 (ot-H m)
/
#= —#ks'—# ke' cos 2 (ot-i-m)
2 /2
#=-###. sin 2 (a t-i-m)
do. s sº .
# =k #4 sin 2 (a t-i- m).
Substituting the series for k in these equations, they become
0 (l (1) (0) (2)
# =-3 peº ("—a ')-; p < *(2a —a — a ) cos 2 (a t + m)
(2) (1) (3)
—# p we’ (2 a -a – a ) cos 4 (a t + m)— &c.
236 THEORY OF THE MOTION OF THE FREE PENDULUM.
/ (0) (1) (1) ) (3
#=-39s tº a + a )— ; p we e' (2 a +a" +d") cos 2 (at 4 m)
(2) (1) (3)
—# p we e' (2 a + a + a cos 4 (a t + m)— &c.
m e2+ eſ?, (0) (2) . . .
gº. -- # pºs: #(a —a ) sin 2 (a t + m)
e?-- eſ?, (1) (3) , .
- p * * :=(a – a ') sin 4 (w t + m)— &c.
6 a. e2 eſ , (0) (2) . .
# =Pe:-a(a – a ) sin 2 (w t + m)
2 (1) (3), .
+po-ºlia (a "-a") sin 4 (a t + n)+ &c.
e?– e'?
Taking on the right-hand side of these equations m-0, their
integration gives the following results:
8 s− —#p s” (a"—a") ot— ºp s” (2 a")—d"—aº) sin 2 ot
—#p s” (20°–a”—a") sin 40 t—&c.
8s'= —#pse' (a"+a")ot–4pgs'(2 a "+ a”--a”)sin 2 ot
—#pes' (2 a”--a” +aº) sin 4 ot—&c.
2 2
8 m- + ps #. (a"—aº) cos 2 ot
– 8
-- ~2 –L eſ?
++ps: + 8 (a"—a") cos 4 ot--&c.
g”—a’”
8 & = —#p **
2 -/
-- 8° 8
—l (1) A, (3)
#P. s/2 (a. a”) cos 4 ot—&c.
Taking these integrals between the limits ot–0 and
ot–7t, we obtain firstly
8 m = 0; 8 oz - 0
which shows that the resistance of the air has neither an
influence on the time of oscillation nor on the azimuth of
the plane of oscillation.
THEORY OF THE MOTION OF THE FREE PENDULUM. 237
We obtain secondly
8 g= —#p s” (a.”—a") ºr; 8 g = —#p g g (a"+ a”) ºr
which equations express the decrease of the greatest and
smallest amplitude during one oscillation.
Since this decrease is only very small, it may be extended
over any number of oscillations in the following way.
Let us denote the initial greatest and least amplitude by
s and g; the same after the lapse of one oscillation by
s, and g’, ; the same after the lapse of two oscillations by
s, and sº, and so on; further, let the respective coefficients
of s, and g’, of s, and s', &c., be denoted by al" and a,” by
a,” and a,”, &c., then we have with sufficient exactitude.
s =s —#p s” (a "—a ") ºr
s' = 3 —#p s sº (a °) -- a. 0) ºr
s, a si —#p s,” (a,”—a,”).T Z
ga-31–3; p 5181 (al"--a,”).T (Z)
ea = 8, -} p s,” (a.”—a,”).T
ga=8/3–#p s,s,(a,"+a,”) ºr
&c.
These equations may be continued to the last observed
oscillation, and by means of them the value of p can be
calculated.
Considering the case, when s'=0, we have
I tr . 2
a"=#|ſ. sin a da;=-t–
Trev 0 T
l "r . 2
Q) = sin a cos2a:da:= —
Oſ, #/. 1. 3. Tr
1 tr . 2
(*) = + - sin a cos4a: da = —
Ol, #:/. . 5. Tr
a"=#ſ." sin a cos 6 acda;= — 2
77 eV () 5. 7. T
&c.
238 THEORY OF THE MOTION OF THE FREE PENDULUM.
whence we derive the series
cos 6a–
e 2 2 2
Ø = ——2 203–2 8400-2
SII). T azeos Ø 3.5:* Ø 5.7.7t
&c.
The convergency of this series will depend on the conver-
gency of the sum of the coefficients, but
I 1 l 1 + ++++++ &c.
—-H −-H — + &c. =
Tā-Fāº; + 3 + + &e –(#-F#-F#-H &c.)
:
hence our series for sin a is converging at least for the
interval from ac-0 to ac-Tr; and it is between these limits
we must take the variable quantity, since we apply our
formulae to each oscillation separately. Now the condition
that s'=0 gives 8 g = 0, and if we substitute the values
2 2
a. *=-f- ; a 9– —
7T. 3
in the equations (Z), we obtain
a result agreeable to the well known theorem on the course
of the pendulum in resisting medium.
We may proceed a step farther. If we suppose sº very
small, the corresponding values of a " and a "differ from
those just derived only by a quantity of the order (#) .
8
and we may therefore assume that the preceding values
hold good when sº is small. Substituting these values in
the equations (Z), we have
THEORY OF THE MOTION OF THE FREE PENDULUM. 239
s'1 = 8 – 3 p 3 &
8'2 = 51–3 p 5, 81
s'a = 82- #ps, sº
&c.
We derive therefrom
8
g/
M. tº- º M. -º tº-
** – 1 8 u – 1
therefore the ratio of any two consecutive greatest ampli-
tudes of oscillation differs twice as much from unity as
the two corresponding smallest amplitudes, a result which
is essentially at variance with that found on the hypothesis,
that the resistance of the air is proportional to the velocity
itself.
Let us finally consider the case when the base of the
conical surface, on which the pendulum moves, is nearly
circular.
Introducing into the expression for k the quantities
Á=# (s--s’); 8'-# (s—s'); h-8' sin 2 m;
l-S' cos 2 m; 2' = 2 + m,
we have
k=p o v&º-F8°–28 (l cos 2 ot—h sin 2 ot)
or, only considering the first power of S',
k=p a £8–p o (lcos 2 ot–h sin 2 ot).
From the differential equations for 8, h, l, c' we get, by
making p-H #Aſ =0, and neglecting the powers and products
of h and l,
- #=-38(3-loos2 ot-hsin 200
:= —#k 8 sin 2 ot—#kh
240 THEORY OF THE MOTION OF THE FREE PENDULUM.
#= # k 8 cos 2 ot—# k l
do." k; tº
*† = — 4 +
di #30 in 2014 hºod
whence, by means of the preceding value of k, we obtain
the following differential equations —
#=-ºpogº-pog(loos2 ot-hsin ot)
#=-| possing ot-aposh
+} p a 3 (lºsin 4 ot-H h cos 4 ot)
#= #p o 8” cos 2 ot—#po 8 l
—#p o S (l cos 4 ot—h sin 4 ot)
d x'
#=–4p o(sin 2 ot-hoos2 ot).
Integrating these equations from a t-0 to a t-T, that is,
for the time of one oscillation, there results
8 & = —#T p 3%
8 h = —#T p 8 h
8 l = —#T p &l
8 o' = 0.
But the initial value of m is m-0, and therefore
h–0, l-S'
whence the second integral becomes
8 h = 0
and, because # = tan 2 m, it follows that
8 m = 0.
TIIEORY OF THE MOTION OF THE FREE PENDULUM. 241
This integral, in conjunction with 8 o' =0, shows that at
the moments of greatest amplitudes the resistance of the
air has neither an influence on the time of oscillation, nor
on the azimuth of the plane of oscillation.
Since for the initial value m = 0, we have le/8", therefore
the integral
8 le —# ºr p 8 l
becomes
88– —#T p 38'
and making use of a similar notation as before, we obtain
8, =8–$7 p 3”; 8,' = 8 – #T p 38'
£8,-81–% T p 31*; 8, -817–3. Trp 8, 8,’
Ás-S,-# trp 8,”; 6, = 8, -# trp 8,8.
These expressions show that (s—s') being small, the
ratio of s to s' undergoes changes during the motion of the
pendulum, a result which differs again from that found on
the hypothesis that the resistance varies as the velocity
itself.
The torsion of the thread.—The force of torsion is pro-
portional to the angle of torsion, which in our case is equal
to the angle through which the pendulum's bob turns in
the rotation about its axis, and is therefore by our pre-
ceding notation fºr dt. Let k denote the momentum of
torsion for the angular unit in circular measure, then, be-
cause the rotation of the pendulum is opposed by the force
of torsion, the momentum of the latter will be in general
expressed by
—k frat.
This is the term to be added to the right-hand side of
the first equation of (L), so that, taking again A= B, we
R
242 THEORY OF THE MOTION OF THE FREE PENDULUM.
have, with regard to the correction on account of the re-
sistance to twisting,
d r_
C Jº T k fºr dt
taking for shortness
R=fºr dt
we have
d” R_ _k R
d; T TTC’
the integral of which is
R= c cos o' t-i-c' o't
o' = A/ l.T
C
and where c and c' denote the two arbitrary constants
which are to be determined by the initial values of R and
d. R
7 º’
Let the initial angular velocity of the pendulum's rota-
tion on its axis be denoted as in (A1), viz., by m/, and sup-
pose that for the time t-0, the torsion is—t; then
where
c= —T; o'c' = m/
and we obtain
ſmſ .
R= -r cos o' t--tº, sin o' t.
º Cº)
This equation shows that the rotation of the pendulum
on its axis is changed by the force of torsion into an oscil-
lating motion, the greatest amplitude of which is
m/?
+A/º
o”
THEORY OF THE MOTION OF THE FREE PENDULUM. 243
The same expression gives by differentiation, since r-
d R.
d t
, the following:
*= o' T sin oſt-H m/ cos o' t.
This is the quantity which must be substituted for the
value of m/ in the equation (FI), where u’=m/ . so that
now the value of A' becomes
/ /
p'-9% Tsino tº cos of t.
We will now proceed to investigate the variation of the
arbitrary constants arising from this value of pſ.
From the equation (C2) we have, neglecting pº,
v=o #44 º'
whence by squaring and omitting p",
v°s”= o” g” + p^ os s.
Taking as before sing=s, the first equation (G1) gives,
with omission of the constants and those terms multiplied
by p. pſ
2 k=|a' os &
and the second equation gives on the same conditions ºc'-0,
so that only that term of the second equation (FI) which is
multiplied by aſ enters the value of k", viz.,
k'-# Aſ 0%.
Substituting these values of ic and k' in the first system of
differential equations for 8, 8', m, and 2, then their integrals
will express the additional effect of torsion on the motion
R 2
244 THEORY OF THE MOTION OF THE FREE PENDULUM.
of the pendulum. But before making this substitution,
we must derive the following expressions.
By differentiating the equations for k and k' we have
d k_d aſ
#=#-os-
dº I dº an 1, a d6
# = }} &#96.
Since we must here take
d^k oss'
Tºº 62
we get
d du/
d Jº de' – a du/ , d. 6 /
6 : ; – 3 dt 60s; +/# oss'.
By means of these values the expressions for the
auxiliary quantities D and E give
D= —pºſ o a sº
d 6 . . d uſ
E= ’ 6 - + 4. P 62
* " I r * in
whence the differential equations are transformed into
** = w - si e' d'A' F1 –
g; = & e sin 2 (****)++...+ [1-cos 2 (a t + m)]
*::=-34 in 2(****)++ 4* [1+cos 2 (a trºl
d : à M. m 4 & d ?. 77
d m_, ee' e e' d uſ .:
g; =k -acos 2 (***)+3=-aº; in 2 (erº)
e” + e” d w
4 w (sº-sº) dº
d a , 1 ...“
; :=#1. –3 pa
+ e” º
—a cos 2 (wt + m)— sin 2 (a t + m
where, according to a preceding result,
, C o' T
/
sin o' t + º cos o' t.
THEORY OF THE MOTION OF THE FREE PENDULUM. 245
The value of o' = A/ # depends chiefly on the in-
tensity k of the force of torsion, and the latter again
depends on the cross-section of the thread. But we are
always able to choose the thickness of the thread so that
o' may be many times smaller than of and on this supposi-
tion the preceding differential equations show that by the
integration no terms can arise which are multiplied into the
time, and more, that such terms can only be then generated
when o' =260 contrary to our Supposition. It is further
evident that of all the terms arising from the integration
those must be the greatest which have the small divisor
-o'. But these latter terms can only be generated from the
first term of the equation for 8 oz.
Therefore we take
d x
ºf T
/ C of T
— 1 o / 1.
Aw = } sin o' t-i- #
#
/
º cos of t.
The integration of this equation gives by determining the
constant, so that for t=0, also 8 a-0, the following result,
C m/
2 o'
&a=ºsinº ot+ sin ot,
Of all the expressions arising from the torsion of the thread
this is the only one which may become perceptible. Its
maximum effect produced on the change of the azimuth of
the plane of oscillation is
C T . O a n”
3. At 3 AA/ " + .
and differs therefore essentially from that produced by the
earth’s rotation, which has no maximum.
246 THEORY OF THE MOTION OF THE FREE PENDULUM.
If the pendulum is at perfect rest before its judicious
liberation, the initial torsion will be very small and may
be even zero. In this case the maximum of the change
amounts to
C mſ
2 o'A'
In order that this quantity may not become great, we must
take care that the velocity m/ be as small as possible.
For illustration we take the numerical example given
in our first approximation (p. 188), where we found
# uſ T-162" w.
But in this expression we have
, - Cº. : T-".
Al AT’ (*)
therefore by substitution
Cºn' – 162" wo
2 A T Tºr
Let us take the case where 2 was 1, and 100 o’=o, then
#, := sº = 43 minutes
which is a quantity disturbing the motion of the pendulum
very considerably.
It is therefore advisable to relinquish Mr. Faucault's
mode of suspension, and to adopt one or the other described
in Chap. 7. Part I., where the thickness of the suspending
wire excludes all possibility of its torsion.
247
CHAPTER II.
THEORY OF THE GYROSCOPE.
HE theory of the gyroscope may be founded upon the
theorems concerning the apparent forces in relative
motion; but these theorems not being generally known
the author * has preferred to base his researches upon the
consideration of absolute motion.
The principal parts of the gyroscope (fig. 29) are a ring
IFIG. 29.
* M. Yvon Willarceau.

248 THEORY OF THE GYROSCOPE.
Aſ movable about a fixed axis O2/ within which a solid
of revolution A turns about its axis Oz, which is con-
strained in the motion of the ring. Let the axes of the
ring and disc, intersecting in the common centre of
gravity O, have any inclination to each other denoted by
m, but let us admit that they are principal axes re-
spectively. One of the three rectangular axes a, y, z,
must always be made to coincide with an axis of rotation
by which means the positive direction of rotation and of
moments is easily determined.
Describe with centre O a sphere, and through the same
point a fixed plane, which cuts the sphere along the great
circle NN'ay (fig. 30). Draw in this plane two rect-
FIG. 30.

THEORY OF THE GYROSCOPE. 249
angular axes a, and y, and perpendicular to it an axis 2.
Let us assume the fixed plane parallel to the terrestrial
equator which we suppose to be immovable during the
duration of the experiment. The points ac, y, z, where
these axes intersect the sphere, are therefore : the first, the
vernal equinox; the other, 90° farther in the direction of
the earth's rotation; the third, the north pole. In each
of the equatorial planes of the two movable axes 2, and 2/
draw two other movable axes perpendicular to each other
and denoted by ac, y, and a,’,ºy/. The equatorial plane
of the disc cuts the terrestrial equator in a point N
distant from the axis of a by an angle Na:- ºr, and makes
with the plane of a y an angle = 0 on the negative side
of the axis of 2. Denoting by p the distance of the axis
of a, from the point N, we have N y,<=90+ ft. The pole
of the equator of the disc is in 2, and its distance from
the axis of 2 is measured by the angle 6 which always
lies between zero and 180°, while the angles Jr and p
have no limits.
The same letters with accents express the same quan-
tities in relation to the ring. The polar distance 2 2/=6'
is constant ; the plane joining the poles 2 and 2ſ is con-
strained in the diurnal motion, and the spherical angle
a: 2 2ſ is the right ascension R of the axis z/ of the ring.
Denoting by a the angular velocity of the earth’s motion,
and by dt the element of time we have
d R
— F . . . . . . . . (1
For brevity's sake we term the plane passing through
the axes of the ring and disc, the plane of the ring, and
that passing through the axes of the ring and earth,
250 THEORY OF THE GYROSCOPE.
the horary plane, denoting by y the angle between these
two planes.
In order to determine the position of the disc with
regard to the ring, we produce the plane of the latter
to meet the equator of the former, in M suppose, de-
noting by c, the angle M2A,. It is now our object to
express the circular quantities c. and y, which determine
the positions of the disc and ring in functions of the time.
Let us first find expressions for the position of the
movable axes with regard to each other:
a'ſ being the pole of M 2/2/, the arc joining 2, and aſ
will be perpendicular to it, therefore a, 2, aſ = 90°–2,
whence
cos (a, aſ)=sin & . . . . . . . . (2)
Joining a, and yſ the triangle a, 2, yſ where the angle at
2' = 180°–2, will give, because the side 2, y',-90°,+ m,
cos (a, y/)= — cosm cos & . . . . . (3)
In the same manner we obtain from the triangle a, 2, 2/
cos (a,2/)= — sin m cos & . . . . . (4)
Completing the triangle a/ 2, y, and remembering that
the angle at 2,- 2, we shall have,
cos (y, alſ)=cos & . . . . . . . . (5)
and from the triangle y, 2, y/ where angle at 2,-90°–2.
cos (y, y/)=cosm sin & . . . . . . (6)
Lastly, from the triangle y, 2, 2,' where likewise angle at
2,- 90°– c.
cos (y, 2/)=sin m sin c. . . . . . . (7)
Besides, we have directly,
THEORY OF THE GYROSCOPE. 251
cos (2, asſ)=0
cos (2, yſ)= -sin m . . . . (8)
cos (2, 2/)= cosm.
The arcs N 2 and N2, being quadrants, the angles at 2 and
2, in triangle N 22, are right angles, and therefore the
sum N 2 a.--a; 22,-90°, or denoting by 8 the angle 2,22',
ap-H R–8=90° . . . . . . . (8 bis)
and substituting s for angle 22, 2/we have g-H N2, M=
90°, or
ºp–2=90°–s . . . . . . . . . (9)
The triangle N'22', having likewise two right angles,
gives firstly
*]" - 90° - R e G o s & Cº © gº º (1 O)
then y +M 2/N’=90°; but because M2ſ N' =90°–$', it
follows
y=$' . . . . . . . . . . . (11)
The value of ſº from equation (8bis)
op=90°–R+8 . . . . . . . . (12)
compared with that of lº' in (10) gives
*—ºft'=8 . . . . . . . . . (13)
The following systems of equations are derived from the
properties of the spherical triangle 22, 2/:
sin & sin 6=sin m sin y (14)
sin 6' sin y=sin 6 sin s
cos n cos 6'--sin m sin & cos y=cos”
cosm cos 6+sin m sin 6 coss-cos 6 × . . . (15)
cos 6' cos 0--sin 6' sin 6 cos Á-cos m
252 THEORY OF THE GYROSCOPE.
—cos y cos g + sín y sin a cosm-cos Á (16)
—cos Á cosy--sin & siny cos 0 = cose
sin 6 cos y +sin 6 cosm coss-cos 6 sin m
sin 6 cos g + sin 6' cosm cos y=cos 6' sin m
The differential equations of the rotation of a solid
body about its centre of gravity, whatever its motion of
translation may be, are the following —
*P_(R_ -º
sin 6' cos Á-Hsin m cos 6 coss-cos m sin 6 }
(17)
~\
*q_fr_ — ſº
B;-(C–A)rp=0 } . . . (18)
d r tº-
cº-(A–B)p q=R
J
— ci in a dº — d 6 )
p=sin ºp sin 6 dº cos p dº
d Jr. d 6
q=cos p sin ºf +sin ºp dº } (19)
_ dº d"), --
7" = dź — cos 0; J
where A, B, C denote the moments of inertia about the
principal axes a, y, z, which intersect in the centre of
gravity; p, q, r the components of angular velocity of
rotation, and P, Q, R the moments of external forces
with regard to the same axes.
The disc under consideration being a body of revo-
lution about the axis of 2, the moments of inertia about
the other two axes are equal, id est
B=A . . . . . . . . . . . (20)
With regard to the moments of the forces which act
upon it, we must in the first instance remark that gravity
THEORY OF THE GYROSCOPE. 253
must be eliminated, because the force which results from
it acts through the centre of moments. The disc receives
from the ring motions, the resultant moment of which we
will denote by A. Neglecting the effect of friction, and
admitting a perfect symmetry about the axis of the disc,
it follows that the forces acting parallel to the axis upon
the pivots are reduced to one force, the direction of which
passes through the axis of revolution, and has therefore
no moment. Likewise neglecting the resistance of the
air, there remain forces perpendicular to the surfaces of
the pivots, and the common direction of all those forces
passes through the generating axis of the whirling body.
The axis of the resultant moment p is therefore situated
in the equatorial plane of the disc. Let X denote the
angle which this axis makes with that of ac, taken in a
direction from ac, to y, ; then we have
P= p, cos X.
te D
Q=A sin A . . . (21)
R= p, cos 90°=0
This last value reduces the third equation of (18) by
means of (20) to # =0; whence denoting by m a certain
Constant we have
*=Tv . . . . . . . . . . . . (22)
By means of the equations just established, the two first
of (18) are transformed into
A #4 (c-A)n = coax
. (23
A}} -(C–A) ap=a inx (23)
254 THEORY OF THE GYROSCOPE.
The equations of the motion of the ring may be written
down at once by accentuating all the terms which enter in
the equations (18) and (19), with the exception of the
time t. We have therefore from equation (18)
dp/_ / – (Y /-. TX/
Air (B’—C) q’r'-P º
Bº-(c-AOrp'-Q } . . . (24)
cº-(A–B)p'ſ-R
d t ~
and from equation (19) we obtain, because 6' = constant,
p'=sin ºp' sin 6' gº
q'=cos p' sin 6' º
, dº — dº' .
7" - d t COS 6' 77 °
but by means of equations (10) and (1) we have
** =–o
d : T . . . . . . . . (25)
Moreover from (11) @' =y; hence by substitution
p' = –0 sin 6 sin y
g’= —a sin 6' cosy
dy & e & (26)
7’’= #-Fo cos 6'
therefore by differentiation
% = –0 sin 6 cos y #,
d q' º ... ... dy
# = +o sin & sinº º, } . . . (27)
dr" | dºy
# T d #2" l
THEORY OF THE GYROSCOPE. 255
The moments of the forces which the ring receives are
of two kinds. Firstly, the moments of the forces equal
and opposite to those which it exerts upon the disc; se-
condly, the moments of forces which the supports exert
upon the axis of the ring itself. The latter may be re-
duced to a single moment uſ, the axis of which is situated
in the equatorial plane of the ring, and makes an angle M
with the axis of a/taken in the direction from aſ to yſ.
Therefore
P’ = –P cos (a, asſ)—Q cos (y, a,')+// cos A/
Q = — P cos (a, yſ)—Q cos (y, y/)+/w/ sin Aſ
R’= —B cos (a, 2/)—Q cos (y, 2/).
Taking the values of P and Q from (21) and making
further substitutions by means of the equations (2) to (8),
we have
P’ = —p, cos A sin c. — A sin A cos & +p'cos A/
Q = + a cos A cos & cosm—a sin A sin & cosm-i-p' sin M
R’= +/w cos A cos & sin m—p sin A sin c. sin m.
If we substitute these values, as also the preceding,
determined by (26) and (27), into the system of equations
(24), we obtain the following expressions —
Aſo sin 6' cosy # +(B'—C') o sin 6'cosy (# + Co. COS 9) ~
= — a sin (2 +A) + p.' cos A/
/ -- ~ : tº dy 7 – A/ !-- Al/ 2.3 dy
Bosing in wº +(C’—A') o sin 6' sin y (; +ocos ?)
= p, COS m cos (« +A) + p/ sin Aſ
d”y
N/
} (28)
C CŞ2 —(A’—B') o” sin” 6' sin y cos y = u sin m cos (a+x),
256 THEORY OF THE GYROSCOPE.
The first two of these equations may be written
f ~3 tº te (B^+C'—A') dy
p' sin A' = a sin 6' sin ºy dt ||—u cosm cos(x+x)
U+(C’—A')ocos 6'. (29.
ſ (B'—C'—A’) dy`
p'cos A/= a sin 6’ cosy dt | +p sin(x+x);
U--(B'—C)ocos 6') J
they serve to determine p' and X' when all the other
quantities relating to the motion of the disc are given.
In order to obtain expressions for the motion of the
system of two bodies, we proceed in the following Way.
Multiplying the first equation of (23) by cosa, the second
by sin o., we have by subtraction
dºp sin .d4 gº tº
A (cosa; sin zº)+(C A) m (g coso, + p sinc.)
=p cos (2+3) . . . . . . . . . (30)
and by a similar method
i., ... dºp d q gº in z. –
A (sina; + COS or #)+(c A)n (g sin 2–p cos 2)
=A sin(x +A) e tº e º ºs e e tº e e (31)
These equations are adapted to give pº and o. --X in
functions of other quantities supposed to be known. The
first may also be used to form the principal equation of
the problem; for the equation (30), being multiplied by
sin m and subtracted from the third equation of (28), gives
, d” y / /\ .2 cºv, 2 *
C 772 —(A’—B') o” sin” 6 sin y cosy
tº in 2°4– dp = 0 (32
+ A sin m (in 4% cosa #) (32)
—(C–A) sin m 'm (g cos & H-p sin c.)
THEORY OF THE GYROSCOPE. 257
This would be the equation from which, if we chose, we
had to eliminate all the variable quantities with its de-
rivatives except y; but for the sake of symmetry we will
proceed otherwise.
By means of the values in (19) we find those of the
following expressions, which are contained in the equations
(30), (31), and (32):-
d 6
in 2. — –2 in a * 1 sin(*-
g cos 2 + p sin &= cos ($ a) sin 6: 4 sin(? *) iſ
q sin &–p cos 2= —sin (p-2) sin 9% +cos (b-a) º
But by the equation (9)
ºp—a = 90°– s;
therefore
q coso, + p sin o. = sia, in 9% toº.”
dt dt
ſº in 92% +sin,” (33)
q SIn c. – 9 COS c. = -cos s sln dº +sins;
By means of the equations (12) and (1) we have the
value of º determined thus:–
dºſe d &
ºff {-º-º-º: dº - Cº. e. te * & C G o (34)
and in order to obtain the value of # We have recourse
to the first equation of (14) and to the third of (15),
Writing them as follows:–
sin 6 sin &=sin m sin y
COS m
sin 6/ —cot 6'cos 6.
sin 6 cos &=
S
258 THEORY OF THE GYROSCOPE.
They give, by differentiating them —
:... a d8 , ... d 6 sº dy
cos S sin 6 7; +sin & cosó † = sin m cosy ºf
dt
—si :- a d8 d 6 . A d 6
sin 8 sin 6 dº +cos(3 cos 6 dº =cot 6' sin 6 dº
multiplying the first by cos Á, the second by (—sin 8), we
get by addition —
d&_
d; T
but the first equation of (15) gives by differentiation,
d 6
sin 6 sin m cos Á cosyº-cost. sin & sin 6 dº
i., a dº —s: tº :... ... dy
sin 6: =sinn sing sinyº; . . . . . . (35)
whence, by substitution—
e d 6 — cº º º ſ d y
SIIl 6; =sin m (cos & cos y—sin 8 sin y COS 6 alſº
or by means of the second equation of (16)
in a "8— . dry g
SID. 9: =-sm n cost fi : . . . . . . (36)
Substituting the values (34), (35), and (36) into the
equations (33), we obtain —
tº - e sin m cos s , :- — ; º tº dy
cos & + p sin & = — a sin 6 sin s— * * : * (sing sin 6—sin 6 sin Y) ºf
Q ſp sin 6 ( y) dt
º º º ... . sin 6 sin y \ dy
sin & –p coso. = o sin 6 cos s--sin (or 8 + SIn 5 –––– ) ---.
Q ſo + Sln m sin {} dt
These expressions are reduced by means of the several
equations of (14) to —
q cos 2 +p sin &= -o sin 6 sin s
dy (37)
q sin c. –p cos & = +o sin 6 cos s -- sin m 77
\-
THEORY OF THE GYROSCOPE. 259
Now, differentiating these two equations, we have
d . d e da, T
cos a #4 in a #– (a sin a -p cosº):
d. sin 6 sing
d t
; , , ..., 00 Q d p . . g. (38)
sin a #-cos a # + (a cos a + p sin a);
d. sin 6 cos a i., a d”y
== +&inn # D
The second equations of (14) and (15) give respectively
sin 6 sing=sin 6' sin y
/
cos 0. —cot m cos 6
In 77
sin 6 cos ge
whence, by differentiation,
*
d. sin 6 sin a º d's w d
* : * * = cos & sin 6 : +sin & cos 0 < .
d t d t d t
— cºn A/ dry
= sin 6 cos y ºf
º (39)
d. sin 6 cos s** *** =–sins sin 9°5. +cos, coso ºff
d t d t dt
. A d 6
amºng cot n sin 9;
.
Multiplying the first of these equations by cos s, the
second by (–sin s), we have by addition
sin 60's =sin 6' cos y cos a dy — cot m sin s sin 9%
d t dt
and, substituting the value of (35)
sin 6 # =sin 6 (cos y cos s—sin y sing cos w)%
S 2
260 THEORY OF THE GYROSCOPE.
which by means of the first equation of (16) is reduced to
in a“ – – in a dy
sin 6; = sin 6 cos 3;. . . . (40)
We must now find an expression for the differential
co-efficient ** which is contained in the equations (38).
d t
The equation (9) gives
d p-do. = — d g :
whence
d x dºp + d's .
dº, T & " dº ’
and by means of equation (22) and the third of (19) we
have
d x — d iſ d g .
Jº = n + cos 6 dº + T :
hence, by substitution,
d” – o – — tº * - /h/ dy
dº = m-o cos 6 sin 6 (sin m cos 6 coss--sin 6’ cos 8).”
or by means of the first equations of (15) and (17)
# = n –o cos 0–cos m # |
dy
d :
With regard to the constant quantity m, we must remark
that it is not directly given us by observation, but the
preceding equation will furnish us with the means to ob-
tain it. Let us suppose that the ring is at rest with
regard to the surface of the earth, making with the horary
plane an angle yo, and that we communicate to the disc
an angular velocity relatively to the plane of the ring ex-
= n –o (cosm cos 6' + sin m sin 6'cos y)—cos m
º
THEORY OF THE GYROSCOPE. 261
pressed by w; then this quantity will be the initial value
Of #. and the equation (41) furnishes the following ex-
pression for m in functions of w and yo:—
m=w--o (cosm cos 6'--sin msin 6'cosy)=w-Ho cos 60 (42)
This equation shows that the velocity of rotation about
the axis of the disc is equal to the relative velocity of the
disc, the ring being at rest, increased by the component
of the velocity of the earth about the axis of the disc,
the ring likewise considered fixed.
Substituting the values (39), (35), and (41) into the
equations (38) we have
*q ...sin.” P- (, — * - *7)/asin.-
cos 44; +ina%-(n a cosé cosmº)(asina pcos 2)
—o sin 6'cos 7%
in, "4– *P–– (, – º dy tº
sin a ri-cosa;; = (n o cos 6 cosmº)(a cos o-Hpsin c.)
tº ſº dy , º, ... dº y
+o sin 6' cosm siny º + Sin "dº
By means of these values the equations (31) and (30)
are transformed into
usin (a+x)=[on-A(e COS 64 co-wºº)] (q sin a-p cos a)
º d y
— Aa sin 6/ COs y : :
(a) S1 77;
u cos(a+A) = [on-A (e COS 6 + COS w}})] (g cos a + p sin a)
d2 y
* :- - d y tº
— Aw sin 6' cosm sin y : * – A -— a
77 77. sin 737.
262 THEORY OF THE GYROSCOPE.
Substituting the values (37) we have
#3)
usin (a + A)= [on-A (acos 0+cos 7%)] (* sin 6 cose + sin "...
g d y
—Aw sin 6' cos y : Z.
73;
z
Al COS (a+x)--[on-A (* COS * cosmº)] a sin 6 sin e
tº ... ... d’Y : ... dº y
–Aw sin 6' cosm sin y : — A sin m *, *;
"sin Y; " ...
and, developing and arranging —
a sin (a + X)=(Cn-Ao cos 6) a sin 6 coss
+ [Cn sin m—Ao(sin m cos 0 + sin 6’ cos y
2
+ cosm sin 6 coss)] # —A sin m cos m #%
p cos(a+A)= -(Cn–Ao cos 6) o sin 6 sin s
+Ao cosm (sin 6 sin s—sin 6' sin y) #
—A sin m #
By means of the second equations of (14) and (17) these
expressions are reduced into
~
+ (Cºv–2 Aſo cos 6) sin m #
p sin (a + X)=(Cn-Ao cos 6) o sin 6 coss
º d ºv”
– A sin m cos n +% . (43)
p cos (2 + N)= -(Cn–Ao cos 6) a sin 6 sin y
d”y *
dº?’
If we put for cos 6 its value from (15) and for sin 6 cos s
that derived from the third equation of (17), the for-
—A sin m
THEORY OF THE GYROSCOPE. 263
mulae (43) will give p, and 2+\ in functions of y and its
two first derivatives.
In order to form the final equation, we eliminate the
quantity p, cos(a+X) between the third equation of (28)
and the second of (43), thus we obtain
d” y
(A sin” m + C) O! #2 +(Cn–Ao cos 6) to sin 6'sin m sin y
—(A’ –B') o” sin” 6 sin y cos y=0;
then replacing cos 6 by its value (15), and transposing,
we have
2
(A sin” + cº- —(Cn-Acosm.ocos 6')osin 6 sin m siny (44)
+(A sin” m + A'—B') o” sin” 6'sin y cosy.
For brevity’s sake we put
g_ Cn — A cos m. o cos 6' ... ...: g
- = e o sin 6' sin
C, A sin” m + C' 77
g_A sin” m + A*–B' , , ;
28 a TTA sing m-EC/T a sin” 67
Where g denotes the accelerating force of gravity, and 8
some abstract number, which will always be very small,
the preceding formula is then written
2
#=}(-sin 7428 in , cos") . . . . (45)
and we have at the same time
tºº 9 A sin” m + C'
To sin 67 sin ºn Cº-Acos m.o. cos 37° (46)
8= | * sin 6/ A sin” m + Aſ —B'
– 5
sin 7 Co-Acos m.o. cos 67'
If +yo be the value of y, for which % becomes zero,
264 THEORY OF THE GYROSCOPE.
we get by multiplying the equation (45) by 2d), and then
integrating—
dy” 2
dº? =#4 [cos y-cos yo–8 (cosº y—cosº yo)]
Ol'
#= + A/# (cos y-cos yo)[1–8 (cos y + cos yo)] (47)
* d
The case where y becomes imaginary, for a value %
=0 is excluded from our considerations, therefore y will
always be a real angle, and because 8 is very small, the factor
1–8 (cos y + cos yo) will always be positive; hence the
only condition for % to be a real quantity is that the
factor (cos y—cos yo) and the constant (a) have the same
sign. Now, as the angles 6 and m are supposed to lie
between zero and 180°, and the term. A cos m. o cos 6' is
very small compared with Cn, the sign of (a) will be
identical with that of (n). When therefore the component
on of the rotation about the axis of the disc is positive,
we have COS Y > COS Yo,
and the values of y follow in the order
Yo, 0,-Yo, 0,-Fyo, 0, . . . . &c.,
but if the motion takes place in an opposite direction, that
is, m being negative, we have
COS y < COS Yo,
and the values of y succeed each other as follows:–
Yo, T, 27-yo, Yo, T, 2 T-Yo, . . . . &c.,
The absolute value of # is the same for two equal
THEORY OF THE GYROSCOPE. 265
values of y with contrary signs; hence the motion of the
plane of the ring is an oscillation about the horary plane,
and the angular velocities are equal in positions sym-
metrical with regard to this plane. Neglecting the term
affected with 8 in equation (47), we find that the expres-
sion coincides with the differential equation for the motion
of the pendulum of length a . The oscillations of the ring
follow in this case the same laws as the rotation of the
pendulum about the vertical. We will now consider sepa-
rately the two cases of n being positive or negative.
Let n be positive and
ë-cos y, £oacos Yo . . . . . . (48)
whence
sin y dy= —d #,
and
1
dry= + –
7–rº-H.
The equation (47) gives therefore
20 274–
A/ºat=#v== (£—fo) [1–8 (#-F#,)] (49)
Let us denote by N the value of the radical on the right
side of this equation, and Suppose
(#—£o) [1–8 (£--á)]=(1-#") z*. . . (50)
so that we have
d #.
N= (1–?”) 2 e e º & O (51)
then the equation to be integrated becomes
2g di-2.É 52
A/* di = j . . . . (52)
266 THEORY OF THE GYROSCOPE.
From the equation (50) we derive
ë (2°–8)= —#--Z. . . . . (53)
Differentiating the equation (50) we obtain
[1 + 2 # (2°–8)] d £=2 (1–?)2 d z;
whence by means of the equations (51) and (53)
d #_d 2.
NT Z. '
and therefore
* 2 g., , dz
A/#at=%. . . . . (54)
Observing that if neglecting the terms affected with 8
the value of Z” is greater than + š,”+ 8, 2*--2*=(; £o + 2*)”,
a quantity essentially positive, we may assume
Z*=v°4-2 v cos 2 c. z*-i-24 . . . (54 bis)
In order that this value may agree with the preceding,
we must put
v°= 1 (1–28 #,)”
2 v cos 2 k=#0–8 (1 + š,”)
whence
2 v-1–28 #,
_{o —8 (1 + š,”)
cos 2 k = I-28 E. . . . . (55)
from which we derive
1-3,4-3 (1-2 %-Hä”)_1-8, 1+8(1-8)
in 2 ... —
**=-ºr-gs; 2 1–28 to
or because
§o-cos ‘Yo
THEORY OF THE GYROSCOPE. 267
& e c 2 l
sing cesinº. 1, 1 tº sin” #3%
# 70
1–28 cos
and ‘Yo
º • -: 1 + 28 sin” #
sin k= + sin # #"Yo
+ SID # Yo 1–28 cos yo (56)
Taking
2= Wv tang 4% . . . . . (57)
we have by differentiation
de-ºv–%. . . . . . (68)
cos” # p
the value of Z” (54 bis) is then written
Z*=v” (1+2 cos 24 tang”3%--tan" ºp)
_ v” {* º º
T cosº, | cos' ºp--2 (1–2 sin” k) sin” #4 cos” #4 +sinº *]
2
º [(coº #4 +sin” #$)*—4 sin” k sin” #4 cos” #]
and if we write c”= sin” k, we have simply
U e
Z= + cosºft 1—cº sin” b . (59)
whence by means of equation (58)
#=+— dq,
— — 6O)
Z 2 v v v^1—c.” sin” (p (60)
and by means of this value and equation (54)
1 -
dt = A/#. ==*= . (6)
2 v 1–2 ö cos yo g v1 – c’ sin” (p
We have here suppressed the double sign, because the
element of time, dt, is essentially positive, and we are
going to determine the angle p, so that dº may also be
268 THEORY OF THE GYROSCOPE.
positive. For this purpose we derive from the equations
(50), (55), and (57)
l, rh — (#–£o) [1–8 (É--ào)]
tan º-E A/2 (1–?) (1–28 fo)
or by means of equations (48)
tan ; b = + 1 A/* (cos y–cos yo) [1–8 (cos y + cos %)]. (62)
1–28 cos yo
SIn y
This value may also be written:

tan # p = + #./* # (ºyo – y) sin # (Yo-H 'y) [ 1–26 QOS # (Yo-H y) COS 36-7)]
sin y 1–26 cos Yo
We have seen that the values of y succeed each other in
the order
+ yo, 0, - 'Yo. 0, + Yo, 0, - 'Yo . . . . etc.
Taking sin yo to be positive, let us write down the cor-
responding values of tan # b, with the signs which they
take in the interval of two consecutive values of y, and
we shall have the series:
0, + o , – 0, -- o , – 0, 4-, oo, - 0, . . . . etc.
the corresponding values of , b are
T
0, #, T, 3:27, 5, 87, . . . . etc.
and those of ‘p :
0, Tr, 2 Tr, 3 T, 4 Tr, 57t, 67t, . . . . etc.
The passage from Yoto – Yo, and conversely, corresponds
to a variation of b, which is equal to 2 tr. If we take
sin yo to be negative, we obtain similarly an ascending and
continuous series of values of p.
Integrating the equation (61), we count the time from
*s
THEORY OF THE GYROSCOPE. 269
that moment when y=yo, and take simultaneously p equal
to zero; thus we have
t= 1 V; F (c, p) (63
TTE A / ... • I' (C2 (b). . .
2 v1–2 ö cos Yo 9. , ºf )
In order to obtain the duration of the simple oscillation
between the limits +yo and -yo, we must make p-2 Tr;
but since
F (c. 27)=4F (e. º).
we have, denoting by T the duration of oscillation,
2 gasms
T= A/; F (c. :) . . . (64)
w/1–28 cos yo 9 2
When we neglect 8, and suppose yo infinitely small, we
are led to the common formula of the pendulum :
9.
Let n be negative.
Sin m and sin 6' being positive, and a very small com-
pared with m ; the value of a in the first equation of (46)
becomes negative. In this case the expression (cos y—
cos yo) must also be negative, so that the equation (49) is
changed into:
- *g di- + dë (66)
– Q, w/(1 —£") (#6–8) [1–8 (£o-F#)]
By comparing the right side of this equation with that
of equation (49), we see that we pass from one to the other
by changing the signs of the quantities é, éo, and 8, whence
we must change y into (180–Y) and yo into (180–06), so
that the formulas (56) and (62) are transformed into
270 THEORY OF THE GYROSCOPE.

1–28 cos” # yo
c=sin k = + cos ? Y.
1–28 cos yo (67)
1 A/ 2 (cos yo–cosy)|1–6(cosyo-H cosy)]
4 d = -- –– 0 0
tang # 5 + siny 1–28 cos To
The values of y, supposing, for instance, yo to be positive,
follow in the order
'Yo, T, 2 T-ſy, T, Yo, T, 2 T-Yo, T, yo. . . . . etc.
The corresponding series of the values of tan ; p is
0, + o ,-0, + oo, 0, + oo . . . . etc.
and the values of p corresponding to it are
0 T 27ſ 37 4T 57 . . . . . . etc.
so that the general value of t (63) is transformed into :
l A/ – OM,
= -— e. F 3. tº 8
t 2 v 1–28 cosyo 9 (c, b) . (68)
The duration of an oscillation corresponding to the
limits yo and (2 T-yo) of y, or to the limits zero and 2 T
of b, is expressed by
2 v/ — Cº, T
2– =#. F(, ).
T v 1–2 ö cos yo 9 C ..) (69)
Neglecting 8, and supposing 'yo infinitely small, the
quantity c, becomes equal to unity, and the function
F (l, º) is infinite. The plane of the ring therefore being
placed without velocity in coincidence with the horary
plane, rests in this position, when m is negative; but the
equilibrium is unstable, for the least change of the
position, or, in other words, the smallest value of yo makes
the value of T finite.
THEORY OF THE GYROSCOPE. 271
In order to compute 2, let us add the equations (41) and
(42), which gives—
d a
d t
whence we derive
=w–o sin 6 sin m (cos y —cos yo)—cos w;
2=w t—y cos m—a sin 6 sin mſ (cosy—cos yo) dt.
In order to effect the integration indicated in the last
term, let us first consider the case where (n) and (a)
are both positive. We derive from the equation (62)
by means of (55)
cos y—cos yo-v sin”y tan” # q +8 (cos”y–cos” yo) (71)
and since # = cos y, the equations (53), (57) and (59)
give :
-3-F Z – ; cosº b + v v^l-cº sin”?
v tan” # ºp-8 v sin” # p—ö cos” # p
cos y =
Since we must have cos y = cos yo for p = 0, the radical
sign must be taken positive, for only then will the preced-
ing equation, by substituting the value of v, give the result
—#-F# (1–2 8 cos yo)
— 8
We have, therefore, in general
cos y =
= COS Yo
v V 1 — c” sin” (p —# cos” # 4,
v sin” # ºp-8 cos” # ºp
COs y =
The term to be integrated being affected with a factor
(o) which is extremely small compared with (w), we may
neglect 8 in the differential expression, which reduces the
value of (v) to , and the equation (71) gives,
ſcos y—cos yo) dt = }ſsin” y tanºpólt.
272 THEORY OF THE GYROSCOPE.
and the value of cos y itself becomes
w/1—c” sin” p-cos” #4,
cos y = tº
sin”; p
from which we derive
sin" + b-cos' 44–(1—c” sin” (b)+2 cos” # 5 v 1–6* sin” ºb
sin" （, ‘b
_(-cos’ p-sin” #6)–1 + 4 cº sin” #4 cos” #4 + 2 cos” # p 271–64 sin” p
sin” #4,
cos” k e —5–F–F–F
= 2 sinº ; (2 c” sin” # p— 1 + v 1–6* sin” #),
sin” y=
and the integral in question becomes
-ms-m-m-
1 w/1—c” sin” (p
ſcos y—cos yo) dt=2 *—ſº a +ſ # * *
Substituting the value of dt from (61), neglecting the
term in 8, and taking for brevity’s sake
A= v1–c” sin” (p,
we have
7 a. I dip_ ſ” dº
ſcos y—cos yo) dt = 2 e-A/. [ſº ſº
The first of the last two integrals may be written
ſ 1 dº /*1 + cos b dº
2 sin” #4, sin” p A
I d£1. cos ºf dºp
-*-*-
TJ sin; d. A J sing 3 &
In order to effect the integrations indicated on the right
side of the equation, let us differentiate the expression
A cot p- v1–c” sin” (p. cot ‘p
THEORY OF THE GYROSCOPE. 273
thus we have
d. A. cot p- — w l—cº sin” p. º – “** dº,
sin” ºf yi−2 sinº q,
_1–c” sin” b--cº sin” p cos” $. dq,
sin” p v 1–6° sin” ºb
1—c” sin” (b
= - . d (p
sin” p v1–c” sin” (p
_ _ _1_d b_i...a sin” b
=-ºx tº dº.
whence we derive
I d b_2 / sin” $ 44,
sin: j : A Tº A d'h-A cot ºb.
Let us now take
ac=c sin ºp
whence we have by differentiation
da;= c . COs ºf . d b,
Or, cosº. dº-º:
all cos ? dº 0. da;
sin” p A a " v 1–a.”
from which we conclude that
*cos d d (p c 27–3.
* Y- . Y = — wººl —
ſº. A ; v1–0.
=–é–
sin ºp'
We have therefore
ſº 1 d (b_ I dºl. cos b dd,
2 sinº ºff AT sin” (p. A sin” (p. A
=& ſº * dº–Acotº——º-
eſ A $—A cot (p sin ºp
=&ſº dºp- º (1+cos (b).
274 THEORY OF THE GYROSCOPE.
We have further by direct integration
—ſ dq, cos （; b_1 + cos (b
2 sin” ºf Tsin TTsin à
Hence by addition:
I dº dºp =eſºta 1 + cos ?/1—A
Jºãº #5 A iſ, sing º' ** dº sin ºp ( )
In order to avoid the indefinite expression which pre-
sents itself in the case of sin ºp being equal to zero, we
multiply the numerator and denominator of the last term
by (1+A), thus:
1 +cos b (1 l-Hcos b , cº sin” b 2 a sin (b. cos” #5
Hº-Hº (1–4)= tº ge = 20° ——‘––
sin ºp sin ºp 1 + A 1 + A
So that we have
* S-2 2 sin ºp cos” # ºp
w-cºat=2&t-ev'ſ ſ”, “a *
ſco y-cos yo) C* U – C 9. A $+ 1 + A
Substituting this value into the equation (70), and sup-
posing that for t=0 we have &=0, y=yo, p=0, we obtain
as the final result :
a =(w–2 o sin 6 sinn cº, t—(y—'yo) cosm
+osing in nºvſ. [ſº tº dº (72)
9. 0 A 1 + A
In the case of (a) or (m) being negative, it suffices to
change the signs of (a) and (o), and computing c and ſh
by means of equation (67).
The formula (72) shows that the angle & undergoes,
besides periodical variations, others which are proportional
to the lapse of time, their magnitude depending on the
amplitude of oscillations.
275
CHAPTER IV.
NEW THEORY OF ROTATION.
THE theory of rotation of a rigid system is strictly
deduced from the elementary laws of motion, but the
complexity of the motion of the particles of a body freely
rotating about a point, renders the subject so intricate
that it has never been thoroughly understood but by the
most expert mathematicians. Many who had mastered
the lunar theory came to erroneous conclusions on this
subject. Even Newton has chosen to deduce the dis-
turbances of the earth’s axis from his theory of the mo-
tion of the nodes of a free orbit, rather than attack the
problem of the rotation of a solid body.
The method by which Poinsot has rendered the theory
more manageable is by a judicious introduction of ideas,
chiefly of a geometrical character; and further progress is
only to be made by proceeding from one distinct idea to
another, instead of trusting to symbols and equations.
It has been shown in the preceding chapters that the
earth is endowed with rotation on an axis which has itself
a conical motion. On account of the complexity of these
two movements, they have been illustrated by a description
of Burr's apparatus for imitating the precession of the
equinoxes (page 145), but in order to obtain a full under-
standing of them a perusal of Poinsot's new Theory of
Rotation will be the most conducive.
T 2
276 NEW THEORY OF ROTATION.
It may, therefore, be of service to many readers to give
at the conclusion of this work an outline of that theory,
which is most remarkable, both for the lucidity of its
reasoning and the importance of its results.
A simple rotatory motion about a fixed axis is easily
conceived; all the points of the body describe circles about
that axis with an angular velocity, which is the same for
all the points at any moment.
But any number of such rotatory motions may be given
to the system at once.
If the axis of these rotations meet in a point, the
resultant of all the rotatory motions will be a simple
rotation about an axis, determined by a theorem analogous
to that of the parallelogram of forces (see page 91).
If the axes are parallel to each other, the resultant will
be a single rotation equal to the sum or difference of the
components, except they are equal in magnitude but
opposite in direction.
The latter case may be termed a couple of rotations.
The result of such a couple is merely one motion of
translation in the direction perpendicular to the plane of
the couple, and measured by its moment, that is by the
product of one of the rotations multiplied by the distance
between their parallel axes.
The motion of any system about a fixed point is generally
considered as a series of simple rotations about a certain
axis, passing through that point, but continually changing
its position; it being supposed to remain at rest for an
instant, whence it is termed the instantaneous axis of
rotation. This definition, though exact, requires the fol-
lowing additional explanation, in order to give a distinct
idea of such motion.
NEW THEORY OF ROTATION. 277
The rotation of a body on an axis which continually
changes its position about a fixed point, is the same as the
motion of a cone with its vertex at that point, and rolling
On the Surface of another fixed cone with the same vertex.
The varying line of contact of these two comes represents
the instantaneous axis, which therefore moves within the
body and in absolute space. If the motion of the body is
given these two cones may be found, and vice versa.
Thus, in order to produce the exact rotatory motion of
the earth, the circumference of the base of the rolling cone
must have the same ratio to that of the base of the fixed
cone, as a day to the time of a complete revolution of the
equinoxes.
If a body has two motions at the same time, one of
translation, the other of rotation, each may be considered
separately, but the most distinct idea of its movements
will be obtained by comparing the complex motion with
that of a solid screw turning within its hollow companion.
With regard to the forces producing motion, it is obvious
that a force applied in a direction passing through the
centre of gravity of a body, will effect a motion of trans-
lation with a velocity measured by this force divided by
the mass of the body; and that a couple of forces applied
to a body in a plane, perpendicular to one of its three
principal axes, will cause its rotation on this axis with an
angular velocity equal to the moment of the couple
divided by the moment of inertia about this axis.
The three components of a couple are three rotations
directly proportional to the couples generating the latter,
and inversely proportional to the moments of inertia
about the three axes. The axis of the resulting rotation
coincides, therefore, in direction with a diameter conjugate
278 NEW THEORY OF ROTATION.
to the plane of the couple in an ellipsoid with axes
inversely proportional to the square roots of the moments
of inertia of the body about the same axes. The con-
struction of such an ellipsoid about the centre of gravity
of the body, and upon its three principal axes, is always
conceivable, and being of the greatest importance in the
theory of rotation, it has received the name —the central
ellipsoid.
Supposing, now, that a body at rest is solicited by a
couple of forces, in any plane passing through its centre of
gravity, this plane being considered a diametrical plane of
the central ellipsoid, we see that the instantaneous axis
to which this couple gives rise, is the diameter conjugate to
the plane of this couple in the ellipsoid, and it is evident
that the angular velocity will be measured by the moment
of the couple estimated perpendicular to this diameter, and
divided by the moment of inertia of the body about the
same diameter.
Since a couple may always be transferred in a plane
parallel to its own, without altering its effect on the body,
we may always assume that the plane of the couple of
impulse, instead of passing through the centre, is a tangent
plane to the surface of our ellipsoid.
If, therefore, a body is acted upon by a couple situated
in any tangent plane of the central ellipsoid of this body,
the instantaneous pole of the rotation to which this couple
gives rise, is exactly at the point of contact.
Conversely, if a body rotates about any diameter of its
central ellipsoid, the couple by which it is acted upon lies
in the tangent plane passing through the pole.
The rotation about an instantaneous axis, which, by
hypothesis, is not a principal axis of the rotating body,
NEW THEORY OF ROTATION. 279
generates centrifugal forces in its particles which are not
in equilibrium, and thus give rise to an accelerating couple.
The effect of the latter is to impress the body at every
instant with an infinitely small rotation, which is to be
compounded with its actual rotation.
Taking two straight lines, the one of which represents
the axis and magnitude of the couple of impulse, and the
other the axis and magnitude of instantaneous rotation,
then the accelerating couple due to the centrifugal forces
will be represented in magnitude and direction by the
surface of the parallelogram described upon the two lines.
The accelerating couple tends to turn the body about
the diameter conjugate to the plane, passing through the
axis of the couple of impulse and the instantaneous axis;
this diameter is, therefore, at the same time conj ugate to
the latter and to its projection on the plane of the couple,
whence we conclude that the axis about which the centri-
fugal forces tend to turn the body is in the same plane
with the couple of impulse.
Taking two lines, one representing the infinitely small
rotation, the other, the actual rotation, and completing the
parallelogram, it becomes obvious that the end of its
diagonal remains always at the same height above the
plane of the couple.
It follows, therefore,
1. During the whole course of motion the angular
velocity is proportional to the radius drawn from the
centre to the instantaneous pole, on the surface of the
central ellipsoid. *
2. The plane of the couple considered as tangent
through the pole, remains always at the same distance
from the centre of this ellipsoid.
280 NEW THEORY OF ROTATION.
This centre being fixed in absolute space, and the plane
of the couple always remaining parallel to itself, therefore,
this plane continually touching the central ellipsoid at the
instantaneous pole of rotation, is always one and the same
plane fixed in absolute space.
The central ellipsoid rolls without sliding on this plane
with which it is continually in contact.
The series of points of contact considered upon the
surface of the ellipsoid trace the motion of the instan-
taneous pole within the body; and the same points con-
sidered upon the fixed plane, trace its motion in absolute
space. We can, therefore, determine these two curves,
and may consider them as the bases of two conical Sur-
faces with common vertex, one of which, movable with the
body, rolls upon the other which is fixed in space.
In order to determine the first curve, it is only neces-
sary to find the series of points where the ellipsoid would
be touched, by a movable plane which remains always at
the same distance from the centre of the ellipsoid, or, in
other words, which would touch at the same time this
ellipsoid and a concentric sphere with a radius equal to
the given distance.
It is obvious that this curve is an oval of double cur-
vature, which, like the ellipse, has four vertices, where it
is divided into four equal and symmetrical parts. These
four vertices are the points where the radius vector and,
consequently, the angular velocity of rotation attains its
maxima and minima values. The maximum takes place
when the instantaneous pole passes through the two
vertices which lie in the mean principal section of the
ellipsoid, and the minimum when it passes through the
other two vertices of the curve.
NEW THEORY OF ROTATION. 281
The second curve may be considered as being generated
by the former rolling about the centre upon the fixed
plane of the couple. It is therefore a plane curve which
goes round the projection of this centre in equal and
regular wave lines corresponding to the equal and sym-
metrical arcs of the generating oval. It is a kind of cir-
cular curve, but with a variable radius, and which un-
dulates between two concentric circles, touching alternately
the one and the other.
If the angle at the centre corresponding to two con-
secutive summits of the equidistant waves is commen-
surable with four right angles, the curve is limited in
all directions after a certain number of revolutions, and
the instantaneous pole by which it is described, returns to
the same point in the body as well as in absolute space.
But, in the contrary case, the curve is infinite, and the
pole which passes periodically through the same point in
the body, cannot simultaneously return to the same point
in absolute space.
The polar equations to these two curves, described at
the same time by the same point, are identical.
The rolling cone, the surface of which has the first
curve for a base, is merely a right cone of the second
degree; but the fixed cone upon which it rolls, is a
transcendental come, the surface of which undulates about
the fixed axis of the couple; it is also a kind of a right
and circular cone, but with a grooved surface.
The compound term polodia will be an appropriate
name for either of these curves; the one described within
the body may be called the relative polodia, the other
described in space, the absolute polodia; or, in order to
282 NEW THEORY OF ROTATION.
distinguish them by their particular form, the first may be
simply termed a polodia, the second a herpolodia.
These two curves depend on four given quantities, viz.
the three principal radii of the central ellipsoid which are
given by the nature of the body, and the height of the
centre above the tangent plane of the couple which is
given by the direction of the couple of impulse.
In the particular case where this height is equal to the
length of the mean radius of the ellipsoid, the polodia
becomes an ellipse, the plane of which passes through this
mean radius; and the herpolodia becomes a spiral, with
two branches turning in contrary directions about the
same fixed centre. In this particular case of the motion
of bodies, the instantaneous pole of rotation is always a
new point in the body as well as in absolute space, though
the length of the spiral is finite and equal to half the cir-
cumference of the rolling ellipse by which it is generated.
If the distance of the centre from the tangent plane is
equal to one of the two extreme radii of the ellipsoid, which
can only happen on a single point of the surface, the
polodia and herpolodia are both reduced to one and the
same point, and the instantaneous pole remains immovable
in the body and in space during the whole course of the
motion.
The same takes place when the plane of the couple is
a tangent exactly at the point which is the mean pole of
the central ellipsoid of the body.
If the body belongs to the species of those which have
two of their principal moments of inertia equal to each
other, in which case the central ellipsoid is one of revo-
lution, the polodia becomes a circle about the axis of this
spheroid, and the herpolodia is another circle about the
NEW THEORY OF ROTATION. 283
fixed axis of the couple. In all bodies of this kind the
motion is that of a right cone with circular base rolling
upon another fixed cone likewise right and circular. If
the circumferences of the two circles are not commen-
Surable, as is generally the case, the instantaneous pole
can never return to a point which coincides at once with
the same point in the body and with the same point in
absolute space.
The most simple case of all is that where the central
ellipsoid of the body is a perfect sphere; for, however the
couple may be applied, the axes of rotation and of the
couple coincide, and the instantaneous pole remains
immovable in the body and in absolute space.
Having examined the nature of the two curves which
are described by the pole and the differential equations
of which are identical when referred to the arc and radius
vector emanating from the centre of the ellipsoid, we may
consider the velocity with which the instantaneous pole
describes at the same time the one curve and the other,
and the velocity which this point has itself in moving
from or approaching to the centre. This is the fluxion
of the radius vector, and, consequently, of the angular
velocity of rotation. We may likewise find the angular
motion of the pole about the fixed axis of the couple of
impulse. It is, therefore, easy to determine the remark-
able points where these different velocities pass through
their maximum or minimum. This will happen at the
alternate summits of the waves of the herpolodia.
After that we may simplify the equation to the her-
polodia, in referring it to a radius vector emanating from
its proper centre taken in the plane of the curve, and to
an angle which this radius describes about the same centre.
284 NEW THEORY OF ROTATION.
If we want to find the formulae for calculating the
place of the body at the end of any given time, we must
express the angular velocity of rotation in functions of
the time which will be effected by a first integration, and
will give the place of the instantaneous pole at the sur-
face of the central ellipsoid. Afterwards we integrate the
preceding equation of the herpolodia, which will give the
place of the pole on the fixed plane of the couple. By
means of these two integrations, the proposed problem is
entirely solved; that is, we are enabled to assign the place
in space where the body ought to be at the end of any
given time. We need only suppose the ellipsoid brought
into contact with the fixed plane, so that it touches it at
the point which we have determined on its surface and at
the point which we have determined on the fixed plane,
then by this operation the central ellipsoid and conse-
quently the body itself will be found exactly in that
position in absolute space to which it is carried by its
proper motion at the end of the given time.
We may vary these determinations in many ways,
taking other variables relatively to the position of the
body; but whatever be the co-ordinates employed, the
expression for these quantities in functions of the time
will always require two integrations which essentially
depend on elliptic transcendentals.
In the particular case where the height of the centre
above the fixed plane is equal to the mean radius of the
ellipsoid, and where in consequence thereof the polodia
becomes a simple ellipse and the herpolodia a spiral, the
difficulty disappears, the whole process is reduced to the
ordinary rules depending merely on the integration of
exponential and logarithmic functions.
NEW THEORY OF ROTATION. 285
Finally for all bodies where the central ellipsoid is one
of revolution, and where, therefore, all becomes uniform
and circular, no integration is necessary for determining
the place of the body at the end of any given time.
When the plane of the couple of impulse is so situated
that the instantaneous pole coincides with one of the prin-
cipal poles of the central ellipsoid, it will remain sta-
tionary, so that the instantaneous axis, the axis of the
body and that of the couple, always coincide in one and the
same straight line immovable during the whole course
of the motion. The principal axes of the body are there-
fore, all three, permanent axes of rotation; but there is
a remarkable difference between them with regard to the
stability which each of these axes may show in the ro-
tation of the body.
If the instantaneous pole coincides with the major or
minor pole of the ellipsoid, and is then removed from it
a little distance by the impulse of a small extraneous
couple, it will not easily move farther, it will describe its
polodia about the same pole from which it has been
removed.
But it is otherwise when the instantaneous pole coin-
cides with the mean pole of the ellipsoid, for the slightest
derangement will remove it farther, describing its polodia,
either about the major pole or about the minor pole,
according as this accidental derangement of the instan-
taneous axis, causes to increase or diminish the distance of
the tangent plane of the couple from the centre of the
ellipsoid. If the derangement is such that this distance
does not vary, which happens on the surface along the two
particular ellipses intersecting at the mean pole, then the
instantaneous pole will describe the particular ellipse
286 NEW THEORY OF ROTATION.
upon which it has been moved, or rather the half of this
ellipse till it reaches the mean pole opposite, which is the
greatest derangement that can happen to the body; now
if the instantaneous pole has been carried to the other
half of the same ellipse, it will immediately return to the
mean pole whence it started, which is the least derange-
ment possible of the rotation of the body.
There exists an exceptional case where the instantaneous
axis being removed from the mean axis where it originally
was, comes immediately nearer to it instead of moving
farther; but in all the other cases it will describe an
elliptic cone, either about the major or the minor axis, or
trace the plane of one or the other of the particular
ellipses intersecting at the mean pole, and we infer, there-
fore, that the rotation of the body about its mean axis is
unstable.
The rotation is only stable about the axis of the greatest
or least moment of inertia of the body, but we cannot
herefrom conclude that it is equally stable for these two
axes, because if one of them differs only little from the
mean axis, it will not have much more stability than the
latter.
In order to obtain a distinct idea of this stability, and of
what constitutes its measure for each of the two extreme
axes of the ellipsoid, let us imagine the surface of the
latter to be divided into four parts or elliptic lunes, by
means of the two ellipses, the planes of which intersect in
the mean axis. The mean poles are therefore situated at
the intersection of these two ellipses, the major poles are
in the centres of two lunes, and the minor poles in the
centres of the two supplementary lunes.
Now if the instantaneous pole of rotation coincides with
NEW THEORY OF ROTATION. 287
the mean pole of the ellipsoid, it is obvious that in con-
sequence of a slight displacement, it will fall into one or
the other of the two lunes in question, and describe its
polodia about the one or the other principal pole of the
ellipsoid. Further, if the displacement of the instan-
taneous pole takes place upon one of the two ellipses
themselves, it will describe the half of this ellipse till it
reaches the mean pole opposite, and being thus carried to
the other half of the same ellipse it will immediately
return to the mean pole whence it started. The result,
therefore, is, as stated before, that the mean axis has no
stability.
If the instantaneous pole coincides with the major pole
of the ellipsoid, any displacement within the extent of the
adjacent lune will not alter its path round the same major
pole; and since this constitutes the stability of the major
axis, we may say that the magnitude of that lune is in
some way the measure of it. Similarly the surface of the
supplementary lune is the measure of the stability of
rotation about the minor axis.
Now if one of these two axes differs little from the mean
axis, the lune corresponding to it will be very small, and
its supplementary lune very great, therefore the axis
slightly differing from the mean axis has very little sta-
bility, while the other possesses it in a high degree.
It is therefore not accurate to say, as is generally done,
that if the instantaneous axis is a little removed from the
principal axis which corresponds to the least or greatest
moment of inertia of the body, it will move very little
farther and only make small oscillations during the whole
course of the motion; because if the moment of inertia
relative to this axis differs little from the mean moment,
288 NEW THEORY OF ROTATION.
the instantaneous pole can in consequence of a slight
derangement, leave its small lune to enter the adjacent
lune, there describing its polodia about the other axis;
or even if the displacement is limited within the small
lune, it can describe there a narrow and very elongated
polodia, and therefore make very great oscillations about
the principal pole from which it has been removed.
In those bodies where one of the extreme moments
of inertia differs little from the mean moment, and where,
therefore, the central ellipsoid of the body is nearly one
of revolution about one of its axes, the stability of the
rotation is only true for this axis.
This is the case with the earth, the rotation of which
is very stable about its actual axis; but there would be
very little stability in a rotation about the third axis,
which, as we know, differs very little from the mean
axis.
We may now seek to determine the motions of the
poles of the central ellipsoid themselves, the velocity with
which they turn round the fixed axis of the couple of
impulse, and the velocity with which they sink and rise
relatively to the plane perpendicular to this axis, by means
of which they receive their movements of precession and
nutation. We may also investigate the nature and pro-
perties of the three curves which the projections of these
three principal poles trace in the same time upon the fixed
plane, and we shall find without difficulty, many remark-
able theorems on the motion of bodies, of which the
following are some examples: —
If we draw from the centre upon the three principal
axes of the body three equal lines, we shall find that the
sum of the three variable areas described by the projec-
NEW THEORY OF ROTATION. 289
tions of these equal lines on the plane of the couple is
proportional to the time.
If instead of taking three equal lines on the principal
axes we take on them three lines proportional to the
square roots of the moments of inertia, we shall find that
the sum of the areas traced by the projections of these
unequal lines is also proportional to the time.
If we consider the three principal poles of the central
ellipsoid, we shall find that the sum of the squares of their
distances from the axis of the couple of impulse is con-
stant;
And that the sum of these squares multiplied respec-
tively by the moments of inertia of the body is also
constant during the whole course of the motion.
If we consider the curves generated by the projections
of the three principal poles upon the fixed plane, we shall
find that these curves are of the same species as the her-
polodia, traced by the instantaneous pole of rotation.
In the general case one of the two poles, major or
minor, which forms the centre of the polodia, describes a
curve with equal and regular undulations about the same
centre. The superior summits of the one correspond to
the superior of the other, and the inferior to the inferior.
Meanwhile the two other poles describe also curves re-
gularly undulated, but so that when the one passes the
highest points, the other passes the lowest points in the
waves of its curve.
In the particular case where the polodia is an ellipse,
and the herpolodia a spiral, the three poles of the body
describe also spirals. There exists in this case a certain
diameter within the mean plane of the central ellipsoid,
which always remains perpendicular to the fixed axis of the
II
290 NEW THEORY OF ROTATION.
couple, describing the plane of the latter with a uniform
motion, so that the whole movement of the body consists
in turning on this particular diameter with a variable
Velocity, while this diameter itself describes uniformly a
circle in absolute space.
If the central ellipsoid is one of revolution, the pole of
the figure will describe a circle like the instantaneous
pole. Taking arbitrarily two other poles on the equator
separated by a quadrant, their projections will describe
two equal herpolodias with a common centre.
Finally, the general motion of a body may be repre-
sented by that of an elliptic cone, rolling upon the plane
of the couple of impulse with a variable velocity and
sliding upon it with a uniform velocity.
The principles just laid down must be well understood
in order to obtain clear and distinct ideas, even of the
most elementary parts of rotatory motion, but their
greatest advantage consists in affording easy demonstra-
tions for the movements of precession and nutation, thus
simplifying and sometimes rectifying the most difficult
theories.
291
NOTE.-Page 167.
TO DERIVE EXPRESSIONS FOR THE MOTION OF FALLING AND PRO-
JECTED BODIES FROM THE DIFFERENTIAL EQUATIONS FOR THE
MOTION OF THE SIMPLE PENDULUM.
Since, in the case of descent or projection, the accelerating
force arising from the suspension of the pendulum vanishes, we
substitute
pt-0
in the equations (C), page 167; thus we get for the motion of
free moving bodies, with regard to the rotation of the earth, the
following—
d’é ... , d m º
d; 2 T 2 m sinº #:
d” m in a 35– d &
Žiž T 2n sinº #: 2 cosº, (1.)
d” & d m
dº? T 2 m cost ºf 4-g J
The first integration gives
# =–2 nm sing + k ~
dm — 2 n & sin ºp-2 n & cos (p + k'
d; T q, ‘p + k -(2.)
% = 2 nm cos (p-H gt-H k"
dt ~
where k, k', k" denote the arbitrary constants. Multiplying the
first of these equations by sin p, the third by (—cos ?) we have,
by adding the products,
d 8 -. d& tº
aſ: sing-i, cosp=–2 n n-gt cosp +2ng
U 2
. 292 NOTE.
where
2nq=k sin p-k" cos p
Substituting this value in the second equation of (1.), we
obtain
d?m
d? =–4m”n–2ngt cos (p +4m°q
which gives by integration
m=q+ q' cos 2 n t-i-g" sin 2n t— # cos ? - (3.)
Substituting the first differential of this expression in the
second equation of (2.), there results
§ sin ()–: cos p =h-q" sin 2nt--q" cos 2 n tº . (4.)
where /
h= — #. cos p — #.
Multiplying the first equation of (2.) by cos p, the third by
sin p, then the integration of the sum of these products gives—
§ cos p + & sin (p=c + c't +} gi” sin (p . . (5.)
where c, the new arbitrary constant, and
c’= k cos (p + k" sin (p
From the equations (4) and (5) we derive
# = p + c't cos p—q' sin p sin 2nt
+q" sin p cos 2 n t-i-4g t” sing cos @ . (6.)
&=p'+ c't sin ºp--q' cos p sin 2 m t
—q" cos p cos 2 n t-F# gt” sin” p . . (7.)
where
p = h sin (p + c cos p
p'= —h cos (p + c sin ºp
The equations (3,6 and 7) are the required integrals containing
the six arbitrary constants q, q', q", p, p", c', three of which are
determined by taking 5-0, m-o, &= 0 for t=0; thus we
obtain
p= — q" sin ºp; p’=q" cos ?: q= -q'
NOTE. 293
and the integrals are transformed into
£=c't cos p-q" sing sin 2nt — q" sin ºp (1—cos 2nt)
+#g t” sin ºp cos p
m= —q' (1—cos 2nt)+q" in 2nt-Écos, > (8.)
£=c't sin (p + q' cos p sin 2nt-i-g” cosºp (1—cos 2nt)
+} g tº sin”q, º ~!
The origin of co-ordinates is by this determination transferred to
the point where the motion of the body commences.
We must now distinguish the two cases for falling and pro-
jected bodies. In the first case the initial velocity of the body is
zero, whence
'—a o'—o. 2"– 9
and the integrals are, therefore,
:= = #. sin ºp cos p (1—cos 2 nt)++g t” sin ºp cos p
m= 9 cos p sin 2nt — 9t. cos p
4 m? 2 n.
&= -2 cosºp (1-cos 2nt)+} g tº sin” p
4 m?
The heights at our disposal for experiments on falling bodies
are comparatively so small that without an appreciable error
sin 2nt-2 n t—# nº tº
1—cos 2nt=2 n°t”—g nºt
and the expressions for the co-ordinates become, therefore,
§s—# gºn” tº sing cos p
m=–# gnt” cos p } (9.)
{=} gt”—# gn” tº cos” (p
which are identical with Gauss' Formulae (page 163) and show
that the deviation towards the south is not sensible, while a
sensible deviation takes place towards the east from the vertical
of the original point of descent,
294 * NOTE.
In the second case, where the body is projected, let us denote
by a the initial azimuth of the direction of the motion, and by v
and v' respectively, the horizontal and vertical components of
the initial velocity, then -
v cos a-c' cos p—2ng' sin (p
v sin a-2 no"— 4-cos
4"– jºcosº
—v’=c' sin p+2 n g’ cos p
whence
c'e-v cos a cos ? – v' sin ºp
y
P 19 g 29
= ~ COS a Sin q + - COS
Q 2 n. p 2 n. p
g” – à sin a + #. cosº.
Substituting these values in the equations (8.), we get
&=vt cos a cos” p-v't sing cosp
y
g; snº cos ? sin 2nt
+;cosa sin”% sin 2nt-i-
* , º *
—gain a sing (1—cos 2n) -#. sing cos p (1— cos 2nt)
+#gt” sing cosºp
1/ º v'
n= g; cosa sin º (1—cos 2nt)+ g; cos ? (1— cos 2nt)
+: in a sin 2nt-F#, cosº in 2nt—4: cosº.
27, 47,2 2 n.
&=vt cos a sing cosp—v't sin” p
-; cos a sing cos p in 2nt-É cos”q sin 2nt
1/ ©
+ gain a cos ? (1—cos 2nt)+ #. cosº (1–cos 2nt)
+4 g tº sin” p.
Sufficiently accurate values for sin 2nt, and cos 2nt, are ob-
NOTE. 295
tained by taking the first terms in their respective expansions,
and substituting them, there results
$=yt cosa—wnt” sin ºp sin a
m=vt sin a + v nt” sing cos a+v'n t” cos p
&=–v't-H$gt”--wnt” cos q sin a.
Taking the vertical component of the initial velocity
v'-o
the expressions for ś and m may be written
=vt cos(a+nt sin p)
m=vt sin (a+nt sin ºp)
whence we infer, that in consequence of the earth's rotation,
horizontally projected bodies always deviate from the vertical
plane of their initial direction towards the right-hand side of the
observer.
Taking the horizontal component of the initial velocity
1/EO
the expressions for the coordinates become
§ –o
m=v'n tº cos p
&= -v't +} g tº.
This value of & shows that the projectile, after the lapse of the
time
_2 v'
7" - —
9
returns to the horizontal plane from which it was thrown; for
this moment, therefore, we have
y^3
g”
70,
cosºp
whence we infer, that in consequence of he earth's rotation,
296 NOTE.
projected bodies fall back to a point which lies to the west from
the original point of projection.
The respective deviations of projected and falling bodies
take, therefore, place in contrary directions, and those of the
former far exceed in magnitude those of the latter. Their ratio
may be determined by substituting in the formula just found,
and in the second of (9) the values &
º-º-º-º-º-º 2s
9 8 7 9.
where s denotes the greatest height of the ascending or descend-
ing body. Thus we obtain respectively
m = 8s ** n cos ?
9
n=–ºs ** n cosº.
3 9.
Whence for the same heights and latitudes, the westerly de-
flection of a body projected vertically upwards is twelve times
greater than the easterly deflection of a falling body.
I.ONDON
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