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University of California • Berkeley
Purchased as the gift of
The Friends
OF THE
Bancroft Library
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^C^, CU^ (JIm^oX Xcc^ Jj.J%v<^ y~Sc(^^^^ Sc^^tt^
MiiCANiaUE ci^LESTE.
MECANIQUE CELESTE.
BY THE
MARQUIS DE LA PLACE,
PEER OF FRANCE ; GRAND CROSS OF THE LEGION OF HONOR ; MEMBER OF THE FRENCH ACADEMY, OF THE AC.4DEMT
OF SCIENCES OF PARIS, OF THE BOARD OF LONGITUDE OF FRANCE, OF THE ROYAL SOCIETIES OF
LONDON AND GOTTISGEN, OF THE ACADEMIES OF SCIENCES OF RUSSIA, DENMARK,
SWEDEN, PRUSSIA, HOLLAND, AND ITALY ; MEMBER OF THE
AMERICAN ACADEMY OF ARTS AND SCIENCES ; ETC.
TRANSLATED, WITH A COMMENTARY,
NATHANIEL BOWDITCH, LL. D.
FELLOW OF THE ROYAL SOCIETIES OF LONDON, EDINBURGH, AND DUBLIN ; OF THE PHILOSOPHICAL SOCIETY HELD
AT PHILADELPHIA ; OF THE AMERICAN ACADEMY OF ARTS AND SCIENCES ; ETC.
VOLUME I.
BOSTON:
FROM THE PRESS OF ISAAC R. BUTTS ;
BILLIARD, GRAY, LITTLE, AND WILKINS, PUBLISHERS.
MDCCCXXIX.
DISTRICT OP MASSACHUSETTS, TO WIT : District Clerk's Office.
Be it remembered, that on the sixteenth day of October, A. D. 1829, in the fifty-fourth year of the Independence of the United
States of America, Nathaniel Bowditch, of the said district, has deposited in this office the title of a book, the right whereof he claims
as Proprietor, in the words following, to wit : " Mteanique Celeste. By the Marquis De La Place, Peer of France ; Grand Cross of the
Legion of Honor ; Member of the French Academy, of the Academy of Sciences of Paris, of tlie Board of Longitude of France, of the Royal
Societies of London and G6ttingen,of the Academies of Sciences of Russia, Denmark, Sweden, Prussia, Holland, and Italy ; Memlier of the
American Academy of Arts and Sciences ; etc. Translated, with a Commentary, by Nathaniel Bowditch, LL. D., Fellow of the Royal
Societies of London, Edinburgh, and Dublin ; of the Philosophical Society held at Philadelphia ; of tlie American Academy of Arts and
Sciences ; etc." In conformity to the Act of the Congress of the United States, entitled, " An Act for the encouragement of learning, by
securing the copies of maps, charts, and books, to the authors and proprietors of such copies, during the times therein mentioned :" and
also to an Act, entitled, " An Act, supplementary to an Act, entitled, An Act for the encouragement of learning, by securing the copies
of maps, charts, and books, to the authors and proprietors of such copies, during tlie times therein mentioned ; and extending the benefits
thereof to the arts of designing, engraving, and etching historical and other prints."
JNO. W. DAVIS, Clerk of the District of Massachusetts.
IJVTRODUCTION BY THE TRANSLATOR.
The object of the author, in composing this work, as stated by him in his preface, was
to reduce all the known phenomena of the system of the world to the law of gravity, by
strict mathematical principles; and to complete the investigations of the motions of the
planets, satellites, and comets, begun by Newton in his Principia. This he has accomplished,
in a manner deserving the highest praise, for its symmetry and completeness ; but from the
abridged manner, in which the analytical calculations have been made, it has been found
difficult to be understood by many persons, who have a strong and decided taste for
mathematical studies, on account of the time and labour required, to insert the intermediate
steps of the demonstrations, necessary to enable them easily to follow the author in his
reasoning. To remedy, in some measure, this defect, has been the chief object of the
translator in the notes. It is hoped that the facility, arising from having the work in our
own language, with the aid of these explanatory notes, will render it more accessible to
persons who have been unable to prepare themselves for this study by a previous course of
reading, in those modern publications, which contain the many important discoveries in
analysis, made since the time of Newton. It is expected that the reader should be
acquainted with the common principles of spherical trigonometry, conic sections, and those
branches of the fluxionary, or differential calculus, usually found in elementary treatises on
this subject, in this country ; and as frequent use is made of the rules for the products of the
sines and cosines of angles, he, it was thought expedient to collect together, at the end of
this Introduction, such formulas as are of frequent use. The demonstrations of these formulas
may be found in most treatises of trigonometry, and some of them occur in the notes on this
work ; the methods in which these demonstrations may be obtained, as well as those of the
common problems of spherical trigonometry, are also briefly pointed out, in the appendix,
at the end of this volume, which may be referred to, in cases where it may be found
B
INTRODUCTION.
necessary. Some of tlie most important theorems in conic sections are also demonstrated
in the second book.
The notation of the author has been strictly adhered to, and the double parentheses, which
he has used to denote the partial differentials, have been retained, though at present many
mathematicians reject them.
For the sake of a more easy method of reference, to any particular part of the work, or
to any single formula, the marginal numbers are inserted. These are frequently referred to,
in the translation, and in the notes. The introduction of these numbers is the only alteration
which has been made in the original work. In other respects it will be found, that the
translation has been as nearly literal, as is consistent with a faithful interpretation of the sense
of the author. These marginal references might supersede the use of those made by the
author, in a few of the most important formulas, but it was thought best to retain them,
because they might possibly be referred to, in quoting from the original work. It must be
observed that in citing a single formula, the marginal reference will be found on the same line
with the formula ; but in referring to a particular sentence, or paragraph, it will generally be
on the middle line of it.
As the author has supposed the quadrant of a circle to be divided into 100 degrees, each
degree into 100 minutes, each minute into 100 seconds, &;c., and has applied the usual marks
° ' " &;c., to these quanthies ; it has been found convenient, in the notes, when the
sexagesimal division is used, to employ the letters d, m, s, he, to denote degrees, minutes,
seconds, he, of the common sexagesimal notation; so that 1000" is equivalent to 324*.
This distinction will be adhered to throughout the work.
The notes were written at the time of reading the volumes, as they were successively
published. The translation was made between the years 1815 and 1817, at which time the
four first volumes, with the several appendices and notes, were ready for publication. Soon
afterwards, the American Academy of Arts and Sciences liberally offered to print the work
at their expense, but this proposal was not accepted. One of the reasons for not printing it
at that time, was the expectation that the author would publish another edition, in which he
might modify the first volume, by the introduction of the matter contained in the appendix to
the third volume, depending on the improvements made by Mr. Poisson, in the demonstration
of the permanency of the mean motions of the planets ; and might also correct the second
volume, on account of the defects in some parts of the theory of the calculation of the
INTRODUCTION. vii
attraction of spheroids, and make other alterations, on account of the improvements in the
calculation of the attraction of an ellipsoid, first pointed out by Mr. Ivory.
The notes are adapted in some respects to the state of the elementary publications on
scientific subjects in this country, and a greater number have been given, than would have
been necessary, if the elementary principles of some of the methods, used by the author,
had been in common use in our schools and colleges. They might in some cases have been
abridged, by small alterations in the original work, but it was thought best to adhere strictly
to the method of the author.
It may be advisable for a young person, in reading this volume for the first time, to pass
over the eighth chapter of the first book, which treats of the motion of fluids, being rather
more difficult than the rest of this volume ; he may also pass over the fourth and sixth
chapters of the same book. After reading the second book, which contains all the most
interesting principles of the motions of the heavenly bodies, he can return with additional
force, to these chapters, before entering on the calculation of the figures of the heavenly
bodies in the second volume.
Since this work was prepared for publication, there have been printed in England, two
translations of the first book, with notes, by Mr. Toplis and Dr. Young, which were seen
before this volume was printed, and occasional use has been made of them. It is understood
that Mr. Harte is now printing a translation, but no copy of it has yet been received in this
part of the country.
The second volume of this translation is now in the press, and will be published in the
course of tlie next year. These two volumes will finish the first part of the work, which
may be considered as forming a complete treatise in itself. If it should be found expedient,
the whole work will be printed, in five or six volumes, as soon as it can be done with
convenience, taking sufficient time to ensure typographical accuracy, in the execution,
and the whole will probably be completed in four or five years. This time has been
considered necessary on account of the laborious occupation of the translator, which affords
him but little leisure to attend to the revision and publication of the work.
VI" INTRODUCTION.
The following formulas are much used in the course of this work. They are to be found
in most treatises on Trigonometry, and may be demonstrated by the method given in the
appendix to this volume.
[1] Int. sm.^ z= — — Acqs. 2 z — 1 ^, Radius =1.
[2] " sin.^ z= — — . < sin. 3 z — 3 . sin. z i ,
[3] " sin.^^ = ^.jcos.4^ — 4.cos.2z + i.^|,
1 C 5.4 )
[4] " sin.5 z = -.< sin. 5z — 5 . sin. 3 ^ + r^ • sin. z > ,
1 ( , 6.5 ^ , 6.5.4)
[5] " sm.6 2:= — - . j COS. Qz — 6 . cos. 4 2; + r"^ • cos. 2z — t • i ^ ^ S '
[6] » cos.2^ ^ \'\ ^°^' 2 ^ + 1 1 »
[7] " cos.^^r = -.} cos. Sz-{-o . cos. z > ,
[8] " cos.'*^ = -.\ COS. 4 2r + 4 . cos. 2 ^ + 2 • J^^ ^ »
1 ( . .5.4 7
[9] " cos.^sr = 04 • ] COS. 5 z + 5 . cos. 3 ^ + ^ • cos.2; > ,
1 ( , <5.5 ^ , , 6.5.4)
[10] » cos.^z = ^ • 1 cos. 6^ + 6. cos. 4 ;2 + f;^ • cos. 2 r + ^ . — — j ,
mi » sin.;? = ii =1^ , Hyp.log.c=l.
2./=^ , —2. /ITT
[12] " 003.;? = -^ '^ ,
[13] » c' * *^^^ = cos. z + /m . sin. z,
[14]
—2.1/—! / T •
c = cos. z — \/— 1 . sin. z,
INTRODUCTION.
\cos.z-{-)/—l.s\n.zl'' = c ' ~ = COS. nz -[-;/— i -sin. nr,
fcos. z — \/IIi. sin. z}" = c " = COS. nz — y/IIl . sin. n z,
sin. a . sin. b = i . cos. (a — 6) — ^ . cos. (a + J),
sin. rt . COS. h = i . sin. (a + ^) + 2 • sin. (a — b),
sin. cr . cos. & = ^ . sin. {a -\- b) — ^ . sin. (6 — o),
COS. a . COS. 6 = i . cos. (« + i) + I • cos. (a — b),
sin. (a -[- ^) = sin. a . cos. b -\- cos. a . sin. i,
sin. (a — ^) = sin. a . cos. b — cos. a . sin. 6,
COS. [a-\-b) = cos. a . cos. b — sin. a . sin. b,
cos. (a — b) == COS. a . cos. b -f- sin. a . sin. b,
sin. a + sin. b = 2 . sin. ^ . (a -}" ^) • cos. i . (a — i),
sin. ft — sin. 5 = 2. sin. | . (a — b) . cos. I • (cf "I" ^)}
cos. a -\- COS. 6 = 2. cos. ^ . (a -}~ ^) • cos. i . (a — J),
cos. a — COS. 6 := 2 . sin. | . (a -f- ^) • sin. ^ . (& — a),
, , ,. tanff. a + tanff.5
tang, (a + 6) = - — —— — -^ ,
° ^ ' -' 1— tang.a.taug.6 '
tang, a — tang, i
tang, (a - &) =i:p^„— t^i;^ ,
^ 2 . tang, a
tang. 2 o = - — ; — ,
° 1 — tang.2 o
^ 2. tang, o
Sin. 2 a =
l-|-tang.2o '
sin. 2 a = 2 . sin. a . cos. o,
COS. 2 a = cos.^ rt — sin.^ 0,
COS. 2 a = 1 — 2 . sin.^ a,
COS. 2 a = 2 . cos.^ a — 1 ,
sin. n 1
tang, a = = ,
" COS. a cot. a
COS. a
1
sec. a.
— - = sec.^ a = 1 -i- tang.^ a,
cos;
C
[15] Int.
[16] "
[17] "
[18] "
[19] »
[20] "
[21] "
[22] "
[23] "
[24] »
[25] »
[26] "
[27] »
[28] "
[29] "
[30] "
[30^ »•
[30"] "
[31] "
[32] "
[33] »
[34] "
[34^ "
[34"] '♦
[34///J >,
[35]Int
[41']
n n
[44] »
[47] »
[48] "
INTRODUCTION.
sin. a — sin. 6 tang. ^ (a — b)
Ein.a-{-sin.b tang.i(a-|-li») '
sin. a — sin. 5 , , ,v
[36] » , -=tang.^.(a — 6),
sin. o + sin. 6 i / i r\
[37] " ■ -^ - = tang.i.(a + 6),
sin. a + sin. b , / 7 \
[38]" -^= - = — cot.|.(a— Z*),
cos.a — cos.o
[393 » cos^o-cos^t _ ^^ i.{a + b). tang. ^ . (6 - a),
COS. a -|- cos.o
1 COS. b n . J
[40] » — - = tang.2i6,
1 -]- COS. 0
sin. a ^
[41] " —I =tang.Jc,
"■ ■' l-j-cos.a
1 + COS. a -
— -, = cot. i a,
sm. a
Bin. a
mil. u- , -I
[42] » :; = cot. ^a,
^ ■' 1 — COS. a
1 — COS. a ,
[42'1 » — : = tang, i a,
*■ ■■ sm. a
[43] " sin.5; = z — j-^+j^^7475-lX3,X-iX7 + ^''-' [GO^^^J'
1. 2. S"" 1.2.3.4.5 1.2.3.4.5.6.7
r2 24 26
COS. - =1- f-, + r2-X4 - 1X3:4-^76 + ^^^ t607e],
[45] » tang, z = z + ^ z^ + ^% z^ + 3T5 ^' + ^c.,
,-46-, » z = arc. (sin. y) =y + iy^ + ^% f + &^c.,
z=arc.(cos.x) = (l— ^)+i-(l— ^') + To-(i— ^) + ^c-'
z = arc. (tang, i) = f — J ^^ + i <' + &c.,
[49] " dz = d. (arc. sin. 3/) = ^^^ ,
dz = d. (arc. cos. a?) = ^jzzTx '
[50] "
rgji » (^2; = J. (arc.tang.O=jq:^
df
INTRODUCTION
d . sin. z=:dz . cos. Zj
d . COS. z= — dz . sin. «,
//r
c?.tang.2: = ^^- = dz. (1 +tang.22r),
e.= 1 _|_^_{_ ^^+_|_ + _|^+&c., Hyp.log. c= 1, [607c],
22 23 2*
a^ = 1 + a: . log. a + ^g . (log. af + j-f-^ . (log. af, [607&],
<; . c* = c? 2: . c%
hyp.log. (1 -\-x) = x — ^x^-\-ix^ — ^x^-j-hc,
tZ.hyp. log.y=^,
sin. [z -\- a) = sin. z -\- a , cos. z, neglecting a^, a', &c.,
COS. (z+a) = COS. 2: — a.sin. z, ibid.
In the notes may be found several formulas, definitions, &«;., some of which will be often
referred to, namely.
Change of rectangular co-ordinates from one system to another, [171a — 172t].
Composition and resolution of rotatory motions, [230r — 231c].
Conic Sections, [378a— 379e, 603a, 726—750].
Curve of double curvature, [256].
Differentials, partial, complete, exact, [13a — 14aJ.
Elliptical Functions of Le Gendre, [82a],
F'{c,<?)=f-^y £-.(c,(p)=/d9.A.(c,(p),
/• do?
n . in, c,cp] = i .
The last of these functions is inaccurately printed in [82a], the factor A . (c, 9), ought
to be in the denominator.
XI
[52]Int
[53]
>j
[54]
)»
[55]
jj
[56]
»
[56^
M
[57]
»
[58]
»
[59]
n
[60]
n
[61]
»
xu INTRODUCTION.
Equations of a right line, [196', 196"].
" cycloid, [102fl].
" plane, [19c, rf].
" spherical surface, [19e].
Integral formulas, fdd.sin.^d, [84a — e}.
Linear functions, [1 25a].
Mechanical powers, [114a, Stc]
Plane triangle CAM, in the second figure page 292,
[G9]Int. AM''=:CM^ — 2 CM, AC. COS. AC M+ A C^, [471].
Principle of the least squares, [849^].
Radius of curvature, [53a, 6, c].
Spherics. The fundamental theorem,
[63] " cos.^G = cos..^^.cos.J5 G-}-sin.^B.sin.jBG.cos.^5G,
[172i], corresponding to a spherical triangle ABG, is used in the appendix, page 729, kc,
to demonstrate the most useful propositions in spherical trigonometry.
Theorems of Maclaurin, [607a].
" Taylor, [617].
" La Grange, [629c].
Variations. Principles of this method, [36a — Jc],
CONTENTS OF THE FIRST VOLUME.
Preface, plan of the work. xxiii
FIRST BOOK.
ON THE GENERAL LAWS OF EQUILIBRIUM AND MOTION.
CHAPTER I. ON THE EaUILIBRIUM AND COMPOSITION OF FORCES WHICH ACT ON A MATERIAL
POINT 1
On motion and force, also on the composition and resolution of forces, [1 — 17] . . . . § 1, 2
Equation of the equilibrium of a point acted upon by any number of forces, in any directions
whatever, [18]. Method of determining, when the point is not free, the pressure it exerts
upon the surface, or upon the curve to which it is subjected, [19 — ^26]. Theory of the
momentum of any force about an axis, [29] § 3
CHAPTER II. ON THE MOTION OF A MATERIAL POINT 23
On the laws of inertia, uniform motion, and velocity, [29"] § 4
Investigation of the relation which exists between force and velocity. In the law of nature they
are proportional to each other. Results of this law, [30 — 34'"] § 5j 6
Equations of the motions of a point acted upon by any forces, [37] § 7
General expression of the square of the velocity, [40]. The point describes the curve in which
the integral of the product of the velocity, by the element of the curve, is a minimum, [49^]. § 8
Method of computing the pressure which a point, moving upon a surface, or upon a curve, exerts
on it, [54]. On the centrifugal force, [54'] § 9
Application of the preceding principles to the motion of a material point, acted upon freely by
gravity, in a resisting medium. Investigation of the law of resistance necessary to make the
moving body describe a given curve. Particular examination of the case in which the resistance
is nothing, [54^—67" ] §10
D
XIV CONTENTS OF THE FIRST VOLUME.
Application of the same principles to the motion of a heavy body upon a spherical surface.
Determination of the time of the oscillations of the moving body. Very small oscillations are
isochronal, [67'"— 86] §11
Investigation of the curve which is rigorously isochronal, in a resisting medium ; and particularly
if the resistance be proportional to the two first powers of the velocity, [86" — 106]. . . § 12
CHAPTER in. ON THE EaUILIBRIUM OP A SYSTEM OF BODIES 71
Conditions of the equilibrium of two systems of points, which impinge against each other, with
directly opposite velocities. Definition of the terms, quantity of motion of a body and similar
material points, [106' — 106'"] § 13
On the reciprocal action of material points. Reaction is always equal and contrary to action.
Equation of the equilibrium of a system of bodies, from which we may deduce the principle of
virtual velocities, [114']. Method of finding the pressures, exerted by bodies, upon the surfaces
and curves upon which they are forced to move, [117] §14
Application of these principles, to the case where all the points of the system are rigidly united
together, [119] ; conditions of the equilibrium of such a system. On the centre of gravity : method
of finding its position ; first, with respect to three fixed rectangular planes, [127] ; second, with
respect to three points given in position, [129] § 15
Conditions of equilibrium of a solid body of any figure, [130] §16
CHAPTER IV. ON THE EQUILIBRIUM OF FLUIDS 90
General equations of this equilibrium, [133]. Application to the equilibrium of a homogeneous
fluid mass, whose external surface is free, and which covers a fixed solid nucleus, of any
figure, [138] §17
CHAPTER V. GENERAL PRINCIPLES OF THE MOTION OF A SYSTEM OF BODIES QQ
General equation of this motion, [142] § 18
Development of the principles comprised in this equation. On the principle of the living force,
[144]. It takes place only when the motions of the bodies change by insensible degrees, [145].
Method of estimating the alteration which takes place in the living force, by any sudden
change in the motions of the system, [149] § 19
On the principle of the preservation of the motion of the centre of gravity, [155']. It takes place
even in those cases, in which the bodies of the system exert on each other, a finite action, in an
instant, [159"]. §20
On the principle of the preservation of areas, [167]. It takes place also, like the preceding
principle, in the case of a sudden change in the motion of the system, [167'"]. Determination
of the system of co-ordinates, in which the sum of the areas described by the projections of the
radii vedores, upon two of the rectangular planes, formed by the axes of the co-ordinates, is
nothing. This sum is a maximum upon the third rectangular plane ; it is nothing upon every
other plane, perpendicular to this third plane, [181"] § 21
The principles of the preservation of the living forces and of the areas take place also, when the
origin of the co-ordinates is supposed to have a rectilineal and uniform motion, [182]. In this
case, the plane passing constantly through this origin, and upon which the sum of the areas
CONTENTS OF THE FIRST VOLUME. XV
described by the projection of the radii is a maximum, continues always parallel to itself,
[187, &.C.] The principles of the living forces and of the areas, may be reduced to certain
relations between the co-ordinates of the mutual distances of the bodies of the system, [189, &c.]
Planes passing through each of the bodies of the system, parallel to the invariable plane drawn
through the centre of gravity, possess similar properties, [189"] §22
Principle of the least action, [196']. Combined with that of the living forces, it gives the general
equation of motion § 23
CHAPTER VI. ON THE LAWS OF MOTION OF A SYSTEM OF BODIES, IN ALL THE RELATIONS
MATHEMATICALLY POSSIBLE BETWEEN THE FORCE AND VELOCITY 137
New principles which, in this general case, correspond to those of the preservation of the living
forces, of the areas, of the motion of the centre of gravity, and of the least action. In a system
which is not acted upon by any external force, we have, Jirst, the sum of the finite forces
of the system, resolved parallel to any axis, is constant ; second, the sum of the finite forces,
to turn the system about an axis, is constant ; third, the sum of the integrals of the finite
forces of the system, multiplied respectively by the elements of their directions, is a minimum :
these three sums are nothing in the state of equilibrium, [196'", &c.] § 24
CHAPTER VII. ON THE MOTIONS OF A SOLID BODY OF ANY FIGURE WHATEVER 144
Equations which determine the progressive and rotatory motion of the body, [214 — ^234]. § 25, 26
On the principal axes, [235]. In general a body has but one system of principal axes, [245"].
On the momentum of inertia, [245'"]. The greatest and least of these momenta appertain to the
principal axes, [246"], and the least of all the momenta of inertia takes place with respect to
one of the three principal axes which pass through the centre of gravity, [248']. Case in which
the solid has an infinite number of principal axes, [250, &c.] § 27
Investigation of the momentary axis of rotation of the body, [254"] . The quantities which determine
its position relative to the principal axes, give also the velocity of rotation, [260'*]. . . § 28
Equations which determine this position, and that of the principal axes, in functions of the time,
[263, &.C,] Application to the case in which the rotatory motion arises from a force which does
not pass through the centre of gravity. Formula to determine the distance from this centre to
the direction of the primitive force [274]. Examples deduced from the planetary motions,
particularly that of the earth [275*] ^29
On the oscillations of a body which turns nearly about one of its principal axes, [278]. The motion
is stable about the principal axes, whose momenta of inertia are the greatest and the least ; but
it is unstable about the third principal axis, [281'«] § 30
On the motion of a solid body, about a fixed axis, [287]. Determination of a simple pendulum,
which oscillates in the same time as this body, [293'] § 31
CHAPTER VIII. ON THE MOTION OF FLUIDS. .... ,n^
Equations of the motions of fluids, [296] ; condition relative to their continuity [303'"]. . § 32
Transformation of these equations ; they are integrable when the density is any function of the
pressure, and at the same time, the sum of the velocities parallel to three rectangular axes, each
being multiplied by the element of its direction, is an exact variation, [304, &c.] Proof that this
condition will be fulfilled at every instant of time, if it is so in any one instant, [316]. . § 33
XVI CONTENTS OF THE FIRST VOLUME.
Application of the preceding principles to the motion of a homogeneous fluid mass, which has a
uniform rotatory motion, about one of the axes of the co-ordinates [321] § 34
Determination of the very small oscillations of a homogeneous fluid mass, covering a spheroid,
which has a rotatory motion, [324] § 35
Application to the motion of the sea, supposing it to be disturbed from the state of equilibrium,
by the action of very small forces, [337] § 36
On the atmosphere of the earth, considered at first in a state of equilibrium, [348]. Its oscillations
in a state of motion, noticing only the regular causes which agitate it. The variations which
these motions produce in the height of the barometer, [363' v] § 37
SECOND BOOK.
ON THE LAW OP UNIVERSAL GRAVITATION, AND THE MOTIONS OF THE CENTRES
OF GRAVITY OF THE HEAVENLY BODIES.
CHAPTER I. THE LAW OF UNIVERSAL GRAVITY DEDUCED FROM OBSERVATION 339
The areas described by the radii vectores of the planets in their motions about the sun, being
proportional to the time, the force which acts upon the planets, is directed towards the centre
of the sun, [367] ; and reciprocally, if the force be directed towards the sun, the areas described
about it, by the planets, will be proportional to the time ^1
The orbits of the planets and comets being conic sections, the force which acts on them, is in the
inverse ratio of the square of the distances of the centres of these planets from that of the sun,
[SSC]. Reciprocally, if the force follows this ratio, the described curve will be a conic
section, [380'^] § 2
The squares of the times of the revolutions of the planets, being proportional to the cubes of the
great axes of their orbits ; or, in other words, the areas described in the same time, in different
orbits, being proportional to the square roots of their parameters, the force which acts upon
the planets and comets, must be the same for all the bodies placed at equal distances from the
sun, [385] § 3
The motions of the satellites about their planets exhibiting nearly the same phenomena, as the
motions of the planets about the sun ; the satellites must be attracted towards their planets,
and towards the sun, by forces reciprocally proportional to the square of the distances, [388] § 4
Investigation of the lunar parallax, from experiments on gravity, supposing gravitation to be in the
inverse ratio of the square of the distances, [391]. The result obtained in this manner, being
found perfectly conformable to observations, the attractive force of the earth must be of the
same nature as that of the heavenly bodies § 5
General reflections on what precedes : they lead us to this general principle, namely, that all the
particles of matter attract each other in the direct ratio of the masses^ and in the inverse ratio of the
squares of the distances, [391'*', &c] § 6
CONTENTS OF Tlffi FIRST VOLUME. XVll
CHAPTER II. ON THE DIFEKENTIAL EaUATIONS OP THE MOTION OF A SYSTEM OP BODIES,
SUBJECTED TO THEIR MUTUAL ATTRACTIONS 261
Differential equations of this motion, [398 — 400] § 7
Development of the integrals of these equations which have already been obtained, and which
result from the principles of the preservation of the motions of the centres of gravity, of the
areas, and of the living forces, [404 — 410"] § 8
Differential equations of the motions of a system of bodies, subjected to their mutual attractions,
about one of them considered as the centre of their motions, [416 — 418]. Development of the
rigorous integrals of these equations, which have been obtained, [421 — 442] § 9
The motion of the centre of gravity of the system of a planet and its satellites about the sun, is
nearly the same as if all the bodies of this system were united at that point ; and the system
acts upon the other bodies nearly as it would in the same hypothesis, [451'''] § 10
Investigation of the attraction of spheroids : this attraction is given by the partial differentials of
the function which expresses the sum of the particles of the spheroid, divided by their distances
from the attracted point, [455'"]. Fundamental equation of partial differentials which this
function satisfies, [459]. Several transformations of this equation, [465, 466] §11
Application to the case where the attracting body is a spherical stratum, [469] : it follows that a
point placed within the stratum is equally attracted in every direction, [469'"] ; and that a point
placed without the stratum, is attracted by it, as if the whole mass were collected at its centre,
[470^. This result also takes place in globes formed of concentrical strata, of a variable
density from the centre to the circumference. Investigation of the laws of attraction, in which
these properties exist [484]. In the infinite number of laws which render the attraction
very small at great distances, that of nature is the only one in which spheres act upon an
external point as if their masses were united at their centres, [485']. This law is also the only
one in which the action of a spherical stratum, upon a point placed within it, is nothing, [485"]. § 12
Application of the formulas of § 11 to the case where the attracting body is a cylinder, whose
base is an oval curve, and whose length is infinite. When this curve is a circle, the action of a
cylinder upon an external point, is inversely proportional to the distance of this point from the
axis of the cylinder, [498']. A point placed within a circular cylindrical stratum, of uniform
thickness, is equally attracted in every direction, [498"] § 13
Equation of condition relative to the motion of a body, [502] § 14
Several transformations of the differential equations of tlie motion of a system of bodies, submitted
to their mutual attractions, [517 — 530] § 15
CHAPTER III- FIRST APPROXIMATION OF THE MOTIONS OF THE HEAVENLY BODIES ; OR THEORY
OF THE ELLIPTICAL MOTION 321
Integration of the differential equations which determine the relative motion of two bodies,
attracting each other in the direct ratio of the masses, and the inverse ratio of the square of the
distances. The curve described in this motion is a conic section, [534]. Expression of the
time, in a converging series of sines of the true motion, [543]. If we neglect the masses of
the planets, in comparison with that of the sun, the squares of the times of revolutions will be
£
XVUl CONTENTS OF THE FIRST VOLUME.
as the cubes of the transverse axes of the orbits. This law extends to the motion of the
satellites about their primary planets, [544"'^] §16
Second method of integration of the differential equations of the preceding article, [545—558]. § 17
Third method of integration of the same equations ; this method has the advantage of giving the
arbitrary constant quantities in functions of the co-ordinates and of their first differentials,
[559-597] §18,19
Finite equations of the elliptical motion ; expressions of the mean anomaly, of the radius vector,
and of the true anomaly, in functions of the excentric anomaly, [606] § 20
General method of reducing functions into series ; theorems which result from it, [607 — 651]. §21
Application of these theorems to the elliptical motion. Expressions of the excentric anomaly,
[657], the true anomaly, [668], and the radius vector of the planets, [659], in converging series
of sines and cosines of the mean anomaly. Expressions in converging series, of the longitude,
[675], of the latitude, [679], and of the projection of the radius vector, [680], upon a fixed plane
but little inclined to that of the orbit § 22
Converging expressions of the radius vector, [683], and of the time, [690], in functions of the true
anomaly, in a very excentric orbit. If the orbit be parabolic, the equation between the time and
the true anomaly will be an equation of the third degree, [693], which may be resolved by
means of the table of the motions of comets. Correction to be made in the true anomaly
calculated for the parabola, to obtain the true anomaly corresponding to the same time, in a very
excentric ellipsis, [695] § 23
Theory of the hyperbolic motion, [702] §24
Determination of the ratio of the masses of the planets accompanied by satellites, to that of the
sun, [709] § 25
CHAPTIIR IV. DETERMINATION OP THE ELEMENTS OP THE ELLIPTICAL MOTION 393
Formulas which give these elements, when the circumstances of the primitive motion are known,
[712 — 716']. Expression of the velocity, independent of the excentricity of the orbit, [720].
In the parabola the velocity is inversely proportional to the square root of the radius vector,
[720"] §26
Investigation of the relation which exists between the transverse axis of the orbit, the chord of
the described arch, the time employed in describing it, and the sum of the extreme radii
vectores, [748, 750] § 27
The most convenient method of obtaining by observation the elements of the orbit of a comet,
[753", &c.] §28
Formulas for computing, from any number of observations, taken near to each other, the
geocentric longitude and latitude of a comet, at any intermediate time, with the first and
second differentials of the longitudes and latitudes, [754, &c.] § 29
General method of deducing, from the differential equations of the motion of a system of bodies, the
elements of their orbits, supposing the apparent longitudes and latitudes of these bodies, and the
first and second differentials of these quantities, to be known, at a given instant, [760, &c.] § 30
CONTENTS OF THE FIRST VOLUME. XIX
Application of this method to the motion of comets, supposing them to be acted upon only by the
attraction of the sun: it gives, by an equation of the seventh degree, [784], the distance of the
comet from the earth. The mere inspection of three observations, made within very short
intervals of each other, will suffice to discover whether the comet is nearer to the sun, or
farther from it, than the earth is, [780'*] § 31
Method of finding, as accurately as is necessary, by means of three observations, the geocentric
longitudes and latitudes of a comet, and their first and second differentials, divided by the
corresponding powers of the element of the time, [787'] § 32
Determination of the elements of the orbit of a comet, when we know, at a given instant, its
distance from the earth, and the first differential of this distance divided by the element of the
time. Simple method of allowing for the excentricity of the orbit of the earth, [788 — 800]. § 33
When the orbit is a parabola, the greater axis becomes infinite, and this condition furnishes another
equation, of the sixth degree, [805], to determine the distance of the comet from the earth. § 34
Hence we may obtain various methods of computing a parabolic orbit. Investigation of the
method from which we may expect the most accurate result, and the greatest simplicity in the
calculation, [806—811] § 35, 36
This method is divided into two parts : in the first, is given a method of approximation, to find the
perihelion distance of the comet, and the time of passing the perihelion, [811" — 820] ; in the
second, is given a method of correcting these two elements, by three distant observations, and
then deducing from these the other elements, [820'" — 832] § 37
Accurate determination of the orbit, when the comet has been observed in both of its nodes,
[833—841] §38
Method of finding the ellipticity of the orbit, when the ellipsis is very excentric, [842 — 849]. § 39
CHAPTER V. GENERAL METHODS OF FINDING THE MOTIONS OF THE HEAVENLY BODIES, BY
SUCCESSIVE APPROXIMATIONS 4'J'5
Investigation of the alterations which must be made in the integrals of differential equations, to
obtain the integrals of the same equations increased by certain terms, [850 — 859]. . . § 40
Hence we derive a simple method of obtaining the rigorous integrals of linear differential equations,
when we know how to integrate the same equations deprived of their last terms, [861 — 871"]. § 41
We also obtain an easy method of computing the integrals of differential equations, by successive
approximations, [872 — 875] § 42
Method of eliminating the arcs of a circle, which occur in these approximate integrals, when they
do not really exist in the rigorous integrals, [876 — 892] § 43
Method of approximation, founded on the variations of the arbitrary constant quantities,
[897-912] §45
CHAPTER VI. SECOND APPROXIMATION OP THE CELESTIAL MOTIONS ; OR THEORY OF THEIR
PERTURBATIONS kqa
Formulas of the motions in longitude and latitude, and of the radius vector in the disturbed
orbit. Very simple form under which they appear, when only the first power of the disturbing
forces is noticed, [913—932] § 46
XX CONTENTS OF THE FIRST VOLUME.
Method of finding the perturbations in a series arranged according to the powers and products of
the excentricities and of the inclinations of the orbits, [933— 948], . ; §47
Development of the function of the mutual distances of the bodies of the system, on which their
perturbations depend, in a series. Use of the calculus of finite differences in this development.
Reflections upon this series [949 — 963] § 48
Formulas for computing its different terms, [964 — 1008] § 49
General expressions of the perturbations of the motion in longitude and in latitude, and of the
radius vector, continuing tlie approximation to quantities of the order of the excentricities and
inclinations, [1009—1034] § 50, 51
Recapitulation of these different results, remarks on farther approximations, [1035 — 1036"]. § 52
CHAPTER VII. ON THE SECULAR INEaUALITIES OF THE MOTIONS OF THE HEAVENLV BODIES. 569
These inequalities arise from the terms which, in the expressions of the perturbations, contain
the time without the periodical signs. Differential equations of the elements of the elliptical
motion, which make these terms disappear, [1037 — 1051] § 53
In taking notice only of the first power of the disturbing force, the mean motions of the planets
will be uniform, and the transverse axes of their orbits constant, [1051' — 1070^^^]. ... § 54
Development of the differential equations relative to the excentricities and to the position of the
perihelia, in any system of orbits in which the excentricities and mutual inclinations are
small, [1071—1095] §55
Integration of these equations. Determination of the arbitrary constant quantities of the integral,
by means of observations, [1096 — 1111] § 56
The system of the orbits of the planets and satellites, is stable, as it respects the excentricities ;
that is, these excentricities remain always very small, and the system merely oscillates about
its mean state of ellipticity, from which it varies but little, [1111'" — 1118] §57
Differential expressions of the secular variations of the excentricity and of the position of the
perihelion, [1118^— 1126^] §58
Integration of the differential equations relative to the nodes and inclinations of the orbits. In
the motions of a system of orbits, which are very little inclined to each other, the mutual
inclinations remain always very small, [1127 — 1139] § 59
Differential expressions of the secular variations of the nodes and of the inclinations of the
orbits ; first, with respect to a fixed plane ; second, with respect to the moveable orbit of one
of the bodies of the system, [1140—1146] § 60
General relations between the elliptical elements of a system of orbits, whatever be their
excentricities and their mutual inclinations, [1147 — 1161] § 61
Investigation of the invariable plane, or that upon wliich the sum of the masses of the bodies of
the system, multiplied respectively by the projections of the areas described by their radii
vectores, in a given time, is a maximum. Determination of the motion of two orbits, inclined
to each other by any angle, [1162 — 1167] § 62
CONTENTS OF THE FIRST VOLtJME. XXI
CHAPTER VIII. SECOND METHOD OF APPKOXIMATION TO THE MOTIONS OF THE HEAVENLY BODIES.
This method is founded on the variations which the elements of the motion, supposed to be elliptical,
suffer by means of the periodical and secular inequalities. General method of finding these
variations. The finite equations of the elliptical motion, and their first differentials, are the
same in the variable as in the invariable ellipsis, [1167' — 1169^"] § 63
Expressions of the elements of the elliptical motion in the disturbed orbit, whatever be its excentricity
and its inclination to the plane of the orbits of the disturbing masses, [1170—1194]. . . § 64
Development of these expressions, when the excentricities and the inclinations of the orbits are
small. Considering, in the first place, the mean motions and transverse axes ; it is proved,
that if we neglect the squares and the products of the disturbing forces, these two elements are
subjected only to periodical inequalities, depending on the configuration of the bodies of the
system. If the mean motions of the two planets are very nearly commensurable with each
other, there may result, in the mean longitude, two very sensible inequalities, affected with
contrary signs, and reciprocally proportional to the products of the masses of the bodies, by the
square roots of the transverse axes of their orbits. The acceleration of the motion of Jupiter,
and the retardation of the mean motion of Saturn, are produced by similar inequalities.
Expressions of these inequalities, and of those which the same ratio of the mean motions may
render sensible, in the terms depending on the second power of the disturbing masses,
[1195—1214] §65
Examination of the case where the most sensible inequalities of the mean motion, occur only
among terms of the order of the square of the disturbing masses. This very remarkable
circumstance takes place in the system of the satellites of Jupiter, whence has been deduced
the two following theorems,
T%e mean motion of the first satellite, minus three times that of the second, plus tunce that of the
third, is accurately and invariably equal to nothing, [1239'"].
The mean longitude of the first satellite, minu^ three times that of the second, plus ttoice thai of the
ihird, is invariably equal to two right angles, [1239^].
These two theorems take place, notwithstanding the alterations which the mean motions of the
satellites may suffer, either from a cause similar to that which alters the mean motion of the
moon, or from the resistance of a very rare medium. These theorems give rise to an arbitrary
inequality, which differs for each of the three satellites, only by its coefficient. This
inequality is insensible by observation, [1240 — 1242^] • ... § 66
Diflerential equations which determine the variations of the excentricities and of the perihelia,
[1243—1266] §67
Development of these equations. The values of these elements are composed of two parts, the
one depending on the mutual configuration of the bodies of the system, which comprises the
periodical inequalities ; the other independent of that configuration, which comprises the secular
inequalities. This second part is given by the same differential equations as those which we
have before considered, [1266', 1279] § 68
A very simple method of obtaining the variations of the excentricities and of the perihelia of the
orbits, arising from the ratio of the mean motions being nearly commensurable ; these variations
F
634
XXll CONTENTS OF THE FIRST VOLUME.
are connected with the corresponding variationa of the mean motion. They may produce, in
the secular expressions of the excentricities, and of the longitudes of the perihelia, sensible
terms, depending on the squares and products of the disturbing forces. Determination of these
quantities, [1280—1309] § 69
On the variations of the nodes and of the inclinations of the orbits. Equations which determine
their periodical and secular values, [1310 — 1327] § 70
Easy method of obtaining the inequalities which arise in these elements from the ratio of the mean
motions being nearly commensurable ; they depend on the similar inequalities of the mean
motion, [1328—1342] § 71
Investigation of the variation which the longitude of the epoch suffers. It is upon this variation
that the secular variation of the moon depends, [1343 — 1345] § 72
Reflections upon the advantages, which the preceding method, founded upon the variation of the
parameters of the orbits, presents in several circumstances ; method of deducing the variations
of the longitude, of the latitude^ and of the radius vector, [1345'^», &c.] § 73
PREFACE BY THE AUTHOR.
Towards the end of the seventeenth century, Newton published his
discovery of universal gravitation. Mathematicians have, since that epoch,
succeeded in reducing to this great law of nature all the known phenomena
of the system of the world, and have thus given to the theories of the
heavenly bodies, and to astronomical tables, an unexpected degree of
precision. My object is to present a connected view of these theories,
which are now scattered in a great number of works. The whole of the
results of gravitafton, upon the equilibrium and motions of the fluid and solid
bodies, which compose the solar system, and the similar systems, existing
in the immensity of space, constitute the object of Celestial Mechanics,
or the application of the principles of mechanics to the motions and figures
of the heavenly bodies. Astronomy, considered in the most general manner,
is a great problem of mechanics, in which the elements of the motions are
the arbitrary constant quantities. The solution of this problem depends,
at the same time, upon the accuracy of the observations, and upon the
perfection of the analysis. It is very important to reject every empirical
process, and to complete the analysis, so that it shall not be necessary to
derive from observations any but indispensable data. The intention of this
work is to obtain, as much as may be in my power, this interesting result.
I hope, in consideration of the difficulty and importance of the subject, that
mathematicians and astronomers will receive it with indulgence, and that
they will find the results sufficiently simple to be used in their researches.
XXIV PREFACE BY THE AUTHOR.
It will be divided into two parts. In the first part, I shall give the methods
and formulas, to determine the motions of the centres of gravity of the
heavenly bodies, the figures of those bodies, the oscillations of the fluids
which cover them, and the motions about their centres of gravity. In the
second part, I shall apply the formulas found in the first, to the planets,
satellites, and comets ; and I shall conclude the work, with an examination
of several questions relative to the system of the world, and with an
historical account of the labors of mathematicians upon this subject. I shall
adopt the decimal division of the right angle, and of the day, and shall refer
the linear measures to the length of the metre, determined by the arc of
the terrestrial meridian comprised between Dunkirk and Barcelona.
FIRST PART.
GENERAL THEORY OF THE MOTIONS AND FIGURES OF THE HEAVENLY BODIES.
FIRST BOOK.
ON THE GENERAL LAWS OF EQUILIBRIUM AND MOTION.
It is my intention to give in this book the general principles of the equi-
librium and motion of bodies, and to solve those problems of mechanics which
are indispensable in the theory of the system of the world.
CHAPTER I.
ON THE EaUILIBRITJM AND COMPOSITION OF FORCES WHICH ACT ON A MATERIAL POINT.
1 . A BODY appears to us to be in motion, when it changes its relative situa-
tion with respect to a system of bodies supposed to be at rest ; but as all
bodies, even those which appear in the most perfect repose, may be in motion ;
a space is conceived of, without bounds, immoveable, and penetrable by the
particles of matter; and we refer in our minds the position of bodies to the
parts of this real, or ideal space, supposing the bodies to be in motion, when
they correspond, in successive moments, to different parts of this space.
The nature of that singular modification, by means of which a body is
transported from one place to another, is now, and always will be, unknown ;
it is denoted by the name of Force, We can only ascertain its effects, and Forw.
the laws of its action. The effect of a force acting upon a material point, or
particle, is to put it in motion, if no obstacle is opposed; the direction of the
force is the right line which it tends to make the point describe. It is evident,
that if two forces act in the same direction, the resultant is the sum of the
two forces ; but if they act in contrary directions, the point is affected by the
2 COMPOSITION OF FORCES. [Mec. Cel.
difference of the forces. If their directions form an angle with each other, the
force which results will have an intermediate direction between the two
proposed forces. We shall now investigate the quantity and direction of this
resulting force.
For this purpose, let us consider two forces, x and y, acting at the same
moment upon a material point ilf, in directions forming a right angle with
each other. Let z be their resultant, and (3 the angle which it makes with the
direction of the force x. The two forces x and y being given, the angle ^ and
the quantity z must have determinate values, so that there will exist, between
the three quantities x, z and <5, a relation which is to be investigated.
Suppose in the first place that the two forces x and y are infinitely small,
and equal to the differentials dx, dy. Then suppose that x becomes succes-
sively ^a:, 2dx, 3dx, 8tc., and y becomes dy, 2dy, 3dy, &c., it is evident that
the angle 6 will remain constant, and the resultant z will become successively
dz, 2dz, 3dz, &c., and in the successive increments of the three forces x, y
and z, the ratio of re to 2: will be constant, and may be expressed by a func-
tion of 6, which we shall denote by 9(0) ;* we shall therefore have x = z.(p(d),
in which equation we may change x into y, provided we also change the angle
6 into 6, 1! being the semi-circumference of a circle whose radius is unity.
Now we may consider the force x as the resultant of two forces a/ and a;",
of which the first x! is directed along the resultant 2:, and the second a/' is
perpendicular to it.f The force a:, which results from these two new forces,
* A quantity z is said to be a function of another quantity a?, when it depends on it in any
manner. Thus, if z, y be variable, a, &, c, Sic. constant, and we have eithel* of the following
expressions, 2' = aa; + &» 2; = aar^ + 6a;4-c; 2: = a% 2; = sin. a a?, &;c. z will be a function
[lo] of X ; and if the precise form of the function is known, as in these examples, it is called an
explicit function. If the form is not Imown, but must be found by some algebraical process,
it is called an implicit function.
f (2) For illustration, suppose the forces x and y to act
at the point A, in the directions AX, AY, respectively,
and that the resultant z is in the direction A Z, forming with
AX, AT, the angles ZAX=6, ZAY='^—6. Then
Y
V ■•5'r \
^'-^ \
as above, we have « = «.(? {f), y = z.cp (- ^ j. Draw
EAF perpendicular to A Z, and suppose the force x in
the direction ^ Xto be resolved into two forces, xf, a/', in the
E
[1]
p]
I. i. <§!.] COMPOSITION OF FORCES.
forms the angle 6 with the force a/, and the angle 6 with the force ar" ; we
shall therefore have
and we may substitute these two forces instead of the force x. We may like-
wise substitute for the force y two new forces, i/ and y", of which the- first
is equal to - in the direction z, and the second equal to — perpendicular to
z ; we shall thus have, instead of the two forces x and y, the four following :
x^ y^ xy xy
~z' ~z' T' T'
the two last, acting in contrary directions, destroy each other ;* the two first,
acting in the same direction, are to be added together, and produce the
resultant z ; we shall therefore havef
x'^-f^z^', [3]
whence it follows, that the resultant of the two forces x and y is represented
in magnitude, by the diagonal of the rectangle whose sides represent those
forces.
directions AZ,AE, respectively, so that the angle ZAX=6, anil XAE=~ — 6. Then, in
the same manner in wliich the above values o(x, y, are obtained from z, we may get a/= x.(p{6);
a/' = a? . (J) f J — 6\ If in these we substitute the values <p {&) = -; 9 ( |^ — ^ J =^, deduced
from the above equations, we obtain a/ = - ; a/' = ^. In like manner, if the force v, in
Z X •'
the direction AT^ be resolved into the two forces y', y", in the directions AZ, AF,
making the angle YAZ=. ^—6, YAF= 6, we shall have y' = y.(p (-— ^) ; y" = y • <P (^) ;
which, by substituting the above values of (p^|— A (p(^), become 2/== -, y"—~, asabove.
* (3) For, by the preceding note, the force a/' = ^, is in the direction AE, and the
force y" = __, is in the opposite direction .41^, and as they are equal they must destroy
each other.
t (4) The sum of the two forces a/ = - , ^ = ^ , in the direction A Z, being put equal
to the resultant z, gives - +"- =5r, which multiplied by z becomes a^-{-f=z\
4
[4]
[5]
COMPOSITION OF FORCES.
[Mec. Cel.
Let us now determine the angle ^. If we increase the force x by the
differential d x, without varying the force y, that angle will be diminished by
the infinitely small quantity d6f now we may conceive the force dx to be
resolved into two other forces, the one dx' in the direction z, and the other dx"
perpendicular to z ; the point M will then be acted upon by the two forces
z-\-dx' and dx", perpendicular to each other, and the resultant of these two
forces, which we shall call z\ will make with dxf' the angle d 6 ;f we shall
thus have, by what precedes,
dx^'^2f.(pf-—'dn;
consequently the function (p( - — dn is infinitely small, and of the form — kd6,
k being a constant quantity, independent of the angle 5 ;t we shall therefore
have
* (5) The resultant of the forces x, y, is, by hypothesis, in
the direction A Z, and, by increasing the force x by dx, the
forces become equal to z in the direction A Z, and dx in the
direction A X, and the resulting force z', must evidently fall be-
tween A Z, AX, on a. line as A G, forming with AZ an infinitely
small angle ZAG, represented by dL Then the force dx, in
the direction A X, may be resolved into two forces, the one doif
in the direction AZ, the other dec" in the direction AE, and as
this last force is inclined to AX by the angle XAE = '^ — 6, we shall have as above
dx"==dx.(f{-^ — M; or by substituting the preceding value of (p( 2 — V"^^^' ''^~~z~'
f (6) This angle is equal to GAE.== '^—dd ; and if the force z' in the direction A G
is resolved into two forces in tiie directions A Z, AE, the last will (by tiie nature of the
function cp) be represented hy z' .cp( -^ — dd\.
X (7) Because (p r- — d&\ contains only die quantities ^, <?^, but does not explicitiy con-
tain ^. Moreover, the function (p ( 1^ — di\ being developed m the usual manner, according
[6]
[9]
f. i. §1.] COMPOSITION OF FORCES.
^ differs from z but by an infinitely small quantity ; again, d x!' forming with
dx, the angle ^, we shall have*
thereforet
^ —ydx - . -^^1"^
d6=-^-^-. ., r.„„ r.., ,-. [7]
If we vary the force yhy dy, supposing x constant, we shall have the corre-
sponding variation of the angle ^, by changing in the preceding equation re
into y, y into a:, ^ into ^, which givesj p") ^' '' '"? ^^
J. xdy
supposing, therefore, x and y to vary at the same time, the whole variation of
cc d ly v doc
the angle ^ will be — ^ — | — ; and we shall have
sr
Substituting for z^ its value 3^ -{-if, and integrating, we shall have§
to the powers of d6, by Taylor's Theorem [617], or by any other way, will be of the form
A — k.dd-{-J(/ .d(P — etc. — A;, /fc', etc. being constant quantities, dependant on the first, second,
etc. differentials of (p {^nr). By this means, do/' [4] will become d3cf' = z' .{A — kdd-{-^d(P —
etc.]. Now it is evident, tliat when dx = 0, the quantities da/' and dd must also vanish;
and the preceding expression will, in this case, become 0 = z .A, or ^ = 0. Substituting this
value of A, we get generally djc"= 2/^ — k d 6 -\- Jcf d (f^ — etc.} ; and by neglecting the second
and higher powers of d &j it becomes as above, dx" = — kdd .z'.
* (8) As in note 5.
f (9) By putting the two values of do/' [5, 6,] equal to each other, and deducing
therefrom the value of d 6.
J (10) As d is changed into - — 6, the differential dd changes into — dd.
§ (I J ) By die substitution of ar^ + ^ for z^, the equation becomes — ^—7^ — = kd6'j
xx-\-yy
"{^
or, as It may be written, -— - =:kdL for the differential of the numerator of the first
2
6 COMPOSITION OF FORCES. [Mec. Cel.
P being an arbitrary constant quantity. This equation, combined with
QC^-\-lf=zz^, gives
[10] x=z.co^.(k^-{-f).
It remains to determine the two constant quantities k and p ; now if we
suppose y to be nothing, we shall evidently have z = x, and 5 = 0; therefore
cos. p = 1 , and x = z . cos. k &. If we suppose a: = 0, we shall have* z = y,
and 6=P=^'7(; cos.k6 being then equal to nothing, k ought to be equal tof
2n-}-l, n being a whole number, and in this case, x will be nothing when-
ever 6 is equal to — — -— - ; but x being nothing, we evidently have 5 = i* ;
therefore 2?i+l=:l, oi n = 0, consequently
[11] x==z,cos.6.
tionoftwo
Forces.
composi- Whence it follows that the diagonal of the parallelogram, constructed upon
the right lines which represent the two forces x and y, represents not only
the quantity, but also the direction of their resultant. Therefore we may,
for any force, substitute two other forces which form the sides of a parallelo-
gram, of which the proposed force is the diagonal ; whence it is easy to infer
that a force may be resolved into three others, forming the sides of a rectan-
gular parallelopiped, of which the proposed force is the diagonal.!
member indicated by the sign d, being taken, and the numerator and denominator multiplied
by x^, it becomes identical with the proposed. The integral of this equation is, (by 51 Int.)
arc. (^tang. l\ — k6-\-p, or ^ = tang. (^ 5 + p). Hence f—x'^.{ tang.^ (A; 5 + p) } ,
and^^ = y^+^^ = a.^{l + tang.MA;^ + p)}=^-^^^^
x = z . cos. {kd-\-p). This calculation might have been much simplified if the Author had
supposed 00 constant, or dx = 0.
* (12) Because the line A Z then falls upon A Y.
f (13) Because the cosine of any uneven multiple of ^* is equal to nothing.
I (13a) In any parallelogram, AB CD, whose diagonal is A C,
vjehRveAB = A C.cos.BA C; andif^ C = z,BA C==6,
this will become AB = z. cos. 6, or hjlU] AB = x. In like
manner we find B C = y; consequently the forces x, y, are equal
to the sides of the parallelogram whose diagonal is z.
Li. ^2.] COMPOSITION OF FORCES. 7
Let «, b, c, be the three rectangular co-ordinates of the extremity of the
right line which represents any force, whose origin is the same as that of
the co-ordinates; this force will be expressed by the function v/«^+&^+c^ ^^^'^
and by resolving it into directions parallel to the axes of a, b, c, the partial
forces will be expressed respectively by these co-ordinates.
Let «', b', c', be the co-ordinates of a second force ; ft + a', 6 + 6', c-f-c', will
be the co-ordinates of the resultant of the two forces, and will represent the
partial forces into which it may be resolved parallel to the three axes ; whence
it is easy to conclude that this resultant is the diagonal of the parallelogram
constructed upon these two forces.
In general a, b, c; a', 6', c' ; ft", 6", c" &c. ; being the co-ordinates of any ^^p^i;^
number of forces ; ft+ft'+ft"+&c. ; 6 + 6'+6"+&c.; c+c' + c"+&c. ; wiluS"'
be the co-ordinates of the resultant, whose square will be the sum of the Qj/n
squares of these last co-ordinates ; we shall thus have the magnitude and
position of this resulting force.
2. From any point of the direction of a force *S, taken as the origin of this
force, suppose a right line s to be drawn to a material point or particle M ;
let X, y, Zj be the three rectangular co-ordinates which determine the position
y'ii
— ^ -^
Again, suppose a rectangular parallelepiped to be formed upon
the base AB C D, having the diagonal A c, and the equal and ^ c/
parallel edges A a, Bb, C c, Dd, perpendicular to the plane
ABCD; forming the rectangular triangles ADd, Adc. Then if
the force « in the direction Ac he represented by A c, it may, by
what has been just said, be resolved into the two forces Ad, dc,
perpendicular to each otlier, and the former force, may be resolved, as
above, into the two forces .4 D, Dd-, consequently the force Ac, represented by the diagonal,
maybe resolved into the three forces AD, Dd, dc, corresponding to the sides of the
rectangular parallelopiped. The rectangular triangle ADd gives A d^ == A D^ -{- D d^,
and the rectangular triangle Adc gives A(? = A^-\-dc^ ; hence by substituting Ad^, we
have A(P = AD^-^Dd^-\-dc^, which, by putting AD = a, Dd=b, dc = c, gives as [Ha]
above Ac^V a^-\-h^-\-(?. K the forces AD,Dd,dc, are supposed to be respectively
equaltoa + a'+a"+&c.; 6 -f- J'-|- 6"-|- &c. ; c+c'+c"-}- &c. ; the corresponding force
A c must be found in the same manner, by taking the square root of the sum of their
squares.
COMPOSITION OF FORCES.
[Mec. Cel.
of the point M, and a, 6, c, the co-ordinates of the origin of the force, we shall
have*
[12] ' s = ^(x — ay + (y — bf+(z-^cy.
Reaoiution Jf wc rcsolvc thc forcc S into three others in directions parallel to the axes
of Forces. f
of X, y, z ; these partial forces will be, by the preceding article,!
[13]
S.
(^— «) . ^^ (y— ^)
s.
ox
s.
s.
[12a]
[13a]
Partial
Diftlren-
tials.
* (136) Let Khe the origin of the co-ordinates,
A the origin of the force S, c the place of the point
M. Draw, as in the last note, the lines AD, Dd, dc,
parallel to the axes x, y, z, represented by ^X, KY,
KZ, respectively, and complete the parallelepiped
ABCD abed. Continue the lines ba, BA, cd,
C D, tiU they meet the plane YK X, in the points
e, E, /, F. Draw the lines fF, eE, to meet the
axis KX perpendicularly in JH, G. Then, by the
above notation, the co-ordinates of the point A are
KG = a, GE = b, EA = c. The co-ordinates
of tlie point c are KH=x, Hf-=y, fc = z, and A c = s. From this construction it
follows that AD=:EF=GH=KH—KG=x — a; Dd=Cc = Ff=Hf—
HF=Hf—GE = y — b; D C or dc=fc—fd=f c — EA = z — c, substituting
these values in Ac = VAD^-\-Dd^-\-d(^, found as in the last note, it becomes
s=:\/{x — af + {y—bf + '{^—cf, as in [12].
f (13c) If the line A c represents the force S, it might be resolved, as in note 13a, into
three forces, AD, D d, dc, parallel to the axes x, y, z, respectively ; consequently these
AD
three forces mil be represented by S . — — ; iS .
Dd
d c
Ac ' Ac
values of Ac, AD, Dd,dc, given in the last note, become S
S.— ; which, by substituting the
-a)
s
jy—fi). g (z— c)
respectively, as in [13]. They may be put under a different form, by means of the partial
differentials or variations of s. The partial differential of a quantity denotes its differential
supposing only part of the quantities of which it is composed to be variable. Thus the
partial differential of s = V {x — a)^ -\-{y — by -\-{z — c)^, taken relative to
X. IS
[x — a)dx
--^^ , X only being considered variable, this is usually denoted by
s/[x-af-\-{y-hf-^{z-cf
■—\ dx including the quantities considered as variable between the parentheses. In the same
Symbol
I. i. ^2.] COMPOSITION OF FORCES.
(—), ( — \ ( — ) expressmg according to the usual notation the coefficients
of the variations dx, Sy, 8z, in the variation of the preceding expression of s.
If, in like manner, we put s' for the distance of M from any point of the
direction of another force S', taken as the origin of that force ; ^''ij-j^ will
be the part of this force resolved into the direction parallel to x, and in the
same manner for others ; the sum of the forces S, S', S", &c., resolved into
directions parallel to x, will be 2 . *S. f — J ; the symbol 2 of finite integrals, 2.
denoting the sum of the terms '^ - ij-ji ^' ' \J~p ^^'
Let V be the resultant of all the forces S, S', &c., and u the distance of
the point M, from a point in the direction of this resultant taken as its
origin ; ^* (t") will be the expression of the part of this resultant resolved
into a direction parallel to the axis of a: ; we shall therefore have, by the
preceding article,*
[136]
manner tlie partial differential of s taken relative to y is denoted by f — - Wy; Stc. When
the differential is taken supposing all the variable quantities x, y, z, to be noticed, it is
called the complete differential. The term variation is used above instead of differential. The
difference between these expressions is fully pointed out in note ( 1 7^). A partial or complete Differen-
variation is found in precisely the same manner as a partial or complete differential, changing
the characteristic d into 5, so that the partial variation of s relative to x, is
V^^/ )r(^af-\-{y — hf-\-{z — cf *
hence ( — - ) = ; and in like manner { — - ) = ; (-— ) = . These ri3cl
being substituted in the three first expressions [13] give the three last values of [13].
* {\3d) The formulas [14, 15] necessarily follow from the principles proved in [11"]
and [13]. By multiplying the three equations [14, 15] respectively by 5x, Sy, Sz, and
adding tliem together, we have
Now it is evident tliat the complete variation of u is equal to the sura of its partial variations
3
10 COMPOSITION OF FORCES. [Mec. Cel.
We shall have in like manner
By multiplying these equations respectively by ^x,^y,^z, and adding them
together, vre shall have
[16] V.6u = ^.S.^s', (a)
This equation exists whatever be the variations 5 rr, ^y,5zj and it is equivalent
to the three preceding. If its second member is an exact variation* of a
function tp, we shall have
consequently
[17] prYi!i\ /^?
6x J \^^ ,
that is, the sum of all the forces S, S', &c., resolved into directions parallel
to the axis of ar, is equal to the partial differential (y- )• This is generally
the case, when these forces are respectively functions of their distances from
their origin to the point M.f In order therefore to compute the resultant of
all these forces, resolved into a direction parallel to any right line, we must
find the integral i: ./. S.ds, and calling it (p, we shall consider it as a function
of X, and of two other right lines perpendicular to each other, and to the line
relative to a?, y, 2r, that is 5M = f-r— ) . (5a? + f — j .(5y-{- (t~) '^^j ^^so
which, being substituted, give V . 6 u = 1 . S . S s, [^16'].
Exact Dif- * (13e) An expression is said to be an exact variation or differential, when as it then
exists its integral is possible. Thus xdy-\-ydx is an exact differential, because its integral
is X y. But xdy-\-2ydx is not an exact differential, because no finite quantity can in
general be found, whose differential will produce that expression. The same remarks will
apply to expressions of any order of differentials. Now having as above J^.8u = S(pj
whatever be the variations 8x, 8y, Sz, we shall obtain by noticing only tlie variation S x, the
expression [17].
f (13/) If S is a function of s, the quantity fS.Ss is evidently integrable ; and if S' is a
function o( ^,fS' .8 s' wiUbe integrable, he. ; and in this case the sum of all these quantities,
fS . 8 s,fS' . 8 s', Sec, or 2 .fS .8s is integrable, or is an exact variation.
I.i.^3.] PRESSURE AND EQUILIBRIUM. 11
X ; the partial differential ( t^ j will be the resultant of the forces S, S', &c.
resolved into a direction parallel to the right line x.
3. If the point M is in equilibrium, by means of all the forces which act Eciuauon
upon it, the resultant will be nothing, and the equation (a) will become ''"'™"
0 = :^.S.6s; (b) ti^l
which shows that in the case of the equilibrium of a point, acted upon by
any number of forces, the sum of the products of each force by the element
of its direction is nothing.
If the point M is forced to remain upon a curved surface, it will be affected preswie
■T •■• upon
by the reaction of the surface, by a quantity which we shall denote by R, Swi.
This reaction is equal and directly opposed to the pressure which the point
exerts against the surface ; for if we suppose the point to be acted upon by
the two forces R and — R, we may conceive that the force — Ris destroyed
by the reaction of the surface, and that the point M presses the surface with
the force — R ; now the pressure of a point upon a surface is perpendicular
to it, otherwise it might be resolved into two forces, the one of which would
be perpendicular to the surface, and would be destroyed by it ; the other
parallel to the surface, and by means of which the point would not have any
action upon that surface ; which is contrary to the hypothesis ; putting therefore
r for the perpendicular to the surface drawn through the point M, and
terminated at any point of its direction, the force R will act in the direction
of this perpendicular ; we must therefore add R. 6r to the second member of
the equation* (6), which will thus become
0 = i:.S.6s + R.6r; (c) [19]
* (14) To illustrate this by a simple example, we shall suppose \jp
that the point M is forced to move upon a curve line FD G, and """"^^v.,^^ \
that all the forces act in the plane of this curve ; these forces being ...™.rrrrnrr~^r^^?=^
S,S',S",inthedirections^D(=s),^D(=/), CD{=s"), ^ ^^^^^^^Jo"'^
producing a pressure — R upon the curve m the direction UTi E, ''^ J
which, by what is said above, must be perpendicular to the curve ' ^
at the point D. This pressure must be destroyed by the reaction jR of the curve in the
direction ED. We may therefore suppose the curve FD G to be taken away, and the
body to be acted upon by the forces S, S', S", R, in the directions AD,BD, CD, ED,
or 5, «', s", r, respectively. In this case tlie equation [16] will become
V.Su=SJs-^S'.6s'-\-S"Js"+R.Sr.
12
PRESSURE AND EQUILIBRIUM.
[Mec. Cel.
^R being then the resultant of all the forces S, S', &c., it will be perpen-
dicular to the surface.
If we suppose the arbitrary variations 5x, 5y, Sz, to appertain to the curved
pondicuiar guj-face upou which the point is forced to remain : we shall have by the nature
face. r ^ T ^
of the perpendicular to that surface, Sr=0* which makes R.5r vanish from
Equation
of the per
If we suppose die point to be kept in equilibrium by means of these forces S, S', S" and the
resistance of the curve, the resulting force T^ vnW be nothing, and we shall have
or by including under the symbol 2, all the forces S,S',S ", it will become 0 = 'S.S,Ss-j-R.§r,
as in [19].
* (14a) Let G^ D Z. jP be the proposed surface, to which
tlie line HD E is perpendicular, at D ; JE being the origin of
the force R, put ED = r, and let D Lhe infinitely small, and
join E L ; then as DL is perpendicular to ED, we shall have
EL=V ED^-\-DLP, and by neglecting the infinitely small
quantity of the second/ order DL^, we shall have EL^^ED,
consequently in this case the variation of ED, which is equal to EL — ED, would be notliing,
tliat is ^r = 0. This would not be tlie case if ED was inclined to the curve, as is the case
with the line MD ; for by drawing the lines MD, ML, and letting fall on ML the perpen-
dicular Dl, we have nearly ML—MD = Ll = LD. cos. ML D, which is of the same
order as LD, except cos. MLD = 0, wliich excepted case corresponds to that where DM
falls upon DE, or where DM is perpendicular to the surface. Therefore the equation Sr = 0
is the equation of tlie perpendicular to tlie surface.
(14&) I shall in this and the three following notes, investigate the
equations of a right line, a plane, and a spherical surface, which will
frequently be wanted in tlie course of this work. Let AcL be a
right line, the origin of whose co-ordinates is A, putting AD = x,
theongiii. j)^-—y^ dc = z for the rectangular co-ordinates of any point c
of this line. The projection of AcL upon the plane AD da,
corresponds to the diagonal A d, and, at whatever point of the line
AL\he point c is taken, the angle DAd will be the same, and by putting its tangent equal to
A, and observing that by trigonometry Dd = AD. tnng. DAd, we shall have y = Ax. In
like manner by projecting Ac upon the plane ^5 CD, and putting tang. DAC = B,we shall
have D C=AD. tang. DA C, or z = Bx. These values of y, z, give z= Cy, putting
[19a]
Equation
of a line
passing
tiirougii
y'li
Al
[195]
B
C= - , which last equation might be found like the two former, by projectmg the line ^c upon
the plane CDdc, and putting tang. dDc=C. Hence the equations of a right line passing
Li. §3.]
PRESSURE AND EQUILIBRIUM.
13
the preceding equation : therefore the equation (5) takes place also in this
case, provided that one of the three variations Sx, Sy, 5Zy be exterminated by
means of the equation of the surface ; but then the equation (6) which, in
/i^
■B.a-B',
through the origin of the co-ordinates are
i/ = Ax; z = Bx; z=Cy.
When the line does not pass through the origin of
the co-ordinates, as is the case in the annexed figure,
where that origin is K, these equations will be varied a
little. For in this case the co-ordinates of the point c are
KHz=^ X, Hf= y,fc=z'y those of the point A are
KG = a, GE=b,E A = c,whence AD = x — a,
Dd=y — 5, dc = z — c, as in note (136). These
being substituted in the three preceding equations,
Dd= AD. tsn^. DAd; D C = ADAmg.DA C;
dc=:Dd .tang. dD c, give y — b = A . (x — a);
z — c = B.{x — a); z — c=C.{y — b) ; which, by
transposing the terms 6, c, of the first members, and putting b — A.a=.A', c
c — C .bz=.C' give the general equations of a right line
y = Ax-\-A'', z^Bx+B; z=Cy+C';
which, like [196'] are equivalent to two distinct equations, and any two of the co-ordinates, as
y, z, are given by means of the third.
(14c) Let C be the origin of the co-ordinates,
C if X the axis of a;, C GY that of y, the axis of z
being perpendicular to the plane of tlie figure, and let
the plane whose equation is required be CDbc,
intersecting the plane YC HB in the right line
CAD. From any point b of this plane let fall the
perpendicular bB upon the plane YC HB. Draw
BDE perpendicular to CD ; jB if perpendicular to
CX; HA parallel to D E, HE parallel to A D. Then the co-ordinates of the point b are
CH=x, HB = y, Bb = z. And if weipnt ihe angle AC H=HB E=zs, and the angle
bDB, which denotes the inclination of the planes YCHB, CDbc, equal to (p, we shall have
in the rectangular triangles CAH, BEH,DBb, the following equations,
AH{=DE) = CH. sin. ACH=x.sm.s; BE = HB .cos. HBE = y .cos. s;
Bb — BD.tAng.bDB.
The two former equations make B D =BE— DE=iy. cos. s — a? . sin. e ; which, being
substituted in the last, gives Bb = z={y. cos. s — a? . sin. s) . tang. 9 ; and by putting
4
[19b']
General
Equation
of aright
line.
[196"]
[196'"]
14 PRESSURE AND EQUILIBRIUM. [Mec. Cel.
general, is equivalent to three equations, furnishes only two distinct equations,
which may be obtained by puting each of the co-efficients of the two remaining
[i^ differentials equal to nothing. Let u==0 he the equation of the surface, the
Equation
of a plane "~"~ • — — — — —
surface
through ^°^' ^ *^S- 9 = B; — sin. s tang. 9 = ^, we obtain the following equation of a plane passing
the origin, through the origin of the co-ordinates
[19c] z=iAx-\-By.
If the plane do not pass through the origin of the co-ordinates, and we put a, b, c, for the
co-ordinates of the point C, measured in the directions x, y, z, respectively, we must, in the
above equation, change x, y, z, into x — a, y — 5, z — c, respectively, as is evident by
oenorai proceeding as in note (135). Substituting these values, we get 2^ — c=zA{x — a)-\-B{y — 5) ;
ofapiane. or 0 = Ax-}-By — ^ + (c — Att — Bb) ; which, for the sake of symmetry, may be multiplied
by — C, putting A' = —C'A, B'==—C'B,D'=—C'{c—Aa-'Bb), and it becomes
[I9d] 0 = A'x-\-B'y-^C'z-iriy,
which is the general equation of a plane surface, and when it is compared with the general
form w = 0, assumed in [19'], we shall find that the function w corresponding to a plane
surhce is A' x-\-B'y+C'z-\-D'.
[I4d) The equation of a spherical surface, the origin of whose ,y
rectangular co-ordinates is at the centre of the sphere, is easily
Equation ° . . .
of a computed by supposing A to be the centre of a sphere whose radius is
r', and c to be any point of its surface, so that Ac^=r'', the rectangular
co-ordinates of the point c being A D = x; D d{:=. C c)=.y ;
dc{=DC) = z. Then by [lla], A <p-AD''-\- D d^-\- dc", X
which in symbols is r'^ ^= x^ -{- y''- -\- z^ ', consequently the equation of
this surface may be thus ^M-itten,
[19e] 0z=a;'^+/--|-;^2_/2, or 0 = /2_^__y2_^2;
This is a particular case of the equation of the surfaces of the second order, given in Book III,
§ 2, [1363], and is what in [19'] is called m= 0 for a spherical surface, u being the function
a? -\- y^ -\- z^ — ^^'
(14e) From the two preceding examples we may perceive the metliod to be used in
finding the equation of any surface ; and it appears from the formulas [19e?, e,] that the
equation of a plane or a spherical surface gives one of the co-ordinates, as z, by means of the
other two^ x, y ; and the same remark will apply to any other surface. If we represent,
therefore, as in [19'], the equation of this surface by w= 0, u will be a function of x, y, z.
The differential, or rather the variation of this function will correspond to tlie infinitely small
plane which touches the proposed surface in the point whose co-ordinates are x, y, z ; and
[19/] the equation of this plane will be of the form 6u = A' Sx-\- B' 5y-{- C dz = 0 ; in which
the rectangular co-ordinates of the plane are S x, S y, 8 z, parallel to x, y, z, respectively, the
origin being at the point of the surface corresponding to x, y, z, where the variations ^x,6y^
8 z, are nothing.
a
e/
1
|.«
/h.
1
..y.
/
\
7
I. i. (^3.] PRESSURE AND EQUILIBRIUM. 16
two equations <5 r = 0 and* 6u = 0 will exist at the same time, which requires
that
6r=N.6u, [19"]
N being a function of x, y, z. To find this function, let a, 6, c, be the
co-ordinates of the origin of r, we shall havef
r = \/(x — ay+(y — 6)^+ (z — cf] [20]
(14/) From the variation of the equation of a surface 5u = 0j we may deduce the
value of one of the variations of the co-ordinates, as S z, by means of the other twOf
Sx, Sy. Thus for a sphere in which the radius / is constant, [19e], Su=:0 becomes
0 = x6x-{-y6y-\-z8z, wliich gives S z, by means of 5 a;, S y, at any point of the
surface whose co-ordinates, x, y, z, are known. On the contrary, the variations of the equations
of B. ri^htlme, \_l9h"'], ore Sy^= A. Sx ', 8z = B.Sx; Sz=C.Sy; which give any ft(?o of
tlie variations, as (5 y, 5 z, by means of the remaining one, 5 x.
* (15) It follows from [19/], that 5m==0, is in general of tiie form Q=A'.^x-\-B'Jy-^C'Jz',
and the variation of r [20] put equal to notiiing is of tlie form 0=.^' .6x-\-B" .6y-\-C" .^z;
w3', B', C, A'\ B", C", being functions of the co-ordinates x, y, z, and constant quantities,
independent of 6x, Sy, 8 z. Now if these equations were not multiples of each other, we
might, in general, by the usual rules of extermination, find any two of the quantities, as Sy, S z,
in functions of the other, S x, so that we should have Sy = A.Sx'f Sz = B.Sx; ^,5, being
functions of A', B, C, A", B", C". These equations correspond to those of any infinitely
small right line whose origin is at the point of tiie surface, whose co-ordinates are x, y, z, as
is evident by writing 8x, Sy,8z, for x, y, z, in the two first equations of a right line [196'],
by which means they become like the preceding ; and it is evident tiiat this right line must be
a tangent to the surface, because by hypothesis, 8x, 8y, Sz, correspond to that surface. It
would tiierefore follow, if 8r = 0 is not a multiple of Su = 0, that tiie point could not be
moved, except in tiie direction of that line, thus putting a limit to tiie direction of the motion,
even when there is none by the nature of the question, and when the point is left free to move
in any direction upon tiie proposed surface ; therefore we must necessarily have 5 r equal to a
multiple of 8 u, which may be denoted hy 8r = JV*. 8 u.
t (15a) This value of r is equal to tiiat of 5 [12], found as in note (1 ob). Its partial vari-
ations found as in note (13c), give (|^) =1=1^; (il) = ^5 ^il) = lZ±j-the
sum of whose squares is (~j + (^J + (^-Sj = if:zll+^I^l±j^.:zfL ; and as die
numerator and denominator of the second member are equal to each other, the first member
vnU be equal 1, tiiat is (j^^ -f ^^J + ^-^.J = 1. Agam, since by hypotiiesis we
have . .= A-. a ., its partial variations give (|^) = JV. (||) ; g) =;v. (^) ,
16 PRESSURE AND EQUILIBRIUM. [Mec. Cel.
whence we deduce (^) + (^) + ij^= ^ ^ consequently
Vjz)''^^^ ' \t~\ "^^^ ^^™ °f whose squares is
which, by means of the preceding equation, becomes l=j'V^. ) (t— ) +(-r-) +(i— ) ( j
1
as in [2 1]. Hence JV*= > ^ — , : consequently 5 r = JV . 5 m
becomes 5 r = > --==--7 — ~'=^ , ; and
R.Su
R.5r — '
which, by using the value of X, [22], becomes R .dr = X.5u. This, substituted in [19],
produces [23].
To illustrate the above method of finding the value of JV, we shall give an example, in
which the proposed surface is a sphere, corresponding to the equation [19e],
u = x^-{-y^ -{-z^ — 7^ = 0;
in which the radius r is given, or constant ; hence f — j=2a?; ^ — j = 2y; {—-\ = 2z;
[25a] therefore \/^ (^)\(^)\(^^ which,
substituted in the above expression of JV, gives JV= -^5— ; and this, substituted in the assumed
\u
value 5 r = JV*. 5 w, becomes 5 r= — -. This might also be found in the following manner.
<i r
The proposed surface being spherical, the line r drawn perpendicular to it must pass through
the centre of the sphere, which was taken for the origin of the co-ordinates, and if we take this
centre for the origin of the force R, and of the line r, we shall have a = 0, 6 = 0, c = 0, r = /,
x5x-\'y6y'\'z6z
and the formula [20] will become r= Voc^-\-'f-\-z^ ; whence 6r= «/'^x~g 1 g' J
or
"♦• A '*» , I , ni X 1 # , I , V f\ 'y
5r= ; but the preceding expression of u = x'^-\-f-\-z'^ — r* gives
I. i. §3.] PRESSURE AND EQUILffiRIUM. 17
therefore by putting
R
\/
6u\^ /SuV'.f^uV' ' [22]
ox y "^ Vy/ V^/
[221
[23]
the term RJr will be changed into
R.5r = 'K.6u ;
and the equation (c) [19] will become
in which equation we ought to put the coefficients of the variations 6x, Sy,
6 2, separately equal to nothing, which furnishes three equations ; but they are
only equivalent to two equations, between x, y, z, because of the indeterminate
quantity X, which they contain. Therefore, instead of exterminating one of
the variations Sx, Sy, dz, from the equation (6) [18], by means of the differential
equation of the surface, we may add to it this last equation multiplied by an
indeterminate quantity x, and then consider the variations of 6x, Sy^ 8z, as
independent. This method, which results from the theory of extermination,
combines the advantage of simplicity in calculation with that of showing the
pressure — R which the point M exerts against the surface.
Suppose this point to be contained in a canal of simple or double curvature,*
it will suffer a reaction from the canal, which we shall denote by k, and this
du = 2xSx-\-2ySy-{-2zSz, or -^ = , which is equal to the preceding
expression of S r, hence 5r= -— , as above. The terms a, b, c, were put nothing for the
salce of simplicity, otherwise we might have put aSx-\-bSy-\-cSy = Oj which equation is
easily deduced from 5r = 0, 8u = 0.
* ( 1 5b) The intersection of a plane wiih any curve surface forms in general a curve line.
Thus a plane cutting successively a cone in certain directions Mill produce the three conic
sections, and in particular cases will also form the triangle and the circle. Curves thus
produced will be wholly in the same plane. But if two curve surfaces intersect each other,
their intersection may form a curve line whose points are not in tlie same plane, and then
it is called a curve of double curvature. As a very simple example of this last species of curve of
curves, we may mention that formed by the intersection of two right cylinders with circular curvature,
bases, whose axes intersect each other at right angles. The curve traced upon the surface of
the greatest cylinder by the other is a curve of double curvature. As all tlie points of a curve [25^1
of double curvature appertain to both the generating surfaces, whose equations are m = 0,
M = 0, the equations of this curve must be defined by the equations of those surfaces, as is very
evident.
18 PRESSURE AND EQUILIBRIUM. [Mec. Cel.
will be equal and directly opposite to the pressure which the point exerts
against the canal, and its direction will be perpendicular to the side of the
canal : now the curve formed by this canal is the intersection of two surfaces,
whose equations express its nature ; we may therefore suppose the force k
to be the resultant of the two reactions R and i2', which the point M
suffers from each of these surfaces ; for the directions of the three forces Rj
R and k, being perpendicular to the side of the curve, they must be in the
same plane. Putting therefore ^r, 6 r\ for the elements of the directions of
the forces R and J?', which directions are perpendicular to the surfaces
respectively, we must add to the equation (6) [18] the two terms R.^r, R .6r',
which changes it into the following,
[24] 0 = ^.SJs+R.^r+R.hr'. (d)
If we determine the variations 6x, Sy, 5z, so that they shall appertain at
the same time to the two surfaces, and consequently to the curve formed by
the canal, 5 r and 5 r' will vanish, and the preceding equation will be reduced
to the equation (b) [18], which therefore still takes place in the case where the
point M is forced to move in a canal ; provided that two of the variations S a:,
8y, 8z, are exterminated by means of the two equations which express the nature
of this canal.
Suppose that w = 0, and w' = 0, are the equations of the twO; surfaces whose
intersection forms the canal. If we put
R
[25]
[26a]
X =
R
' v/(^
5y J \^^ /
the equation (d) [24] will become*
[26] 0 = 2.>S.<55+X.5w+X'.(5w';
*• (I5c) The reasoning used in finding [23] may be used for [26]. For in the same
manner in which R.Sr is introduced in [19], we may introdnce RJr-\-R' J r' in [24] ; and
for the same reason that Sr was put =0 in note (14a), we may here put Sr, Sr' equal to
nothing, the lines r, /, bemg perpendicular to the canal. The assumed values of X, X' in
[25], being similar to [22], lead to the result [22'], i2.5r = X.5w, Jl'.(5r' = X'.5M', as in
note (15a). These, substituted in [24], give [26.]
Li. <§3.] PRESSURE AND EQUILIBRIUM. 19
in which the coefficients of each of the variations 5x,5y, S z, ought to be put
separately equal to nothing ; we shall therefore have three equations, by [261
means of which we may determine the values of x, x', which will give the
reactions R and R of the two surfaces ; and by combining them we shall
have the reaction k of the canal upon the point M, consequently the pressure
which this point exerts against the canal. This reaction, resolved parallel to
the axis of X, is equal to* B.(Q+i?.(^); or x. (^^) + x' . (Q :
the equations of condition u = 0, u' = 0, to which the motion of the point
M is subjected, express, therefore, by means of the partial differentials of
the functions, which are equal to nothing because of these equations, the
resistances which the point suffers, in consequence of the conditions of its
motion.
From what has been said, it follows that the equation of equilibrium (b)
[18] is generally satisfied if the variations 6x, <5y, 6z, are subjected to the
conditions of equilibrium. This equation furnishes the following principle.
" If we vary by an infinitely small quantity the position of the particle M, in
such a manner that it may remain always upon the surface or upon the curve
along which it would move, if it were not wholly free ; the sum of the forces
which act upon it, each multiplied by the space which the particle describes
according to the directions of the forces, is equal to nothing in the case of
equilibrium, "t
* (I5d) The sura of the reactions jR, R', multiplied by the elements of their directions,
are jR . 5 r -f- -R' • ^ ^ ; hence by formula [17] this reaction resolved in a direction parallel to
the axis of x/isR. f — j -\-R' .(—)j ^d by substituting hr R.5r, R' .S r', their values,
[26a], "k.Su, X' . 5 m', it becomes X . f—j -j- X' . f — j, as above.
f (15e) The in6nitely small space described in the direction of any
one of the forces, must be considered dis jtositive, if the motion tend to
increase the distance of the body from the origin of that force, but
negative if it tend to decrease it. As an example of the formula [18]
0 = 2.5.55, let tliere be three forces, S, S', S", originating at the
points A, B, C, at the distances A M{= s), B M{= «'), CM{= s"), *^
from the moveable body M respectively, and acting upon the body in the directions of those
lines. Then the preceding expression will become 0 = <S . 5 5 -j- <S' . 5 ^ + S" . 5 s". Now
20
[27]
PRESSURE AND EQUILIBRIUM.
[Mec. Cel.
The variations (5 a:, 6y^ bz, being supposed arbitrary and independent, we
may, in the equation (a), [16], substitute for the co-ordinates a:, y, z, three
other quantities which are functions of them, and then put the coefficients of
the variations of these quantities equal to nothing. Thus let p be the radius
drawn from the origin of the co-ordinates to the projection of the point M,
upon the plane of x, y, and ta the angle formed by p and the axis of x, we shall
have*
X = p. COS. zs ; y=p' sin. -ui ;
if the body be moved from M to any otlier point m, through die infinitely small space Mm,
and we let fall from m the perpendiculars ma, mh, mc, upon the lines AM, B M, CM,
continued if necessary, the variations of the lines A M, B M, C M, wiU be represented by
Am — A M, B m — B M, Cm — C M, which, by neglecting infinitely small quantities of
the second order, become Ma, Mb, — Mc, respectively; the negative sign being prefixed
to the last because OJIf decreases. These being substituted for Ss, Ss', Ss", in the preceding
expression of 2 . S . ^ 5, it becomes S .Ma-{- S' .Mb — S" .Mc, which by formula [18]
ought to be equal to nothing in the case of equilibrium. And that this equation really takes
place is easily perceived by geometrical considerations. For Ma = Mm . cos. a Mm;
Mb = Mm . COS. b Mm ; M c = Mm . cos. C Mm; which being substituted in the pre-
ceding value of 2 . S . S s, it becomes
2. S .Ss = Mm. \S. COS. a Mm -\- S' .COS. bMm — S" .cos. CM m].
But by formula [11] the quantities S. cos. a Mm, S'. cos. bMm, — S" .cos. CM m,
represent the parts of the forces S,S', S", resolved in tlie direction
Mm ; the latter having a different sign from the two former, because
it has an opposite direction ; now as the body is by hypothesis in
equilibrium, the sum of these forces must be nothing ; therefore
S . cos. aMm + S' . cos. b Mm — S" . cos. CMm = 0,
consequently I, .S .Ss = 0, as above.
* (1 6) Let ^ X ^ be the plane oi x,y; W the projection of the
place of tlie particle M upon that plane ; the co-ordinates of the point
W will be .^ X= a:, X W=^ y. Draw the line A W= p, and put the
angle XA W=zi, then we shall have
AX = A W.cos.XA W; XW=A W.sm.XAW;
which, by substituting the symbols x, y, p, «, become as in [27].
Substitute these in (12) and we get s = \/ {p.cos.-a—af-^ip .sin.ni—bf -{-{z — cf, and
the formula [16] may be considered as containing p, *, instead of x, y. The partial difl'erential
of this equation, taken relative to trf, will then be as in [28]. Now it appears from the equations
[14, 15,] that the force F resolved into three forces, in directions parallel to the axes cc,y,z.
Li. §3.] PRESSURE AND EQUILIBRIUM. 21
by considering, therefore, in the equation (a) [16], M, 5, s', &c., as functions
of p, *, and z ; and comparing the coefficients of 5 «, we shall have
— • ( T— ) is the expression of the force V resolved in the direction of the
element pS-a. Let V be the force V, resolved in a direction parallel to the
plane of x and y, and p the perpendicular let fall from the axis of z upon the [281
direction of V parallel to the same plane ; will be a second expression*
or parallel to the rectangular elements Sx, Sy, 8z, are represented byF'.f-^j; 7^. fy- j; [28a]
P^.( — ]; and as the axes of x, y, z, are arbitrary, we may put any other rectangular
elements in place of S x, S y, S z. Thus instead of the rectangular elements 8x, Sy, parallel
to the axes of x, y, we may take, in the plane of these two co-ordinates, the element
Sp= W a, upon the continuation of the line A W', and tlie element p(5ts= Wh, perpen-
dicular to A W, corresponding to the variation of the angle ztf, represented by W A h = d'af;
and we may then use the elements 5p, p5 tf, (J^r, instead oi 8x, 6y,8 z^ and the expression of
Fi resolved in directions parallel to the rectangular elements 5p, p^ts, 6 z^ will become
^' (l7/' ^' (t^^)' ^' \^r ^^ ^^ evident, by changing 8x, 6y, into 5 p, p 5 ts, in
the formulas [28a]. By bringing the term p from under tlie parenthesis, the force in the
direction of the element p^-m becomes — . ( — ), as above.
* (16a) Suppose the figure to be similar to that in the last note,
with the addition of the line P W, representing the projection of the
direction of the force V, upon the plane of x, y, and let the line
AP=phe drawn perpendicular to P W. Then the force V in
the direction parallel to P W, may be resolved into two forces, in
the directions parallel to Wa, Wb, of which the last, in the direction
parallel to fVb, is equal to V . cos. 6 Wc, [1 1], or F'.cos. WAP;
and as cos. WAP =—-=-, this force in the direction ^6 will be equal to - — as
above. Putting this equal to tlie expression of the same force found in [286], we get
— — = 7 • ( ^ )• Multiplying by p, it becomes as in [29.]
6
[286]
22 PRESSURE AND EQUILIBRIUM. [Mec. Cel.
of the force F, resolved in the direction of the element p5«; therefore we
shall have
Momen- If wc supposc thc forcc V to be applied at the extremity of the perpen-
force^about dicular p, it will tend to make it turn about the axis of z ; the product of
this force, by the perpendicular, is what is called the momentum of the force
[293 ^j about the axis of z ; this momentum is therefore equal to F. ( -^ ) ; and it
follows from the equation (e) [28] that the momentum of the resultant of
any number of forces, is equal to the sum of the momenta of these forces.*
* (166) The partial variation relative to -a being taken in the formula [16] gives
[29a] j;r /'_^^ =^,S. (-AX and by substitution in [29] we obtain pV = 1.S. (t~\ in
which the first member p V represents the momentum of the resulting force V about the
axis of z, [29'], and the second member is the sum of the momenta of all the forces S, S',
he, about the same axis. For the force S resolved in a direction parallel to the element
pSzijis- .(- — ], [2SZ>]. This multiplied by p, gives, as in [29'], the momentum of this
force about the axis of z, equal to S. f — J ; in like manner the momenta of the forces S',
S", he, are S' . (-r~\ *^" • ("T" )' ^^■' ^^^ ^^ ^""^ ^^ ^^ ^^^^ momenta is represented
by 2 . S . (j^j ; and this, for the sake of reference, is inserted m [29], though it is not so
placed in the original,
I.ii. <^4.] MOTION OF A MATERIAL POINT. 23
CHAPTER II.
ON THE MOTION OF A MATERIAL POINT.
4. A POINT or particle at rest cannot give itself any motion, since there is
no reason why it should move in one direction rather than in another. When
it is acted upon by any force and afterwards left to itself, it will continue to
move uniformly in the direction of that force, if not opposed by any resistance.
This tendency of matter to continue in its state of motion or rest, is what is
called its inertia. This is the first law of the motion of bodies. loetus.
That the direction of motion is in a right line follows evidently from this,
that there is no reason why the point should deviate to the right rather than
to the left of its first direction ; but the uniformity of its motion is not equally
evident. The nature of the moving force being unknown, it is impossible to
know, a priori, whether this force is constantly retained or not. However,
as a body is incapable of giving to itself any motion, it seems equally incapable
of altering the motion it has received, so that the law of inertia is at least
the most natural and simple that can be imagined ; it is also confirmed by
experience ; for we observe upon the earth that motions continue longer
in proportion as the opposing obstacles are decreased ; which leads us to
suppose that the motion would always continue if these obstacles were
removed.
But the inertia of matter is most remarkable in the motions of the heavenly
bodies, which, during a great many ages, have not suffered any sensible
alteration. We shall therefore consider the inertia of bodies as a law of
nature ; and when we shall observe any alteration in the motion of a body,
we shall conclude that it has arisen from a different cause.
In uniform motions, the spaces passed over are proportional to the times ;
but the times employed in describing a given space are longer or shorter
according to the magnitude of the moving force. This has given rise to the
idea of velocity, which, in uniform motion, is the ratio of the space to the veiocuy.
24 MOTION OF A JNIATERIAL POINT. [Mec. Cel.
time employed in describing it ; therefore, s representing the space, t the
time, and v the velocity, we have
[29"] '^~1'
Time and space being heterogeneous quantities, cannot be directly compared
Unit of with each other ; therefore an interval of time, as a second, is taken for the
space, . ^ .
tiTwity. ^^*^ ^^ *^™® ' ^^^ ^ given space, as a metre, is taken for the unit of space ;
then space and time are expressed by abstract numbers, denoting how many
measures of their particular species each of them contains, and they may then
be compared with each other. In this manner the velocity is expressed by
the ratio of two abstract numbers, and its unit is the velocity of a body, which
describes one metre in a second.
5. Force being known only by the space it causes a body to describe in a
given time, it is natural to take this space for its measure ; but this supposes
that several forces acting in the same direction would make a body describe
a space equal to the sum of the spaces that each of them would have caused
it to describe separately, or, in other words, that the force is proportional to
po/tToHa?' the velocity. We cannot be assured of this a priori, owing to our ignorance
reiocTty. of thc uaturc of the moving force : we must therefore again have recourse to
experience upon this subject ; for whatever is not a necessary consequence of
the little which we know respecting the nature of things, must be the result
of observation.
Let V be the velocity of the earth, which is common to all the bodies upon
its surface ; /the force by which one of these bodies Mis urged in consequence
of this velocity, and let us suppose that v=f.(p(f), expresses the relation
between the velocity and the force ; 9 (/) being a function of /, to be
determined by observation. Put a, h, c, for the three partial forces, into
which the force /is resolved, parallel to three rectangular axes. Let us then
suppose that the body M is acted upon by another force /', which may be
resolved into three others a!, h', d, parallel to the same axes. The whole forces
acting on the body in the directions of these axes will be « + «', 6-f 6', c+c';
putting F for the single resulting force, we shall have, by what precedes,*
[30] F = S/I^dy + (6 + hj + {c +?/.
* (16c) Using the figure as in (I la), the forces AJ) = a^a!, Bd=h-\-h', dc = c-\-<ft
the resultant Ac will, as in (11a), be equal to \/^a + a7 + (^ + ^T + (c+c'f» »' ^'
I.ii.§5.] MOTION OF A MATERIAL POINT. 25
( n -4- n'\ TT
If we put U for the velocity corresponding to F; - — —^ — will be this
velocity resolved in a direction parallel to the axis of a ; hence the relative
velocity of the body upon the earth, parallel to that axis, will be ^^ — :f^ 7-,
or (a -{-a') .cp (F) — « 9 (/). The greatest forces which we can impress on
bodies upon the surface of the earth, being much smaller than those with
which they are affected by the motion of the earth, we may consider «', 6',
c', as mfinitely small in comparison with f; we shall therefore have*
r, J. , aa'-\-hb' -j-ccf ,, .-j^. ,j,.,aa'-\-bb'-\-ccf , . j,. ..,.
F=/H X__E_; andf 9(i^) = <p(/)H --j-^— - Hf) '•> ^^^^
(p (f) being the differential of <p (f) divided by df. The relative velocity of
M, in the direction of the axis a, will therefore become
a'.cp(f)+j.laa' + bb' + cc'l.^'(f), m
Its relative velocities in the directions of the axes b and c, will be
b'.'?(f) + t{aa' + bb' + cc'l.^'(f);
'' [33]
c'.^{f) + j.{aa! + bh'-{-cc'},^'(f).
The position of the axes a, 6, c, being arbitrary, we may take the direction
And we shall have AcAD:: velocity C7 in the direction A c, to the corresponding velocity
resolved in tlie direction A D, which is therefore equal to - — , as above. In like
manner the velocity of the earth in the direction AD is — . Hence their relative velocity
is „ U — ; which, by putting U=F(p(jy, r=/. (p(/), becomes
(a -j- a') . (p (F) — a (f) (/), as above.
* (ICrf) The expression [30], neglecting a'^, b'^, cf^, on account of then- smallness,
becomes F= ^ {a^ -{- IP -\- c^) -{- 2 {a a' -\- b b' + c~d) = i^ p _^ 2 {a a' -\- b b' + cd),
extracting the square root, still neglecting a'^, b'^, (/^ we obtain F [31].
t (I6e) This expression of (f){F) is easUy deduced from the general development of
(p{t-\-a)y according to the power of a, by Taylor's theorem, [617], retaining only the two
first terms 9 (<) + « . ~% or 9 (0 + « • ?' (0. and putting <=/,« = ^^^l'±l^^
7
[34]
26 MOTION OF A MATERIAL POINT. [Mec. Cel.
of the impressed force, for the axis of «, and then h' and c' will vanish, and
the preceding relative velocities will become
«'-f*(/)+y-*'C/)|; j-c'-v'(f); j-a!.9'(f).
If 9 (f) does not vanish, the moving body, by means of the impressed force
«', will have a relative velocity, perpendicular to the direction of that force,
provided b and c do not vanish ;* that is, unless the direction of this force
coincide with that of the motion of the earth. Therefore if we suppose a
spherical ball at rest upon a very smooth horizontal plane, to be struck by the
base of a right cylinder, moving horizontally in the direction of its axis ; the
relative apparent motion of the ball would not be parallel to that axis, in all
the positions of the axis with respect to the horizon : this furnishes therefore
a simple method of discovering by experiment whether cp (f) has a sensible
value upon the earth ; but, by the most exact experiments, the least deviation
is not perceived in the apparent motion of the ball from the direction of the
impressed force ; whence it follows that upon the earth, 9 (/) is very nearly
nothing. Its value, however small it might be, would be most easily perceived
in the time of vibration of a pendulum, which would vary if the position of
the plane of its motion should alter with respect to the direction of the motion
of the earth. Now, since the most accurate observations do not indicate any
such difference, we may infer that 9' (/) is insensible, and it may be
supposed equal to nothing upon the surface of the earth.
If the equation 9 (^f) = 0 exists for all values of/, (p (/) would be constant,
and the velocity would be proportional to the force ; it might also be
proportional to it if the function c? (f) was composed of more than one term,
since otherwise 9' (/) could not vanish unless / was nothing ;t we must
* (17) There is one case not noticed by the author, namely, when the motion of the
earth is in a plane passing through the origin of the co-ordinates perpendicular to the axis of
a ; for then a = 0, the relative velocities in the directions parallel to the axes h, c, will be 0,
and in the direction parallel to a will be a' cp(f). This omission does not however affect the
general reasoning of the author, nor the correctness of the conclusion he has drawn.
f (17a) If <p(/) was composed but of one term, as a/"*, a being a constant quantity, it
would give cp' (/) =.maf'^~^, which would become 0 either when m=:0, or when m^ 1
and/= 0. The first case gives 9 (/) = a, whence 9' (/) = 0, for all values of/. If 9 (/)
was composed of more than one term, as «/"*+ a'/*"', it would give
<p'(/)==ma/— i+Wa'/-'-S
Lii. <§6.] MOTION OF A MATERIAL POINT. 27
therefore, if the velocity is not proportional to the force, suppose that in
nature the function of the velocity, vv^hich expresses the force, is composed
of several terms, w^hich is nov^ise probable ; and that the velocity of the
earth is exactly that which corresponds to the equation <?' (/) = 0, which is
contrary to all probability. Moreover, the velocity of the earth varies at
different seasons of the year : it is about a thirtieth part greater in winter
than in summer. This variation is yet more considerable, if, as everything
appears to indicate, the solar system itself has a motion in space ; for according
as this progressive motion conspires with that of the earth, or is opposed to
it, there must result, in the course of the year, great variations in the absolute
motion of the earth ; which would alter the equation we are treating of, and
the ratio of the impressed force to the absolute velocity which results from it,
unless this equation and velocity are independent of the motion of the earth :
however no sensible alteration is perceived by observation.
We have thus obtained from observation two laws of motion ; namely, the
law of inertia, and that of the force proportional to the velocity. They are the
most natural and simple that can be imagined, and without doubt have their
origin in the nature of matter itself ; but this nature being unknown, they
are, as it respects us, facts deduced from observation, and are the only ones
which the science of mechanics derives from experience.
6. The velocity being proportional to the force, the one of these quantities
may be represented by the other, and all we have previously established
respecting the composition of forces may be applied to the composition of [34^
velocities. Hence it follows, that the relative motions of a system of bodies
acted upon by any forces, are the same, whatever may be their common
motion ; for this last motion resolved into three others parallel to the three
fixed axes, increases by the same quantity, the partial velocities pf each of
the bodies, parallel to these axes ; and as their relative velocity only depends
upon the difference of these partial velocities, it must be the same, whatever
be the common motion of all the bodies : it is therefore impossible to judge
and this might be nothing if m>- 1 , »»'>► 1, either when /= 0, or/= f j^T^T^. Hence
we see that the only case in which (p' (/) is nothing and / indeterminate, is when <p (/) is a
constant quantity a, and v =.f. 9 (/) = af.
28 MOTION OF A MATERIAL POINT. [Mec. Cel.
of the absolute motion of the system of bodies of which we make a part, by
the appearances we observe in them, and this is what characterizes the law
of the proportionality of the force to the velocity.
It follows also from § 3, that if we project each force and its resultant upon
a fixed plane ; the sum of the momenta of the composing forces, thus projected
about a fixed point taken in this plane, is equal to the momentum of the
projection of the resultant :* now, if from this point, we draw to the moving
Vector, body, a radius which we shall call the radius vector, this radius projected
upon the fixed plane would describe upon it, by means of each force acting
separately, an area equal to the product of the projection of the line which
it would cause the moving body to describe, by half the perpendicular let fall
from the fixed point upon this projection if this area is therefore proportional
* (nJ) This is proved in note (16Z>).
f (17c) Let A^Whe the plane oi x, y, A the fixed point taken
upon that plane, W the projection of the place of tlie body, A W the
projection of its radius vector, W w the projection of the space it vi^ould
describe in the time <Z ^ by the force S if it acted alone upon the body,
fZw2= the element of the area W Aw described in the same time,
4/2 jB = * the perpendicular let fall upon WwB, and rf the value of
the force S resolved in a direction parallel to the line Ww ; then the
force tf may be taken for the velocity in the direction Ww [34'], and this velocity multiplied
by the time d t gives the space W w described in that time, hence Ww^d.dt. This
multiplied by half the perpendicular A B gives the area A Ww, or dA^^dt.dic, whose
integral taken relative to t gives the area described in that time A = ^t . tf *, supposing A
to commence with t, and observing that in this integration the force tf and the perpendicular *
are constant. Now the quantity tf * is equal to the momentum of the force 5 about the fixed
point A [29'], and if we put this momentum equal to m, we shall have A = ^t .m. In like
manner, if we put A', A", he, for the areas, and m', m", &;c., for the momenta corresponding
to the forces S', S", he. ; we shall have A'=it.m'; A"=.^t .m" ', Sec, and the sum of
all these is 2 . .^ = ^ ^ . 2 . m. In like manner. A, being put for the area which would be
[34o] described about the same axis in the time t by means of the single force V [28'] which is die
resultant of all the forces S, S', S", he, resolved in a direction parallel to the plane of x,y;
we shall have as above A, equal to the product of ^ i by the momentum p V [29] of die
force V about that axis, ot A, = it.jpV'. Now p V is equal to 2 . <S . ( — ^ j [29], which
last expression represents the Sum of the momenta of all the forces S, S', Sic. about that
axis ; and this momenta we have before put equal to 2 . m, or p V =:'L .m, therefore
1. ii. ^ .7.] MOTION OF A MATERIAL POINT. 29
to the time. It is also, in a given time, proportional to the momentum of [34"]
the uroiection of the force ; hence the sum of the areas which the projection Descrip-
J^ ^ lion of
of the radius vector would describe, by means of each force acting separately, "«^«-
is equal to the area that the resultant would cause it to describe. Hence it
follows that if a body is at first projected in a right line, and is afterwards
acted upon by any forces directed towards the fixed point, its radius vector
will always des(;ribe about this point, areas proportional to the times, since
the areas which these last forces would cause the radius vector to describe
would be nothing.* Inversely, we must conclude that if the moving body
describes about the fixed ])olnt, areas proportional to the times ; the resultant
of the new forces acting upon it must be always directed towards that [34'"]
point.f
7. Let us now consider the motion of a point acted upon by forces, which,
like gravity, appear to act continually. The causes of this force, and of the
similar forc(^s which exist in nature, being unknown, it is impossible to
discover whether they act without intermission, or their successive actions
are separated by ins(uisible intervals of time ; but it is easy to prove that
the phenomena ought to be very nearly the same in both hypotheses ; for
if we represent the velocity of a body upon which a force acts incessantly, [34'"]
by the ordinate of a curve whose absciss represents the time ; this curve, in
the second hypothesis, will be changed into a polygon of a very great number
A^ = ^ti:.m, wlilcli, by substitution in the preceding value of 2. A, gives A^-=^ .A. [346]
Hence it follows tlmt the sum of all the areas which would he described by each force acting
separately is equal to the area A^ which would be described by means of the resultant V,
and as this area A^ is equal to ^ i . 2 . m, it must be proportional to the time of description.
* (17^) If the direction Ww of any force rf of, the last note passes through the point A,
the perpendicular A B = '!f would become nothing, therefore the momentum m of Uiis force,
which is equal to tr C, would also be nothing, consequently 2 . m would not be affected by this
force, and as the area A, described by the resuhing force was shown m tiie last note to be
equal to it .^,m, tliat area will not be affected by any force passing through the point A.
f (17e) By [34&] we have A^ = ^t .i:.m, tiierefore, if A be proportional to t, the
quantity 2 . m must ])e constant, and tlie momentum of any new force must be nothing, which
takes place only wlien the perpendicular A B is nothing, that is, when the direction of the
force W w passes tiirough tiie fixed point A.
8
[34 ''i]
30 MOTION OF A MATERIAL POINT. [Mec. Cel.
of sides, and, for that reason, it may be considered as coinciding with the
curve.* We shall, with geometricians, adopt the second hypothesis, and
suppose that the interval of time which separates two consecutive actions of
[34 V] any force is equal to the element of time dt, the whole time being denoted
by t. It is evident that we must suppose the action of the force to be greater
in proportion as the interval which separates the successive actions is
increased, in order that the velocity may be the same at the end of the same
time t : the instantaneous action of a force ought therefore to be supposed in the
ratio of its intensity and of the element of the time during which it is supposed
to act. Therefore, if we denote this intensity by P, we ought to suppose, at
the beginning of each instant d t, that the body is urged by a force P.dt,
and moves uniformly during this instant. This being premised ;
We may reduce all the forces, which act upon a point M, to three forces,
P, Q, R, in directions parallel to the three rectangular co-ordinates x, y, z,
* (I'T/) Let the times be measured on the absciss AB F, in
which are taken the equal intervals B C, CD, D E, he.
Suppose the velocities corresponding to the points B, C, D, he,
to be represented by the ordinates Bb, C c, D d, he. Complete
the parallelograms B C yh, C DSc, &ic. Then if tlie forces act
incessantly during the intervals B C, CD, &;c., the velocities will j^ J3 CCT} E F'
gradually vary, and the general expression of the velocity will be
represented by the ordinate of a regular curve hcde Sic. drawn tlirough the proposed points
h, c, d, &c. On the contrary, if the force act instantaneously at the points corresponding to
B, C, he, the velocity through tlie interval B C would be equal io Bb', at the point C it
would instantaneously become C c, and would remain the same during the interval C D, when
it would become Dd, &;c., so that the irregular figure byc^dse, &;c. would be the limit of
the ordinates representing the velocities. In both hypotheses the acquired velocities B h,
C c, &;c., at the points B, C, &tc., are equal, so that the velocity computed for any point, as
E, by either hypothesis is the same ; and if the intervals B C, CD, &:c. are taken infinitely
small, and equal to d t, the velocity corresponding to any portion of the line A F, computed
in either way, cannot differ but by an infinitely small quantity of the order cy, dS, &;c. ;
therefore we may use either hypothesis at pleasure. Again, it is evident that if the intervals
of time B C, C D, he. should be decreased, the instantaneous forces acting at B, C, D,
fee. must be decreased in the same ratio. For if the interval was C C, the velocity
corresponding to the point C would be C 8', and its increase at C would be 6' d', which is
to 5<Z as C C to CD, and the increment of the velocity would be as the intensity of the
force P multiplied by the element of the time dt,ox P.dt as above.
I. ii. <^ 7.]
MOTION OF A MATERIAL POINT.
31
which determine the position of this point ; we shall suppose each of these
forces to act in a contrary direction to that of the origin of the co-ordinates^
or, in other words, that these forces tend to increase the co-ordinates. At the
beginning of the next instant d t, the body acquires in the direction of each
of these co-ordinates, the increments of force, or of velocity, P.dt, Q,dt,
R.dt. The velocities of the point M, parallel to these co-ordinates, are
—^ — ^, — ; because in an infinitely small moment of time, they may be
dt dt dt
supposed uniform, consequently equal to the elementary spaces divided by
the element of the time. The velocities of the point at the beginning of the
second instant of time, are therefore
% + P.dt;
'i+Q.dt;
dz
dt
-\-R.dt\
or
^^j^d.^~d.^-\-P.dt',
dt dt dt ^
Tt^'^'dt "^'Tt^^'"^^'
dz . J dz
-r--\-d . -;-•
dt ' dt
•d.^+R.dt;
at
but in this second instant, the actual velocities of the point parallel to the
,. ., , dx , -, dx dy , , dy dz , -, dz
co-ordmates x, y, z, are evidently -yj + d.-j:^; 3T+^*3Ti 37 + "'Tri
dt
dt dt
dt ' dt
dt
the forces — d.~-\- P .dt, — d .-^+ Q.dt, — d .-j- -{-R,dt, ought
at at at
therefore to be destroyed, so that the point iW would be in equilibrium, if acted
upon by these forces only. Therefore, if we denote by 6x, 6y, S z, any
variations of the three co-ordinates x, y, z, which must not be confounded
with the differentials d x, dy, d z, representing the spaces described by the
point parallel to the co-ordinates during the instant d t* the equation (&) § 3 [1 8]
[34 ^»3
[35]
[36]
[36']
[36"]
* {\^g) We shall here explain in a geometrical manner the
principles of tlie raetliod of variations, so far as it may be necessary
to understand tlie computations made in the present worlc. Let
A Bb C be tlie ortliographic projection of any curve upon tlie
plane of x, y, so that the co-ordinates of any poiQt B BieEF=:x,
F B = y. The co-ordinates of tlie point h of the same projection
of the curve, infinitely near to B, will be represented, according -^
32
[37]
MOTION OF A MATERIAL POINT.
[Mec. Cel.
will become
If the point Mbe free, we must put the coefficients of 6x, (hj, Sz, separately
to the usual differential noiaiioi), hy Ef—x-{-dx, fb = y-\-dy, and if we draw B§, parallel
[366] and equal to Ff, tbe line B^ — Ff— d x, and iih = dy, will represent the differentials of the
absciss E F, and of the ordinate FB respective!)' ; these diQerentUds being the differences
of the co-ordinates of two consecutive points B, b, of the same curve A Bb C. But if the
nature of the proposed curve be cha)iged In an arbitrary manner, so tliat its projection may
become A' B' U C infinitely near io A B b C, the points B, b, being changed into B', b', the
co-ovdiiiates of (he point B will be then changed into E F', F' B', corresponding to the point
[36c] B', which co-ordinates aie represented hj E F' =i x-\-5x, F' B' =:y-\-Sy, and the changes
in tlie values of the co-ordinates are called the variations ; thus F F' =:^ B B" := S x, is the
variation of the absciss E F, and B' B" = 5 y, h the variation of the ordinate FB. In like
manner, if we draw the ordijiate b' f par;'llel to bf, and let fall upon it the perpendiculars
&/3', J5/3", B' b", the variations of the oi'dinates Ef, fb, will be represented by //, and b' ^'
respective]}. From ibis ex))lanation of the term variation, it is evident that tlie variation of
any function of x, y, z, &;c., is found by cljnnging x, y, z, Sic, into x-\-6 x, y -{-^y, z-\-6 z,
&:c., respectively, and subtracting the formei" value from the latter, neglecting as in the
diff'erential calculus, the powers and products o( S x, § y. 5 z, he, so tliat the variation of any
function Is found in the same manner as its differential, using the sign 6 instead of d.
We may proceed from the point B to b' in two different
ways. First from B to B' by the method of variations,
then from B' to b' by the differential of the cui-ve A' B.
Secondly, from B to & by the differential of the curve AB,
then from & to b' by the method of variations. The
comparison of these two methods furnishes a very important
theorem in the doctrine of variations. To avoid a compli-
cation of letters we sliall put dx=x, dx = x', tlien
E F= X, Ef= X -{- X, E F' = X -\- xf ; now by proceeding as in the first method, we
have B'b", or F' f equal to the differential of E F', along the curve A' B', tlierefore
B' b" = F'f = dx + dx', this added io E F'==:x-{-8x, gives Ef == x-{-dx-{-dx-\-dx'.
And by tlie second method we have b ^' or //' equal to the variation of J5J/, therefore
h^'=ff = Sx-{-§x, ihhsiddedto Ef=x-\-dx ^wes Ef'=x + dx-{-6x-{-6x. Putting
these two expressions of Ef equal to each other, we get
x-\-8x-\-dx-{-dx'=:x-\-dx-\-Sx-\-Sx,
-^^; 7.
I'
c
■■r¥ /•''':
y^-R
--^rrc
"';
J.
A.
r
\B' fi
-■\a"
E as
FF' £ £' H h
I.ii.§7.]
MOTION OF A MATERIAL POINT.
33
equal to nothing ; and by supposing the element of the time d t constant, we Equation.
shall obtain the three differential equations
ddx
~d¥
= P;
ddy
~d¥
= Q;
ddz
U
R;
motion of
a point.
[38]
this, by neglecting the terms in each member which destroy each other, gives dx' = Sx,
consequently the differential of the variation of x is equal to the variation of the differential
of the same quantity, and if we substitute for a/, x their values 8x,dxj it becomes d6x = 6dx, [36rf]
consequently the characteristic dS may be changed into 6d, or the contrary. What is here said
relative to the axis of x may be easily proved in the same manner for that of y or z, which,
however, is evident of itself, since the axis of x may be changed into that of y or z, the names
being arbitrary, so that dSy = Sdy', d8 z = 8 dz. This theorem might be generalized, [3fie]
but it will not be necessary in the present work.
Another tlieorem of great importance is tliis. The variation of the integral of any quantity
U is equal to the integral of the variation of the same quantity, or in symbols S .fU=f. <5 U.
This is easily proved, for if we substitute x -{- S x, y -\- S y, z -\- S z, (or x, y, z, respectively in
U, and caU the result U', we shall have S .fU=fU' —fU=f{U' — U), and as [36/]
U' — U=8U, this becomes 5 .fU=f. S U, consequently tlie characteristic 8 ./ may be
changed into /. 5, in the same manner as Sd was changed into dS. This agrees with the
geometrical consideration of the subject in the follo^\-ing paragraph.
It may not be amiss to explain in a geometrical manner tlie
value of an expression of tliis form 5 .fMdx, which frequently
occurs. For greater simplicity we shall suppose Sx = Oj M=y;
then the lines B B', b b', will fall on the continuations of the
ordinates FB,fb, so that EF being =x, and FB^y,we
shaWhave Ff=dx, BB'=Sy, ^b = dy, dSx = 0; and ^ ^ ^ ^ [3%]
the element of the area BFfb = ydx, that of BB'b'b = Sy.dx.
Taking the integrals of these expressions, supposing them to be limited by die ordinates GAJV^
HCC, they wiU give the areas G A C H, AA'C'C, namely GACH=fydx',
AA' C C=f5y .dx; now it is evident that the latter area is the variation of the former,
and as the variation of fydx is denoted by S ,fy d x, we shall have the area
AA' C C = S.fydx=fSy.dx.
The identity of these two expressions is also a consequence of the preceding theorem that
the characteristic Sf may be changed into /5, for by that means the first expression S .fydx
becomes/. 8 {yd x), and as 5 e? a? = 0, this is evidently equal iof8 y.dx. From this simple
example we may obtain a better idea of the unport of such expressions as 8. fydx, f 8 y.dx,
he, dian could be done wiiliout considering the subject geometrically.
9
34 MOTION OF A MATERIAL POINT. [Mec. Cel.
If the point M be not free, but forced to move upon a surface or a curve
line, we must, by means of the equations of the surface or curve, exterminate
from the equation (/) [37], as many of the variations 6x, Sy, 6 z, as there
are equations in this surface or curve, and then put the coefficients of the
remaining equations separately equal to nothing.
8. It is possible to suppose in the equation (/) [37], that the variations
,Sx, 8y, 6 z, are equal to the differentials d x, d y, ^2, respectively, since these
differentials are necessarily subjected to the conditions of the motion of the
An expression of the form fM.Sdx maybe reduced so as to contain Sx without its
differential. For, by putting 6dx=:dSx = dx', (cc' being as above = 5 x), khecomes/M.dx'j
which, by integrating by parts, is equal to Mx^ — fdM.xf, as is easily proved by
[36h] differentiation. Resubstituting for x' its value Sx,it becomes fM.Sdx = M.§x — fdM.dx.
We might add a constant quantity to the second member to complete the integral, so as to
render it nothing at the Jirst point ^, (Fig. page 35), where it commences, the co-ordinates
of which point we shall call x^, y^ z^, those of the last point C of the integral being x^^, y^^, z,, j
the values of M corresponding to the points A^ C, being respectively M^, M^^. Hence the
[36i] complete integral fM. dSx = M. ^x — M^.Sx^ -\-fd MJx, and the whole integral comprised
between the points A, C, is fM .dSx = M^ .6x^^ — M^.S x^ -\-fd M. S x, the term affected
with the sign / being taken within the same limits. If M is constant, dM=0, and
fM.dSx = M.Sx„—M.5x,. In like manner fM.d6y = M,,.Sy,,—M,.5y,-j-fdM.5y,
and if M=l, and dM=0, f.d5y = Sy^^ — §y^. The import of this integral maybe
explained geometrically, supposing ^^ = 0. Then BB'=:5y, and
if we draw B' e parallel and equal to B b", and suppose the curve
A' B'" V" C" to be such that the intercepted parts of the ordinates
BB"\ h"h'\ C C", may be equal to A A', we shall evidently
have d.Sy=b" b' — BB'z=b'e, and f.dSy is equal to the sum
of all the lines b' e comprised between the points A', C, and this sum is evidently equal to
C C" == C C'—C C" = C C' — AA' = Sy^^ — 8y,, as above. We might extend
these remarks to a much greater length, but what we have said will suffice for all the purposes
of the present work, and we shall conclude by observing that tlie calculus of variations is of
great importance in finding the form of functions like fy 6 x, having the property of a
maximum or minimum ; which is obtained by the usual principles of the maximum or
[36/k] minimum, by putting its variation S .fydx=zO, or the area AA'B'C'C equal to
nothing.
I. n. § 8.]
MOTION OF A MATERIAL POINT.
36
moving particle ili.* Making this supposition, and then integrating the
equation (/) [37], we shall have
df
c-\-2.f(P.dx+Q.dy-{-R.dz);
d x^ -\- d y^ -\- d z'-
c being an arbitrary constant quantity. -^ is the square of the
velocity of M,\ which velocity vve shall denote by v ; supposing therefore
that P.d x-{-Q.dy-\-R.d z is an exact differential of a function ?, we shall
have
This case takes place when the forces acting upon the particle M are
functions of the respective distances from the origins of these forces to this
[39]
[39']
[39"]
[40]
* (17A) If the point M was compelled to move in a
curve whose projection is ABC, the curve A' B' C,
depending on the variations, might be supposed to coincide
with ABC, and we might take tlie arbitrary variations
Sx, Sy, such that the point B' would fall in b, and then we
should have 5x = dx, 5y = dy, and the projection upon the
plane of x, y, would, upon similar principles, give Sz = dz.
2
Substitute these in [37], multiply by -— , and ti-anspose -E 6-y
the terms depending on P, Q, R, it becomes
2rfx , dx , 2dy j dy , .2 d z , dz
FF'
.d.
^r-^-^+^Tr-^-^=2(^-^^+^-^2/+^-^^)'
dt dt
whose i„.e,.al gives [39], because ..(^y = ^^ ... ^; ^{ff ^Hf -^ -fr
\dtj dt dt
f {ill) If, in the figure of note (13&), we suppose .4? c to be infinitely small, and the points
A, c, to represent two consecutive points or places of the body ; the ordinates of the point
A being KG = x,GE = y, EA = z ; those of the point c, KU= a?+ dx, Hf= y + dy,
fc=iz-\-dz, we shall evidently have AD = dx, Dd — dy, dc=zdz, Ac = ds, and
Ac'^AD''-i-Dd^Jrdc^ = dx^-\-df-{-dzM\lal ^nds\nce^ = Yehcky v, we shall
have as above ^ = i;2^ which, substituted in [39], gives [40]. Moreover it
appears from the figure, that tlie cosines of the angles which the element of the curve d s makes
with lines drawn parallel to the axes x, y, z, are represented by ~, ^, ^, respectively.
[405]
36 MOTION OF A MATERIAL POINT. [Mec. Cel.
particle, which comprises almost all the forces in nature. For S, S\ &c.,
representing these forces, and 5, s', &;c., being the distances of the particle
M from their origins ; the resultant of all these forces, multiplied by the
variation of its direction, will be, by § 2 [16], equal to ^.SJs; it is also
equal to PJx-\-Q.6y-{-R.8z; therefore we shall have*
[41] P.5x-{-Q.6y-^R.8z = :s^.S.5s;
and as the second member of this equation is an exact differential, the first
member must be so.
From the equation (g) [40] it follows, 1st. That if the particle M is not
[411 acted upon by any forces, its velocity will be constant, because then 9 = O.f
This is easy to prove in another way, by observing that a body moved upon
a surface or a curve, loses at the contact with each infinitely small plane of
the surface, or each infinitely small side of the curve, but an infinitely small
[41"] part of its velocity of the second order. | 2d. That the particle M, in passing
* (17A;) By hypothesis tlie forces S, S', S", he, acting in the directions s, s', 5", &;c.,
are equivalent to the three forces P, Q, R, acting in directions parallel to the axes x, y, z,
respectively. Now supposing, as in ^ 2, V to be the resultant of the first named forces, and
u its direction, we have V.Su — S.S.SsllQl; and as F is also by hypothesis equivalent
to the forces P, % R, we have by the same formula [16] V.Su = P .6x-{-Q^Jy-^R.8z,
hence P.5x-{- Q^.Sy^RJz = X.S.Ss, the second member of which is by note (13/) an
exact variation of a function, consequently P.Sx-{-^.Sy-\-R.dz is an exact variation of
a function 9, or
[40c] &(p = PJx-{-Q.Sy-'s-R'^Z'
f (111) If the particle is not acted upon by any forces, we shall have P = 0, Q = 0,
R = 0,[S4^''],\nddcp = P.dx-i-^.dy-{-R.dz [39"] would then become d(p = 0, whose
integral 9 — constant, may be put 9 = 0, including this constant term in the quantity
c [40].
J (18a) Thus if a body move with the velocity z in the ^^_^i
direction B A, and at the point A be compelled to change -_ ^^--"""^ 1
its direction to the line A D, and we contmue JJ A to C,
and put the infinitely smaU angle BAC=6, we shall have, (by 1 1 , 34') the velocity in the
d\rect[onAD = z.cos.6=zz{l—i6^+hc.), by 44 Int.; this differs from the original
velocity z by the quantity ^d^.z of the second order, as is stated above ; and the loss of
velocity on an infinite number of such lines, or on the whole curve, would be an infinitely small
quantity of the first order only.
I. ii. «^8.] PRINCIPLE OF THE LEAST ACTION. 37
from a given point, with a given velocity, towards another given point, will
have acquired, upon arriving at this last point, the same velocity, whatever [4i"]
be the curve which it may have described.*
But if the particle is not forced to move upon a determinate curve, the
curve which it describes possesses a singular property, which had been
discovered by metaphysical considerations ; but which is in fact nothing
more than a remarkable result of the preceding differential equations. It
consists in this, that the integral fvds, comprised between the two extreme ^onue*
points of the described curve, is less than on every other curve, if the body action.
be free ; or less than on every other curve described on the surface upon [41'"]
which the particle is forced to move, if it be not wholly free.
To prove this, we shall observe that P.dx-\-Q'dy-\-R,dz being
supposed an exact differential, the equation (g) [40] givesf
u5tj = P.5a; + Q.5y + i?.52:; [42]
the equation (/) [37] of the preceding article therefore becomes
rt r 7 dx , , -, dy , , 1 dz -^ .
Q = ^x.d,-r-\-^il.d.-r-\-^z.d.- vdt.6v. r«,n
dt ^ ^ dt ^ dt l*3J
* (186) Suppose the values of v, 9, corresponding to the first point of tlie curve, to be
t)', 9', those to the last point v", 9". The equation [40] at the first point will become
t;'2=c-[-2 9', whence c = v'^ — 29', which being substituted in [40] gives generally
v^ = tj'2 — 2 9' -j- 2 9 ; hence at tlie last point of tlie curve we have v"^ z=v''^ — 2 9' -}- 2 9". [40(f]
Now 9 [39"] is a function of S, S', he, s, s', SiC, wliich quantities are given at the first
and last points of the curve, consequently 9', 9", must be given, and 1/ is also given, by
hypothesis, therefore the value of v" must be the same whatever be the curve described.
That is, we can determine the difference of the squares of tlie velocities at two points without
knowing the curve described by the body, and this curve might become a right line in the
case where a body should fall fi-eely towards a point to which it is attracted by a force varying
as any function of the distance. We must however always observe that the tlieorem [40c?]
would not hold true if P.dx-{-Q.dy-^R.dz was not an exact differential d 9, and it
would not generally be an exact differential if it contained terms depending upon the particular
curve described, as might, for example, be the case if the curve produced a particular resistance
or firiction.
f (18c) Taking the variation of [40], dividing it by 2, and substituting for 5 9 its value
P.5x-^ q.5y-{-R.Sz [40c], we get [42]. Substituting now in [37] the value of
P.Sx-]-q.6y-\-R.6z [42], we obtain [43].
10
38 PRINCIPLE OF THE LEAST ACTION. [Mec. Cel.
Put ds for the element of the curve described by the particle, we shall
have*
144] vdt = ds; ds = Vd¥+dy^-{-dz';
therefore
[45] 0 = 5x.d.-^-i-6y.d.-r + ^z.d.~ — ds,8v; (h)
Taking the differential of the expression ds relative to the characteristic <5,
we shall have
TAti^ d s . ^ dx . ^ , dy . ^ . dz , ^
[46] -r-.^.ds=--A,dx4-^.^.dy-\---.6.dz.
dt dt ^ dt ^ ' dt
The characteristics d and 5 being independent, we may place them at
pleasure the one before the other ; we may therefore give to the preceding
equation the following form,t
, , d.\d x.^x-\-dy.hy-\-dz.^z\ , , dx , n dy , , dz
•- ^ dt dt ^ dt dt'
g
* {I8d) Tlie expression vdt = ds, follows from the equation v = - [29"] by changing
t into dt, s into ds, the velocity v being esteemed uniform during the time dt. The equation
d s = Vdx'^-\-dy^-{-dz^, is deduced, by putting Ac=ds in [40 a]. The substitution of
vdt = ds in [43] gives [45]. Taking the variation of d s^ = dx^ -{-d]f -\-d z'^ [44], and
dividing it by 2dt, gives [46].
f (19) Substituting —=zv in the first member of [46], It becomes like the first of
[47]. Tlie second member of [46] may be transformed by observing that
dx ,, d.{dx.6x) dx dy d.jdyJy) . , dy
dt dt dt ^ dt ^ dt *' dt
~.Sdz= /' — (iz.d.-^; as is easily proved by developing the first terms of
dt dt dt
the second members, these, substituted in [46], give [47]. Again, by the equation [45], we have
^x.d . — -\-8y .d.-^-{-5z.d.~ = ds.8v, this being substituted in [47], it becomes
(t Z CI Z CL Z
V .8ds= • t ^- ^+ y- y I ^- ^5 — ds.Sv. Transposing the term dsJv, the first
dt
member becomes v .5d s-\-d s .S v, which is evidently the variation of vds, or 8.[vds),
hence we obtain [48], whose integral, changing the characteristic fS into 8f, as in [36/],
gives [49]. If we suppose as in [36A] that x„ y^, z„ are the co-ordinates of the first point of
[491
I. ii. §9.] PRINCIPLE OF THE LEAST ACTION. 39
Subtracting from the first member of this equation the second member of the
equation (h) [45], we shall have
' ^ dt
This, being integrated with respect to the characteristic d, gives
5. /«;<? 5 = constant H ■ ^^ • ^^^^
If we extend the integral to the whole curve described by the particle, and
suppose the extreme points of the curve to be invariable, we shall have
5 .fv ds = 0; that is, of all the curves which a body could describe in ^49^
passing from one given point to another, when subjected to the forces P, Q,
R, it will describe that in which the variation of the integral /y£?5 is nothing,
consequently that in which this integral is a minimum.
If the particle moves along a curve surface, without being acted upon
by any force, its velocity will be constant,* and the integral fvds will
become vfd s ; therefore, the curve described by the particle is then the
shortest that can be traced upon the surface, from the point of departure to
the point arrived at.
9. We shall now investigate the pressure of a particle upon a surface on
which it moves. Instead of exterminating from the equation (f) [37] of '*3?/
§ 7, one of the variations &x, Sy, 6z, by means of the equation of the surface,
the curve, and a?^^, y,^, z,^, those of the last point, and take the integral [49] so as to be
nothing at the first point, it will be generally expressed by
^ ^ _ . dx .8 x-\-dy.Sy-\-dz.6z (dx,.6x,-\-dy,.6y,-\-dz..Sz,) . , ■, ■, • ,
8lvds) = ^^— ^J i — — \^ - — -, and the whole integral
J^ dt dt
becomes i.r(.rf.)^''"'-^'^'+''^---^^"+'"^-^^"'-'''"'-^"' + ''-'';^''-+'''-^'''. If the
•'^ ' dt dt
extreme points of the curve corresponding to A^ C, [Fig. page 32], are fixed or given, the
variations <J x,, ^yn ^ ^ii ^ ^ni ^ Vm ^ ^//' ^^^ ^^ nothing, therefore the second member of
the preceding equation will be nothing, consequently 5 .fv d s = 0, which corresponds to
its minimum, as is observed [36A;].
* (19a) This is shown in the remarks which follow the formula [41']. When v is constant
5 .fv d s=^0, becomes v.Sfds = Oj or Sfdszt=zO, andas/rfs = «, this becomes Ss==Oj
corresponding to the minimum value of s.
As an example of the application of the principle of the least action, we may mention the
manner of deducing from It the laws of reflection and refraction of light. Thus if a ray of
Pressure of
upon a
surface.
40
[50]
PRINCIPLE OF THE LEAST ACTION.
[Mec. Cel.
we may, by ^ 3, add to that equation the differential equation of the surface,
multiplied by an indeterminate quantity,* — \dt, and we may then consider
the variations 5x, 5y, 5z, as independent. Let w = 0 be the equation of the
surface ; we must add to the equation (f) the term — x.du.dt, and the
pressure of the point against the surface will be by § 3, equal to
light proceed from a luminous object to the eye of tlie
observer, its path is a straight line in conformity with the
principles of the least action [49"]. Again, if the ray
proceed from the luminous point L, and be reflected from
the plane surface MRJV at R, in the direction R E, to
the eye of the spectator at E ; the space passed over,
s==LR-\~RE, ought by this principle [49"] to be a
minimum, the velocity v being supposed constant. From this we may easily deduce the
equality of the angles of incidence and reflection, P RL, PRE; the line P R, as well
as LM, EJV, being perpendicular to MJV. For, if we put LM = a, EJV= b, MJV= c,
MR = x, RJV=c — X, we shall get s = VaP-{-a^-\-V(c — a;)^-f-62, whose variation,
relative to x, being put equal to nothing, gives
V^xS-f-aa
or
MR
TTr
Rj\r
RE '
v/(c — x)2-l-62
whence sin. M L R = sm. R E JV, or sin. P R L= sin. P R E ; therefore in the case of
reflection, the principle of tlie least action would make the angles P RL, PRE, equal.
If the velocity on the line LR is equal to m, and on the line RE is equal to n ; the
principle of die least action [49'], 8 .fv d s=: 0, would require that the variation of
mV oi^-\-a^-\-n v (c — a?)^ -f- b^ should be nothing, corresponding to the minimum. Proceeding
with the calculation as above, we should get m . sin. P RL = n . sin. PRE, and this is the
same as the usual law of refraction, supposing the point E to fall below MJV upon the
continuation of the line E JV oi the present figure, and that the ray of light enters the refracting
medium at the point R situated in the line J\IJV, which separates die two mediums, where
die velocity of the ray is changed from m to w. In this case the sine of the angle of incidence
is to the sine of refraction in the constant ratio n to m.
* (19J) The second member [37] corresponds to 2.S.Ss of [18] and [23] and to tliis
term is added "k.Su in [23] on account of the equation of the surface, and it is shown in the
remarks following [23] that the pressure the particle JW exerts upon the surface is — R. If
we had changed the sign of X and added in die equation [23] die term — XiJm, the sign of X
I. ii. § 9.] PRESSURE UPON A SURFACE. 41
Suppose at first that the particle is not acted upon by any force, its
velocity v will be constant;* then as vdt = ds, the element of the time dt
being supposed constant, ds must also be constant, and the equation
(/) [^^J' augmented by the term — 'k.Su .dt, will give the three following
equations ;t
d^ \dxj' ds"^ \dyj d^ \dz
whence we deduce
^ I / fduy . /duy./duy v'^V{ddxf-\-{ddyf-^{ddz)\
but d s beine: constant, the radius of curvature of the curve described by the Radius of
~ •' Curvature.
particle is equal toj
df^
^^{ddxf + iddyf-^iddzf ' ^^^
in [22] would be changed, and that expression [22] would give for the pressure — R the
quantity X . ^ (^- j + (^-j + [j;) as in [50].
* (19c) As is shown in [41'].
f {I9d) By hypothesis the forces P, % R, are nothing; substituting these in [37] and
adding, as above, the term —~\.8u.dt, which by [14a] is equal to
it becomes, by dividing hy dt,
Substituting for dt^ its value deduced from [44] —^, and putting llie coefficients of
^x, 5y, Sz, equal to nothing, gives the equations [51], whence >..(-~)=^ .d dx ;
\dx/ ds^
fdu\ v'^ /du\ «2
' \dv) ^^ d^ y'" \dz) ^^ d^ Squaring each of these equations, adding
them together, and taking tlie square root of the sum, we get the equation [52].
J (19e) Let Fa J cE (see figure on page 42) be the proposed curve, ah, he, two
infinitely small and equal parts of it, considered as right Unes, whose centre of curvature is C,
11
42
[54]
PRESSURE UPON A SURFACE,
and by putting this radius equal to r we shall have
[Mec. Cel.
that is, the pressure of the particle against the surface is equal to the
squares of the velocity, divided by the radius of curvature of the curve it
describes.
[5:3a]
G
c
\
^.If
\ \ /\
\ \^^
\y
^b
__^.^
^a
F
/li
making Ca=Cb=Cc. Continue the line ab to A, making
bA = ah = bc, equal to the element of tlie curve ds supposed ji
constant. Then, by construction, the angle cb A is equal to
a Cb, and as the triangles a Cb, cb A, are isosceles, they must v
be similar. Hence Ac:bc::ab:Cb, or in symbols Ac:ds::ds:r, ^
whence r = — - . Let G a' V A! c' H be the axis of x, which „'
Ac ' a
in general is in a different plane from C ac. Upon this let
fall the perpendiculars a a', b b', A A\ c c', then a' b' = d x,
b' c' = d X -\- d d X, and since by construction abs=sb A, we
have b' A' = a' b' = d x, therefore A' c' = d d x. That i^ to
say, the projection of the line A c upon the axis of x is equal to
d d X. In the same way we may prove that the projection of
the line Ac upon tlie axis of y is d dy, and its projection upon
the axis of z is d d z. If we therefore, upon .^ c as a diagonal,
describe a rectangular parallelopiped, whose sides AD, D d,
d c, are respectively parallel to the axes of x, y, z, we shall have AD^=d d x, D d = ddy,
dc= dd z. The lower figure was drawn separately from the other to enlarge it, so as to
avoid confusion in the lines. Now we have as in [Ha] A c = \^ A D^-\-Dd^ -}-d(^,
and by substituting the preceding values o( A D, D d, d c,
Ac
B
This, substituted in r=
Ac'
V{ddxf + {dd yf +{dd zf.
gives
rf«2
V{ddxf']-[ddyf-\-[ddzf
ds'^
being substituted in the second member
as in [531. This value of -
^[ddxf-{-[ddyf-\-{ddzf ^
of [52], produces the expression [54]. The first member of which is by [50], equal to the
pressure of the point against the surface, consequently that pressure is also equal to the second
member — of the same expression.
I. ii. <^9.] CENTRIFUGAL FORCE. 45
If the particle move upon a spherical surface, it will describe the circum-
ference of a great circle of the sphere, which passes by the primitive direction
of its motion ; since there is no reason why it should deviate to the right
rather than to the left of the plane of this great circle : its pressure against
the surface, or in other words, against the circumference it describes, is
therefore equal to the square of its velocity, divided by the radius of this
circumference.
Suppose the particle to be attached to the extremity of an infinitely thin
thread, void of gravity, whose other extremity is fixed at the centre of the
sphere ; it is evident that the pressure exerted by the particle against the
circumference will be equal to the tension of the thread, if the particle were
wholly supported by it. The effort of the particle to stretch the thread,
and to move from the centre to the circumference, is what is called the ^tuo/co.
centrifugal force ; therefore the centrifugal force is equal to the square of the
velocity, divided by the radius. [54']
In the motion of a particle upon any curve whatever, the centrifugal force
is equal to the square of the velocity, divided by the radius of curvature of
the curve ; since the infinitely small arch of this curve coincides with the
circumference of the circle of curvature ; we shall therefore have the pressure
of a particle upon the surface it describes, by adding to the square of the [54"]
velocity, divided by the radius of curvature, the pressure arising from the
forces which act upon the particle.
If the curve is situated in a plane surface, tliat sui-face may be taken for the plane of two
of the co-ordinates, as y, z, and tlien x, dx, ddx, may be neglected in [44, 53a], which
will become
d^^dy^ + dz", ''*'
The differential of Jr', supposing always ds constant, gives 0^dy.ddy-}-dz.ddz, whence
ds .ddz
whence V {ddyf^ {d d zf = —jr-- Substituting this in r [536], it becomes
ds.dy
'"""TrfT' [53c]
which will hereafter be used.
44
CENTRIFUGAL FORCE.
[Mec. Cel.
In the motion of a particle upon a surface, the pressure arising from the
centrifugal force, is equal to the square of the velocity, divided by the radius
of curvature of the curve described by the particle, and multiplied by the
[54'"] sine of the inclination of this circle of curvature to the tangential plane of
the surface :* by adding to this pressure what arises from the action of the
other forces acting upon this particle, we shall have the whole pressure of
the particle against the surface.
We have proved [54] that if the particle is not acted upon by any force,
its pressure against the surface is equal to the square of its velocity, divided
by the radius of curvature of the described curve ; the plane of the circle of
[54iv] curvature, that is, the plane which passes through two consecutive points of
the curve described by the particle, is in this case perpendicular to the
surface.f This curve, relative to the surface of the earth, is what is
[546]
* (20) To illustrate this, suppose the particle to move upon
the surface AEDF of a right cone, whose axis is CB, and
to describe, with the velocity v, the circumference of the circle A.^
A E D F, whose radius r = B D is perpendicular to C B.
Continue BD to i?, making DH= — , draw HG per-
pendicular to C D. Then, by [54], while the particle is at
D, moving towards F, its centrifugal force / in the direction
B D, will be represented by /= D H= — , and this may be resolved into two forces
DG, GH, of which the former, being parallel to the side of the cone, produces no pressure,
the latter, GH, represents the actual pressure P=GH. Now if we put J equal to the
inclination of the circle AEDF upon the plane which is a tangent to the conical
surface at -D, it is evident that /= angle B D C= angle G D H. But by trigonometry
GH=DH.sm.GDH, or in symbols P=/. sin./=- .sin./, as above. What is
here said of a cone will evidently apply to any other surface, supposing always the axis B C
to be drawTi through the centre of tlie circle of curvature perpendicular to that circle till it
meet in C tlie plane which is the tangent of the proposed surface at the point B.
t;2
f (20a) Comparing the expression of the pressure [54] — with that in [546], we
get - = - . sin. /, whence 1 =sin. /; tlierefore the inclination /must be a right angle.
I. ii. § 1 0.] PROJECTILES. 46
called the perpendicular to the meridian,* and we have proved in § 8 [49"],
that it is the shortest which can be drawn between two points upon the
surface.
10. Of all the forces which we observe upon the earth, the most remarkable
is that of gravity ; it penetrates the inmost parts of bodies, and were it not
for the resistance of the air, it would cause all bodies to fall with an equal
velocity. Gravity is nearly the same at the greatest heights to which we can
ascend, and at the lowest depths to which we can descend : its direction is
perpendicular to the horizon ; but in the motic«is of projectiles, we may t^'^
suppose, without sensible error, that it is constant, and that it acts in parallel
directions, on account of the shortness of the curves which they describe, in
comparison with the circumference of the earth. These bodies being moved
in a resisting fluid, we shall call € the resistance which they suffer. This
resistance is in the direction of the element ds of the described curve. We
shall also put g for the force of gravity. This being premised,
Let us resume the equation (/) [37] § 7, and suppose that the plane of
X, y, is horizontal, and the origin of z at the highest point ; the force § will
produce, in the directions of the co-ordinates ar, y, Zj the three forcesf
dx dy dz
-^•dl' -^-Ts' -^-dl' [«")
therefore we shall have, by ^ 7 [34"'],
* [20b) The properties of this curve are shown in [1897] &;c.
f (21) This follows from the principle of the decomposition of forces [11]. Tlius, in
the figure page 7, a force 6 in the direction A c, may be resolved into three rectangular
forces, €.~ — )ii,^-~7~j ^•T"? which, by putting, as in [40a], AD = dx, Dd = dy,
dc=dz, Ac=ds, becomes as in [54'^'], the negative sign being prefixed, because the
resistance tends to decrease the co-ordinates. Adding the gravity g to the last of these
forces, we get P, Q, R, [34''], as in [55]. Substituting these in [37], it becomes as in
[56] ; and if the body be wholly free, it will not be necessary to introduce terms like X S u,
X' 8 u', [26], but we must put the coefficients o( S x, S y, S z, separately equal to nothing,
and by that means we shall obtain the three equations [57].
12
46 PROJECTILES. [Mec. Cel.
and the equation (f) [37] will become
If the body is wholly free, we shall have the three equations
at as at as at ds ^
The two first give*
r«cai dy J ^^ dx T dy _
dt dt dt dt
whence by integration, dx=fdy, f being an arbitrary constant quantity.
This is the equation of a horizontal line ; therefore the body moves in a
vertical plane. Taking this plane for that of x, z, we shall have y = 0;
the two equations
[59] 0 = d.~-{-^.^,dt; 0^d.^+^.~.dt — gdt
dt ^ ds ' dt ^ d s ^
will give, by supposing d x constant,!
rflni ^ ds.d^t _ d^z dz .d^t , ^ dz ,, ,,
* (216) Multiplying the first of the equations [57] by — ^, the second by — — , and
dx d 1/
adding these products, we get [58]. If, for brevity, we put —=af, — = y, it becomes
i/ d 3/ of d 1/ ^
\/dc(f — x'dy' = 0 -f dividing by 1/% we obtain ^ ^ = 0, whose mtegral is 7=/j
/ being the constant quantity required to complete the integral. Replacing the values of x', i/,
it becomes -—-=/, or dx=fdy, as above. The mtegral of this equation is x=fy-\-ff
ay
f being another constant quantity. This represents the equation of a horizontal right line,
since the vertical ordinate z does not occur in it, and it corresponds to the first of the
1 f ...
equations [196"], by putting ./2 = — , ^= — —. We may consider this line as the
J J *
vertical projection of the path of the body upon the plane of a?, y ; and, as this projection is
a right line, the body must evidently move in a vertical plane.
f (22) Developing the terms d.-^, ^-TTT' ^" E^^l' ^^ S®^ C^^] ^7 ^ ^^^y
small reduction. The value of § [60] substituted in the second equation [60] gives
gdfi=zd^z[60''\.
I. u. <§10.] PROJECTILES. 47
whence we deduce
taking its differential, we get 2^6?^.<f i = ^2 ; by substituting in it for ^t its
value -r— , [601, and for J ^ its value , we shall have
€ ds.S?z
g ~~ 2.{d^zf
This equation gives the law of the resistance €, which would be necessary to
make the projectile describe any given curve.
If the resistance be proportional to the square of the velocity, § will be
d^
proportional to h . -7-^, h being constant in case the density of the medium
is uniform. We shall then have [60']
6 h . d^ h .d^
g gdt^ d^ z
2d-^z'
d^ z
therefore* h.ds=^ , which gives by integration
* (23) Comparing the two values of —, [61, 62], gives hds=^-—-,ov 2hds = -r^,
whose integral is, (by 59 Int.), 2^5 = log. d^z, to which must be added a constant quantity,
which, for the sake of homogeneity, may be put — log. 2 a .dar^f and we shall then have
2 A « = log. - — —J-. Multiplying the first member by log. c = 1 , it becomes 2hs log. c,
or log. c , hence log. c =log.^-^;-^, >r c =§;^;^-- Multiplying by 2 «,
we get [63]. The integral of this equation has been obtained by putting dz=p dx, whose
differential is d^z = dp.dx, also ds=V dx^ -{-d'if^=dx v\~-\-j^. Substitute these in
[63] multiplied hy dx. VI -f-^ = ds, it becomes dp . V 1 +^ = 2 a . c ' ds, whose
integral is A. {^Kl+^+ log. (p + /l4-p2)}=-.c +-oT> ^^ may be proved by
taking the differential and reducing, h being a constant quantity added to complete the
mtegral. Substitute in this the value of — . c deduced from the preceding equation [63a],
[61]
[e2]
d'^Z ^ 2A,
^^ = 2a.c , [63]
[63a]
48 PROJECTILES. [Mec. Cel.
a being an arbitrary constant quantity, and c the number whose hyperbolic
logarithm is unity. If the resistance of the medium be supposed nothing, or
^ = 0 ; we shall have, by integration, the equation of the parabola*
[64] z = ax^-\-hx-{-e\
b, c, being arbitrary constant quantities.
The differential equation <f 2: = ^ <?^^ will givef df = — .da^, whence
we deduce t=x.\/ \-f. Suppose that x, 2, and ^, commence
together, we shall have c=0, /' = 0, conseqently
[65]
^ /2 a „ ,
t=x.\/ — ; z = ayr-\-bx\
.... . , a 2As dp.\/l4-p^ dp.[/l4-p^ dp
wluch gives successively, - c = \^, — ; = =^, . , we get
dx=. '^
in which the variable quantities are separated, and we may, by the usual methods, obtain its
integral, and we shall thus have x in terms of p. This value of cZ a; being substituted in
d z =p dXf ds = dx. v\ -f-p^ , will also, by integration, give z and s in terms of p, and by
means of the quantity p, the path of the trajectory may be determined.
dz
[64a] * (24) Put A = 0 in [63] and it becomes <^2: = 2a.<?a?^, or J.— =2«.rfa?, whose
integral is ■^- = 2aa? + &, whence d z^=2 a x d xA-l d x. Again integrating we obtain
dx
[64]. If we alter the origin of the co-ordinates, putting ^r = 2/ -}- ^ — i • —•> and
a; = a;' — —, the equation [64] becomes sZ + e — l,~—z=a\x' — — j +^f«' — 2^)^"^'
[645] which, by developing and reducing, becomes :!! = ax'^j the well known equation of a
parabola.
f (25) Putting the values oid^z [60', 64a] equal to each other, we get gdf=2a.daP,
or dt = dx, \/ — , whose integral is t = x. \/ — +/» or, as in [65],
t=x.y/^, whence x=t.^^^.
This value of a?, substituted in z [65], gives [66].
I.ii.<^10.] PROJECTILES, 49
whence
2 V 2a
[66]
These three equations contain the whole theory of projectiles in a vacuum ;
and it follows from them that the velocity in a horizontal direction is uniform,
and the vertical velocity is the same as would be acquired by the body falling [66'J
down the vertical.*
If the body fall from a state of rest, h will vanish, and we shall have
Motion of
, , _ bodies fall-
the acquired velocity therefore increases in proportion to the time, and the ^^^J'^^
space increases as the square of the time. '^'^"^'
It is easy, by means of these formulas, to compare the centrifugal force gafrorce
with that of gravity. We have shown before, [54'], that v being the velocity ^^j^t^^^
of a body moving in the circumference of a circle whose radius is r, the
centrifugal force will be — . Let h be the height from which the [^'1
body ought to fall to acquire the velocity v ; we shall have, by what
(25a) The horizontal velocity is evidently denoted by — ; and the differential of the first of
dx
d X M X fir
the equations [65] gives — = \/ — , which is constant because g, 2a are given quantities.
dz
The vertical velocity is evidently — , which we shall caU » j and the differential of [66],
dz . . I 1 i / S
gives ~{==^v)=gt-\-b. 1/ — . If the projected velocity, resolved in a vertical [G7a]
direction, be v\ the preceding equation, when <=: 0, will become v' ^h\ / —, conse-
quently «=^<-|-i;'; and if this projected velocity «' be given, the vertical velocity v will [676]
be the same at the end of a given time t, whether the body be projected obliquely or
vertically, supposing the vertical velocity of projection v' to be the same in both cases. If
r' = 0, the preceding value of v' = b.y/~^, gives 6 = 0, a, g-, being finite. This
value of hj being substituted in [66, 67a], gives [67.]
13
60 PENDULUM. [Mec. Cel.
precedes,* v^=:z'2.gh, hence— =^.—. If ^ = ^r, the centrifugal force
becomes equal to gravity g ; therefore if a heavy body be attached to one
end of a thread, and the other end be fixed to a horizontal plane, the tension
of the thread w^ill be the same as if it v^^ere suspended vertically, provided
that the body be made to move upon the plane, with the velocity it would
t^'i have acquired in falling from a height equal to half the length of the
thread.
Motion of 1 1 • Let us now consider the motion of a heavy body upon a spherical
sXrkai surface. Putting r for the radius of the spherical surface, and fixing the
surface. Qriglu of thc co-ordluatcs x, y, z, at its centre, we shall have [19e]
[67'"] r^ — 0^ — / — 2^ = 0;
this equation, compared with w = 0, gives u = 7^ — s^ — if — z^ ; adding
therefore to the equation (/) § 7 [37] the function S u, multiplied by the
indeterminate quantity f — >.dt, we shall have
[68]
0^8x. i d.—^^XxJi l+SyA d.^+2'Ky.dt l+SzA d.~-[-2\z.dt—gdt I ,
in which equation we may put the coefficients of Sx, 6y, Sz, separately equal
to nothing [26'], which will give the three following equations :
dt
[69] 0 = cZ.^+2xi/.(i^; > {A)
0==d.~+2>.z.dt—gdt.
The indeterminate quantity x shows the pressure which the moving body
V
* (25&) When t?' = 0, we shall have v=gt [67J], whence ^ = -, which, being
[67c] substituted in 2; [67], gives 2; = ^.-, 'or 2gz==^iP, and by changing z into A, it
becomes as above,
f (25c) In the same manner as in note (19J) ; observing also that P = 0, Q=0, and
I. ii.<§ll.] PENDULUM. 51
exerts against the surface. This pressure is by ^ 9 [50], equal to
consequently it is equal to 2 x r.* Now by § 8 we havef
c-\-2gz= — --jf^ — ' [70]
c being an arbitrary constant quantity; by adding this equation, to the
equations {A) divided by d t, and multiplied respectively by x, y, z ; then
observing that the differential equation of the surface, is 0 = xdx-]-ydy-\-zdZt
we shall havet
0 = xddx-{-yddy+zddz-\-da^-{-df + dz^; [71]
we shall find§
^ c + Sfi-z
If we multiply the first of the equations (A) [69] by — y, and the
[72]
* (26) The pressure is X . ^ (^£j + (^)V {^J, [50], and by [25«], the
terms under the radical are equal to 2 r, therefore the pressure is X . 2 r, or 2 . X r, as
above.
f (27) This equation is the same as [39], putting P = 0, Q = 0, R=g^ whence
2.fRdz = 2gz.
J (27a) The differential of the equations of the surface 0 = oc^-\-if'-]-z^ — r^ [l^e]
gives 0 = xdx-\-ydy-\-zdz, and the differential of this is as in [71]; dividing by d ^, it
ddx , ddy , ddz , d3^+dy^-\-dz^ ,., . , . ,
becomes 0 = x .-—:r-jry'~r:r-r^ '-r^rn t^ > which is used m the next
. dt^ ' ^ df^ ^ rf<2 ' rff2
note.
^ (276) The sum, found by addmg [70] to the equations [69] multiplied respectively by
X y z .
dt dt dt
I ^ ddx , ddy , ddz , ,„ / o ■ « • nv . dx^4-dv^+dz^
From which, subtracting the equation at the end of the last note, it becomes
c-{.2gz = 2>.{o^+f + z^)-gz, j72„3
or c + 2gz = 2\r^, because ar^-]-f-\-z^ = r\ [67'"] . This, divided by r, gives [72.]
52 PENDULUM. [Mec. Cel.
second by a-, we shall have, by adding these products, and mtegrating their
sum,*
xdy — ydx
[73] ^ ^ ^JL_=.,';
c' being another arbitrary constant quantity.
The motion of the point is therefore reduced to these three differential
equations of the first order,
xdx~\-ydy^= — zdz\
ry^l xdy — ydx = c'dt;
Squaring each of the two first equations, and adding them together,
we get
[75] {x' + y^).{dx'-\-df) = c'^df-{-z'dz';
if we substitute, instead of x^'-{-y^, its value r^ — z^ [67'"], and, instead of
doc^-\- difi d z^
— -j-^-^i its value c-^2gz — -— -, [70], we shall have, by supposing that
the body recedes from the vertical,!
[76] dt =
[77]
\/ (^r'' — z^).{c + '2gz) — d''
the function under the radical, may be put under the form
[76'] {a — z).(z—b).(2gz+f),
«, 6, /, being determined by the equationsj
2g.{r^ + ab)^
_2g.{r'' — a^—ab — b^)
a + 6
jt z^- \iTii. • ^ 1 dy , dx ^ xddy — yddx .
* (27c} 1 his sum is 0 = x .d .-j y ' d . -—, or 0 = ~Jr j whose integral
. , xdy— ydx .
IS cf = — —~ — , as in [73].
f (27c/) This requires that z should decrease while t increases, consequently, d t being
supposed positive, dz must be supposed negative, as in [76].
J (28) The expression [76] gives the velocity in tlie direction of tlie vertical x, or
I. ii. §11.] PENDULUM. 63
We may therefore substitute for the constant quantities c and c', other
constant quantities a and 6, of which the first is the greatest value of z, and [77']
the second is its least value.* Then puttingf
sin. ^= IX 7, [78]
_. = — — )-v^T- g^J ^ -^^^ ^jg velocity must evidently be nothing at the
dt r
highest and lowest points of the curve, at which points we shall suppose the values of z to be
d z
b and a respectively : therefore z= h, and z = a, ought to make — equal to nothing,
dz
consequently these values ought to malie the numerator of that value of — equal to nothing ;
so that «, 5, must be roots of the equation (r^ — z^) . {c -{- 2 g z) — c'^= 0 ; tlierefore,
a — z = 0, and z — 6 = 0, must be factors of that equation, and by division we shall find,
that the other factor must be of the form 2 g" «-{-/= 0, so that
(r^-z^). {c + 2gz)-d^ = {a-z) . (z-b) . (2^^+/).
Developing botli sides of this equation, and putting the coefficients of z% z and the constant
terms equal to each other, we obtain the three following equations : — c=2g. {a-\-b) — f;
2r^g ==/. {a-{-b) — 2 gab; r*'c — cf^ = — fa b. The value of /, deduced from the
second of these, gives the first of the equations [77]. Substituting this in* the first of the
preceding equations, and changing the signs of all the terms we obtain
which, by reducing to tlie common denominator a-{-b, gives the second of the equations [77].
Lastly, by substituting tlie values of c, /, in the last of the preceding equations
r^ c — (/^ = — fab, or €f^ = r^c-^fab,
it becomes c'^ = -^^. j (r^ — a^— ab — b^) . r^- -{- {r^ + ab) . abl , which, by a small
reduction, is easily put under the same form as the third equation [77].
* (28a) This is proved in the preceding note.
f (29) This expression squared and reduced, gives
z = a — {a — b) sin.^d = a{l — sm.^ 6) -\- b sm.^ 6 = a . cos.^ 6 -\- b sin? 6 [78o]
as in [80]. The diflferential of tlie first value of z gives — J;s= 2 (a — b). sin. 6 . cos. 6 . d6.
The same equation also gives
a — z=(a — i) .sin.^d; z — b = (a—b) . {I —sin.^6) = {a — b) .cos.^6 ;
2gz-i-f=2g.{a—{a — b)s\n.^6]-\-f
14
54 PENDULUM. [Mec. Cel.
the preceding differential will become
[79] dt- rV2.{a+h) d^
[79']
V/^gTf(a + &)2 + 7^— 62 j v/i— ^sin.^2
by putting
^2_ gg— 6^
The angle ^ gives the ordinate z by means of the equation [78a],
t^^l 2: = a . cos.^ ^ + h sin.^ ^,
and this ordinate 2, divided by r, gives the cosine of the angle which the
radius r makes with the vertical. Let ^ be the angle which the vertical
plane passing through the radius r, makes with the vertical plane passing
through the axis of x ; we shall have*^
^®^^ a; = V/"^^^T^ COS. ^ ; 3/ = v/'i^^"^ . sin. « ;
from which we getf
[82] xdy — ydx=(r^ — 2^).<?tz(;
Substituting in this last, the value of / [77] and reducing, it becomes
O 1^0 («+6)2+r2— 62_(a2_59).sin.24 (« -|- fc)2 -|. ^ — fcS .
2^«+/=2^. '--^ — p^ '- =2g. ^^^ .{\—fsm?&).
Hence the factor of — rdz [76] which was assumed, in [76'], equal to
1
V/(a_2).(2-6).(2^z+/),
becomes
V/2:{a + b)
2.(a — 6).sin. ^.cos. d.v/g"-{{a + *F + ^^ — &^} . V/l— T^sin-S^,
this multiplied by the preceding value of — tZsr and by r gives ^^ ['^9].
* (29a) In the figure page 7, let .^ be the centre of the sphere and c a point of its
circumference, whose co-ordmates 2uce AD = x, Dd = y, dc= z,we shall evidently have
in the rectangular triangle Adc,Ad^=A(P — d(?,OT:Jld= Vr^ — z^ ; this is the quantity
called p in [27], and by substituting this for p it gives [81], observing that in this case
■a = angle DAd.
f (30) This expression is found most easily by dividmg the value of y [81] by x, which
gives -=^ tang. Trf, whose differential is — ^—^ — = — —. This multiplied by the square
of a; [81], that is, by 0?== (r^ — z^) . cos-^-itf, produces the expression [82].
I. ii. § 1 1 .] PENDULUM. 5^
the equation xdy —ydx^ c' dt [74] , will therefore give
by substituting for z and d t, their preceding values in 6, we shall have the
angle zs in a function of 6 ; and we may thence obtain, at any time, the two
angles 6, *, which are sufficient to determine the position of the moving
body.*
* (30a) These angles 6 ondzi may be obtained, very easily, by means of the Tables of
elliptical integrals, computed by Le Gendre, and published in the third volume of his
Exercices de calcul integral, in which he uses the following abridged symbols.
6=V/l— c2 ; A(c,<p) = V/l— c2.sin.2(p;
A(6,<p) = V/l-&^sin.^<p ^{^^<^)=f^y [82a]
E (c, 9) =/fZ (? . A (c, (p) ; n (n, c, ?) =y ^
A(c,9)'
d(p.A(c,(p)
-f- n . sin.2 <p
The functions F, E, IT, being called by him elliptical functions of the Jirst, second, and third
species, and when these integrals are taken bet^veen the limits 9=0 and 9=:90'', they are
denoted by F' (c), E' (c), n' (n, c) respectively. The values of the functions F, E, have [826]
been computed for each degree of the arch 9, from 0 to 90** , and for the various values of
the modulus c, from 0 to 1. By means of these tables all calculations of these mtegrals are
much facilitated.
r.\/2.(g+&)
Putting for brevity ; — ^Z:^^, changing also fl into 9, and y into c, to
^9
conform to Le Gendre's notation; the formula [791 will become, dt=h-— — -, whose
integral is
t = h.F{c.(p), [82c]
tiierefore, when t is given, we may obtain 9, (or 6 of La Place), by means of the tables of the
integrals of the first species.
Substituting this value of J < in [83], and observing that — — - may be put under the form
1(1,1-) - (^ ( 1 , 1 } hd(p
now z =: a — (a — b) . sin.^ 9 [78a], hence
da=^.5--r 1 - + I ].±^
2r (r-j-a— (a — 6).sin.29 ' r— a+(a — 6). sin.2 9 > A(c,9)
56 PENDULUM. [Mec. Cel.
Time of ^6 shall call a half-oscillation of the body the time it takes to pass from
3Er ^^® greatest to the least value of z ; which time we shall put equal to ^ T,
.urfaco. To compute it, we must integrate the preceding value of dt from ^==0 to
and if we put n = - ^-^\ n' = ^i, ^ = _J±__ ^ _ __^__ -,. becomes
r-f-a' r — a' 2r.(r+a)' '" ^r.{r^^' " Oecomes
eZ« = m.--, ,4? + W. ^
(l + n.sm.2(p).A(c,(p) ' (l+n'.sin.s^). A(c,<?) '
whose integral is
f 82rfJ * =?= »i . n (w, c, 9) -f m' . n (n', c, <p).
Therefore ^ may be found by means of two integrals of the third species.
In a semi-vibration of the pendulum 9 varies from 0 to 90*^ [S3a], and if we suppose -a/ to
be the value of ^, corresponding to 9 =90'' , the integral commencing with 9 = 0 we shall
have
[82^1 ^ = OT . n' («, c) + m' . 11' (n', c).
These definite integrals of the third species may be reduced to those of the /r*^ and second
species, by the formulas in pages 137, 141, Vol. I, of Le Gendre's work.
Thus, by putting w = — 1 -}- ^^ • sin. ^'^ , we have by formula (m') page 141,
[82/] n'(n,c) = r{c) +^i^- . ^ l^Y\c) . F(&, ^')~E'(c) .Y{U>)^Y'{c) .F(6,^) \ ,
and by putting w' = cot.^ ^ in formula (A/) page 1 37 of the same work, we shall have
[8%] n(<c)=^i^°|^.^|+!^.A(^^).F(c)+F(c).F(6^^^^ | .
These, being substituted in •zs^, give its value in functions of the first and second species,
which may be found from the Tables.
John Bernoulli, in Vol. Ill, page 171 of his works, remarks that the motion of a pendulum
of this kind, in which a and h differ but little from each other, and neither of them vary much
from the whole length of the thread, by which the body is suspended, may be made to
represent, beautifully, the progressive motion of the moon's apsides. For the projection of
the patli of the body upon the horizontal plane will be nearly an ellipsis ; and in the time 2 T
of a double oscillation, the longer axis of this ellipsis will have passed over an arch, which is
equal to four times the excess of the angle z^ above a right angle ; so that in every successive
vibration of the pendulum at the arrival of the body at its highest point, corresponding to the
extremities of the longer axis, this axis will have moved about the centre of the ellipsis, in a
manner wholly similar to the progressive motion, observed in the moon's apsides, in the
successive revolutions of that body about the earth.
I. ii. §11.] PENDULUM. 67
6 =z-iir* ir being the semi-circumference of a circle whose radius is unity,
we shall by this means findf
* (31) These limits are easily deduced from [78], by substituting in it the greatest and
least values of z, which are a, h, [77'] ; for the values of sm. 6 [78] corresponding are
1/ — — - = 0, and 1/ r = J , which give, at the limits, ^ = 0, and 6 = ^ if.
f (32) By developmg the radical /- — — - . ^ , [79], by means of the binomial
theorem, we shall have
[83a]
dt= ^V/2.(a + 6)
The integral of this, taken between the limits ^ = 0, 5 = J- -r, gives J T. This mtegral may integral
be found by substituting the values of sin.^^, sin.^ 6, &;c. (Int. Form. 1, 2, &tc.) or by the
following formula,
fd6. sin." 6=—-. cos. 6 . sin.«-M + 'H—i-.fd & . sm."-^ ^ ; [84a]
which is easily proved to be correct by taking the differential of tlie whole, and reducing by
means of cos.^ 4=1 — sm.^ 4, n being any number whatever. Now at the limits when
4 = 0, or 4 = 1*, the term without the sign / generally becomes nothing, and if we take
the integrals with those limits we shall have, n being an integer greater than 1 ,
n 1
/<Z4. sin.'*4= /fZa.sin.«-2 4 ; [845]
and Sis fd6 = — , we shall have, when n = 2,
fd6.sm.^6 = ±fd6=l.'^ [84c]
If n = 4, the preceding formula gives fd6. sin.^ 6 = ^.fd6. sin.^ 6 ; substituting the value
of /df 4 . sin.~ 4, it becomes
fd6.sm.U = ^^.^. [84^
Putting n = 6, the formula gives fd 6 . sin.^4 = ^.fdd. sm.'* 4, and, by using the value of
fd&. sin.^ 4, it becomes
/rf«.sin.V=^.J, to. [8^,
16
58 PENDULUM. [Mec. Cel.
Suppose the point to be suspended from the extremity of a thread without
mass, the other extremity being fixed. If the length of the thread is r, the point
will be moved exactly as if it was upon the interior of a spherical surface ;
Pendulum, and it will in this manner form a pendulum, in which the greatest deviation
from the vertical will be measured by an angle whose cosine is -. If we
suppose that when it is in this situation, the velocity of the point is nothing ;
it will oscillate in a vertical plane, and we shall have, in this case, « = r,
7^ = —^ — •* The fraction — — is the square of the sine of half the greatest
angle which the pendulum makes with the vertical ; the whole duration T
of the oscillation of the pendulum will therefore be
,. T=^.v/=;.i •+GT-(^)+GT:)X'i^)V(^:;.(9^)'+s.. ].
Time of 7
of asfrnpie If thc oscillatiou is very small, -— — will be a very small fraction, which
pendulum, 2 T
whose
'^^f^l'tjf ' may be neglected, and then we shall have
beiug g.
The law of continuation being manifest. Substituting these in the preceding integral of d t
representing J- T, it becomes '
•-;;=:.l=^5=-S;+G)-:-+G-:?)'''HGS)'-''l+'-i
whence we can easily deduce the value of T [84].
* (33) The value a ;= r being substituted in the expression of f [79'J, it becomes
r^—^ 1, J 1 • r2_&2 - r2 — 62 r—h , Now the
(,+,). + ,. ^feF» orbydevelopmg, ^^^Tft " i7. FF^ " 2 r ' ^^ ^^"^^•
cosine of tlie greatest angle which the pendulum makes with the vertical being -, its versed
gine is — , and (by 1 Int.) the square of the sme of half this angle is equal to the half of
r
this versed sme or—. Substituting these values of a and 7 in [84] we obtain [85],
^ T
/ 2r (a + 6) t /^2rr-|-2r6 ,. , .
observing that the factor \/^T^z:^ becomes \/ ^T^^^Tb' '"^^"^ ''
equal to 1 .
I. ii. <512.] TAUTOCHRONOUS CURVE. 69
very small oscillations are therefore isochronal, or of the same duration,
whatever be the length of the described arch ; and we may, by means of
this duration, and of the corresponding length of the pendulum, determine
the variations of the intensity of gravity, in different places of the earth.
Let z be the height through which a body would fall by the force of
gravity in the time T\ we shall have, by ^ 10 [67],* 2z = gT^, conse-
quently 2: = J^ -71^ , r ; we shall therefore have with great precision, by means [86^
of the length of a pendulum vibrating in a second, the space through which
gravity would cause a body to fall in the first second of its descent. It has
been found, by very exact experiments, that the length of such a pendulum,
vibrating in a second, is the same, whatever be the substance of the oscillating
body ; hence it follows that gravity acts equally upon all bodies, and that in
the same place, it tends to impress upon them the same velocity, in the same
time.
12. The oscillations of a pendulum not being perfectly isochronal, it is Tamo-
interesting to know the curve upon which a heavy body ought to move, curve,
to arrive in the same time at the point where its motion ceases, whatever
may be the length of the arch which it shall describe from its lowest point.
To solve this problem in the most general manner, we shall suppose,
conformably to what really takes place in nature, that the body moves in a
resisting medium. Let s be the arch described from the lowest point of the
curve ; z the vertical absciss counted from that point ; d t the element of the
time, and g the force of gravity. The retarding forces along the arch of the
curve will be ; First, gravity resolved in the direction of the arch ds, and which
is therefore equal tof ^ . — - ; Second, the resistance of the medium, which we [86"]
* (34) This is deduced from [67], by writing Tfor t. Substituting the value of T [86]
we obtain 2; = ^ ir^ r.
f (34a) The cosine of tlie angle formed by the elements ds, dz, [406] is
•-— = COS. cAB, (Fig. page 7.)
and the force of gravity g, acting in the vertical dz, is resolved in a direction along the arc ds,
by multiplying it by this cosine, hence it becomes g . — [86"].
60 TAUTOCHRONOUS CURVE. [Mec. Cel.
shall denote by <p(-^)j -t: being the velocity of the body, and 9(^)»
any function whatever of this velocity. The differential of this velocity*
will be by § 7 equal to — ^ . _f . <? ^ — . 9 (~\ .dt; we shall therefore have,
supposing dt constant,
^ dds , dz , /d s\ , ..
df ' ^ ds ' \dt
* (35) The forces of gravity and resistance, computed as above, give the whole force P',
d z / ds \
acting in the direction of the element ds of the curve, hence P' = — g . cp ( y^ ) ;
the negative sign bemg given to the terms, because the forces tend to decrease that element.
Now from the formula [381 we have — — - = P, or d ,--— = P dt, in which —— is the
^ -' dt^ dt dt
velocity in the direction of the element d x, and P is the force in that direction. This
equation would take place, whatever be the direction of the arbitrary axis a?, and we may
assume generally, that the increment of velocity in any direction is equal to the force in that
ds .
direction, multiplied by the element of the time ; and as -j- is the velocity in the direction
d s dds
d s, we shall have d . -r- = P' d t, or 0 = -j^ P', which, by substituting the preceding
value of P', becomes as in [87].
(35a) We may apply formula [87] to the computation of the velocity of a body projected
directly upwards, along an inclined plane, which will be wanted in note 39. In this case the
curve becomes a right line inclined to the horizon by a given angle /, and we shall evidently
have -7— = sin. J, so that if we, for brevity, put g' = g > sin. /, we shall get
^.^ = ^.sin. J=^, therefore 0 = — ^ + ^ +(p(^j, [87].
If we suppose the resistance to be notliing, it will become 0 ="T7r"^^'°'^~7rr ^^ — <§^^^»
whose integral is — = v' — g' t ; v' being the constant quantity added to complete the
integral, and as -,— represents the velocity v of the body, we shall have v= v' — g' t, hence
it is evident that v' will be the initial velocity of the body when ^ = 0. If it be required to
find the time T, in which the whole motion v will cease, we must put, m the preceding
equation, f = T, and v = 0, and we shall have, 0=v' —g' T, hence T=^. Consequently
[87a] die time T, m which the whole motion would be destroyed, is directly proportional to the
initial velocity v'.
JV
I.ii. <^12.] TAUTOCHRONOUS CURVE, 61
Suppose
<^)=-^+"-'-$^* ^=+(^)= '^'
put also 4^' (5') for the diflferential of 4. (5') divided by d s', and 4^" (5') for the
differential of ^^'(5') divided by ds' \ then we shall have
ds ds' , ,.
^"^^ ^ [88]
dds dds' ,, , ,x , dsl^ w' / /\
and the equations {i) [87] will becomef
^ dds' ds' ds'^ i^"(s')-\-n.\-\>'{s')Y } . g.dz ,,.
ds'^
the term multiplied by -7^ may be made to disappear by means of the
following equation
0 = 4."(5') + n.[+'(5')r; [90]
from which by integration we obtainf
4.(5')=l0g.|^.(5' + ^)i|=5,
* (36) The solution is given for a much more general form of resistance in [103].
f (37). Take the first and second differentials of tlie assumed value 5 = 4> {s') [87'], and
d s dds
divide them respectively by dt, df, we shall obtain — -, —r-^, [8S]. The assumed form
°^ '^ dJ ^^^^ changes the equation [87] into 0 = -^ +^. — + m.~ + n. —-.
d 3 dds
Substitute in this the values of — , -j^ [88], and divide by 4^' (*')' "^^ shall obtain the
formula [89].
J (38) Transposing tlie second term of the assumed formula [90], and multiplying by
d s' Jj" (s'] . d s'
— - it becomes ' =i — n . ■^' {s') . ds', whose integral is (59 Int.)
log. 4-' {s') == — 71 . 4^ (*') + constant.
Multiplying the first term of the second member by log. c = 1 , and putting the constant term
A"
equal to log. - we shall get
[911
1///N ^'^ — n4(«0
^ {') = - -C ""' ^ [876]
16
62 TAUTOCHRONOUS CURVE. [Mec. Cel
h and q being arbitrary constant quantities. If we suppose 5 and s' to
[9l'j commence together, we shall have hq ^ =1 ; and if, for greater simplicity,
we put A= 1, we shall have
[92] 5' = c"' — 1,
c being the number whose hyperbolic logarithm is unity : the differential
equation (I) [89] then becomes
[93, o='S'+--^+»'^-^-a+^T.
Supposing 5' to be very small, we can develop the last term of this
equation, in a series ascending according to the powers of 5', which will be
[93'] of this form* ks' -\-l s''-\- &c. ; i being greater than unity ; and then this
Multiply this by n c^^^' ds', and it becomes c" "^ ^* ^ . n 4-' (s') .ds' ^hJ'd /, whose
integral is c** ' ' = k^ {s' -\- q) ; h^q being the constant quantity, added to complete the
integral. Extracting die root n, we obtain c^^^' = h{s'-\-q)'', whose logarithm gives
4, (s') [91]. If for 4 (s') we substitute its assumed value s [87'], the preceding expression
1
will become (f = h{s' -}-q)''. Now by h}^othesis [91'], s' = 0 when s = 0 ; these values
1
being substituted in this equation, we get 1 = Ag"*, and if we suppose A = 1, it will make
1
q = l, therefore the preceding equation will be c =(5' + !)" ore ==s' + l> whence
y = c""* _ 1 , [92]. The same value of A = 1 makes [87&] become 4' (s') == 1- . c"~" "^ ^* \
and by substituting in the second member for 4 (s') its value s [91], it becomes
[87c] ■^'(s') = -.r''' = -^=—^, [92],
and this, being substituted in tlie last term of [89], changes it into [93], neglecting the
coefficient of -^ ; which vanishes by reason of the assumed equation [90].
■ * (39) The slightest attention will make it evident that the tautochronous curve, whose
horizontal ordinate is y, and vertical ordinate z, both Commencing at the lowest point of the curve,
dz
as their origin, must, at that point, be parallel to the horizon ; or in symbols, that -^ = Oj when
^ = 0, y = 0, s == 0. Moreover, the radius of curvature r= -^~ [53c] must at that
I. ii. §12.] TAUTOCHRONOUS CURVE. ^^
equation becomes
dfi * dt ^ '
point be a finite quantity. For if the curve, at its lowest point, be inclined to the axis of y,
the body would at first move on an inclined plane, and if projected successively with
infinitely small but different velocities, the times would, by [87a], be as the initial velocities
nearly, and therefore not equal as the problem requires. But at the lowest point of the
curve, where -^ ^ 0, we have by [536] d^^dy''-\-dz^=^df or ds^dy, hence
1 ft /f 7
— = , which must at the lowest pomt be a finite quantity ; since, if r were infinite, the [94a]
curve would become a horizontal line at its lowest part, and if r = 0, the curvature would be
infinite, which could not be the case for this curve. Suppose now the general value of —
for any point whatever of the curve to be denoted by — = a . 5" -}- ^ • ^ + 7 • 5* + &ic. in [946]
which the terms are arranged according to the magnitudes of the exponents a, 6, &c. a being
dz
less than &, S less than c, &tc. Then since at the lowest point — = 0 and s = 0 ; a,b,Cy &ic.
dz .
must in general be positive. Again, this assumed value of - — gives, by taking its differential,
(dshems constant), ^— - = a . a . s^—^ -f- § . 6 . s^~^-4- Stc. Now to make this finite when
s = 0, as is required above [94a] it is necessary to put a = 1 . For if a ^ 1 , it becomes 0,
and if a <1 1, it becomes -, when s = 0. Therefore we shall have [94&],
-_ = a.5 + €.s'4-y.5*4-&C. [94c]
b, c, he. being greater than unity.
Again, c**=l -\-^ [92], its logarithm, divided by n, is 5^— . < s' — ^s'^-\-k,c. >
d^
(55 Int.), its differential gives ds = — (1 — s' -\- he.) Substitute tliese values of s and d s
in [94c] and we shall obtain an expression of the form — ■ = a' .s' -\-^' . s" &c. in which the
exponent of the first term «' of the second member is unity, and the other exponents exceed
unity. Lastly, if we multiply the expression by n^ ^ ( 1 -f- «')^ we shall obtain, for
a similar expression, of the form Ic ^ -\-ls''-}- he. i being greater than unity, which is the
form assumed in [93'].
[95]
[95bl
^ TAUTOCHRONOUS CURVE. [Mec. Cel.
m t r \
This equation multiplied by c 2 . ) cos, 7 ^ + \/ — 1 • sin. yt > , and then
^^"^ integrated, becomes, by supposing 7 =\/ k — ,*
mt r ^ C ds' / m / \ )
c 2 .|cos.7^+v/ — l.sin.7^^.|-^+(^— — 7.V/--1J.5'^
/m t r \
s'\dt.c~^. < cos. 'yt-\-V — 1 . sin. yt > — &c.
By comparing together the real and the imaginary parts of this equation,
* (40) Having an equation of the form
dds' , ds', J , , ^
[95a] 0 = -— +m.— + ^.' + Q,
mt 1/™^
if we multiply it by c 2 ^ 4 ^t^[i ^in become,
The integral of the first member is
•^W—k
4
C
^s is easily proved by taking its differential, which is
or, by reduction,
^+'^?-*r^^+(i-\/"-*)''^i'
,^+<v/"-;
Hence we have the integral of [95a]
Now if for \X ? — ^ we substitute its value \/—l'\/ ^— ^ =7 • V''— ^» [^^^'l ^<^
for / its value cos. yt-{- \/^^ sin. 7 <, [13 bit.], we shall obtain [95], observing
that Qiacludesthe terms Z, &ic.
0 = 1 ,f^\ dt,c ^ . sin. yt-{- &c.
the integral being taken from ^ = 0, to t=T. Supposing 5' to be very
* (41) This expression is deduced from that part of [95] connected with \/ — 1 , by
putting it equal to nothing and dividing it by {/ — 1.
f (42) The exponent i being greater than unity, [93'], the term s" will be infinitely less
than /, if ^ be supposed infinitely small ; therefore, the second member of [97] must be
nothing in comparison with the first ; hence
c V
mT
Dividing this by the factor c » s', we get [98], and this, divided by ^ cos. 7 T, gives
T
5'. J — . sin. 7 T — y . cos. 7 T > = 0.
0==tang.7T-^,[99].
17
[96]
I. ii. <^12.] TAUTOCHRONOUS CURVE. ^^
ds'
we shall have two equations, by means of which we may exterminate -j- ;
but it will suffice here to consider the following :*
c 2 .— .sm./^+c 2 ,s . < -.sm.yt — y.cos.yt > = — l.Js^dt.c 2 sm.yt — oic.
the integrals of the second member being supposed to commence with t.
Put T equal to the value of t at the instant the motion ceases, when
d s
— - = 0 ; we shall have, at that time,
at
C2 .5'.<- sin. 7T — 7.cos.7T'>= — l.fs''dt.c 2 .sin. 7^ — &c.
If we suppose s' to be infinitely small, the second member of this equation
will vanish, in comparison with the first,t and we shall have
0=-. sin. 77^ — 7.cos.yT'; [98]
whence we deduce
tang.7r=^; [99]
and as the time T is, by supposition, independent of the arch passed over,
this value of tang. 7 T takes place for any arch whatever ; we shall therefore
have for all values of 5',
[97]
[100]
66 CYCLOID. [Mec. Cel.
small, this equation will be reduced to its first term, which cannot be satisfied
m t
except by putting / = 0; for the factor c~2~.sin.7^ being always positive
from ^ = 0 to t=T, the preceding equation is necessarily positive* in that
interval. The curve cannot therefore be tautochronous except we have
dz
^.^.(l+s'7 = &s';
Equation
of the i«i« r t • /•
ehfo"nou8 which gives for the equation of the tautochronous curvef
Curve. 7 7
[102] gdz = ^^.n—c-^^).
In a vacuum, and when the resistance is proportional to the simple power of
the velocity, n is nothing, and the preceding expression becomes the same as
Cycloid, the equation of a cycloid, J
[lOU"] gdzz=ks ds,
mt
— ■■ - w J
* (43) The part c ^ is evidently positive for all real values of -^, we have therefore
only to examine the sign of the term sin. y t. Now from [99, 94'] we have
[100a] ^^"S->'^ = ^=V/ ^~^-
To render this expression of tang. 7 T possible, it is necessary that the unknown quantity 4 k
should be positive and equal to, or exceed, the known quantity m^ ; and if we suppose 4 /fc to
be increased from w^ to 00 , its sign would always remain positive, so that 7 t would never
exceed a right angle. Hence we easily perceive tliat 7 1 must be less than a right angle, and
m t
its sign must therefore be positive, consequently c ^ . sin. 7 1 must be positive. Therefore
TO t
the equation 0 = lfs'^.dt.c^ . sin. 7^-f-^c. [100] cannot be satisfied except by putting
Z = 0. In the same manner we may prove any other following term of the series
dz
k s' -{- 1 . s' -{- &£c., assumed in [93'] for n^g • jr • (1 + ^'T to be nothing, so that we shall
have, as m [101], n^g'^,'{^+ «T = ^«'-
f (44) Substituting m [101] the value of s' [92] we get
w^ ff . —^- .(?"*== ;fc . (c"* — 1 ) .
° n c^ d s ^ '
IMultiplying by — '■ , and reducing, it becomes gdz = (1 — c""*) as in [102].
" . " T
d s d s'^
1(45) The general expression of the resistance assumed in [S7'] is ^'Tr + ^'T^'
If we suppose the term depending on the second power of the velocity to be nothing, it vdll
I.ii.^12.] CYCLOID. 67
It is remarkable that the coefficient n of the part of the resistance,
proportional to the square of the velocity, does not enter into the expression
of the time T* and it is evident, by the preceding analysis, that this
make » = 0. Now for all values of n we have c~^= 1 — n s -\- ^ . rP' s*- -{- &tc. (56 Int.),
1 C — Its
hence =s — ^n^-\- &c. ; the second member of which becomes s when n = 0,
n
and the formula [102] becomes, in tliis case, g d z = ks ds, whose integral, supposmg
z and s to begin together, is gz=-^ksr^. This, as we shall soon show, is the equation of a
ler oi wxiose geiierauiig uucie is ; " '' ^ "
equation becomes
cycloid, the diameter of whose generating circle is r-r-; and by putting -— = 2r, this
Qrz = sK [102a]
A Cycloid is a curve GBfU, formed by jj a
the motion of a point of the circumference of
a circle bfa, while it rolls on the straight line
e F
G
GAH as a base. Tliis moving point falls on the ^
base at H and G, aiid is at its greatest height at B. The perpendicular B A being equal to
the diameter 2 r of the generating circle BFAorbfa. Through /draw the ordinate
feFE parallel to the base. Put
Ef=y,BE=zbE = z,aichBF=Qichbf=Aa = Ff=p,
andjPE= v2rz — z z. Then, since by construction fE = Ff-\-F E, we get, for
the equation of the curve, y =p -f- y2 rz — zz. Its differential is
rdz — zdz rdz
2rrfz — zdz
hence rfy=/-- — , and as the numerator and denominator can be divided by
y ^ T Z 2 Z
V/2r — 2r, it becomes dy=idz\/ ^^^^. Susbtituting this m d ^ =:: \/ df^dz^
[53&] we get ds = dz\/ — = dz . z~-' \/2r, whose integral is s = 2.;?^ V^ ,
z and s commencmg together. The square of tiiis is «^ = 8r5r, which agrees witii the
equation of the tautochronous curve before found [102a].
* (46) The time T is deduced from the formula [1 00a]
tang. 7r=-L==l / -^—1
which does not contain n.
^^ TAUTOCHRONOUS CURVE. [Mec. Cel.
expression would be the same, if we should add to the preceding law of
ds^ ds*
[102"] resistance, the terms* p . ^-3-+ ^--rr + ^^*
In general let R be the retarding force in the direction of the curve ; we
shall havef
[103] . 0=:~ + jR.
a r
s is a function of the time t, and of the whole arch passed over, therefore that
arch is a function of t and s. Taking the differential of this last function,
we shall have a differential equation of this form,
d S T7-
[104] ^=F;
F being a function of t and s, which ought to be nothing by the condition of the
problem, when t has a determinate value, whatever be the length of the whole
[104'] arch passed over. t Suppose, for example, that V=^S.T; aS being a function
* (46a) If we suppose part of the resistance to be as the third power of the velocity and
d s d s d s
to be represented by i? • j^ , this, by substituting the value of :t7- = -77 • 4-' (*') C^^]
(t Z (t Z (Z z
would have introduced into the equations [S9, 93, 94] the term p . -r^. ] 4^' (*') ( 5 hut
1 ds'
4*' (s') = — [87c] and -j- will, as in [95, 96], be represented by quantities dependmg
71 yj. ~p S J (It
on the first or higher powers of s', therefore the preceding quantity by which p is multiplied,
will depend on s'^ or higher powers of s'. But such terms produce nothing in the equations
[98], &;c., for the same reason that the term depending on s'» produced nothing. Therefore
d s^
the term p . -— - produces no alteration in the expression of the time T [98], and the same
d s'^
would be the case with terms like q . -— -, Stc. as is observed above.
f (47) This retarding force R is supposed to be the combined effect of the resistance of
the medium and the force of gravity.
J (48) Let the whole arch described be a', the time of description = f. This time does
not vary from any change in the value of a', by the conditions of the problem, so that if any
part of that arch described in the time t be represented by s, this arch s will be a function of
a' and t, consequently a' must be a function of s, t, which we shall denote by a' = ■^ (5, t).
To determine the velocity v of the body at the end of the time t, we may take the differential
I.ii. §12.] TAUTOCHRONOUS CURVE. 69
of s only, and T a function of t only, we shall have*
dds rp dS ds fy d T dS d^ ^ AT
but the equation ~=zST, gives T, therefore — — is equal to a function
(it Oi t
d s d s^ / ds \
of -^-r-, which function we shall denote by -^2~T^ * ^ ( "q77 ) ' ^'^ ^^^^^ ^^^^^
.SJi' ' x«.....xwxx .,^ ..... .^ ^j SKdf" ^ \Sdt
therefore have
dds
~d¥
d^ ( dS , J ds\ ) o
Such is the expression of the resistance which corresponds to the differential
equation — = ST; and it is easy to see that it comprises the case of the
of tlie preceding expression of a', supposing it to be constant, and we shall get,
whence -7— = — j; i. The first member -r- is evidently equal to v. and the
dt /d.^{s,t)\ dt / 4 ,
ds
second member is a fimction of s, t, denoted above by F, therefore — = f^, or v = V
At the end of thethne f, when the body has described the whole arch a', the velocity v will
be nothing, and m this case V^= 0, as above.
* (49) The value of V=^S T, substituted m ^ = F, gives ^ = S T, whose
dt dt
y.rr • 1 ■ dds d.[ST) . ,
ditlerenual is — - = — ^^ — , and by considering 5^ as a function of s, and 5 as a
c *• c ^ -. X. dds ^ dS ds , „ dT , ^ V ds
funcuon of t, .t becomes -J^=T.- .~ + S .-, hni T= -=---, [104, 104-]
, dds dS ds^ , „ dT , ^ ^ ds
hence ^^ = ^^ . __ + 5. — as m [105]. Again, since T =- ^^ is a function of t,
we shall have t=^ function of (^*-), and as ^= function of t, we shall also have
— = function of {~j\ which, being assumed as above equal to ^^ . 4. (— ), and
substituted in the preceding value of — ^, or— i2[103], becomes as in [106].
18
^^ TAUTOCHRONOUS CURVE. [Mec. Cel.
resistance proportional to the two first powers of the velocity, multiplied
respectively by constant coefficients.* Other differential equations would
give different laws of resistance.
* rdQn\ Tf «ro ,.„+ Cf .as 1 /' »* ^ X / '^^ \ ^ :.„:.„_ ^,_^ga» qj.
(49„) If we put S = ^', +(^) = 5.(il-)-', u gives
ds
the
— = aS, which, substituted in [106], makes
two first powers of the velocity — .
(495) It may be observed that John Bernoulli published in the Memoirs of the Academy
of Sciences of Paris, for 1730, and afterwards in the third volume of his works, a curious
paper on the Tautochronous curve, both in the ascending and descending branches. He
justly remarks that there is a limit to the whole length of tlie curve, and mentions as a proof
of it, the familiar instance of the cycloid, which is the tautochronous curve when the
resistance is nothing [102']. For if a body, falling freely by gravity along the whole
cycloidal arch, would, in tlie time T, arrive at its lowest point, with the acquired velocity F";
we might project the body upwards, along the arch, from its lowest point, with any velocity
less than /^, and the force of gravity would destroy the whole velocity, in the time T,
whatever might be the length of the part of the arch described. But if the projected
velocity should exceed F, the whole cycloidal arch would be passed over in a time less tlian
T; so that this particular cycloid would cease to be tautochronous with such a projected
velocity. In the preceding calculations it is not necessary to suppose the whole curve to be
in the same plane ; for, without altering tlie investigation, we may suppose tlie curve to be of
double curvature, taking care not to have any abrupt bend, and keeping every point of the
curve in the same horizontal plane, in which it was originally placed," when the whole curve
was extended in a vertical plane. For example, the curve might be bent round a vertical
cylinder, keeping the extremities at their former height. It bemg evident tliat, in this case,
the spaces passed over, the times, the velocities, and the resistance would not be varied, for
the reasons mentioned in note 18a.
I.iii.^13.] EQUILIBRIUM OF A SYSTEM OF BODIES. 71
CHAPTER III.
ON THE EaurUBEIUM OP A SYSTEM OF BODIES.
13. The most simple case of the equilibrium of several bodies, is that of
two material points or particles which impinge against each other, in opposite
directions, with equal velocities ; their mutual impenetrability evidently
destroys their velocities, and reduces them to a state of rest.
Suppose now that a number m of similar contiguous particles, are arranged
in a right line, each having the velocity u, in the direction of this right line ;
and that a number m' of similar contiguous particles are arranged on the same
right line, each having the velocity u' directly opposite to u, so that the two
systems shall strike directly against each other. It is required to determine
the ratio of u to u', that there may be an equilibrium at the instant of
impact.
For this purpose we shall observe that the system m, having the velocity
w, would be in equilibrium with a single particle having the velocity m u, in
an opposite direction ; for each particle of the system would destroy in this
last particle, a velocity equal to w, consequently the whole number of particles
m would destroy the whole velocity mu. Therefore we may substitute, for this
system, a single particle, moving with the velocity mi«. In like manner we may
substitute for the system mf, a single particle having the velocity m' u' ; now the
two systems being supposed in equilibrium, the two particles which have been
substituted for them, ought also to be in equilibrium, which requires that their
velocities should be equal ; we have therefore, for the condition of the
equilibrium of the two systems, mu = m'u'. [106']
The mass of a body is the number of its material particles, and the product auantuy
of motion.
of the mass by the velocity is called the quantity of motion, this is also what
is understood by the force of a body in motion. To maintain the equilibrium
between two bodies, or two systems of points, impinging against each other
72 EQUILIBRIUM OF A SYSTEM OF BODIES. [Mec. Cel.
in opposite directions ; the quantities of motion, or the opposite forces, ought
to be equal; consequently the velocities ought to be inversely proportional to
the masses.
fcty, '^^® density of a body depends upon the number of material points or
vo£ Particles contained in a given space. To obtain the absolute density, it
would be necessary to compare the mass with that of a substance without
pores ; but as no such substance is known, we can obtain only the relative
density of a body ; that is, the ratio of its density, to that of a given
substance. It is evident that the mass is in a ratio compounded of that of
the magnitude and density ; putting therefore M for the mass of a body, U
its magnitude, and D its density, we shall have, in general,
[106"] M=DU;
in which we ought to observe that the quantities M, D, U, express the ratios
to the unity of each species, taken as a measure of those quantities.
In what has been said, it is supposed that bodies are composed of similar
material particles, and that they differ only by the relative positions of these
particles. But the nature of bodies being unknown, this hypothesis is at
least precarious, and it is possible that there may be essential differences in
the ultimate particles. Fortunately the uncertainty of this hypothesis does
not affect the science of mechanics, and we may use it without fear of error,
ri06"'l provided we understand, by similar material points or particles, such as would
be in equilibrium, if they impinged against each other with equal velocities,
in opposite directions, whatever might be their nature.
14. Two material particles, whose masses are m and m', cannot act upon
each other, but in the direction of the right line which connects them together
It is true, that if the two particles are connected by a line passing over a
fixed pulley, their reciprocal action cannot be in the direction of this line.
But we may suppose the fixed pulley to have, at its centre, a mass of an
[I06iv] infinite density, which reacts upon the two bodies m and m', whose action on
each other may thus be considered as indirect.
Let p be the action which m exerts upon m' by means of an inflexible
right line without mass, which is supposed to connect them. Conceiving this
line to be affected by two equal and opposite forces p and — p ; the force — p
will destroy in the body m, a force equal to p, and the force p of the right line
Balance.
I. iii. ^ 14.] LEVER. '^^
will be wholly communicated to the body ?«'. This loss of force in m,
occasioned by its action on m', is what is called the reaction of m' ; thus, in Eeacuon.
the communication of motion, the reaction is always equal and opposite to [loc*]
the action. It appears by observation, that this principle exists in all the
operations of nature.
Suppose two heavy bodies m and m' to be attached to the extremities of an
horizontal line, inflexible and without mass, which can turn freely about one ,
' LiCTert or
of its points. To conceive of the action of these bodies upon each other,
when they are in equilibrium, we must suppose the right line to be bent, at
its fixed point, through a very small angle, so as to form two right lines,
making at that point an angle which differs from two right angles but by an
infinitely small quantity w. Let/, /', be the distances of m and m' from the
fixed point ; by resolving the gravity of m, into two forces, the one acting
upon the fixed point, the other directed towards m', this last force will be*
* (50) To illustrate this, let DAC be the
bent lever, A its point of suspension, C, D tlie
extremities, to which m, m' are attached ; the line ^'
CD being horizontal. Draw the vertical lines ' \
AB,CE, meeting D C, dmd DA (continued), in B and E. Then A C=f,AD=f',
C AE = u. Supposing the angle w to be infinitely small, and neglecting its second and
higher powers, we shall have CB=f, DB=f', DC=f-{-f', C £=/ w ; this last
line being nearly equal to the arch of a circle, described about A as a. centre, witli the
radius^ C. The similar triangles D C E, D B A give D C:DB:: CEiAB; hence in
symbols, AB= 777^. Now the weight m acts at C, in the direction E C, parallel to A JB,
with the force of its gravity mgj which may be represented by A B. This may be resolved
into two forces A C, C B -, of which the first is destroyed by the reaction of the point of
C B
support A ; the other, in the direction C B, is equal to w^g- . -y^J and, by substituting the
above values oi AB, CB it becomes, "^^ J" . In a similar way, by changing /into/',
m into mf, and the contrary, we obtain the force of the weight mf, acting at D, resolved in the
direction D B, — — — , which agrees with the above. Putting these two expressions
ividing by
19
equal to each other and dividing by ^ 1; ' we get [106^*1.
74 EQUILIBRIUM OF A SYSTEM OF BODIES. [Mec. Cel.
' -^, , g being the force of gravity. The action of m' upon m will
likewise be — ; putting these forces equal to each other, on account
of the equilibrium, we shall have
[106 vi] inf= ^'f ;
which gives the known law of the equilibrium of a lever, and shows also,
how we may conceive of the reciprocal action of parallel forces.
We shall now consider the equilibrium of a system of particles m, m',
m", &c., impelled by any forces, and reacting on each other. Let f be the
distance of m from m' ; /' the distance from m to m" ; /" the distance from m'
to m", &c. ; jp the reciprocal action of m on m' ; p' that of m on m" ; p" that
of m' on m", &c, Moreover, let 7n S, m' S', m" S", &c., be the forces which
act upon m, m', m", &c. ; and s, s', s", &c., the right lines drawn from their
origins to the bodies m, m', m", &c.* This being supposed, the particle m
may be considered as perfectly free, and in equilibrium, by means of the force
mS, and the forces which the particles m', m", &c., impart to it ; but if the
particle m be forced to move upon a surface, or a curve, we must add to these
forces, the reaction of the surface or curve. Let 5 5 be the whole variation
of s ; S^f the variation of f, supposing nt to be at rest ; 6^f' the variation of
/', supposing m" to be at rest, &c. ; R, R the reactions of the two surfaces,
which by their intersection form the curve upon which the particle m is forced
to move ; and 6r,6r' the variations of- the directions of these last forces.
The equation (d) § 3 [24], will give
[107] 0 = mSJs+p.sJ+p'.5j'-\-kc.-}-RJr+R'Jr'+kc.
In like manner m' may be supposed perfectly free, and in equilibrium by
means of the force m' S', the actions of the bodies m, m", &c., and the
reactions of the surfaces upon which it is forced to move, which reactions we
* (51) To illustrate this, let m, mf, ml', be the particles ;
cm, dm', c" ir^', the curves upon which they are forced to
move ; o, o', o", the origins of tlie forces S, S', S". Then
om = s, o'm' = s', o"m"=s"; mm' =^f, mm" =f',
m'm" =/". The forces jR, R, act at w ; and R", R", at m', o'
in directions perpendicular to the surfaces whose intersections form c m, dm!
Liu. §14.] EQUIUBRIUM OF A SYSTEM OF BODIES. "^^
shall denote by R" and i2"'. Let <5 5' be the variation of s' ; S^J the variation
of /supposing m to be at rest ; S^f" the variation off" supposing m" to be at
rest, &c. ; and <5r", 6r^" the variations of the directions of /?", R" ; the equilibrium
of m' will give
0 = m'S'Js'+p.Sj+f,Sj" + kc.+R",6r" + R"Jt"'. [108]
We can form similar equations relative to the equilibrium of m", ml", &c. ;
and by adding them together, observing that*
^f=^f+iJ'^ ^f = if + ^J''^ &C. [109]
8fj 8f, &c., being the whole variations of /, /, &c. ; we shall have
0 = 2. m.»S. 55+2. jp. (5/4-2. i2.<5r; (k) [iiO]
in which equation the variations of the co-ordinates of the different bodies of
the system, are wholly arbitrary. It should be observed, that by the equation
(a) § 2 [16], we may substitute, instead of mS .Ss, the sum of the products
of all the partial forces acting on m, by the variations of their respective
directions. The same may be observed of the products m' S'Js', m"S".5s", &c.
If the bodies m, ml, inl', &c., are firmly connected together in an invariable
manner; the distances/,/',/", &c. will remain constant; and we shall have
for the conditions of the connexion of the parts of the system 5/= 0 ;
6/' == 0 ; <5/" = 0, &c. The variations of the co-ordinates in the equation
(k) [110] being arbitrary, we may make them satisfy these last equations,
and then the forces p, p', p", &:c., which depend upon the reciprocal action ,
of the bodies of the system, will disappear from that equation. We may also
* (51a) This follows from the known principle, that the complete differential, or
variation, is equal to the sum of all the partial differentials, found by supposing each quantity
separately to vary. Thus if a?, y, z are the rectangular co-ordinates of m, x\ y', x! those
of m', we shall have their distance [12 or 118],
f=\/{^ — xf + (y - yf ^[^ — zf- [109a}
Now if for brevity we put A = — - — , B= — - — , C = — - — ; its complete variation
J J •/
will be, [1096]
Its partial variation, supposing only the body m to vary, is <5/= — A S x — B ^y — C5z;
and, if m' only vary, we get ^„f=- A^x' -{-B-^l/ -\-C8s^ ', whence we get 5/== ^,f-\- f^„ f
as above. In the same way we get df [109], he.
76 EQUILIBRIUM OF A SYSTEM OF BODIES. [Mec. CeL
make the terms RSr, R'Sr', &c., disappear, by subjecting the variations of the
co-ordinates to the conditions necessary to satisfy the equations of the
surfaces, upon which these bodies are forced to move; the equation (k) [110],
by this means, will become
[110"] 0=±-s.mS.5s; (/)
whence it follows, that in the case of equilibrium, the sum of the variations
of the products of the forces, by the elements of their directions, will be
nothing, in whatever manner we may vary the position of the system,
provided that the conditions of the connexion of its parts be observed.
This theorem, which we have obtained in the particular case in which the
bodies are connected together in an invariable manner, is true, whatever be
the conditions of the connexion of the parts of the system with each other.*
To prove this, it is sufficient to show that by subjecting the variations of the
co-ordinates to these conditions, we shall have in the equation (k) [110].
[Ill] 0 = 2.^.5/+2.i2.(5r;
now it is evident that Sr, 6r', &c., are nothing in consequence of these
conditions [19«] ; it therefore only remains to prove, that by subjecting
the variations of the co-ordinates to the same conditions, we shall have
0 = 2. p. <5/.
Suppose the system to be acted upon only by the forces j9, p', jo", &c., and
[111'] that the bodies are made to move upon the curves, that they would describe
by means of these conditions. Then, these forces may be resolved into the
following, namely, one part, q, q', q", &;c.,t directed along the lines f-,f'if"^
* (52) The meaning of this, in an analytical point of view, is that the equation [110"],
2 . m (S . 5 5 = 0, takes place in all cases of equilibrium, provided as many of the variations
are exterminated as there are conditions in the proposed system. For as La Grange has
observed, in his JVLecanique Analytique, " Each equation of condition is equivalent to one
or more forces, applied to the system, according to given directions ; so that the state of
equilibrium will be the same, whether we employ the consideration of these forces, or that of
the equations of condition." We may observe that the equation [1 10"] is used, in tlie rest of
the work, in the case where we actually have S f= 0, 8f' = 0, he. The equation [116]
being used in other cases.
■{■ (52a) In the reasoning [110'] the forces j?,p', &;c. represent the reaction of the bodies
upon each other, which are supposed mutually to destroy each otlier. Here p, p', &;c.
I.iii. §14.] EQUILIBRIUM OF A SYSTEM OF BODIES. 77
&c., which mutually destroy each other, without producing any effect on the [mi
described curves ; another part T, T", T"', &c. perpendicular to these curves ;
and lastly, the remaining part in the directions of the tangents of these
curves, by means of which the system would be moved. But it is easy to
perceive that these last forces ought to be nothing ; for the system being
supposed to submit to them freely,* they could neither produce a pressure
upon the described curves, nor a reaction of the bodies upon each other ;
they could not therefore produce an equilibrium with the forces — p, — p',
— y, &c. ; q, q', c/', &c. ; T, T', &c. ; and they must therefore be nothing ; [iii"]
consequently the system must be in equilibrium by means of the forces — p,
— p'j — p", &c.; q, q', q", &c. ; T, T", &c. Let 6i, 5i', &,c. be the variations
of the directions of the forces T, T', &c. ; we shall have, by means of the
equation (k) [110],
0 = ^.(q—p)Jf+^.TM; [112]
but the system being supposed to be in equilibrium by the forces q, q'^ &c.,
without producing any action on the described curves,! the equation (k) [110]
represent the total forces exerted upon each body, exclusive of tlie reaction or pressure of
the curves ; and the object of tlie author is to show that tlie decomposition of these total
forces, produces forces, which are equivalent to the reciprocal action of the bodies above
treated of ; also that these and the remaining parts of the forces will balance each other.
Thus let the bodies m, mf be situated at tlie points m, m', and
suppose tlie total force p acts upon m, in the direction m m\
while m is subjected to move upon the curve m D, whose tangent
is m C. Upon m m! take mB=p, AB = q; and since, by
hypothesis, this last force is destroyed, by the reaction of the otlier
bodies ; the remaining force will be m A, which may be resolved
into the forces A C=T, perpendicular to the tangent ; and m C
in the dtfection of the tangent. This last force must be nothing,
for reasons stated by the author [111"]. Lastly, we may remark that if any of the forces
g', g', Sec, were in any particular instance equal to notliing, it would not affect the above
demonstration.
* {52b) The system being at liberty to move in the respective du-ections of these
tangential forces, would do so, unless it were held in equilibrium, by equal and opposite
forces; so that the sums of the opposite forces acting upon tliese bodies must be equal, and
these forces will vanish for all the bodies.
t (52c) Because, by hypothesis [111'], the forces q, ^, Sic. mutually destroy each other
in the system.
20
78 VIRTUAL VELOCITIES. [Mec. CeL
will also give, 0 = 2. qJf; therefore the above expression [112] becomes
^113] 0 = :^.p.8f—2.T.6i.
If we take the variations of the co-ordinates, so as to satisfy the conditions
of the described curves, we shall have 6i=0, 6i' = 0^ &c. [19rt] ; and then
we shall have
[114] 0 = 2. p. 6f;
and as the described curves are themselves arbitrary, being only subjected to
the conditions arising from the connexion of the parts of the system ; the
preceding equation will take place, provided these conditions are satisfied ;
and then the equation (k) [110], will be changed into the equation (/) [110"].
This equation is the analytical expression of the following principle, known
by the name of the principle of virtual velocities.
^^ If we vary, by an infinitely small quantity, the position of a system of
[114'] bodies, subjecting the system to the conditions which ought to be satisfied, the
Virtual sum of the forces which act upon the several bodies, multiplied each by the
space that the body to ivhich the force is applied describes in the direction of
that force, must be equal to nothing, when the system is in equilibrium.^
55*
[114a]
* (52d) For the purpose of illustration, and to show the manner of using the principle of
virtual velocities [1 10"], we shall apply it to the investigation of some of the elementary
•propositions in mechanics.
First, Let A C A' he an inflexible straight line, void of gravity, situated in a horizontal
position, with the weights m, mf, attached to the points A, A' ; the rod being fixed at O, as a
centre of suspension, so that it can move about that centre in a ^
vertical direction, as in the common balance or steelyard. Put f — - /^
C A = a, C A'=a', then if the weights m, mf be an equili- -S
brium, and the rod be made to revolve about C, through an ^ ^
infinitely small angle B C A=r. B' CA' = u, so that the weight m may ascend through the
vertical space AB=a .u; the distance of this weight from the centre of the force, which
is in this case the centre of the earth, will be increased by the quantity Ss = a .u. In like
manner the distance of the weight m! from the centre of the earth, will be decreased by the
quantity A' B' =^a' . w, therefore d s' = — a' .w, the negative sign being prefixed, because
the distance of the body m', from that centre, is decreased by this motion. The principle of
virtual velocities [110"], becomes in this case,
OT.S.6«+m'.5".<5«' = 0,
I. iii. § 14.] VIRTUAL VELOCITIES.
This principle not only takes place in the case of equilibrium, but it
assures the existence of the equilibrium. For, suppose the equation (/) [110"]
to be satisfied, and that the particles wi, m', &c., acquire the velocities
79
and as the force of gravity, acting upon both bodies, is the same, we shall have S= S',
therefore,
m.Ss-\-m' .Ss' = 0. [1146]
Substituting the above values of Ss, S /, it becomes, m .a.u — m' . a' . w = 0, hence
m.a= m' . a'y which is the usual formula of the balance [106''].
It is evident that what is here stated, relative to the action of gravity, may be applied to the
consideration of any other forces, acting at the extremities of a straight lever A C A' in
directions perpendicular to the arm of the lever, and in the same plane. In Uiis case, instead
of the forces S m, S' m', representing the gravit}', S, S', acting upon the bodies m, m', we may
take the equivalent forces P, P', acting upon the extremities of tlie lever, and we may put the
formula [11 4a], under the following form,
P.6s-{-P'.Ss' = 0. [lUc]
Second. In the preceding calculation, the line A C A' was supposed to be horizontal ;
but if it be inclined to the horizon, by an angle f, the lines A B, A' B' would be inclined to
tlie vertical, by the same angle s. In tliis case, the vertical ascent of the body m, in moving
from A to jB, would he 6s = AB . cos. s = a . w . cos. s. In like manner
6 s' :^ — A' B' . cos. f = — a' .u . cos. s.
Substitute these in [1145], and reject the common factor u . cos. s, we obtain, as above,
m .a=m' . a'. It is easy to apply the same principles to the action of any forces, applied
to the extremities of the lever A A', in any directions.
Third. Instead of supposing tlie balance to be a straight line, as in the two preceding
examples, let it be bent at C, so as to form tlie oblique angle A C A'. Using the same
notation as before, we shall have A CB==A' CB' = w, CA = a, C A! = a', AB = a.u;
A! B^ = a' .u; these two last lines being perpendicular to 2? CI)
CA, CA', respectively. Draw the horizontal line D C U,
and upon it let fall tlie perpendiculars A D, A' If, meeting
the horizontal lines B b, B' b' in b, b', respectively. Put
the m^les DC A = BAb=C, jy CA'=B' A'b'=C'. ^
Then the vertical space passed over by the body m, while
moving from A to B, will be m
Ab = AB . cos. C = a . u . cos. C,
hence 5s = a .u. cos. C. In like maimer, the vertical space passed over by the body to',
in moving from A' to B, will be A'b' — A' B' . cos. C = a' . w . cos. C, hence
8 s! = — a! .u . cos. C j
[114rf]
80
VIRTUAL VELOCITIES.
[Mec. Cel.
v^v', &c., by means of the forces mS, m' S', &c., acting upon them. The
system will be in equilibrium, by means of these forces and the forces
— mv, — m'v', — 7n"v", &c. ; put(5«;, Sv', &c., for the variations of the
the sign — being prefixed, because the distance from the earth's centre is decreased by the
motion. Substituting these in [1 146], and dividing by w, we get
m .a . COS. C = 171' . al . cos. C ;
but a. COS. C= CD, a', cos. C" = CD', therefore, m . C D= m' . C Uf which is the
well known principle of the bent balance.
Fourth. Let C J be an inclined plane. Draw the horizontal line IE, and the vertical
line C E. Put the angle of inclination of the plane to the horizon C I E= I. Suppose
two bodies m, m', to be connected together, by tlie flexible thread A C A', void of gravity,
passing over the vertex of the triangle ICE', so that the body m' may be at liberty to move
along C I, while the body m moves in the vertical C E. It is very easy, in tliis case, to
apply the principle of virtual velocities, when the bodies m, m',
are in equilibrium. For, if we suppose the body m to move
tlirough the infinitely small space AB,va.2i vertical direction, its
distance from the centre of the earth will be increased by
AB=5 s ; and, during this motion, the body m' will slide down
the line CI, through an equal space. A' B' ^^ 5 s. To find the
corresponding vertical distance A' h', passed over by this body
m', we may draw B'h', A! V , parallel to IE, CE, respectively; and we shall have
A! y ^A!B' . sin. A! B' h' ^A! B' . sin. / = 5 5 . sin. /, therefore 5 s' = — (5 5 . sin. /;
the negative sign being prefixed, because the distance of the body m', from the earth's
centre, is decreased. Substitute these values of 5 s, Ss', in [1 14&], it becomes,
m .S s — mf . 5 s . sin. J= 0,
whence, m = m' . sin. /; which agrees with the usual rule for the equilibrium of bodies upon
an inclined plane.
Fifth. In the motion of a screw, let the power P be supposed to act at the extremity A
of the horizontal lever C A ; the direction of this force being horizontal and perpendicular to
the arm of the lever ; the screw turning about a vertical axis, jj s B JL
passing through the point C, perpendicular to the plane of the
figure ; and raising the weight P', so that the extremity of the
lever may describe the circumference of the circle AEFA = c,
in the same time that the weight P' is raised, through a vertical
height i, equal to the distance of tlie threads of the screw. Now
the tangent A D being drawn perpendicular to C A, we may
take upon it any point D, as the origin of the force P, so that
I. iii. ^ 14.] VIRTUAL VELOCITIES. ^^
directions of these last forces ; we shall have, by the principle of virtual
velocities,
but, by hypothesis, we have, 0 — ^.m.SJs; therefore 0 = :^.m.v6v. As
the variations &v, Sv', &c., ought to be subjected to the conditions of the
system, we may suppose them to be equal to vdt, v' dt, &c., and we shall
then have 0 = 2.wi;^, which equation gives* v = 0,if = 0, &c. ; therefore
the system will be in equilibrium by means of the forces m *S, m' *S', &c.
The conditions of the connexion of the parts of a system may always be
reduced to certain equations between the co-ordinates of its different bodies.
Let w = 0, m' = 0, u" = 0, &c. be these equations; we may, by § 3 [26],
add to the equation (/) [110"], the function X(5?f-fx'5w'4-&c., or s.x.^t*;
X, x', &c., being indeterminate functions of the co-ordinates of the bodies ;
this equation will thus become
0 = 2.m»S.55-f-2.x.5«; (hf) [116]
in this case, the variations of all the co-ordinates will be arbitrary, and we
AD = s; and we may consider the infinitely small part of it A J5, which is common to this
tangent and to tlie circle, to represent the variation of s, therefore Ss = — A B ; the
negative sign being prefixed, because, wliile the extremity of the lever moves, from A to J5,
the distance, from the origin of the force D, is decreased. It is evident, that dm-ing the
h Jl Ti
motion from Ato Bj the weigjit P' will be raised through the space — '- =Ss'. Sub-
stitute these values of 5 s, 5 j/, in [114c], and reject the common factor AB, it becomes
— P -]- P' . - = 0, or P = P' . -, which is the usual formula for the screw,
c c
Sixth. In the case of a compound pulley, in which a power P is applied, to raise a weight
P' vertically ; if we suppose the power to act at the end of the cord, while the weight is
supported by n parts of the same cord, each bearing an equal part of the weight, so that
while the weight P' is raised through the vertical height d /, the power P is depressed by n
times that quantity, we shall have ds = — nJs'. These values being substituted in [1 14c],
give — Pn.Ssf -{-P' .5sf = 0y hence P' =^Pn, which is the usual rule for computing the
force of a pulley.
♦ (53) Each term m v^, w! t/^, &;c. of the equation 2 . m ir* = 0, is positive, and to
render their sum nothing, we must necessarily have mt;^=0, mV'2 = 0, Stc. whence v = 0,
f/ = 0, Sic. m, m', he. being positive.
21
82 EQUILIBRIUM OF A SYSTEM OF BODIES. [M6c. Cel.
may put their coefficients equal to nothing, which will give an equal number
of equations, by means of which we may determine the functions x, x', &c.
If we then compare this equation with the equation (k) [110], we shall
have
[117] 2.x.5M = 2.p.(5/+2.i?.(5r;
whence it will be easy to deduce the reciprocal actions of the bodies w, m',
&c., and the pressures — R, — E, &c., which they exert against the surfaces,
upon which they are forced to remain.
oflTlrm ^^* ^^ ^^ ^^ bodies of a system are firmly attached together, its position
ciei'firmiy may be determined by any three of its points, which are not in the same
connocteU
^therr*"^ right line. The position of each of these points depends on three co-ordinates,
which produce nine indeterminate quantities ; but the mutual distances of
the three points being given and invariable, we may, by means of them,
reduce these quantities to six others, which, substituted in the equation
(/) [110"], will introduce six arbitrary variations ; putting their coefficients
equal to nothing, we shall have six equations, which will contain all the
conditions of the equilibrium of the system ; we shall now develop these
equations.
[1171 Let x^ y, z be the co-ordinates of m ; af, t/, zf those of m' ; x", y", sf' those
of m", &c. ; we shall have*
/ = xTJ^- xf + (jj -yf + {z' -zf ',
[118] /' _ \/ (x"-.xf+{if — yj + {^' — zf ;
/" = V/ (a;" ^ xj + (/ - 'i/f + (2:"- ^)^ ;
&c.
If we suppose
[119]
5a;==5ar'= 5a;" = &c. ;
6y=iS'i/ = Sy"= &c. ;
8z = S2f = 5zf' = kc.;
* (53a) The value / is tlie same as in [109a] ; /', /", &c. are of the same form,
changing of, i/,2^ into a/', y", z", &ic. The variation of/ [1096] is
8f=A{^x' — Sx)-{-B{Sy'—Sy)-\-C{S::f — Sz),
which, by means of the equations [119], becomes 5/= 0. In like manner we get Sf = 0,
Sf" = 0, &z;c. as above.
I.iii.<^15.] EQUILIBRIUM OF A SYSTEM OF BODIES. 83
we shall have
6/=0, ^/' = 0, <5/" = 0, &c.; [ii&l
the requisite conditions will therefore be satisfied, and we shall have, by means
of the equation (/) [110"], the following:*
0 = ..m.S.(i^); o = ..,«.S.(g); 0 = ..».S.(ii); (m) Um
we shall thus have three of the six equations, which contain the conditions of
the equilibrium of the system. The second members of these equations
are the sums of the forces of the system, resolved in directions parallel to the
three axes x, y, 2, [13] ; each of these sums ought therefore to be nothing in [119"]
the case of equilibrium.
The equations Sf=0, 5/' = 0, 5/' = 0, &c., will also be satisfied if we [ll9i']
suppose z, z'j z". &c. to be invariable, and putf
6x'=y'.^'si ; ^y= — x'J-a;
&c.
6 TA being any variation whatever. Substituting these values in the equation
(I) [110"], we shall havej
0 = ..».S.J,.(Q-x.(ii)5. tm,
* (535) The values [n9] being substituted in [1 10"], developed as in [14a], it becomes
Putting as in [SG'] the coefficients of 5 a:, Sy,Sz, separately equal to nothing, we obtain [119"].
f (54) Substitute in 5/ [1096], the values [120], also 8 z=0, 6 z' = 0, he. [119"], it
becomes Sf=Szs . ^A .{1/ — y) — >J5.(a/ — x)\, and since, by [109 a, J], a/ — x=f.^,
y' — y=f .B, it may be changed into Sf=f.6-a[AB — A Z?| = 0. The same takes
place with 8f', Sf", &c. because all these expressions are symmetrical.
J (55) The co-ordinates z, z", &;c. being invariable, the part of the equation [110"]
depending upon the body m,ym}lhem.S.(-T-j5x-{-m.S.{-T—j.6y; or, by substi-
tuting the values of Sx,Sy^ [120] S-a .} m . S ,y .(—-j — m.S.x,(j-\>. In a
similar manner the terms depending upon m! are,
34 EQUILIBRIUM OF A SYSTEM OF BODIES. [Mec. Cel.
It is evident that we may change, in this equation,, either x, a/, x% &c., or
[121'] y, ij^ y, &c., into Zj 2!, 2f\ &,c. ; this will give two other equations, which,
connected with the preceding, will furnish the following system :
[122^ The function 2 . m . 6* . ?/ . ( — ) is, by § 3 [29'], the sum of the momenta of
all the forces, parallel to the axis of x, to make the system turn about the
axis of z. Likewise the function :s.,mS .x A-—] is the sum of the
VyJ
momenta of all the forces parallel to the axis of y, to make the system turn
about the axis of z, but in a contrary direction to the first forces ; the first of
fi22"] the equations (n) [122], therefore indicates, that the sum of the momenta of
the forces, relative to the axis of z^ is nothing. The second and third of
these equations indicate, in like manner, that the sum of the momenta of the
forces is nothing with respect to the axis of ?/, or x. Uniting these three
conditions to the former [119'"], namely, that the sum of the forces, parallel
to these axes, is nothing relative to each of them ; we shall have the six
conditions of the equilibrium of a system of bodies, invariably connected
together.
If the origin of the co-ordinates is at rest, and invariably attached to the
system, it will destroy the forces parallel to the three axes, and the
conditions of the equilibrium of the system, about this origin, will be reduced
to the following, that the sum of the momenta of the forces, to make the
[122'"] system turn about these three axes, is nothing relative to each of them.
We shall suppose the bodies m, m', m", &c., to be acted upon by no other
force than gravity. Its action is the same upon all the bodies, and we may
[I22iv] suppose its direction to be the same through the whole extent of the system ;
The sum of all the similar terms of the equation [110"], being divided by 5ttf, becomes as
in [121], which is the same as the first of the equations [122]. The other two equations
are found in precisely the same manner, changing tlie co-ordinates as in [121'].
[123]
Liii. §15.] EQUILIBRIUM OF A SYSTEM OF BODIES. ^^
therefore we shall have*
S=S' = S" = Slc.;
the three equations (n) [122], will be satisfied, whatever be the direction of
5, or, in other words, whatever be the direction of gravity, by means of the
three following equations :t
0^2. ma;; 0 = ^.my ; 0:=2.m2; (o) [^24]
the origin of the co-ordinates, being supposed fixed, will destroy the three
forces S .( — j.'s.jn; S . l-r~].^.m; S .(—j.i.m; parallel to each
* (55a) The action of gravity being the same upon all the bodies, makes
S=S'=S" = &Lc.;
as in the first of the equations [J 23]. If we refer to the figure in page 8, we shall have
[l^^J' (l~ = =^ — =cos.cAD, (-— )="^ =— -=cos.c^a,
Vox/ s Ac \6y/ s Ac '
/6s\ z — c AB ^ „
I T—]^= ^— — ==cos. cAB;
\dz J s Ac '
therefore \t^\ \T~\ \T^\ represent the cosines of tlie angles which the line s makes [^-3«]
with lines drawn parallel to the axes of x, y, z, respectively. In like manner (t^\ (i^\
\67j' ^^P'*^^^"^ *^ cosines of tlie angles, which the line s' makes, with lines dra^vn
parallel to the same axis, and so on for the other lines, s", s'", he. Now since the lines
s, s', &c. are parallel [122'^, we shall have, (|^) = (^j~\ ^ &c, (t.\ = fl'l) = &^.,
t (56) The forces S, S', S", &tc, being equal [123], as well as (^\ (^Y &c.
\Jy)^ V^/' ^'' \JzJ' ( IT' )' ^^ese quantities may be brought from under^the sign 2,
22
86
CENTRE OF GRAVITY OF A SYSTEM OF BODIES. [Mec. Gel.
[124T
[124"]
of the three axes ;* by the composition of these three forces, they will
produce the single force aS . 2 . m, which is equal to the weight of the
system.
This origin of the co-ordinates, about which we have supposed the system
to be in equilibrium, is a very remarkable point, because if this point is
sustained, and gravity acts only upon the system, it will remain in equilibrium,
in whatever situation we may place it about this point, which is called the
Gratuy?*^ ccw^rc of grttvlty of the system. The position is found by the condition, that
if any plane whatever be made to pass through the centre of gravity, the sum
of the products of each body, by its distance from that plane will be nothing.
[124'"] Yox this distance is a linear function of the co-ordinates of the body .r, y, z,t
in the equations [122], which will then become
S.(|l)...»y_S.(^)...«.= 0;
S.(if).x.™.-S.(il)...». = 0;
S.(^)...»y-S. (?!)... ». = 0;
these three equations are evidently satisfied by means of the equations [124].
* (57) These forces are similar to those used in [119" &c.], bringing the same
[124o] terms from under the sign 2, as in tlie preceding note. Now ^•\j^\ ^ xTvr
S.(^~\^ [13], represent the force S, resolved in a direction parallel to x,y,z, and the
composition of these three forces will again produce the single force S. IMultiplying all these by
2.W, it will follow that the three forces S.^y^j.^.m; S.f— j.S.mj S AjA .2.7w,
in the directions parallel to x, y, z, will produce the single force »S . 2 . m, in the direction of
the origin of that force.
f (57«) Suppose the body m to be placed at m,
in tlie annexed figure, (which is the same as that in
page 13), upon the continuation of the line Bb, so
that its rectangular co-ordinates may be CH=:x,
HB = y, B m= z, and let the co-ordinate B in be
intersected in h, by a plane CDbc passing through
the centre of gravity of the system C, the ordinate
B b being denoted by the accented letter z'. Then
I. iii. <^15.] CENTRE OF GRAVITY OF A SYSTEM OF BODIES. 87
and by multiplying this distance by the mass of the body, the sum of these
products will be nothing, in consequence of the equations (o) [124.]
To determine the position of the centre of gravity, let X, Y, Z, be its three
co-ordinates, referred to a given point ; x, y, z, the co-ordinates of w, referred
to the same point ; x', ^, z', those of m', and in the same manner for the rest,
the equations (o) [124], will give*
0 = 2.m.(a:— X); [125]
but we have 2.m.X=X.2.m, s.w being the whole mass of the system ;
hence we shall have
^ 2. ma?
A = — . [126]
2.ffl '■ ■■
In like manner,
^ 'L.my ^ y 2.ffi2r
2 . m ' 2 . wi '
[127]
the general equation of the plane [19c] z' = ./2 a? -|- -S y, gives
Now if from the point m we let fall, upon the plane CDbc, a perpendicular^, this
perpendicular, or distance of the body m from the plane, will be equal to bm multiplied by
the sine of tlie inclinaSon of 6 m to that plane ; and this inclination is evidently equal to the
angle Db B, whose complement b DB was named (p in [19i"'], hence
p = bm . sin. Db B = {z — Ax — By) . cos. 9, [125a]
which is linear, or of the first degree, in x, y, z, as was observed above. Accenting the li. ear
, • Function.
letters J?, z, x, y, with one accent, for the body m', and with two accents for m , &c., we obtam
p^ = {z' — Aa/ — By') . cos. 9, p" = {z" — A a/' — B f) . cos. <?, he.
IMultiplying diese respectively, by to, m', to", &;c., and adding tliese products together,
we get
• 2 . mp = COS. q> .1 .mz — A . cos. <p . 2 . to a? — B . cos. 9 . 2 . to y.
Which, by substituting the values of 2 TO a:, 2TOy, 2toz, [124], becomes 2 . wp = 0, as
in ri24"1.
* (57J) In the equations [124], the co-ordinates, x, y, z, are referred to tlie centre of
gravity of this system, [124"], but if we count them from another point, which would make
the co-ordinates of tliat centre X, F, Zj it is evident, that the co-ordinates of the body,
referred to that centre, would be x — X, y — Y, z — Z, which are to be substituted in
[124], for x,y,z; and the first equation [124] becomes as in [125], which gives
2 . TO a: = 2 . TO X,
and as X is the same for all the bodies to, to', &c., we may put 2 . to X = X. 2 . to, hence
2 . TO a; = X . 2 . TO, as above.
88 CENTRE OF GRAVITY OF A SYSTEM OF BODIES. [Mec. Cel.
Therefore the co-ordinates X, Y, Z, correspond but to one point,* consequently
there is but one centre of gravity of a system of bodies. The three preceding
equations give
[128] X^ 4- ya I ^2 = i^'^^T+{^'^yy + {^-mzf
which may be put under this form,t
[129] X' + Y'+Z' = ^•^•(^^+y^ + ^^) ^.mm'.l{x'-xf + (y'-yY + {z'-zf]
the finite integral 2.mm'.;(a:'— a:/ + (?/'— ?/7+ (2'— z)'} expressing the sum
of all the products similar to that under the sign 2, formed by combining all
the bodies, two by two. We shall therefore have the distance of the centre
of gravity, from any fixed point whatever, by means of the distances of the
bodies of the system from the same fixed point, and from each other. By
* (57c) Because the equations [126,''127], give but one value of X, one of Y, and one
ofZ.
t (58) Both these expressions of X^ + Y^ + Z^, [12S, 129], are symmetrical in
X, y, z, x', y', z', &ic. To prove therefore their identity, it is only necessary to show, that
the coefficient of any one of these quantities, as x, is the same in the second members of both
these values. This requires that the coefficient of x should be the same in both members
01 ^r-= ;: sT; 5 or
[129a] (2 . m a:)^ = 2 m . 2 m a;"2 — 2 m mf [xf — x)^.
Substituting tlie values of 2 mx, 2 w, &;c., and retaining only the terms multiplied by a?,
we get
(2 . mxY = [m x-\-m' x' -\-m" x" -[-&;c.)^ = m^ x'^-\-2nix [m! x' -\-m," a!' -\-hLc)',
[1296] 2wi.2ma:^=(m + w' + &z;c.) . {m.x''^-]-m' x'^-\-k.c.) =w? x'^ -{-ma^ {m! +m" -{-kjc.) *
— l-mm' [x' — xj^ = — mm' [x' — a?)^ — m m" [x" — a?)^ — &z;c.
=: — mx^ [m! -\-m" -\-hc.)-\-2mx[m' x' -\-m" x" -\-hc.';
hence
1 .m ."Z .map' — 2 .mm'(x' — xY ^ m^ x"^ -\- 2 m x (^ m' xf -\- m" x" -{- he),
which, being equal to the development of (2 m xy [1296], proves that the coefficients of x,
in both members of [129a] are equal.
It maybe observed, tliat the quantities which occur in the second member of [129], are
the squares of the distances of the bodies m, m', he. from the origin, represented by
'^+y^+ -s^j ^'^ + y^ + ^% &z^c. [19e], and the squares of tlieir mutual distances /,/', &tc.
[118] ; as is observed in [129'].
I. iii. §16.] EQUILIBRimi OF A SOLID BODY. ^^
determining in this manner the distance of the centre of gravity from any [129']
1T11- ... l^'l. New me-
three fixed points whatever, we shall have its position m space ; winch j^od of ^^^
furnishes a new method of determining it.* oraluyf
The name of centre of gravity has been extended to the point determined
by the three co-ordinates X, Y, Z, of any system of bodies, whether they
are acted upon by gravity or not.
16. It is easy to apply the preceding results, to the equilibrium of a solid Equiubn-
body of any figure whatever, by supposing it to be formed of an infinite «»«'J'>«^y-
number of particles, invariably connected together. Let d m be one of these
points or infinitely small particles of the body ; x.y^z the rectangular ^29"]
co-ordinates of that particle ; P, Q, R the forces acting upon it, in directions
parallel to the axes x, y, z; the equations (m) [119"], and (n) [122] of the
preceding article will becomef
0=fP.dm; 0=fQ.dm; 0=fR.dm; [130]
0==f(Py—Qx).dm] 0=f(Pz—Rx).dm; 0==f(Ry—Qz).dm ; [i3i]
the sign of integration f refers to the particle d m, and must be extended to
the whole mass of the solid.
If the body be so fixed, that it can only turn about the origin of the [131']
co-ordinates, the three last equations will be sufficient for its equilibrium.
* (59) These three fixed points may be considered as tlie angular points of tlie base of a
triangular pyramid, whose vertex is the centre of gravity ; and it is evident, that when the
base is given, the vertex may be found, by means of the length of tlie tliree lines, drawn from
those angular points to the vertex.
f (59a) Substituting for the forces )S. (-T^ J, S .(-r—), S. (.--), [124a], their values
P, Q, R, respectively [129"] ; also putting dm for 7n, and / for 2. This changes [119"]
mto [130], and [122] into [131].
23
90 EQUILIBRIUM OF FLUIDS. [Mec. Cel.
CHAPTER IV.
ON THE EaUILIBRIUM OF FLUIDS.
17. To obtain the laws of the equilibrium and of the motion of each of
the particles of a fluid, it would be necessary to ascertain their figure, which
is impossible ; but as these laws are required only for the fluids considered in
a mass, the knowledge of the figure of the particles becomes useless. Whatever
may be these figures, and the dispositions which result in the separate
particles, all fluids, taken in a mass, must present the same phenomena, in their
equilibrium, and in their motions ; so that the observation of these phenomena
will not enable us to discover anything respecting the configuration of the
particles of the fluid. These general phenomena depend on the perfect
Mobility mobility of the particles, which yield to the least pressure. This mobility
is the characteristic property of fluids ; it distinguishes them from solid
bodies, and serves to define them. Hence it follows, that to maintain the
equilibrium of a fluid mass, each particle ought to be held in equilibrium, by
maidl. means of all the forces acting on it, and the pressure which it sustains from
the surrounding particles. Let us now investigate the equations resulting
from this property.
For that purpose we shall consider a system of fluid particles, forming an
infinitely small rectangular parallelopiped. Let x,y,z, be the three rectangular
co-ordinates of that angle of the parallelopiped, which is nearest to the origin
of the co-ordinates ; d x, d y, dz the three dimensions of the parallelopiped ;
p the mean of all the pressures upon the different points of the face dy .dz
of the parallelopiped, nearest to the origin of the co-ordinates ; and p' the
same quantity relative to the opposite face. The parallelopiped will therefore
be urged, by these pressures, in a direction parallel to the axis of or, by a
force equal to {p — p') .dy.dz. p' — p is the diflerential of p^ considering
X only as variable ; for, although the pressure p' acts in a contrary direction
[131"]
Equilibri
Liv. §17.] EQUILIBRIUM OF FLUIDS. ^1
to p, yet the pressure upon a particle of fluid being the same in all directions,
y — p may be considered as the difference of two forces, acting in the same
direction, at an infinitely small distance from each other ; therefore we shall
have, p'—p = r^V tZa:; and* (p—p').dy.dz = — (^.dx.dy.dz. [131"!
Let P, Q, R, be the three accelerating forces, which also act on the fluid
particle, parallel to the axes of x, y, z ;t if we call the density of the paral-
lelepiped p, its mass will be ^. dx.dy .dz, and the product of the force F
by this mass, will be the whole resulting force which tends to move it ;
consequently the mass will be urged in a direction parallel to the axis of x,
by the force 5pP — l-^\^,dx.dy.dz. In like manner it will be urged [131**]
in directions parallel to the axes of y and 2, by the forces
^fQ^f^X.dx.dy.dz', and \ fR — (-~\\ . dx .dy .dz\ [131*3
* (60) Let DEFG HIKL be the inSnitely small rectangular parallelepiped, the
co-ordinates of its angular point D being C A=x, AB = y, B D^ z,ks sides D H= dx,
DF=dy,DE=dz; area of tlie parallel faces DEGF, HIKL = dy.dz. Now
the pressure upon the face of D E G F is p, in the direction parallel to DHorx, and tending
to increase x ; p being, in general, a function of x, y, z. Therefore the parallelopiped is pressed
in tlie direction parallel to D H or x, by the force p .dy .dz. Now if x were increased
by d X, without varying y, z, the point for which the pressure is computed, would be changed
from D to H, and we should obtain the pressure at the point H, from the preceding value of
p, which would become p' =p-\-(—-\ d x, by the common principles of the differential
calculus, the direction of the pressure being the same. But as fluids press in every direction,
the face HIKL must be press^ backwards, towards the
origin of x, by the force p' .dy.dz ; the difference of tliese cf
two forces, {p' — p) . dy .dz or ( -—- Wz y .dz represents ^y
the whole pressure, suffered by the parallelopiped, in the
direction H D, and as this tends to decrease x, the negative
sign must be prefixed, and it becomes
— {£)'doc. dy.dz,
as m [131'"].
f (60a) ThesQ, forces are supposed to tend to increase the co-ordinates.
A.
\B
K
W
Jbc M
92 EQUILIBRIUM OF FLUIDS. [Mec. Cel.
we shall therefore have, bj means of the equation (b) ^ S [18],
or^
[133] 6p = p.{PJx+Q.Sy-JrR.Sz].
The second member of this equation ought to be, like the first, an exact
variation,! w^hich gives the following equations of partial differentials :
[134] /d.pP\_fd.pq\^ /d.pP\_fd.pR\^ /d.pq\_/d.pR^
dy J \ dx J \ dz J \ dx J \ dz J \ dy
whence we deduce
-> 0 = P.(4i)-Q.O+i..(f)-P.(^-)+Q.(^)-ie.(4f)4
* (606) Substituting in [132], for (^) 5 x + (^^ ^ y + (^^ d z, its value, S p,
[J 4a], It becomes as in [133].
■f (61) The second member of [133] being an exact variation of j>, gives {—-j = pP,
[133a] /._L. j=p Qj (— ) =pR' The differential of the first being taken, relative to y, and that
of the second, relative to x, the first members of both expressions will be (t~-7— ) j hence
(-— — j = (— li— ?j. In a similar way, the other equations [134] were deduced from
( — —\ ( f- ). These ai-e the well known equations of condition, of the integrability
\dxdzj \dy dz/ ^
of a function p of tliree variable quantities x, y, z.
J (62) Developing the three equations [134], and transposing the terms to one side,
we get
-a+^e-f)-^•(^)-^C-!H'
]\Iultiply the first by R, the second by — Q, and the third by P, and add these products
tosrether; the coefficients of the terms ( — ), (-r-), {-;—), will vanish; and the rest,
Vrfx/ \dy/ \dzj
divided by p, will become as m [135].
I. iv. §17.] EQUILIBRIUM OF FLUIDS. ^^
This equation expresses the relation which ought to exist between the forces
P, Q, Rf to render the equilibrium possible.
If the fluid is free at its surface, or in any parts of its surface, the value of
p will be nothing in those parts ; in which case we shall have* ^p = 0,
provided we take the variations Sx, Sy, 5z, so as to appertain to this surface ;
therefore, by fulfilling these conditions, we shall have
o=P.6x+Q'^y + R'^^' [136]
Let 6u = 0 be the difierential equation of the surface, we shall havef
P.5xi- Q.6y-{-R.6z = -k.5u, [137]
X being: a function of x, y, z : hence it follows, from §j 3, that the resultant -ofTqii"
of the forces P, Q, R, ought to be perpendicular to those parts of the surface H^^^^,
where the fluid is free. J [137']
Suppose that the quantity P.5x-{-Q.Sy-{-R.Sz is an exact variation,
which is the case by ^ 2, when P, Q, R, are the result of attractive forces.^
Put this variation equal to ^ 9, or
S(p = P.5x-\-Q.5y-[-R.5z, [137"]
* (63) For if the pressure p, in the direction of the tangent of the surface, is of any
magnitude, the fluid would yield to that pressure, in tliose parts of the surface where it is free,
and this motion would continue till tlie particles had assumed the state corresponding to
8p = 0, and then [133] would change into [136].
f (63a) From Su = 0, and the formula [136], we obtain [137], as [19"] was found in
note 15.
f (64) The three forces P, Q, R, acting in directions parallel to x, y, z, maybe reduced
to one force V, acting in the direction r; so tliat by the formula [16], we should have
P.8x-\-Q^.8y-\-R.5z=VJr,Bndth\s,hy means of [!36], becomes V.8r = 0; hence
in general, F" being finite, we shall have Sr=0. Now Sr cannot be equal to nothing, unless
the line r be drawn perpendicular to the surface [19a]. Therefore the resultant of the
forces P, Q, R, must be perpendicular to the surface, in those parts where the fluid is free.
This is also evident of itself. For if the resultant of the forces, acting upon a particle of
the fluid, at the surface, was not in the direction of the normal, it might be resolved into two
forces, the one in the direction of the normal, the other in the direction of the tangent, and
this last would, as was observed above, cause the particle to move, on the surface, and
destroy the equilibrium.
§ (64a) As is shown in note 1 3/.
24
^ EQUILIBRIUM OF FLUIDS. [M6c. Cel.
and we shall have
[137'"] 5j? = p.5<p;
therefore p must be a function of ^ and cp -* and since the integration of this
equation gives <? in terms of p, we shall also have p expressed in a function
of p. Consequently the pressure p is the same for all particles of the same
density ; therefore dp is nothing, relative to the surfaces of the strata of the
fluid mass, in which the density is constant, and as it respects these surfaces
we shall havet
[138] O^P.5x+Q.5y-\-R.5z.
* (65 j This is evident, because the equation (lp = p5 (p, contains the variations 6 p
and (5 9, which could not be integrated, unless p was a function of p, 9, and any constant
quantities.
f (65a) By hypothesis dp or 6p = Q. If we substitute tliis in [133], and divide by p,
we shall get [138], and from this last equation we find, as in note 64, that the result of the
forces P, Q, R, is perpendicular to the level surface at that part.
It follows from what has been said in this chapter, that no heterogeneous mass of fluid
can remain in equilibrium, unless each level stratum be homogeneous throughout its whole
, extent. This is the only condition required, when the fluid completely fills a vessel, which is
closed on every side; but if any part remain open, it is also necessary, [137'], that the
resultant of all the forces, at that part, should be perpendicular to the surface.
As an example of the use of tlie formula [138], we may apply it to tlie investigation of
the form of the level strata, when the force, acting upon the particles of the fluid, is reduced
to one single force S, tending towards the origin of the co-ordinates x, y, z, and varying, as
any function of the distance s, of the particle from that origin. This force, resolved in
fl37a] directions parallel to the axes x, y, z, will be S . -, S .—, S.-, respectively, as in [13a] ;
observing that, the force »S is supposed to be situated at the origin of the co-ordinates, and
we may therefore put a = 0, 6 = 0, c =: 0. Substitute these for P, Q, R, in the equation
of the stratum [138], and it will become — . (x S x -{- y S y -\- z 5 z) = 0, or
x8x-{-ySy-\-zSz = 0;
whose integral is a?^ -j- y^ -|~ ^^ = constant. This corresponds to a spherical surface [19e],
[1375] the constant quantity being equal to the square of the radius s. Therefore the level strata
will, in this hypotliesis, be concentrical spherical surfaces.
The equation [136] might also be applied to the computation of the figure of the upper
surface of a fluid, contained in a vertical cylinder, open at the top, and revolving uniformly
about its vertical axis z, with the angular velocity n. In this case the centrifugal force,
I.iv. <^17.] EQUILIBRIUM OF FLUIDS. ^^
Hence it follows, that the resultant of the forces acting upon each particle
of the fluid, when in a state of equilibrium, is perpendicular to the surfaces
of those strata ; which, for that reason, are called level strata^ or level [138']
surfaces. This condition is always fulfilled, when the fluid is homogeneous
and incompressible ; since then the strata, to which this resultant is perpen- [138"]
dicular, are all of the same density.
Therefore, to support the equilibrium of a homogeneous mass of fluid,
whose exterior surface is free, and which contains within it a fixed solid
nucleus, of any figure w^hatever, it is requisite, and it is suflicient ; First, ri38"q
that P .^x-\- Q .8y-\- R. 6z should be an exact differential ; Second, that
the resultant of the forces acting on the exterior surface should be perpen- [i38W]
dicular to the surface, and should be directed towards the inner part of the
fluid.
arising from the rotation, may be considered as an actual force applied to the particles. Now
if p be the distance of a particle from the axis of the cylinder, its rotatory velocity will
be n p, and its centrifugal force [54'], being equal to the square of the velocity, divided
by the radius, will be n^ . p. This force is in tlie direction of the radius p, or A W, (Fig. 2, [138a]
page 20) ; it may be resolved into two forces, parallel to the ordinates w2 X:= a?, X W= y,
and will be represented hy P = n^ . x, Q = w^ . y, [11] ; these forces tending to increase
the co-ordinates. Moreover, the force of gravity g tends to decrease tlie ordinate z, so that
R = — g. Substituting these in the differential equation of the surface [136], it becomes
n^ .{x5x-{-ySy) — g Sz = 0, whose integral is - . n^ . (o::^ -j- /) — g -2= constant. If
we suppose x, y, z to commence together, the constant quantity will be nothing, and by putting
n^ = 2ga, this equation will become z = a (x^ -f y^), but ^ -{- ^2= p^ [27], therefore the
equation of the surface will be z = a . p^. This is the equation of a parabola [646].
Therefore the figure of the upper surface of tlie fluid is that of an inverted parabolic conoid.
This subject is treated of in a different manner in [323a].
It may be observed that tlie preceding values of P, Q, jR, satisfy the equation of condition
[135], since each term of that equation vanishes.
96 MOTION OF A SYSTEM OF BODIES. [Mec. Cel.
[138']
CHAPTER V.
*
GENERAL PBINCIPLES OF THfi MOTION OF A SYSTEM OF BODIES.
18. We have, in ^ 7, reduced the laws of motion of a point, or particle,
to those of its equilibrium, by resolving its motion, at any instant of time,
into two others, one of which remains in the next instant, and the other is
destroyed by the forces acting on that point ; the equilibrium between these
forces and the motion lost by the body, has given us the differential equations
of its motion. We shall now make use of the same method, to determine
the motion of a system of bodies m, m', w", &c. Therefore let m P, m Q,
mR, be the forces which act on the body m, parallel to the axes of its
rectangular co-ordinates x, y, z ; m' P', m'Q'^m'R the forces which act on w',
parallel to the same axes,* and in the same manner for the others ; and let
uoc d ti d z
t be the time. The partial forces m . --, m.—i ^--n of the body w,
^ at at at
at any instant, will become, in the next instant, f
dx , -, dx 7 dx , rt 7^
m.^r- 4-171 .a .~ m.d .-r--\-mP ,dt
dt dt dt ^
[139J m.-^+m.d.-^-'—m.d.-^ + mQ.dt
dz , -J dz , J d^ , -n ^4
dt ^ dt^ dt
and as the following forces only are retained,
r.^ni dx , , dx dy , i dy dz , j dz
140 yn.-r+m.d.-^; m.-r+m.d.~; m,^--{-m.d.^-;
dt ^ dt^ dt ^ dt dt ' dt
* (66) It may be observed that these forces are supposed to tend to increase the
co-ordinates.
1 t (^^«) I'h^ reasoning in this article is like that in page 31. The expressions [139],
being like [36], [140] are like [36'J, &c.
(P) [1411
I. v.§19.] MOTION OF A SYSTEM OF BODIES. 97
the other forces
at cLt at
will be destroyed. By marking successively, in these expressions, the letters
m, a:, y, z, P, Q, i?, with one accent, two accents, &c., we shall have the
forces destroyed in the bodies m', m", &c. This being premised, if we
multiply these forces respectively by the variations (5a:, by, <5 z, 5a/, &c. of
their directions, the principle of virtual velocities explained in ^ 14 [114'],
will give, by supposing d t constant, the following equation :
&c., or*
0=:s.m.&. J ^-P J +^.mJy. J ^-Q j +^.m.iz. J ^-iJ J ; (P) [i4si
we must exterminate from this equation, by means of the particular conditions Equauon
of the system, as many variations as we have conditions, and then put the ^°y"^m
coefficients of the remaining variations separately equal to nothing ; we shall °
thus have all the necessary equations, to determine the motions of the different
bodies of the system.
19. The equation (P) [142] contains several general principles of motion,
which we shall now proceed to develop. We shall evidently subject the
variations 5 a:, 5y, 5z, «5 a:', &c. to all the conditions of the connexion of the
parts of the system, by supposing them equal to the differentials dx, dy,
dz, dx', &c. This supposition can therefore be made, and the equation
(P) [142] will give, by integration, f
:£.mS-^^±^^^±^=c + 2,2.fm(P.dx+Q.dy+R,dz); (Q) [143]
c being the constant quantity to complete the integral.
* (66&) The equation P [141'], is again repeated in [142], though it is not in the
original. This was done because it is most commonly referred to under this last form.
t (66c) The equation [143] is found in the same manner as [39], in note I7A.
25
96 MOTION OF A SYSTEM OF BODIES. [Mec. Gel.
If P, Q, R, are the result of attractive forces, directed towards fixed points,
and the mutual attractions of the bodies on each other ; the function
[143^ ^.fm.(P.dx-i-Q.dy+R.dz)
will be an exact integral. For the parts of this function, depending on
attractive forces, directed towards fixed points, are, by ^ 8,* exact integrals.
The same is true with respect to the parts, depending on the mutual
attractions of the bodies of the system upon each other. For, if we denote
by y, the distance from m to m', m'F the attraction of m' upon m ; the part
[143"] of m.(P.dx-\-Q.dy-\-R.dz), depending on the attraction of m' upon m,
will, by the article just named, be equal tof — mm' . F. df, the differential
df being taken supposing only the co-ordinates x, y, z to be variable. But
the reaction being equal and contrary to the action, the part of
m:,{F' .dx!^q.d]l -\-E ,d^\
depending on the attraction of m upon m', is equal to — mm' .F. df, supposing
only the co-ordinates x', ?/, z! to be variable in f\ the part of the function
[143"J 1 . 7n . (P . d X -{- Q . d y -{- R . d z), depending on the reciprocal attraction of
m and of m', is therefore — mm' . F , df, all being supposed to vary in f
This quantity is an exact differential, when jPis a function ofy, or when the
attraction varies as a function of the distance, which we shall suppose to be
[143 -v] always the case ; the function i:.m.(P.dx-\-Q.dy-\-R.dz) is therefore an
exact differential, whenever the forces which act upon the bodies of the
system, result from their mutual attraction, or from attractive forces, directed
towards certain fixed points. Put therefore for this differential
[143'] dip=:P.dx + Q.dy + R.dz;
* {66d) As was proved in note 13/.
•j- (66e) Suppose the force m! F, in the direction /, to be resolved into three forces
P, Q, R, parallel to the axes of x, y, z, respectively; we should have, by formula [16],
— m! F .6 /= P,Sx4-Q^.Sy-\-R.Ss. The negative sign being prefixed to 5f, because
the force m' F tends to decrease/. This, multiplied by m, gives
m{P.Sx-{-q.8y-]-R-Sz)= — mm'.F.Sf;
consequently, the part of the general formula m .{P. dx-{- Q^.dy-\-R,dz), [143"], must
evidently be of the form — mm! . F . df.
I.v. §19.] LIVING FORCE OF A SYSTEM. 99
and let v be the velocity of m ; v' that of m' &c. ; we shall have*
2.m?j^ = c + 2(p. (R) [144]
This equation is analogous to the equation (g) ^ 8 [40] ; it is the analytical
expression of the principle of living forces. The product of the mass of a '^J^l'"^
body by the square of its velocity is called its living force, and the principle
just mentioned consists in this ; that the sum of the living forces, or the
living force of the vrhole system, is constant, if the system is not urged by
any forces ; and if the bodies are urged by any forces whatever, the sum
of the increments of the whole living force is the same, whatever may be
the curves described by the bodies, provided the points departed from and
arrived at are the same.f
This principle takes place only in those cases where the motions of the The living
bodies change by insensible gradations. If these motions suffer sudden minUhV
..... by sadden
changes, the living force is diminished, by a quantity, which may be deter- '^'''"'ses.
mined in the following manner. The analysis which led to the equation
(P) [142] of the preceding article, will give, instead of it, the following : J
_ ( Sx dx , Sy dy . 5z dz ) ^ ^
0 = ..m.^^.A.^ + ^.A.^+_.A._^_..^.(F.5x+Q.^3,+ii.fe); [145]
"■% ^■% ^•^' being the differentials Of ^, g, g, from one
instant to another ; which differences become finite, when the motions of the
bodies receive finite alterations in an instant. We may suppose, in this
equation,
6x = dx^A.dx\ ^y=dy-{-A.dy; 5z = dz + A.dz; [146]
* (6(1/) The formula [144] is deduced from [143] in the same manner as [40] from
[39] in note 17 i.
f (67) Tliis is demonstrated as in note 16b.
t (67a) The terms j^, j^, &c., which occur in [142], are given in [139], under
the form ^ • JJ^ d . ■^, he. If in these we change d into A, because the differentials of
the velocities ^, ^, &c. are finite, they become A.~, A.^,&c. and [142]
at at "^
changes mto [145].
100
LIVING FORCE OF A SYSTEM.
[Mec Cel.
[147]
[147']
[147"]
[148]
because the values of dx, dy, dz, become in the following instant, dx-{-A.dx,
dy-^A.dy, dz-{-A.dz. These values of 5x,6y,5z, satisfy the conditions
of the connexion of the parts of the system ; hence we shall have
C /dx dx\ dx . /dy , ^ dy\ ^ dy . fdz . . dz\ dz >
( V^^ dt) dt^\dt^ dtj dt^\dt^ dtj dt^
— ^.m.{P.(dx + A.dx) + Q.(dy-i-A.dy)i-R.(dz + A.dz)],
This equation ought to be integrated like an equation of finite differences
as it respects the time t, whose variations are infinitely small, as well as those
of X, y, 2, x\ &c. We shall denote by 2 , the finite integrals resulting from
this integration, in order to distinguish them from the preceding finite
integrals relative to all the bodies of the system. The integral of
m P . (dx -i-A .dx)
is evidently the same as/w . P . dxf therefore we shall havef
constant = ^.m.
(dx^+dy'^+dz^)
+ 2^ . s . m
■s(4:
+ (^.^)+(^
dt^ ' ' ^ V 'd^J ' \ ' dt
2 . ^ ./ . 7n . (P . d X + Q . d y -{- R . d z) ;
^■m
[147a]
c cc
* (68) Put for brevity dx-\-A.dxr=£!:!.dx, then the
integral of m .P . [d x-^-A.dx), relative to the characteristic 2^,
becomes ^,.m .P.A' . dx. The value of this expression may be
conceived of, by supposing the curve bed to be of such a
nature that to any absciss A C = x, the perpendicular ordinate
C c may be w . P. For by talcing the infinitely small quantity ^
B CC'C" JO
C C" = ^ .dx, and drawing the ordinate C" c", the space Ccd' C" will be
m .P .A' .dx,
and the sum of all these elements relative to the characteristic 2^, will represent the whole
curvilinear space Bbc C; so that by taking the integral between the limits x = AB, and
x=AD,'we shall have 2^ . m . P . A' . rf a: == space B b d B. In a similar manner, by taking
C C' = dx, and drawing the ordinate C d, we sliall have the space Ccd C ■=m.P . dx,
and its integral relative to /, taken between the same limits, will, by the usual rules of
integration, give /m. P. <Za;= space jB6f?jD, whence we have S^.m.P.A' .dx=fm.P.dx,
as in [147"].
f (69) JMultiplying the equation [147] by 2, and changing the arrangement of the terms,
we shall find
— 2.2.m. {P.{dx-i-A.dx) + q.{dy-\-A.dy)-]-R.{dz + A.dz)l.
[149]
I. V. ^ 19.] LIVING FORCE OF A SYSTEM. 101
denoting therefore by v, v', v", &c., the velocities of w, m', m", &c., we shall
have
2.mt^ = constant~.,.2.m. [ (A.^)V(A.i^)V(A.^^^
The quantity contained under the sign 2^, being necessarily positive,* it is
evident, that the living force of the system is diminished by the mutual
fi or fi 11
action of the bodies, whenever any of the variations A.-—, A.— ^, &c.,
at at
become finite, during the motion of the system. The preceding equation
furnishes a very simple method for determining this diminution.
At each sudden change of motion in the system, we may suppose the
velocity of m to be resolved into two others, one of which v remains in the
following instant ; the other V is destroyed by the action of the bodies ; now
the velocity of m being 1a^~ — ' ^^^'^^^ ^^^ resolution [40a], and
becoming afterwardsf '^(<^- + ^■<^''f + (iy+±dyT + i^' + ^-"^f, jt
at
is [149']
Now
/ rfar , dx\ dx /dx . d x\^ /dx\^ rd x\^
whose integral relative to 2^ is f — ) ; and the similar terms relative to y, z, furnish the
/dyY /dz\^
^^^^^ \di) ' \dj) ' -^S^^' ^^ integral of — 2 .i: .m . P . {dx -\- A. d x), relative to
the characteristic 2,, is by [147 a], equal to — 2.:E.f.ni.P.dx, and tiie similar terms
in y, 2, produce — 2.2./.m.Q.<Zy, —2.:s.f.m.R.dz', with these reductions tiie
integral of the preceding equation, relative to 2,, becomes as in [148], and by substituting for
j^ Its value IT , &z;c. [40a] it changes into [149],
positive.
(70) Because each term of the expression, as (a. jrj , is a square, consequently
t (71) This expression of the velocity at tiie second instant, is of the same form as that
in [40a], changing tiie elements d x, dy, dz, corresponding to tiie first mstant, into
dx-{-A.dx, dy-\-A.dy, dz-{-A.dz, correspondmg to tiie second instant.
2^
[152]
102 LIVING FORCE OF A SYSTEM. [Mec. Cel.
easy to perceive that we shall have*
the preceding equation can therefore be put under this form,
[151] 2. mt?^ = constant — 2 .2.m V^-{-2^.f.m(P, dx-{- Q,dy + R ,d z.)
20. If in the equation (P) § 18 [142], we suppose
6x"=^x+^x'/ \ ^f=^y + ^y!'; 8z"=6z+6z;';
&c.
by substituting these variations in the expression of the variations 6f^ Sf,
8f", &c. of the mutual distances of the bodies of the system, whose values
are given in § 15 [118] ; we shall find that the variation 6x, 6y, Sz, will
disappear from these expressions. If the system is free, that is, if no one of
its parts has any connexion with foreign bodies ; the conditions relative to
the mutual connexion of the bodies, will depend only upon their distances
from each other, and the variations 5x, 8y, Sz, will therefore be independent
of these conditions ;t whence it follows, that, if we substitute the preceding
* (7 la) The primitive velocity of the body m, in a direction parallel to the axis of x, is
-— , which after the first instant becomes -— ^ — , consequently the loss of velocity, in
at at
^ ^ ^x
that direction is — -r^ — . In a similar manner the losses of velocity in the directions
parallel to the axes of y and z, are respectively j — , — -^ — , and tlie sum of the
Civ (Z V
squares of these expressions is, as in [40a], evidently equal to tlie square of the whole loss of
velocity, or F^, as in [150]. The substitution of this, in [149], gives [151].
f (72) As the system is supposed to be wholly unconnected with any foreign body, we
can suppose each one of the bodies to be moved through an equal space, in a parallel
direction, without producing any change in tlieir relative situations, or in their mutual actions
upon each other. Therefore we may suppose each of tlie bodies to be moved through tlie
arbitrary spaces 5x, Sy, Sz, parallel to the three axes x, y, z, respectively ; and tliese
spaces may be varied at pleasure, without affecting the relative situation of the bodies j or, in
other words, without affecting the vakies of Sxf, Sx", he. 8yJ, 8yJ', he. Szf, Sz", &c.
[152]. Therefore, if we substitute the values of [152] in the equation [142], it will not
generally be satisfied, unless the coefficients of these arbitrary quantities 8 x, Sy, S z^ are put
separately equal to nothing, hence we obtain the three equations [ 1 53].
I.v. ^20.] MOTION OF THE CENTRE OF GRAVITY. 103
values of (5a/, hj, 6z', Sx", fee, in the equation P [142], we ought to put the
coefficients of the variations of 5 a:, <5 y, 8 z, separately equal to nothing ;
which will give these three equations.
Let X, Y, Z be the three co-ordinates of the centre of gravity of the system,
we shall have, by % 15, [126, 127],
^^2^ y^^y Z = ^; [154]
2.OT ' 2.m 2.m
consequently*
^~~d? 27^"' df 2.m ' d'lP 2.m '
therefore, the centre of gravity will move in the same manner, as if all the
bodies m, m', m", &c., were collected in that centre, and all the forces which
Motion of
act upon the different bodies of the system, were directly apphed to the %^^^^^
whole mass collected in that centre.
If the system is affected only by the mutual action of the bodies upon each
other, and their reciprocal attractions ; we shall have
0 = 2. mP; 0 = 2.7rtQ; 0 = 2. mi2; [155']
[155]
[155']
^ wi X T" itt d d X
* (73) The second differential of X=— , is ddX=— , which, multiplied
, 2.TO . ddx ddX „ o ■> • ^- i.- • r^ ^ /ddx nAncon
by -—— , eives 2 . m . -—-==——-. 2 . m. Substituting this m 0 = 2 . m . -j-- Jr ) [1 53 1,
■' dP ° rf<2 dfi ° \ dfi / ■-
• 1, .. ddX „ 1- , J. .J J t. ^ • r. d^^ 2.mP
It becomes 0 = — -- . 2 . wi — ^ .mP, which, dmded by 2 . m, gives 0= -—
dfi ^ ° d1^ j,.m
[155]. The two other equations, in Y, Z^ are found in the same manner. Now if all the
bodies were collected in the centre of gravity, and all the forces applied to it, as above, the
sum of all tlie forces in the direction parallel to the axis of a?, would be 2 . m P. This
divided by the sum of tlie masses, 2 . m, would give the accelerative force, acting upon one
particle, equal to — '- , wliich is what is called P in [38]. Therefore the first of the
equations [38], for finding the motion of a particle, is similar to the first of [155]. In like
manner the second and third of the equations [38] become like those of [155] ; conse-
quently the motion of the centre of gravity is found by the same equations as that of a single
particle of the mass 2 . m, collected at the centre of gravity. The motion of the centre of [I55a]
gravity will, therefore, be exactly tlie same, as that of the congregated mass, supposing the
forces to be applied at that centre, in the manner mentioned above.
104 MOTION OF THE CENTRE OF GRAVITY. [Mec. Cel.
For, if we put p to denote the reciprocal action of m upon m', whatever be
its nature, and/ for the mutual distance of these two bodies ; we shall have,
by means of this action only,*
[156]
[158]
„ p.(x — x') „ p.(y — i/) p p.{z — z!)
/ / /
whence we deduce
[157] 0 = mP + m'P'; 0 = mQ + m'Q; 0=:mR-{-m' R ;
and it is evident, that these equations exist, even when the bodies instanta-
neously exert, upon each other, a finite action ; so that their reciprocal action
must disappear from the integrals 2 . w P, 2 . m Q, 2 . m J?, therefore these
integrals will become nothing, when the system is not acted upon by
extraneous forces. In this case, we shall havef
^~ df ' ^~ df ' ^~~d¥'
* (74) Using the figure in page 8, let the body m, whose co-ordinates are x, y, z, be at
A ', the body m' whose co-ordinates are x', y', z', be at c, and the distance A c ^f. Then
the force p, which we shall suppose to act upon the body m' in the directions A c, would
produce a force in a direction parallel to A D, represented by p . — — = p . — -— , as
Ac f
is evident by the first of the formulas, [13] ; this force is what is called above mf I^.
From the same formula it follows, that the force p, acting upon the body m, in the direction
c A, would produce a force in a direction parallel to D A, represented by
AD (x'—x)
[156a] P'Al-^=^'~r''
(x x')
or in other words, a force in the opposite direction A D represented by p . — —j and
called above m P. Adding this to the preceding value of m! P', the sum becomes nothing,
as in the first equation [157], and the two otlier equations [157], are found in the same
manner for the other axes y, z.
f (74a) The equations [158] are deduced from {1 55], by substituting the values [155"].
The first integrals of [158], are jj = b, rfT~ ^'' Jt^^"' ^^^ square root of the
sum
of the squares of these is y/(^£J+ (™J-f (^-^J==V'b^-\-V'-\-b"^ the first
I. V. §20.] MOTION OF THE CENTRE OF GRAVITr. 105
and by integration,
X=^a + hU Y = a'+h't, Z = a!' + h"t\ [159]
c, 6, «', 6', «", &", being arbitrary constant quantities. By exterminating the '^"^'
time ^, we shall have an equation of the first order, between X and Y, or
between X and Z ; whence it follows, that the motion of the centre of gravity
is rectilineal. Moreover, its velocity being equal to
1/
l^Yo-mViri^^'
.T;+^Tr;+^^)' '"^^
or to vh^-\-h'^-\-h"^^ it is constant, and the motion is uniform.
It is evident, from the preceding analysis, that this permanency in the
motion of the centre of gravity of a system of bodies, whatever be their
mutual action, exists even in the case, where some of the bodies lose
instantaneously, by this action, a finite quantity of motion.*
[159"]
member of which represents the velocity of the centre of graAaty, as is shown in [40a],
therefore that velocity is equal to the constant quantity \/h'-\-b'^-\-h"^. Taking the
integrals of — =&, kc, we obtain the expressions [159]. The value of ^, deduced from
the first, being substituted in the second and third, gives Y, Z, in equations of the form [159a]
F=./2 X-\-A'', Z^B X-{-B', which are the equations of a right Ime [19S"], therefore
the motion of the centre of gravity must be in a right line.
* (75) That the uniform motion of the centre of gravity, is not disturbed by a sudden
change of the motions of some of the bodies of the system, arising from their mutual
attractions, or impact, &;c. may be proved, by means of the equation [145], in which this
sudden change of motion is supposed to take place. For, by substimting, in tliis equation,
the values of Sx',Si/, S z', Sx", he. [152], and putting, as in note 72, the coefficients of
Sx, Sy, 6 z, equal to nothing, we get
d X
The first of these equations gives 'S. .m. A .jj='L.mP. Now the differential of the first
of the equations [154], divided by — - is ^ .m.~ = ^-£.'Si.m, and its differential relative
2,-in at at
. „ dx dX
toA, is2.m.A.— =A. — .2.m. Substituting this, in the former equation, it becomes
A.^.2.m=2.mP, or a/-^=^; and in Kke manner, A. 11=?:^, and
27
106
PRESERVATION OF AREAS.
[Mec. Cel.
[160]
21. If we put*
y .5 X
6x =
6v =
y
— X . S X
5x/,
6x' =
i/ .Sx
r^/y' .
{-Sx;;
Sx"
f.Sx
+ Sx;'; &c. ;
y
y/^
y ' ' y
y "' " 3/ "' " y
the variation S x will again disappear from the expressions! of ^/, Sf'^ 5f",
A.-—=— . But when the system is subjected only to the mutual action of the bodies
upon each other, we shall have [155"], 2.wP = 0, 2.mQ = 0, 2. mR = 0 ; hence
dX . dY _ dZ
'dT
of finite differences A, give —-=h, ■—- = 1', —— = 6". Integrating these, relative to ^, we
d t d t d t tjtj
obtain X= a^ht, Y=a'-\-h't, Z= a"-^h"t, as above.
A . ^TT- =0, A . - — = 0, A . — — = 0. The integrals being taken, relative to the characteristic
* (7G) For the sake of symmetry, the value
6x= \-sx,,
y
which is not in the original, is here inserted, supposing
6x^=^0. It may be observed that in the formulas [ 1 60],
the whole system is supposed to have an angular rotatory
motion, equal to — , about the axis of r ; so that for any
one of the bodies, as m', whose co-ordinates projected upon the plane of x, y, are C'A = cd,
Am! = yi, and distance from that axis C'm'=s', this rotatory motion would be represented by
5x . .
the arch C m' = s' . — . This would increase the ordinate x' by the quantity
y
Y
rm
y^
\
(y^'
,'■""
y^
y
^' >^
(J'
A
1
1 y
AD = BC
s y
and would decrease the ordinate t/ by the quantity B m' = C m' X -
x'A:
These
ai"e the first terms of Sx', Si/, [160]. Those of S x", Sy", are found in the same manner,
or by merely adding another accent to the letters x', y', he.
f (77) The expressions of /,/',/", Stc. [US], and the assumed values of the variations
Sx, Sxf, Stc, Sy, Si/, he, [160], being of a symmetrical form, it will only be necessary to
prove that Sx disappears from any one of the quantities Sf, Sf, Sf", &ic., as Sf". Now
the value of Sf", found as in [109&], supposing z', z!', invariable, is
l,X^'-x').{Sx"-.Scd)-^l,{f-]/)'{^f-^^\
[161]
[ie2]
I. V. .§21.] PRESERVATION OF AREAS. 107
&c. ; supposing therefore the system to be free, the conditions relative to the
connexion of the parts of the system, affecting only the variations 5/, 5/',
&c., the variation 5 a: is independent and arbitrary ; therefore by substituting,
in the equation (P) § 18 [142], the preceding values of 5 a:, <5a/, <5a/', &c., [1601
hy, 6i/, 6y", &c., we ought to put the coefficient of 5 a: equal to nothing ;
hence we get*
whence we deduce, by integrating relative to the time t,
c^z.m.^^-jf-^ + ^.f.m.iPy-Qxy.dt:
c being an arbitrary constant quantity.
We may, in this integral, change the co-ordinates y, «/, &c., into z, 2f, &c.,
provided that the forces R, R\ &c., parallel to the axis of z, are substituted
instead of Q, Q', &c., which are parallel to the axis of y ; hence we get
c' being another arbitrary constant quantity. We shall have, in like
manner,
c" being a third arbitrary constant quantity.f
which, by substituting the above values of 5 a/, 5 a;", 5 y, 5 y", and retaining only the terms
multiplied by 5 a:, becomes — \ {a!' — of) . (y" — r/) + (y" — i/) . ( — oi/'-\-oi/) > , which is
evidently = 0, because the terms between the braces mutually destroy each other.
* (77a) Substituting the values [160] in [142], retaining only the terms multiplied
by 5 a;, putting the coefficient of 5 a; equal to 0, and multiplying by — y, we get [161],
whose integral, relative to dt, gives [1 62]. The same method of reasoning, applied to the
co-ordinates x, z, combined together gives [163], and applied to those of y, z, gives [164].
f (776) If we put cf= — c,^ and d'=c,, the equations [162, 163, 164] may be put [161a]
under the followmg more symmetrical form,
c = ^.m}—^-^-^^:s:.f.m.{Py—qx).dt, (162al
c,= -S..my ^~^ ^^-f-2./.m. {Qz — Ry).di. [163a]
c^, = i:.m.- ^- '-{-2.f.m.{Rx—Pz).dt. [164a]
108 PRESERVATION OF AREAS. [M^c. Cel.
Suppose the bodies of the system to be affected only by their mutual
action upon each other, and by a force directed towards the origin of the
co-ordinates. If we put, as above, p for the reciprocal action of m on m', we
shall have, by means of this action alone,*
[165] 0 = m . {P y — Qx) + m! . {F 1/ — q x') ;
consequently the mutual action of the bodies will disappear from the finite
integral i^ .m . (P y — Qx). Let S be the force which attracts the body m
towards the origin of the co-ordinates ; we shall have, by means of this
force alone,t
— S.x —S.y
[166] p^-—=i==. Q^—= .
therefore the force S will disappear from the expression Py — Qx, and if
In which each equation may be derived from the preceding one, by taking in these three
[165o] ggj-ies of letters, c, c^, c^^ ; x,y,z; P, Q, -^ j the next letters in order, observing to begin
the series of letters again, when it is required to change tlie last terms c,^, z, R, which
become respectively c, x, P.
* (78) By [156a] it appears that this force ^ produces the forces m P = p . -^^—r- i
m'P=p. — -r— , parallel to the axis of a?; and in a similar manner, m Q = p. — - — ,
(y'—y)
m' ^ =p -; — , parallel to the axis of y. These give
«» • (^ 2/ — Q *) = 7 • ^ y • (^— ^')— ^ • (y— y') I = J • (^y — y *')'
and m' . [T'lf — Q' a;') = ^ . j y' . {x' — x) — ctf . {y' — y) \=-. ( — x /+y a/), and the sum
of these two is m . [Py — (^x)-\-m' . [P' ?/ — ^ x') = 0, since the terms of the second
member mutually destroy each other.
f (78a) The force S, in the direction of the origin of the co-ordinates, may be resolved
into the forces S .-, S.—, S. - [137a], parallel to the co-ordinates a:, y, 2:, respectively;
and since s = Vx^-\-y^-{-z^, [137&], tlie two former forces become as in [166], the
negative sign being prefixed because this force is supposed to decrease tlie co-ordinates.
Substituting these values of P, Q, [166], in Py — Q^x it becomes 0. Noticing, therefore,
only the mutual action p, of the bodies on each other, and the force S, we may neglect
P.q.R,in [162, 163, 164], and they will become as in [167].
I. V. §21.]
PRESERVATION OF AREAS.
109
we suppose the different bodies of the system to be affected only by their
mutual attraction, and by forces directed towards the origin of the co-ordinates,
we shall have ^ ^*'"^ - ^'
(xdz — zdx)
(xdy — ydx)
dt '
c'=i.m.-
dt
dt '
(^)
[167]
If the path of the body m be projected, on the plane of a;, y, the
differential — ^~ — , will be the area described in the time dt, by the ^ ^
radius vector drawn from the origin of the co-ordinates,* to the projection
B F
i.DE.BF.
* (79) In the adjoining figure let A be tlie
origin of the co-ordinates, CD the projection
of the part of the path of the body m, on the
plane of x y, described in the time d t. Draw
the ordinates C B, DEF; also, C E parallel
to B F, and join A E. Then the triangle
A C D, described in the time d t, is equal to
AED—AEC—DEC.
But
AED=i.AF.DE=i.{AB+BF).DE; ^
AEC=i.BC.CE = i.BC.BF; DEC=^.DE.EC
Hence,
ACD = i.DE{AB-^BF)—i.BC.BF—^.DE.BF=i.{DE.AB—BC.BF),
which, by substituting AB = x, BF= dx, B C = y, D E=dy, becomes
ACD = ^.[xdy—ydx).
This might have been simplified a little, by neglecting wholly the infinitely small triangle
C D E, of the second order.
If the angle BAC==zi, and A C=p, we shall have a;=p . cos.-nf, y = p. «i». zi, [27],
and the area dA, of the infinitely small triangle A C D, may be found by describing about
the point ^ as a centre, with the radius A C, the circular arch Cc=p.d'&, to meet
^ D in c. This arch, multiplied by ^.^ C = i. p, gives the area of the triangle A C c, or
•^ CD=i.f^ .d-si, which is to be put equal to the value above found ; hence
dA = ^.p^ .dvi = ^.(xdy — ydx).
In Uke manner, since CD= \/CE^+DW= \/~Dj'-fc^, we shaU have in symbols.
CD^Vdx'-^df'^Vdf+f.d^^.
28
[167a]
[1676]
[167c]
110 MOTION OF A SYSTEM OF BODIES. [Mec. Cel.
[167"] ^^ *^ ' *^® ^^^ ^^ these areas, multiplied respectively by the masses of these
Princi le ^^^i^^j ^^ therefore proportional to the element of the time d t ; whence it
preJrvV follows, that in a finite time this sum is proportional to the time. In this
t'on of . 1 • • 1 r 1 • r
areas. cousists thc prmciplc 01 the preservation oj areas.
The fixed plane of a:, y, being arbitrary, this principle exists for any plane
whatever ; and if the force *S^ is nothing, that is, if the bodies are subjected
only to their action upon each other, and to their mutual attraction, the
origin of the ordinates will be arbitrary, and we may take this fixed
point at pleasure. Lastly, it is easy to perceive, by what precedes, that
this principle holds good even in the case in which, by the mutual action of
[167""! ^^^ bodies of the system upon each other, sudden changes are produced in
their motions.*
The velocity of the body m being v, the space C D, described in the time d t, will
he V d t. If we multiply this by the perpendicular jp, let fall from «^, upon the tangent at C,
or on the continuation of the line C D, it will give another expression of the area of the
triangle A C D, represented by ^.p.vdt. Putting this equal to the former value
\X dij ij d x\
^.[xdy — ydx), we ^et pv.dt=x dy — y d x, hence mv.p = m. . Now
mv .p represents the momentum of the body m, about the axis of z, [29'], and if we put
this equal to M, we shall have
[167rf] M=m^-^^^~-^^.
dt
Accenting these with one accent for the body m', two for the body m", he. and taking the
( y* H qj 77 // T" I
[I67c] sum of all, we shall have 2 . ./If = 2 . wi . ^^ — ^-rr — ' J hence, from the first of the equations,
at
[167], we get 2 . Jli =:; c. A similar result might be obtained relative to the other axes.
Thus it appears, that the principle of the preservation of areas is equivalent to the
supposition, that tlie sum of the momenta of all the bodies, about any axis, which is nothing
'• •^■' in the case.oC equilibrium, is constant, in the case of motion here treated of.
* (80) Substituting the values 8 of, S x", &;c., Sy'.S f, he. [160], in tlie equation [145] ;
then putting the co-efficient of 8 x equal to nothing, and multiplying by — y, we shall get
0=2. m. — ^ + 2.m. (Py — Q^x) .dt;
dt
in which the terms 2 . m . {P y — Q^x).dt vanish, when the bodies only act on each other
by impulse or attraction, or are impelled by forces tending to the centre of the co-ordinates
[165, 166]. Hence we have 0=2 . m . ^^l^^II^— -Y This is to be integrated as an
I. V. § 21.] CHANGE OF CO-ORDINATES. 1 1 1
There is a plane with respect to which the quantities c', c" are nothing, [I67iv]
and on this account it is interesting to know it : for it is evident that this piane in
° _ ^ which the
supposition of c'=0, c" = 0, must tend greatly to simplify the investigation ^^^^f^^^
of the motions of a system of bodies. To determine this plane, it is multiplied
•^ -"^ by the
necessary to refer the co-ordinates x, y, z, to three other axes, having the ^SJ^
same origin as the preceding. Therefore let 6 be the inclination of the mum.
required plane, which is formed by two of these new axes, to the plane of
X, y ; and 4^ the angle which the axis of x forms with the intersection of [167 »]
these two planes, so that - — (3 may be the inclination of the third new
axis to the plane of x and y, and - — \ may be the angle which its
projection upon the same plane, makes with the axis of ar, * being the semi-
circumference.
To assist the imagination, suppose the origin of the co-ordinates is at
the centre of the earth, and the plane of a:, y^ is that of the ecliptic, the
axis z being the line drawn from the centre of the earth to the north pole
of the ecliptic ; suppose also that the required plane is that of the equator,
and that the new third axis is the axis of rotation of the earth, directed
towards the north pole ; ^ will then be the obliquity of the ecliptic, and \ [167 »«]
the longitude of the fixed axis of a:, counted from the moveable vernal
equinox. The two first of these new axes will be in the plane of the
equator, and by putting cp for the angular distance of the first of these axes
from that equinox, 9 will represent the rotation of the earth counted from
A [167WI ]
that equinox, and -^^^ will be the angular distance of the second of
equation of finite differences, upon the principles adopted in [147']. From which it would
follow, since the differentials o{x,y,t, are infinitely small, that we may put A.{xdy)
for x.Ady, and A. {y dx) for y . Ad x -, hy which means, the preceding expression will
become, 0 = 2 . w . A . ( — V whose integral, relative to the characteristic A,
or 2.^, is c=2i.m. ~ ; as above. In like manner we may find the other two
equations [167].
112
MOTION OF A SYSTEM OF BODIES.
[Mec. Cel.
Principal thesB RXGS fioin the same equinox. We shall name these three new axes
the principal axes.* This being premised,
[I67'"i] Let x^, 2/^, z^, be the co-ordinates of m referred, Jlrst, to the line drawn
from the origin of the co-ordinates, to the vernal equinox, the positive values
of X being taken in the direction of that equinox ; second, to the projection
of the third principal axis, upon the plane of x, y ; third, to the axis of z ;
we shall havef
x = x^. cos. ^-{-y, . sin. 4- ;
[168] y = y^.cos.4^ — x^.sm.-^;
z = z.
* (81) Suppose that BFPE represents the ecliptic, or
plane o( x,y', CAIX the axis of a:, C D Y the axis of y,
the axis of z falling above C, perpendicularly to the plane of
the figure. Also BOP the plane of the two principal axes
^i/n Viii^ ^he part BOP being supposed helow the plane of the
figure, and making with it an angle F B 0=6, and the angle
A CB=^^. Let C Ghe the first principal axis x^i, CjfiT the
second, or axis of y^, the third being drawn through C
perpendicular to the plane of the equator B G HP, and
falling above the plane of the figure, its projection on this plane
being on the line CF, drawn at right angles to CB, making,
with the axis of x, the angle F C j1= — — 4'. Lastly, cp is
the angle B C G, which the first principal axis C G, makes with CBj; the angle, which the
second principal axis C H makes with the same line C B, is -^ -j-ip.
f (S2) Let K be the projection of the place of the body m upon the plane of the
ecliptic, in the above figure, the co-ordinates of this point being either CI=x, IK=y', or,
C L=x^,LK=y^. Through L draw L M parallel to KI, and L JV parallel to C I, to
meet K I produced in JV. Then in the right angled triangles C ML, KJYL, we have the
angle MCL = J\'KL = ^. Hence C M= C L. cos. M CL = x, . cos. 4.; •
M L = IJ\'= C L.s'm. M CL=x^. sin. ^-j
KJV= KL . cos. JVKL = y,. cos. 4. ; L JV= MI=KL. sin. JVKL = y, . sin. ^.
Substitute these in a? = C /= C M-{- MI, y=K 1= K J\ — 7 JV, and they become as
in [1G8]. As the axis of z is not changed, we shall have z = z,, [168].
I. V. ^21.]
CHANGE OF CO-ORDINATES.
113
Let x^^, y^^, z^,, be the co-ordinates referred, ^rs^, to the line of the vernal
equinox ; second, to the perpendicular to this line, in the plane of the equator ;
third f to the third principal axis ; we shall have*
*^/ =^ **'// »
y, = Vn • COS. ^ + z^^. sin. a ;
z^ = z^i . COS. t — y^^. sin. L
Lastly, let x^^^, y^^^, z^^^, be the co-ordinates of m, referred to the first, second
and third principal axes, we shall havef
^// = ^///-cos. 9 — ^^,,.sin.(?;
y. = y,u ' COS. 9 + a:^„ . sin. 9 ;
z = z .
II III
* (83) In this part of the computation the place of the body
is supposed to be projected upon the plane of tlie solstitial colure
F C O, (Fig. page 1 12), which we shall suppose to be the
plane of the annexed figure. The point of projection being k,
its co-ordinates may be either Ci=y^, ik== z^; or CI = y„,
lk = Zi,. Draw Z m re parallel to ik^ and kn parallel to C i.
Then in the right angled triangles Cml, Ink, we have the
angle lCm = nlk=6, hence
C m = CI. COS. I C m = y^^. cos. 6 ;
Im = CI . sin. I C m = y^^ . sin. 6 ;
In^lk . COS. kln = Zii. cos. ^ ;
kn = mi = lk . sin. kln = z^^ . sin. 6.
Substitute these In y^ = Ci = C m -\- m i, and z^ = ik = mn = ln
[169]. The axis of x^ not being altered we get x, = x^, [169].
f (84) In this case, the axes in the plane of tlie equator, are
changed fiom CB, CO to C G, CH, (Fig. page 112, 113). The
ordinates in the first case being CT=x,,, Tq=y,^, and in the second
CR = x„„ Rq = y,„. Draw R S perpendicular to C T, and R U
equal and parallel to ST. Then, in the right angled triangles
CSR, qUR, we have the angle RCS = RqU = (p', hence,
C S= CR .COS. R C S=x,^^.cos. cp;
RS{=TU) = CR. sin. RCS=x,„.sm. 9j
qU=Rq. cos. RqU= y^,, . cos. 9 ;
RU=ST=Rq.s\n.RqU=y,,,. sin. <p.
Substitute these in
x,.,= C T=CS-ST, y„=Tq =qU-{- TU,
they become as in [170]. The axis of z,^ remaining unchanged gives z^ =z [170].
29
[169]
[170]
/ »?, we get y,y z^
114 MOTION OF A SYSTEM OF BODIES. [Mec Cel.
Formula, Whexicc It Is casj to conclude*
for the
therlctan- X = X ^^^ . {cOS. ^ . sifl. 4^ . Slll. (p + COS. 4^ . COS. (p}
f alar axes. . . •, .
-\- y^^^ . [cos. & . sm. %}. . cos. 9 — cos. -^^ . sin. <P j+ 2,,, . sin. 4 . sin. 4 ;
[171] y ==a?^^^.[cos. ^ . cos. 4>. sin. (p — sin. 4^ . cos. ?}
+ y^^^ . [cos. & . COS. 4^ . cos. (p + sin. 4. . sin. 9}+ z^^^ . sin. a . cos. 4. ;
z = z^^^ . cos. ^ — y^^^ . sin. ^ . cos. 9 — a:^^^ . sin. & . sin. 9.
Multiplying these values of ar, y, 2^, respectively by the coefficients of x^^^
in those values, and adding these products together, we shall obtainf
r — ar . f COS. 6 . sin. -l . sin. cd + cos. 4^ • cos. 9 \
[172] '" ^ ^ . . ^ ,
+ 2/. [cos. <5. COS. 4- • sin. 9 — sin. 4- • cos. 9 J — z . sm. 6 . sin. 9.
* (85) Substituting in x, [168], the values x^, y,, [169], it becomes
x=x^, . COS. 4^ + sin. 4^- [ 2///- cos. 6-\-z,^.sm.&] =^„ . cos. 4^ -j- 2/// • cos. d . sin. 4* -|- 2;^^ . sin. 5 . sin.4',
and by substituting in this last the values of x,,^ y,,, z^, [170], we obtain
a:=cos. 4- . [ Xiii . cos. 9 — y^ . sin. 9 1 -f cos. 6 . sin. 4- -{y,,, • cos. 9 +a?,^^ . sin. 9} + ^/// • sin. 6 . sin. 4',
by reduction it becomes as in [171].
In a similar way we may find y ; or, more briefly, by changing, in this value of x, sin. 4^
into cos. 4^, and cos. 4^ into — sin. 4^ ', for these changes being made in the values of x [168],
it becomes — x^ . sin. -^-{-y^ . cos. 4^, which is equal to the value of y, [168], therefore the
same changes being made in x, [171], it will become the same as y, [171]. Lastly, the
values z, z^, z^^, [168, 169, 170], give successively
z = z^^z^f. COS. 6 — y,, . sin. & = z^^^ . cos. 6 — sin. 6 . [ y^^^ . cos. 9 -|- x^^^ . sin. 9 j
= z^ff . COS. 6 — y,„ . sin. 6 . cos. 9 — x^^, . sin. 6 . sin. 9, [171].
f (86) If we for brevity put
Aq = COS. 6 . sin. 4- • sin. 9 -|- cos. 4' • cos. 9,
Ai = cos. 6 . cos. 4^ . sin. 9 — sin. ^ • cos. 9,
w^a = — sin. & . sin. 9 ;
Bq = COS. 6 . sin. 4^ • COS. 9 — cos. 4- • sin. 9,
J9j == COS. 6 . COS. 4^ • COS. 9 -j- sin. 4^ • sin. 9,
JBg = — sin. 6 . COS. 9 ;
Cq = sin. 6 . sin. 4>j
Ci = sin. d . COS. 4^,
Cg = cos. 6.
[171a]
I.V. §21.] CHANGE OF CO-ORDINATES. l^^
In the same manner, if we multiply the values of x, y, z, respectively by
the coefficients of y^.^ in these values ; and also by the coefficients of z^,^, we
[^72a]
The equations [171] will become
x=Aq. X,,, -\- Bq . y,„ + Co . «„„
y=A' ^,„ + ^1 • y,n + ^1 • ^///'
z = A^. x,„ + B^ . y,„ + Ca . z,„.
If we put r for the distance of the body m from the origin, we shall have, by [19e],
r2=a:2_j_yO._|_^. ^j^^ ^^^^^2 _|.y^ 2_|_ ^^^2^ whence x'^^y^^z'' = x„f^y„J-^z„f.
Substituting the above values of x, y, z, [172a], we shall get an identical equation in
^///> ViiP ^iin in which the coefficients of xj, yj, z,,f, in both members, must be 1 , and the
coefficients of the products of a?^^„ y„„ z,„^ must be nothing ; whence we obtain,
Bi-\-B^^Bi = \,
^0.^0+^1.^1 + ^2.-^2=0, [172fc]
•^0 • ^0 + •^l • ^\ + .^2 . Cg == 0,
5o-C'o + ^i.Ci + 52.C2 = 0.
Multiply the equations [172a], by A^^ A^, Aq, respectively, and add the products, we shall
get by means of the equations [1726], the value of x,^^, [1'72]. In like manner, multiply
the same equations [172a], by Bq, jBj, B^, respectively, and add the products, we shall get
y^^^ [173]. Lastly, multiply the same equations by Cq, C^, Ca, respectively, and add the
products, we shall get, by tlie same reductions, tlie values of z^^^, [174], all of which are in
the following table ;
^M = A 'X-\-Ai.y-}-A2.z,
y,„ = B^.x + B^.y-\-B^.z, [172^j
^///=^^-^+C^i-y+ ^^^^^
If we substitute these values in the above equation x^ -\- y^ -]- z^ ==■ x^f -\- y^^f -\- zj, we
shall obtain an identical equation in a?, y, z, in which the coefficients of o?^ , y^ , 5^ , are
1, and the coefficients of the products of a?, y, z, are nothing, whence we get the following
equations, similar to [1726],
A' + ^o'+Co^=l,
^i2+B,2+C,2=l,
•^0 • "^l + -^0 • -^1 + C'o . Ci =: 0,
•^0 . •^2+ ^0 • ^2 + ^0 • ^2 = 0,
^1.^2 + 5l.J?2+Ci. C2 = 0.
The quantities A^^ A^^ &c. represent the cosines of the angles, formed by the axes
[172dJ
116 MOTION OF A SYSTEM OF BODIES. [Mec. Cel.
shall successively find
[173] 2^^^^ =a;.fcos. ^ . sin. 4- . COS. (p — cos. -1. sin. ?}
-f-y .{cos. Kcos. N^. COS. (? -|- sin. ^. sin. <pj — z . sin. ^. cos. (p ;
[174] 2^^^ = a:, sin. ^. sin. 4- + 2/ • sin. ^. COS. 4- + 2:. COS. ^.
These different transformations of the co-ordinates will hereafter be very
[172e]
[172/]
[172s-]
of X, y, z, witli those of x^^^, y^i, z^^^ ; so that if we represent, by {x, y^^^), the angle formed
by the axes x, y^^^ and in lil<e manner for the angles formed by die other axes, we shall have,
^0 = cos. (a?, a;^,J ;
-So = cos. (a:-, yj;
Co = cos. {x,zj;
A = COS. {y, xj ;
Bi = COS. {y, yj ;
Ci = COS. (y, zj ;
^2= COS. {z,xj',
Bz= COS. {z,yj;
C2=cos. {z,z,,,);
[172h]
To prove this, we shall refer to the adjoined figure, in which
C X, C Y, C Z, are the rectangular axes of x, y, z, and
C X-iii the axis of x^^^. Let M be the place of the body m,
whose co-ordinates are CA = x, ./3 B = 7/, parallel to C y,
&nd BM = z, parallel to C Z. From the points A, B, M,
let fall, upon C X^^^, the perpendiculars A a, B b, Mm ;
then C m will be the value of x^^^, corresponding to tlie body
at M. Hence a:^,, = C m= C a-\- ab-\-hm. Now by
the principles of the orthographic projection, we have
C a= C A . cos. (x, x^i) = x . cos. {x, x^^ ; (^ JL JC
ab = AB. cos. (tj, x^) = y . cos. (//, x^^^, and bm = B M. cos. {z, x,,^ = z . cos. {z, x„,).
Substituting these in the preceding value of x^^^, we get
^/// = ^ • COS. {x, x^^,) + y . cos. (y, xj -f z . cos. {z, x,^;).
Comparing this value of x^,^, with the first of the equations [172c], which is identical with it,
we get the values of Aq, A^, A^ [l''^/]- I" ii^^c manner, using the axis of y^^^, instead of
C Xiip we get the values of ^O) -^u -^2 ; and the axis of z^n gives the values of Cq, C^, Cg.
If we divide die value x,i^, ['"^-o"]? hy r =^V x'^ -\- ■f ~\- z^ , observing, that by the
principle of orthographic projection, -^ represents the cosine of the angle, which the line r
makes with the axis of x^,,, or cos. (r, x^^) ; in like manner - = cos.(a?, r), - = cos. (y, r),
- = cos. {z, r), we shall get,
cos. (r, x„) = cos. (a?, r) . cos. {x, x„) -\- cos. (y, r) . cos. (y, x,,) + cos. (2r,r) . cos. (2:, x,„).
I. V. <§21.]
CHANGE OF CO-ORDINATES.
useful. By marking, at the top of the letters, x, y, 2, x,^^, y^^^, 2^,,, one accent,
two accents, &tc., we shall have the co-ordinates corresponding to the bodies
?»', m", &c.
Hence it is easy to deduce, by substituting c, c', c", instead of
2.m.
{xdy — ydx) ^
dt
2.m.
[x dz — zdx) ^
dt
2.m.
{ydz — zdy)
dt '
117
[174']
This is a well known formula of spherics, which will be used here-
after. For the sake of illustration, we may refer this formula to the
arches of a spherical surface xyzr x^, described about the centre
C, with the radius unity ; since the angles {x, r), (y, r), {z, r),
(r, Xii)i [x, a?^/J, &c. will correspond respectively to the arches x r,
yr, zr, &;c.
In Fig. page 112, the arch A G, drawn upon the spherical surface
ABG, whose radius is unity, is the same as the angle {x, x^^, so that we
shall have, as in [172/], cos. A G = cos. (x, x^^^ = Aq. If we makff^
use of the value of .^q [171a], and rcsubstitute ■\> = AB, cp = BG,
d= angle AB G, we shall obtain the following fundamental theorem of
spherics, . '(}
COS. AG = cos.ABG.sm. AB.sin. B G -{■ cos. A B . cos. B G,
from which all other formulas of spherics may be deduced.
It would have been easy to derive from this formula, and other well known formulas of
spherics, the whole of the values [171a]. Thus since by [172/],
Bq = COS. {x, y^^) = COS. arch A H,
in Fig. page 112, it is evident that in die triangle ABH, we shall have the same data, as in
the preceding example, except that hr B G = 9, we must use (p increased by a right angle,
and this evidently changes w^ointo Bq[\1\(i\.
Before concluding this note we may observe, that the values x^,,, y^^^, z^i^ [172 — 174],
might be derived from those of x, y, z, [171], by changing in those formulas x, y, z, into
•^///j V/iP ^i,n respectively, and making the changes of 9 into -v^, 4^ into <p, and 6 into — 6,
which would follow from this supposition, as will easily appear, from the situations of those
axes, as marked down in Fig. page 112. Observing also that if the inclination 6 is taken
positive, as it regards the plane BFP, it must be considered as negative as it regards the
plane BOP.
30
[172t]
118 MOTION OF A SYSTEM OF BODIES. [Mec. Gel.
[175] 2 ^ ^ J^///-^y///---y///'^^///} ^ ^ ^ COS. ^ — c' . sin. ^ . COS. ^ + c" . sin. ^ . sin. 4. ;*
at
. m .i— ^^i^ — ^ — '— — ^ = c . sin. ^ . COS. (p+ c'. { sin. N^ . sin.(p+ cos. 5 . cos. 4^ . cos.? \
at '
[176]
[1756]
dt
dt
+ c". { cos. vp . sin.(p — cos. ^.sin.-^. cos.cp \ ;
* (87) The differentials of tlie values Xi^^, y^^^, [172c], being taken, considering a:, y, z,
■^///j y/// ^s variable, give
dXiii= AQ.dx-\-Ai.dy-\-A'i.dz ;
<? y^,, = Bo . f? « + ^1 . <? y + ^2 . <; z ;
substituting these values in x^^^dy^^^ — y^^^ d x^^, it becomes
^mdy,—y,,dx,^,
[175a] =(^0 .Bi — Ai.Bq). {xdy—ydx)-\-{A^ .Bq — Aq .B^).{zdx—x dz)-\-{Ai .B^ — A^ .Bi ).{ydz—zdy)
Now the values [1 71 a], give
^1 . Bq = — sin. 6 . COS. & . cos. 4^ . sin. 9 . cos. 9 -f- sin. 6 . sin. 4- • cos.'^ <p,
— A2. Bi = sin. d . cos. 6 . COS. 4^ . sin. 9 . cos. 9 -1- sin. 6 . sin. 4^ . sin.^ 9,
whose sum is
Ai .B% — A-i. Bi = sin. & . sin. 4^ . (cos. 9^ + sin.^ 9) = sin. d . sin. 4- = Co, [i 71a].
Therefore we shall have in general Cq = Ai . B^ — Aq. Bi, and from the perfect symmetry
of the formulas [171«, 172a], this formula would exist if we changed the axes of x, y into
those of y, z, respectively, which would have the effect to increase the index of the letters
by 1 , rejecting 3, when the index exceeds 3, so that from derivations of this kind, we
shall obtain the following system of equations.
Ao=Bi. C^ — B^.Ci; Bo= Ci.A^—C^.Jii; Co= Ai.B^^d^.Br,
[175c] Ai = Bq.Co^Bo.Cs; J5i=C2. .^0—^0.^2; Ci = A^.Bo — AoB2',
*4a = Bo . Ci — Bi . Co J Bq = Co . ^1 — Ci . jIq > Cq = Ao . Bi — Ai . x>o ;
which may also be easily proved, by substituting tlie values [171a]. Hence the above
expression [175a] becomes
[175d] a?/,/d!y/// — 2////^^///= Cz.{xdy — y dx) -\- Ci.{z d x — xdz)-\r Co.{y d z — zdy).
The inspection of the formulas [172c], shows that we may change a:^^^, 3/^^^, z^^^, into
y///j ^iin ^,,,1 respectively, without changing the values of a?, y, z, or altering the indexes of the
letters A, B, C, provided we change A, B, C, into B, C, A, respectively ; that is, we must
put the letters one term forward, beginning the series again when we come to the last term ;
I. v.\5>21.] PRESERVATION OF AREAS. 1^^
2.mJ^''''^^'''~'^''''^^'' ==— c.smJ.sm.(p+c^jsin.vl..cos.(p— cosAcos.q.sin.cp}
dt L ''j
+c". { C0S.4. cos.(p + cos.a. sin.^. sin.<p \ .
since the same three equations [172c], will exist after these changes. We may therefore
make the same changes in [175<ri, which will give
y^Jz„ — z,Jy„^A^,.{xdy — ydx) + Ai.{zdx — xdz)^A^.{ydz — zdy\ [175e]
and by a similar process,
z„,dx,„^x,Jz„=^B^.{xdy-^ydx)-YBi.{zdx — xdz) + B,.{ydz--zdy) [175/]
If we now multiply these expressions [175J,e,/], by — , then mark, at the top of the
letters, x, y, z, x„^, y„„ z„^, one accent for the body m', two accents for the body m", and take
the sum of all these products, using as before the characteristic 2, and putting also as in
[167],
(xdv—vdx) (ydz — zdy) „ {zdx—xdz)
observing that c, = c", c^, = — </, [161a] we shall get
2 . m . ^'^'"^y'"-y'"^'="'^ = c . C2 + c,, . Ci + c, . Co ; [^75h]
dt
2 . TO . 7- = C . ^ii-\- C,, . ^i-\- C, . ^0 f
at
[175t]
S.m.^^^^'^j^^^^^^ = c.B^-^c„.B,-\-c,.Bo. [175k]
Substituting the values [171a], and putting c^^ = — d,c,=i c", it becomes as in [175 — 177]. [VJU]
It may be observed that the formulas [175d, e,/] may be very easily found from
geometrical considerations. For if the body to move from a point whose co-ordinates are
X, y, z, and whose radius vector, drawn from the origin of the co-ordinates, is
r = Var^ -{-y^-\- s^ ,
[19e], and at the end of the time dt arrive at a point whose co-ordinates are x-{-dx,
y-\- dy, z-\- dz, and radius vector r-\-dr, the angle included between the radii r and
r-\-dr, being d-a^ the described area will be ^i^dzi, [1676]. Now it follows, from the
principles of orthographic projection, that if this area be projected on any other plane, the
projected area will be equal to the described area ^r^du, multiplied by the cosine of the
inclination of the two planes. But the inclination of two planes, passing through the origin
of the co-ordinates, is evidently equal to that of tlie two lines drawn through the origin
perpendicular to these planes. Suppose now that a line R is drawn through the origin,
perpendicular to the plane of the described area, included by the radii r, r -j- dr ; the angle
formed by the line R and the axis of z, will be represented by {z, R), [172e], and the area
120 MOTION OF A SYSTEM OF BODIES. [Mec. Cel.
If we determine 4- and 6, so that we may have*
c" c'
[178] sin. 6 . sin. 4^ = — . sin. 6 . cos. 4 = — ;
V/ c2 + c^ + c"2 ' ^ c2 + c^ + c"2
[179]
which gives
we shall have
COS. 6 =
V/ c^_|-c^-|-c"2
^ ^ , k^^y.-y.-l^J ^ v/7T^?^^ ;
[180] s.w.-^^^^S^^^— -==0;
2.m. ^^ _U,
^ r^ . (Z'ztf, projected upon the plane of x, y, which is perpendicular to the axis of z, will be
represented by ^ t^ . d zi . cos. {z, R) ; and this, by [167«], is =^{xdy — ydx),
therefore we shall have i^ .dzi . cos. (z, R) = {x dy — ydx). Changing in succession
X, y, z into y, z, x, respectively, we get
r^ .dzs. cos. {x,R) = {ydz — zdy); r^ .dvi. cos. (y, R) = {zdx — xdy),
also r^.d-a. cos. {R, x,,) = y,„ d z,,, — z,„ d y,,). Now if in the formula [172A], we change
r into R, and multiply the whole by r^ .dta^we shall get,
r^ .dvi . cos. (if?, x,!")
z=.r^ .d-tii. \ COS. («, R) . cos. {x, x,,) + cos. {y,R) . cos. {y,Xii) + cos. (z, jR) . cos. (2;, x^) \ ,
substituting the values just found, it becomes
y,Jz,„—z,„dy,,,=^{ydz—zdy)cos.{x, x„)-\-{zdx—xdy) . cos.{y, xj+{xdy—ydx) . cos.{z,xJ,
and by using the values A(^, Ai, A^ [1'72/], it becomes
y,,d z,, — z„d y,,,:=Ao(y d z— z dy) -J^Ai{z d X — X dy) -{-A2{x dy — yd x),
being the same as in [175e], and from this the others [l'75d, /], may be derived, as above.
* (88) If we put, as in [161a], c' = — c^^, (^' = c/, also for brevity,
c c
The equations [1 78], will become sin. 6 . sin. 4 == -, sin. 6 . cos. 4- = -j the sum of whose
squares is sin.^ 6= ^' ^"-, whence cos. ^ = \/l — sin.^^ = -. Multiply these diree
equations by m, and use the values, Co , Ci, C2 , [171a], we get c, = m Co, c^^=m Ci,
c = mC2. Substitute these in the second members of the equations [175i, A;], and they
will become respectively, m(./22 C2 + ^1 Ci + v2o Co ), w (i?2 C2 + Bi Cx +5o Co ,) which.
[181]
I.v.§21.] PRESERVATION OF AREAS. 121
The values of d and of c" are therefore nothing with respect to the plane of
X and w , determined in this manner. There is but one plane which [1801
/// >j III' 1 11
possesses this property ; for, by supposing it to be that of x and y, we shall
have*
^,jri. ^—^ -^ = c . sm. 6 cos. <? ;
dt
2 . ^ . {y»''^^"—='"'-^y'"^ = — c . sin. 6 . sin. <p.
dt
By putting these two funct'ons equal to nothing, we shall have, sin. ^ = 0 ;
whence it follows, that the plane of x^^^, y,^,, coincides, in that case, with the
plane of x, y.
The value of 2 . ^ . i^^^J=^^^, being equal to V/c^+C^+c'^
whatever be the plane of x and y ; it follows that the quantity c^+c'^+c"^,
will be the same whatever be that plane, and that the plane of x^^^, y///» ^^^^^
in the preceding manner, is that which renders the function
•^- di
a maximum. Hence this plane has the following remarkable properties, ^'jf the^
namely, First, that the sum of the areas, traced by the projections of the Trc™!^
radius vector of each of the bodies, multiplied by its mass, is a maximum ;t [ISI'I
[i8n
Propertiei
by [1726], are notliing, as in [180]. The same values of c, c„ c,,, being substituted in the
second member of [ 175A], it becomes w . ( Cg C2 -f- C^i C^i + ^"0 Co ), which, by [ 1 726], is
simply m, or Vc^ -j- (/^ -f- c"^ , as in [1 80].
* (89) These equations are obtamed by putting c' = 0, c" = 0, in [176, 177]. If we
now find the value of 6, which will render the second members of [181] nothing, that is,
0 = c . sin. 6 . cos. 9 ; 0 ^ — c . sin. 6 . sin. 9, it will give ^ = 0. For the sum of the
squares of these equations becomes, by putting cos.^ 9-|-sin.2 9=^ 1, 0 = c^.sin.^ 6^
whence sin. ^ = 0, and ^ = 0, c being finite and 9 indeterminate.
f (89a) Upon any plane taken as that of x, y, we have 2 » m . ~ — - = c, and c
must be less than the quantity vc^-)-c'^ -f"c"^ J except c'=0, c" = 0, and as this quantity
V c^ 4" c'''^-\-cf'^ , is constant for every system of planes, it is evident that the maximum value
of c will be obtained, by putting c' = 0, c" = 0.
31
[182]
122 MOTION OF A SYSTEM OF BODIES. [Mec. Cel.
[181'"] Second, that the same sum, relative to any plane perpendicular to the
preceding, is nothing, since the angle 9 remains indeterminate. We may,
by means of these properties, find this plane at any time, whatever variations
may have taken place in their relative situations, by the mutual action of the
bodies, in the same manner as we can easily find, at all times, the position
of the centre of gravity of the system ; for this reason, it is as natural to
take this plane for that of x, y, as to take the centre of gravity for the
origin of the co-ordinates.
^in^jpies 22. The principles of the preservation of the living forces and areas,
wea7fa"ke tako placc also vrhen the origin of the co-ordinates is supposed to have a
place
wS^hls rectilineal uniform motion in space. To demonstrate this, put X, Y, Z, for
earand"" thc co-ordluatcs of this moveable origin, referred to a fixed point, and
uniform
motion, suppose that
x' = X+x;; y' = Y+y;; 2f = Z+z;;
&c. ;
x^, y^, z^, x', &c., will be the co-ordinates of m, m', &c., referred to the
moveable origin. We shall have, by this hypothesis,*
[183] ddX=0; ddY=0; ddZ=0;
but, by the nature of the centre of gravity, we have, when the system is
free,t
0 = :2.m.{ddX-{-ddx^}-':E.m.P.df;
[184] 0 = ^.m .{d dY-^ddy^l — :s.m.Q.df;
0 = 2.m.{ddZ+ddz^]^:^.m.R.df;
* (90) The velocity of this origm being rectilineal and uniform makes — • = constant ;
J v J y
[182a] TT = constant; — = constant ; the differentials of these being taken, supposing d t constant,
gives <i(iX=0, ddY=0, ddZ—0.
f (91) X, Y", Z, which have been usually taken for the co-ordinates of the centre of
gravity, are supposed above to be the co-ordinates of the moveable origin, we shall therefore,
in this note, put X', Y\ Z', for the co-ordinates of the centre of gravity referred to the
I. r. §22.] PRESERVATION OF LIVING FORCES AND AREAS. 123
the equation (P) ^18 [142], will therefore become, by substituting 5X-\-Sx^,
SY+iy^j &c., for Sx, Sy, &c.,*
C ddx, r. ) , r ( ddy, ^ ) , r ( (?<?2^/ r» 7
5! • fit X ^ Wl t/
fixed point of origin, we shall then have by [126, 127], X' = , Y' = — ,
^ . ° "^ *- -" 2.m 2.W*
_,, 2 . TO z
2.*»
; substituting the values [182], we get
2.m ' 2.m ' 2.m
Multiplying by 2 . w, and taking the second differentials, we obtain,
rfrfX'.2.7n = 2.m. {ddX-\-ddxy,
ddY'.^.m = 2.m.{ddY-\-ddy,)', [1836]
but by accenting X, F, -^, in [155], we have
d d X' . :s . m = :e . m . P . d f^ ;
ddlt' .:s.m^^.m. Q.df',
ddZ' ,2.m==:s.m.R.df^.
Substitute these and we shall obtain, by transposition, the formulas [184].
* (92) Since x = X-{-x^, [182], and ddX=0, [183], we have
ddx = ddX-{-ddXj = ddx^j
and in a similar way ddy=.ddy^, d d z =-. d d z^, &£c. These, together with
Sx = SX-\- Sx^f Sy =:8Y-{-Sy^, he.
being substituted in the equation [142], it becomes
+..™.(.r+.,,).(i?^-Q) + ..».(.z+..,).(^-4
and as 5 X, 5 Y, 5 Z, are common to all the bodies, they may be brought from under the
sign 2, which gives
But l.m.{ddX-\-ddx^)—S.m.P.df=0, and rf£ZX= 0, [184, 183], hence
we get 2 . m . (—rr^'- — P) = 0, and in a similar manner 2 . w . ( ^' — Q j =0,
2.TO.f-j— ^ -Rj=0; these quantities are the co-efficients of S X, SY, SZ, in the
124 MOTION OF A SYSTEM OF BODIES. ' [Mec. Gel.
which is exactly of the same form as the equation (P) [142], if the forces
P, Q, R, depend only on the co-ordinates x^, y^, z^ x', &c. By applying the
preceding analysis, we can deduce, from this equation, the principles of the
preservation of the living forces and areas, with respect to the moveable
origin of the co-ordinates.
If the system is not acted upon by any extraneous forces, its centre of
gravity will have a rectilineal uniform motion, as has been shown in ^ 20
[159'] ; by fixing, therefore, at this centre, the origin of the co-ordinates
X, y, z, these principles will always subsist. X, Y, Z, being then the
co-ordinates of the centre of gravity, we shall have, by the nature of the
point,*
[186] 0 = 2. m.a:^; 0 = 2.^.3/^; 0 = 2. m. 2^;
hence we getf
[185']
[187]
(xdy — ydx) (XdY — YdX.) , (x.dy, — v.dx)
dt dt ' dt ^
preceeding equation, therefore those terms must vanish, and the resulting equation will be
0 = ..».^.,.(^-P) + ..».^,,.(^-Q)+..»..z,.(^-4
as in [185].
*
(93) By putting X' = X, F = F, Z' = Z, in [183a], we obtain
^^ 2.m.(X+a:,) ^ ^ ^2.m.(r+y,) ^ - Z = ^•'"•^^+^/)
2;.m' 2.m' 2.m'
and as X, Y, Z, are common to all the terms, we may bring them from under the sign 2,
making = = A, Cstc, consequently Jl= A -j — -, or 0 = -;
i;.m 2.m -i ^ 2.m 2.m
or 0 = 2 . m a;^, and in a similar way, from the other two equations, we get 0 = 2 . m y^,
0 :;= 2 . m z^. The differentials of tliese equations divided by d t, are
[186a] «=^^-^-^'' 0 = 2. m.^', 0 = 2. m.^',
which will be used hereafter.
(x £?'U— — t/ d X^
f (94) Substituting the values of x, y, z, [182], in m . -; , it becomes
({XdY— YdX) , dY dX . ^dij, ^d x, . {x,dv,—y,dx,)}
"*• 1 Tt +'''^-y'dT + ^IT-^dT + ^d-t 5*
Marking these letters with one, two, Sic, accents, we obtain the corresponding equations for
w', m", &;c., their sum gives
I. V. §22.] PRESERVATION OF LIVING FORCES AND AREAS. 126
(da? + df'-\-dz') (dX^-{-dY^ + dZ^-) , (dx^-^dy^+dzj') ^ r.^.
dr dr «r
hence the quantities resulting from the preceding principles, are composed, [188^
First, of quantities which would exist if all the bodies of the system were
united in their common centre of gravity ;t Second, of quantities relative to the
(xdy—vdx) {XdY—YdX) , dY „ dX ^
, .^ dy, ^r dx, , fx,dy. — y,dx\
' dt dt ^ \ dt P
the factors X, Y, j— , -r— , being the same for all tlie terms, are brought from under the
sign 2, as in the preceding notes. Substituting the values [ISG, 186a], it becomes as
m [187].
* (95) Substituting the values of a?, y, z, &;c. [1 82], in "L . m . [d ot^ -\- d y^ -\- d z^), we
obtain
(rfa:2+<i7y2-f-rfz2) r i^dX-\'dxf'\-{dY-]-dyf-\-{dZ-\-dy)'i
2.m.
= ^ ra \^dX^dxf + {dY-^dy;f-\-{dZ-^dy)'i >
di^
Developing the second member, and bringing X, Y, dX, d Y, from under the sign 2,
we get
^•^- -J^ = JT^ .2.m + 2.m. —
+ 2.cZX.2.m.^' + 2.cZr.2.m.^' + 2.(iZ.2.m.^% ^^^^
dt dt dt
which, by means of tlie equations [18Ga], becomes as in [188].
f (95a) If all the bodies were situated in the centre of gravity of the system, we should
have X=a; = a/ = a;" = &ic.; Y =3/ = y'=y" = &;c. ; Z=z — z=s^' = Uc.',
and the quantities a?^, a:/, &;c. ; y^, y/, &;c. ; z^, r/, fee., [182], would vanish. Therefore the
first members of the equations [187, 188], would become, respectively,
[XdY-YdX) (rfX2 + rfY2 + rfZ2)
^•'^- Tt ' ^•'"- 1[^ '
and by bringing the terms X, Y, Z, from under the sign 2, they become
[XdY—YdX) , (dX2+rfy24-rfZ2)
. 2 . ffz, and ^^ ' ■ . 2. m ;
dt ' rff2 ^."»,
which are like the first terms of the second members of [187, 188], as is observed above.
Again, if the centre of gravity is at rest, we shall have -—=0; —- = 0; -—=0, [1 82a],
and the first members of [187a, 188a], will become like the last terms of the second
members of the equations [187a, 188a] or [187. 188].
32
12^ MOTION OF A SYSTEM OF BODIES. [Mec. Cel.
Centre of Centre of gravity supposed at rest ; and as the former quantities are constant,
Gravity ,
"o"/t^hr^ we perceive the reason why these principles exist with respect to the centre
C' of gravity. By fixing, therefore, at this point, the origin of the co-ordinates
X, y, z, x', &c. of the equations (Z) [167] of the preceding article, they will
always subsist ; whence it follows, that the plane passing always through
that centre, and relative to which the function 2 . m . l^L_^Zll_^ is a
at
maximum, remains always parallel to itself, while the system continues in
motion, and that the same function, relative to any other plane perpendicular
to the preceding, is nothing.
The principles of the preservation of the areas and the living forces may
be reduced to certain relations between the co-ordinates of the mutual
distances of the bodies of the system. For the origin of x, y, z, being
always supposed to be at the centre of gravity, the equations (Z) [167] of
the preceding article, may be put under the form*
[189]
( (cc' — x).(dy'-—dy) — (y'—y).(daf — dx) >
I dt y
I dt )
* (96) As some doubt of the accuracy of these equations has been expressed by a
writer in an eminent scientific European journal, from a misconception of their meaning,
we shall enter into some detail for illustration ; and for brevity, shall put,
[xdy—ydx) {o^dj[—^[d^ __ [x" df —y" dif') _ ^„
Jt == ^' rf < "~ ^ ' IT" — ^ ' ^•
[x! — x).{d'(i — dy) — {)i —y).{dx' — dx) _ -, ^
dt ~ — L^»'^J.
[1886] {x"-x).{dy"-dy)-{f-y).{dx"-dx) ^ ,
dt
{^'-jf).[dy"-dj/)-{f-y').{dx"-dx') _ ^^
dt L > J» •
Then tlie first of the above equations, by substituting for c its value [16.7], becomes
{xdy—ydx) , ^ jx' — x).{dy'—dy)—{y'—y).{dx'—dx) }
^•^- Tt — '^'nt=^'^^-l Yt 5'
which, if there be only two bodies m, m', becomes {m C -\- m' C) . {m -f- m') = m m' .[x, a'].
I.V. <^22.] PRESERVATION OF LIVING FORCES AND AREAS. *27
')We may observe that the second members of these equations, multiplied
hy dt J express the sums of the projections of the elementary areas, described [^^^1
by each right line connecting any two bodies of the system, of which the
If there be three bodies m, mf, m", it is
{mC-\-m' e'-\- m" C") . (m + m' + m") ^mm! . [a?,a/] + mm!' . [x, a/'] + m' m" . [a/,a/'].
If there be four bodies m, w', m", m'", it becomes,
{mC + m'C' + m" C" + m'" C") . (m + m' + m" + m'") = mm' .lx,x'^-{-m m" . [x, a/']
+ m m'" . Ix, x'"'] + m! m" . \x', a/'] + m' m'" . [x\ x'"^ -\- m" m'" . [x", a/"],
and thus for any greater number of bodies. Observing that each body, in the second
member of this equation, is supposed to be combined with all the others but once, and that
the whole number of bodies being w, the number of terms in that second member is
-^ -, as is evident by the usual rules of combination. Similar remarks may be made on
the second and third equations, [189]. Having thus explained the import of these equations,
we shall now proceed to tlie demonstration. On account of the symmetry of these
equations, we might limit ourselves to the consideration of two bodies only, as m, m\ but for
the reason above named, we shall notice the other bodies »»", w/", &c.
We shall first prove that we have identically,
[xdy—ydx) C[if—x).{dy'—dy)—{y'—y).{d3^—dx)
2.m.-
'-^'^'^^ y m y mn,' ^ {^-=^)-{dj/-dy)-{j/-y).{d^-dx)^
— . 2 . m = 2 . mm . < >
dt I dt ^
, mdy mdx
-\-'L .mx .H. — - — 2 .my .1. . ,
' dt ^ dt'
whatever be the origin of the co-ordinates ; that is, whether the origin be at the centre of
gravity or not. Now as both members of this equation are symmetrical in m, m', m", he. it
is only necessary to prove that the terms multiplied by any one of the quantities m, to', to", &c.
as m, is the same in both.
The second member of [189o], being developed, becomes
mm' [x, a/] + TO m" [a?, a/'] -j-;fiic.
I / I / / I 0 \ /fn,dy-\-m'dy'-\'SLC.\ . , , , o \ /'mdx-\-mfdx'4-&.c.\
-}-{mx-\-mx +hc.). i^ jf j—{m y-\-m'i/ + &c.) . i^ ^7~^ )»
of which the part having the factor m is
m^
-\-m. <X. ^— ! -^—^ — y.- ; — ■ ! '- \.
' ( dt ^ dt i
■^'"* |^.K^ + »»"a^" + &c.)— ^.(toY + to"/ + &ic.)| ;
[189o]
128 MOTION OF A SYSTEM OF BODIES. [Ivlec. Cel.
one is supposed to move about the other considered as at rest, each area
being multiplied by the product of the two masses which are connected by
the right line.
If we apply the analysis of ^21 to the preceding equations, we shall find
I'll 1 • T 1 1 , / ( n ,T I {xdi/-{-xfdy — ydxf — tj dx) )
m which the terms multiplied by m m are mm . < [x, x\ -\- - — ^— ' ^^—r: ( »
and by substituting the value of [a?, a;'], [1S8&], it becomes
, C [xf — x).[d'\/ — dy) — {■}/ — y).{dx! — dx)-\-xd'if -{-x' dy — ydx! — ^dx >
mm . ^ •■ •■ - ^ ;
or by reduction mm' . \ ^—■^L:zy.-±1±LJJJZA. — I } eq^al to mm' . {C -{- C). In like
manner, by adding one, two, Sic. accents, to the letters m', C, we obtain the parts depending
on m", m'", &;c., which will be mm".{C+ C"), mm"'.{C+ C"), kc; consequently,
the part of the second member of [189a], having the factor m, is
m2C + wim'(C4-C')+mm"(C+C") + &c.
= m C (m + m' + m"-{- he.) + m {m' C + m" C" + fee).
(x d y y d'X^
AgaiQ the first member of [189«], 2 . m . .H.m, being developed, is ;
{mC + m' C' + m" C" + &c.) . (w + m' + m" + &c.),
the part of it, having the factor m, is
mC.{m-\- m' + m"+ &ic.) + w . {m' C' + m" C" + &c.) ;
and as this is equal to the expression of the second member of the same equation [189a],
just found, it will follow, that the equation [lS9a], takes place, for any origin of the
co-ordinates; and by fixing the origin at the centre of gravity, we shall have, as in [186a],
0 = 2. m. — ; 0 = 2. m. — . Substitute these in riS9a], it becomes like the first of
d i d t
tlie equations [189]. The second is easily derived from the first, by changing y, y', &;c.
into s, 2^, kc. ; and the third is obtained from the second, by changing x, x', &,c. into
y, y, &;c. If, as in [161a], we put c' = — c^^, c" = c^, the three equations [189], may be
placed in the following, more symmetrical form.
[1896]
[189c]
[189rfJ
I. V § 22.] PLANE OF THE GREATEST AREAS. 129
that the plane passing constantly through any one of the bodies of the system,
and relatively to which the function
C dt ' 3
is a maximum,'*' preserves its parallelism, during the motion of the system,
[189e]
* (97) Using the values [172c] for the body m, and adding one accent to a?, y, z, &c.
for the body »»', and two accents for m", &tc., we shall get
y.;-y«/ = ^o.(^-^)+^i-(y'-y)+^2.(^-^);
and by [175^], we have
{x.,dy„-y„,dxj {xdy-ydx) {zdx—xdz) [ydz-zdy)
dt —^^' di ^^1- dt ^^° Tt •
Now by comparing the values of x„„ y,,^ z,„, [172c] with these values of xj, — o?,^,, yj — y„fl
zj^ — z,„, it is evident that we may substitute in this equation, x,/, — x,,, for x,,,, yj, — y,,, for
y///' ^/// — ^111 ^o^ -^//z' ^ — * ^^^ '^' y — y ^^'^ y-> ^- because the quantities A^^ A^, &ic. are
not affected by these changes. Hence, by making tliese substitutions in [189e], and multiplying
by mm we shall get,
mm . < ; >
I dt s
^ {xf —x) .[dy' —dy)—{y[ — y).{dif — dx) )
= C2.mm.| J
A.r rnm' <^ i^ -^) ■ Jd ^ -dx)-{x^ -x) .{dzf - dz) }
O-r »,«,' ^ {y'-y).{d2f-dz)-{:^-z).{dy'-dy) )
If we change the co-ordinates of m, m', into those relative to any two other bodies of the
system, we shall obtain similar expressions for them. Taking the sum of these equations,
and substituting in the second member c . 2 . m, c^ . 2 . »n, c,, . 2 . m, for
we shall get,
^ ^^/ ^ i^J-^J-idyJ-dyJ-{y,:-yJ.{dxJ-dxJ >
2. mm . < ; >
I dt s
= C2 . c . 2 . m + Ci . c,, . 2 . m + Co . c, . 2 . m, ^89/]
and by changing the letters x^^^, y^^,, z,,^ Cg, C^, Cq, we shall get the following equations, in
the same manner as [175e, /] were obtained from [175</],
V ^«,' UyJ-yJ-idzJ-dzJ-{zJ-zJ.{dyJ-dyJ ^
2. mm . < ; ^ >
( dt S
= A2>c.:si .m-{-Ai.c^^.i: .m~{- Aq.Cj.S .m;
33
[lesfer]
130 MOTION OF A SYSTEM OF BODIES. [Mec Gel.
and that this plane is parallel to the plane passing through the centre of
( 00 d y III 91 n. oc\
gravity, and relatively to which the function 's..m.- — ^—r-- — - is a
maximum,* wq shall also find, that the second members of the preceding
[189^'] equations are nothing, for all planes passing through the same body perpen-
dicular to the plane just mentioned.
The equation (Q) ^ 19 [143], may be put under this form,t
^.mrnl. \ ^ 2 [ = constant— 2.2. m .:s .f,mm' . F .df;
[190]
[189A] ^.mm'A ^'''''-''''^•^^<'-^''''^-^<-''''^-^^<-'^''''^ \
which are similar to the equations [175A, i, A;], and agree with them, by writing x^l — x^n,
y„l—y,n^ V — ^///» c.-L.m, c,.:s.m, c,,.2.m, for x^^^, y^,„ z,,,, c, c,, c,„ respectively;
consequently the results obtained from the equations [17 5 A, i, k] or from their equivalent
expressions [175 — 177], are equally applicable to the equations just found. Therefore, the
second members of the two last of these equations are rendered equal to nothing, by
assuming for 6 and 4^, precisely the same values, as are required to make the second members
of the equations [176, 177], equal to nothing. For, if we change c, c, c", into c.2.m,
c'.H.m, c".'S.m, respectively, the equations [178], by which ^, ■vj' are determined, will
remain unchanged, rejecting the factor 2 . m, common to the numerator and denominator.
* (98) This evidently follows from the calculation in the preceding note, where it is shown
(x d V -~" V d x\
that the same values of d, 4", render each of the expressions ^ .m. — — '■ — j
y rr^rr,' V W - x) .[d^f -dy) -[jf -y) .{dod - d x) >
^,mm . I - ^,
a maximum.
f (99) Multiply the equation [H3], by 2 . m, and substitute for
•s.,f.m.{P.dx-\-q.dy-\-R.dz),
its value — ^ .f.m m' .F.df, deduced from [14.3"], the system not being acted upon by
any extraneous forces, [185'], it becomes,
{dx^-\-dy^4'd:!^) ^ i^ , r, i j-
n89i] 2 . m . 2 . m .-^ ^2 ==c.'S. .m — 2 .s .m .^ ./.mm .F . df,
of which the second member is the same as in [190], supposing the constant quantity to
be C.2.W?, and the first members of the same equations, [190, 189ij will be found by
developement, to be equal. For we have identically
nsdk] i:.m .s.m .dx^ = 'S.mm' . {dx — d x)^ -\-(s . m . d x)^, or
(m + m'+ m" + he.) . {m .dx^ -\- m' . dx'^ -\- m" . dx"^ + he.)
[189/] =mm'.{dx—dx)^+mm".{dx" — dxf-}-he.-\-{m.dx + m'.dx'+hc.)^;
I.v. §23.] PRINCIPLE OF THE LEAST ACTION. 131
which contains only the co-ordinates of the mutual distances of the bodies
from each other, and in which the first member expresses the sum of the
squares of the relative velocities of the bodies of the system about each
other, combining them in pairs, and supposing one of the two to be at rest,
each square being multiplied by the product of the two masses corresponding
to it.
[1901
23. Let us no# resume the equation (R) ^19 [144]. By taking its
differential relative to the characteristic 5, we shall have*
^.m.v5v = :^.m.(P.5x+Q.dy-{.R.Sz); [191]
the equation (P) ^ 18 [142], thus becomes
C d 00 d y d z i
/ CLZ Ct Z Ct Z J
since the terms multiplied by m, in the second member of this last expression, are
m.{m{dx'—dxf^m''.{dx''—dxf-{-hc.+m.dx^-\-2.dx.{m'.dx'-\-m".dx'' +&;c.)},
which by reduction becomes m.^d x^ . (m! -\-m" -{- &c.) -{- {m . d ar^ -\- m . d x'^ -{- he.) ] ,
and this is evidently equal to the terms multiplied by m, in the first member of [] 89Z],
therefore, the equation [189>5;], is identical. In like manner, by changing a/, x, into i/, y,
and zf, z, we obtain,
2 .m.:s.m.dy^=^:£.mm' .{di/ — dyY -^ {^ .m . dyf;
:s . m .:s . m . d z^ == :s . mm! . {d z' — dzf~\-{^.m.d z)^.
The sum of these, divided by d ^, gives identically,
{dx^-{-drf-]~dz^) , C{d3/ — dxf-^{dj/ — dyf+{dz' — dzf)
dpi I d& 5
Now the origin of the co-ordinates being supposed at the centre of gravity, we shall have, by
[186a], 2.OT.-- = 0, 2.m.— =0, 2.wi.-i^0; all which, substituted in
[189t], gives the equation [190]. Lastly, it is evident, by the reasoning in [40a], that the
—d
~dt
H J — ^ fj ft*
relative velocity of m' about m, resolved in directions parallel to a?, y, z, is -^ ■ j [1905]
dy' — dy dz' — dz i /• ,
— j^ — 5 — ry — ; the sum ol whose squares gives, as in [4Ca], the square of the whole
of that relative velocity, as in [190'].
* (100) By substituting, in the differential of [144], divided by two, for S(p, its value [143'],
S<p==:E.m.{P.5x-{- Q^.5y-\-R.Sz). [190c]
^^2 MOTION OF A SYSTEM OF BODIES. [Mec. Gel.
let dshe the element of the curve described by m ; d s' that described by m',
&c. ; we shall have [44]
[193] ^^t = ds; v'dt=ds'; &c. ;
[1931 ds = V dx'+df'-^dz' ;
whence we deduce, by following the analysis of § 8,*
nan „ ^ / i \ i (d x.Sx4~dy .Sy4-d z .Sz)
i^^i ^ .m .S .(v d s) = 2 ,m .d. — - — ^— !— r ^.
^ ^ dt ^
By integrating this equation with respect to the characteristic d, and
taking the limits of the integrals to correspond to the extreme points of the
curves described by the bodies m, m\ k,c., we shall have
.,-„ ■ ^ , ^ ^ , (dx.5x-\-dy.6y4-dz.Sz)
[195] S.6 . r.mvds = constant + 2 . m . — —^ — ^— ^ ^- ;
'^ dt ^
the variations ^x, 6y, 6z, &c., and the constant term of the second equation,
correspond to the extreme points of the curves described by m, m', &c.
Hence it follows, that if these points are supposed invariable, we shall
havef
[196] 0 = :s,6.f.mvds;
which shows that the function ^.f.mvds, is a minimum.! In this consists
* [101] By substituting vdt = ds, v' dt = ds', he. in the equation [192], it becomes
Cr ■I'^^i^ 1 dy , . ^ d z ^ , f,
0 = 2. m. <()x.d.- \-oy . d.-^4-d z .d .—- } — ^.m .ds .ov,
I dt^^dt' dt ^
which corresponds to the equation [45]. Again, the formula [47], which was easily deduced
from the variation of ds^, being multiplied by m, and added to the similar equations in m',
m", he. gives,
C , \dx.5x4-dy.8y-\-dz.5z\ , . , dx . j dy . , dz}
I dt ' dt '' dt of)
Adding this equation to the preceding, transposing 1 .m .ds .8v, and puttmg 8 .{vds) for
V ,d6s-{-ds .6v,\\. becomes as in [194], whose integral is [195].
[.,
f (102) This will appear by the reasoning at the end of note 19.
[196a]
X (103) It has been remarlced that the maximum, or minimum, of the expression of the
living force of a system of bodies 2 . m . t>- , corresponds to that state of the system, in which
it would remain in equilibrium, by means of the accelerative forces acting upon it, supposing
I.v.§23.] PRINCIPLE OF THE LEAST ACTION. 1^3
the principle of the least action in the motion of a system of bodies ; which
principle, as we have proved, is a result, deduced by mathematical principles, [IQ^T
from the primitive laws of the equilibrium and motion of matter. We
[19Cc]
[196rf]
the system to be placed directly in that situation without any velocity. This would follow,
from the expression of the living force [144], c + 2(p, which, by the usual rules of the
differential calculus, would become a maximum, or minimum, when
d^ = ^.m.{P.dx^q^.dy-^R.dz)=0, [143^], [1966]
Now from the principle of virtual velocities, we shall have, in the case of equilibrium,
[41 , 11 0"], 'Z .m.{P .6x~\- (^.^y-{-R.^z) = Q; Sx, 8y, S z, being arbitrary variations,
satisfying the conditions of the system ; and since all these conditions are satisfied by putting
6x = dx, Sy = dy, Sz = dz, we may substitute these values in the preceding equation
of equilibrium, and it will become ^.in.{P.dx~\-Q.dy-\-R.dz) = 0, which corre-
sponds to the maximum or minimum of the living force, [1 9G6]. A well known example of this
principle occurs, in tlie case of a heavy homogeneous cylinder, rolling upon a horizontal
plane j the sum of the living forces of all its particles wiU be a maximum, or minimum, when
either extremity of the conjugate, or transverse axis of the ellipsis, touches the horizontal
plane ; because the cylinder would remain in equilibrium, in either of tiiose states, if it had
no velocity. The equilibrium would be stable when the conjugate axis touches the plane ;
but unstable when die transverse axis touches the plane. In the former case the centre of
gravity would be at its lowest point, in die latter case, at its highest point. If a system of
bodies is held at rest in a situation very near to its state of equilibrium, as would be the case
with the cylinder just mentioned, supposing the end of the conjugate axis nearly to touch the
horizontal plane, the system would, generally, upon being left free from restraint, endeavor
to attain this point of equilibrium, and in doing diis, the particles of the system, from being at
rest, would, by the mere action of the accelerative forces, acquire a very small velocity,
which would increase the living force, in approaching the point where the equilibrium
would take place. Therefore the maximum of the living force generally corresponds to the [196e]
point of stable equilibrium. The contrary takes place when the system is placed at rest, in a
situation inSnitely near to the state of unstable equilibrium ; since tiie system, if left to itself,
would, on account of its being in an unstable state of equilibrium, generally endeavor to
recede from die point of equilibrium, and in doing this, the velocity of the particles, and die
whole living force, from being nothing, would become finite, while removing fiom the situation
of unstable equilibrium ; therefore this state must generally correspond to the minimum of the
living force.
K the system be acted upon only by gravity, g, which we shall suppose to act upon die
bodies, in tiie direction of the lines z, z', kxi., drawn to the centre of the eardi, which may
be considered as parallel; we may put P=0, Q = 0, R==gj and we shall have,
34
1^^ MOTION OF A SYSTEM OF BODIES. [Mec. Cel.
also find, that this principle, combined with that of the living forces,
gives the equation (P) § 18 [142],* vs^hich contains all that is necessary
d(p='Z.m.gdz=^g.'2..mdz, whose integral is (p = c -}- ^ . 2 . m «. If we put Z
for the distance of the centre of gravity of the system from the centre of the earth, we shall
have 'S, .mz = Z .1 .m, [127]. Substitute this in the preceding value of cp, it becomes
(^=c-\- Z .g .'2 .m, and as g* . 2 . m is a constant quantity, it is evident that tlie maximum,
or minimum of (p, must correspond to the maximum or minimum of Z. Hence it follows,
that the system will be in equilibrium, when the centre of gravity is at its lowest or highest
point. The former case occurs in the catenarian curve, the latter in an arch or bridge
composed of small globules resting upon each other, in the form of an inverted catenarian
curve.
* (103a) Having, as in [44], ds^= dx^-\- dy^ -\- dz^, if we take its variation relative
. d s .
to ^, then divide hy 2 dt, and substitute — =:v, it becomes,
.J [dx.d5x-\-dy .d^y-{-dz.d(i,z)
dt '
which multiplied by m, gives
. , [dx .d5x-\-dv .d5y~\-dz.dSz)
m.v.8ds = m. dt ^'
adding to these one accent for the body mf, two accents for the body m", he, and taking the
sum of all these equations, we get
. , C dx ., , dy ^, , d z ^ ")
2.m.v.6ds = 2.m.< -r— .odx + -r— .0 dy -\--— .od z > .
Idt ' rff ^ ' dt y
/dx
— .6dx, its value
d X ct X
— .Sx — f. (ix.d . -r— , &;c. which is easily proved by differentiation, it becomes,
(t z a V
<:{dx.Sx-\-dy.6y-\-dz.5z) /Y. dx j dy . ^ j dzS}
[mg] ^.f.m.vJds=^.m.\' ^^^- ^-y(^^-^-rf7+^^-^-^ + ^^-^Ws-
The principle of the least action ^ .(i .f.mv . ds = 0, [196], by expanding the differential
relative to 5, is 0 = 1 ./.{mv . d8s-\-m§v.ds), whence
J, .f.mv . d S s = — 2.m(5v.«?5,
or since ds = vdt, l.f.mv.d5s = — II .f.m . v 5v . dt ; and the principle of living
forces gives, as in [191],
— ll.f.m.v6v.dt = — ll.f.m..dt.{P.8x-{- q.5y-\-R.Sz).
Substituting this in the preceding equation, we get
— l.f.m.dt.{P.8x+q.8y + R.8z)
({dx.Sx-\-dy.(hj-\-dz.8z) f( dx j ^V \ s j^^W
I.v.^23.] PRINCIPLE OF TOE LEAST ACTION. l^^
to determine the motions of the system. Lastly,* it k evident that this
principle also exists, when the origin of the co-ordinates is in motion,
and by reduction this becomes
((dx.6x-\-dy.Syi-dz.Sz)}
^,rn. I j-^ ^
The terms of the first member correspond to the extreme points of the curves described by
the bodies, that member is therefore constant, as is evident by the reasoning in note 19, and
its diiferential relative to d being takenj it wiU beconae nothing, and we shall finally obtain,
0 = 2.m.^(^rf.^ —P.dty8x-{-(d.^—q.dtySy-{-{d.^ — R.dt\Szl,
which is the same as the equation [142].
* (1036) The meaning of this proposition is, that if v^, r/, v", &£C. are the velocities of
the bodies m, m', m", &;c., referred to the origin of the co-ordinates, supposing that origin to
be in motion, and v^dt = ds^, v^ dt^dsj, v" dt=ds", kc., we shall have,
0=1.6./ .mv^ds^, similar to [196]. For by using the values of x, y, z, x/,y',z', &lc.
[182], and supposing the motion of the point of origin to be uniform and rectilineal, we shall
have, as in [185],
From this equation, which is exactly similar to [142], we may easily deduce one similar to
[143], by putting S x^ = dx^, 6y^ = dy^, he., and integraung, which gives,
^^^,iM±M±^ll). = ,^2.:E.f.rn.{P.dx^+q.dy, + R.dz,),
or, by putting as in [144], the first member equal to 2 . m . v/^, and supposing
1.m.{P .dx^-\- q.dy^-\-R.dz^)^d(p,
an exact differential ; also taking the variation relative to 5, we shall have, as in [191],
li.mv,Sv, = Z.m.{P.8x,-{-q.Sy^-{-R.Sz^).
This, substituted in the equation [196?>], multiplied by dt, gives
0=2.m.\8x^.d.^ + Sy.d.^' + 8z.d.^^]—:E.mdt.v8v,
(. ' dt ' ^' dt ^ ' dt ) ' ''
or, by putting v^d t = ds^,
0 = Z.m.Ux,.d.~^ + Sy,.d,^^-'+Sz.d.p]-S.mds.Sv.
i ' dt ' ^' dt ' ' dt ) ' '
Again from the variation of ds^^=d xj^ -\-dyf -{-d zf, we obtain as in [47], or rather
as in [196jg-],
J..m.v^.8ds;=J,.m.y '^ ^'^ ^'^ ' '^^Sx,.d.jf—Sy^.d.-^—Sz^.d.-^^^. [I96k]
[196h]
[19Gi]
136 MOTION OF A SYSTEM OF BODIES, [Mec. C61.
ri96"i provided that its motion is rectilineal and uniform, and that the system is
free.
Add together the equations [196i, ^], transpose the term — S .mds^ .6v^, and put
S (v^ d s^) for v^.d 6 s^-{-ds, .8 v^, and we shall obtain,
2 .m.5. {v^ds^)^=l.m.-
dt
1 • . 1 • -, f /> 7 1 ^ {dx..8x.-\-dy,.Sy,4-dz,.6z,)
whose integral is ^ .o .f.mv^ds^ = constant + 2 . m . — '- — — — -[. — ■ — -, the
variations in the second member correspond to the extreme points of the curves, and when
these are invariable, as it respects the moveable origin of the co-ordinates, we shall have
at these points, 5 a;^ = 0, 5 y^ = 0, 6 z^ = 0, &c. hence 1 .8 .f.mv^dSf = 0, which is
similar to [196].
I. vi. §24.] IN ALL RELATIONS OF FORCE AND VELOCITY. ^^"^
CHAPTER VI.
OF THE LAWS OF THE MOTION OP A SYSTEM OF BODIES, IN ALL THE RELATIONS MATHEMATICALLY
POSSIBLE BETWEEN THE FORCE AND VELOCITY.
24. We have observed, in § 5, that there are an infinite number of
methods of expressing the force by the velocity, which imply no mathematical
contradiction. The most simple is that of the force being proportional to the
velocity, and we have seen that this is the law of nature. It is according to
this law, we have explained, in the preceding chapter, the differential
equations of the motions of a system of bodies ; and it is easy to extend the
analysis we have used, to all the laws mathematically possible between the
force and velocity, and to present thus, in a new point of view, the general
principles of motion. For this purpose, suppose that F being the force and
V the velocity, we may have
F=q>(v); [196///^
(p (v) being any function whatever of v ; put (p' (v) for the differential of 9 (v)
divided by dv. The denominations of the preceding articles being used,
the body m will be urged parallel to the axis of ar, by the force 9 {v) . — .* [igeiT]
d X / d i2? \
In the following instant, this force will become 9 (^) • ^ — h <? • ( 9 W • ^ j » or [igg ,]
(p(2j).-^-|-(?.( — ^.-=- ), because -— = 1;, r40a]. Now P, (^iR-, being the
as \ v at J at
forces which act on the body m, parallel to the axes of the co-ordinates ; the
system will be, by ^ 18 [141], in equilibrium, by means of these forces, and
* [1036] Found as in note 34a.
35
O
138 LAWS OF MOTION OF A SYSTEM, [Mte. Cel.
of the differentials rf/^.^Y d/^."-^), <i/^.^\ taken
with a contrary sign ; we shall have, therefore, instead of the equation (P)
[142] of the same article, the following :
[197] 0=..m.^...S^{^4.?^)-Prf<|+.,.(rf{^.?^)-Q*i+&.H.(§.*-^)-i?*i^;(S)
d cc d ti d z
which differs from it only in this respect, that — , -~, — , are multiplied by the
function -^, which, when the force is proportional to the velocity, may
be put equal to unity. But this difference renders the solution of the
problems of mechanics very difficult. However we may deduce, from the
equation (»S) [197], principles analogous to those of the preservation of the
living forces and of the areas, and of the centre of gravity.
If we change ^ x into d x, 5y into dy, ^z into 6? z, &c., we shall have*
* (104) Put Sx = dx, §y = dy, 5z = dz, in the equation [197], and develop the
, /dx <p(v)\ . , . ddx (p {v) . dx ^ /(p{v)\ . , „,
terms like d.[—- . — into the lorm — -— . \- -—.d .1 &;c. we shall have
\dt V / d t V dt \ V /
^ /-{dx.ddxA- dy.ddy-j-dz.ddz) <P (v) , {dx^-\-difi-\-dz^) , /<?(v)'
\ :- — = . ; . d
A -P ™ y dt V ' ■ dt
0 = 2 .W. <
d — P.dx.dt—q.dy.dt — R.dz.dt
but dx^ -{-dy^ -\-dz'^=:^v^ . dt^, [40a], and its differential gives,
dx.ddx-\-dy.ddy-\-dz.ddz=vdv.dt^,
hence by substitution,
0=2 .m. j — .----^——.dJ-^-j—P.dx.dt—q.dy.dt—R.dz.dti,
11 1 . • r 7 /'P('^)\ • J 1 dv.cp'{v) dv.<p{v) . ,
and by substituting lor a . ( 1 its development — , it becomes
rvdv.dt^ (?{v) v^.dpi dv.(p'{v) v^.dt^ dv.<p{v)-
0 z= ^ m < ^^ ^ rff* V dt ' v^
) — P .dx.dt— Q.dy .dt — R.dz .dt
in which the first and third terms destroy each other, and the expression becomes, by
dividing hj dt, 0 = 'E .m . v dv . cp' {v) — H . m . {P . d x -\- Q . d y -\- R . d z), which being
integrated, gives, as in [199],
:2 .f.m.v dv .cp' (v) = -E .f.m. {P . d x -\- Q.dy-\-R.dz) + constant.
I. vi. §24.] TN ALL RELATIONS OF FORCE AND VELOCITY. ^^^
consequently,
2./ .m.vdv. (p'(v) = constant+s. f.m. (P. dx+Q.dy+R.dz). [1991
Supposing i.m.{P .dx-\-Q.dy-\-R.dz) to be an exact differential, and [199']
equal to d x, we shall have,
2./.w.«?^i^.9'(^) = constant+X; {T) [200]
which is similar to the equation (R) ^19 [144], and becomes identical, in
the case of nature, where (p'(v)=l. The principle of the preservation of
the living forces would therefore take place in all possible laws which might ^^^j^f
exist between the force and velocity, provided that we define the living force j.^^^
of a body by the product of its mass, by the double of the integral of its
velocity multiplied by the differential of the function of the velocity which
expresses the force.
If in the equation (^S) [197] we suppose (5a/ = 5a:+^^/ ; ^y' = ^y-{'^yl i [200*']
Sz' = Sz-j-Sz' ; 6x"=5x-{-6xl' ; &:c. [152], we shall have, by putting the
coefficients of 6x, <5 «/, ^ z, separately equal to nothing,*
These three equations are similar to those of § 20 [153], from which we
have deduced the preservation of the motion of the centre of gravity, in the ofG^vhy*
case of nature, where the system is subjected to no other forces than the
action and mutual attraction of the bodies of the system. In this case,
2.mP, 2.mQ, 2.9wi?, are nothing,* and we shall have
dx o(v) dv Q>(v\
constant = 2. m.-y-.-^^; constant = 2. wi.—.-^:
at V dt V
. , dz cpM ^^^
constant = ^.m. — . --^-.
dt V
[201]
Motion of
Centre
* (105) For the reasons mentioned in note 72.
t (106) As is proved in [155"]. Substituting these in [201], and integrating, we
get [202].
[203]
140 LAWS OF MOTION OF A SYSTEM, [Mec. Gel.
d x 0 (vS d X
[202'] m, JlJ. is equal to m.(p(v).—,^ and this last quantity is the finite force
Ci/Z 1) (L S
of the body, resolved in a direction parallel to the axis of x ; the force of a
body being the product of its mass by the function of the velocity which
expresses the force. Therefore the sum of the finite forces of the system,
resolved in a direction, parallel to any axis whatever, is then constant,
whatever be the relation of the force to the velocity ; and what distinguishes
the state of motion from that of rest, is that in the last case this sum is
[202"] nothing.f These results are commoa^to all the laws mathematically possible
between the force and velocity ; but it is only in the law of nature, that the
centre of gravity moves with a uniform rectilineal motion, t
Again, let us suppose in the equation (S) [197],
Sx^y^ + Sxr, 6x' = ^-^ + 6x:; Sx" = y^^+6x:'; &c.
y y y
— x.8x , . , , — xf.Sx , „ , o
6y = J^Sy^; 6^= J^Sy^; &c. ;
the variation <5 x will disappear from the variations of the mutual distances,
/,/', &c., of the bodies of the system, and from the forces which depend on
. d X .0(v)
* (107) This is found by putting, as in [44], ds for v dt, in m . —^ whicli malces
it become m . (p . (i?) . — which, as in [196*^], represents the force of the body resolved in
a direction parallel to the axis of x.
f (108) The sum of these forces, resolved in a direction parallel to any axis, must
evidently be nothing, in the case of equilibrium ; since, if this was not the case, the system
would have a motion, in consequence of these forces.
{ (108a) When the centre of gravity has a uniform rectilineal motion, we shall have, as in
[158, 159], — r— - = 0, which, substituted in the second differential of the value of X, [154],
gives 2 . m . (Z . -r— = 0. Now the first equation [201], by putting, as above, 2 . ?» . P=0,
becomes ^ .m .d .{-—. — ^) = 0, which cannot in general become identical with the
\dt V / °
9 (v)
preceding, except we have constant, or <p {v) proportional to v, which is the law of
nature.
I. vi. Ǥ 24.] IN ALL RELATIONS OF FORCE AND VELOCITY.
these quantities.* If the system is free from external obstacles, we shall
have, by putting the coefficient oi 6x equal to nothing,t
whence by integration!
141
[204]
[205]
we shall likewise have
[206]
\ at y V
c, c', c", being arbitrary constant quantities.
If the system is subjected only to the mutual attraction of its parts, we Pre.erva-
shallhave, by§21, [165], s.m. (Pi/— Qa:) = 0, s.w.(Pz— i2a:) = 0; a?^^'
^.m.{Qz — Ry)=0. Again, jw . fa;.— ^ — y.— j.-^, is the momentum t^^l
of the finite force with which the body m is urged, resolved in a direction
parallel to the plane of x, y, to make the system turn about the axis of z ;§
* (109) This is shown in note 77.
f (110) The reasons for putting this coefficient equal to nothing, are fully explained
in [160'].
f (111) Integrating by parts, the expresssion x .d . (—-. — ^ j, it may be put under the
(dy (p{v)\ /dy (p{v)\
— . j — f dx • { TT • j, as is easily proved by taking its differential ; also
sum of these WD integrals, is ^, (^.^) /£i.gW) equal to '"'^-"""'■^
because the terms, under the sign of integration, destroy each other.
<§, (112) Suppose a plane, A C D B, to be drawn through Y
the place D, of the body m, parallel to the plane of x, y, to meet
the axis oi z'mA. Through A draw A C,AB, parallel to the ^
axes of X, y, respectively, and complete the parallelogram
A CBB. Then it is shown [202'], that the body m is urged A.
1^2 LAWS OF MOTION OF A SYSTEM, [Mec. Gel.
the finite integral 2 . m. (^ ^T!^ ^ ).ii!2 is therefore the sum of the
\ at J V
momenta of all the finite forces of the bodies of the system, exerted to make
[206"] it turn about the same axis ; this sum is therefore constant. It is nothing
in the state of equilibrium ;* there is therefore the same difference between
these two states, as there is in the sums of the forces parallel to any axis.
In the law of nature, this property indicates, that the sum of the areas
described about a fixed point, by the projections of the radii vectores of the
[206'"] bodies is equal in equal times, but this constancy in the described areas does
not exist in other laws.
If we take the differential of 2 .fm . 9(2)) . ds, relative to the characteristic ^,
we shall have
[207] ^ . 2 .fm .(p(v) .ds = :s. fm . 9 (1;) . 5 <? 5 + 2 .fm .6v.(f> (v) . ds ;
but we havef
,, doc.§dx.-\-dy .6dy-\-Sz .Sdz 1 ( dx ^ , , dy , , , dz ^ , )
[208] 5ds^ -^—^ — ^^ = -• I -r'd.5x+^.d.6y-{-—-.d.6z } ;
■■ ^ ds V ( dt ^ dt ^ ^ dt y
we shall therefore find, if we integrate by parts,
-f- 2 .fm . 6 V . cp (v) . d s.
in the direction B Db,hy the force m .(p{v) . — , and m the direction C Dc,hy the force
m .cpv . -— . Multiplying these forces by the corresponding perpendiculars AB = y, and
ds i. J o
A C==x, let fall upon the directions of the forces, they will give, by [29'], the momenta
. . vdx , , . xdy - 1 ■ ' r J : 1 j^ (p{v) ydx
m .o(v) .- , and in.cp(v) . -— , or, by substituting tor d s its value vdt, m. . -— -,
^ ' ds ^ a s V a I
j)i . fxL . - — y^ and as these forces tend to give to m a different motion about the axis of z,
V dt V
(x dv — v dx) 'P('w)
we must take their difference, or m . - — —-^ — ■ . — -, for the resultmg momentum, as
dt V
above.
*(ll2a) In like manner as in [122"].
f (113) The variation o{ ds^ = dx^ -\- dy^ -^ dz^, [44], relative to the characteristic 6,
being divided by 2ds, gives the first value o^ 8ds. The second is deduced from it, by
[212^
I.vi. <^24.] IN ALL RELATIONS OF FORCE AND VELOCITY. 1^3
The extreme points of the curves described by the bodies of the system,
being supposed fixed, the term without the sign /disappears in this equation ;
we shall therefore have, by means of the equation (S) [197],*
8.2:fm.<p(v),ds = :^.fmJv.<?'(v).ds—:s.fmdt.(P.6x+QJy+RJz); [210]
but the differential of the equation (T) [200], being taken relative to <5,
givesf
^.fm.6v,(p'(;v),ds=^:s.fmdt.(P,dx + Q.6y + R.Sz); [2ii]
we therefore have
0 = 5.2 .fm . 9 (y) . d s. [2121
This equation corresponds to the principle of least action in the law of ^^^^
nature. m.(p(v) is the whole force of the body m ; therefore the principle
amounts to this, that the sum of the integrals of the finite forces of the
bodies of the system, multiplied respectively by the elements of their
directions, is a minimum : in this form, it corresponds to all laws mathemati-
cally possible between the force and velocity. In the state of equilibrium,
the sum of the forces multiplied by the elements of their directions is
nothing, by means of the principle of virtual velocities, [114'] ; what
distinguishes, therefore, in this respect, the state of equilibrium from that of
motion, is, that the same differential function, which is nothing in the state [212"]
of equilibrium, gives, by integration, a minimum in the state of motion.
putting ds = vdt, [44]. Substituting this in the first term of the second member of [207],
and then integrating by parts, it produces the two top lines of the second member of [209].
* (114) Tlie integral of the equation, [IQT'], gives,
-../»4.....(^.'4>)+.,...(£f.^) + .....(-.m)|
= — i:.fm.dt.{P.Sx-\-q.Sy-}-R.Sz),
which being substituted in [209], gives the equation [210], rejecting, as above the term
without the sign /.
f (1 15) If we put fvdv.<f/(v) = -^ (v), the equation [200] would become,
'E.m.-^{v) = constant + X,
and its differential relative to the characteristic S, would give 1 .m.Sv . ( -^ — j = 5 X.
But the assumed value of 4> (v), gives -^ — = v .cp' («), hence 2 . m . 5 « . v . <p' (») = 5 X.
Multiplying this by d t, and in the first member, putting v dt = ds, [44], and in the second
for 5 X, its assumed value [199'], (5X = 2 . m . (P . 5a;+ Q . 5y4-i2.(5z), we get [211],
which being substituted m [210], gives [212].
144 MOTION OF A SOLID BODY. [Mec. Cel.
CHAPTER VII.
OP THE MOTIONS OP A SOLID BODY OF ANY FIGURE WHATEVER.
26. The differential equations of the progressive and rotator}' motions of
a solid body, may be easily deduced from those we have given in the fifth
chapter ; but their importance in the theory of the system of the world,
induces us to develop them to a greater extent.
Suppose a solid body, whose particles are urged by any forces whatever.
[212"'] Let X, y, z be the rectangular co-ordinates of its centre of gravity ; a: + ^'?
^ H- y, z-\- 7^ the co-ordinates of any one of its particles denoted by d m, so
that a/, 2/j ^ may be the co-ordinates of this particle, referred to the centre
[2i2iv] of gravity of the body. Moreover, let P, Q, i?, be the forces which act on
the particle, parallel to the axes of x^ y^ z* The forces destroyed in the
particle d m, at each instant, in directions parallel to those axes, will be, by
^ 18 [141], supposing the element of the time dt to be constant,!
-(^Jf+iiiydm + P.dt.dm;
P"] -(^Jy+^.dm + q.dt.dm;
/ddz-\-ddz'\ , , n J. J
— ( ^ \,dm-\-R.dt.dm.
* (116) These forces are supposed, in this computation, to tend to increase the
co-ordinates.
f (117) These expressions are easily deduced from the similar quantities, [141] ;
changing m into d m, and a?, y, z, into a; -{- a?', y -f- ^ , « -j- z', respectively.
I. vii.§25.] MOTION OF THE CENTRE OF GRAVITY. 14^
Therefore all the particles, urged by similar forces, ought mutually to be in
equilibrium. We have shown in § 15 [119'"] that, for this to be the case, it
is necessary that the sum of the forces parallel to the same axis should be
nothing ;* which gives the three following equations :
S_(ill+li^.dm = S.Pdni;
S.hf±/^.dm=S.Rdm;
the letter S being a sign of integration relative to the particle d m, which ^n^^-
integration ought to be extended to the whole mass of the body. The
Symbol
variable quantities x, y, z, are the same for all the particles ; so that we may ^
place them without the sign S ; therefore by putting the mass of the body ^4*1
equal to m, we shall have,t
^ ddx , ddx ^ ddy , ddy „ ddz , ddz
We have also, by the nature of the centre of gravity,!
S.x'.dm = 0; S.y'.dm = 0; S.z'.dm=:0; ^6]
* (118) This follows from what is proved in [119"]. Prefixing therefore, the sign S to
each of the three forces [213], parallel to the axes of x, y, z, respectively, and putting
them separately equal to nothing, then transposmg and dividing by d t, we obtain the
formulas [214].
f (118a) Since x is independent of the sign S, we may bring it from under the sign, and
by this means iS^.— J—. (?m will become .S.dm, and as S. <?m= m, we get
^ ddx ddx rn*i-M
^•-^2-«awi=-^2- • ?w, [215], and m like manner we obtain the similar expressions in
y and z, [215].
t (119) These are deduced from the equations [124], by writing d m for m, S for 2, and
changing x, y, z, into x', ?/, z', respectively. The second differential of any one of these
equations, as S.xf.dm=0, divided hyd^,'isS. ~ . <? m = 0. For if we denote by
37
1^6 ii7A.iM0TI0N OF A SOLID BODY. [Mec. Cel.
hence
[217] S.-j^.dm = 0; S.-j^.dm = 0; S.-j^.dm^O;
therefore we shall have
[218]
d dec
m.~^ = S , P dm
^o'J'Sr m,-j-^^ = S.Qdm } ; (A)
motion of
the Centre /J /J v
"^«''^'"^- m.^^ = S.Rdm
d^
these three equations determine the motion of the centre of gravity of the
body; they correspond to the equations § 20 [155], relative to the motion
of the centre of gravity of a system of bodies.*
We have shown in § 15 [122] that to maintain the equilibrium of a solid
body, the sum of the forces parallel to the axis of a:, multiplied respectively
by their distances from the axis of z, less the sum of the forces parallel to the
axis of y, multiplied by their distances from the axis of z, is equal to nothing ;
we shall therefore have,t
^•S(-+-')<^^^^i^ (1)
^S,\{x-\-7!).q-^{y-^^).P\,dm',
[219]
dm, dm,, dm,,, 8ic., the particles of the body; x', a;/, a?/, &;c. the corresponding co-ordinates,
in the direction of the axis a?, we shall have
S .of .dm::=cd .dm-\- xj .dm,-\- x/i .dmii-\- &c.
The second differential of the second member of this equation being divided by d t^ is
,3 .dm-\- ' . dm,-{- &;c., which is evidently of the form S . . d m, [217],
Substitiiting [215, 217] in [214], we get the formulas [218].
* (119a) It follows from these equations that the motion of the centre of gravity is the
same as if all the forces were applied to it which act upon the whole body, in like manner as
has been proved in [155'].
f (120) This equation is easily deduced from the first of the equations [122], by writing
S for 2, dm for m, a? -j- a/, y -\-i/f z-{-z', for x, y, z, respectively ; also for 'S' • (-7— ),
I. vii.'§25.] MOTION OF THE CENTRE OF GRAVnT. 14.7
now we have*
S,(x,ddy — y.ddx).dm = m.(x,ddy — y,ddx); [230]
likewise
S.(Qx^Py)^dm = x,S.Qdm — y.S.Pdm; [22i]
lastly we have
S.(afddy-\-xddi/ — i/ddx — ydda^).dm=ddy.S.xf dm — ddx.S.i/dm
+x.S.dd^,dm—y.S.dda^.dm; t^^
and by the nature of the centre of gravity, each of the terms of the second
member of this equation is nothing ;t the equation (1) [219] will therefore [232']
become, by means of the equations (A) [21 8], J
S.ij—\ S.{-r-\ which, by formulas [13], represent the forces acting on a particle
parallel to the axes of a?, y, z, their equivalent values, for this case, as they are given in
[213], connected with the factor dt .dm.
* (121) The second members of this, and of the two following equations, are easily
deduced from the first members, by bringing x, y, from under the sign S ; and putting
S .dm = tn, as in note 1 18a.
t (122) Because S.a/.dm=0, S.y'.dm = 0, [216], and S.ddi/ .m = Oj
S.ddx' .m=0, [217].
J (123) Performmg the multiplications, iadicated in the first member of the equation
[219], it becomes,
fxddy—yddx\ /3fddi/—}/ddaf\ /3/ddy-\-xddj/—7/ddx—j/ddif\ ,
^\—dW—)''^'''-^^'[ rf^ )-dm + S.i^ — ydm.
Of the three parts, into which this is divided by the sign S, the last is nothing, [222^], the
first is equal to m . f- — ^^ — ^Y [220], and if in this we substitute the values [218],
weshaUfind m.^- — ^^ — -\ = x . S. qdm—Y .S .P dm-, hence that first
member becomes x . S.Qdm^y.S.Pdm + S. ^^JiM^^l^^^ In ji^g manner the
second member of [219], becomes x .S . (^dm—y .S .Pdm-\-S .{(^xf -^Pt/).dm.
Reject x.S.Qdm — y .S.Pdm, common to botii members, and we shall obtain [223].
[224]
Symbol
1^^ MOTION OF A SOLID BODY. [Mec Cel.
taking the integral of this relative to the time t, we shall have
'/■"" the sign of integration / refers to the time t.
Whence it is easy to conclude that if we put
S.f(Qx' — Py').dt.dm:=N;
[225] S.f(Ra/ — Pz'),dt.dm = N';
S.f(Ry'—Q2f) .dt.dm^N";
we shall have the three following equations :
[226] S\ ^ ydm=N'; ) ; (B)
These three equations contain the principle of the preservation of areas, and
[226^ are sufficient to determine the rotatory motion of the body about its centre
of gravity. When combined with the equations (A) [218], they determine
completely the progressive and rotatory motions of the body.*
If the body is forced to move about a fixed point, it follows from § 15
[226"] [122'"], that the equations (B) [226, &c.], are sufficient for this purpose ;
but then we must place the origin of the co-ordinates x'^ y', z', at that point.
26. Let us now consider particularly these equations, supposing this
fixed origin to be at any point whatever, whether it be the centre of gravity
Its integral relative to dt, evidently gives [224]. Substituting, in this assumed value of JV,
[225], we get the 6rst of the equations [226]. The others are found in like manner, or by
changing ^ into s^, and afterwards x' into y', he.
* (123a) The equations [218], serve to determine the co-ordinates of the centre of gravity,
X, y, Zy upon which the progressive motion depends ; and [226] will give the values of the
co-ordinates x', y\ z', referred to that centre, from which may be found the rotatory motion
of the body about that point. Moreover, the remarks made in [155', 159', 167"], relative to
the centre of gravity of a system of bodies, may also be applied to the case of a solid
body.
I. vii^26.] PRESERVATION OF AREAS. 149
or not. We shall refer the position of each particle to three axes, perpen-
dicular to each other, and fixed in the body, but moveable in space. Let 6
be the inclination of the plane formed by the two first axes upon the plane
oi X, y ; <p the angle formed by the line of intersection of these two planes, [226'"]
and by the first axis ; lastly, let 4^ be the complement of the angle which
the projection of the third axis, upon the plane of x, y, makes with the axis aim of'
of X.* We shall call these three new axes, principal axes, and we shall
* (124) This change of co-ordinates is precisely the same as that in [167^ &c.], writing [227a]
a/, i/f «', for Xj y, z, and x", y", 2", for x^^^ y,,,, z,^^, respectively. In this case, by referring
to the figure page 112, C X will be the axis o( of, C Y that of y', the axis of 2/ being above
C, perpendicular to the plane of the figure ; C G is the axis of x", C ff the axis of y", the
axis of z" being drawn above C, perpendicular to the plane B OP, so that its projection upon
the plane of the figure shall fall on C F, the part of the iplane BOP, falling below the plane
of the figure. The angles FBO=&, ACB = ^, GCB = <p.
It is of importance to notice particularly, the different kind of axes mentioned in this
chapter, which are of frequent use throughout the rest of the work. If the origin of the
co-ordinates is supposed to be fixed in space, the rectangular axes x, y, z, will also be fixed,
but if their origin is supposed to be in any way connected with the body, and to move with it,
these axes will continue to pass through the moveable origin always retaining situations
parallel to their original directions. The co-ordinates of the centre of gravity of tlie body
[212'"], being x, y, z; and those of any particle d m being x-\-xf, y + y*, z-]- z',it wiU
follow, as in [212'^], that x',y',z', are the co-ordinates of the particle dm, referred to
three axes, drawn through the centre of gravity, parallel to the axes x, y, z. On the
contrary, the directions of the axes x", y", z", [226'''], vary with the motion, being fixed in
the body and moveable with it ; the situation of these axes, relative to the axes a/, t/, s/,
being determined by means of the variable angles 6, ■^, cp. So that the place of any particle Remark*
dm, may be determined two ways ; first, by means of the variable co-ordinates a/, y', z', ^ J"^
corresponding to that particle ; or, second, hy the constant co-ordinates a/',y", «", correspond- x",y",xf\
ing to the same particle, taken in connexion with the variable angles 6, -v^, 9, which determine
the positions of the axes of a/', y", z". We must therefore, in finding the differentials of
^j y\ ^ •, [227], suppose a/', y", «", to be constant and ^, 4^) <Pj to be variable, as is done [2276]
in [230&, &tc.]
Before closing this note, it may not be amiss to remark, that in the calculations [175 — 181],
the angles ^, 4^, <?> were supposed to be constant, the object being merely to change the
system of co-ordinates, x, y, 2;, into another system, x„„ y„„ z,,„ enturely similar, and in which
the directions of the axes of x„„ y,,,, z^^^, should be invariable.
38
150 MOTION OF A SOLID BODY. [Mec. Cel.
[226iv] denote by x'\ y", z!\ the three co-ordinates of the particle d m, referred to
these axes ; we shall have, by § 21,*
a! = ar" .{cos. 6 , sin. 4' • sin. 9 + cos. -^I^ . cos. 9}
+ «/' .{cos. d.sin. 4^.cos. (p — cos. 4^. sin. <?}-{- zf' .sin. ^. sin. •>|^;
[227] 1/ = x" . {cos. 6 . cos. -^ . sin. (p — ■ sin. ^j^ . cos. 9}
+ y . {cos. () . COS. 4. . COS. (p + sin. 4^ . sin. (p}+ 2:" . sin. a . cos. 4^ ;
zf == 2f' . COS. 6 — y" . sin. d . cos. 9 — x" . sin. 6 . sin. (p.
By means of these equations we can develop the first members of the
equations (B) [226], in functions of 6, 4., and 9, and their differentials. But
we may simplify the calculation considerably, by observing, that the position
of the three principal axes depends on three constant quantities, which may
always be determined so as to satisfy the three equations
[228] S,x"y".dm=0; S.af'2f',dm=0; S ,f z" . dm=^0.
Putt
S.(f^-\-2f''),dm = A;
[229] S.(x"^ + 2f'^).dm = B;
S.(x"^ + f'').dm = C;
[230]
and for brevity!
^ (p — 6? 4^ • cos. 6 = pdt ;
d ■\' . sin. ^ . sin. (p — d& , cos. cp = qdt;
d 4^ . sin. 6 . COS. cp-\- d6. sin. (p = rdt ;
* (125) These values of x', y', z', are deduced from those of x, y, z, [171], by writing
a/, y, zf, for a?, y, z, and x\ f, z!', for x,,„ y,,„ z,,^, &;c., as in [227a].
f (125a) If we put pfor the distance of a particle dm, from the axis of z'\ we shall
have as in [27], p2= a/' 2 -[-?/' 2^ and the expression of C. [229], becomes C = S .f.dm;
so that C represents the sum of the products of each particle d m, by the square of its
distance from the axis of z", and this is what is called in [245'"], the momentum of inertia of
the solid body about the axis of z". In like manner, B represents the momentum of inertia
[QSX7c] about the axis of 1/', and ./2 the momentum about x". This is analogous to the definition
[29'] ; for the velocity of a particle d m, revolving about the axis «", at the distance p, will
be proportional to p, and the corresponding force acting at the end of the lever p, in a
perpendicular direction, its momentum will he p^ .d m, [29'], and the sum, for the whole body,
S . p^ . d m, as above.
J (126) The importance of this substimtion, for abridging and simplifying the calculations,
will be seen in § 28 [259], where it will be proved that p, q, r, are proportional to the
I. vii. <5 26.] ROTATORY MOTION. ^^^
the equations (B) [226] will become, by reduction, as follows :* Eqnatiow
Rotatorj
^.g. sin. 5. sin. (p + 5 r. sin. 5. COS. (p — C p. cos. 6 = — N; ^
cos.>l..{^9.cos. ^. sin. 9 + -B r . COS. ^ . COS. 9 + Cp. sin, 6]
-\-sm.^. {Br. sm.cp — Aq. cos.(f>}=— N' ; ^; (C) [231]
COS. ^.{Br . sin. 9 — Aq. cos. 9}
— sin.-4>.{^g.cos.^.sin.9+5r.cos.5.cos.9+Cj7.sin.fl} = — N"
cosines of the angles formed by the momentary axis of rotation, and the three principal axes.
These quantities p, q, r, might have been found, a priori, being the co-efficients of a/', y", «",
in the equations 0=paf' — q z", 0=pf — rz'\ 0 = gr y" — r a/', [256, 257, 258],
computed by putting dx'=0, dy' = 0, dsf = 0, [256a], but we shall, in the notes
on this part, follow precisely the method of the author, and, on account of the importance
of the subject, shall give the calculations at full length, usmg however the abridged
expressions [171a].
* (127) For greater symmetry we shall put JV*'= — JVg, JV*"= JV*i, and the equations
[226], by altering the order of the two last equations, will become, [228aJ
[229a]
Under this form any one of the equations can be derived from the preceding, as in [165a],
by taking the next letters in order in the two series a/, y', «', ; JV, JVi, JVg* ^ ^^ "se the
values [171a], the co-ordinates a/, yf, 2:', [227], will become,
a/=.Z?o.a/' + Bo.y'+Co.z",
y = ^i.a/' + A./+Ci.^', [230a]
s' = ^2.a?" + 52.y'+C2.<s".
The differentials of these equations, supposmg a;", y, 2", constant, and &, •<^, 9, variable,
[227&], will be
Ja/==(Z./3o-a/'+rf5o.y" + <?C'o.2^,
dy' = dA^.x"-^dB^.f-]-dC^.s!', [2306]
ds! = dA^.^'-{-dB^.y"-\-dC^.z'\
Substitute tiiese m [229a], neglecting the products x" y", x" 2", / z", which produce nothing
152 MOTION OF A SOLID BODY. [Mec. Cel.
By taking the differentials of these three equations and supposing ^|. = 0,
in the result, inconsequence of the equations [228], and we shall obtain.
S.dw.\(;^^'^^~'^^^'^'\o:!>^ I fBodB,~^,dBo\^„^ ^ /Co;rfC2-C2rfeo\ ,,|^ ^^^
K we take the differentials of Aq, Bq, Cq, he. [171a], supposing^, 4-, 9, variable, and
substitute the values p, q, r, [230], we may obtain the following system of equations,
dAo = {Bop—Cor),dt, dBo=={CQq — AQp).dt, d Cf^^iA^r^B^q) .dt,
[230rf] dAi^{B,p—C^r).dt, dB, = {C^q — ^ip).dt, d C^ = {A^r — B^q) . dt,
dA^=^{B^p—'C^r).dt, dB^^z (C^q — A^p) . dt, d C^=^{d^r— B^q) .dt,
For, if we take the differential of Aq, [171a], and afterwards substitute the coefficients
Cq, t/3i, Bq, [171a], it will become,
[230cJ dAo = — d& . sin. 9 . Cq -\- d -^ . Ai -{- d cp . Bq.
Now if we multiply the first and third of the equations [230], by Bq and — Cq, respectively,
and add the products we shall find,
[230/'] {BqP — Cqt) .dt = — d^. sin. <p . CQ-\-d-]^. ( — Co.sin. d.cds. (p — Bq. cos. {i)-\-BQ.dcp,
and if we substitute, in the coefficient of d-\^, the values [171a], — sin. & . cos. 9 = ^25
COS. ^ = Cg, it will become JSg Cq — BqCq, which is equal to A^, [I75c] j hence the
second member of [230/*], will become like that of [230e], and we shall get
[230fir] dAo = {BqP— Cqt) .dt,
This is the first of the equations [230^/], and the otliers may be found in a similar manner.
But the labor may be much abridged, by observing that if we increase 4-, by a right angle,
it would change the values of Aq, Bq, Cq, [171a], into Ay, B^, C^, respectively, without
altering the values of jp, q, r, [230], and it is evident, from the manner in which [230e,/,^]
were found, that we may make the same changes in [23pg-], by which means it would
[230ft] become dA^ = {Bip — C^r) . dt, which is the second of the equations [230f?]. In like
manner, if we put ■^ = 0, and then increase ^ by a right angle, it will change the quantities
Ai,Bi, Ci, [171a], into A2, B^, Cg, respectively, without altering the values of p, q, r,
provided the terms depending on d 4, are neglected, which can be done in making this
derivation, because these terms vanish from both members of the expressions, d A^^, d B^, d C^,
[230cZ]. For if we notice only the terms depending upon d-\>j in [230], using also the
values, [171a], we shall have pdt^:^ — CQ.d-\^'y qdt= — A^.d-^; rdt= — B^.d-]^.
Substituting these in dA2, d B^, d Cq, [230c?], they mutually destroy each other. Making
therefore, the change of A^, Bi, Cj, into A2, B^, Cg, the expression d Ai, changes into
[230i] dA2={Bii.p — C^.r) .d t, which is the third of the equations [230<?].
163
I. vU. §26.] ROTATORY MOTION.
after taking the differentials, which is the same thing as to take the axis of [asiT
Again, if we increase 9 by a right angle, it will change the quantities A^, Bq, Cq, p, q, r,
[1.71a, 230], into Bo, —A^ C^, p, r, — q, respectively, and the equation [230^], will
become dBo= ( — Aop -\- CQq).dt, which is the fourth of the equations [230(^]. From
this we may derive d B^, d B^, by increasing the index of the letters, Aq, Bq, Cq, as was
done in [230A, i], since the method of derivation, there used, can be applied here without
alteration.
The differential of Co = sin. & . sin. ^]>, [171 a], is
dCQ = d 5 .cos. 6 . sin. -^ -\- d -^ . sin. 6 . cos. ■\', [230A;]
and if we multiply the two lower equations [230], by — Bq, Aq, respectively, the sum of
these products will become,
(^^r — Boq).dt = dd . {Aq . sm. (? ->r Bq , COS. cp) -\- d -^ . sin. 6 . {AqCos, (p — JBq . sin. 9), [230i]
but from [171a], we get, by reduction,
Aq . sin. 9 + 5o . cos. q> = cos. 6 . sin. 4^} Aq . cos. <p — Bq . sin. <p = cos. 4/, [230ni]
hence, the second member of [230Z], will become like that of [230A;], and we shall find
rf Co = (w^o r — Bq q).di. This is the seventh of the equations [230(f], and from it we may
derive d Ci, d C^, by increasing the mdex of the letters A, B, C, as in [230h, {]. It may
be observed that the system of equations [230c?] is symmetrical, either by increasing the
indexes of A, B, C, without changing the letters, or by changing in the two series of letters,
Pj q, r, A, B, C, any letter into the following one of the series, without altering the indexes
of A, B, C.
The values [230fZ], being substituted in the factors of 00"% y"^, z"^, [230c], they will
become as in the first of the following forms, and these may be reduced to tlie second form
by using the equations [175c],
AidAo—AodAi = {{AiBo—AoB,).p-\-{AoCi—AiCo).rl.dt =—{CQp-{-B^r).dt,
BidB,—BQdB,={{B,Co—BoC,).q+{A,Bo—AoB,).p].dt = — {A2q-\-C^p).dt, [230n]
C^dCo—CodC,=\{AoC,—A,Co).r-{-{BiCo—BoC,).q\.dt = — {B^r-{-A^q).dt,
and the others may be found in the same manner, or more simply, by the method of
derivation above used, adding one or two to the index of the letters, rejecting three when the
index is equal to that number. The quantities thus obtained are to be substituted in [230c],
and we shall get,
S.dm.{-{C,p+B,r).a/'^-(A,q+C,p).y"^-^{B,r+A,q).z"^}=:-JV,
S.dm.[ — {Cop+Bor).x"^^{Aoq+Cop).y"^—{Bor+Aos)'Z"^} = —JVi, [230o]
S.dm.^ — {Cip-{-Bir).x"^—{Aiq+Cip).y"^—{Bir+Aiq).z"^^=—JV^,
39
l-^ MOTION OF A SOLID BODY. [Mec. C6I.
ic' infinitely near to the line of intersection of the plane of a/, y, with that
Connecting the terms depending on A^, Bq, &;c., and bringing the quantities p, q, r, from
under the sign S, because they are the same for all parts of the body, we get,
--Aiq.S.dm.{f^+z''^)—Bir.S.dm.{x''^+z''^)--Cip.S.dm.{a/'^+y'^^^
Substituting the values [229], we shall find,
— A^.q.A — B^.r.B—CQ.p. C==— -a;
■^Ao.q.A — Bo.r.B—Co.p. C= — A\ = — A*",
— Ai.q.A — Bi.r.B—Ci.p.C= — JV^ = JV',
and by using the values [171a], connecting the terms multiplied by sin. 4^, cos. •^, they will
become as in [231], the order of the second and third equations being changed.
If we multiply dA^, dA^, dA^, [230<?], by B^, B^, B^, respectively, and add the products
together, the coefficient of pdt, in the sum will be 1, and that of rdt nothing, in
consequence of the second and sixth of the equations [1726], and in like manner we may
obtain the rest of the following system of equations, which are easily proved by the
substitution of the values of dA(„ dAi, Sec, [230fZ], and reducing by means of [172&],
pdt=^Bo.dAo + Bi.dAi-\-B2.dA2 = — Ao.dBo — Ai.dBi — A2.dBz,
[Q30q] qdt= Co.dBo+Ci.dBi-i-C^.dB^^ — Bo.dCo — Bi.dCi — B^.dC^,
rdt=^Ao.dCo + Ai.dCi-{-A2.dC2 = —'Co.dAo—Ci.dAi— C^.dA^.
(127a) The angles ^, 4^, (p, used by the author in computing the rotatory motions of a
solid body, which is at liberty to move in any direction, are peculiarly well adapted to
astronomimal uses, but for other purposes, the following notation has been generally used.
It consists in putting, as usual, x, y, z, for the rectangular co-ordinates of any particle dm o(
the body, and then changing them successively into polar co-ordinates, as in [27]. If we
put p=\/x^-\-y^, and change ts into 9, the expressions of x, y, [27], will become,
x=-p . cos. <p, y = P- sin. 9,
[230r] Taking the differentials of these, supposing x, y, <p, to be variable, we get the values of d a?,
d y, corresponding to a rotatory motion d cp about the axis of z.
dx= — dcp . p . sin. 9, dy = dcp . p. cos. 9,
and by using the values of x, y, we shall find
[230»] dx = — y^dtp, dy = x.d(p.
In like manner the differentials of y, z, depending on a rotatory motion d 4^, about the
axis of X, will be obtained, by changing, as in note 87, x, y, 9, into the letters inmiediately
following y, z, 4^, by which means we shall find,
[230f] dy^= — zd-^^i dz^yd-\>.
I.vii.§26.] ROTATORY MOTION. ^^^
of a/', jj'y we obtain
Lastly, the diiFerentials of z^ a?, depending on a rotatory motion i w, about the axis of y,
will be found, by changing, in like manner, y, z, 4', into 2;, a?, u, respectively, hence we shall
have,
dz=^ — xdo), dx = zdu. [230u]
Connecting together these partial differentials [230^, t, u], we shall obtain the complete
differentials of x, y, z, corresponding to the element of the time d t, namely,
dx = z doi — ydcp, dy = xdcp — zd-^, d z = y d-^ — xdu. [230p]
To find the points of the body in which these variations are nothing, we must put dx=0,
d y = 0, dz = 0, hence
0 = z d u — ydcp, 0=xdcp — z d-\^, 0 = y d -^^ — xdu. r230u)l
Which equations may be satisfied for various values of x, y, z, corresponding to the points of
the body which remain at rest, during the rotatory motions, dcp, d-^, d u, about the axes
- rf<p j-i d(p du
Zy a?, y. It we put — = C, jj = xj, — = A, these equations may be put under the
forms z=Cy, z = Bx, y = Ax, which are the equations of a right line passing
through the origin of the co-ordinates, [196']. In all parts of this right line, we shall have
dx = 0, dy = 0, dz = 0, and this line will therefore be the momentary axis of revolution, L^^^l
corresponding to these three angular motions dcp, d-^,du.
The two first of the equations [230u;], give y = z .-^, x = z . — . Substituting
a 9 d <p
these in r = V^ + y^ + z% [19e], putting also d 6 = \/dcp^-\- d-\.^-\-d w^, we shall get
I /rf4.2 rfw2 d& dcp z z
r = z.y/ J^+-^+l = ^-^' hence— = -; but-, represents, as m page
116, the cosme of the angle formed by the lines r, z, represented by cos. (r, z), therefore
dcp
— ==cos. (r, z). Changing successively z, cp into x, ■\>, and y, u, we get,
— = COS. (r, z), — = cos. (r, x), — = cos. (r, y). [230z]
Hence it follows that the rotatory motions dcp, d-\^, d w, about the axes z, x, y, respectively,
are equivalent to a single rotatory motion about the momentary axis r, the situation of this
a:ris with respect to the axes z, x, y, being determined by means of the angles (r, z), (r, x),
(r, y), which depend on the equations [2 SO^r). The actual angular velocity, about this
momentary axis, may be found as in [259", 260], by considering the motion of a particle
situated in the axis of z, at the distance 1 from the origin, so that x==0, y = 0, z=l.
In this case the equations [230v], ^yedx=du, dy = —d^, dz=:0, hence,
\/da^-\-df-{-dz''= Vdoy'-\-d^^= s/d^—dcp^
[2%]
= (;^.^/ 1— ^ = d^./l— cos.2(r,5r)=<Z5.sin.(r,«).
[231a]
1^^ MOTION OF A SOLID BODY. [Mec. C61.
d ^ . COS. &.(Br. COS. cp-j-Aq, sin. ?) + sin. d,d.(Br. cos. <p + Aq, sin. 9)
— d . (Cp , cos. 6)==z — dN;
[232] ^"^^ C-^*'- sin. <p — J g. COS. 9) — c^^.sin. ^. (B r .cos. cp -\- A q . sin. (p)
+ cos. 6.d .(Br. COS. 9 + ^9. sin. 9) + <? . (C^ . sin. 6)= — d N' ;
d.(Br . sin. 9 — Aq.cos. 9) — ^4-. cos. ^. (Br. cos. 9 + ^ 9 . sin. 9)
^Cp.d4>-sin.d=^dN".''
This represents the motion of the proposed particle, [40a], in the time d t, and its distance
from the axis of rotation being evidently equal to sin. (r, z), [260"], its angular motion will be
obtained by dividing d 6 . sin. (r, z) by sin. (r, z) j therefore tliis angular motion will
be dL
Hence it appears, that the same rules which prevail in the composition and resolution of
rectilineal motions, are analogous to those in the composition and resolution of angular
rotations. For the angular motion dd = Vd 9^ -j" ^ 4'^ ~1~ '^ '^^ about tlie momentary axis
[2316] ^j may be resolved into three angular motions <? 9, d-\^, du, about the rectangular axes
z, X, y, the cosines of the angles which the axis r makes with z, x, y, being represented by
-r-, -7^, — , as in [230^;]. In like manner, the three angular motions dcp, d -^^ du,
[231c]
Composi-
about the axes z, x, y, may be composed into one, represented hy d 6 = Vd (p^ -{- d ■^^ -{- d w^,
and the situation of this axis with respect to the axes z, x, y, may be determmed by means of
tton^'and tho equatious [2302;], which give the cosines of the angles which the momentary axis of
resolution . • i i
of rotatory rotation makcs With the axes z, x, y.
motions. *^
Hence the motions of any solid body, which is at liberty to move in any direction, may be
resolved into a progressive motion of tlie centre of gravity [218], and a rotatory motion about
r231rfl ^ momentary axis passing through that centre. The motion of the centre of gravity may be
resolved into three progressive motions, parallel to tlie thi'ee rectangular axes x, y, z, [218],
and the rotatory motion may be resolved into three rotatory motions about these axes. The
converse of this is also true, that the three motions of the centre of gravity, in the directions
parallel to the axes x, y, z, may be composed into one single progressive motion of that
r231el centre ; and the three rotatory motions about the axes x, y, z, may be resolved into one
rotatory motion about the momentary axis.
* (128) Put
L = B r. sin. 9 — Aq . cos. 9,
[232a] U = Br . cos. (p-{-Jlq. sin. 9,
JW= A q . cos. ^ . sin. (p-\-Br . cos. & . cos. 9 + (^ P • sin. 6 =^ L' cos. 6 + ^P • sin. 6.
[233]
I.vu.§26.] ROTATORY MOTION. ^^^
If we put
these three differential equations will give*
dp' + ^^~f\ q' r' .dt = d N . cos.6-^d N' . sm.6 ;
^^'4.L^I^.r'/.<Zi==-.(^Ar.sin.^4-cZiV'.cos.0.sin.(p+^i*/''.cos.(p; \; (D) [234]
C/ Jo
dr' + ^^^^^.p'(^.dt=—(dN, sinJ+dN', cosJ). cos,<f>^dlV". sin. 9.
and the equations [231] will become,
L' . sin. 6 — Cp . COS. fl = — JV,
JJf.cos-vH-jL.sin. 4> = — JV', [232i]
L . COS. 4. — Jf . sin. 4. = — A*",
whose differentials are,
J^.cos.^.jL' + ^-J^'-sin.^ — d. {Cp .cos. ^)= — dJV,
dM. COS. 4. + <Z L . sin. 4/ — ^4. . Jlf . sin. ■\> -\- d-\, . L . cos. 4. = — <? JV', [232c]
<?I/.cos.4' — t^.^. sin. 4^ — rf 4^ • i' • sin. 4^ — d-\^ .M. cos. 4^ = — dK",
the first of these equations is the same as in [232], and if we put •\> = 0f sju. 4' = 0,
cos. 4- = 1, [231'], in the two last, they become,
d-\..L-\-dM= — dN'', dL — d-^.M^—dJV". [232rf]
Substitute M [232a], and its differential
dM= — dd. sin. 6. L'-\-cos.6.dL' + d.{Cp.^ 6). [232e]
and we shall find,
d-^.L— d 6. sm. 6. L' + dL' .cos.6-\- d.{C p. sm.6) = — dJV'y
dL — d-\>. cos. &.L'—Cp.d-\.. sin. & = —dJV", ^^-^
which are the same as the second and third equations [232].
* (129) Multiplying the first of the equations [232c] by — cos. 6, the first of [232/], by
sin. 6, and adding the products, we shall get by reduction,
— d 6 . L'-j- rf 4' • sin. d .L+eos.4 , d .[( Cp . cos.^) -fsin. d.d.{Cp.s\n.6)=dJV. cos.4 — d JV' mi J, [234a]
but
cos.^.dJ.(Cj3.cos.^)=C.cos.5.(£?p.cos.^^ — p dd. sin J)=C dp. cosM — Cp . rf^ . sin.d . cos. ^,
sin. &.d.{Cp. sin. 6)= C . sin. 6 . {dp . sin. & -{-pd6 . cos. 6) = Cdp. sin.^d -{-Cp.dS. sin. 6 . cos. ^,
whose sum is,
cos.^.<Z.(Cp.cos.^) + sin. fl.£^.(Cp.sin.^)= C d p . {cos.^ 6 -{- sm.^ 6) == C d p. [234c]
40
1^8 MOTION OF A SOLED BODY. [Mec Cel.
These equations are very convenient for finding the rotatory motion of a body
when it turns very nearly about one of its principal axes, vv^hich is the case
with the heavenly bodies.
Substitute this in [234a], it becomes,
[23id] —d 6 , L' -{- d 4. . sin. 6 . L-Jr C d p = d JV . COS. 6-- d JV' . sin 6.
Resubstitute the values of L, L', [232a], it becomes,
[2Sie] Br .{d-^.sin.6.sin.(p—d&.cos.cp) — Aq.{d-^.sin.6.cos.<p-\-dLsin.^)-\-Cdp=:dJV.cos.e — dJV'. s\n.6).
The first member of this equation, by the substitution oi qdt, rdt, [230], becomes,
Brq.dt — Aqr.dt-\- C dp, and from [233], we get
[234/] i> = ^; q = -^; r=-;
therefore, the equation [234e] finally becomes,
^^^. g' r'.dt + dp' = dJV. cos. 6-^dJV'. sin. d,
as in the first of the equations [234],
Again, multiplying the expressions of — dJV, — dJV', [232c,/], by sin. d, cos. ^,
respectively, and adding the products, we shall obtain by reduction,
[23ig] dL'-\-d-\>,L. COS. 6 — sin.6.d.{Cp . cos. ^)-l-cos.fl .d.{Cp. sin. 6)= — dJV. sin. 6 — dJS/"'. cos.d,
but,
— sin. 6.d.{ Cp . cos. 6)== — C. sin. 6 .{dp .cos.d — pd6.sin.6)= — Cdp.sin,6.cos.&-\- Cp.dS.sin.^6,
COS.6 .d .{Cp .sin.6)=C .C0S.6 .{dp .sin.d-\-p dd .cos.6)=C dp. sin. 6 .cos.6-{- C p .d6 cos.^ 6,
whose sum is — sin. d .d .{Cp . cos. &) + cos. 6.d.{Cp.sin.6)=Cp.dd,
hence the equation [234^], becomes,
[234f] dL'-{-d^.L.cos.6-\-dd. Cp = ^{dJV.sin.d-i-dJV'.cos.6).
Multiply this by sin. 9, and — dJV", [232/], by — cos. 9, and take the sum of the
products, the second member is evidently equal to the second member of the second
equation, [234], and the first member becomes,
[234ft] {dL'. sin. 9 — tZL.cos.(p)4-^4/.cos.5.(L.sin. cp-\-L' . cos. q>)-\- Cp . {d-^. sin. 6 . cos. <p-{-dd. sin.(p).
The difierentials of L, L', [232a], are,
dL = Bdr . sin. cp — Adq. cos. cp-^-dcp. {Br . cos. (p -{- Jl q . sin. 9),
dL'=Bdr . cos. <? -\- Ad q .sin. (p-\~d(p .{ — Br .sin.(p-{-Aq. cos. 9),
which being multiplied by sin. 9, cos. 9, we shall obtain by reduction,
dL'.sin.cp — d L. COS. cp = A dq — Br.d(p,
f234m] dL'.cos.<p + dL.sin.<p = Bdr'\-Aq.d(p;
moreover the values of L, L', [232a], give by reduction,
L . sin. cp-\-L' . cos. q)=Brf
P^"5 L.co3.9-.Zi'.sin.9==— ^9.
[234A]
[2341]
'///»
* (130) These values of a/', y", 2f", may be deduced from those of a?,,,, y,,„ z^
[172, 173, 174], by writing, as in [227a], of, y, 5/, a/\ f, s", for x, y, r, a.,,^ y,„ z,,^
respectively.
t (131) Substituting the values of d\ y", [235], in the first members of [236], and
reducing by putting cos.^ 9 + sm.^ 9= 1, we obtain the second members of [236].
[236]
I. vii. % 27.] PRINCIPAL AXES OF ROTATION. 169
27. The three principal axes to which we have referred the angles ^, 4'*
and (p, deserve particular attention ; we shall now proceed to determine their
position in any solid body. The values of ar', y, 2' of the preceding article
give, by § 21, the following :*
a/' = a/ . (cos. 5 . sin. 4^ • sin. <p + cos. 4^ . cos. 9)
+ ij . (cos. 5 . COS. 4' . sin. 9 — sin. -4 . cos. 9) — 2' . sin. ^ . sin. 9 ;
y" = a/ . COS. ^ . sin. 4' • cos. 9 — cos. 4- • sin. 9) j235j
+ 2/ . (cos. ^ . cos. 4^ • cos. 9 + sin. 4^ . sin. 9) — 2:' . sin. ^ . cos. 9 ;
2!' =:^ x! . sin. 0 . sin. 4^ + 2/ • sin. 6 . cos. -^-{-2^ . cos. 6.
Whence we deducef
a/' . cos. 9 — y" sin. 9 = a/ . cos. 4' — ^ • sin. ^]> ;
x" . sin. 9 + y . cos. 9 = ^' . COS. ^ . sin. 4^ + y . cos. d . cos. 4 — z' . sin. 6.
Substitute the first equations of [234m, n], and the last of [230] in [234A;], it becomes,
Adq — Br .d(p-{-d-^. cos. 6 .Br-{- Cp r .dt,
in which the coefficient of Br is — [dcp — d-^ . cos. fl), or — p d t, [230], so that the
preceding expression becomes Adq — B rp .dt-\- C pr .dt, and this, by means of
//^ D\
[234/], is dq-\ —--.r'p'.dt, which agrees with the first member of the second
equation [234], which is therefore correct.
Lasdy, multiply [234i], by cos. 9, and — dJ\"\ [232/], by sin. 9, and add the products,
the second member will be like that of the last of the equations [234], and the first member
of the sum will be,
(JZ/'cos.9-|-^L.sin.9)-}-c?4'-cos.5.(L.cos.9 — L'.sin.9) — Cp.{d-]^.sm.6.sm.(p—dd.cos.(p)y
and this by means of the second equations [234m, n, 230], becomes,
Bdr-\-Aq.d(p — d-], .cos.d .Aq — C pq.dt,
and the coefficient of Aq being dcp — d-^. cos. dz=pdt, [230], it may be put equal to
Bdr-\-Aqp.dt^Cpq.dt, and this, by means of [234/], is dr'-{-^'^~'^ .p'^ .dt,
as in the last of the equations [234], which is therefore correct.
[237]
}^^ MOTION OF A SOLID BODY. [Mec. Cel.
Put
S.xf'i/.dm=f; S.x'z'.dm=g; S.iJ z'.dm = h;
we shall have*
COS. (p.S.x" zf' .dm — sin. (p . S .y" z" .dm = (a^ — b^) . sin. 6 . sin. -^ . cos. 4^
+/. sin. 6 . (cos.^ -I — sin.^ ^^) + cos. 6 . (g . cos. ^ — h. sin. vj.) ;
[238] sin. (f>.S,(jif'z".dm + cos. ? . S. f z" . dm
= sin. 6 . COS. 6 .(«^.sin.^4' + &^.cos.^4' — c^+2/. sin. 4^ . cos. 4-)
+ (cos.^ & — sin.^ 0 • {§■ • sin. ^-{-h. cos. 4.) ;
by putting the second members of these two equations equal to nothing, we
shall have,t
* (132) The first of these equations is found by multiplying the value of z", [235], by
that of a/' . COS. (? — y" . sin. (p, [236], and hy dm,', tlien integrating relative to S. The first
member will evidently agree with [238] ; the second member is
S . ^x' . sin. & . sin. 4^ +/ • sin. <} . cos. -^-{-z' . cos. d| . {«' . cos. 4' — ^ • sin. 4^} • d m,
or by reduction
S . \ {af^ — ^y'^).sin.5.sin.4'-cos.4^-j-a?'y.sin.5.(cos.^4 — sin.^4')+cos.5.(a;V.cos.4> — ^yV.sin.4') | .dm,
and as 6, 4'> ^re not affected by the characteristic S, they may be brought from under that
sign ; and putting S . x'^ .dm = a^ ; S .1/^ . dm, = b^, he, as in [237], we shall obtain
the second member of that equation under the required form. In a similar way the second
equation [238], is found by multiplying the value of z", [235], by that of
x" . sin. (p -j- y" . cos. (p, [236],
and by d m, which produces as above, by prefixing the sign S,
S.lx'.sm.d. sin. 4^+^ . sm. 6 . cos. 4'+^^. cos. 6jJx'. cos. ^.sin.4>+y'. cos.^.cos.4. — zf- sin.5 \.dm,
equal to
S . {sin. 6 . cos. 6 . {x^ . sin.24. + 1/^ . cos.^ ■;. — z'^ + 2 x' xf . sin. 4. . cos. 4.)
+ (cos.^ 4 — sin.^ ^) . {od z . sin. 4> + y' ^r' . cos. \)\ .dm,
which is reduced to the required form, by putting S .x^ .dm= a^, &c.
f (133) The second member of the first equation [238], put equal to nothing, and
divided by cos. 6, using tang. 6 for ^^, will give tang. 6, [239]. The second member of
COS. ^
the second equation, [238], put equal to nothing, substituting sin. 6 . cos. 5 = ^ sin. 2 ^ ;
cos.2 & — sin.2 6 = cos. 2 &, (31, 32 Int.), then dividing by cos. 2 6, and putting tang. 2 &,
for ^^°' , will give the value of I tang. 2 &, [239],
I.vu.^27.] PRINCIPAL AXIS OF ROTATION. ^^1
A. sin. ■I' — ^.cos. '"I'
tang. 6 = -^^ __ ^2^ ^ ^^^ ^ ^ ^^^^ ^ _^^^ ^^^^2 ^ _ ^.^2 ^^ , ^^^
c, . g-. sin. -4. 4- A. COS. 4. .
^^^^S-^^ — c2_a2.sb.24. — 62.cos.24. — 2/.sin.4..cos. + '
but we have [29 Int.]
ci . tang. 6
* tang. 2 ^ = 5—3- ;
^ ° 1 — tang.2 a
Making these two expressions of i tang. 2 a equal to each other, and
substituting in the last, for tang, a, its preceding value in 4^ ; then putting, for
brevity, tang. 4. = w ; we shall obtain, by reduction, the following equation
of the third degree :*
* (134) Put for brevity, to, JV, for the numerators, and «, JB, for the denominators of the
n JV
values of tang. 6, and } tang. 2 6, [239], we shall have tang. &=-; i tang. 2 a = — .
n
These being substituted in [240], it becomes -^ = — ^ = ^__o« Multiplying by
■^~^
£.(e2—w2), and reducing, we get 0 = JVTO2 + e.(nE — A'e); but from the values of
n, e, JV, Ej we obtain,
niJ = A.sin. 4^. {c2 — a^ . sin.24' — &2.cos.2 4^ — 2/. sin. 4' • cos. 4^}
— g-.cos. 4^. {c2 — a^. sin.24. — b^ . cos.2 4' — 2/. sin. 4^ . cos. 4-}
— 'Ke=- — g . sm. 4^ . { (a^ — J^) . sin. 4' • cos. 4* +/• (cos.^ 4^ — sin.^ 4^) \
— h . COS. 4^ • { (a^ — b^) . sin. 4^ . cos. 4^ +/• (cos.^ 4^ — sin.^ 4/) },
adding these together we get nE — JV*e, and, by connecting the similar terms, it becomes,
h (?■ . sin. 4^ — A a2 . sin. 4* • (sin.^ 4. + cos.^ 4-) + A 6^ . sin. 4^ . ( — cos.^ 4^ -|- cos.^ 4^)
•\-f g . sin. 4^ . (2 cos.2 4' — C0S.2 4^ -j- sin.2 4^) -{-g c? . cos. 4* • (sin.* 4^ — sin.* 4.)
■\-g 62.COS.4' • (cos.24'+sin.24') — ^c2.cos.4'+A/.cos.4'. ( — ^2sin.2 4^ — cos.24'+sin.24')»
which is easily reduced to
h<? . sin. 4- — A a*. sm.-\>-\-fg. sin. 4' +5"^ • cos. 4* — g<^^ ' cos. 4* — A/, cos. 4)
and by putting for sin. 4^ its value cos. 4* • tang. 4^, or u. cos. 4^} it becomes,
TO £ — JVe = COS. 4. . J(Ac2 — A a* +/^) . M +^ 62__^ c2 — A/j.
Again, the value sin. 4> = m . cos. 4^) substituted in to, e, JV, they become,
« = cos.4..{Am— ^5 ; e = cos.2 4.. } (a*- &2).w +/.(!— M*)};
JV= COS. 4''{^" + A] ; substituting these in 0:=J^n^ -\-e .{nE — JVe), and rejecting
the common factor cos.*4'j we obtam the equation [241].
41
[240]
[241]
[242]
^^2 MOTION OF A SOLID BODY. [Mec. Cel.
This equation having at least one real root, it is evidently always possible to
render the two following quantities at the same time equal to nothing,*
COS. cp.S.af' zf' .dm — sin. (p. S.y" z" . dm;
sin. (p. S.x" z" .dm-\- cos. c?.S .y" zf' , dm ;
consequently, the sum of their squares (S.x"zf' . dmy + (S .f zf' .dmf^O,
which requires that we should have separately,
[243] S.x^'z".dm = 0; S .f zf' . dm=0.
The value of u gives that of the angle 4-, consequently that of tang. 6, and
thence the angle 6, [239]. We must now determine the angle 9, which is
to be found by means of the condition S.a/'i/' .dm = 0 ; which yet remains
to be satisfied. For that purpose we shall observe, that if we substitute in
S .af'if .dm, for re", if, their preceding values ; it will become of this formf
H. sin. 2 9 + -£' • COS. 2 9, H and L being functions of the angles & and 4^,
and of the constant quantities a^ ¥, ^tf^gi h ; putting this expression equal
to nothing, we shall have
[244] tang. 29 = ::^.
Equations
mint'the Thc thrco axes determined by means of the preceding values of 6, ^|^, and 9,
Rot'auon. satlsfy thc three equations [228],
[245] S.x"y".dm = 0; S.x"zf'.dm = 0; S.fz".dm = 0;
* (134a) As this value of u renders the second members of [238] equal to nothing,
their first members [242], must also be equal to nothing. The squares of these last added
together, putting cos.^ 9 -f- sin.^ 9=1, give the equation
{S.a/'z" .dmf-{-S,{y"z" .dmf=.0,
abovementioned ; both terms being squares, their sum cannot be nothing, unless we have
separately the two equations [243].
f (135). If in the expressions of x", y", [235], we connect together the terms multiplied
by sin. 9, and those multiplied by cos. 9, and put for brevity,
H' = x' . cos. 4 . sin. 4- + y' . cos. & . cos. 4* — ^ • sin. d, i' = a;' . cos. -4 — 1/ • sin. 4?
they will become a/' = H' . sin. 9 + ^' • cos 9 ; y" = II' . cos. 9 — L' . sin. 9, which,
multiplied together, give a/' y" = {H'^ — L'^) . sin. 9 . cos. 9 -{-H'L' . (cos.^ 9 — sin.^ 9),
or (31, 32 Int.), x" y" = ^ . {H'^ — L'^) . sin. 29 + H'L' . cos. 2 9. Multiplying this by
dm, and prefixing the sign of integration S, then putting 5 . J {H'^ — L'^) .dm = H;
S.H'L' .dm = L, we get S . cc" y . rf m = Jf. sin. 2 9 + i^ . cos. 2 9, as above.
[245']
[245^]
I. vii. §27.] MOMENTUM OF INERTIA. ^^^
The equation of the third degree in «, seems to indicate three systems of
principal axes, similar to the preceding ; but we ought to observe that u is
the tangent of the angle formed by the axis of ar' and by the line of intersection
of the plane of a/, 'if, with that of a/', / ; now it is evident that we may
change any one of the three axes a/', f, ^\ into any other of them, and still
the three preceding equations [245], will be satisfied ; the equation in w,
ought therefore also to determine the tangent of the angle formed by the
axis of a/ with the line of intersection of the plane of a/, i/, either with that
of a/', \j\ or that of a;", 2", or /, z". Therefore the three roots of the
equation in u are real, and appertain to the same system of axes.
It follows, from what has been said, that in general a solid has but one
system of axes which has the property treated of. These axes have been
called principal axes of rotation, on account of a property peculiar to them,
of which we shall hereafter treat.*
The momentum of inertia, or rotatory inertia of a body, relative to any JJ^^f-
axis, is the sum of the products of each particle of the body, by the square R^j^^y*'
of its distance from that axis. Thus the quantities A, B, C, are the
. . [245"']
momenta of inertia of the solid we have just considered, relative to the axes
a/', y, and 2", [227c]. Let us now put C for the momentum of inertia
of the same solid relative to the axis of 2:' ; we shall find, by means of the
values of a:' and y of the preceding article,!
C' = A, sin.^ d . sin.2 q?-\-B. sin.^ & . cos.^ 9 + C* . cos.^ & ; [246]
* (135a) The property here mentioned is treated of in [280*''], where it is proved that if
the body begin to turn about one of the principal axes, it will continue to move uniformly
about that axis, and from this property they are called principal axes.
f (136) Square the values of cd, y, [230a], multiply the sum by dm, prefix the sign S,
neglect the products depending on x" y, a!' z", y" «", on account of the equations [245],
we shall get
which by means of the three first equations, [1726], A^ -\- A^ = 1 — A^, &;c. becomes,
C' = S.dm \{l—Ai) . a:"2 + [l—B^^] . f^ + (1 — C^^) .z"^]. m^]
Put2s = A-\-B-\-C, and from [229], we shall obtain s= S .{a/'^-{-y"^-{-z"^).dm, [246i]
subtracting from this each of the equations [229], we get,
S.a/'^.dm=^s^A, S.y"^.dm = s — B, S .z"^ .dm = 5— C, [246c]
164 MOTION OF A SOLID BODY. [Mec Cel.
The quantities sin.^d . sin.^cp, sin.^ ^ . cos.^ 9, and cos.^^, are the squares*
of the cosines of the angles which the axes of a/', ?/', 2:", make with the axis
of 7^ ; whence it follows, in general, that if we multiply the momentum of
inertia relative to each principal axis of rotation, by the square of the cosine
[246'] of the angle it makes with any other axis whatever, the sum of these three
products will be the momentum of inertia of the solid, relative to this last
axis.
The quantity C is less than the greatest of the three quantities A^ B, C,
[246"] and exceeds the least of them ; the greatest and the least momenta of inertia
appertain therefore to the principal axes.f
r^iQm Let X, Y, Z, be the co-ordinates of the centre of gravity of the solid,
referred to the origin of the co-ordinates, which we shall fix at the point
about which the body is forced to turn, if it be not free ; a;' — X, 3/ — Y,
2! — Z, will be the co-ordinates of the particle dm oi the body, referred to
and from the substitution of these in [246«], we find,
C' = {l^Ai).{s-A)-^{l^Bi).{s-B)^{\^Ci).{s-C)
But from \\12d\, {3 — A^^ — B^^— Cg^) . 5 = (3 — 1) .s = 25, and {A-{-B+ C)=25,
therefore the two first terras of the preceding expression destroy each other, and we finally
get,
[246dJ C'=:A.A^^-\-B.B^^+C.C^^ = S.{a/^-\-y'^).dm,
which, by using the values of ./^g, B^, Cg, [Hla], becomes as in [246].
* (137) This is evident from [246^;], observing that the values of ^2. Aj ^'3, [172/,
171a], became, by changing x, y, z, x^^^, y^,,, z,,,, into a/, y, zf, a;", 1/', 2;", respectively, [227a],
[246e] i^g = cos. (2/, x") = — sin. ^ . sin. 9 ; -Sg = cos. (2/, 3/') = — sin. 5 . cos. 9 ;
C2 = COS. {2^, z") = cos. 6.
f (138) For, if in the general expression of C, [246], we substitute for A, B, C, the
greatest of those quantities, for example A, the result would evidently exceed C, because
each of its terms is positive. Now this result would be,
[247a] *^ • { sin.2 6 . (sin.^ 9 -f- cos.^ 9) + cos.^ 6]=zA.[ sin.^ 6 + cos.^ 6]= A.
Hence C is less than the greatest of the quantities A, B, C. And in a similar way, by
taking A for the least of the quantities A, B, C, we may prove that C exceeds the least of
the quantities A, B, C.
I. vii. ^ 27.] MOMENTUM OF INERTIA. ^^^
its centre of gravity ; the momentum of inertia, relative to an axis parallel
to the axis of z', and passing through the centre of gravity, will therefore be,
S.{{x'—Xf-{-{^^Yf].dm', [247]
now we have, by the nature of the centre of gravity, S.afdm^mX;
S .1/ dm = mY [154] ; the preceding momentum therefore becomes*
— ?W.(X^+Y') + 5f.(a/^+y^).cZ7». [248]
Thus we shall have the momentum of inertia of the solid, relative to the axis
which passes through any point whatever, when the momenta are known
with respect to the axes which pass through the centre of gravity. It is
evident also, that the least of all the momenta of inertia corresponds to one [248^
of the three principal axes passing through that centre.f
Suppose that by the nature of the body, the two momenta of inertia A
and B are equal, we shall have,t
C" = J . sin.2 & + C . cos.^ 6 ; [249]
making therefore 6 equal to a right angle, which renders the axis of z'
perpendicular to that of z" [246e], we shall have C = A. The momenta of [249']
inertia relative to all the axes situated in the plane perpendicular to the axis
* (139) Developing the expression S . \ {cd — Xf -\- {y' — Y)^] . dm^ it may be put
under the form,
S.{x'^ + y'^).dm + S.{X^-{-Y^).dm — 2S.{3/X+i/Y).dm. [2476]
Now
S ,{X^+ Y^) , dm = (X^-\- Y^) . S . dm = {X^^ T^) .m;
— 2S.x'Xdm = —2X.S.a/dm,
and by [154], S.a/dm = mX, hence — 2S.x'Xdm = — 2mX^; in like
manner — 2 S. y' Ydm = — 2 m F^ ; these being substituted in [2476], the whole
expression becomes, — m . (X^ -f- Y^) + jS . (a/^ + t/^) . rf m, as in [24S].
f (140) For the momentum of inertia about the axis of z is S . (x'^ -\-y'^) . dm,
and about a parallel axis passing through the centre of gravity is
S , {a/^ -{- y'^) . dm — m . {X^ -{- Y% [248].
Hence the latter must generally be the least, and as the least momentum corresponds to one
of the principal axes [246"], the proposition becomes manifest.
f (141) PutjB = ^, in the general expression, [246], and cos.^(? + sm.^(p= 1, we
shall get [249]. An ellipsoid of revolution, about the axis of «", is a figure of this kind.
42
[251']
[252]
^^^ JMOTION OF A SOLID BODY. [M^c. Gel.
t)f ^\ will then be equal to each other. But it is easy to prove that we shall
have, in this case, for the system of the axis of 2:", and of any two axes
whatever, perpendicular to each other and to this axis, the following system
of equations.
[250] S.x'i/ .dm^O\ S.x'2f' .dm = 0; S.y'z" .dm = 0;
for, if we denote by x", y", the co-ordinates of a particle dm of the body,
referred to the two principal axes, taken in the plane perpendicular to the
axis of 2f\ and with respect to which the momenta of inertia are supposed to
be equal, we shall have*
(351) S , (x'"" + z"^) . dm = S . (f^ + z""") , dm ;
or simply S . x"^ . dm = S . y"^ . d m ', now by putting s equal to the angle
which the axis of x' makes with the axis of x", we shall havef
ccf = x" . COS. s + 2/' . sin. s ;
y' = y" . COS. s — a;".sin. s;
we therefore have
1^53] S. xy. dm= S. x"y". dm . (cos.^s — sin.^s) + S. (y"^—x"^) . dm . sin. s . cos. s = 0 ;
* (142) The momentum of inertia relative to the axis of y" is S . {x"^ -\-s/'^) . dmj
[229,227c], and relative to the axis of x", is S .{1/'^ -\- z"^) .dm; putting these equal
evidently gives S . x" ^ . d m = S . y"^ . d m.
f (143) The values of x', 1/, are fovmd, hy using the figure in page 112, changing the
co-ordinates x, y, x^, y^, of the point K, into x', y', x", y", respectively, and putting •\j= s,
by which means the two first equations, [168], become as in [252]. Multiply these values
of a?', 2/, together, and the product by <Z m, prefixing the sign S, we obtain [253],
S . a^Y • '^wi^ (cos.2 s— sin.2 s) . 5. a;"/ • ^^» + sin. s . cos. s . S: . (/2_a/'2) . <?TO ;
observing that sin. ?, cos. s, being common to all points of the body, may be placed without
the sign S. The second member of tliis equation becomes nothing in consequence of the
equations S.x"y".dm = Q, [245], and S .a/'^.dm=S.y"^ .dm, [251^, so that we shall
have S .x'y' .dm=0, as in [250]. Again, multiply the values [252] by z" . dm, and
prefix the sign S, we shall find,
S.x' z" .dm = COS. s .S.x" z" .dm -{-sm.s. S .y"z" .dm,
S .y'z" .dm = cos. s . S.i/' z" .dm — sin. s. S. x" z" .dm,
the second members of these equations become nothing, by means of the equations [245],
therefore we shall have S . a;V . <d m = 0, S.y' «" .dm.=^ 0, as in [250].
[253al
1. vu. ^ 28.] MOMENTARY AXES OF ROTATION. ^^'^
we shall find in a similar manner S. a/ zf' .dm = 0 ; S .1/ zf' . dm= 0 ; ^253f]
therefore all the axes perpendicular to that of z" are then principal axes, and
in this case the solid has an infinite number of principal axes.
If we have, at the same time, A= B = C we shall have in general*
C = A; that is, all the momenta of inertia of the solid will be equal ; but
then we shall have generalljf
[253*1
S.xy .dm = 0; S .of zf .dm = 0 ; S.i/zf.dm^O; [254]
whatever be the position of the plane of x', ij, so that all the axes will then
be principal axes. This is the case of the sphere ; we shall see hereafter
that this property appertains to an infinite number of solids, of which we shall
give the general equation.}
[254']
28. The quantities p, q, r, which we have introduced in the equations [254"]
(C) ^ 26, [231], have this remarkable property, that they determine the j,^^g„^^_
position of the real but momentary axis of rotation of the body, with respect Routlon?
* (144) Putting A = B= C, in the general expression of C, [246], it becomes equal
to A, as in [247a].
f (145) From A = B,-we deduced S . of y' .dm = 0, in [253a]. In like manner, by
putting A= C, we should get S . a/ z' . dm=r- 0 ; and B= C would give
S.y'z' .dm=0.
We might also prove llils by means of a/, y, z', [230a]. For, if we multiply those values
of a/, j/, together, and their product by d m, preSxing the sign S, neglecting the quantities
[228], we shall get,
S.x'y'.dm=AoA.S.x"^.dm-\-BoBi.S.f^.dm+ C^ Ct$.z"^.dm, [254a]
butfrom A = B=C, [229], we get S .x"^ .dm = S .y"^.dm= S . z"^ .dm.
Substituting these in [254a], we 6nd,
S.xy.dm = {AQAi + BoBi-i-CoCi).S.x"^.dm,
and tiiis, by means of the fourtli equation, [172c?], becomes S.x'i/.dm=0. In like
manner we might find the other two equations [254].
X (146) In Book V, § 2, [2940].
[255]
16^ I MOTION OF A SOLID BODY. [Mic. Cel.
to the three principal axes. For we have, relative to the points situated in
the axis of rotation,*
[254'"] dx' = 0; d7/ = 0; dz' = 0;
and by taking the differentials of the values of x', y', z', ^ 26 [227], making
sin. 4^ = 0, after taking the differential,! which can be done, since we may
fix at pleasure the position of the axis of x' in the plane of x', y', we shall
have
d af = x" .{d -^^ . cos. & . sin.? — d<p.sin.(p]-\-y".{d-]^. cos. ^ . cos.? — dc?, cos.qt]
+ z" .d4''Sin.6=0;
di/ = x" .[d^. COS. 6 . COS. (p — d 6 . sin. & . sin. 9 — d^. cos. 9}
~{-y".{d^. sin. cp — d(p .COS. ^.sin. 9 — da. sin. ^ . cos. (?}-\-z" .d6 .cos.6=iO;
dzf = — x" .{da . COS. 6 . sin. cp-{- d(p. sin. & . cos. cp]
— ^' .{d6 . COS. 6 . cos. (p — dcp . sin. & . sin. 9} — 2!' .da . sin. ^ = 0 ;
If we multiply the first of these equations by — sin. 9 ; the second by
cos. a . cos. (p ; and the third by — sin. 6 . cos. 9 ; we shall have by adding
themf
[256] 0=px" — qzf';
d 3/ di/
* (147) If a point, whose co-ordinates are x', y\ z', be at rest, the quantities -y— , — ^,
rfz' .
— -, which represent the velocities in directions of the axes x, y, z, respectively, must be
(t t
nothing, consequently, dx' = 0, dy'^^O, dz'=0. Substituting in these, the values
of x', y', z', [227], we must suppose &, 4, (p> to be variable, and x", y", z", constant, because
any particle of the body retains always the same relative position to the principal axes
x", y", z"j as was observed in [2276].
f (148) By this substitution, the differential of any expression of the form
J\l . cos. 4^ -|- -^ • sin. 4^)
becomes dM-}- L .d-^, as is shown in [232<Z], therefore, in finding the differential, we
may change sin. 4^ into d 4^, and in the terms multiplied by cos, 4^, put cos. 4- = 1 , and
take the differential of the terms in which this substitution is made.
J (149) The equations [255], are the same as [2306] which are to be put equal to
nothing, [254'"], and then the values [230<Z], are to be substituted. The first of these
equations arising from dx' = 0, is,
0 = x".{B,j,-Cor)-\-y".{Coq — ^oP)+'^''{'^or'-Boq)
I. vii. <^28.] MOMENTARY AXES OF ROTATION. 169
If we multiply the first of the same equations by cos. cp ; the second by
COS. 6 . sin. 9 ; and the third by — sin. d . sin. 9 ; we shall have by adding
them
0=pf---rz"; [257]
Lastly, if we multiply the second of the same equations by sin. 6 ; and the
third by cos. 6 ; we shall have by adding them
0 = qf — raf'. [258]
This last equation evidently results from the two preceding ;* therefore the
three equations dx' = 0, di/ = 0, d z' = 0, are reduced to these two
equations, which appertain to a right line forming with the axes of ar", y", zf',
the angles whose cosines aref
Sjfj^fj^^ \/fJ^fJ^jSi ^^^y2_|_^
[259]
the values ^^ = 0, rf^/^rO, give similar expressions, and by arranging them according to
the order of the letters A^ B, C, we get the following system of equations,
0 = Ao.{rz"-py")+B,.{pa/'-qz")+Co.{qf-rx"),
0 = A,.{rz"-py")+B, . {paf'-qz")+C, . {qf-rx"), [256a]
0 = A,.{rz"-py") + B,. {px" -qz") + C, . {qf-rx"),
Multiply these equations by Aq, A^, Aq^ and add the products ; the coefficient of rz'' — py^\
becomes 1, and the others vanish, in consequence of the equations, [J72J], hence
r«" — py"=0, as in [257]. Multiplying the same equations by Bq, Bi, Bq^ the sum of
the products, reduced in the same manner, becomes pa/' — qz"=0, as in [256]. Lastly,
the same equations, being multiplied by Cq, Cj, Cg, and the sum of the products taken, is
qj^' — j.x" = 0, as in [258], It may be observed that the factors given above by the
autlior are the same as Aq, A^, A^, Bq, &;c. [I71a], putting 4. = 0.
* (149a) From [256], we get z" =J~, and from [257], z!' — ^, hence ^~ ==^,
which is easily reduced to the form 0 = qy" — r a/', [258].
f (150) The equations [256, 257, 258], may be easily reduced to the form of the
equations [196'], which correspond to those of a right line passing through the origin of the
co-ordinates. Let this line be A c, (Fig. page 7), whose origin is A, the co-ordinates of
the point c being AD = x", Dd^=Aa = y", dc = AB=z'\ and the cosines of the
angles, which the line A c makes with the axes of a/', fy", «", will be represented by
43
^"7^ MOTION OF A SOLID BODY. [Mec. C61.
this right line is therefore at rest, and forms the real axis of rotation of the
bodD^.
Rotatory To obtaitt the rotatory velocity of the body ; let us consider the point of
the axis of z", which is distant from the origin of the co-ordinates by a
qyuantity equal to unity. We shall have its velocities parallel to the axes of
[259^ y,'^ 2/^ and z\ by making ar" = 0, j/'=^0, 2"= 1, in the preceding expressions
of dx', di/, dz\ [255], and dividing them by dt'^ which gives for these
partial velocities,
d-h . , dd — d& .
[2601 -rrr • sm» 4 ; -— . cos. 4 ; — r— • sm. 6 ;
^ ^ dt dt ' dt ^
therefore the whole velocity of that point is^
~dt
[3601 y/ di.^^d^^.sm.^6 ^ ^ (fj^j^.
[258a] ~, 1^, ~, and since A c^=A D^ + D d^ + dc^ = x"^ + f^ + z"^ [11a],
these cosines will be represented by
t^^'*! /x'^4-/2-f2"2' v/a:"2-|-y'24-z"2' y/ay'2-|-y"2-|-z«2'
TTf'
Q 2 T %
respectively. Now the equations [256, 257], give x" ■= — , y = — , hence
v^^-^^3^^':^±±±^.
[259&]
[259c]
therefore, the preceding cosines will be
q ^ ?
^^2-1-524-^2' vV4^52+r2' v5h^T»^'
a8;in [259]. Hence we find that the sines of the same angles will be,
V/p2-|-r2 vV+92 v/g2 4-r2
y/^2l[-g2^r2 \/^2-|-52^r2 ^p2^g2.|.^
and the tangents of the same angles will be
v/pH^ y/^+g^ y/g^+r^
[259aj , , >
q r p
as is evident from the common rules of trigonometry.
* (151) This expression is found as in [40a], by taking the square root of the sum of the
squares of the partial velocities, [260], and reducing by putting cos.^ 6 -f- sin.^ ^ = 1, by
which means it becomes, ^ ^^^"^^ '^'"— -. But if we take the sum of the squares of
a t
I. vii. <§ 28.] ROTATORY MOTION. ^"^^
Dividing this velocity by the distance of the point from the momentary axis
of rotation, we shall have the angular velocity of rotation of the body ; now
this distance is evidently equal to the sine of the angle which the real axis
of rotation makes with the axis of 2f', the cosine of which angle is*
— ^ ; we shall therefore have V^ p^ + ^^ + r^ for the angular rQeo^i
velocity of rotation.
Hence we perceive that whatever be the motion of rotation of a body,
about a fixed point, or a point considered as fixed ; this motion can be no [260'"]
other than rotatory motion about an axis fixed during an instant, but which
may vary from one instant to another. The position of this axis, with
respect to the three principal axes, and the angular velocity of rotation,
depend upon the variable quantities p, q, r, the determination of which is
very important in these researches, and as they denote quantities independent
of the situation of the plane off xf, 2/, they must be independent of that
situation.
the vdlues'of qdt, rdt, [230], and reduce it by putting sin.^(p-f-cos.^<p = 1, we shaD
get d6^ + d^^.sm.^6={f-\-r'^).dt^, hence,
v^llf±li!:!!^=: /^qr^.
♦ (152) The sine corresponding to this cosine is equal to — , [259c], and as
the distance of the point, assumed in [259'], from the origin of the co-ordinates is 1, its
r/o2_L,a
distance from the momentary axis of rotation will be — Tf we divide the
\/p^'\-q^-{-r^'
velocity v^M" ^ [260'], by this distance, we shall obtain the angular velocity of rotation,
Vp^ -\-q^ -\-r^i which for brevity we shall call a. [260a]
f (153) The terms p, q, r, have been shown, [259], to be proportional to the cosines of
the angles, which the momentary axis of rotation forms with the principal axes, and as these
axes are wholly independent of the plane arbitrarily assumed, for that of a/, y', it is evident
ihotpj q, r, cannot depend on the position of this plane.
^"^^ MOTION OF A SOLID BODY. [Mec. Cel.
29. Let us now determine these variable quantities, in functions of the
time ty in the case where the body is not urged by any external forces. For
this purpose, let us resume the equations (jD) ^ 26 [234] between the
variable quantities p', q', r', which are in a constant ratio to the preceding.
[260'] The differentials d JV, d N', d N", are in this case nothing ;* and these
equations give, by adding them together, after having multiplied them
respectively by p\ q\ r',
[261] 0=p' dp' -\-(J dq' -\-r' dr' '^
and by integration,
[262] /2 + 9'2 + r'2 = A;^
k being an arbitrary constant quantity.
The equations {D) [234], multiplied respectively by 2AB.p', 2BC.q\
and 2AC.r', and these products added, give, by integrating their sum,t
[263] AB.p'^ + BC.q'^ + AC.r'^ = H^;
H being an arbitrary constant quantity ; this equation comprises the principle
of the preservation of the living forces, f We may deduce from these two
integrals.
C.{A — B)
* (154) As is evident by putting the forces [212'^], P=0, Q=0, R = 0, in the
[2606] differentials of the expressions [225], which makes ^^"=0, dJV' = 0, rfJV" = 0, and
then the equations, [234], become
[261a] dp'i-^-^^.^r'.dt = 0, d q' + ^-^-^ ./ p\ dt = 0, dr'+^-^^.p'q' .dt=0,
which being multiplied by p', (f, r', respectively, and the products added, the coefficient of
p' q' r', becomes nothing, and the sum is 0 =p' dp' -\- q d q' -\-r' dr'y as in [261].
f (155) Using the equations [261a], of the last note, and multiplying them as above
directed, the coefficient of p' q' r, becomes 0, and the sum of these products is
2AB.pdp'+2BC.^dq' + 2AC.r'dr' = 0,
its integral is [263].
f (156) To prove this, we shall use the figure page 112, taking C G for the axis of a?",
C H for the axis of y", C X for the axis of x\ C Y for the axis of i/, and we shaD
suppose the momentary axis of rotation to be the axis of z', perpendicular to the plane of the
I. vii. <^29.] ROTATORY MOTION. ^'^^
thus we shall have ^ and / in functions of the time t, when p' shall be
determined ; now the first of the equations (D) [234] gives*
therefore
ABC, dp'
at— ■ _. zr= ..__ ; [266]
^lAC.I<^ — H-^ + A.{B—C}.p'^.{H^ — BC.k^ — B.{A — C).p'^]
figure BAP', and since a =*^^ + 9^ ~f" ^» [260a], is the velocity of a particle revolving
about the axis of z', at the distance 1, the velocity of the particle dm, whose distance from
that axis is \/oi/^-\-y'^, will be a .\/xf^-{-y'^ ', therefore the expression of the living
force, corresponding to this particle will be a^ . {x'^-\-y'^) . dm. Its integral relative to the
characteristic S, gives the whole living force,
a^.S.{x'^-{-y'^).dm=a^,{A,A^^ + B.B2^+C.C2'^l, [246<fj,
in which Aq, B.2, Cq, represent, as in [172/, 227a], the cosines of the angles which the
axis of s/ makes with the axes of x", y", z", respectively, so that we shall have, from
[259, 260a],
consequently,
a^.S.{x'^ + y'^).dm=aK(±l- + ^^+^£-^==Aq^ + Br^+Cp^.
Substituting the values of p, q, r, [234/], it becomes,
^'S'~C" ABC '
and by the principle of the living forces this ought to be a constant quantity as — ; hence
ABC
this becomes as in [263]. Lastly, from [262, 263], we obtain the values of ^^7^% by the
usual rules of algebra.
* (156a) The equations [234], are reduced to the form [261a]. The first of these gives
dt, [265], and this, by means of q\ /, [264], becomes as in [266].
We may also find c? ^ in terms of q' or r', by using the second or third of the equations
[261a]. By comparing these three equations together, it appears that they will not be
altered if we change in the two series of letters, p', qf, r, A, B, C, each letter into the
following one of the series, beginning the series again when we come to the last terms /, C.
44
^^74 MOTION OF A SOLID BODY. [Mec Cel.
J266'] an equation which is integrable oiilj in the three following cases, J5 = ^
B = C, A = C*
And the same takes place in the formulas [263 — ^266], derived from [261a] ; hence we get
from [2GG] the two following expressions,
[266a] d t =
V/ \BAk^—'m-\-B.[C—A).q'^l.\m — CAk^—C.{B—A).q'n
[2666] dt = -- -^^^'^ .
V/jCBfea — H2 4-C.(^ — l{).r'2|.jH2_^£A;2_^.(C— JB).r'2|
These may be used when A=^B^ or B = C, to prevent the formulas [264,265]
becoming — , or indeterminate.
* (157) The general integral of the equation [266] depends on Le Gendre's elliptical
functions [82a]. In the three cases mentioned by the author, [266'], this integration can be
done by means of circular arcs and logarithms. There are also three other cases, not
mentioned by him, in which this integration is possible, by the same method ; namely, when
fP = ^ C . ^^ H^ = AB.k\ or n^ = BC. k^. We shall examine these cases
separately. First, when B = A, the expression [266a], becomes,
, ABCdcf
at =■
^\BA.'k^—H^-\-B.{C—A).q'^^ .\/m — CA.k^
This formula is used in preference to [266], because the denominators of [264, 265]
. , „ . ABC - BA.k^—m -„ . -
vanish. Puttmg — — =tf? and — ^-77; — 57"=^ > it becomes
\/[H^—CA.k^).B.[C—A) B.{C—A)
/T // //
dt== — . If a and b be both real quantities, we shall have for its integral
t=a. hyp. log. (5' + ^j;^^^
as is easily proved by differentiation and reduction. If the coefficient of q'^, in [266a],
should be negative, it might then depend on circular arcs, but it is not necessary to go into
this investigation. Second, When B= C, or A=C, the coefficient of p'^ in one of the
factors of the denominator, [266], vanishes, and the equation may be put under the form
:, a and b being constant quantities, differing from the preceding. Its integral, when
^fe2_p'2
a is real, and J 6 positive, is f = a . arc. f sin.- j -{- constant. The other cases where
6 6 is negative, or a imaginary, which may depend on logarithms, are easily computed.
Third, When H^=AC.k% or H^ = BC. B, the terms independent of p'\ in one of
the factors of the denominator, [266], vanishes, and the term p'^ comes from under the radical,
Lvii.^29.] ROTATORY MOTION. ^^^
The determination of the three quantities y, ^, /, introduces three
arbitrary constant quantities, namely, H^, h^, and that introduced by the ^^^
integration of the preceding equation. But these quantities give only the
position of the momentary axis of rotation upon its surface, or relative to the
three principal axes, and its angular velocity of rotation. To obtain the real
motion of the body about the fixed point, we must also find the position of
the principal axes in space ; which would introduce three new arbitrary
constant quantities, depending on the primitive position of these axes, and [26e*^
this would require three new integrations, which, combined with the
preceding, would give the complete solution of the problem. The equations
(C) §26 [231] contain three arbitrary constant quantities, N, N', N" ; but
they are not wholly distinct from the arbitrary constant quantities H and k. [266'»]
For, if we add together the squares of the first members of the equations
(C) [231], we shall have*
which gives 1^ = N"" -\- N"" + N"\
and that equation may be put either in the form d <= — ~~» whose integral is
p's/lf^p"^
t=a. hyp. log. — — ;
• 1 r 7. ^ab.dp' , . , , , , 6— v/62__p'2 ,
or, in the form dt=: . whose mtegral is < = hyp. log. — ^— , o being a
p' y/62— p^ b + v/i2— p'2
real quantity. It was not thought necessary to notice the different cases arising from the
negative values of b^. When H^ =AB .k^, the two factors of the denominator become
divisible by Ar* — p'^, the one being
{A C — AB) . [Ic'—p'^ the other {AB—B C) . {k^'-p'^)y
J 1. • ^ j-i ABC , , r, , , ^fk.djf
and by putting 2/A; = — — , the value of of becomes di=-jz ~t
\/{AC—AB).{AB—BC) k^—p^
whose integral is t=f. hyp. log.
k — p
♦ (158) The sum of the squares of A*', JV", [232&], reduced by putting
cos.^ 4. 4- sin.^ 4^ = 1 > is
consequenUy, N^ -\- N'^ -['Xi'"^^U -i- J^P -\- N^ Also, the sum of the squares of
I'^'S MOTION OF A SOLID BODY. [Mec. C^l.
The constant quantities N, N', N", correspond to c, c', c" of ^ 21 ;* and
[2671 the function it.Vp'^-{-q'^-{- r'^ expresses the sum of the areas described
during the time t, by the projection of each particle of the body upon the
plane on which this sum is a maximum, multiplied respectively by the mass
of each particle. N' and N" are nothing relative to this plane ; by putting
therefore the values found in ^ 26 [231], equal to nothing, we shall havef
M= L'.cos.&-\- Cp.sm.d, JV= — L' . sm. d -\- C p . cos. 6, [232a, 6], reduced by
putting cos.2^ + sin.2^=l, is M^ -{-JV^=L'^-Jr C^ p% hence
Again, the sum of the squares of L, L', [232a), reduced by putting cos.^ (p -f- sin.^ <p = 1,
is L2 + L'2 = ^2 ^2 _|_ ^2 ,.2^ therefore, JV^^JV'^-{- JV"^ = A^q^-{- B^ r^ -f C^p*,
and this, by the substitution of the values [233], becomes p'^ + j'^ + r'^, as in [2G7],
and then from [262] we get,
P =y 2 _|_ 2'2 _|_ ^2 ^ jvrs + JV'2 + JV"K
[2676]
* (159) The forms of the expressions JV, JV', JV*", [226], are precisely similar to those
of c, c', c", [167], and we may derive from the former, similar results to those derived from
the latter, in [ISO, 181'], namely, that the plane corresponding to JV'= 0, JV" = 0, will
be that of the maximum of the areas mentioned in [267']. Moreover the quantity
i.<?<./c2 + c'2 + c^
deduced from [181'], will become, for this case,
i.dt. \/JV2-f JV'2 4-JV"2 = ^.<;^.v//2 + ?'2+r'2, [267].
This expression represents the sum of the areas described in the time d t, by each particle
of the body, projected upon the plane of maximum areas, and multiplied by the particle.
Its integral relative to the time ^, is ^t. Vp^+ q^-\-r'^, observing that the terms under the
'- * radical are, in the case under consideration, equal to the constant quantity k, [267J]. It may
be observed that the words, " multiplied respectively by the mass of each particle," were
accidentally omitted in the original.
f (1 60) Put JV" = 0, JV" = 0, in the two last equations [232&],' and they will become,
L . sin. ■^-\-M. cos. 4^ = 0, L . cos. -^^M . sin. 4. = 0. Multiply these by sm. >|^,
COS. 4', respectively ; the sum of the products, reduced by putting sin.2 4'-f~ cos.^4'== 1>
becomes L = 0 ; substituting this in either of the equations, we shall get, M= 0. Using
the values [232a], these equations, L = 0, M= 0, become as in [268]. We might also
have deduced these equations, by putting JV'=0, JV" = 0, in [267a], which becomes
L^ -f- M^ = 0, and when Lor Mis not imaginary, this must give, L = 0, M=0.
I.vii.^29.] ROTATORY MOTION. 177
[268]
0 = Br . sin. <p — Aq . cos. cp ;
0 = Aq , COS. 6 . sin. 9 + ^ »" • cos. 6 . cos. 9 + Cp . sin. ^,*
whence we deduce
COS. e =
sin. d. sin. (p =
-9'
y/y^^2_j_/2' [369]
sm. ^ . COS. (p = — —
By means of these equations, we shall know, in functions of the time, the
values of & and 9, referred to the fixed plane, which we have considered. It
only remains to find the angle 4-, which the intersection of this plane and that
of the two first principal axes, makes with the axis of a/ ; which requires
another integration.
The values of q and r ^ 26 [230] givef [270]
d -^ . sin.^tf =zqdt . sin. & . sin. (p-\-rdt. sin. d . cos. (p ;
* (160a) Substitute the values [233], in [268], and they will become,
/.an. 9 — g^. COS. 9 = 0, [268a]
{/ . COS. 9 -|- j' • sin. 9) . COS. 6 = — jp! . sin. 6, [2686]
To the square of [2686] add the square of [268a], multiplied by cos.^d, reducing the sum
by putting sin.^ 9 + cos.^ 9=1, we get
(/2_|_ ^2) , cos.25 ==y2 , sin.2d = p'2 , (1 -_cos.2^),
or (p'^ + 9'^ + »^^)- cos.^ ^ =y^ whence we get cos. ^, as in the first of the equations
[269], also y = cos. ^ .v/p^HYM"^- This being substituted in [2685], we get from
division by cos. 5,
r'.cos.9 + g'.sin.9 = — sin.d . Vp'2_|.^2_|_/2, j-ggg^-j
IVIultiply this by — sm. 9, and [268a] by cos. 9, add the products and reduce, by
putting sin.2 9 -\- cos.^ 9 = 1, we get — 5' = sin. ^ . sin. 9 . Vp'^+^'^+r'^ which is the
same as the second of the equations [269]. Substitute this in [268a], and divide by sin. 9,
we find — / = sin. ^ . cos. 9 . \/p'2_|_^2_j_/2^ ^l^^^jj jg ^^ j^j ^f ^^ equations [269].
t (161) IVIultiply the values of qdt, rdt, [230], by sin. fl. sin. 9, sin.d.cos.(p,
respectively, add the products and put cos.2 9-f-sin.2 9 = l, we shall obtain [270].
Substitute in this q=-j, ^—~^^ [234/], also the values of sin. «. sin. 9, sin. ^. cos. 9,
45
^"^^ MOTION OF A SOLID BODY. [Mec. Cel.
whence we deduce
[271] ^ ^^kdt.(Bg'^ + Ar'^)
Now we have by what precedes [262, 263].
[272] 5'^ + /^^ = ^^-/^; Bq"-\-Ar^'=^'~'i^-P";
we shall therefore have
^ JiBC.{k^—p'^)
If we substitute instead oi dt its value found above [266], we shall have
4^ in a function of p' ; the three angles ^, (p, 4-, will therefore be determined in
functions of the variable quantities p', ^', r', which are themselves determined
in functions of the time t. We shall know therefore, at any instant whatever,
the values of these angles with respect to the plane of x', ?/, which we have
considered, and it will be easy, by the formulas of spherical trigonometry,
[273'] to deduce from them the values of the same angles referred to any other
plane, which will introduce two more arbitrary quantities, which, combined
with the four preceding,* will make the six arbitrary quantities which the
complete solution of this problem requires. But it is evident that the
consideration of the plane just mentioned simplifies the problem.
The position of the three principal axes upon the surface of the body being
supposed known, if we know also, at any instant whatever, the position
of the real axis of rotation upon that surface, and the angular velocity of
[273-^1 rotation, we shall have, at that instant, the values of p^ q, r, because these
values, divided by the angular velocity of rotation, express the cosines of the
[269], and for sm.^ 6, its value J, To_l^> deduced from that of cos. 6 [269], and
it will become,
^_|-/2 —B^.dt — Ar'^.dt
The substitution oi h = v/^^F^I^^T^, [2676], will give [271].
* (162) These four quantities are H, k, and the two constant quantities introduced by the
integration oi dt, d 4/, [266, 273].
[273"']
I. vii. <§29.] ROTATORY MOTION. ^'79
angles which the real axis of rotation forms with the three principal axes ;*
we shall therefore have the values of y, 9', / ; now these last values are
proportional to the sinesf of the angles which the three principal axes form
with that plane of a/, y, upon which the sum of the areas of the projections
of the particles of the body, multiplied respectively by their masses, is a
maximum ; we can therefore determine, at every instant, the intersection
of the surface of the body by this invariable plane ; therefore the position
of this plane can be found by the actual conditions of the motion of the
body.
Suppose that the rotatory motion of the body was caused by a force
striking it in a direction not passing through its centre of gravity. It will
follow from what we have demonstrated in ^ 20, 22, [155', 188'], that the
centre of gravity will acquire the same motion, as if this force was applied [273iT]
directly to it, and that the body will have, about this centre, the same
rotatory motion, as if the centre was immoveable. The sum of the areas
described about this point, by the radius vector of each particle, projected
upon a fixed plane, and multiplied respectively by these particles, will be
proportional to the momentum of the impressed force projected on the same
plane ; now this momentum is the greatest when referred to the plane which
passes through the direction of the force and the centre of gravity ; this
plane is therefore the invariable plane. If we put f for the distance of the ^^^^
* (163) This angular velocity is Vy + ^^-f^ = a, [260a], which is supposed to be
O T P
given. The cosines of the angles abovementioned are — , — , — , [2595]. These being
known, we may obtain from them the values of p, q^ r, and thence by [233], the values
of/, 5',^.
f (164) The sines of the angles which the three principal axes a/', y", «", make with the
plane of cd, 1/, are evidently the same as the cosines of the angles which the same axes
a/', y", z", respectively make with the axis of z', and by [246e], these are respectively,
— sin. ^ . sin. (p ; — sin. 6 . cos. 9 ; and cos. 6; but in [269], it is shown that these quantities [273a]
are represented by
g' / y
VVHYH/^' \/p'2-^5'2_^/2' V^p'S-^^S-j-Za'
which are to each other in the same proportion as the quantities j', /, p', as above.
'^^ MOTION OF A SOLID BODY. [Mec. CeL
line of the primitive impulse from the centre of gravity ; and v the velocity
[273'i] it impresses on this point ; m being the mass of the body, mfv, will be the
momentum of this force, and by multiplying it by ^ t, the product w^ill be
equal to the sum of the areas described during the time t ;* but this sum,
by what precedes, is ^ ^ . V^ y^-' + g'^ + r'^ [267'], therefore we shall have
■yhoomtytm ,
t2743 V/y'+^' + r'2 = m./y;
If at the commencement of the motion, we know the position of the
principal axes with respect to the invariable plane, or the angles 6 and 9 ; we
shall have, at that time, the values of y, q', /,! consequently those of
[274'] p, q, r ; we shall therefore have, at any instant whatever, the values of the
same quantities.
fve°Mu" This theory may serve to explain the double motion of rotation and
ma^'Z revolution of the planets by a single primitive impressed force. Suppose the
produced
by one
* (165) Let the force exerted, in giving motion to the body
m, be represented by a mass m', moving with the velocity v', and
therefore with the force m' v', in the direction AB D, which line
is perpendicular to the line C D =/, drawn from the centre of
gravity C, of the body m. Then, by the nature of the centre
of gravity [155'], this must cause the same progressive velocity v, in the centre of gravit)'^ C,
of the body m, as if the whole force was applied at that centre, and as the whole force m! v',
is supposed to be exerted upon the body m, we shall have m' v' = m v, [273'^]. Again, if
we put AB = v't, for the space passed over by the mass mf, in the time t, before impact, the
corresponding area described about the point C, would be the triangle ABC = ^v tf, and
this multiplied by the mass m', produces ^m' v' .tf, or ^mv . if, and by the principle of the
preservation of areas, [167'"], this must remain the same after impact, or be equal to the
quantity it .\/;^^-\-q'^-\-r'^, [267'], corresponding to the plane of maximum areas, it
being evident that this plane must be the same as that passing through the centre of
gravity of the body m, and the line of direction of the primitive force [273'']. Putting
therefore these two expressions equal to each other, and dividing by the common factor
^^, we get [274].
f (166) The values p', q', /, may be deduced from the known values of 6, 9, w,/, v, by
means of the formulas [269, 274], and from the formulas [233], we may find p, q, r.
These values of y, 5^, r', at the beginning of the motion, which serve to determine the
constant quantities of the integrals of [266, 266a, 2666].
I.Tii. §29.] PROGRESSIVE AND ROTATORY MOTION. 181
planet to be a homogeneous sphere, whose radius is i?, and that it revolves
about the sun vt^ith an angular velocity U ; r being supposed to express its [274"]
distance from the sun, we shall have v ^r U; moreover, if we suppose that
the planet moves in consequence of a primitive impressed force, whose
direction passes at the distance f from the centre of gravity, it is evident
that the body will also revolve about an axis perpendicular to the invariable
plane ; considering this axis as the third principal axis,* we shall have ^ = 0;
consequently ^ = 0, r' = 0 ; we shall therefore have j)' = mfv, or
Cp = mfr U. [274"']
But in the sphere, we havef C = f . m i2^ ; therefore
* (167) This supposition may be made, because aU the axes of a sphere are principal
axes, [254']. In this case the axis of revolution z", being perpendicular to the invariable
plane of, y', the inclination of this plane to that of x", y", or 6, [226'"], must be equal to
nothing. This being substituted in the two last of the equations [269], gives g^ = 0,
^ = 0; hence p' = mfv, [274], and by substituting y= Cp, [233], and v = rU,
[274"] we shall get Cp = mfr U, [274'"].
f (168) Let Cbe the centre of the sphere, C P the axis of z",
C Eq the plane o( cc" y", P E Q^q P a spherical surface described
about the centre C with the radius CP=C E^=R, P E, P Q,
P q, quadrantal arches drawn through P, perpendicular to E Q, the
two last being infinitely near to each other ; A B, a b, arcs of circles
parallel to Q q, forming the infinitely small parallelogram A Bab,
whose area is A B X Aa, and if on ibis base we erect a
parallelopiped whose height, taken in the direction of C A, is d R, its solidity will be equal to
the element dmo( tlie mass of the sphere, hence dm = ABxAaXdR. Put the
angles E C q = s, ACP=p', then the arc Eq = R.s, qq = R,ds,
AB= Qq. sin. A C P=R.ds. sin. p', Aa = R .d p'. Substituting these we get
dm = R^dR.ds.dp' .sin.p', whence m=f ff R^ dR. ds . dp' .sin. p'. The
integral relative to s being taken from 5 = 0, to 5 equal to the whole circumference 2 *, gives
m = 2 -TT .//jR2 dR.dp' sin. p'. Again, fdp' . sin. p', taken from p' = 0, to p' = cr,
is fdp' . sin. / = 1 — cos. / = 2, hence m^Ant.fR^dR, this, integrated from
[275o]
[275t]
i? equal 0, to R=R, gives m^ — .R^, which is the well known theorem for the solidity
46
[276]
1S2 MOTION OF A SOLID BODY. [Mec. Cel.
which gives the distance / of the direction of the impressed force from the
centre of the planet, corresponding to the observed ratio between the angular
velocity of rotation p, and the angular velocity of revolution about the sun U.
As it respects the earth, we have -^ = 366,25638 ; the parallax of the sun*
[275'] gives — = 0,000042665, consequently /= yi^ . i2 very nearly.
The planets not being homogeneous, we may consider them as being
formed of spherical and concentric strata of different densities. Let p be the
density of one of these strata, whose radius is i?, p being a function of R ;
we shall havef
^_ 2m fp.RKdR
2~'fp.R\dR'
m being the whole mass of the planet, and the integrals being taken from
iJ = 0, to its value at the surface ; we shall therefore have
[277] f=^P- fp-R'-d^
^ S'ru'fp.R^.dR'
of a sphere. To obtain the momentum of inertia C, [229], about the axis C P, we must
multiply the particle dm, by the square of its distance j1 D from the axis CP, [245'"], and
take the integrals as before, and as AD=R.sin. p',
[275c] C =fff dm.AD^ =fffR' dR.ds.dp' . sin.^/ = 2ir .f/R^dR. dp' . sm.^p'.
Now by [84a], fdp' . sin.^' p' = — ^ . cos.p' . sm.^p' + f ./ dp' . sm. p', consequently
fdp' . sm.^p' = — ^ . COS. p' . sin.^ p' — f . cos.y + f , the integral being supposed to
commence with p' = 0, this when p' = ir, becomes |, consequently
[275rf] C = ^-.fR'dR=~.R^
and as m = -— .R^, this becomes C=— .mP?.
o 5
* (169) The parallax here used is 8% 8, whose natural sine is nearly equal to 0,00004266.
f (170) The elements of the integrals of m, 0, found in the note [168], are to be
multiplied by p, and the integrations being made relative to s, p', which are independent of i?,
Q
[276a] we shall get as in [275&, rf], m^Ant.f^.R^.dR, C= -^.f ^ .R^ .d R. Dividing
this last by the former, and multiplying the quotient by m, gives C, [276]. Substitute this
in [274'"], and it will give/, [277].
I.vii.§30.] OSCILLATORY MOTIONS. 183
If the strata nearest the centre are the most dense, as it is natural to suppose
is the case, the function /' „,' , „, will be less than f i?*,* the value of/ [27r]
J p .nr .alt
will therefore be less than in the case of homogeneity.
30. We shall now compute the oscillations of the body, in the case where
Oscilla-
tion of &
I
it turns very nearly about the third principal axis. We may deduce them *°' ''
from the integrals we have obtained in the preceding article ; but it is more ^^^"^
simple to deduce them directly from the equations (D) §26 [234], The
body not being impelled by any forces, these equations become, by substi-
tuting, for p', gf, r', their values Cp, Aq, B r,t
dp + ^-^^^.qr.dt^O;
dq-\-^^-^^—^.rp,dt = 0; [278]
dr + ^^^:^.pq.dt^O.
[278o]
*(I71) Whenp=l, fp.RKdR = \R^ s^nd f p .R^ .dR = i R^,
fp.mdR " '
Suppose now the whole mass of the planet to remain unchanged, but to become denser
towards the centre, by the removal of some of the particles from the outer parts towards the
centre. The differentials of m, C, [276a], bemg
dm = A'K.pR^.dR, dC = ^.pR'^.dR, [2786]
we shall have d C=%R^ .dm^ and C = %fR^ .dm. Now if any particle dm is
carried from the surface towards the centre, to increase the density of the parts near the
centre, the radius corresponding to this particle must be decreased, consequently, R^ d m, and
fR^ .dm, or C, must be decreased, therefore the quantity mtiR 2 ' [276a], must
be decreased, m being constant, and it must be less than f R% found above.
1(172) Put in [234], dJ\r=0, d JV' = 0, dJV" = 0, [260&]. Substitute the
values p', ^, /, [233], divide the results by C, w2, Bj respectively, and we shall
obtain [278].
^^^ MOTION OF A SOLID BODY. [Mec. Cel.
The solid being supposed to turn very nearly about its third principal axis ;
[278'] q and r will be very small quantities,* whose squares and products may be
neglected ; which gives dp = 0, consequently^ is constant. If in the two
other equations we suppose
[279] q = M. sin. (nt-\-y)\ r = M' . cos. (n t + y) ;
we shall havef
[280]
"^-P'V AB~ ' ^^^ -—Jyj^'Y B.{C-By
M and 7 being two arbitrary constant quantities. The angular velocity of
[280'] rotation will be V p^ -\- f -{■ 11^ [260"], or simply^, by neglecting the squares
of q and r ; this velocity will therefore be nearly constant. Lastly, the sine
* (173) In this case, the angles formed by the axis of rotation and the axes of a/', or 1/',
must be nearly equal to a right angle, consequently their cosines must be very small ; now
by [259] these cosines are represented by — — — — consequently
gf, r, must be much smaller than jt>. If we therefore neglect the product q r, in the first of
the proposed equations, [278], it will become dp=0, whose integral is jo = constant.
f (174) If we substitute the assumed values q, r, [279], in the two last of the equations
[278], supposing p to be constant, we shall get, by rejecting the factors cos. {nt-{- 7),
sin. {nt-\- 7),
whence we may easily deduce tlie values n, M', [280], which will evidently give for q, r,
[279], values satisfying the proposed equations, and containing two arbitrary constant
quantities. This solution is only a particular case of a much more general form given in
[1089]. We might also have obtained this solution, by taking the differential of the second
equation [278], substituting in it the value d r, deduced from the third of the equations [278],
and using for brevity the value of n, [280], by which means it would become
which is of the same form as [865'], whose solution, [864a], is as in [279]. This value
of q would give r, by means of the second equation [278].
I. vii. § 30.] OSCILLATORY MOTIONS. 186
of the angle formed by the real axis of rotation and the third principal axis,
wUlbe* .^^2+?. [280-]
p
If, at the commencement of the motion, we have ^=0, and r=0, that
is, if the real axis of rotation coincides, at that instant, with the third bieprojti-
principal axis, we shall have M= 0, M' = 0\ q and r will therefore always p^ri?.cia*i
, , , Axes of
be nothing ; and the axis of rotation will always coincide with the third ^°»**'°"-
principal axis ;t whence it follows, that if the body begin to turn about one of [280"']
the principal axes, it will continue to revolve uniformly about that axis. This
* (175) By r259c1 this sine is — — :, or nearly •
f (176) The radical expressions which occur in [280] being supposed free from
imaginary quantities, \i q = 0, and r = 0, when ^=0, the expressions [279] will become
which cannot generally be satisfied for all values of A, B, C, except by putting M=0,
and M' =■- 0. These being substituted in the general values of q, r, [279], they
become nothing, as in [280"']. The case where n becomes imaginary is treated of in
[28r, &tc.].
In this demonstration the values of q, r, are supposed absolutely correct, but as they are
only approximate values, obtained by supposing p constant, it may not be amiss to give a
loose accurate demonstration. To do this, we may multiply the two last of the equations
[278], respectively, by 2A.{A—C).q, and 2B.{B—C).r, and take tiieir sum,
A.{A—C).2qdq-i-B.{B—C),2rdr = 0,
whose integral is A. {A— C) .q^-\-B .{B— C) .i^ = 0, die constant quantity being 0,
because at the beginning of the motion g- = 0, and r = 0. The same equation would also
result by the extermination of p'% from the equations [262,263], using the values [233].
Now if C be greater than A and B, the quantities A — C, B — C, will both be negative,
and if q, r, are real, q^, r^, will be positive, and the preceding equation,
A,{A—C).f + B.{B—C). 7^=0,
cannot be satisfied except by supposing the general values of q, r, to be ^ = 0, r = 0. In
like manner, if C is less than A and 5, tiie quantities A — C, B — C, will be positive, and
the preceding equation cannot be satisfied except by putting g' = 0, r = 0. Lastiy, when
C falls between A and B, the quantities A — C, B — C, will have different signs, and
then it will not necessarily follow that we must put q = 0, r== 0, to satisfy that equation,
"47
186 MOTION OF A SOLID BODY. [Mec Cel.
remarkable property of the principal axes, has caused them to be named
[280i^] principal axes of rotation : it appertains exclusively to them ; for if the real
axis of rotation is invariable at the surface of the body, we shall have*
L '1 dp = 0, d q = Oi dr := 0 ; the preceding values of these quantities then
become,
f38n {B—A) - {C—B) „ (A—C)
^L__._^.r9 = 0; ^—-.^,rp = 0', ^ \pq=^0.
In the general case where A, B, C, are unequal, two of the three quantities
[281'] p, g, r, are nothing in consequence of these equations, which requires that the
real axis of rotation should coincide with one of the principal axes.f
If two of the three quantities A, B, C, are equal, for example if A = Bj
rasi'^ the three preceding equations are reduced to these, rp = 0, pq = 0; which
may be satisfied by supposing only p = 0. The axis of rotation is then in
a plane perpendicular to the third principal axis ;t but we have seen, in
because it would be satisfied by putting r = q. 4/ ' — —, in which the radical
\/ —^ -^, is a real quantity ; this corresponds to the case of unstable equilibrium,
mentioned in [281'"]. These results agree with those found above, upon the supposition
that p is constant.
* (177) When the real axis of rotation is invariable^ the angular velocity of rotation,
[281a] which is v p^ -\- (^ -\- 'fi = a^ [260a], must be constant, by the principle of the preservation
of the areas ; and the cosines of the angles, which this axis makes with the three principal
q T V
[2816] axes must also be constant. These cosines are represented by -, -, — , [259], and since
they are constant, their differentials must be put = 0, hence dp = 0, dq==0, f?r = 0.
Substituting these in [278], we shall get [281].
f (177a] Thus if p, r, were nothing, the expressions of the cosine of the angle, formed
Q
by the axis of x", and the momentary axis of rotation, v/ 24- 24- 2' P^^]> ^ould
become 1 ; therefore these axes would coincide.
J (178) Because by [259], the cosine of the angle, formed by the axis of sf', and the
P
axis of rotation is y '. ^ , ^, which being equal to nothing, that angle is a right angle.
vP t5' T^
I. vii. § 30.]
STABLE AND UNSTABLE EQUILIBRIUM.
187
§ 27 [249'], that all the axes situated in this plane are then principal
axes.
Lastly, if we have at the same time A=B = C, the three preceding
equations will be satisfied, whatever hep, q, r ; but then by § 27 [254'], all
the axes of the body are principal axes.
Hence it follows, that the principal axes alone have the property of being
invariable axes of rotation ; but they do not all possess this property in the
same manner. The rotatory motion about that axis whose momentum of
inertia falls between the two others, may be sensibly troubled by the slightest
cause ; so that there is no stability in this motion.
A system of bodies is said to be in a stable state of equilibrium^ when an
infinitely small derangement of the system can produce only an infinitely
small change in the positions of the bodies, by making continual oscillations
about the situation of equilibrium. This being premised, suppose that the
real axis of rotation is infinitely near to the third principal axis ; in this case
the constant quantities M and M' [279] are infinitely small ; and if w is a
real quantity, the values of q and r will always remain infinitely small, and
the real axis of rotation will not deviate from the third principal axis but by
quantities of the same order. But if n be imaginary, sin. (nt-\- y), and
COS. (nt + 7)j will then become exponential quantities ;* and the values of q
and r might then increase indefinitely, and at length cease to be infinitely
small ; there will then be no stability in the rotatory motion of the body
[281'"]
[281 i']
[281 V]
Stable
Eqnilib-
lium.
[281 vi]
[281 vii]
Unstabl«
Eqoilib-
rium.
*(179) If in
sin. (nt-^y) =
y.v/HT
(Form. 1 1 Int.) we put — =: a, or
— -. y^ — =T-» it will become
2/— 1 4a'
4 o
Now if n be imaginary, and equal lo n' . \/—i, n being a real quantity, the preceding
expression will become sin. (n < -|- y) = a . c -|-__ . c , and as the exponent of c"
increases with the time, this quantity may become indefinitely great. The same takes place
with cos. (n t -\r y), as is easily proved in the same manner, using (Form. 12 Int.).
188 MOTION OF A SOLID BODY. [Mec, Cel.
about the third principal axis. The value of n is real, if C is the greatest,
[28l^"i] or the least, of the three quantities A, B, C; for then the product
(C — A.).(C — B) is positive; but this product is negative, if C falls
between A and B ; and in this case, n [280] is imaginary ; therefore the
[28 1«] rotatory motion is stable about the two principal axes whose momenta of
inertia are the greatest and the least ; and it is unstable about the other
principal axis.
Now to determine the position of the principal axes in space, we shall
suppose the third principal axis to be nearly perpendicular to the plane of
[281 ^] x'l y'i so that & may be a very small quantity, whose square can be neglected.
We shall have, by ^ 26,*
[282] <?(? — d^=pdt\
which gives by integration
[283] 4- =9 — pt — s,
6 being an arbitrary constant quantity. If we then put
[284] sin. ^ . sin. 9 = 5; sin. & . cos. <j: = 11 ;
the values of q and r of ^ 26,t will give, by exterminating rf^l^,
d s du ,
[285] --pu==r; _+^s = -5;
* (180) In the first equation of [230], we may substitute 1 for cos. 6, neglecting the
square and higher powers of 6, (44 Int.) and it becomes as in [282].
f (181) Take the differentials of s and w, [284], substitute 1 for cos. 6, m for sin. fl. cos. 9,
s for sm.6 . sin. 9, and divide by d t, we shall obtain,
ds d & . , dcp du dd d(p
— = — . sm. © -4- M . — : — == — . cos. 9 —"S . -: — ;
dt dt ^^ dt' dt dt ^ dt'
substractpw from the first, add ps to the second, and put — — P=-T7i [282], they will
ds d & . , d-L du , dd d-L .
become— »m=-— . sm. 9 + a. --^: tt +7'* = tt-cos.9 — s.—-} the second
dt ^ dt dt dt -^ dt dt
members of which are equal to the values of r and — q, deduced from qdt^ r dt, [230] ;
substituting tliese values in the preceding equations, they become as in [285].
I.vii. §30.] COMPOUND PENDULUM. 1^^
whence by integration*
s = p.sin.(pi+x) — . sin. (nt + y) ;
u = ^. cos, (pt+y^) ^ — . COS. (w ^ + y) ;
p and X being two other constant quantities ; the problem is thus completely
resolved, since the values of 5 and u give the angles 6 and 9 in functions of
the time,t and -^ is determined in a function of <?. and t. If 3 is nothing, [28^1
* (182) If we substitute in [285], the values of r, q, [279], these equations will be
satisfied by the assumed values of s, u, [286], as may be easily proved by substituting in the
coefficients of cos. {nt-{- 7), and sin. {nt-\- 7), the values of n, M', [280], which renders
these coefficients nothing. We may also find the equations [286], by a direct method by
means of formula [865], in the following manner.
Take the diflferential of the first of the equations [285], supposing, as above, p to be
constant. Substitute in this, the value of — , deduced from the second of these equations,
d d 8 d T
and we shall get -\-p^ s — — -\-p q = 0. But the values q, r, [279], give
— ^^+pq=={M'n + Mp).sm.{nt + yl
and from [280], we get M'n = — Mp . \ hence
M'n-{-Mp = '^'^^~^.Mp,
which being put equal to a K, the preceding equation will become
-j^ -{-p^ 5 + a jBT. sm. (n ^ + 7) == 0.
which is of the same form as [865], whose solution is given in [865&, 870', 871], changing
y, a, 6, m, s, 9, into *, p, ^, n, 7,X, respectively, so that firom [8656, 871], we get,
« = p . sin. (p <+X) +^^3^ . sin. (« ^ + 7).
But from the value of n [280], we get r? — p^ = — p^ . — ^ -^ hence
Lo_^ = 77~j ^d the preceding value of s becomes as in [286]. Substitute this in
«==-.]t7- — r iy [285], using r [279], and reducing as above, it becomes as in [286].
t (183) Having s and w, [286], we obtain ^, 9, fi:om the equations [284], and then v|^,
from [283].
48
i^ MOTION OF A SOLID BODY. [Mec. Cel.
the plane of a;', ?/, becomes the invariable plane, to which we have referred
the angles ^, (p and 4- in the preceding article.*
[286"] 31. If the solid body be free, the analysis of the preceding articles will
Compound ffivo tho motloH about its centre of gravity : if the body be forced to move
Pendulum. ^ , . . .„ ^ J J
about a fixed pomt, it will give its motion about that point. It now remains
to consider the motion of a body which is forced to move about a fixed
axis.
Let a;' be this fixed axis, which we shall suppose to be horizontal. In this
[286'"] case, the last of the equations (jB) ^ 25 [226] will be sufficient to determine
the motion of the body. Suppose also the axis of y to be horizontal, and
the axis of z! vertical, and directed towards the centre of the earth. Lastly,
suppose the plane which passes through the axes of y, z\ passes also
through the centre of gravity of the body, and that an axis is drawn from
this centre of gravity to the origin of the co-ordinates. Let ^ be the angle
that this last axis makes with the axis of z' ; and if we put y[\ z!\ for the
co-ordinates of any particle referred to this new axis, we shall havef
[287] y == y . cos. &-\-z!' , sin. d ; 2! = z!' . cos. ^ — f . sin. ^ ;
* (184) Wtien ^ = 0, the general values of s, u, [286], become
• 5 = — -— . Jkf.sin. (n^ + 7), M = — — -. J»f' .COS. (wf + r),-
Cp ^p
or by substituting the values q, r, [279], s = ——-^, ^~~c — ' ^^ ^^ "^^°^ ^^ values
— & . -y
s, u, [284], p, q', /, [233], they become sm. 6 . sm. (p = —r-, sm. 6 . cos. <P== — tj
p ji
which agree with the values of sin. ^ . sin. 9 ; sin. 5 . cos. 9, given in [269], for the invariable
plane, neglecting the small quantities </^, r'^, in comparison with p'^.
f (185) In the adjoined figure, let A be the origin of the
cordinates, G the centre of gravity of the body, ADZ' the axis
oi z', AY that of 3/' ; ABG the axis of z", and B C parallel to
that of y''. Then the co-ordinates of a particle C are either
AD = z', DC = i/, or AB=z", BC = f. Through
B draw B K parallel to 5;', and B I parallel to y. Then we ha¥e:
BCK=BAI=&,
I. vii. ^31.] COMPOUND PENDULUM. ^^^
whence we deduce
S .dm . (y"^-{- 2f'^) is the momentum of inertia of the body relative to the
axis of af : let this momentum be C. The last of the equations (B) § 25 [2881
[226] will give*
\dt^J dt
Suppose that the body is acted upon by gravity only ; the values of P, Q, of
§ 25 will be nothing, and R will be constant, which givesf
// TV"
^£^= S. Rt/. d m = R. COS. 6, S. f .dm + R. Sin. 6. S.z". dm. [290]
The axis of z" passing through the centre of gravity of the body, makes
S.y" . dm = 0 ; and if we put h for the distance of the centre of gravity
AI=AB. cos.BAI= z" . cos. 6 ; BI{= KD) = AB. sin. BAI^z" .sm.6;
CK= B C . COS. B CK= f . cos. 6 ; B K{= DI)^BC. sin. B CK^ f . sin. d.
Substitute these in ^ = DC==CK-{- KD, z! =AD = A I — D /, they become
as in [287], whose differentials, considering d only as variable, are
di/ = dd.{z".cos.d — y". sin. 6) =zsfd 6, a.ndd2^= — d&.{y". cos. 6 -{-:/'. sin. 6) = — i/ .dd,
hence y' dz' --z' di/ = — d6 . {z'^ -}-y'^)==-^dd .(xf'^ -j-f^). This multipUed
by -^, and mtegrated relative to S, gives the value of S . (- — ~ — -\ as in [288].
* (186) By, taking its differential and dividing by dt, having first substituted the value of
S. (^-^^^YT^ .dm, of [288], and that of C [288'].
f (187) Take the differential of the third of the equations [225], and divide it by dt,
dJV"
putting Q = 0, we obtain — — = S.Ry' .dm, [290]. Substitute in this the value of i/,
[287], and bring the terms R, 6, from under the sign S, because they are the same for all
dt
diN"'
the particles, we get the second expression of — r— , [290]. The values S .y" .dm = 0,
and S.z" . dm = mh, are easily deduced from [127], hence we get the value of — j — ,
[291], which, being substituted in [289], gives [292].
192 MOTION OF A SOLID BODY. [Mec. Cel.
of the body from the axis of a:', we shall have S .z" . dm =mh, m being
the whole mass of the body ; therefore we shall have
dJV"
t^^l — = — =mh.R. sin. & :
at
[292]
[292Q
consequently
ddd — mh .R . sin. 6
Let us now consider a second body, all whose parts are united at a single
point, at the distance / from the axis of a/ ; we shall have, with respect
to this body, C ^=m! P, m' being its mass ; moreover, h being equal to Z,
we shall have*
[293] ~^==-.^.sm.6.
These two bodies will therefore have exactly the same motion of oscillation,
if their initial angular velocities, when their centres of gravity are in the
vertical, are equal, and alsof
[293'] l=—i-
mh
* (188) This is found by substituting ^ = Z, and C= m! P, in the general expression of
[292], changing also m into m'.
t (189) Substitute Z=— , in the equation [293], corresponding to the simple
pendulum, and it becomes identical with the expression of , [292], corresponding to the
compound pendulum. IMuItiply this by 2d 6, and put for brevity p= ^ — , it
[293a] becomes ' — = — ^d6 . sin. 6, whose integral is -— = a + 3 . cos. 6 ; a being an
arbitrary constant quantity, which may be determined by means of the angular velocity
— , when 5=0, and if this quantity be the same in both pendulums, the angular velocity —
will be the same in all situations. Lastly, the value of dt bemg found from the preceding
. .„ • 1 . . /* ^^
[29361 equation, it will give, by integration <=/— .
I. vii. §31.] COMPOUND PENDULUM. ^93
The second body, we have just mentioned, is a simple pendulum, whose
oscillations were computed in § 11 [84, 86] ; we can therefore always
compute, by this formula, the length Z of a simple pendulum, whose vibrations
are isochronous with those of the solid just considered, and which may be
considered as a compound pendulum. In this manner the length of a simple
pendulum vibrating in a second, has been ascertained by observations made
with compound pendulums.
49
194 MOTION OF FLUIDS. [Mec. Cel.
CHAPTER VIII.
ON THE MOTION OP FLUIDS.
32. We shall make the laws of motion of fluids depend on those of their
equilibrium ; in the same manner as we have deduced, in Chapter V, the
laws of motion of a system of bodies from those of its equilibrium. Let us
therefore resume the general ^equation of the equilibrium of fluids, given
in ^ 17 [133].
[294] Sp = p,lP.5xi-Q.5y-}-R.Sz};
the characteristic 6 refers only to the co-ordinates x, y, z, of the particle,
[294^ and does not affect the time t* When the fluid is in motion, the forces
which would retain the particles in equilibrium, are, by ^ 18 [141, 142],
supposing d t constant,
p /ddx\ ^ fddy\ „ fddz
[295] ^-\-dj)^ ^"W^V' ^^\Ji-
we must therefore substitute these forces,t instead of P, Q, i2, in the
preceding equation of equilibrium. Putting
[295'] 6V==P.6x + Q.^y^R.^z,
* (190) As in [36"], where Sx, 8y, Sz, are arbitrary variations of x, y, z, independent
of the time t.
f (190a) These forces, as in note 60a, are supposed to tend to increase the co-ordinates.
[295a] Moreover, the quantities ~^, --J^, -—^-, being partial differentials relative to t, are
included in parentheses in [295], the reasons for which are more fully stated in the
note 197.
Lviii. §32.] MOTION OF FLUIDS. 196
which we shall suppose to be an exact variation, we shall have
important .
this equation is equivalent to three distinct equations ; for the variations ^f'the"
... i.^y,. , ■, Motion of
6x, Sy, 6z, being independent, we may put their coeincients separately equal ^Fiuid.
to nothing. [296^
The co-ordinates x, y, z, are functions of the primitive co-ordinates and
of the time t ; let a, b, c, be these primitive co-ordinates, we shall have* V^Q^
Substituting these values in the equation (F) [296], we may put the
coefficients of 6a, 6b , 6c, separately equal to nothing ; which will give three
equations, of partial differentials between the three co-ordinates x, y, z, of
the particle, its primitive co-ordinates «, 6, c, and the time t.
It now remains to fulfil the conditions arising from the continuity of the
fluid. For that purpose, we shall consider, at the commencement of the
motion, a rectangular fluid parallelopiped, whose three dimensions are d a,
d6, dc. Denoting by (p) the primitive density of this particle, its mass will [297T
be (p) . da . d6 .dc. We shall call this parallelopiped (^) ; it is easy to
perceive, that at the end of the time t, it will become an oblique parallelopiped ;
for all the particles situated at first on any face of the parallelopiped (A),
will continue in the same plane, neglecting infinitely small quantities of the
second order ;t all the particles situated on the parallel edges of (A), will
* (1906) The co-ordinates x, y, z, being functions of a, b, c, t, their complete variations
relative to 6 are as in [297], observing that the characteristic 6 does not affect t, [294'].
f (191) This is analogous to the principles of the differential calculus. For if the
extreme points of an in6nitely small arch d s o( a. curve be given, the intermediate parts of
this arch d s are supposed to fall on the straight line joining these two extreme points,
neglecting quantities of the second order. In like manner on a curved surface the
intermediate parts between the parallel edges of an infinitely small part of the surface may be
196
MOTION OF FLUIDS.
[Mec. Cel.
continue to be placed on equal and parallel right lines. Let us call this new
[297'] parallelopiped (B), and suppose through the extremities of the edge formed
by the particles, which composed the edge d c of the parallelopiped (A),
we draw two planes, parallel to the plane x, y.. By prolonging the edges of
(B) till they meet these two planes, we shall have another parallelopiped
[297'"] (C), contained between them, which is equal to {B) ; for it is evident that
one of the two planes cuts off from the parallelopiped (JS) as much as the
other adds to it.* The parallelopiped (C) will have its two bases parallel
to the plane of a:, y ; its height comprised between its bases, will evidently
considered as being on the plane joining these parallel edges, neglecting quantities of tho
second order. And it is evident that the same must take place in the case under consideration.
For the forces, acting on the different parts of any face of the parallelopiped, differ from
each other only by infinitely small quantities, which vary gradually, from one point to another
of the face, and the effect produced must be as above stated.
* (192) To illustrate vfhaX is here said,
we have given the annexed figures, in which
CAaX IS the axis of a:, C Y the axis of y,
to which JIB, ah are parallel, and BD,hd
which are supposed to be perpendicular to
the plane of the figure, are parallel to the
axis of z ; C being the origin of the co-
ordinates ; D H I K G the rectangular
parallelopiped (A), at the commencement
of the motion ; dhiJcg the oblique parallelopiped (J?) representing the situation and form
of (A) at the end of the time t', the parallelopiped (B) is
described in a separate figure, upon a larger scale, so as
to make the letters and lines of reference more legible,
and to this figure we must occasionally refer in the rest
of this note. Then the co-ordinates of the point D are
CA = a, AB = b, B D = c; those of the point d
are C a = x, ab = y, b d = z; the sides of the
parallelopiped (A) are i>fZ"=da, DF=db,
DE=dc, and its solidity is the product of these three sides, and as its density is (p), its
mass must be (p) . d a . d 6 . d c, as in [297']. Suppose now a plane to be drawn through
the point e, parallel to the plane xy, ov C ah, it will cut the edges gf, kl, ih, conUnued if
necessary, in the points g', J(f, i', respectively ; and a similar parallel plane being drawn
I.vm.§32.] MOTION OF FLUIDS. ^^^
be equal to the diflferential of z, taken on the supposition that c only varies ;
which gives ( ^ j . d c for this height.* [297i']
We shall have its base, by observing that it is equal to the section of (J5),
by a plane parallel to that of a:, y ; let us call this section 0. The value
of z will be the same for all the particles of which it is composed, and we ^^^"^
shall havef
rcj
through d, will cut the same edges in the points /', Z', h' ; these parallel
planes will thus form another parallelopiped dKiKg'f'd, which for ^
greater distinctness, is given separately on its proper scale ; this is the
solid called ( C). Now it is evident that the part of the solid [B)
included between the planes eiJcg, ei'Td ^, must be equal to the part included by the
parallel planes dJilf, d h! Z'/', and as the former is taken from [B), and the latter added, to
make ( C), it follows that the parallelopipeds {B) and ( C) must be equal.
* (193) The height of the parallelopiped (C) comprised between its two faces, drawn
parallel to the plane of a?, y, must evidendy be equal to the difference of the elevations of the
points <Z, e, above that plane ; that is, it must be equal to the value of z, corresponding to the
point e, less that corresponding to the point d. Now z is evidently a function of a, 6, c, t,
representing the ordinate h d, corresponding to the point d ; and by changing in it, c into
c-\-dc, we obtain the value of the ordinate corresponding to the point e, which is therefore
z-\- (—\ .dc; for by making this change in the ordinate c, corresponding to the point D,
we obtain the ordinate corresponding to the point E, which last point, at the end of the time t,
falls in e. The difference of these values ^ -}~(;t~) • d c, and z, namely, [ — ) . d c, is
therefore, the required height of the parallelopiped, as in [297'''].
f (194) Though zis'm general a function of a, b, c, t, whose complete differential is
''-(^)-^'"+(^)-^'+(^)-<'^+(^)-". [^]
yet in the present instance the term depending on dt is to be neglected, because the object is
to find the value of d z, at the same instant of time t, in different parts of the parallelopiped
(B) or (C), so that we must put dt=0, and then the points of the parallelopiped (C), in
which dz = 0, will correspond to the equation [298].
50
[298']
l^S MOTION OF FLUmS. [Mec. Cel.
Let Sp and 6q he two contiguous sides of the section (s), of which the first
is formed by the particles of the face db .dc of the parallelopiped (A), and
the second is formed by the particles of the face d a . d c. If through the
extremities of the side Sp, we suppose two right lines parallel to the axis of
X to be drawn, and prolong the side of the parallelogram (s) which is parallel
to 5p, until it meets these lines ; they will intercept between them another
parallelogram (x) equal to (s)* and the base of which will be parallel to the
axis of X. The side 5p being formed by those particles of the face d6.dc
[298'], which have the same value of z [297'] ; it is easy to perceive, that
the height of the parallelogram (x) is the differential of y, taken by supposing
a, Zj t, constant, which givesf
[299]
d r axis of x
* (194a) The section (s) is represented by the parallelogram
dh' I'f d of the figure C, drawn separately in the annexed figure.
The side f'd = Sp, the side h' d = 6q. Through the points
djf'f draw the lines dn, f m, parallel to the axis of a?, meeting
I' h' in m, n, and forming the parallelogram d n mf, which is here
called (X), whose base is equal to n d, and height is the perpendicular /' r, let fall from/'
upon n d ; and it is evident from the construction that the parallelograms (X), (s) are equal.
f (195) In these equations I have placed an accent on the letter d connected with c,
[299a] in order to distinguish the term d' c, found in this part of the calculation, from the side D E
of the parallelogram (A), which is denoted by d c. It being evident that D E may be
increased or diminished at pleasure, without changing the value of the height fr of the
parallelogram (X), which is represented in [299], by d y = (-—j . d b -{- (—j . d' c; no
term depending on the differential of a being introduced, because all the particles situated on
the line/' d, appertain to the plane of the face d egf of the solid (B) or (C), on which the
value of a is constant ; neither is there any term depending on dt introduced, because at the
same moment that the point (D) arrives at d, the particles of the face DE GF,
corresponding to the line df, arrive at their proper places on that line. The relation
of d & to d' c, is determined by the condition that z is constant, or that d 2; is nothing, [297'],
for all particles situated on the line d /' ; this gives the second of the above equations [299],
Lviii. §32.] MOTION OF FLUTOS. 199
whence we deduce
C fdy\ fdz\ fdy\ fdz\ I
,^. l[jb)'\rc)''\d-cj\db)l [300]
ay- -. . a o ,
\rc)
this is the expression of the height of the parallelogram (x). Its base is
equal to the section of this parallelogram made by a plane parallel to the
axis of X ; this section is formed by the particles of the parallelopiped (A)
which correspond to z and y constant ; its length is therefore equal to the
differential of x, taken on the supposition that z, y, and t are constant ; which
gives the three equations*
putting also, as in the first of the equations [299], t and a constant ; or, in other words,
rejecting the terras da,dt, from d z=0, [298a]. From this second equation [299], we get
(-)
d'c = ~pZ.d6,
this being substituted in the first equation [299], gives the final value of dy, [300], which
is proportional to d b, or to the side D JP of the parallelogram (A), and is independent of the
sides DH=da, DE = dc.
* (196) In these equations are put d^b, d,c, instead of db, dc, to distinguish them from
the sides dh, dc, or D F, D E, of the parallelogram (A) ; for the same reason that d' c
was accented in the last note ; it being evident that the length nd of the. parallelogram (X)
is proportional to J) fl, or d a, and that this length does not vary by increasing or decreasing
DF, D E, or db, dc. The first of these equations, [301], is the value of dx, the second
that of dy=0, the third that of dz=0 ; dt being, as in the last notes, rejected. The
values of d^b, d^c, being found from the two last, and substituted in the first, give the
required value of d x. This may be found more simply by multiplying the first, second,
and third equations, [301], respectively by the following factors,
/dy\ /dz\ /dy\ /dz\ /dx\ /dz\ /dx\ /dz\^
\d b) ' \d c)~'\d7) ' \db) ' \d7) ' \Tb)^\db) ' \d7) '
/dx\ /dy\ /dx\ /rfy\
\dh)'\dc) \dc/'\dbj'
[2996]
200
MOTION OF FLUIDS. [Mec. Cel.
Put for abridgment
[302]
/'dx\
(dy\ (dz^
'\db)'\dCy
V /'dx\
1 \daj
l-Cn-
/dz\
\dh^
) + '
/'dx\
\dh)
(dy\
/dz
' \dC',
/dx\
\dh)'
/dy\ /dz\
'\daj' \dcj
•+©
•ri!V(
^dz\
\dh)
rdx\
\dc)'
m-
/dz"-
\da,
we shall have
da; = .
^.dflj
[303] /dy\ (dz\___rdy\ /dz^
which is the expression of the base of the parallelogram (x) ; the surface of
this parallelogram will therefore be _! '. . This quantity also
expresses the surface of the parallelogram (?) ; multiplying it by ( -— j.dc,
[303'] we shall have ^ . da .db . dc, for the magnitude of the parallelopipeds (C)
and (B). Let p be the density of the parallelopiped (A) after the time t ;
we shall have for its mass
[303"] p.^.da.d6.dc;
putting this equal to its original mass (p).da.d6.dc [297'], we shall have
[303'"] P(3 = (p) (G)
important for thc cQuation relative to the continuity of the fluid.
Equation ■*• •'
of the
Tfthf^ 33. We may give to the equations (F) [296], and (G) [303% forms
which will be more convenient in some circumstances. Let u, v and v be
[303 ivi the velocities of a fluid particle, parallel to the axes of x, y, z, respectively ;
we shall have*
and adding them together; the coefficient of da in the second member, becomes equal
to the quantity denoted by ^, [302], while those of d^h, d^c, vanish; hence
which gives for dx the same value as in [303]. This multiplied by the height dy, [300],
gives the area of the parallelogram (X) or (s), and this multiplied by the height of the
parallelopiped (C), which by [297'^] is ( j^j . d c, gives its solidity ^.da.db.dc, [303'].
* (197) The co-ordinates of any particle of the fluid, which were represented by a, J, c,
[296"], at the commencement of the motion, when ^ = 0, and by x, y, «, when t=-ty
I. viii. §33.] MOTION OF FLUIDS. 201
/"dxS /dy\ /dz\
U;==^' U;='' U;=^'
[304]
/i*i**x /dvX , /du\ , /du\ , /^du\
Taking the differentials of these equations, supposing u, v, v, to be functions
of the co-ordinates of the particle a;, y, z, and of the time t ; we shall
have
(ddx
~d^
/ddy\ /dv\ , fdv\ ^ fdv\ , /dv\
V77^;=U;+^-U;+"-V^J+^A^;' [305]
[296"], become respectively x-\-udt, y-{-vdt, z-j-vdt, [306'], when t is increased [305o]
to t-\- dt. In this notation the co-ordinates a, b, c, of any particular particle, do not vary nemarks
with the time, but differ for different particles, and they serve merely to denote the o?dinl^
primitive situation of the particular particle of the fluid, whose motion is to be considered. ^^1/^^*
Again, since the co-ordinates, x, y, z, of the particle, corresponding to the time <, are ' '
increased during the following instant d t, by the quantities
dx=udt dy = v dt, dz = vdt, [305i]
it follows that the velocity of the particle, resolved in directions parallel to these axes, will be
represented by
d X dy dz ron/t«i
J7=«' 7^="' U-"' '*^'
but we must observe, that in taking the differentials of x, y, z, the quantity t only was
considered variable, and since x, y, z, [296"], are denoted by functions of a, b, c, t, the
. d X dy dz
precedmg expressions — , — , — , must be considered as the partial differentials of
a?, y, z, relative to i, and ought, in conformity to the usual notation of the author, to be [305rf]
included in parentheses, as in [304]. For the same reason the quantities -r-r-, -r-^,
■^— -, [142], were included in parentheses, in [295], it being evident, from what has been
said, that they are partial differentials relative to the time /. Moreover, the expressions of
«, V, V, [304], must be considered as functions of a?, y, z, t, since for the same value of ^,
these velocities will vary from one particle to another ; and for the same co-ordinates x, y, Zj
the velocities will vary from one instant to another ; so that in taking the differential of any
one of the equations [304], as for example, (-^ j =u, we must consider m as a function of [305ej
61
202 MOTION OF FLUIDS. Mec. Gel.
Second The equation (F) [296] of the preceding article will thus become
form of the
To obtain the equation relative to the continuity of the fluid ; suppose in
the value of P [302] of the preceding article, a, b, c, to be equal to x, y^ z;
and X, y, z, equal to x-\-udti y -\-vdt, z -\-y dt, respectively ; which is
[306'] equivalent to the supposition that the primitive co-ordinates «, 6, c, are
infinitely near to x, y, z, we shall have*
.=i+.q(^:)+(g)+(Q|;
and the equation (G) [303'"] will become
X, y, z, t ; and x, y, z, as functions of a, b, c, t. Therefore if we take the partial differential
of ( -T— j = M, relative to t, it will become, according to the usual notation of partial
differentials,
(ddx\ /du\ . /d u\ /dx\ . /d u\ /dy\ , /d u\ /dz\
-d^)=[dT) + [d^) ' [dT) + {d^) ' [dT) + {j7) ' [dT)'
Substitute in this the values of f — j, [tt)^ (tt)' C^^"^]' and it will become as in the first
of the equations [305], The expressions {-ji^)i ("t^)> [305]> are found in the same
manner, from (■-—j=v, f— j = v, [304]. These being substituted in [296], it
becomes as in [306].
* (198) By changing a into a?, and x into x-\-udt, the expression (-r-) becomes
[306a] (— — -i )> and as x, y, z, t, are considered as independent variable quantities, this
becomes ( ;r- ) = 1 + ( t— ) -dt. In like manner
I.viu. §33.] MOTION OF FLUIDS. 203
If we consider p as a function of a:, y, z, and t ; we shall have*
the preceding equation will therefore become
»=©+(^)+(^)+(^)= _ (^ . ,^l
which is the equation relative to the continuity of the fluid, and it is easy ^""^^^
to perceive that it is the diflferential of the equation (G) [303"] of the
preceding article, taken relative to the time /.f
[309]
The remaining terms or factors of which ^ [302] is formed are of the order d t, thus
/dx\ (d.{pc^dt)\_fdu\ /dx\_/d.{x-^udt)\_/du\
Therefore all the terms of 3, except the first {'i~) • {jz) - \j~)i ^^ of the order dt^
or d t^, and may be neglected ; and this first term gives
which by developing and neglecting the terms multiplied by d f^j becomes
as in [307]. This bemg substituted in [303'"], gives [308].
* (199) Supposing (p) to be a function of a?, y, z, f, and p to be a similar function of
x-\-udt, y -\-vdt, z -\-y dt, t-{- dt, we shall have by developmg according to the
powers of d t, by Taylor's theorem [607, — 612],
neglecting the terms of the second and higher powers of d t, and by transposing all the
terms of the second member, except the first, it becomes as in [309]. This value of (p) being
substituted in [308] divided by d t, it becomes,
which is the same as the equation [310], developed by writing
1(200) The differential of the equation [303'"], isp.dp + (3.rfp=0, because (p) [3io5]
[297'], is a constant quantity, not varying with the time, and its differential is nothing.
204 MOTION OF FLUIDS. [Mec. Cel.
The equation {H) [306] is susceptible of integration, in a very extensive
case, namely, w^hen u^x~\-v^y-\-\^z is an exact variation of a:, ^, 2 ; p
[SKy] being any function ivhatever of the pressure p. Let this variation be Sep,
so that
[310"] 8cp = u8x-\-vSy-\-y8z ;
the equation (H) [306] v^^ill give*
Now by [307], when dt = 0, ^ becomes = 1 . Subtracting this value of p from that in [307],
corresponding to the time t-\-dt, we get the change of the value of p during the time d t,
or dP = ^^ •\{;r-)~h['T~)-\~\-T-){' ^ like manner from [308], we get the value
of p, corresponding to dt = 0, which is (p) ; subtracting this from the general value of p,
deduced from [309], we get the value of
substituting these in [3106], we get,
in which ^ [307], may be put equal to 1, by neglecting terras of the order d t^, and then the
equation becomes precisely of the same form as [310a], which in the last note was shown to
be equal to [310].
[311a]
* (201 ) The expression of 5 9 [310"], gives, u = (^\ v= C^X v= f~\
The partial differential of the first of these expressions, relative to t, is
\dtj \dtSx) \8xdt)'
and by putting f -— j = (p', it becomes f — j = f -r — j. In the same manner
(il) = (lll^ = (m, and }p-UC-^. Hence
the second member of which is evidently equal to S<p' or ^'{y~)^ observing that the
characteristic 8 does not affect the time tj [294']. Thus we shall have.
I.viii.§33.] MOTION OF FLUIDS. 205
whence by integration relative to S,
We must add to this integral an arbitrary constant quantity, which is a
function of t ; but this may be considered as included in the function 9.* [312^
This function <p gives the velocities of the fluid particles, parallel to the axes
of X, y, and z ; for we have [3126]
» = ©= -O' -©•
Again, since S cp [310"], is an exact differential we have
/d(p\ /d(p\ /d(p\
(p being a function of the independent variable quantities x, y, z, t. Taking the partial
differential of this value of v, relative to x, we get.
[3126]
/dv\ /ddcp\
\dxj \dxdy) X^ dy
and in a similar manner l-—\==(-—\. These being substituted in
it becomes u .\hx. i~\ + ^y • (^) -\-^z . (1— ) [ , which is evidently equal to
M 5 M = J . 5 . w^j and by substituting the value of «, [312&], we get.
We may proceed, in the same manner, with the terms of [306], multiplied by v and by v, or we
may obtain the same result, much more simply, by changing, m [312c], u into v, and x into y,
and the contrary, hence we shall get,
In like manner, changing in this v into v, and y into z, and the contrary, we get
The sum of the expressions [313a, c,d, e], constitutes the second member of [306], which
is by this means, reduced to the form [311].
* (202) As the characteristic (5 does not affect ^, the integral of u^X'\-v5y-\-v6z=-6<p^
[310"], taken relative to this characteristic, may be completed by adding an arbitrary
52
206 MOTION OF FLUIDS. [Mec. Cel.
The equation {K) [310], relative to the continuity of the fluid, becomes*
^3,43 \dtj~^\dxj\dccj'^\dyj\dyj~^\dzj\dzj
thus, relative to homogeneous fluids, we havef
« = (-J)+(^)+(^>
We may observe that the function u.Sx-{-vJy-\-v.Sz [310"] will always
be an exact variation of x, y, z, if it be so during one instant. For if we
[315] suppose, at any instant whatever, that it is equal to 5 9 ; in the following
instant, it will be|
function of t ; and as a?, y, z^ t, [306a], are supposed to be independent variable quantities,
this function of ^, will not affect the values of (~j, (;t— )> (t~)j i" the expression
[312] ; and <p may therefore be supposed to contain the arbitrary function of t, [312'], which
is required to complete the integral of [311].
* (203) This is easily deduced from [310], developed as in [310a], using the values
[313], which give
^,, /du\ /ddcp\ /dv\ /dd(p\ /rfv\ / d d cp \
^^'^'^ [d^)=[-d^} WJ==V"i?-> [dinKT^J'
f (204) The fluid being homogeneous, its density p is constant, therefore rf p = 0.
Substitute this in [314], and divide by p, we get [315].
J (205) The ordinates x, y, z, being supposed to remain unaltered, but the time t, to
increase by the differential d t, new particles of fluid taking the place of those which formerly
corresponded to x, y, z, the partial velocities u, v, v, will become u-\-(-—).dt,
v-{-(y-j.dt, v-{-(-—j.dt, respectively, for these new particles ; consequently the
expression uSx-\-v5y-\-vdz, will become,
which, by substituting, for u8x-\-vdy-\-vSz, its assumed value ^9, [310"], becomes
as in [316].
Lviii. ^33.] MOTION OF FLUIDS. 207
it will therefore be, in this last instant, an exact variation, if
be an exact variation at the first instant ; now the equation {H) [306] will
give at this instant*
consequently the first member of this equation will be an exact variation in
X, y, z ; therefore if the function uJx-\-v.Sy-{-v.6z be an exact variation
at one instant, it will also be an exact variation in the next instant ; and it
will therefore be an exact variation at all times, if it be so at any instant.
When the motions are very small, we may neglect the squares and
products of u, V, and v ; the equation (H) [306] then becomesf
therefore, in this case, u . 5x -{-v . Sy -{-y .8z is an exact variation, if as
* (206) The equation [306], by substituting the values computed in [312c, d, e] becomes
which by transposing the last terra, becoraes as in [317] ; and as p is a function of p, by
hypothesis [310'], the term — is an exact variation, and every term of the second member
of the expression [317] is an exact variation, consequently the first member of [317] is also
an exact variation.
f (207) In this hvpothesis the terms u . \-f-\ ^-(j— )» ^' ^® ^ ^^ neglected, bemg
of the order of m^, ?r^, &«. Neglecting these terms, the expression [306] becomes like
[318]. The first member of which being an exact variation, its second member must also
be an exact variation, and by putting it equal to 5 . ( j— ), we shall get 8V =5.f — — j,
whose integral relative to the characteristic <5 gives [319]. This assumed value of
multiplied by d t, and integrated relative to d gives 5 cp = u d^x -\- v d^y -{• v (tz, since
6x, 8y, 6Zf are not afiected by the time t, [294'].
208 MOTION OF FLUIDS. [Mec. Cel.
we have supposed [310'], p is a function of p; still calling this variation
5 9 [310"], we shall have
and if the fluid is homogeneous, the equation of continuity will become [315]
Equation
unduia- These two equations contain the whole theory of very small undulations in
tions of 1 n't
homogene- nomogeueous fluids.
OU9 Fluids.
34. We shall now consider the motions of an homogeneous fluid mass,
which has a uniform motion of rotation about the axis of x. Let n he the
angular velocity of rotation, at a distance from the axis, which we shall take
[320"! for the unity of distance ; we shall have* v = — nz\ v=:ny; the equation
(H) [306] of the preceding article, will therefore becomef
[•321] ^ = SV+n\{y6y-}-z5z};
which equation is possible, since its two members are exact diflerentials.
The equation (K) [310] of the same article, will become J
* (208) The angular rotation about the axis of x, in the time d t, is ndt [320'], and this
is called d -^ in die expressions oi d y, d z, [230^], which, by this means, become
dy = — nz.dt, dz= ny .dt. Substitute these in dy = vdt, dz = ydt, [3055],
and we shall get v = — nz, y = ny, [320"].
[321a] t (209) The values u = 0, v = — nz, v = ny, [320"]^ give f — j = — n,
(■—\ = n, and all the other partial differentials, which occur in the second member of
[306] vanish. This equation will therefore become
8 V — = — 8y.n\-\-8z.nv = — n^ . [y 8y -{-z § z],
as in [321].
X (210) In the equation [310], developed as in [310a], substitute f -^ j=0, ^-^ j=0,
(j- ) = 0, [321a], and multiply by dt; in this manner we shall obtain the equation [322].
f (211a) This is proved by reasons similar to those in note 64. The equation of the
surface of a fluid, [323], having a rotatory motion about the axis x, would agree with the
result of a calculation made in note 65a, page 95, by a different method.
63
[322^
I.viii.§34.] MOTION OF FLUTOS. 209
and it is evident that this equation will be satisfied, if the fluid mass be
homogeneous. The equations of the motion of the fluid will then be
satisfied, consequently this motion will be possible.
The centrifugal force at the distance \/ y^-\- z^ from the axis of rotation,
is equal [34'] to the square of the velocity n^. ('f-\- z^) divided by this
distance ;* the function r? . (y^y -\- z^z) is therefore the product of the
centrifugal force by the element of its direction ; therefore by comparing the
preceding equation of the motion of the fluid with the general equation
of the equilibrium of fluids, given in ^ 17 [133] ; we find that the conditions
of the motion now treated of, are reduced to those of the equilibrium of a [322"]
fluid mass, urged by the same forces, and by the centrifugal force arising
from the rotation, which is evident from other considerations.
If the external surface of the fluid mass is free, we shall have ^p = 0, at
this surface, consequently
0 = bV-\-n'.{y^y-\-z^z)\ [323]
whence it follows that the resultant of all the forces acting on each particle
of the external surface, must be perpendicular to that surface ;t it ought also
to be directed towards the interior of the fluid mass. These conditions [323n
being satisfied, a mass of homogeneous fluid will be in equilibrium, even if
we suppose it to cover a solid body of any figure whatever.
The case just examined is one of those in which the variation
u^x-{-v^y-\-Y^z
This equation will be satisfied if the fluid be homogeneous, or p = constant, because all its
partial differentials [322], would vanish. The equations [321, 322] being satisfied, the
motion will be possible without any internal change m the situation of the particles.
* (211) Let r be the distance of a particle of the fluid from the axis of z, we shall have [322a]
r^ = j/2 _|_ 2;2^ whose variation gives r^r = y^y-\~z5zf The centrifugal force of this
particle is r^r, [138a], this being multiplied by the element of the direction 5 r, becomes
n^.rSr, or n^ . {y5y-\-zSz), as in [322']. Multiplying this by the density p, and
adding it to the second member of the equation of equilibrium [133], we get, by using S V,
[295'], the same expression as in [321].
[3226]
210 MOTION OF FLUIDS. [Mec. Cel.
[323"]
is not exact ; for this variation becomes* — n. (zSy — y^z); therefore in
the theory of the tides we cannot suppose that variation to be exact ; since
it is not so in this very simple case, in vrhich the sea has no other motion
than that of rotation common both to the earth and sea.
S5. Let us now determine the oscillations of a fluid mass, surrounding a
spheroid, having a motion of rotation n t about the axis of x ; supposing it to
be deranged but very little from the state of equilibrium, by the action of very
[323"'] small forces. At the commencement of the motion, let r be the distance of
a particle of the fluid from the centre of gravity of the spheroid which it
[3231"]
covers. This centre of gravity we shall suppose to be at rest. Put ^ for
the angle which the radius r makes with the axis of x ; and * for the angle
which the plane passing through the axis of x and this radius, makes with
the plane of x, y.\ Suppose that at the end of the time t the radius r
*■ (212) Substitute in u6x-\-v6y-\-vbz, the values [321 «,] w = 0, v==. — nz,
v=ny, and it becomes — nzSy-{-nydz. Now, upon the principles explained in note 61,
it appears that if PSy-\-RSz, is an exact variation of a function of y, z, we ought to have
In the present example P = — n z, R = ny, and as n is constant, ( -r— ) = — n,
( T~ ) ""^ ^ ' ^"^ ^^ these quantities are not equal, the expression — nzdy -\-ny 5 z, is not
an exact variation.
f (213) To illustrate this we may refer to the annexed figure,
in which C is the centre of gravity of the spheroid, CXthe axis
of a;, C Y the axis of y ; the axis of z being drawn through C,
perpendicular to the plane of the figure. Suppose a particle of the
[323a] fluid, whose motion is to be con^dered, to be at the point D', when
^ = 0 ; and at the point D, when t = t. Draw U B', D B,
perpendicular to the plane of the figure x, y, and B' A', B A, -^C: JJ
perpendicular to the axis CX. Then the co-ordinates of the proposed particle, when
i = 0, will be, CD'=:r, angle jyCE'=d, D'A'B'=^', and when i=t, they
will become CD = r-\-as, DCE = ^-\-au, DAB=r.nt-{-vi-\-av, or CA = x,
AB = y, BD = z. Now C A= C D .cos. D C E; AD= C D .sm. D C E ;
AB = AD.cos.DAB; BD= AD .sm. DAB ; whence
I. viii. ^35.] MOTION OF FLUIDS. 211
becomes r-\-as, 5 becomes ^ + aw, and « becomes nt-\-'a-\-av ; as, au, and [323T]
av being very small quantities, whose squares and products we shall neglect ;
we shall have
x= (r-\-as) . cos. (d-{-au) ;
y = (r + a 5) . sin. (^ + « w) . cos. (n t -\- is -\- a v) ; [324]
•2 = (r 4- « 5) . sin. (d-{-au) . sin. (n t -{- -a -}- a v) ;
If we substitute these values in the equation (F) § 32 [296], we shall have,
by neglecting the square of «,*
AB=CD.sm.DCE.cos.DAB; B D= C D .sm.D C E .sm.DAB ; ^3235]
which, by substituting the values of CD, D C E, DAB, give x, y, z, as in [324].
It may be observed that a u represents nearly the motion of the particle in latitude, and
a r its motion in longitude [347'"], from a meridian of the earth, which has the angular
motion n t in the time t.
* (214) In the notation here used, the quantities a and n are constant ; r, 6, -a, take the [323c]
place of a, b, c, [305a], and are constant in the differentials relative to the characteristic d ;
but 5, w, V, are variable, and take the place of the quantities x, y, z, [305a]. The
characteristic ^, as usual, does not affect t, [294']. Put now for brevity,
p = (r -[- a «) • sin. (d-j-att) v=^nt-\-'a-\-av, [324a]
and the expression of y, z, [324] will become,
y = p . COS. T, 2r = p . sin. t. [3246]
The variation of z, and its second differential being taken, we shall find,
5 z = 5 p . sin. T -|- p 5 T . cos. t,
ddz = {dd p — p<? T^) . sin. T -\-{2df.d'r-\-^.dd'r) . cos. r.
Multiply these two expressions together, and put A for the coefficient of sin. r . cos. t, in the
product, we shall get,
6z.ddz={ddf — ^ .dv^) J ^ .sm^v -\-{2 .d^ .dv -\- ^ .ddr) . ^ Jv .cos.^ v-\- A. wa.v .cos.'r. [324rf]
From this we can easily obtain Sy . ddy, by putting ^ -n: -\- r {or r, J * being a right angle.
This changes sin. r into cos. r, and cos. r into — sin. r. By this means z changes into y,
[3246], and the preceding expression [324J], becomes
8y.ddy={ddp — p.d'r^).8p,cos.^r-\-(2.dp.dr-\-p.ddr).p.Sr.sin.^'r — .^.cos. t. sin. t. [324c]
Add together [324<Z, e], and reduce by putting sin.^ r -\- cos.^ r = 1, we get
8y.ddy-{-Sz.ddz = {ddp — p.d7^).5p-\-{2.dp.dr-{-p.ddr).pSr. [324/]
If we now put
p' = r + a 5, t' = ^ -f- a M, [324^]
the values of x, p, [324, 324a], will become
x= p' . COS. t', P = p' • sin. t', [324A]
[324c]
212 MOTION OF FLUIDS, [Mec. Cel.
13.] »^.^q(S)-2n.sin...cos...g)|
General
Kquation
for all
parts of
the Fluid,
in Motion.
2 p
[324fc]
which are similar to y, z, [324J], and may be derived from them by changing y into a?,
z into p, and accenting the letters p, r. Making these changes in [324/], it will become, by
this principle of derivation,
[324i] Sx.ddx-{-5p.ddp={ddp'—p'.d'r'^).Sp'-}-{2.dp'.d'r' + p'.ddr'),f'.S7',
and since d'r' = adu, dp'=ads, we may neglect dr'^ and dp .d'/, which are of the
order aP ; then adding [324i], thus corrected, to [324/*], and rejecting the term 6 p .ddp,
which occurs in both members, we shall get
8x.ddx-\-Sy.ddy-\-Sz.ddz= — pd'r^.Sp-\-(2.dp.d'r-\-p.ddr).p.Sr
■i-ddp' .Sp'Jf-p'^.dd.r'.Sr'.
If we now suppose the differentials to refer to the time t only, the first member of this
expression, being divided by d t^, will be equal to the second member of [296], and it will
S p .
therefore be equal S V , and if we add to both members the variation of p^, or 2 p . 5 p,
multiplied by \ w^, the first member of the sum will be "^ • "5 • (p^) + ^ ^ j which
is the same as the second member of [325], and the second member of this sum will be
and it now remains to be proved that this is equal to the first member of [325].
If we neglect terms of the order a^, we shall get from [324a, g], noticing the remarks
in [323c],
/rfT\ , (dv\ /ddt\ /ddv\
w;= ^+"- kdi) ' \-d¥)="'\-dt^) ^
/ddp'\ /dds\ /dd'T'\ /ddu\
Substituting these in [324Z], it becomes,
-2.n.(^).pip + 2p«T.«.J(^).sin.«+r.cos.«.(^)|+«P^jT.(^_,^)
[324i]
[324m]
[324n]
4
I. viii. ^ 35.] COVERING A SPHEROID. 213
at the external surface of the fluid, we have 6p = 0; moreover, in the state [325 ]
Equation
of equilibrium, we have* riomatth^
2 Sorface.
0 = — .5. {(r-{- as) . sm.(6 + au)}^ + (6 V) ; [326]
(<5 F) being the value of ^ F corresponding to this state. Suppose the fluid [326'1
in question to be the sea ; the variation (<5 V) will be the product of gravity
multiplied by the element of its direction. Let g be the force of gravity,
a y\ the elevation of a particle of the fluid above its surface of equilibrium, [326"]
which we shall consider as the true level of the sea. The variation (^ V)
will increase by this elevation in the state of motion, by the quantityj [326'"]
and as every term of this expression is of the order a, we may neglect the terms of the
order a in 5p, 5t, 5p', 5t', and we shall get from [324a, ^], and [323c]
p = r . sin. ^, (5 p = 5 r . sin. ^-\-rh^ . cos. d,
p' = r, Sp':=Sr, . [324p]
Substituting these in [324n], it becomes,
— 2a.n.{ — ) . \r5r ,sm.^ 6 -{-1^ .56 .sin. 6 .cos. 6^
-^2r .sm.6 .StS .an .} ( — j.sm.6-\-r .C0S.6 .(—] i
this, by connecting together the terms depending on 5 6, 5 «, 5 r, becomes as in the 6rst
member of [325], agreeing with the above.
* (2] 5) In the state of equilibrium, «, v, s, are constant, and their differentials relative
to t are nothing, which makes the first member of the equation [325] vanish, and at the
external surface 5p=0, and 5 F becomes {5 V), [326'] : these bemg substituted in [325],
it becomes as in [326].
f (215a) It may be observed that the quantity y is here wholly different from the
rectangular co-ordinate y, [324], but as this is not used in the rest of the chapter, it cannot
produce any ambiguity or mistake.
X (216) The function 5 V, [295'], represents the sum of all the forces acting upon a
particle of the fluid, multiplied each by the element of its direction. These forces may be
composed into one single force ^, [16], acting in the direction of a line r", which we may
suppose to be drawn towards the origin of that force. This origin is very near to the origin
64
21^ MOTION OF FLUIDS, [Mec. Cel.
— ag . <5y ; because gravity acts nearly in the direction of a«/, towards the
[326iv] origin of that line. Then denoting by aSV, the part of «5F depending on the
new forces which in the state of motion, act on the particle, and which
depend either on the changes in the attractions of the spheroid arising from
that state, or on any external attractions ; we shall have at the surface
[327] 6V=(5V) — ug,8y + a.dV'.
The variation -x- -^ -{(r -j-as) . sin. (6-j-au)]^ is increased by the quantity
[327] an^ .6y .r: sin.^<), by means of the elevation of the particle of water above
the level of the sea ; but this quantity may be neglected in comparison with
71 T
[327"] — ag .5y, because the ratio — , of the centrifugal force at the equator, to
[327'"] gravity, is a very small fraction, equal to ^is^.* Lastly, the radius r is very
of the co-ordinates, or the centre of the earth ; so that the direction of the line r", and that
of the radius r-\-as, ov r' [334], differ but very little, and the quantity g is nearly equal to
the gravity g at the earth's surface. Now from the formulas [295', 16] we obtain
[327a] S V= — ^ . 6r", the negative sign being prefixed, because the force g tends to decrease r",
instead of increasing it, as is supposed in [295a].
The co-ordinates of the particle, upon the momentary surface of the sea, are at the end
of the time t, represented byr-f-a*j ^ -\- au, nt -\-zi -\- av, [323^], which may, for
[32761 brevity, be denoted by r, 6', z/, respectively, as in [334]. The corresponding co-ordinates
at the point of the surface of equilibrium, treated of in [326"], will be r' — ay, 6', ■s/, and
if the same force g', acted at this point, and in the same direction, the formula [327a], would
become for this point, [8V) = — g' • ^ • {r' — «2/) = — ^•^^ + «^'<^y' This, by
substituting the value of S F, [327a], and in the very small terms multiplied by 8y, putting
g for g', becomes [8V) = S'P^-{-ag . Sy, or 6 V={8 V) — ag .Sy. To which must be
added the quantity a 8 V, [326'""], depending on the difference in the direction and in the
value of tlie force g', at the two points, arising from the change of situation of the attracting
mass in the state of motion, and from the attraction of other bodies, as the sun and moon.
By this means we finally get S V= {8V) — ag.8y-\-a8V^ as in [327].
* (217) For the sake of brevity, let the function,
[327c] -^.(5.[(r + as).sin. (^ + aM)S2 or ^ . 8 . {{y . sm. ^')]\
corresponding to the point of the surface of equilibrium, treated of in [327i], be represented
by [8 JV) ; and the same function, at the corresponding point of the momentary surface,
by {8JV) -\-8JV. Put M for the second member of the equation [325], in the state of
at the mo-
mentary
r^ . <5 a . I i ^1 ]^2n. sin. 6 . cos. 6 . f~\ [ ^^^^^f
Lviii. §35.] COVERING A SPHEROID. 216
nearly constant at the surface of the sea, because it differs but very little
from a spherical surface ; we may therefore suppose ^ r nothing. The [327>v]
equation (L) thus becomes, at the surface of the sea,* Equation
C /"dduX
I [jWj "~
. 3 . ^ • 2.' /ddv\ , ^ . , , /'du\ , 2n.sin.2d /d s\ } [328]
+ ,-.... ^sm.^^f^j+2r^.sm.Kcos.d.^-j + ——-.^-) I
the variations S y and <5 V correspond to the two variable quantities fl and w. [328']
Let us now consider the equation relative to the continuity of the fluid. For
this purpose, suppose at the origin of the motion, a rectangular parallelopiped
motion, and we shall have, M=^{6iN'^-\- 8JSt-{- ^V —. The same notation being [327rfj
used in [326], it becomes 0 = (J A*) + (5 F), whence (5 F) = — (5 JV). Substitute
this in [327], and we shall get 5 V= — (<5JV) — ag .^y -\-a6V 'j therefore the preceding
0 o
value of M will become M= SJST — — ag .5y-\-aS V. Now ^ JV is the increment [327c]
of tlie function ~^ • ^ • [^ • sin. ^|- = n^.r'5r'. sin.^ d', arising from the change of
r' — ay into /, by which means, the variation 5 / is increased by the quantity a 5 y ; so that
we shall have 8JV=an^r^.dy.sm.^6', [327']. This, being compared with the term
— ag.Sy, [327e], is of the order — ; being of the same order as the centrifugal force [327/*]
[138a], is to gravity, or 2^ [1594a]. Therefore we may neglect SJV; and if we also put
8p = 0, as in [325'], the value of M, [327e], will become, M = —ag.Sy-}-aSV. ^327^]
It may be observed, that the quantity _ . 5 . (r . sin. if)% [327c], depends on the
centrifugal force, [322'], and this force might have been included among the forces on which
8 V, [327a], depends, and it would then correspond to the whole force of gravity, g, acting
in the direction r", perpendicular to the surface of equilibrium ; in which case the variation
h r", of the line of du-ection of that force, along the surface of equilibrium, would be
nothing, [19a]. In this view of the subject we also perceive the propriety of neglecting the
term [327'], depending on n^.
* (218) The second member of the equation [325], represented by M in the preceding
note, and reduced to the form [327g], is to be substituted in [325], neglecting 5 r in both
members of the equations, on account of its smallness. Then dividing by the common
factor a, the" equation [325] will become of the form [328]. This last equation corresponds
to the momentary surface of the sea.
216 MOTION OF FLUIDS. [Mec. Cel.
[328"j whose height is dr, width rdt^.sin.^, and length rdL* Let /, 6', td',
be the values of r, 6, ^s, corresponding to the time t. Pursuing the same
method of investigation as in § 32, we shall find, that at the end of that
time, the magnitude of the fluid particle will be equal to a rectangular
— J . d r ;t its width
..si„...jg^).d«+(4^).d'.j,.
* (219) The dimensions of this parallelopiped are found as in
[275a], using the same figure, and changing R, s, p', into r, -a, 6,
r328a] ^° ^^^ CA = r, E C Q^ = t^, ACP = 6; from which
we get jiB=:rd-ss . sin. 6, Aa = rd&, and the height of the
parallelopiped formed on the base A Bb a, is the other dimension
d r. These correspond to the commencement of the motion. At
the end of the time t, the terras r, 6, vs, become /, 6', •»', [328"].
In the equations [329 — 331], the letter d is accented for the same reason as it was done
in [299a, 6].
f (220) These dimensions are easily deduced from those of the parallelopiped (C),
§ 32, in the following manner. The dimensions of the parallelopiped (A) [297'], at the
commencement of the motion, are da, d5, dc; these correspond to the rectangular elements
rd6, r d trf . sin. 6, d r, [328"], respectively. At the end of the time t, a, b, c, become
X, y, z, [305a], and 6, zi, r, become 6', -5/, /, [328"]. Now if, for the sake of brevity, we
put p = r . sin. 6, p' = r' . sin. 6', and [follow, in every respect, the method of calculation
detailed m [297' — 303'"], it will evidently appear that we may change in all these equations,
da, d&, dc, da?, dy, dz;
[329aJ j^^ ^ J ^^ p ^ ^^ ^ ^^ / d d', p' d ^, d r', respectively ;
and by this means, we shall obtain the dimensions of the parallelopiped ( C), in conformity
with the present notation. First, The height corresponding to (—) . d c, [297'^], will
become ( — ) .dr, as in [328"']. Second, The width dy, [300], deduced from the two
equations [299], will, in the present case, be the value of p'd'trf', deduced from the two
following equations, which were obtained from [299], by changing the symbols, as in [329a],
and if we bring the quantities p, p', without the parentheses, they will become as in [329, 330].
-^
1. viii. §35.] COVERING A SPHEROID. 217
exterminating d' r by means of the equation
and whose length is /. i Q^.dj + (j^ . d d + (^ . d^z. | , [330']
exterminating d r and d^ ^ by means of the equations
Supposing therefore
/<7A /dt\ /d^
[331]
/dy\ /d6'\ /dz/\_^/dr^\ /^dd^ (d
\drJ'\d&J \dzij \dr J \dr^J \d6
\jdj'\d^
\d^)~^\I^J'\drJ'\dJj \d^)'\dd)'\d^)'''
[332]
the magnitude of the particle, at the end of the time ^, will be*
p' . r'^ . sin. ^ . d r . d ^ . d « ; [3321
therefore supposing the density of the fluid at the commencement to be (p), General
and at the end of the time ^ to be p ; we shall have, by putting the expressions continuit*
of the mass at these times equal to each other,t &« fom!
P . ^' r'2 . sin. ^ = (p) . 7^ . sin. 6 ; [333]
The value of d' r, being found from [330], and substituted in [329], gives the required
width. Third, The length da?, deduced from the three equations [301], will become,
in the present notation, equal to the value of r'd^, deduced from the three following
equations, by tlie elimination of d, -ss, d^ r.
These, by reduction, become as in [33^, 331], changing the order of the two last equations.
* (220a) If in the value of ^, [302], we make the same changes as in [329a], it will
/2 , sin. 6'
become ^ = -^ ' . ' . p\ using 3', [332]. The same changes being made in the magnitude [329d]
p . d a . d J . d c, [303'], it becomes, by reduction, as in [332'].
t (2206) This is the same as [303'"], multiplied by r^ . sin. 6, substituting fi, [229d].
55
218
MOTION OF FLUIDS. [Mec. Cel.
which is the equation of the continuity of the fluid. In the present case,
[323% 328"],
[334] r' = r-\-as; &' = 6-}-au; zi' ==nt + -a-\-av ;
we shall therefore have,* by neglecting quantities of the order a^
^'= '+«•©+«•©+-©•
Suppose that at the end of the time t, the original density (p) of the fluid
[a35'] becomes (p) + « p', the preceding equation of the continuity of the fluid will
Second • i
form of glVef
tho same
ffquaUon. ^ o 7 / . x \ ^ /'d u\ , /d v\ , U . COS. 6 ) f . .x /'d.T^S
[336]
n. ^ 1 I s /■ \ ^ fdu\ fdv\ . u. COS. 6 } f , . .
* (221) The values of r', 6', ^, [334], give
all the other terms of s' are of the order a, ( — ) = «.( — ); ( -— l = a. ( -— ), &£c.
' \d6/ \d6 / ' \dT^J \d -a/
Therefore by neglecting terms of the order a^, the value of ^', [332], will be reduced to its
first term,
as in [335].
t (222) From r', 6', [334], we get r'^ = r^ . ^ 1 + 2 a . -Y
sin. 6' = sin. (^6 -\- au) = sin. 6 -{- au . cos. &,
Sin i\ / cos A\
(60 Int.), or -T^=il-{-au.~-]. These values, and that of p', [335], being
substituted in the equation of continuity, [333], put under the following form
p.s'./^.-T- r^-(?)=0. it becomes,
[w+-1•^+-(?7)+';•(^)+«•(^:)^^M'+-•fH'+"•Sl^'^w=«'
reducing and dividing by a, it becomes,
wliich, by a slight reduction, is easily reduced to the form of the equation [336]. If the fluid
be homogeneous, and p = (p), we shall have p'=0, and the equation [336], divided by (p),
will become as in [337].
eral form
for homo-
geneous
Fluids.
[337']
I. viii. §36.] OSCILLATIONS OF THE SEA. 219
36. Let us apply these results to the oscillations of the sea. Its mass
being homogeneous, we shall have p' = 0 ; consequently [336] [^^1
/^^A ^fdu\/dv\u. COS. 6 ) [337]
""-ydrj^ •l\d6j^\d^J^ sm.6 5* ™|-
Suppose, conformably to what appears to be the case, that the depth of the
sea is very small in comparison with the radius r of the terrestrial spheroid ;
let us represent it by 7, 7 being a very small function of 6 and ^s, depending
on the law of the depth. If we integrate the preceding equation with respect
to r, from the surface of the solid which the sea covers to the surface of the
sea, ;* the value of 5 will be a function of 6, xs, and t, independent of r,
increased by a small function, which will be, with respect to u or v, of the
same order as the function - ; now at the surface of that solid, when the [337"]
r
angles 6 and * become 6-}- au, and nt -{-■& -\-av, it is evident that the
distance from a particle of water contiguous to that surface, to the centre of
gravity of the earth, varies but a very small quantity in comparison with
a M, or a V, and that variation is of the same order as the product of those [337 "]
quantities au or av by the eccentricity of the spheroid covered by the
sea : the function independent of r, which occurs in the expression of s, is
therefore a very small quantity of the same order ;\ therefore we may in
* (223) This method of integration will be more easily understood after reading the part
included between [843 — 346]. In speaking of the order of the terms depending on r, 7, 5,
in [337"], and in other parts of this chapter, it will be convenient to refer all the linear
measures to the mean radius of the earth, considered as unity, so that we may say indifferently [337a]
7
either that a term is of the order -, or of the order 7.
r '
. f (224) A particle of the fluid at the bottom of the sea, being supposed in its motion
always to touch the solid spheroid, which is very nearly spherical; the value o( ds for a
particle so situated must be very small ; being to au or av, of the order of the eccentricity
of the spheroid, to its mean radius taken as unity. Now this function of 6, zs, t, added in
[337"], to complete the -integral s, being independent of r, must be the same, on all parts of
the radius r, as it is at the bottom of the sea ; and as we have just shown, that s varies but
very little at the bottom of the sea, by changing 6 into 6-\-au, and or into -a-j-av, it follows
that the function of 6, ts, ty here treated of, must be very small, and of the order mentioned
in [337"].
220 MOTION OF FLUIDS. [Mec. Cel.
[337 iv] general neglect s in comparison with u and v. The equation of the motion
E uation ^^ ^^^ ^^^ ^^ ^^"^ surfacc, given in § 35 [328], becomes by this means*
££, ^.,,.j(^)_2».sin...cos...(i^)J ^^^
face of
the Sea.
[338] + ^ . ^ ^ • I sin.^ ^ . (-^J + 2/J . sin. ^ . cos. 6 . ("-^J | =^gjyj^6V' ;
The equation (L) [325] of the same article relative to any point whatever
of the interior of the fluid mass gives in the state of equilibriumf
[339] 0= ^.<5.{(r + a5).sin.(^ + aw)}^ + (<5F) — ^;
2 p
((5 V) and (Sp) being the values of ^ F and- 5p, which, in the state of
[339] equilibrium, correspond to the quantities r -{-as, 6-{- au, and m-^-av.
Suppose that in the state of motion we have
[340] ^F=(<5 F) + «5 F ; Sp = (5p)-^a6p' ;
the equation (L) [325] will give
rQ^n W-(^' — -)? /'dds\ ^ . ., fdv\
* (225) This is the same as the equation [328], neglecting 5, as in [337'''].
■|- (226) In the state of equilibrium, as, au, av, must be constant, and their differentials
relative to t are nothing, therefore the first member of [325] will vanish ; and if we put, as
in [339'], {S V), — ; for <5 F, — ; the second member will become as in [339]. This
equation gives -^ . 8 . {{r -]- as) . (sin. 6-\- au)]^= — (5 F") -| , and by
-* P
neglecting, as in [327'], the variation arising from an^Sy, on account of its smallness, we
may substitute this in the second member of [325], which will make it
This, by substituting the values of S J^, Sp, [340], becomes -\- a6 V — a . — , and
the part of this expression relative to the independent variation S r, is
a . <^ V P_/ > . 5 r,
(^ dr )
p being constant as in [336']. This, being put equal to the term depending on a8r, in the
first member of [325], produces an equation, which, being divided by a8r, becomes as
in [341].
I. viii. ^ 36.] OSCILLATIONS OF THE SEA. 221
The equation (M) [338] shows that n.(-^j is of the same order as y or
s* consequently of the order — [337"] ; the value of the first member of [341']
this equation [341] is therefore of the same order ; so that if we multiply this
value by dr, and then integrate, from the surface of the spheroid which the
sea covers, to the surface of the sea, we shall have V — - equal to a very
small function of the order — , increased by a function of 6, ts, t, inde-
r
pendent of r, which we shall denote by x ;f considering therefore in the [34r']
* (227) Making die coefficients of the independent variations S6j 8zs, in the equation
[338], separately equal to nothing, we shall get,
^.(^)-.„^.s...cos.,.(lf)=-,.(|f) + (i^),
Add the differential of this last relative to <, to the first equation, multiplied by
— 2 . n . sin. & . cos. ^, and let the second member of this sum be represented hy i/ .r^ . sin. 6.
Then divide by r^ . sin.^ 6, and put for brevity 2 n . cos. 6^ a, we shall get
This becomes of the form of the equation [865], by changing y into (tt)> and a Q into
— y ; and the value of y, [870], being multiplied by a or 2 n . cos. 6j will give,
2 n . cos. 6 . (—) = sin. a t .fif .dt. cos. at — cos. a t ./y' ,dt» sin. at ;
the constant quantities, produced by the integration, being supposed to be included under
the signs/ of integration. The second member of this equation being of the order y', the
first member, or 2 n . f — j, will be of the same order. But ^ depends on y, V, and by
note 231, we shall see that V is of the order y, therefore xf is also of the order y; hence we
finally perceive that w . (— ) is of the order y, as in [341'].
f (228) This integration is made as in [337'], and if we put (p (r), for the small function
y s p'
of the order — , mentioned in [341"], we shall have V = X -|- (p (r). r34lol
56
222 o MOTION OF FLUIDS. [Mec. Cel.
[34n.
equation (L) ^ 35 [325], only the two variable quantities d and «, it will be
changed into the equation (M) [338], with this difference only, that the
second member will become ^x.* But x being [341"] independent of the
depth at which the particle under consideration is found ; if we suppose
this particle to be very near the surface, the equation (L) [325] ought
[UV^] evidently to coincide with the equation (M) [338] ; therefore we shall have
5x = dV — g '^y'l consequently t
[342]
'\v'-^l]^=^V'-^g.^y',
the value of <5 V in the second member of this equation corresponds to the
[:342'] surface of the sea. We shall see, in the theory of the tides, that this value
is nearly the same for all the particles situated on the same radius of
the earth, from the surface of the solid covered by the sea, to the surface
61/
* (229) The second member of [325] is in [339a], reduced to the form a 6 V'—a . -^,
[3416] Qj. ci.sJv — - j, and this by means of [341a], becomes a.8x4-a. S.cp(r). Therefore
if we consider only the parts of the equation [325], depending on the two variable quantities
General 6, Si, as is directed in [341"], we shall get, by dividing by a,
B(]uation
fur all
parU or
the Fluid.
[341c]
[341d]
+ r2.5«. i sin.2 6 . (-^^\ +2 n . sin. 6 . cos. 6 . (j^) ]=S\
which is the same as the equation [338], changing its second member, — g -^y -h^ ^> into
8X; observing that the equation [341c] corresponds to any point in the interior of the fluid,
whereas, [338] refers only to its external surface. Now X being a function of 6, -a, t,
independent of r, [341"], it must be the same upon any part of r, either at the surface of
the fluid, or below it, and it must therefore, in all cases, be equal to the value [338], so that
we shall have S\ = — g .6y -\-S V, in which the values 8 y and 5 V\ correspond to
particles at the external surface of the fluid. Tliis agrees with [341'''].
f (230) Noticing only the variations of ^, •cr, we get, from [3416], 8\=8 A V — ->.
Substitute this in [34Irf], and it will become as in [342]. In the last member of the
equation [342], the quantities F', y, correspond to the surface of the sea; in the first
member, to any point of the interior, on the same radius.
223
in which r is taken for the distance of the sun or moon from the centre of the earth,
the polar radius of the earth being unity ; if the earth's radius had been put equal to /,
corresponding to a stratum below the surface, the expression would have been
and the part of V, depending on this, would be,
iy(0) /2 ,3
r r3 r*
which does not sensibly vary for all the particles situated on the same radius, from the surface to
the bottom of the sea, because the variation of r' is only 7, which is very small in comparison
with /, and Z(">, Z(^>, &xj. are independent of /. The second of these forces, [2135"],
arising from the attraction of the shell whose thickness is ay, is computed in [1501], which
varies but very little, for all particles situated on the part y of the radius, from the top to the
bottom of the sea ; the greatest variation being of the order ^ • « y, in comparison vnih the
whole attraction of the earth, and it may therefore be neglected. The author has given a
short note on this subject at the end of Book XIII.
t (231a) This arbitrary function is added to complete the integral, and as usual, it must
be independent of the variable quantities ^, w, r.
[342"J
I. vui. -^Se.] OSCILLATIONS OF THE SEA.
of the sea ;* therefore we shall have, relative to all these particles,
-E. ;= g ,5y ; which gives p'=pgy, increased by an arbitrary function,
P
independent of 6, -a and r ;t now at the level surface of the sea, the value
of «^' is equal to the pressure of the small column of water ay, which rises ^342*^
above this surface, and this pressure is equal to ap.gy; hence we shall
have, in all the interior of the fluid mass, from the surface of the solid which
the sea covers, to the level surface of the sea,
p' = pgy; [342i']
therefore any point whatever of the surface of that solid is more pressed
* (231) It is shown in Book IV, § 1, [2130'— 2135"], that the value of a <5 V, consists
of two parts, the one arising from the attraction of the sun and moon, &£C. on any particle of
the fluid ; the other from the attraction on the same particle by an aqueous stratum, whose
interior radius is /, and exterior r' -{- ay; r' being the radius of the earth, corresponding to
the state of equilibrium, and r'-\-ay, that in the state of motion. The part aV, depending
on the first of these forces, computed in [2134], is equal to
^(0) „m 7(3)
224 MOTION OF FLUIDS. [Mec. Cel.
than in the state of equilibrium, by all the weight of the small column of
water, comprised between the surface of the sea and the level surface. This
excess of pressure becomes negative, in those parts where the surface of the
sea falls below the level surface.
[342;^] It follows from what we have said, that if we notice only the variations of
6 and trf, the equation (L) [325], will change into the equation (M) [338],
for all the particles in the interior of the fluid mass. The values of u and
V, relative to all the particles of the sea, situated on the same radius of the
[342vi] earth, are therefore determined by the same differential equations : hence if
we suppose, as we shall do in the theory of the tides, that at the
commencement of the motion, the values of u, (-j-)i v, ( — ), were
the same for all the particles situated in the same radius ; these particles will
[342 "i] remain in the same radius during the oscillations of the fluid.* The values
of r, u, v, may therefore be supposed nearly the same on the small part of
the radius of the earth, comprised between the solid covered by the sea, and
the surface of the sea ; therefore by integrating, with respect to r, the
equation [337],
we shall havef
[.344, 0 = r^s-(r-s)+r^r.{{^) + {l^) +
u . COS. 6 )
sin. & 5
M . COS. d
sin. 6
* (232) The quantities u, v, at the commencement of the motion, being supposed to
change into m', v', at the end of the time dt; we should have, by the usual rules of the
differential calculus, «' = m + (-^ j .dt, v=:v-\- (-^j . d t. Now m, v, [323^], are
the same for all particles situated upon the same radius ; and if we suppose, as in [342'"],
that at the commencement of the motion (tt)» ijrp ^^^^ ^® ^^^^ ^°^ ^^ ^^^® particles,
we should have also u' and v' the same for all the particles, and these values of u', i/, would
therefore, [323^], correspond to the same radius, at the end of the time d t ; and for the like
reasons, the same would happen at any successive instant.
f (233) Multiply [343] by d r, and integrate it, supposing
' l\d6/ \dT!iJ sm.6 >
to be constant throughout the length y, of the radius, we shall get [344]. The term — (r^s),
I.viii. §36.] OSCILLATIONS OF THE SEA. 225
(r^ s) being the value of r^ 5, at the surface of the spheroid covered by the
sea. The function r^s — (r^s) is nearly equal to r^ .{s — (5)} + 2 r y. (5) ;
(s) being the value of 5 at the surface of the spheroid ; we may neglect [344]
the term 2r 7. (5), on account of the smallness of 7 and (s) : we shall thus
have
r^s—(t^s) = r^.[s — (5)]. [345]
Now the depth of the sea corresponding to the angles ^ + a m, and
n t -{- zi -{- a V, is* y+a.{5 — (s)] , if wc take the origin of the angles [^^ST
6, and nt-\-vij at a fixed point and a fixed meridian upon the surface of the
earth, which may be done, as we shall soon show ; this depth will bef
y + ai^.f-^j + azrf.f— ^j, increased by the elevation ay of the fluid particle [3^
at the surface of the sea above its level ; we shall therefore have
S-W=y + «-(^)+«-(@- 1346)
The equation relative to the continuity of the fluid will therefore become J
by putting r — 7 for the value of r, at the bottom of the sea, becomes — r^ (*) + 2 r y . (s),
neglecting 7^, as in [344'.]
* (234) The depth 7 corresponds to the angles 6 and « or ni-\-'a, and when these
increase by au, av, the elevation at the upper surface increases by as, [323^], and at the
bottom by a (s), [344'], hence the whole depth becomes y -{-as — a (s).
f (235) 7 is a function of &, -sJ, corresponding to the surface of equilibrium, and these
quantities, in the state of motion, become ^ + "^5 zi-\-av. Developing this function
according to the powers of a m, a v, neglecting their squares and products, on account of their
smallness, it becomes as in [345"] ; this depth is to be augmented by the elevation a y, of the
particle above its natural level. The expression thus found, being put equal to the former,
7 -f- a . [s — {s)l, [345'], it becomes, by neglecting 7, common to both sides of the equation,
and dividing by a, the same as in [346].
f (235a) Substitute [345] in [344], divide by r^, and add tiie equation [346], we shall
get, by rejecting s — (s), from botii members of tiie equation, the following expression,
51
226 MOTION OF FLUIDS. [Mec. Cel.
We may observe, that in this equation, the angles 6 and nt-{-vi are reckoned
[:147'] from a fixed point and from a fixed meridian upon the earth, and that in the
equation (M) [338] these angles are reckoned relative to the axis of x, and
to a plane which, passing through that axis, will have about it a rotatory
motion equal to n ; now this axis and this plane are not fixed on the surface
of the earth, because the attraction and the pressure of the incumbent fluid,
ought to alter a little their position upon this surface, and vary a little the
r347*'l ^tatory motion of the spheroid.* But it is easy to perceive, that these
alterations are to the values of au, av, in the ratio of the mass of the sea to
that of the terrestrial spheroid ; therefore, in order to refer the angles Q and
n ^ + ^) to a fixed point and meridian on the surface of the spheroid, in the
two equations (M) [338], and (N) [347] ; it is only necessary to vary u
and V by quantities of the order — and —, which may be neglected ;
[347'"] we may therefore suppose, in these equations, that cu and av are the motions
of the Jiuid in latitude and longitude.
We may also observe, that the centre of gravity of the spheroid being
supposed immoveable [323""], we must transfer to the particles of the fluid,
[347i^] in an opposite direction, the forces with which that centre is urged by the
-« «-di)+-(^)=(^)' -' -(E)+-a=(^> "-e
/d.Yu\ /d.yvS yu.eos.Q
by substituticm and reduction, we get, y== — ( 1 — ( .^ 1 — — -r^- — , as in [347J.
* (236) The co-ordinates 6, zi, of a particle, at the beginning of the motion, become
8 -{-au, and nt -j-vi -{-av, at the end of the time t, [323''] ; and the rotatory velocity of
the particle instead of being n, will be n + a- (ttJj as evidently appears, by taking the
differential of the angle nt -{-vi -\- av, relative to t. This change of velocit}"- "^'[jrji
would produce in the whole fluid mass quantities of the order " • ~ • ( TT )j when compared
with the motion of the whole spheroid ; and we may neglect such quantities as in [347"].
A similar variation arises from the change of 6 into 6-{-au. It may be remarked that
instead of saying, as above, [347'], that the angle nt -{--a, is reckoned from a fixed meridian
on the earth, it would be more correct to neglect n t, and call the angle simply w, but this
does not affect the reasoning.
I.viii.§37.] OSCILLATIONS OF THE ATMOSPHERE. 227
reaction of the sea ; but the common centre of gravity of the spheroid
and the sea does not change its situation by this reaction ; it is therefore
evident that the ratio of these forces to those with which the particles are
urged by the action of the spheroid, is of the same order as the ratio of the
mass of fluid to that of the spheroid; consequently of the order - ; we l^"^"]
r
may therefore neglect them in the calculation of <5 V.
[347^]
37. Let us now consider in the same manner the motions of the
atmosphere. In this research, we shall neglect the consideration of the
variation of heat, in different latitudes and at different heights, as well as
all the irregular causes of agitation, and we shall only notice the regular
causes which act upon it as upon the ocean. We shall therefore suppose the
sea to be surrounded by an elastic fluid of uniform temperature ; and we
shall also suppose, conformably to observation, that its density is proportional r347vin
to the pressure. This supposition makes the height of the atmosphere
infinite ; but it is easy to prove that at a very moderate elevation, its density
is so small that it may be regarded as nothing.*
This being premised, we shall put s', u', v', to denote, for the particles of [347^"']
the atmosphere, the quantities which were named 5, u, v, for the particles
of the ocean [323"] ; the equation (L) § 35 [325], will become
+ «..«. I si„...(^r)+2-in..cos..(|) +^-^.© i
= ^.5.J(r + a5').sin.p + aw')}' + ^^— — •
/
* (237) The formula for computing the density p, [355], rejecting the term — , on
_r'
I
account of its smallness, and putting g=g^, [349', 353'], becomes p = n . c . Now
if, for an example, we suppose / to be equal to lOZ, which, as will be seen in the next note,
—10
is about 55 English miles, it will become p = n.c ; and as c = 2.71828, we shall find [348a]
p = -" , which is extremely small, as is observed above.
/dv'\ ) General
Equation
for all
parts of
du'\ . 2n.sm.^6 /ds'\ ) ^w
228 MOTION OF FLUIDS. [Mec. Cel.
Let us at first consider the • atmosphere in a state of equilibrium, in which
s\ u', and v' are nothing. The preceding equation, in that case, will give by
General . ,
Equation intcgration
ofEquilib-
— - . r* . sin.^ 6 + V — / — = constant.
[349] 2 ' */ p
The pressure p being supposed proportional to the density, we shall put
[349'] p = l.g.p;
g being the force of gravity at a given place, which we shall suppose to be
the equator, and / a constant quantity, denoting the height of a homogeneous
atmosphere of the same density as at the surface of the sea ; this height
is very small in comparison with the radius of the earth, being less than
[349"] rhr^^ part.*
[349'"] . rSp .
Second The integralf / — is equal to IgAog.p; therefore the preceding
General _ ' , , ,
of^Equu equation of the equilibrium of the atmosphere will become
librium. n
[350] / g . log. p = constant -{- V -\- —- . r^ . sin.^ 6,
At the surface of the sea, the value of V is the same for a particle of air, as
for the particle of water contiguous to it, because the forces acting on both
are the same ; but the conditions of the equilibrium of the sea, require that
we should have,t
[351] F+ — 'T^' sin.^ ^ = constant ;
* (238) By Book X, § 9, [8500], the general expression of I, corresponding to the
temperature of x degrees of the centigrade thermometer, is 7974 . { l-|-0.00375.a;|, and
if, for example, we take x = 29, It will become equal to 8841 metres, or 5J English miles,
nearly, which is about j^ V^^ ^^ ^^ earth's radius.
t (239) From Sp=lg.8p, [349'], we get f-^ = Ig f— = h • ^og- P- This
substituted in [349], gives [350].
J (240) The equation [326] corresponds to the surface of the sea. In the case of
equilibrium, and when au=0, av = 0, it becomes 0 = — .S.(r^.sln.^6)-{-SV; its
integral relative to 8 is as in [351]. This, being substituted in [350], we obtain
Ig . log. p = constant, whence p = constant.
[35(y]
I. viii. <^ 37.] OSCILLATIONS OF THE ATMOSPHERE. 229
we have therefore, at this surface, p constant ; consequently the density ^352/1
of the stratum of air contiguous to the sea, is constant in the state of
equilibrium.
If we put R for the part of the radius r, comprised between the centre of ^^^ ^
the spheroid and the surface of the sea, and / the part included between
this surface and a particle of air elevated above it ; / will be the height of
this particle above the surface of the sea, neglecting quantities of the order*
-)■
I
„ ^ , and quantities of this order we shall neglect. The equation [351'"]
±i Third gen-
.-,1 . , eral form
between p and r will givef ^^^^^^
/iJF\ r'2 /ddV\ n^ ofEqui-
* (241) Let FBDH be a meridian of the earth
considered as an ellipsoid of revolution, whose semi-axis
is HI, equal to unity, centre /, equatorial semi-axis IF.
A the place of a particle of air ; ID = R, DA^r',
A /=: r ; AR the perpendicular let fall from A upon
the surface at jB, jBC a tangent to the meridian touching
it in jB. J5 E an arch of a circle described about tlie
centre A, meeting I A in JE; then by §25 of Book HI, [1648], the ellipticity of the meridian,
is proportional to the centrifugal force, consequently the angle BAD is of the same order as
the ratio of the centrifugal force to gravity, or of the order — , [327"], or — , [337o],
therefore B D or B C is of the order — . /. But E C= , nearly, and CD is
g 2 . JIB
nearly equal to the square of B C divided by twice the radius of curvature of the meridian
at B, and is therefore of the same order as the square of B C% divided by 2 R. The sum
of C -E, CD, expresses the difference between A B and /. The first of these quantities is
of the order \£ — — , the other is of the order aJ! L. This last term is that mentioned
2/ 2R
by the audior ; the other is larger, but even this is very small, being in comparison with / of
•he order ^„ or ^, [327"].
f (242) Supposing the values of V, (;t— ), ( , „ ), to correspond to the surface of
the sea, or to the distance R firom the centre, the general value of V, at the distance
58
230 MOTION OF FLUIDS. [Mec. Gel.
the values of F, ( -^ j and ( j being taken to correspond to the
surface of the sea, where we have*
[353] constant = F+ — . i2^ sin.^^ ;
[353'] the quantity — {~r~) — ^^ R - sin.^ & is the gravity at the same surface ;t
R-\-r'j 6 and zi remaining unaltered, will become by development, by the formulas
[607, 608], V+ / . (^ -\-i.r'^. {~~\ + &tc. This is to be substituted for F,
and 72 + r', for r, in [350], neglecting Z^, on account of the smalbess of the term w^ . /^ ;
we shall thus obtain [352].
* (243) This is the same as [351], substituting R for r, as in [351"], to make it
correspond to the surface of the sea.
■}• (244) The function V represents the integral of the sum of the products, formed by
multipl3dng each force acting on the particle, by the element of its direction, [295']. It is
[352a] similar to the function 9, [16, 17], and in the same manner as ( 7— ) was proved in [17], to
be the resultant of all the forces resolved in a direction parallel to x, we shall 6nd that ( ~ )
is the resultant of the forces P, Q, jR, acting on the particle, resolved in the direction r.
Again, by drawing, in the preceding figure, D K perpendicular to die axis, we shall have
nearly, DK=R .sa\. &, and the centrifugal force in the direction J^Ti) is, by [138a],
[3525]
n^ .DK=n^ .R .sm. L This resolved, in the direction /D, is nearly
rfi .R. sin. 6 X -t^Tj or n^ .R . sin.^ & ;
[353a]
therefore the whole force acting at D, in the direction ID, is i — \-\-n^.R . sin.^^, this
force being supposed to tend to increase J D as in note 190a ; now this ought to be equal to
gravity — ^, [358"], the sign — being prefixed, because gravity acts in the opposite
direction D / nearly ; hence ^ = — ( — j — n^ .R. sin.^ &, as in [353']. The last term
of the second member being much smaller than the other, we have nearly {j~) = — <§"'»
whence (—~-\ = — (-r-) •
In [470], it will be sho^vn that for a sphere ^ = -^, r being the radius of the sphere,
J . , r ^dg'\ 2m , ,' /ddV\ 2m 2g'
[3536] and m Its mass; therefore (-7^1 = — —, nearly; hence ( j = — = — , and this
EQUILIBRIUM OF THE ATMOSPHERE.
231
I. viii. § 37.]
we shall denote it by g'. The function (-r-r) being multiplied by the [353"]
very small quantity Z^, we may determine it on the supposition that the
earth is spherical, and neglect the density of the atmosphere relative to that
of the earth ; we shall therefore have very nearly
\drj
m'
[354]
m
being the mass of the earth ; therefore ( j = -^ = -^ ; we shall
y/2
I therefore have* /g. log. p^ constant — 1^^+-^ - ^ '•, w^hence we deduce [3541
R
p= n.c
ig \ r)
[355]
c being the number whose hyperbolic logarithm is unity, and n a constant [355/]
quantity, which is evidently equal to the density of the air at the surface ^"?he°
of the sea.f Put h and h' for the lengths of pendulums vibrating in a second of the At-
' 01 o mosphere.
at the surface of the sea, at the equator, and in the latitude of the particle [355"]
at the surface, where r = R^ becomes -=;-, agreeing with the author, in the original work,
Jo.
except in the sign, which is changed in the present translation, and likewise the signs of the
term -^, in the equations [355, 356], which required the same correction.
R
*(245) Substituting in [352], the value of V-{-~ .R^ .^?&, [353]; also for
+ r' (—\ -{-r? .Rr^ . sin.^ 6, its value — g' /, [353a], and for (-TT-)^ its value ■—,
/a
[3536], it will become /^ . log. p = constant — ^g'-\-~^'g'- Divide this by Ig, and
R
put the constant quantity equal to Ig . log. IT, we shall get log. p = log. n — ;'(1 ^)
which is easily reduced to the form [355].
f (246) Because when r = 0, we shall have from [355], p = n.
2^2 MOTION OF FLUIDS. [Mec. Ce\.
jj -n
[355'"] of air under consideration ; we shall have - = r » consequently*
[356] ^H'V-r)
P= n. c
This expression of the density of the air, shows that a stratum of the same
density is in all parts equally elevated above the sea, neglecting the quantityf
[356] — — 7 ; but in the exact calculation of the heights of mountains, by
observations with a barometer, this quantity ought not to be neglected.
We shall now consider the atmosphere in a state of motion ; and shall
determine the oscillations of a level surface, or surface of the same density,
[356"] in the state of equilibrium. Let a(p be the elevation of a particle of air
above the level surface to which it appertains in the state of equilibrium ; it is
evident that by means of this elevation, the value of 6V will be increased by
[356'"] the variation — 0Lg.5(p; therefore we shall have 6V=(6V) — ag.6cp-\-a5V' ;t
(SV) being the value of S V, which in the state of equilibrium corresponds to
the level surface, and to the angles 6-\-uu' and nt-{-'a-\-av' ; and SV being
the part of ^ F, arising from the new forces which in a state of motion
agitate the atmosphere.
Let p = (p)+«p', (p) being the density of the level surface, in the state of
equilibrium. If we make ^ = y, we shall have§
h' s!
* (247) Deduced from [355], by putting — for — , observing that when the time Tof
re g
the vibration of a pendulum is given, the length of the pendulum will be proportional to the
force of gravity g^ or ^^ [86].
f (248) If in the exponent of c, [356], we change ^' into A + (^' — A), and neglect the
very small quantity 77 — j the expression of p, [356], will become very nearly
p =n . c , which is nearly constant at the same elevation /.
f (249) This equation is precisely similar to [327], changing y into 9, the demonstration
is made as in note 216.
<^ (250) From [349'] we get 5^ = Z^.5p. Substituting p, [356»^],, we find by
neglecting a^,
P (P) + ap' (P)
.(p).,p^-p^,.(p)> Zgj(p) C|K
^ ( (PF > (P) ^ ^ UP)
Putting as in [356''], — = y, it becomes as in [357],
* (251) In the case of equilibrium u'^vf^sf, are constant, and their differentials relative
to t vanish, which makes the first member of [348] vanish, and the second member becomes
as in [358] ; observing that in this case 5 F" becomes (5 V), and p becomes (p), therefore,
p'==0, [356''] ; also /= 0 ; hence [357] changes into _?.=ilJ-M.
f (252) These level surfaces bemg nearly spherical, we shall have, in like manner as in
[327''], 5r=0, and in the same way that [328] was obtained from [325], we may deduce
[359], from [348]. For, by neglecting the term 5r in the first member of [348], and dividing
by o, it becomes identical with the first member of [359]. The second member of [348],
divided by a, becomes -. | -^ .^{(r + as') . sin.(d + aM')r + ^ ^— — 1 5
m which "^ • ^ • ] (^ + « ^) • sin. (^ + a m') > , corresponding to the state of motion,
may be divided into two parts, the one, being its value in the state of equilibrium, is equal
to — {^V)-\ — - — , [358] ; the other depending on the change in the value of this
quantity, arising from the motion of the particles. Now in the state of motion the distance of
the particle from the centre of the spheroid, has increased from r to r-{- as/, the increment
being a «' ; but in tlie same time the motions of the particle in the directions a »', a «', along
69
Lviii. §37.] OSCILLATIONS OF THE ATMOSPHERE. '^^
^^'-i^ + ^^.^y;
now we have in the state of equilibrium*
O=^.3.{(r + «^0»sin.(^ + ai^0r + (^n-^^'J)^^^ [358]
the general equation of the motion of the atmosphere [348], corresponding
to these level surfaces, in which 5r is nearly evanescent, will therefore
becomef
, „ , C . 2 /ddv'\ , ^ . , /du'\ , 2n.sm.^6 /^A ) t359]
+^.a..|sm.^^.f-^) + 2,..sm.a.cos.^.(^-^)+— ^— .(-)^
[358a]
234 MOTION OF FLUIDS. [Mec. Cel.
r359'i " • W being the variation of r, corresponding in the state of equilibrium,
to the variations au', and uv' of the angles ^ and w.
Suppose that all the particles of air, which are situated at the beginning of
the motion on the same radius of the earth, remain constantly on that radius
during the motion which, as we have seen, takes place in the oscillations of
the sea [342'] ; and let us see whether this hypothesis will satisfy the
equations of the motion and the continuity of the atmospherical fluid. For
this purpose it is necessary that the values of u' and v' should be the same
[359"] for all these particles ; now the value of 5 V is nearly the same for all these
particles, as will be seen when we shall hereafter compute the forces from
which this variation results ;* it is therefore necessary that the variations
^ (p and ^ y' should be the same for all these particles, and that the quantities
[359'"] 2nr .^-m . sin.^ 6 . ( ^ ) > and n^ r . sin.^ 6.5.^s' — (s') ] , should be neglected
in the preceding equation.
rg^Qj At the surface of the sea we have [366", 326"], cp = y, ay being the
elevation of that surface above its level. Let us now see whether the
[359 V] supposition of ^ = y, and y constant for all the particles of air, situated upon
the same radius, can subsist with the equation of continuity of the fluid.
the surface of equilibrium increase the height of that point of this surface by a (s'), [359'],
consequently the elevation of the particle above the surface of equilibrium is increased by
a . [ s' — {s') I , hence the variation of the term ~7f • ^ • { (^ + « «') . sin. (^ + " ^') \ ^ will be
nearly a, n^ . r . sin.^ 6.8. [/ — (s') \ ; and by reasoning as in note 211 , it will be evident,
that this expresses the wholcv variation of — - . ^ . ^{r -j-as) . sin. {6 -\-au')l, arising from
the motion of the particles. Its complete value therefore, in the state of motion will be
— (5 F) 4.^-^:-^'-^^ + a . Ti^r . sin.2 6.5. {s'—{s')l.
Substituting this and S V= {S V) — ag . Scp-Jr ocS V, [356'"]; -^ = i^-~S^-\-ag . 6y',
[357], in [358a], it becomes, by reduction, 8J^' — gScp — g5y' -\-rfir .sin.'^ 6. d.^s^ — (0|j
as in the second member of [359].
* (254) This may be shown as in note 23 L
I. viii. ^ 37.] OSCILLATIONS OF THE ATMOSPHERE. 235
This equation by § 35 [336] is*
n ^ ^ ' L/^ ( /'^w'A , /^ A _L «' • COS. a ) ) ... /J.r^A
[360]
whence we deduce
, ( /d.T^s'X , fdu^ , /dv\ , m'.cos.O
[361']
r + a s' is equal to the value of r of the level surface, which corresponds to
the angles ^ + « %', and ■a-\-av\ increased by the elevation of the particle
of air above this surface ; the part of a s' which depends on the variation of
the angles (5 and *, being of the orderf '—^ may be neglected in the
preceding expression of /, therefore we may suppose, in this expression,
5' = ip; if we then make ^ = y, we shall have f— j = 0, since the value [361"]
of (p is then the same relative to all the particles situated on the same radius.
Moreover y is, by what precedes, of the order /, or — ;t the expression
* (255) This is the same as [336], changing 5,w, v, into s\ u', v\ as in [347''"]. IVIultiplying
this by--, we get 0 = - +/• ^^)+ (^)+ — ^+ (^ j , and by
I p'
substituting for —^ its assumed value i/, [356''], we obtain [361].
f (256) By note 241, page 229, the angle BAD, which the perpendicular to the surface
of the ellipsoid makes with the radius ID, is of the order — . This multiplied by a m, will
give the order of the increment of the radius, arising from the motion of any particle of air
along its surface of equilibrium, through the angular space au'. The increment will therefore
be of the order , and this, on account of its smallness, may be neglected, as in
note 217.
d . (r^ f') s* /d «'\
{(257) Developing tlie term -^1 — > [361], it becomes 2. — l~{v~.)) and
by [361"], s^ = (pz=y, and f— _j = f — j = 0, it becomes simply
smce
T^dr r
Now by [347], y is of the order yu,ovyv, so that this term 2 . — , must be of the order
2^^ MOTION OF FLUIDS. [Mec. Cel.
of ?/ will thus become
therefore u' and v' being the same for all the particles which were at the
beginning on the same radius, the value of y' will be the same for all these
particles. Again, it is evident from what we have said, that the quantities
[362'] 2nr .Szi , sin.^ ^ • ( "^ ) » ^^^ '"'^ ^ • ^^^*^^ . <5 . { 5' — (s') } , may be neglected in
the preceding equation of the motion of the atmosphere,* which may then
be satisfied, by supposing u' and v' to be the same for all the particles of air
situated originally on the same radius ; the supposition that all these particles
remain constantly on the same radius during the oscillations of the fluid, is
therefore admissible with the equations of the motion and of the continuity
of the atmospherical fluid. In this case, the oscillations of the different level
^f"thi°" strata are the same, and are determined by these equations,!
Oscilla-
tions of
the level
u.e«v«i o ,, (/c^^m\ a ' . . fdv
strataof iT . h & . { — nT" ) Zn . SlU. & . COS. ^ • -;-
the At- ) \ dt^ J \dt
mosphere. v \ ^^ \
[363]
+ /^ . 5« . ^ sin.^^ . (J^\ + "2.71.^11, 0 . cos. 5. (^\ \ ^^V'—g.di/^gSy;
, J ( /du'\ , /dv^ , u' . COS. ^ )
These oscillations of the atmosphere ought to produce corresponding
oscillations in the heights of the barometer. To determine these, by means
of the former, let us consider a barometer fixed at any height above the
surface of the sea. The height of the mercury is proportional to the pressure
2 . — . w, which is to the other terms of the formula, [361], {-r-\ (-;—), w', of the order -,
r ^ \a 6 / \a TH/ r
and by neglecting it on account of its smallness, the expression [361] will become as
in [362].
* (258) Because /, (s), are small in comparison with u and 'v, and these terms are also
multiplied by n or n^ in [359] ; they are therefore so small that they may be neglected.
f (259) These are the equations [359, 362], neglecting the terms depending on s\ (s),
on account of their smallness, and putting <p = y, [361"].
I. viii. §37.] OSCILLATIONS OF THE ATMOSPHERE. 237
its surface experiences when exposed to the action of the air ; it may r^^^r,
therefore be represented by Ig.p [349'] ; but this surface is successively
exposed to the action of different level strata, vv^hich rise and fall like the
surface of the sea ; thus the value of p, at the surface of the mercury, varies ;
First, Because it appertains to a level stratum, which, in the state of
equilibrium, was less elevated by the quantity uy. Second, Because the
density of the stratum increases in the state of motion, by* a p', or " |'^-
By means of the first cause, the variation of p isf — ^V'tl^p ^^ j '^ ''^^^
the whole variation of the density p, at the surface of the mercury, is therefore
a (p) . i^Y' Hence it follows, that if we put k for the height of the mercury [363"']
in the barometer, corresponding to the state of equilibrium, its oscillations
in the state of motion, will be expressed by the function! "^ — ^ ^ ^^ ; they [363i']
* (261) In [356'^], p = (p) -j- a p', (p) being the value of p, corresponding to the level
surface ; therefore a p' is the increment arising from the state of motion. Using the value
of y, [356''], it becomes a p' = -~-^.
f (262) The density p is a function of r which decreases when r increases, therefore when
the increment of r is ay, the decrement of p will be — ay.f — j. Now the equation
[355], neglecting — on account of its smallness, and putting — = 1, becomes p=n.c
St S
R—r R—r
nearly, and as r'=:r — R, [351"J, we shall get p = lie ' , hence (t^) = — j • c ' ,
R — r
— I — p /^p\ p (p)
or, by substituting the value of c = — ; — f — j = — , or — -, nearly, therefore
/dp\ a.(p).y
J (263) For if the density (p) give the height k, the increment of density a • (p) . ,
must, by proportion, give a corresponding increase of height of the barometer denoted
by "^-(y+y")
60
238 MOTION OF FLUIDS. [Mec. Gel.
are therefore similar, at all elevations above the sea, and proportional to the
heights of the barometer.
To determine the oscillations of the sea and the atmosphere, it is now
only necessary to know the forces which act upon these two fluid masses,
and to integrate the preceding differential equations ; which will be done in
the course of this work.
^0
SECOND BOOK.
ON THE LAW OF UNIVERSAL GRAVITATION, AND THE MOTIONS OF THE CENTRES OF
GRAVITY OF THE HEAVENLY BODIES.
* (264) From [38] we have -^ — P, -jJ- = Q ; but m the computation [34'"],
the forces P, Q, were supposed to tend to increase the co-ordinates ; whereas in the present
case, [363^'], these forces tend to decrease the co-ordinates ; we must therefore put P and Q
[363 V]
CHAPTER I.
ON THE LAW OP UNIVERSAL GRAVITATION, DEDUCED FROM OBSERVATION.
1 . Having explained the laws of motion ; we shall now proceed to
deduce from these laws, and from the phenomena of the motions of the
heavenly bodies, given in detail in the work entitled, " Exposition du
Systeme du Monde," the general law regulating the motions of those bodies.
Of all these phenomena, the elliptical motion of the planets and comets about
the sun, seems the best adapted to this investigation ; we shall therefore use
it for this object, and shall suppose x and y to be the rectangular co-ordinates
of a planet in the plane of its orbit ; the origin of these co-ordinates being in
the centre of the sun. Let P and Q be the forces acting on the planet,
parallel to the axes of x and y, in its relative motion about the sun, these [ses^ij
forces being supposed to tend towards the origin of the co-ordinates. Lastly,
let dt he the element of the time, which we shall suppose to be constant.
We shall have, by Chapter II of the first book,*
o = ~+P; (1)
"* [364]
0 = ^ + Q; (2)
[365]
[365']
240 LAW OF GRAVITATION, [.Mec. CeL
If we add the first of these equations, multiplied by — y, to the second
multiplied by x, we shall have
^ d.ixdy — ydx) , ^ _,
0 = — A— J-^A—^ + a: . Q — y. P.
It is evident that xdy — ydx is double the area which the radius vector of
the planet describes about the sun in the instant dt [167«] ; this area is
proportional to the element of the time, according to the first law of Kepler ;
so that we shall have
[366] xdy — ydx = cdti
c being a constant quantity ; the differential of the first member of this
equation must therefore be nothing ; hence we shall find*
[367] x,Q—y.P==0.
It follows from this equation, that the forces P, Q, are to each other in the
ratio of x to y ; consequently the resultant of these forces must pass through
the origin of the co-ordinates, or in other words, through the sun's centre.f
[367'] Moreover, the curve described by the planet being concave towards the sun ;
it is evident, that the force which causes it to describe this curve tends
towards the sun.
negative, and then, by transposition, we shall get [364]. JMultiplying the first by — y, the
second by x, and in their sum putting d.{xdy — ydx) for xddy — yddx, we
shall obtain the equation [365].
* (265) The differential of [366] being nothing, it reduces [365] to [367].
f (266) Let iS be the origin of the co-ordinates, or the
centre of the sun ; P the centre of the planet, S X the axis
of X, and P X the perpendicular let fall on it from P, making
SX=x, PX = y. On P X, take P E equal to Q, and
draw E D parallel to /S X, to meet P S m D ; then from the
similar triangles P X S, P E D, we get
PX{=y):SX{=x)::PE{=q)
but from [367] we have — ^=P, consequently, DE^=P. The two forces, DE=:=P,
PE^% being composed [11 Sic], form the single force P D = c^ = \/P^-{-q», in
[367a] jjjg direction P S, towards the origin of the co-ordinates S ; the curve described being
concave towards the sun.
n.i. <^2.] DEDUCED FROM OBSERVATION.
241
The law of the areas proportional to the times of description, leads
therefore to this first remarkable result ; namely, that the force which acts t^^"]
on the planets and comets, is directed towards the centre of the sun.
2. We shall now investigate the law according to which this force acts at
different distances from the sun. It is evident, since the planets and comets
alternately approach to, and recede from, the sun, at each revolution, that
the nature of the elliptical motion ought to conduct us to this law. For
this purpose, we shall resume the differential equations (1), (2), [364] of
the preceding article. If we add the first multiplied by d x, to the second
multiplied by d ?/, we shall obtain
- dx.ddx-\-dy.ddy , „, , ^ j .-^^fii
0 = ^j^ ^-{-Pdx+Qdy; [368]
and by integration,
0^^-^:^ + 2.f(Pdx+Qdy), [369]
the arbitrary constant quantity being indicated by the sign of integration.
I Substituting, instead of dt, its value [366] 3/ — y ^^ gj^^^^ ^^ ^j^^ j^^ ^^ ^269']
the proportionality of the areas to the times, we shall find
0 = ^^4^-\-2.f(Pdx+Qdy). [370]
We shall transform, for greater simplicity, the co-ordinates x, y, into a radius
vector, and polar angle, in conformity to the usage of astronomers. Let r
be the line drawn from the centre of the sun to the centre of the planet, or
its radius vector ; v the angle which this radius forms with the axis of x ;
we shall have*
Polar Co-
ordinates.
[370'J
ar = r . cos. v ; ^ = ^ • sin. v ; r^V x^ + 'f \ [371]
* (267) This is evident from the preceding figure, where SP = r, PSX= v, whence
SX=S P . COS. P SX, P X^SP .sm. P S X, which in s3nnbols are the same as the
above values of x,y, [371]. These agree with the values of x, y, in the note page 109,
changing p into r, and w into v. The same changes being made in [167c], it becomes as in
the first of the equations [372], and from [1676], we find, that double the area described
by the radius vector r, in the time dt, is represented by r^ .dv = xdy — ydx, as [•372a]
in [372].
61
2^2 LAW OF GRAVITATION, [Mec. Cel.
whence we deduce
[372] dx'-\-df=r'dv' + dr'', xdy — ydx^r'dv.
If we then denote by 9 the principal force which acts on the planet, we
shall have, by the preceding article,*
[373] P = (p. cos.zj; Q = (p.sin.y; 9=v/ P'+Q";
which givef
[374] P dx-\-Qdy =^(^dr\
we shall therefore have
[375] Q= ^4^^2 + ^f^dr;
whence we deducel
[376] rf^ =-—=======. (3)
r .V — (T — 2 r''/(p rf r
This equation will give, by means of the quadrature of curves, the value of
V in r, when the force 9 is given in a function of r. If this force is unknown,
but the nature of the curve it causes the body to describe is given, then by
taking the differential of the preceding expression of 2f(pdr, we shall have
the following equation^ to determine 9 ;
[377] c' (P ^'\-?^^l ,..
* (268) By [367a], we have PD = cpz= y/psTp^, and
P=:DE = Pncos.PI)E = (p.cos.v, also q = P D.sin. P D E=(p .sm.v,
as in [373].
f (269) Substitute in P dx -\- Q^dy, the values of P, Q, [373], also those of dx, dy,
deduced from [371], it becomes
(p . cos. V . \dr . COS. V — rdv. sin. vl -{-cp . sin. v . ^dr . sin. v -\- r d v . cos. «|,
which by reduction is cpdr . ^ cos.^ v -\- sin.^ vj, or cp .dr, as in [374]. This equation
is the same as [16], putting V= cp, P = S, Q = S', m = r, s = x, s' = y ; the
forces P, Q, being equivalent to 9. Substituting the values [372,374] in [370] we
get [375].
{(270) JVIultiply [375] by r'^.dv^, transpose the terms c^r^dv^, 2r'^ . dv^ .fcpdr,
divide by — c^r^ — 2 r^ .fcp d r, and extract the square root, we shall get [376].
§(271) The equauon [375], divided by 2 is /'P'^'- = — ^— |- • (^ri^) 5 «s
differential divided by dr gives [377].
II. i. § 2.]
DEDUCED FROM OBSERVATION.
243
The orbits of the planets are ellipses, in one of whose foci the centre of the .^^^^
sun is placed : now if in the ellipsis we put*
■a = the angle included between the axis of x and the transverse axis ;
a = the semi- trans verse axis ; [^'^"1
e = the ratio of the excentricitv to the semi-transverse axis ;
aj/ \^
\
* JLF S
C s 3
* (271a) Let APHBhe an ellipsis whose trans-
verse axis is A B, conjugate semi-axis C H, centre
C, foci S,s, vertices A,B, and directrix DE, perpen-
dicular to BA. Then if from any point P of the ^
curve, we let fall upon D JS the perpendicular P E,
and join PS, we shall have SP:P£::e : 1, ebeing j
a constant quantity. This property appertains to all
the conic sections, and it may serve to define them.
In the ellipsis e <C 1 5 in the parabola e=l; in the hyperbola e >> 1. We shall, in the
first place, demonstrate tlie formula [378], by means of this property of the directrix, and
shall afterwards give another demonstration, depending upon the rectangular co-ordinates of
the curve. Put CA=CB = a, SA = D, SB=^2a — D, SP=r, angle
ASP — v—TH. Then (roml^78a]we get SA = e. AD, SB=e.BD. Therefore
SB — SA — e.{BD—AD), or 2.CS = 2e.CA, and in symbols, CS=ae;
also SA=CA — CS becomes <S^ = D = « — ae = a . (1 — c), and
SA a . (1— e)
e
AD =
, the sum of these two last expressions is
a.(l — e) a.(l — c2)
Properties
of the
Conic
Section!.
[378a]
[3786]
[378c]
[378rf]
[378e]
[378/]
SI> = a.(l — e)-|-
SP
subtracting from this PE= — =-, [378a], we shall get SF =
a.(l— c2)— r
and as this [378g-l
r
e e
is evidently =SP . cos. ASP, or r . cos. (u — «), we shall get, by multiplying by e
r e . COS. (u — •cj) = a . (1 — e^)
whence we easily deduce the value of r, [378].
We may also demonstrate the formula [378], by showing that the usual equation of the
ellipsis, referred to the rectangular co-ordinates CF=^x, FP = rj, maybe derived from it.
For in the rectangular triangle SEP, we have P F= S P . sin. P S F,
SF=CF—CS = SP.cos.PSF, or in symbols,
y = r.sin. (v — w), x — ae = r.cos. (« — raf).
If we eliminate r and v — « from these equations, by means of the assumed relation between
r and v — -a, [378], we shall obtain the equation of the curve, corresponding to [378],
expressed in terms of x and y. Now from [378] we get
o.(l— e2)— r a —
[378fc]
r . cos. (r — «):
ae.
[378t]
'^^ LAW OF GRAVITATION, [Mec. Cel,
Polar
^Tan" the origin of the co-ordinates being fixed at the focus, we shall have
Ellipsis. /. - Q\
[378] 1 -\-e . COS. {v — w)*
Put this equal to the preceding value of r . cos. {v — zi), [378A], reject — ae from both
[378A;] members, and multiply the result bye, we shall get ex = a — r, or r = a — ex.
The sum of the squares of the two equations [378A], is evidently equal to r^, and by
substituting the preceding value of r, [378^], we shall get y^ + (* — « e)^ = {a — exY ;
[378i] which, by development and reduction, becomes ^^ = (1 — e^).{aP — x^), and as this
is the well known equation of an ellipsis, it proves that the curve defined by [378]
corresponds to that curve. When a?=0, the ordinate y will correspond to the semi-conjugate
[378m] axis C H= b, hence 6^ = (1 — e^) . a^. Dividing the preceding value of y^ by that of
y2 x^ X^ w2
[378n] 6^, we get — =1 -, or, -— + — =1, which is one of the usual mediods of
expressing the equation of an ellipsis.
If the ellipsis differs but very little from a circle, whose radius is 1, and we put 6=1,
[378o] a=l-{-a, CP = p, angle HCP = &, neglecting a^, the preceding equation [37 8n]
will become x^ . {I — 2a)+y^ = l, hence \/x^-\-y^ = y/l +2 a x2 = I -[- a o:^, and
[378p] since y/x2-|-2/2 = p, and a? = p. sin. d = sin. 5, nearly, we shall have p=l-|-a.sin.^^, nearly.
In using the focus S, we have found ror SP = a — ex, [378A-] ; if the focus s had
been used, e would have been negative, and sP=a-\-ex; the sum of these two expressions
[378^] is SP-\-s P=2 a = AB, a noted property of the ellipsis. When x=0, the preceding
[378r] value of S P=^a — ex, will correspond to the point H, and will become S H=a= C A.
The whole ordinate 2 y, corresponding to the focus S, is called the parameter of the curve.
At this point x becomes a e, [378e], and the value of 2y = 2. v/l — eS. ^a^—x% [378/]
[3785] becomes 2 a . ( 1 — e^) . If we represent this by 2 p, we shall have, p = a.{l — e^) = — ,
[378m].
Upon the diameter A B describe the semi-circle AP' H' B, to meet the ordinates FP,
C H, continued in P' and H'. Put FP'^='i/=^ \/a2— a;2, and since y=-. v/^aZI^,
[378f] [378n], we shall find ?;=-> or Yf^^'ch' ^'^^^ ^^ ^ property sometimes used to
define the ellipsis. This value of y = - .y', gives fydx = -.fy'dx, whence it follows
that the area of the elliptical segment A P F, is to the area of the corresponding circular
segment AF F,Sishto a, also the area of the semi-ellipsis A HB is equal to the area of
I 378m] the semi-circle AH' B multiplied by -, that is ^ -i- . a^ x - = J * . « J, * bemg the
semi-circumference of a circle, whose radius is unity. Therefore the area of the whole
[378i;] ellipsis is * . a & = <»•. a^ . \/r^ [378w].
II. i. § 2.]
DEDUCED FROM OBSERVATION.
246
This equation becomes that of a parabola, when c = 1 and a is infinite ;* [379/1
and it corresponds to an hyperbola, when e exceeds unity, and a is negative.
This equation givesf
dv^
ar.(l— e2) r' t a2.(l— e^)'
[379]
The solidity of an ellipsoid of revolution about the axis AB is represented by * .fy^ dx,
because the area of the circle described by the radius FP, during this revolution is ■n' y^.
4 ^
In like manner * . fyf^dx, represents the solidity of the sphere, -— • . a^, [275&], described
by the revolution of the semi-circle AH'B, about the same diameter. Now smce y= — . y',
we shall have ir . fir^ . dx=:—-. ir. fy'^ . dx^=: — .-^ — =— . a
solidity of the ellipsoid, formed by the revolution of the
ellipsis, about the transverse axis 2 a, is represented
by -—.ah^', and if the revolution be about the
o
conjugate axis 2&, the solidity would be represented
by -^.a^h.
* (272) Substitute D ='a . (1 — e), [378/], in the numerator of [378], it will become
Z).(l + e)
for the ellipsis r = — —
parabola r
Put
e.cos. (« — •n)
in which a =
1, and we shall get the equation of a
D
shall have as before
D
l-f-cos. (t> — TH) '
D.(l+e
1— e
, is infinite. If e exceed unity, we
corresponding . to an hyperbola, in which case
l4-e.cos.(v — xa)'
a = -—■ — , [378/], becomes negative.
Put AF=x', or x = a — a/, in the equation of the ellipsis, [378n], and it will become
1, hence t/2 = — .(2a/ ); and if we substitute for — its
^ a \ a J' a
(a — 3ff . y2
value p, [378s], also -=
l — e
D
, [378/] we shall get for the ellipsis,
f^p.(^2a/^^^.x'^y
In the parabola, where e = 1, it becomes y^= 2p a/.
In the hyperbola, where e exceeds unity, it becomes i/^ = p . ( 2 a;' -j jr — . a/^ j.
t (273) From [378] we get, -= 't^-^^-J"—'^). ^^^q^q differential divided by — dv,
d r e. sin. [v — iJ)
— ^ This being squared, putting sin.^ {v — «) = 1 — cos.^ (v — ©),
[378w]
[379a]
r^dv a.(l— c2)
[3796]
[379c]
[379rf]
[379e]
62
246 LAW OF GRAVITATION, [M^c. Gel.
consequently*
f«»] ' = a.(l_e=)-^!
therefore, the orbits of the planets and comets being conic sections, the force
[SSC] (p will be inversely proportional to the square of the distance of the centre
of the planet from the centre of the sun.
i We also perceive, that if the force 9 be inversely proportional to the square
of the distance, or expressed by — , h being a constant coefficient, the
preceding equation of the conic sections, will satisfy the differential equation
(4) [377] between r and v,-f which gives the, expression of the force, when
h
we change 9 into -^. We shall then have
[380"] h=: f ,
a. (1 — e^)
(lr'2 e2 — e2.cos2(i; — •sJ) ^ i • • • i r i i ,
we eet — — -— = — — . bubstitutins; in the numerator 01 the second member,
e . COS. (v — tjj) = — 1-1 — '- , it becomes e^ — 1 -j '■ ' — ,
or (1 — e^) • ) — 1 H '~~z ( • Dividing the numerator and denominator by
2a a2.(l — e2)
d r2 — "T" ^
1 — e\ we find, — — — -= 'L.- _I , which is easilv reduced to the
r^ di^ a2.(l— e2)
form [379].
* (274) The differential of -^^, [379], divided by dr is ■-— —-] — -.
Substitute this in [377], it becomes as in [380].
f (275) If we substitute ^ = —r i" [377], multiplied by — dr, we shall get by
7-2
C2
[380^
,„j integration -^=^3+-|-.^;—^ I +constant. Put this constant equal to —-^-^j-^^,
c2 c2
and h = — ; divide by — , and it will become by transposition,
dr2 2 1 1
r^difi ar. (l — e2) r2 a2.(l— c2)'
as in [379], which was found above to be the differential of the equation [378], correspond-
ing to the conic sections.
II. i.§3.] DEDUCED FROM OBSERVATION. 247
which forms an equation of condition between the two arbitrary constant
quantities a and e of the equation of conic sections ; the three arbitrary
constant quantities «, e, «, of this equation, will thus be reduced to two
distinct arbitrary constant quantities ; and as the differential equation between
r and v is only of the second order, the finite equation of the conic sections [380"]
will be the complete integral.
Hence it follows, that if the described curve be a conic section, the force
will be in the inverse ratio of the square of the distance ; and conversely, [380^^]
if the force be in the inverse ratio of the square of the distance, the
described curve will be a conic section.
3. The intensity of the force 9, relative to each planet and comet, depends
on the coefficient — -. 5^ [380"! ; the laws of Kepler furnish the means
a . ( 1 — &^) ^
of determining it. For if we put T for the time of revolution of a planet ; [380 v]
the area which its radius vector would describe during that time, would be
equal to the surface of the planetary ellipsis, represented by * . a^ . v 1 — e^ [380"1
[37 8?;], It being the ratio of the semi-circumference of a circle to its radius ;
but by what precedes [365', SQ^I^ the area described during the instant dt
is ^ cdt \ the law of the proportionality of the areas to the times, will
therefore give this proportion.
hence we deduce
[381]
[382]
With respect to the planets, the law of Kepler, according to which the
squares of the times of their revolutions, are as the cubes of the transverse
axes of their ellipses, gives T^ = ]<^.a^, k being the same for all the planets ; [382']
therefore we shall have
C = ^ L; [383]
2 a . (1 — e^) is the parameter of the orbit [37 85], and in different orbits, [383']
the values of c are as the areas described by the radius vector in equal
248 LAW OF GRAVITATION, [Mec. Cel.
times ;* these areas are therefore as the square roots of the parameters of
[383''J the orbits.
This proportion takes place also in comparing the orbits of the comets,
either with each other, or with those of the planets ; this is one of the
fundamental points of their theory, and it agrees exactly with all their
observed motions. The transverse axes of their orbits, and the times of
their revolutions being unknown, their motions are calculated in a parabolic
[383"'] orbit, denoting the perihelion distance by Z), and puttingf c = — '—— ,
which is equivalent to making e equal to unity, and a infinite, in the preceding
expression of c ; we shall therefore have, with respect to the comets,
[383«^] T^ = J(^a^; whence we may find the transverse axes of their orbits, when the
times of their revolution are known. Now, the expression of c [383] gives
[384] ^ = ZI_ ;
0.(1— e2) Ic' '
therefore we shall havej
4*2 1
[385] ^ = ^-^-
The coefficient -j^ being the same for all the planets and comets, it
follows that for each of these bodies, the force 9 is inversely proportional to
the square of the distance from the centre of the sun, and that it varies
[SSST from one body to another, only by reason of these distances ; whence it
* (277) Putting A for double the area described in the time t, we shall have, [365', 366],
dA==cdt, whose integral is A = ct, A being supposed to commence with t. Let A', </,
[383a] be the values of A, c, corresponding to another planet ; then A' ^=cft, consequently,
A:A'::ct:c't::c:cf.
t (278) Substitute a.{l—e) = D, [378/], in [383], and we shall find,
2*. ;/«•{!— «)•(! + «) 2*.v/i).(l-fe)
c = 1 = 1 '
and in a parabola, where e = 1 , [378'], it becomes c = — '-^
J (279) By substituting the value of ^J^^ir. [3S4] in [380].
n.i.§4.] DEDUCED FROM OBSERVATION. 249
follows that it would be the same for all those bodies, supposing them at
equal distances from the sun.
We are thus induced, by the beautiful laws of Kepler, to consider the
centre of the sun as the focus of an attractive force, which extends infinitely
in every direction, decreasing in the ratio of the square of the distance.
The law of the proportionality of the areas described by the radius vector
to the times of description, shows that the principal force acting on the [385"]
planets and comets, is always directed towards the centre of the sun ; the
ellipticity of the planetary orbits, and the almost parabolic orbits of the
comets, prove that, for each planet and comet, this force is inversely
proportional to the square of the distance of the body from the sun ; lastly,
from the law of the proportionality of the square of the times of revolutions,
to the cubes of the great axes of the orbits, or from that of the proportionality [385"']
of the areas described in equal times by the radius vector, in different
orbits, to the square roots of the parameters of the orbits, which law
comprises the preceding, and extends to comets ; it follows that this force is
the same for all the planets and comets, placed at equal distances from the
sun, so that in this case, these bodies fall towards it with the same velocity. [385*"]
4. If from the planets we pass to the satellites, we shall find that as
the laws of Kepler are very nearly observed in the motions of the satellites
about their primary planets, they ought to gravitate towards the centres of
these planets, in the inverse ratio of the square of their distances from those
centres ; the satellites ought likewise to gravitate towards the sun in nearly [385 v]
the same manner as their planets, in order that the relative motions about
their primary planets may be very nearly the same as if these planets were
at rest. The satellites are therefore attracted towards the planets and
towards the sun, by forces inversely proportional to the squares of the
distances. The ellipticity of the orbits of the three first satellites of Jupiter [385^^]
is small, but that of the fourth is very sensible. The great distance of
Saturn has hitherto prevented the discovery of the ellipticity in the orbits
of any of its satellites except the sixth, which is sensibly elliptical. But
the law of gravitation of the satellites of Jupiter, Saturn, and Uranus, is most
apparent in the ratio of their mean motions, to their mean distances from the
centres of their planets. Which ratio for each system of satellites is, that [sssvii]
the squares of the times of their revolutions are as the cubes of their mean
63
250 LAW OF GRAVITATION, [Mec. Cel.
distances from the centre of the planet. Suppose therefore that a satellite
describes a circular orbit, with a radius equal to that of its mean distance from
the centre of its primary planet ; let this distance be «, and T the number of
seconds contained in its sidereal revolution, ir being the ratio of the semi-
'2, n If
[385'i»] circumference of a circle to its radius ; -— will be the small arch which the
satellite describes in a second. If it was not retained in its orbit by the
attractive force of the planet, it would fly off, in the direction of the tangent,
increasing its distance from the centre by a quantity equal to the versed sine
[385«] of the arch -=^, which is,* ; the attractive force causes it
therefore to fall towards the planet by the same quantity. Relative to
another satellite, whose mean distance from the centre of the planet is a',
and T' the time of revolution in seconds, the fall in one second would be
2 a'lt^
[385^] ; now if we put 9 and 9 for the attractive forces of the planet at the
distances a and «', it is evident that they are as the spaces fallen through in
a second ; therefore we shall have
, 2a'K^ 2a''K^
[386] ^'^''-jir--T7V'
' The law of the squares of the times of revolution, proportional to the
cubes of the mean distances of the satellites from the centre of their planet,
gives
[387] 7'^: T'^iia^ia"";
from these two proportions it is easy to deduce
[388] oicp':: — :-^;
therefore the forces 9 and <?' are inversely proportional to the squares of the
distances a and a'.
* (280) The versed sine of an arch of a circle, is equal to the square of the
corresponding chord divided by the diameter, and the chord of a very small arch — =— . is
nearly equal to this arch. The square of this arch, divided by the diameter 2 a, gives the
versed sme , as above.
J.
n. i. §5.] DEDUCED FROM OBSERVATION. 261
5. The earth having but one satellite, the ellipticity of the lunar orbit is
the only celestial phenomenon, which would lead to the discovery of the law
of the attractive force ; but the elliptical motion of the moon is very sensibly
affected by the disturbing forces, which would leave some doubt about the
law of the diminution of the attractive force of the earth, in the ratio of [388']
the square of the distance from its centre. However, the analogy which
exists between this force, and the attractive forces of the Sun, Jupiter,
Saturn, and Uranus, leads us to believe, that it follows the same law of
diminution ; but the experiments on gravity upon the surface of the earth,
afford a direct method to verify this law.
For this purpose we shall investigate the parallax of the moon, from the
experiments of the length of a pendulum vibrating in a second, and shall
compare it with astronomical observations. On the parallel on which the square
of the sine of the latitude is ^, the space through which gravity causes a heavy
body to descend in a second of time, is, according to the observations of the [388"]
length of the pendulum, equal to S"", 65648, as we shall see in the third
book ;* we have chosen this parallel, because the attraction of the earth on
the corresponding points of its surface, when compared with that at the
distance of the moon, is very nearly as the mass divided by the square of the [388"']
distance from the centre of gravity of the earth.f On this parallel, the force
* (281) The formula given in Book III, [2054], for the length of this pendulum, is
0"',739502 + 0'",004208 . (sin. lat.)2, and if (sin. lat.)2= J, it becomes 0™,740905.
Putting this = r, and T = 1", in the theorem T=ir . l/ -, [86], we obtain
g = <if^.r= 7'",31244. The space z, fallen through in one second of time, by the force
of gravity is [67], z = ^g t^, and by putting t= 1", it becomes equal to ig, or 3'",65622,
which differs a little from the above, but it will be unnecessary to revise the calculation,
as the whole is to be considered merely as an approximation.
f (282) That the attraction of the earth is nearly as its mass, divided by the square of
the distance of the moon from its centre of gravity, is proved in [470'"]. Suppose now that
the earth is a homogeneous ellipsoid of revolution, whose polar semi-axis is denoted by J=l,
4flr
its equatorial semi-axis a = 1 + « ; its solidity will be -— . (1 -j- «)", [378t^], and if we
o
put this equal to the area of a sphere, whose radius is p, which is — . p^, [2756] ; we shall
[388iv]
252 LAW OF GRAVITATION, [Mec. Cel.
of gravity is less than that depending on the attraction of the earth, by two
thirds of the centrifugal force, corresponding to the rotatory motion at the
equator ;* this force is — of gravity ; we must therefore increase the
preceding space by its ^ part, to obtain the vrhole space arising from the
attraction of the earth, which on this parallel is equal to the mass divided by
the square of the radius of the earth : we shall therefore have 3™, 66394 for
this space. At the distance of the moon, it ought to be diminished in the
ratio of the square of the radius of the terrestrial spheroid, to the square of
the distance of the moon from the earth ; and it is evident that this is effected
by multiplying it by the square of the sine of the moon's parallax ;t putting
[388vi] therefore x for this sine, corresponding to the parallel under consideration, we
shall have a^ . 3"*, 66394, for the space the moon ought to fall through, by the
attraction of the earth, in one second of time. But we shall see, in the
get p3 = (l-|-a)^, hence p=l-f-§a5 nearly. Putting this equal to the expression of
[388a] p = 1 + a . sin.^ 6, [378p], we get sin.^ ^ = §, hence cos.^ ^ = h ^ being very
nearly equal to the complement of the latitude of the place whose radius is p. Therefore the
mass of the ellipsoid is equal to the mass of a sphere described with a radius equal to tliat of
the ellipsoid, in a latitude whose sine is equal to \/Y'
* (283) Referring to the figure in page 229, we find that the centrifugal force, resolved
in the direction of the radius ID, is n^ . R . sm.^ 6, [352Z»]. At the equator, where
sin. 6=1, it becomes n^ . R. If we suppose this to be -^\-g of the attractive force A
of the earth, at the equator, [1594a], the actual force of gravity g, at the equator, will be
^ = |||../3, hence «^ = f||-^, and the centrifugal force at the equator ^^^ becomes
2-|^ = n^ .R'j therefore the preceding expression of the centrifugal force, in the direction of
the radius ID, will be g^ . sin.^ 6 j and since, in the case now under consideration, [388a],
we have sin.2 4 = §, this will become ^^'% = :^, as in [388'^]. Adding to 3,65548,
[388"], its ^42- part, or 0,00846, the sum becomes 3.66394, as in [388^]. We may
observe that in all the calculations, relative to the figure of the eartli, in this work, terms
of the order a^, are generally neglected, and for this reason the centrifugal force might be
[3886J taken indifferently for -^^-^.A, or ^-^-g, without departing from the usual limits of
accuracy.
f (285) This corresponds with the usual rule for finding the horizontal parallax of any
body, seen from the earth's surface, by saying, as the distance of the observed body from the
centre of the earth, is to the earth's semi-diameter, so is radius to the sine of the horizontal
parallax, nearly.
U. i. §5.] DEDUCED FROM OBSERVATION. 253
theory of the moon, that the action of the sun diminishes its gravity towards
the earth, by a quantity, whose constant part is the ^ part of gravity ;*
moreover, the moon, in its relative motion about the earth, is acted upon by
a force equal to the sum of the masses of the earth and moon, divided by
the square of their distance from each other ; we must therefore diminish the
preceding space, by -^y and increase it in the ratio of the sum of the masses
of the earth and moon to that of the earth ; now we shall see in the fourth
book, that the phenomena of the tides give the mass of the moon equal to
* (286) A student in astronomy, who has not examined the calculations of the lunar
theory, had better pass over this, and assume, with the author, that the decrement of gravity,
arising from the sun's disturbing force is .j^^ part. This may be safely done, as the present
calculation is not used for any other purpose in the rest of the work. After reading the
theory of the moon's motion in Book VH, the subject may be again resumed, and this
decrement of gravity may be investigated in the following manner.
If we represent the masses of the earth, moon, and sun, by M, m, m', respectively, the
quantity [j~)i [499rt], will represent the force acting on the moon m, in the direction of the
radius vector r of her relative about the earth. From the general value of Q, [4806], we
may obtain the mean value, required in the present calculation, by neglecting the terms
depending on the angle v — v', and its multiples, which nearly destroy each other in every
revolution ; we may also neglect the terms depending on the tangent of the moon's latitude s,
on account of their smalbess, by which means [4806] will become, Q^ = u-\-m'y! -\ — ,
and since by neglecting s^ we have m=-, m'= — , [4776, 4779'], we shall get
Q=-+- +773-; hence (^— J=— -. ^ 1 ^^^ ^ . If the sun did not disturb
the motion, or m! = 0, this would become ^. The ratio of the former expression to
the latter is represented by 1 ; therefore the gravity g of the moon towards the
earth, is decreased by the sun's disturbing force, a quantity equal to — '■ — . g, nearly ; and
if we use the mean values r = a, r' = a', [4791], it is — ~ .g. Substitute — =1>
[4795], and — = m^, [4794], it becomes -^ .g-; and since m = 0,0748013, [5117],
it is nearly -^.g = :^-g, as in [388"'].
64
[388c]
[388d]
[388^^
264 LAW OF GRAVITATION, [Mec. Cel.
1 357 59 7
[sssviii] of that of the earth :* therefore we shall have -— - . — ^ .a:^.3'",66394,
•■ ■■ 58,7 358 58,7 '
for the space through which the moon falls towards the earth in one second
of time.
Now, if we put a for the mean radius of the moon's orbit, and T for the
[388i^] number of seconds in the time of its sidereal revolution, „ will be, as
1
we have shown [SSS'''], the versed sine of the arch described in one second
[388=^] of time. This expresses the space through which the moon falls towards the
earth in that time. The value of a is equal to the radius of the earth under
the parallel of latitude just mentioned, divided by x ; this radius is equal tof
6369514™ ; therefore we have
6369514™
[389] a = ;
but to obtain the value of a, independent of the inequalities of the motion
of the moon, we must take for its mean parallax, whose sine is a:, the part
of the parallax which is independent of those inequalities, and which, for
[389'j ^^^ reason, is usually called the constant term of the parallax. Hence, by
['^89'] taking for * the ratio of Sbb to 113, and for T its value 27321 66",t the
* (289) In [2706], the disturbing forces of the moon and sun on the tides, are found to
be nearly as 3 to 1 ; and from this, in [4321], the mass of the moon was found to be -g^^^^,
of that of the earth, which nearly agrees with the above. Further observations induced the
[389a] author, in [4631], to change this into -^^ ; and afterwards in Book XIII, § 9 to y'^, nearly ;
making the force of the moon on the tides to that of the sun as 2,35 to 1, nearly.
f (290) Using the ellipticity ^, given by the author, [2034], we get in [20356], the
polar semi-diameter =6356676"*, and the equatorial semi-diameter =6375709™, their
difference being 19033™. Now the decrement of the radius, in proceeding from the equator
to the pole, bemg nearly as the square of the sine of the latitude, [378p], the decrement
corresponding to the latitude whose sine is \/j', vdll be 19033^X^ = 6344"*, which
subtracted from the equatorial semi-diameter, leaves the radius of that latitude 6369365*",
which is rather less than that above given ; the difference may have arisen from using
another ratio of ellipticity ; this however has but a very littie effect on the result of the
calculations.
J (291) This is the time of a sidereal revolution of the moon in seconds, corresponding to
27''7*43"* ll'',4.
[3896]
II. i.§6.] DEDUCED FROM OBSERVATION. 255
[390]
mean space through which the moon falls towards the earth, will be
2 . (355f . GsegsM™
(113)2. a; . (2732166)2*
Putting the two expressions [SSS""'", 390] of this space equal to each other,
we shall have
,_ 2.(355)2.358.58,7.6369514
~ (11 3)2 . 357. 59,7 . 3,66394 . (2732166)^ ' ^^^^
whence we deduce 10536",2, for the constant term of the moon's parallax, rggj^
under the parallel of latitude before mentioned. This value differs but very
little from* 10540",7, computed by Triesnecker, from a great number of
observations of eclipses and occultations of stars by the moon ; it is therefore
certain that the principal force which retains the moon in her orbit, is the
attraction of the earth, decreased in the duplicate ratio of the distance ; thus r^y,
the law of the diminution of gravity, which, for the planets accompanied by
several satellites, is proved by the comparison of the times of their revolutions,
and of their distances, is demonstrated for the moon by the comparison of
her motion with that of projectiles on the surface of the earth. Hence it
follows, that we must fix the origin of the distances, at the centre of gravity ^ ^
of any heavenly body, in computing its attraction upon bodies placed upon
its surface, or without it ; since this has been proved to be the case with
respect to the earth, whose attractive force is of the same nature as that of
the other heavenly bodies, as we have shown.
6. Hence it follows that the sun, and the planets which have satellites,
are endowed with an attractive force, extending infinitely, decreasing
inversely as the square of the distance, and including all bodies in the sphere
of their activity. Analogy leads us to infer that a similar force exists [391"]
generally in all the planets and comets ; and it may be proved in the following
* (293) The constant term corresponding to the equator in Burg's tables, is by
Book Vn, §26, [5603], equal to 10558",64. The decrement for any other latitude,
(sin. lat.^
according to Burg, [5604], is found by multiplying this by — ' . In the present case,
(sin. lat.)2 = J, hence the decrement is 10",66, and the constant term becomes 10547",98,
differing a few seconds from the above. The calculation of the parallax in [5331], differs
a few seconds from this.
256 LAW OF GRAVITATION, [Mec. C61.
manner. It is an invariable law of nature, that a body cannot act on another,
without experiencing an equal and contrary reaction ; therefore, since the
planets and comets are attracted towards the sun, they must in like manner
attract that body. For the same reason the satellites attract their planets ;
this attractive property is therefore common to the planets, comets and
[391 V] satellites ; consequently we may consider the gravitation of the heavenly
bodies, towards each other, as a general law of the universe.
We have shown that this law follows the inverse ratio of the square of
distances. It is true, that this ratio was deduced from the supposition the
[SQl^i] of aperfect elliptical motion, which does not rigorously accord with the
observed motions of the heavenly bodies. But we ought to consider that the
most simple laws should always be preferred, until we are compelled by
observation to abandon them. It is natural at first to suppose that the law
of gravitation is inversely as a power of the distance ; and we find by
calculation that the slightest difference between this power and the square,
[39i'»] would become extremely sensible in the position of the perihelia of the
planetary orbits,* in which, however, no motions have been discovered by
observation, except such as are very small, the cause of which will be
explained hereafter.f In general we shall see, in the course of this work,
that the law of gravitation, in the inverse ratio of the square of the distances,
represents with the greatest precision, all the known inequalities of the
motions of the heavenly bodies ; and this accordance, taken in connexion
with the simplicity of the law, authorizes the belief that it is rigorously the
law of nature.
Gravitation is proportional to the masses ; for it follows from ^ 3 [385"''],
[39iviii] that if the planets and comets are supposed to be at equal distances from the
sun, they would fall freely towards it through equal spaces in equal times ;
* (294) This is very sensible in the motion of the moon, which would move in a fixed
ellipsis, if the moon was aflected only by the mutual attraction of the moon and earth.
But the disturbing force of the sun, which is about ^^ of that of their gravity towards each
other, [388""], produces a motion of the perigee of nearly 40*^ in a year, as is easily proved
from the value of c. Book VB, § 16, [51 17].
f (295) As in Book VI, § 25, where d^i is determined for tlie planets, and in Book VQ,
§ IC, for the moon, &;c.
n.i. §6.] DEDUCED FROM OBSERVATION. 257
consequently their gravities would be proportional to their masses. The
motions of the satellites about their primary planets, in nearly circular orbits,
prove that the satellites gravitate, like the planets, towards the sun, in the
ratio of their masses ; the slightest difference, in this respect, would be
sensible in the motions of the satellites ; but no inequality depending on this [39l'»]
cause has been discovered by observation.* Hence we see that the comets.
* (296) To point out the effect of this difference in
the attraction, let us suppose that a very small body, or
particle of matter revolves about the sun S, in an elliptical
orbit ab cd, whose transverse axis is a c, and one of its
foci S ; and that another similar body or particle revolves
about iS as a centre, in the circular orbit AB CD, whose
diameter .^ C is equal to a c. Then as the mean distances
from the sun are equal, their times of revolution, by
Kepler's law, will be equal, [382'] ; neglecting the mutual
attractions of the two revolving bodies, and the sun will attract them equally at equal
distances, [SSo'""]. Now if we suppose both the bodies to revolve in the same direction,
the one of tliem being at A, when the other is at a, the distances of the two bodies will
always be of the same order as the quantity A a ; and if .^ « be small in comparison with
S A, as for example ^^ part, the two bodies will be somewhat similarly situated to that of a
primary planet and its satellite ; the primary being at A, B, C, D, when the secondary is
at a, 5, c, d, respectively ; the distances A a, Bb, C c, Dd, being of the same order as
j^ part of the distance S A. The satellite will be in conjunction with the sun at a, in
opposition at c, and in tlie quadratures at b and d; and the same will happen in the successive
revolutions of the bodies. And it may not be amiss to notice, particularly, that in this case L^^la]
one of the bodies would appear to revolve about the other, as a satellite, without being in the
least attracted by it ; the motion being maintained wholly by the sim's attraction. Suppose
now that the action of tlie sun on the planet is less than on the satellite, when at the same
distance, by ■j-^\-^ part ; so tliat instead of falling through the space, or versed sme g", in a
second of time, when at the point A, it should only fall through the space g — x/o o • ^ diis
case, to make the planet continue in its orbit it is necessary to decrease its velocity, in the
proportion of [/g to Vg — xoi^o, or nearly, as 1 : 1 — g oVo J because the versed sines of
small arcs are nearly as tlie squares of the arcs. The time of describing the circle AB CD^ [3916]
must in this case be varied in the inverse ratio of the velocities, and it must therefore be
increased about -^^^ part ; consequently, at the end of one revolution, when the satellite has
arrived at a, the planet will be at A\ a litde short of A ; and this distance will increase in the
65
258 LAW OF GRAVITATION, [Mec. Cel.
planets and satellites, placed at the same distance from the sun, would
gravitate towards it, in the ratio of their masses ; and as action and reaction
are equal and contrary, it follows that they attract the sun in the same ratio ;
[391 »] consequently their actions on the sun are proportional to their masses divided
by the square of their distances from its centre.
The same law is observed upon the surface of the earth ; for it has been
found by very exact experiments, made with a pendulum, that if we neglect
the resistance of the air, all bodies would fall towards the centre of the earth
with an equal velocity. Such bodies gravitate therefore towards the earth,
in proportion to their masses, in like manner as the planets gravitate towards
the sun, and the satellites towards their primary planets. This perfect
conformity in the operations of nature, upon the surface of the earth and in
[39lxii] the immensity of space, proves, in the most striking manner, that the gravity
observed upon the earth, is only a particular case of a general law extending
throughout the universe.
The attractive force of any one of the heavenly bodies does not appertain
[391«"] exclusively to its aggregated mass, for the property is common to each
component particle. If the sun acted only on the centre of the earth,
without attracting each of its particles, there would result, in the ocean,
incomparably greater and extremely different oscillations, from those now
observed ; the gravity of the earth towards the sun results therefore from the
gravitations of all the particles of the earth ; consequently these particles
must also attract the sun, in the ratio of their respective masses. Moreover,
each body upon the surface of the earth gravitates towards the centre of the
earth, in proportion to the mass of the body. It therefore reacts on the
earth, and attracts it in the same proportion. If this were not the case, and
any part of the earth, however small it might be, did not attract the rest of
successive revolutions. At the end of about 1000 revolutions, when the satellite is at a, the
planet will be at C. Thus we see that by only varying the gravity y^oZ P^^^j ^^ would have the
effect to increase the distance of the bodies so much, that they could no longer be considered
as a planet and satellite. If we had not supposed the velocity of the planet to be decreased,
the circular orbit AB C D, would have become elliptical, and its greater axis would have
exceeded AC or ac, consequently the periodical time of revolution would have been
increased, and a similar effect, in the separation of the two particles, would have been
produced.
n.i. §6.] DEDUCED FROM OBSERVATION. 269
the earth, in the same manner as it is attracted ; the centre of gravity of the
earth would be put in motion, by gravity, which is impossible.*
Observations of the heavenly bodies, compared with the laws of motion,
lead therefore to this great principle of nature, namely, that all the particles of
matter attract each other in the direct ratio of their masses, and the inverse [391*']
ratio of the square of their distances. And in this universal gravitation we
perceive the cause of the perturbations of the motions of the heavenly
bodies. For the planets and comets, in obeying their mutual attractions,
must vary a little from the elliptical motion, which they would exactly follow,
if they were attracted only by the sun. The satellites, disturbed in their [391"']
motions about their planets, by their mutual attractions, and by that of
the sun, vary also from these laws. We find also, that the particles of
each heavenly body, united by their attraction, ought to form nearly a
spherical mass ; and the resultant of their attractions on the surface of the
body, ought to produce all the phenomena of gravity. We also perceive that [391""]
the rotatory motion of the heavenly bodies must produce a small change in
their spherical form, by compressing the poles, and then the resultant of the
mutual attraction of the particles, will not pass exactly through their centres [391*^"']
of gravity ; in consequence of which there will arise, in their axes of rotation,
motions similar to those discovered by observation. Lastly, we see that the
particles of the ocean, being unequally attracted by the sun and moon, ought
to have an oscillatory motion, similar to the flux and reflux of the tide. But [391*"]
the development of these effects of the general gravitation of matter requires
a profound analysis. To embrace this subject in the most general manner, •
* (297) To illustrate this, let DBEF be a meridian of the
earth, divided into two unequal parts, B D E, D F E, by a plane
passing through the line D E, perpendicular to the plane of the
figure. Through the centre C draw CAB, perpendicular to D E.
Suppose now the larger part DFE, attracts the smaller part in the
direction B C, with a force represented by F-\-f', and that the
part DBE attracts the larger part with the force F only. These forces will not balance
each other ; on the contnary, the resuUant will be the force f, acting in the direction B C,
consequently, in this hypothesis, the earth would acquire a motion, in the direction B C, by
the mere force of the mutual attraction of its particles, which is absurd.
260 LAW OF GRAVITATION. [Mec. Cel.
we shall give the differential equations of the motion of a system of bodies,
obeying their mutual attractions, and shall investigate such rigorous integrals
as can be obtained. We shall then, in finding the integrals by approximation,
make use of those simplifications which depend on the ratios of the distances
[391 «] and masses of the heavenly bodies ; and shall carry this approximation to
such a degree of exactness, as shall be necessary to determine the phenomena
of the heavenly bodies with the accuracy required by observations.
n.a. §7.] MOTION OF A SYSTEM OF BODIES. 261
CHAPTER II.
ON THE DIFFEEENTIAL EaUATIONS OF THE MOTIOX OF A SYSTEM OP BODIES SUBJECTED TO THEIK
MUTUAL ATTEACTIONS.
7. Let w, m', m", &c., be the masses of the diflferent bodies of the system,
considered as so many points ; let a:, y, z, be the rectangular co-ordinates
of the body m; a/, y, zf, those of m' ; &c. The distance from m' to m
being [118]
V/ (a/-a;)^+ (y'—yyJr(^ - zf, [392]
its action on m will, by the law of general gravitation, be equal to
[393]
If we resolve this force in directions parallel to the axes of a:, y, z, the .^g^
force parallel to a:, in a direction opposite to the origin of these co-ordinates^
will be*
m' .{a/ — x)
{{^'-^f+W-yf + i^'-^W]^ ' ^^^
or
I \d.
-ZT'i K^^'-^fi-iy'-yr + i^-zf}' [^^^
m
dx
We shall in like manner have
1 \d.
sj {^' — ocf^{i^'^yf-\-{7!'—zf> [3^]
dx
* (297a) This is deduced from the formula [13], S,- -, by writing for S and jj, the
values [393, 392] ; also changing x into a/, and a into x. The expression [395] is evidently
equivalent to [394], as will appear by developmg the differential relative to d.
262 MOTION OF A SYSTEM OF BODIES, [Mec. Cel.
for the action of ml' on m, resolved in a direction parallel to that of the axis
of a:, and in the same manner for the rest. Suppose therefore
m .m , m.m
X = -:= ^Z3 — = +
[397]
Vi^-cof + i^f-yf + i^ — zf ^{a^'-.xf + {f-yf+{z"-zf
m'.m:'
+ &c. ;
[397']
X being the sum of the products of the masses m, m\ m", &c., taken two by
two, and divided by their respective distances, — • ( T~ ) ^^ express the
sum of the actions of the bodies m', m", &c., on w, resolved parallel to the
axis of ar, in a direction opposite to the origin of the co-ordinates. Putting
therefore dt for the element of the time, considered as constant, we shall
[397"] have, by the principles of dynamics, explained in the preceding book,*
-, ddx /dx^
We shall likewise have
d f \d yj '
^^^ ^ ddz /d-K\
d t^ \d z)
S" If we consider in the same manner, the action of the bodies w, m", &c., on
of the vf^ . that of the bodies m, m', &c., on w", and in the same manner for the
motions of ' 7777
ofTodlTs rest ; we shall have the equations
referred to
dt^ \dx'J' df \dy J df \d2fj'
[400] o-m" ^^^'-.?^^^. 0-m"^^^'-^f~^' 0-m"^^--f—\
^-"^"d^ \j^)' ^-'""-dw \d/)' ^-'^' dt^ \j7y
&:c.
The determination of the motions of m, unl, rnl', &c., depends on the
integration of these differential equations ; but this has not yet been done
* (298) The equations [398, 399, 400], are found by putting in [38], for F, Q, R, &c.
their values,
_1_ (^ . _L (^ . JL (^ . J- (^\ • Sec
m'\dx)' m'\dyj' m'\dzj' m' ' \d afj '
n. ii. § 8.] REFERRED TO A FIXED POINT. ^QS
completely, except in the simple case where the system is composed of only
two bodies. In the other cases, there have been obtained but a few rigorous
integrals, which we shall now investigate.
8. For this purpose, we shall first combine the differential equations in x,
x', xf', &c., by adding them together, observing that by the nature of the
function x, we have*
, „ , ^ ddx T ,., ^ ddy ., ddz
we shall have 0 = ^.m.-r-^. in like manner 0 = 2.m.-=-4: 0=2.7».-— r-. mi']
dt^ dt^ di^ ^ ^
Let X, Y, Z, be the three co-ordinates of the centre of gravity of the system ; [40i"]
we shall have, by the property of this centre,t
X=- — ; Y=-— ^; Z= ; [402]
2.OT 2.m 2.m
therefore we shall have
^_ddX 0=^^; 0 = ^^. [403]
dt^ ' dfi ' d^ '
whence by integration
X=a + bt; Y=a' + b't; Z =- cd' -\- h" t ; [404]
* (299) This is easily proved by taking any term of X, [397], and computing its effect on the
proposed function. Thus the term . /,,, ,,o , , „ ,,o , ,,, ;:r» affects only the terms
(^) ^^ (^)' the former is (^)= j(^._^"^;^~!Jl(^,_^,p^; the latter
J^')= |(3:^/_3/)2-|-iy^ly^4-(z^^_2^)2)r and the sum of both is equal to nothing. The [401a]
same thing occurs with all the similar terms, consequently,
f (300) As in [126, 127]. Taking the second differential of these equations [402],
divided \ d fi. and substituting in the second members 0 for 2 . m . , 2 . m . — .
2 . m . ^^ , [40r], we get [403], which, being mtegrated twice, gives [404]. Hence
we may prove, as in note 74a, page 104, that the motion of the centre of gravity is
rectilinear and uniform.
[406]
264 MOTION OF A SYSTEM OF BODIES. [Mec. Cel.
a, b, a\ h\ a", 6", being the arbitrary constant quantities. Hence we perceive
[404] that the motion of the centre of gravity of the system, is rectilinear and
Uniform uniform, and that it is not affected bv the action of the bodies of the system
motion of ' -I •' J ^ "
of%Ta"yky upott cach Other ; which is conformable to what was proved in the fifth
system, chaptcr of the first book [159'].
Let us now resume the differential equations of the motions of these
bodies. If we multiply the differential equations in y, y\ -ij'^ &c., respectively
by x^ xf, x", &c., and add these products to those formed by multiplying the
differential equations in x, a/, x!', &c, by — y, — i/, — y, &c., respectively,
we shall have
„ fxddii — yddx\ , , fx'ddij — ifddx^ , «
° = >» • (- it' ) + "^ • ( — -JF—) + ^'=-
--(g)--(^) + -
But the nature of the function x gives*
* (301) This is proved by an analysis similar to that in note 299. For by substituting in
the function y . (—j -{-y'. (^) +^^- — '^'u) — *' ' V^)'"' ^^*' ^^ ^^'^^^ depending
on the term of X, affected by mf . m", which are
they will produce, in the proposed function, the terms,
in which the terras between the braces are easily reduced to tlie form
iy'-f) ' {^"-^') - iy'-y") • i^"-^')^
which is identically nothing. The same takes place with all the other terms of X,
consequently, the function [406], is equal to nothing. Substituting this in [405], and taking
the integral, we get [407] ; changing y into z, and then x into y, we get the formulas [408]
II. ii. § 8.] PRESERVATION OF AREAS AND LIVING FORCES. ^^^
therefore we shall have, by integrating the preceding equation,
We shall find in a similar manner
, fxAz — zdx\
y dz — zdy\ ^
c
[408]
c =i:.m.
iJf y ' Preser-
"■ * y vation
of Areas.
c, c', c", being arbitrary constant quantities. These three integrals comprise
the principle of the preservation of areas, explained in the fifth chapter [408']
of the first book [167].
Lastly, if we multiply the differential equations in re, a/, a/', &c., respectively
by dx, dxf, dx", &c. ; those in y, i/, &c., respectively by dy, di/, dy",
&c. ; those in z, z', &c., by d z, dz', &c. ; and then add all these products
together, we shall have*
_ idx . d d X -\- dy . ddy -\- d z . ddz\ ,
0 = 2 . m . ^ ^ ^ ^—^ ^ — rfx, [409]
and by integration
h = ^.„,.(^Jl±M+i£\-'2y.. [410]
V d t^ J Preserva-
^ ' tion of the
h being another arbitrary constant quantity. This integral comprises the foVw!
principle of the preservation of the living forces, explained in Chapter V, [4io^
Book I, [144].
The seven preceding integrals are all the rigorous integrals which have
hitherto been discovered : in the case where the system is composed of only [4l0"]
two bodies, they reduce the determinations of the motions of these bodies to
differential equations of the first order, which may be integrated, as we
shall show hereafter ; but when the system is formed of three, or of a
greater number of bodies, we must necessarily have recourse to methods of
approximation.
*(301a) Putting for (£) .^^ + (^) -^y + (f^) • ^^ + (^) • rf^+&«^.> its [409a]
value d X.
67
266 MOTION OF A SYSTEM OF BODIES. [Mec. Cel.
r4l0"1 ^* ^^ ^® ^^^ ^^^y observe the relative motions of the heavenly bodies,
it is usual to refer the motions of the planets and comets to the centre of the
On the ^
mitiont of sun, and the motions of the satellites to the centres of their primary planets.
ItTJi^s To compare the theory with observations, it will therefore be necessary to
about one i ./ ' ^
of them, (determine the relative motions of a system of bodies about one of the bodies,
considered as the centre of their motions.
Let Mbe this last body, m, m\ m", &c., being the other bodies whose relative
motions about M are required ; let ^, n, and 7, be the rectangular co-ordinates
of iW ; ^ + x, n 4- y, 7 + z, those of m; ^-\- x', n -f- ?/, 7 + 2^? those of
[410»»] ^/^ ^^ . j^ -g gyjtjejit that X, y, z, will be the co-ordinates of m, referred to M
as a centre ; x', y', z', will be those of m', referred to the same body ; and in
[410 »] like manner for the others. Let r, r', &c., be the distances of the bodies
m, m', &c., from M; so that*
[411] r = V/"^T?T^ ; r' = v/a:'2 + y2_|_^2 .
and suppose
m . m' , m .m"
[412]
V/ {oo'-xf-\-{]f — yf^{^ — zf ^{af'-xf+{f-yf+{z"-zf
+ &c. ;
Then the action of m on Jf, resolved parallel to the axis of x, and taken in
a direction opposite to the origin, will be --j ; that of m' on M, resolved in
[412'] the same direction, will be —73- ; and in the same manner for the rest. We
shall therefore have, to determine ^, the differential equation,!
_ ddP mx
t«31 0 = ^-..-^;
* (302) The values of r [411], are easily deduced from [12], by writing ^, n, 7, for
a,h,c, and ^-\-x, n-f y, y-\-z, for a?, y, z, respectively ; r',r", &:c. are found in the
same manner.
f (303) The action of m upon M is -^. Putting this for S in the first formula, [13],
and also for s, x, a, writing r, ^-{-x, ^, respectively, we shall get — , for that force resolved
n. ii.§9.] RELATIVE MOTION. 267
we shall in like manner have
OddU my
^~d? 7^'
r.^ddj^ mz
[414]
The action of M on w, resolved .parallel to the axis of ar, and taken in a
direction opposite to the origin^ will be* ^, and the sum of the actions [414']
of the bodies m\ m", &c., cm m, resolved in the same direction, will bef
— .( -r- ) 5 therefore we shall havef
m \dxj
dd.{l+x) Mx 1_ /rfXN . f415^
rf<2 -T ^ m'\dx) '
in a direction parallel to x. In like manner, --^, —jf^, &c. will represent the similar
forces of m', m", &c. upon M. The sum of all these is 2 . — . Putting this for P, in
[38], and writing dd^, for ddx the second differential of the co-ordinates of M, we get
J J P m r
■ -rrS.— , as in [413]. In like manner the two last formulas of [38] give those
of [414].
* (304) This force — , is found in the same manner as the force -— , of the last
note, but it must be observed that in the present arrangement of the symbols, the body M
may be supposed to be nearer the origin than any of the other bodies, its attraction must
therefore tend to draw the other bodies towards that centre, and thus decrease the co-ordinates
and as the effect of m on M, was supposed positive, this must be put negative.
f (305) This is proved as in [397'], the value of X, [397] being of the same form as
in [412].
f (306) This maybe deduced from [398], by changing x, y, z, into ^-{-x,11-{-y, /+«,
respectively, as in [410'^], and mstead of the force — . ( t— )> [397'], substituting the value
m \d X /
found in [414'], — .f—-\^~. Substituting in [415] the value of -^, [413], it
becomes as in [416]. In like manner, from [414, 399] we get [417, 418].
268 MOTION OF A SYSTEM OF BODIES. [Mec. Cel.
substituting for — | its value [413] 2 . -j, we shall have
ddx , Mx , mx
1 f-2.
dt'^ r^ T^ m \dx
[416] 0 = ^ + — + ^. ---.(-]; m
ddy My my 1 /^^X'
and in like manner
[417, 0 = ^ + --^ + ..:^-^.^^^^^; (2)
[«8, 0 = ^; + :^ + . "^--lY^^Y (3)
d r r^ r^ wi \d z^
If we change successively, in the equations (1), (2), (3), the quantities m,
X, y, z, into m', a/, ^, 2:' ; m", a/', y, 2:", &c., and the contrary ; we shall
'■ ^ have the equations of the motions of the bodies m', w", &c. about M.
If we multiply the differential equation in ^, by M+2.m ; that in x by
m ; that in a;' by ml ; and in the same manner for the others ; and then add
these products together, observing that by the nature of the function x, we
have*
« = ©+(^)+^-^
we shall findf
,.«„, ^ ,Ti*- , \ dd? . ddx
[420] 0 = (M+2.m).-^ + 2.m.-^;
* (307) Proved as in [401], the value of X, [397], being of the same form as in [412].
f (308) Multiplying [416] by m, we get,
ddx , ^r mx . mx /d'k\
and the similar equation in x', is
, ddx' ,, mxf , mx /rf M
increasing the accents we shall get the similar equations in «", a/", &;c. The sum of all these,
by putting (t~) + (t^) + ^^'^^J [^^^l' becomes,
ddx , -.r i^x I "^*
This added to the product of [413], by JW"+ 2 . ot,
0 = (J)i+2.m)._i~;tf.2.— -2.m.2.— ,
II. ii. §9.] RELATIVE MOTIONS. 269
whence by integration
i = a + bt-^^^^; [421]
a and h being two arbitrary constant quantities. We shall likewise have
^ I I -LI ^ ^ 'tny
[422]
a', b', a", 6", being arbitrary constant quantities : we shall thus obtain the
absolute motion of M in space, w^hen the relative motions of m, m', &c.,
about it, are known.
If we multiply the differential equation in x [416], by
and the differential equation in y [417], by
mx — m . -^Ti-
M-
Likewise if we multiply the differential equation in ar', by
— m' i/ -{-Ti
and the differential equation in i/ by
mx — m. -^rr-. ; [424]
^Y + ^^ J:Z^ ; f425]
^'^_^'.^'^; [426]
and in a similar manner for the rest ; and then add together all these products,
observing that the nature of the function x gives*
becomes as in [420]. Integrating [420], and adding the constant quantity, {M-{- l.m) , 6,
to complete the integral, we get
(Jkf + 2 . 7w) . 6 = (JH + 2 . m) . -i. + 2 . »i . 3^.
at at
IMultiplying this by d t, and again integrating, putting {M-\- 2 . m) . a, for the new constant
quantity, we get, {M-{- 1: . m) . {a -\- h t) = {M-\- ^ .m) . ^-\- X.mx. Transposing the
last term, and dividing by M-{-l: . m, we obtain the value of ^, [421]. By changing 2,, x,
into n, y, and y, z, we find [422] from [414].
* (309) The first of these equations [427], is proved as in [405a,406] ; the second and
third as in [401a], where it is shown that 5 • ffi. (t~] = 0 5 ^d as X, [412], is symme- [427a]
trical in x, y, z, we may change in this x into y or z.
68
270 MOTION OF A SYSTEM OF BODIES. [Mec Cel.
0 = ....(g)-..,.(-);
we shall have*
_ \xddy — yddx\ S.mx ddy , J.. my ddx
[428] 0 = 2 . m . -^ ^—^^- i zr— . 2 . m . -^1- + -Tl^^ — - — • ^ • »»• -r^ ;
* (310) Put for brevity X^ ^i^"' ^= 'M^T^' ^"^^^P^^ ^^^^e by J):/+ 2 . nj,
[428a] andweshallget — JM'X+2 . mo: — X.2 . w = 0, JW" Y— 2 . my+ T. 2.m = 0.
Multiply [416], by — my-^-m Y, and [417], by ?» a? — m X; the sum of these products
will be
-{' ~r-] ^- ( — my-^mY)-{-y.{mx — mX) >
+ ( — my-{-mY).s. — -{-{mx — 7nX).2.-—
Neglecting the two terms — m x y -\- m x y, in the second line, and changing a little the
order of the terms in each line, we get,
+m.{y.'^-x.^\
, ( tnx , ^_ mx , my „ 'my')
Marking the letters m, x, y, r, without the sign 2, successively, with one, two, three, he.
accents, we shall obtain the similar equations in a/, x", a/", &;c. The sum of all these
equations will be,
TA9Qh^ inx , ^r mx , my xr „ „ ^3/
[4286] — 2.mi/.2.— +r.2.m.2. — + 2.ma;.2.^ — X.2.w.2.-^
II. ii. §9.] RELATIVE MOTIONS. 271
the integral of which is*
(xdy — ydx) l^.mx dy , I,, my dx [4001
constant = 2. w.^ — -^ — ^^^7-- — •2-^1.37 + ^ry-r — ^-.2.m.— ; ^^^
dt M-\-^.m dt M-^-'L.m dt
or
,. (xdy — ydx) . ^ , C (x — x).{dy' — dy) — (y' — y).(dxf — dx) ) ...
dt ( dt )
The lower line becomes nothing by means of [427]. The coefficients of 2 . — -,
2 . — , in the second and third lines, are respectively M Y — ^.my -\-Y .1 .m, and
— MX-\-l> .mx — X.2.W, which by [428«] are nothing ; therefore the equation
[4286], will be reduced to the terms in the first line, which are the same as in [428].
*
(311) This is easily proved by taking the dijSerential of the equation [429], divided
by <?<, and comparing it with [428]. The first term 2.m. \ evidently produces
\xddy — yddxi
the first term of [428], 2 . to . -* — —. If we neglect, for a moment, the
constant factors, Jlf -f- 2 . to, dt^^ which occur in the denominators of the two last terms
of [429], divided by d t, they become, — ^,mx.i:.mdy-\-^.my.^.mdXf the
differential of which is
— ^^.mdx.^.mdy-\-l^.mx.^.mddyj-\-ls.mdy.:^ .mdx-\-l.my .1:. mddx]^
which by reduction is — i:.mx.I^.mddy-\-i:.my.^.mddx. Resubstituting the
factors of the denominator M-\-S .m, dt^, it becomes like the two last terms of [428].
Multiply [429] by .M -(- 2 . w, and put c for the product of the constant term, by JW+2 . to,
we shall get,
-, (xdy — ydx) , (xdy — ydx)
c = M.J..mr — ^—^ — ^ + 2.TO.^ — ~-^ — ^.2. TO
dt ' dt
dy . dx
2 .TOa? . 2 . TO .-— + 2 .TO V •2.TO. -— •.
dt ^ ^ dt
Now by [189a], we have identically,
[xdy— ydx) ^_„ ^. ^ {^ — x).{d}/ — dy) — (^—y).{d3f — dx)
2 . TO . : . 2 . TO = 2i . TO TO . <— —
dt I dt
dy dx
+ 2 .too;. 2 . TO.-; 2.TOV.2.TO. .
' dt ^ dt
Substituting this in the preceding value of c, it becomes as in [430]. Changmg in this the
terms relative to the axis x, into those relative to the axis z, it becomes as in [431] ; and in
like manner, by changing the axis of x into the axis of y, in [431], we shall get [432].
[430]
272 ^ MOTION OF A SYSTEM OF BODIES. [Mec. Cel.
c being an arbitrary constant quantity. In the same manner we may obtain
the two following equations :
r^o,-, / HT ixdz — zdx) . , C (x' — x).(dsf — dz) — (s^ — z).(dx' — dx) } ,_.
[431] c'=M.s.m.- --\-^.mm'. { — —^ — ) : (b)
at ( at J
i,T (ydz — zdy) , , C (y' — y).(dz' — dz) — (z' — z).(dy' — dy) ) ,^,
[432] c"=M.^.m.- — r ^+2. mm'. \ ^ — ^^ ^-~ -^^-^ ? ; (6)
at ( at }
c', c", being two other arbitrary constant quantities.
If we multiply the differential equation in x [416], by
rjooT /-» I ^~t ^ » 171 a X
[433] Zmdx — Zm .
[434]
Jlf+2.m '
the differential equation my [417], by
_ , ^ J, .mdy
2mdy — 2m. .. , ^ ;
the differential equation in z [418], by
^ , ^ 2 . mdz
[435] 2mdz — 2m.
JVi+2.m '
in like manner, if we multiply the differential equation in x', by
[436] 2m! dx' —2m' . -^-t— — '
the differential equation in ?/', by
- , ^ . S .mdy
[437] 2^'rfy_2m'.-^-^^;
the differential equation in zf, by
2 7/1 d z
[438] 2m'^z'-2m'.-;^^.^-^;
and in the same manner for the others ; and then add together all these
products, observing that*
0 = ..©; 0 = ..(g); 0 = ..g);
we shall havef
* (312) As in [427], or in [427a].
f(3l3) The product of [416] by 2 mda? — 2m .^——, is
0 = 27ndx.l-^--\-M.-+^.---.{j^)\
^.mdx C ddx ^ X «»a; J_ /^\ >
II. ii. §9.] RELATIVE MOTIONS. 273
_ (dx.ddx-\-dv.ddy-\-dz.ddz) 2.Z.mdx m.ddx
n — 9 V m - ■ — ■ — . 2 .
U_Z.^.W. ^^2 Jtf+2.m dt^
2.J..mdy m.ddy 2.i:.mdz m.ddz ^^ mdr ^, ^^^^^
M-^-L.m' ' df Jli+2.m dt^ r^
and the similar expressions in x\ x", he. are produced by merely accenting the letters.
The sum of all the equations in x, a/, x". Sic., thus formed, will be obtained by prefixing
the sign 2 ; hence, we get
dx.ddx , ^ „, xdx , ^ mx , ^ , ^d'K\
d<3 ' r3 ' r3 \dxj
,2.2.wi.dxC ddx -, mx mx , /f?X\>
H ~ . }—:z.m ,———M.2.-— — 2.m.2.-— + 2.(-— ) f
I'll/. /.^ "*^ 7 • ^ -i -^ 2.WI") ,.,
in which the factor oi 2.2.— -.2.m.rfa?, is <1 — --— ; -—- > , which
r3 (. J)f + 2 . m M-\- 2 . m >
vanishes, because these terms mutually destroy each other; also from [439], 2 [ — ) = 0;
ddx / dx \2
lastly, the term 2 2 . m c? a; . 2 . m . -— , may be written <Z . f 2 .m. -- J ; these values
being substituted, we get
dx.ddx , ^ ,^ xdx ^ , /d\\ ""[^'^'Til
dt^ r3 \dx/ J)f+2.m
Changing successively, x into y, and into z, we shall obtain the similar products formed from
the equations [417, 418], namely,
dy.d dy
0 = 22. m.
rf<2
r3 ^ \dyj {Jlf+2.m).d<2'
rfz.rfrfz-^ zrfz J /^ ^\ d.{'S, .mdzf^
0 = 22.m. — ^t: [-2Jlf.2.m.-- 22.<;2:
■m-
dt^ ^ ' ' ' r3 ' '\dzj {M-{-z.m).dfi'
Adding these three equations together, and reducmg, by putting lor 2 2 . wi . =^-^
T d T d V
its value, [411], 2 2 . m •— ^ = 2 2 . m . — , and for
its value d X, [409a], we shall obtain the equation [440], or as it may be written,
^ {dx.ddx-\-dy.ddy-\-dz.ddz) . {{:z.mdxf-\-[^.mdyf-\-{^.mdzf]
, _ T,« mdr _ ,
+ 2;t/.2.— 2.rfX,
rf<2 (Jtf-|-2.m).rf<2
i ■ '
69
274 MOTION OF A SYSTEM OF BODIES. [Mec. Cel.
which gives by integration
constant = 2 m i^^'+^y' + d^') {{^•md^f+i^.^ndyf+j^^mdzf}
[441] * ' dt^ {M-}-2,m).dt^
— 2.M.2. — — 2x;
r
[442]
QX
I. HJ ^ {da^+dy^-\-dz^) , , \{dx'—dxf^(dy'—dyf + (dz!-^dzf)
d^ ' / dt^ S
— J2M.2.-^ + 2x|.(M+2.m) ; (7)
h being an arbitrary constant quantity. We had already found these
integrals, in the fifth chapter of the first book, for a system of bodies which
act upon each other in any manner w^hatever ; but on account of the
importance of these formulas in the theory of the system of the world, we
have thought it best to give this additional demonstration.
10. These integrals are all which have been obtained. In the present
state of analysis, we are under the necessity of having recourse to methods
r442'i ^^ approximation, making use of those simplifications which arise from the
structure of the system of the world. One of the most remarkable of these
methods depends on the circumstance that the solar system is divided into
several smaller systems, composed of the planets and their satellites. These
systems are so formed that the distances of the satellites from their primary
planets, are much less than the distances of the planets from the sun ; and
[442"] since in this case the action of the sun is nearly the same on the planet and
on the satellites, it follows that the satellites must move, about the planet,
whose integral is evidently of the form [441]. Multiply [441] by {M-\- 2 . w), and put
h for the term produced by the constant quantity of the first member, it becomes,
h = M.S.m. ^- + 2.7n.2.w. ^^
which, by substituting for 2 . 7?i . 2 . w . ft^ ' '^® identical value [lOOrt], becomes
of the form [442].
[443]
II. ii. § 10.] ATTRACTION OF DISTANT SYSTEMS. 276
in nearly the same manner as they would if they were attracted by the planet
alone. Hence we derive this remarkable property, that the motion of the
centre of gravity of a planet and its satellites, is nearly the same as it would [442"]
be if all the bodies were united together in this centre.
To demonstrate this, suppose that the distances of the bodies m, m', m",
. . . . r442'''l
&c., from each other, are very small in comparison with the distance of their
common centre of gravity from the body M. Put
x'=X + x;; ^^Y+y;; z'^Z+zl;
&c. ;
X, Y, Z, being the co-ordinates of the centre of gravity of the system of
bodies m, m', m", &c. ; the origin of these co-ordinates, as well as that of [443']
the co-ordinates x, y, z, a/, 1/, zf, &c., being at the centre of M. It is evident
that x^, y^, z^, x', &c., will be the co-ordinates of w, w', &c., referred to their
common centre of gravity ; we shall suppose these co-ordinates to be very
small quantities of the first order, in comparison with X, Y, Z. This being ^ ^
premised, we shall find, as has been shown in the first book,* that the force
acting upon the centre of gravity of the system, parallel to any right line
whatever, is equal to the sum of the forces which act on the body parallel to
that right line, multiplied respectively by their masses, and divided by the [443"]
sum of the masses. We have seen also, in the same book, that the mutual
actions of bodies connected together in any manner, does not alter the motion
of the centre of gravity of the system ;t and by § 8 [404'], the mutual
attraction of the bodies does not affect this motion ; it is therefore sujfficient,
in estimating the forces acting on the centre of gravity of the system,
to have regard to the action of the body M, which is foreign from the ^^^'"^
system.
The action of M upon m, resolved parallel to the axis of x, and taken in a
* (314) This results from the equations [155], and the remarks which follow imme-
diately after them, taken in connexion with what is said on the same subject, in the note in
page 103.
t (315) This appears from the equations [157], taken in connexion with the remarks
immediately following them.
2'^6 MOTION OF A SYSTEM OF BODIES. [M^c. Cel.
direction opposite to the origin, is* ~ ; therefore the whole force
which acts on the centre of gravity of the system of bodies wi, m', m", &c.,
in that direction, is [443'"]
[444]
[445J
By substituting for x and r their values [443, 411], we have
X 'K-\--x^
If we neglect the very small quantities of the second order, that is, the
[445'] squares and products of the variable quantities x^^ y^, z^, x', &c. ; and put R
equal to the distance V X^ + Y'^ -\- Z^ of the centre of gravity of the system
from the body M\ we shall findf
r4461 X ^X X, ^.X.\Xx,+ Yy, + Zz,]
\ r^ R^~^R^ R'
By marking successively with one accent, two accents, &c., the letters
[446'] x^, yp z^, in the second member of this equation, we shall have the values of
x' x"
— ^, -^; &c. ; but we have, by the nature of the centre of gravity,t
[447] 0 = 2. wa:^; 0^2. mi/^; 0^2.^2^;
* (315a) The force — ~T '^^ found as in [414']. The similar forces on m, m", Sic.
M3^ M x!'
are — "~7"~j "^Z^' ^^' Multiply these respectively by m, m', m", &ic. and add the
products together ; then divide the sum by 2 . w, as in [443'"], and we shall obtain the
expression [444].
f (316) The denominator of [445], putting X^-\- Y^-\-Z^ = R% is nearly
lR^ + ,.^Xx,+ Yy^ + Z.^)]-^=R-\{l + '^'^±^^^
_ 3 ( , S.{Xx,+ Yy,i-Zz,) l __1 3.(XT,+ ry, + Zz,)
hence the value of -, [446], is easily deduced.
J (317) These are the same as the equations [124]. Now if we multiply [446] by m,
B shall
these is
-r m X
we shall have — r-j and by accenting the letters we shall get , Sec. The sum of all
II. ii. § 10.] ATTRACTION OF DISTANT SYSTEMS. 277
hence we shall have, by neglecting quantities of the second order,
iJf.2.—
r^ M.X [448]
2 . m jR3 '
therefore the centre of gravity of the system is attracted in a direction
parallel to the axis of ar, by the action of the body M, in the same manner as [443']
if all the bodies of the system were united in that centre.* The same
result evidently takes place with respect to the axes of y and z ; so that
the forces with which the centre of gravity of the system is urged, parallel
j^ Y JVL.Z
to these axes, by the action of M, will be ^3-, and ^^. [448"]
When we consider the relative motion of the centre of gravity of the
system about ifeT, we must transfer to that centre, in a contrary direction, the
force which acts on the body M. This force, resulting from the actions of
OT, w', m", &c., on M^ resolved in a direction parallel to the axis of x, and
tending to increase the co-ordinates, isf 2 . — ; and if we neglect quantities [448'"]
of the second order, this function will be, by what precedes [448], equal to
X.2.m
-w- ' '""1
In like manner, the forces with which M is urged, by the action of the other
mx X ,1 3.X2 3.XF 3.XZ
which by means of [447], becomes simply, 2 . — -=— . 2 . m. Substituting this in [444], [447a]
we shall get [448].
* (318) The action of the body M on that centre, resolved in a direction parallel to the
axis of X, is found in the same manner as in [414'], where the action of M upon m was
jyix
computed to be — —, and by changing x, r, into X, R, we obtain the force corres-
M X
ponding to a point placed in the centre of gravity of the system ~— .
f (318a) This is proved in [412'], and in [447a], it is shown that 2 . — -=-— . 2 . m,
T^ R^
as in [449], The formulas [450] are found in the same manner, merely changing X
into F, Z.
70
278 MOTION OF A SYSTEM OF BODIES. [Mec. Gel.
bodies of the system, parallel to the axes of y and 2:, and in directions contrary
to their origin, are
[450] — fl3— ' and — R§— •
Hence we see, that the action of the system on the body M, is nearly the
same as if all the bodies of the system were united in their common centre
[450'] of gravity.* By transferring to this centre, and with a contrary sign, the
three preceding forces, this point will be urged parallel to the axes of x, y,
and z, in its relative motion about M, by the three following forces, [448,
449, &c.]
[451] -(^ + 2.^).-^; -(M+2.m).^3; -- (ili-+ 2 . m) . J.
[4511
[451"]
These forces are the same as if all the bodies m, m', m", &c., were united
in their common centre of gravity ;t this centre therefore moves as if all
the bodies were united in it, neglecting quantities of the second order.
Hence it follows, that if there are several systems whose centres of gravity
are very far from each other, in comparison with the mutual distances of the
bodies of any system from each other, these centres of gravity will be moved
in nearly the same manner, as if the bodies of each system were united
in their respective centres ; for the action of the first system upon each
body of the second system, is, by what has been said, very nearly the same
as if all the bodies of the first system were united in their common centre
of gravity ; the action of the first system upon the centre of gravity of the
second, will therefore, by what precedes, be the same as in that hypothesis ;
hence we may in general conclude that the reciprocal action of the different
* (3186) If all the bodies m, m', &c. were collected in their centre of gravity, their
action upon the centre of the body M would be represented by -^— - , and this resolved in
■V" y
the direction parallel to X, will be ' ' , [13] ; this is equal to the expression [44Q],
which represents the sum of all the separate forces of m, mf, &;c. upon M. The formulas
[450] furnish the same result.
f (319) Because the terms x„ y,, z^, x/, Sic. are not found in [451] which is the same
as if oc^, = 0, y^ =.0, &£c.
II.U. '^ll.] ATTRACTION OF SPHEROmS. '^'^^
systems ou their respective centres of gravity, is the same as if the bodies
of each system were united in these centres ; consequently these centres
must move in the same manner as if the masses were thus united. It is
evident that this result takes place, whether the bodies are free, or connected on the
together in anv manner whatever ; because their mutual action has no effect ofa
o •/ system of
on the motion of their common centre of gravity [443'", &c.] onlJnother
The, system of a planet and its satellites acts therefore very nearly on '^'
the other bodies of the solar system, as if the planet and its satellites were Mgiivi
united in their common centre of gravity ; and this centre is attracted by the
different bodies of the solar system, as in this hypothesis.
Every heavenly body is formed by the union of an infinite number of
particles, endowed with an attractive power, and as the dimensions of the
body are very small in comparison with the distances of the other bodies L^^^'-l
of the system of the world, the centre of gravity of the body will be
attracted in nearly the same manner, as if all its mass was collected in this
centre ; and its action on the other bodies of the system will be nearly the
same as in that supposition. Therefore we may, in the investigation of the
motions of the centres of gravity of the heavenly bodies, consider these
bodies as massive points, placed in their centres of gravity. This supposition,
which is very near the truth, is rendered much more exact by the spherical r45ivi]
form of the planets and satellites. For these bodies may be considered as
being formed very nearly of spherical strata, whose densities vary according
to any law ; and we shall now show that the action of a spherical stratum
upon a body placed without it, is the same as if its mass was united at
its centre. To prove this, we shall give some general propositions on the r45ivii]
attractions of spheroids, which will be very useful in the course of this work.
1 1 . Let ar, y, z, be the three co-ordinates of the attracted point, which [Scuon"?
we shall denote by m ; dM a particle of the spheroid, and af, i/, zf, the [45iviii]
co-ordinates of this particle ; if we put p for its density, p being a function of r^gjj,,
a/, j/, 2f, independent of x, y, z ; we shall have*
dM—p.dxf.di/^dzf, [452]
* (320) The attracting particle is supposed to be a parallelopiped whose sides are d od^
di/^d z\ and density p. Its attraction upon the point w, is evidently equal to the mass d M,
.*
'^^^ MOTION OF A SYSTEM OF BODIES. [Mec. Cel.
The action of dM upon m, resolved parallel to the axis of x, and directed
towards the origin of the co-ordinates, will be
[4531 p.dx'.dy'.dz'.{x-^x')
consequently it will be equal to
^7 p .dx' . dy' . dz'
V. dx
putting therefore
[455] ^= / —
J V/(^'-^)'+(2/'-2/)^+(^-^)^
the integral being taken so as to include the whole mass of the spheroid ;
[455] we shall have — l-^ — j, for the whole action of the spheroid on the point
m, resolved parallel to the axis of .t, and directed totvards the origin of the
co-ordinates.
[455"] ^ ^^ ^^® ^"^ ^^ *^^ particles of the spheroid, divided by their respective
distances from the attracted point. To obtain the attraction of the spheroid
upon this point, parallel to any right line whatever, we must suppose V to
be a function of three rectangular co-ordinates, one of which is parallel to
divided by the square of the distance [392], {x — x')^ -}- {y — y)^ + (-^ — *T- '^'his
[455a] being multiplied by ^.^^__ ^yi^^jil^^:^:!^^ gives, as in [394], the force resolved
in a direction parallel to the axis of x, and directed towards the origin of the co-ordinates,
agreeing with [453]. The direction of the force in [455'] is different from that in [393'],
and this is indicated by the term {x — a?), in the numerator of [394], being changed to
[x — x''\ in [453]. This last formula might be illustrated by a figure, as in page 8, in which
tlie attracting particle d M, whose co-ordinates are x, y', z', should be placed at A, while the
attracted point m is placed at c, whose co-ordinates are x, y, z ; so that x', y', z', would be
less than x, y, z, respectively. We may observe that the formula [453] is, by development,
equivalent to [454].
* (321) In finding the integral [455], x, y, z, are considered as constant quantities;
therefore — (t")? deduced from [455], must be equivalent to the whole of the forces
[454], corresponding to the whole mass of the spheroid.
II. ii. <^11.] ATTRACTIONS OF SPHEROIDS. ^81
the proposed right line, and we must then take the differential of this
function relative to that co-ordinate ; the coefficient of this differential, [455'^]
taken with a contrary sign, will be the expression of the attraction of the
spheroid, parallel to the proposed line, and directed towards the origin of the
co-ordinate to which it is parallel*
If we put
p = {{x — xj + {y—^f + {^—m ~^ ; [455i']
we shall have
V=f^.?.dx'.d'i/.dz!. [456]
The integration referring only to x', ij, 2!, it is evident that we shall havef
fddV\ , fddV\ , fddV\ ^ ,,,,,, ( fdd^\ . fdd^\ , fdd^\ }
But we havet
* (322) As V is equal to the sum of the particles of the spheroid, divided by their
respective distances from the attracted point, [455"], the values of V^ must be independent
of the situation of the co-ordinates, which may therefore be changed, so as to have one of
them parallel to the right line, in the direction of which the force of attraction is required to
be computed, and by naming this new right line X, the attraction will be represented by
— (jv)' ^^ ^^ evident from [455'], changing x into X.
f (323) For p, dx', dy', dz', being independent of x, y, z, [45 P^], we shall have
{-—\==f?.dx\dif.d2f.i—^\ and by changing, successively, x into y and z, we
shall obtain similar expressions of f-ry ), \J"^\ whose sum is as in [457].
X (324) ?={{x — x'f -f (y — y[f + {z — z'f } " % [455-], gives
The differential of tliis relative to x, is ( -— j = — (3^ — 3 . (a: — x') . ^^ . (—\ and
by substituting the value of f — j, we shall get (-,— ] = p^ . \ — ^~" ^ -f- '^ • (* — ^'T \ •
Changing, successively, x into y and z, &z;c. we shall find
(^)=^'-S-^-'+3-(y-m. (^') = ^^ {- ^= +3 ■ (—')=)•
71
282
ATTRACTIONS OF SPHEROmS.
[Mec. Cel.
[459]
Important
Eiquation
for com-
puting the
attrac-
tions of
Spheroids
and the
figures
of the
Heavenly
Bodies.
[459'J
[460]
therefore we shall have
0 =
ddV
dx^J
+ (^)+(^> (-)
This remarkable equation will be of the greatest use to us, in the theory of
the figures of the heavenly bodies. We may put it under other forms which
are more convenient on several occasions. For example, suppose we draw
from the origin of the co-ordinates to the attracted point, a radius which we
shall call r ; let ^ be the angle which this radius makes with the axis of x,
and ^ the angle which the plane formed by r and by this axis, makes with
the plane of x and y ;* we shall have
X = r , cos. ^ ; y = ^' sin. ^ . cos. ta ; z= r . sin. & . sin. « ;
The sum of these three quantities is
the second member of which is evidently equal to nothing, because
(^ - a/f +{y — y'f + (^ - z'f = r\ [455-] ;
therefore each term of the integral, [457],
must vanish, consequently, ( -—
, fddV\ , /ddV\ ^ . r^^..T
* (325) In the adjoined figure, let C be the origin
of the co-ordinates, D the attracted point, C A the
axis of a: ; ^ J?, parallel to O F the axis of y ; and
B D parallel to the axis of z ; making C A =x,
AB = y, BD = z, CB = r, the angles
ACD=&, and BAB = ^, Then
CA= CD. COS. A CD,
gives x = r . cos. 6, [4G0],
AD=CD. sin. ACD = r. sin. 6, and in the rectangular triangle DBA, we have
AB^AD.cos.DAB; B D=^AD .sm.DA B ; hence y = r .sin. ^. cos. ^ ;
z = r. sin. 6 . sin. zs, as in [460]. It is also evident that CD = \/CJfi-\-AB^-\-Bm,
cos. A L D == -^7^5
JO jy
im^.DAB=-—,
whence we easily obtain, geometrically, the
expressions of r, cos. &, tang. -sJ, [461], which are also easily proved to be correct, by
substituting the values of x, y, z, [460], in the second members of [461] ; since by
reduction they will .become like the first members of [461.]
II. ii. §11.] ATTRACTIONS OF SPHEROIDS,
from which we get
r = \/ a? -\- y^-{-z^\ cos. 4 =
tang. « =
ddV
1^
Hence we may obtain the partial differentials of r, ^, and «, relative to the
variable quantities x, y, z\ and may thence deduce the values of
( J, f J, in partial differentials of V, relative to the variable
quantities r, 6, ^. As we shall often use these transformations of partial
differentials, it will be useful to recall to mind the principles of this calculation.
Considering V first as a function of a;, y, z, and then as a function of the
variable quantities r, ^, to, we shall have
'dV\ rdr\ . rdV\ (d^_,fdV\ f^\
'dd\ /'d^\
©=(^vr^vr^
^dxj ' \ d6
To obtain the partial differentials ( ;^ )>
only to vary in the preceding expressions of r, cos. ^, and tang, to, [461] ;
taking therefore the differentials of these expressions, we shall have*
d r
d X
= COS.
f d&\ sin.^ /'dvi\
283
[461]
[461'J
[462]
[463]
* (326) The partial differentials of r= v/^+p + z2, [461], relative to x, y, Zj are
/dr\ X X , /dr\ y y
i%h
Z 2 , . ,
: ,===— = Sin. ^ . sin. TO
Again from cos. 6 =
V/x24-1/2 + 22 r
— , we get
\/a:2-|-2/2-|-z2'
r.
= sin. 6 . cos. -m ;
sin. <
V^^+2/^+z^'
and since the
differential of cos. ^ is — d6 . sin. 6. we shall find, by substituting the value of sin. d,
— V/a:2+2/2-|-22
(id =
<^.
hence we shall get its partial differentials, which may be reduced, by observing that
V/^+22 == r . sin. 6, [460].
— V/x2+i/2-p^ 2/2-1-22 \/^+^ _ !i!lj
7-3.- — - ' " — — — ■■ ' '-
\dx)
(x2-f-l/2-]-22)ii
xy
a:2-j-2/2 + 22
/dA_/?±E±E ^y 3 ^ ^y
\dy)~ V/^H^ '(a:24-r/2_|-z2)t~\/^ipi2.(a:2+2/2^22)~~r2.v/^+i2'
[460].
COS. A .cos.'zs
[463a]
[4636]
[463c]
[465a]
[4656]
284. ATTRACTIONS OF SPHEROmS. [Mec. Cel.
hence we get
[464]
dV\ , fdV\ sin J /"dV
= COS. 6 . -t: —
dx / \dr J r \ d 6
We shall thus have the partial differential {~'r~)j expressed in partial
[465c] In a similar way (j^) = -;^/^^^-j-^y^:^^§, which is equal to the preceding
value of (—), multiplied by - = tang.'5r, [461], hence (t^) = — — "^ — --• Again
from [461], « = arc. ftang.- j, its differential, by 51 Int. is
d '
[465rf] ^^_ y __ ydz — zdy
Hence,
^2 2^+za
Vary ' Uy/ 2/2 + 22' \dz) y2-|-z2'
and by substituting the values y, ^r, [460], y^-\-z^, [463c], they will become,
r465g] f^"^ f\ /dzi\ — sin. "ss /d to\ cos. ■»
\dxj ' Vt'y/ r.sin.^ ' \dz/ r.sin.^'
The preceding quantities being substituted in (—- j, [462], and in the similar expressions of
[ — J, ( — j, will give their values under the required form, [464]. It now remains to
obtain, from these, the values of (-j^\ (7^)' \~d^)' ^""^"S ^^^ brevity
f^\ =y>^ f^\ = V", (^^ = V", the formula [462] becomes,
'' e^)-^' &)+^"-(^)+^"'-G^i' ■
whose partial differential relative to x, is
^d
dx^ ) \dx )'\dx)~^ '\dx^)~^\dx J'\dx)
The terms (-T-), ("J^)' (tt)' "^ay be deduced from those of T— \ [462],by
changing F into (^) ' {dj) ' (rf^)' ^^^pectively ; hence
U. ii. ^ 11.]
ATTRACTION OF SPHEROmS.
286
differentials of F, taken with respect to the variable quantities r, ^, w.
Taking again the differential of (-f-ji we shall obtain the partial differential [464]
\dx )~\ dr^ J ' \dx)~^\drddj ' \dx)'^\drdzi) ' \d x )
(dV"\_( ddV\ (dr\ /ddV\ /d &\ / ddV\ /d^\
V dx )~\drdd) ' \dx)~^\d(i^ ) ' \dx)'^\d&d^) ' \dx)
(dV"'\_/ ddV\ fiji\_i_fddr\ /dd\ /ddV\ /dM\
V dx )~\drd^) ' \dx)^\ddd^) ' \dx)~^\di^) ' \d^)
which being substituted in (-%— )j it becomes
/ddV\_fddV\ fdrY./ddV\ fd^^ . /ddr\ /rf «\ 2
'^\dr) ' \dx^)~^\dd) ' ydx^J'^yd-a) ' \d3^)
"T" '\drddj '\dxj '\dx)^ '\drd^J '\dxj '\dxj~^ '\d6d7sj-\dx)'\dx)'
By changing in this, successively, x into y, and z, we obtain ( ——— j ; f ] ; and the
sum of all three of these expressions being put equal to nothing, gives the transformation of
the proposed equation, [459]. We shall examine, separately, the coefficients of each of the
[465g-]
nine terms of which this equation is composed. First, The coefficient of ( ],
Second, The coefficient of (-7-— ), is
\defi /
/ddY /^^Vi /d&Y /— sin.^\2 /cos. ^ . COS. TS \2 / cos. ^ . sin. W \2
[17) + Wj + fcj =\r~T-) + { — ; — ; + ( — ; — )
is
6in.2 ^ -J- cos.2 ^
j-2
sin.2 ^ -[- cos.2 ^ . (cos.2 ■13' -|- sin.2 trf)
Third, The coefficient of (— — r N is
\dvfi J
(d -g\2 /rf ^2 /rf^\2 sin 2 xtf-f cos.2 Tji 1
\dx) ~^\dy)'^\dz) (r . sin. &f r2.sin.2a'
F«„.*, The coefficient of (if) is (i^) + Q + (i^).
\dx)'^\7x^-\- 2+22' C"*^^^]' ^'^d its partial differential relative to x, is
Now we have
/ddr\
\d^)~\
a;2
dx^J \/a:2-|-^2-|-z2 (a;2-f i/24-z2;t r r3
72
[465A]
[465f]
[465A;]
[465Z]
286 ATTRACTION OF SPHEROmS. [Mec. Cel.
[464"] (-——\ in partial differentials of V, taken with respect to the variable
quantities r, 6, and x*. We may find, by the same method, the values of
fddV\ fddV\
\df y \dz^ )'
and in like manner, by changing x into y, and into z, we get
/ddr\ 1. 3/2^ /ddr\ 1 z2
\dW)~^ ^' \dz^)~'^~^'
m, /. , , . . 3 (a;24-2/2a-22) 3 7-2 2 , ^
Ihe sura oi these three expressions is = :;=— , therefore
■I If pi f i<i f
/ddr\ , /ddr\ , /ddr\ 2
and this term becomes — . (— j.
«y.,THecoefficie„.of (^^) is (^i)+(^^) + Q.
and if we take the partial differentials of
/^\ — y/^2ip^ f—\ ^ {—\ —
\dx)~ x^-\-y^-{-z^' \dy)~{x^-{-y^-\-z^) • \/i/2 + z2' l^rfzyl ~(j;2^2^^a:2) . y/^Iir^'
[465a — c], relative to x, y, z, and put afterwards for brevity ^x^-^-y^-^-z^ = r, and
y/^eip^ = s, we shall get,
/dd(i\ 25.r /dd&\ x_ 2xjf aryS /^liA £ 2x22 x^
[465n] (^-— J = -— , V"d^y) i^ ^ r2 s3 ' y rf 22 y) ^2 g ^ ^ ^3 '
[465o]
The sura of these three expressions is
2sx 2 a; 2 x . (i/2 -j- 22) a:.(2/2-|-22) 25x 2x 2 5x x a;
7-25 r* 5 r2 s3 7-4 j-'H s r^ r2 s
COS. ^
r2 . sin. ^'
COS. ^
and using s = r . sin. ^, [463c], it becomes - — ~, therefore this terra becomes
7-2 . sin
in.^ *W^ /*
Si^^A, The coefficient of (^) is (^^) + (^^ + (^^) ;
and if we take the partial differentials of the preceding values [465<? — e], we shall get
/dd'ss\ /dd-^\_^ 2yz /dd-ss\ __ —2yz
[A65p] \Tx^) — ^' {jfJ^Wi^^' Vd 22; — (2/2 + 22)2'
whose sum is nothing, therefore the coefficient of f — j is nothing.
Seventh, The coefficient of 2 . f j is
I. ii. §11.] ATTRACTION OF SPHEROIDS. 287
In this way, the equation (A) [459] may be transformed into the
following.
fddV\
^ ~'\d^)'^ sm.6 '\d6 J~^ sin.2^ ~^^\ dr' J'
and if we put cos. 6 = fj^, this last equation will become* [4651
/dr\ /d&\ , /dr\ /d 6\ , /dr\ /dd\
{17) ■ Wj + l^j • W) + V-) ■ Vz) f«5«l
— COS. ^. sin. ^ (sin. ^ . COS. to) . (cos. d . COS. -to) , (sin. ^ . sin. is) . (cos. ^ . sin. to)
r
cos. ^ . sin. 4
. \ — 1 -j- cos.^TO(-{-sin.^'5j} = 0.
Eighth, The coefficient of 2 . f j is
/rfr\ /(£ TON /rf_r \ /rfTON /rfr\ /rfroX
of which the first term vanishes, because (t— ) = 0, [465e], and the other two become
( sin. "SS) , , . . X COS. "Ef ^ rtf\f -y
(sin. ^ . COS. zi) . ^ r---' + (sin. ^ . sm. -s) . -^— = 0. [465r]
r.sin. 4 r.sin.
is
Ninth, The coefficient of 2 . ( , . . ) is
' \dHiaJ
/dt\ /d^\ /ehi\ /d^ /dd\ /d^\ .
\dx) '\dx)'^\dy) •\dy)~^\dz) 'KdzP
the first term of which vanishes, because (-— ) = 0, and the other two terms become
\d X /
(cos. 5. COS. ■sJ) ( — sin. TO) , (cos. ^ .sin. to") cos. zi „ tvt • in
-^ . ^ 4- . = 0. Now connecting together all [46531
r r.sin.6 r r.sin. ^ <-->-■ l j
these terras of the equation, [465^, Stc] we shall obtain the following transformation of [459].
f._/ddr\ !_ (ddV\ 1 /dd V\ 2 /dV\ cos.^ /dV\
V"rf^y "^rS" V'"7^/~^r2.sin.2d* Wto2 /"T'r * Vrfr/ ' r2.sin. 4 \d & )' t^GSf]
Multiply this by r\ and substitute r . (-^-) for 2 r . (^— ) + r^ . {j~j, which ^465„-|
are easily proved to be identical, by development, we shall obtain the formula [465].
* (327) Considering V first as a function of &, and then as a function of fi, we shall have
/dV\ /dV\ /dfJi-N /ddV\ fddV\ /dfi\Z , /dV\ /ddi>'\ „
Uj = W j • \jj} VTT2- ; = u^j • w + fc; • [-dw)' ^^°'''
\d¥) + 7i^6 '\dd)~ \dl^) ' W/ "^ W/ 1 \d¥) "^ sin. 6'\d6)y t^^^]
288 ATTRACTION OF SPHERES. [Mec Cel.
[466] 0 = ^ i^ ^^\dl^)] \ W^V (dd.TV\ (C)
12. Suppose now that the spheroid is a spherical stratum, whose centre
;ffsphere.'! IS at tho Origin of the co-ordinates ; it is evident that V will depend solely
[466'] ^^ '"' ^"^ ^^^^ ^^^^ contain either f* or to ; the equation (C) [466] will then
become
[467] Q^/dd.rV'
dr"
whence by integration*
[468] F=^ + — ,
r
[469]
[469']
A and 5 being two arbitrary constant quantities. Hence we shall have
'dV\ B
[466,
dr
— i~ — J expresses, by what has been said [455'"], the action of the
spherical stratum upon the point m, resolved in the direction of the radius
y, and tending towards the centre of the stratum ; now it is evident that
the whole action of the stratum must be in that direction ;t therefore
3g^^ NowM' = cos.5, [465'], gives sin. ^ = v/l— (xs ; {----\=z — sin.^ = — ^i_fj.2;
[4666] {j^,}--^^^--^' Hence, --.(^-j=-^, and the preceding
expression becomes \7^ ) • (^ — ^^) — ^ ^'{'7~p which is evidently equal to
( ^ ( ^^~^^^' W/ > ) • This being substituted in [465], gives [466].
* (328) Multiply [467] by (Zr, and integrate, it becomes d.i--—\=A. IMultiplying
TO
again by d r, and integrating, we find r F'= Ar-{-B, whence V=A-\ .
f (329) For the stratum being spherical and homogeneous, its attracting particles must be
equal, and similarly situated on every side of the line, drawn from tlie attracted point to the
n. ii. § 12.] ATTRACTION OF SPHERES. 289
J expresses the whole action of the spherical stratum upon the
point m.
Suppose, in the first place, that this point is placed within the stratum.
If it was situated precisely at the centre, the action of the stratum would be
nothing ;* therefore, when r = 0, we shall have — (— — ] = 0, or [469"]
— = 0, which gives ^ = 0 ; consequently — ("J~ ) = ^' whatever
may be the value of r. Hence it follows, that a point placed within a spherical [469'"]
stratum is not affected by it ; or, which is the same thing, it is equally attracted
in every direction.
If the point m is placed without the stratum ; it is evident that by supposing
the point to be infinitely distant from the centre of the stratum, the action
of the stratum will be the same as if the whole mass was collected in that
centre ;t putting therefore M for the mass of the stratum ; — ( — — ] or [469*^]
-g- will become, in this case, equal to -g-, which gives B ^ M ; therefore
in general, for points situated without the stratum, we shall have
'dV\ M
[470]
dry r^ '
that is, a spherical stratum attracts any point situated without it, as if all the [47(y]
7nass of the stratum was united at its centre.
A sphere being a spherical stratum, the radius of whose interior surface
centre of the stratum; and it is evident there cannot be any attraction, in a direction [469a]
perpendicular to this line, because there is no reason why it should deviate on the one side
rather than on the other. The whole attraction must therefore, be in the direction of the
line connecting the centre of the stratum and the attracted pomt.
* (330) Because the attraction of the particles, situated on opposite sides of the stratum,
would exactly counterbalance each other, and the whole result would be nothing.
f (331) The dimensions of the body, being infinitely small in comparison with the
distance, its whole mass may be considered as collected in a point.
73
290 ATTRACTION OF SPHERES. [Mec. Cel.
(^470"j is nothing, it is evident that its attraction on a point placed at its surface,
or without it, is the same as if the whole mass was collected at its centre.
The same thing takes place also with globes composed of concentrical
strata, varying in density from the centre to the surface, according to any
law, since it is true for each of these strata ; now the sun, the planets and
[470'"] ^jjg satellites may be considered as very nearly like globes of this nature ;
they will therefore attract external bodies in nearly the same manner as if
their masses were united at their centres of gravity ; which is in conformity
to what we have found by observation in 5> 5. It is true that the figures of
[470''^] . .
the heavenly bodies vary a little from a sphere ; but the difference is very
small, and the error resulting from the preceding supposition, is of the same
[470^] order as that difference, as it respects points near their surface ;* but for
[470a]
[4706]
* (332) Let D be the distance of a body or massive particle M, from the centre of
gravity G, of a system of particles, m, m', m", he, and r, r'//', he, the distances of these
particles, m, m', he, from their common centre of gravity G. Then it has been prove4
in [445' — 450'], .that the action of the system, m, m', &;c., upon the body M, would be the
same as if all the particles, m, m!, &;c., were collected at G, neglecting quantities of the
order — . The same would hold true, if the particles, m, m!, &;c., composed a solid
body, or spheroid S, and its attraction on the body M, neglecting terms °of the order — -,
would therefore be the same as if its whole mass were collected in G. Suppose now the
body S to be nearly spherical, so that it may be considered as being composed of an inscribed
sphere, whose radius is p, and centre G, and of an external spheroidal shell, of very small
thiclcness a p, having the same centre of gravity G , a being a very small quantity, depending
on the thickness of this shell, in various parts of its surface. Then by [470"], the sphere
would attract the external particle M, exactly as if all its mass were united in the centre G ;
and the spheroidal shell, by what has been shown, [470c], would attract the same particle M,
as if all its mass were collected in the same centre G, neglecting the part of tlie attraction of
r2
this shell, corresponding to the terms of the order -— , and as the mass of this shell is of the
order a, in comparison with that of the sphere, it follows that this neglected part would be
7-2
of the order a . — -, in comparison with the whole attraction of the spheroid, as is asserted
in [470^'].
Again, since the attraction of the internal sphere is as its mass (x, divided by the square of
the distance of the attracted point from the centre G, if that distance should vary by
n.ii. <^12.] ATTRACTION OF SPHERES. 291
distant points, the error is of the same order as the product of this difference,
by the square of the ratio of the radius of the attracting body to its distance [470^]
from the attracted point ; for we have seen, in ^ 10, that the circumstance
of the distance of the attracted point, renders the error of the preceding [470 ''"j
supposition, of the same order as the square of this ratio. The heavenly
bodies attract therefore, in nearly the same manner, as if their masses were
united at their centres of gravity, not only because their distances from each r^^^^ji,
other are very great with respect to their dimensions ; but also because their
figures are very nearly spherical.
The property posessed by spheres, in the law of nature, of attracting as if
their masses were united in their centres, is very remarkable ; and it is an
object of curiosity, to ascertain whether the same thing takes place in other [47oix]
laws of attraction. For this purpose, we shall observe that if the law of
gravity is such that a homogeneous sphere attracts a point placed without its
surface, as if all its mass was united at its centre, the same result ought to
take place in a stratum of uniform thickness. For if we take from a sphere a
spherical stratum of uniform thickness, we shall form another sphere of a
less radius, which will have, like the preceding, the property of attracting
as if all its mass was united at the centre ; now it is evident that these two
spheres cannot have this common property, unless it also appertains to the
spherical stratum formed by the difference of these two spheres. The
problem is therefore reduced to the determination of the laws of attraction,
by which a spherical stratum, of an infinitely small but uniform thickness, [470 j^]
attracts an external point, as if all its mass was united at the centre of the
stratum.
Let r be the distance from the attracted point to the centre of the
spherical stratum, u the radius of this stratum, and du its thickness. Let & ^ "^
quantities of the order a, in different points of the surface of the spheroid, so that the
distance of one point should be p, and of another point p . (1 -}-«)) the attraction of this
lit' U> Ui
sphere would change from -— to -r--^ — ^ or —- . (1 — 2 a), and it would therefore
p2 p2.(l + a)2 p2 ^ >"
vary by terms of the order a, consequently the attraction of the whole spheroid, upon bodies
placed on or near its surface, would vary by terms of that order, as is asserted in [470^].
This subject is fully treated of in Book III, particularly in <^ 25, [1647].
292
ATTRACTION OF SPHERES.
[Mec. Cel.
be the angle which the radius u makes with the right line r ; « the angle
[470^"] which the plane passing through the right lines r and u makes with a fixed
plane passing through the right line r ; the element of the spherical stratum
[470«"] ^jjj i^g* u^du .dzi , d& , sin. 6. If we then put / for the distance of this
element from the attracted point, we shall havef
[471] f^ z=r^ — 2ru. COS. 6-\-u^.
[471'] Let US denote by <?(/), the law of attraction at the distance /; the action
of the element of the stratum, upon the attracted point,t resolved in a
* (333) In the adjoined figure, which is similar to that
in page 181, M is \he attracted point, situated on the
continuation of the diameter p C P} C the centre of the
splierical stratum PEpR; PEp the great circle from
which the angle 'si = E C Q^ is counted ; P Aa Qp,
P Bb qp, two great circles drawn through the poles P, p,
infinitely near to each other ; E Q^qR S E the great
circle, whose poles are P, p', AB, ah, arcs of circles
parallel to Q ^. CA=u, CM=r, angle JiCM=&,
Q^q = udzi, AB = ud 'Si .sm. 6, jla=r.ud& ; hence
the space ABba=u^.d'^.d&.smJ } this multiplied by
the thickness of the stratum d u, gives its mass,
u^du.dzi.dd. sin. ^ as in [470^^'"].
f (333a) In the plane triangle CAM, which is
here drawn separately from the preceding figure, if we
let fall from A, the perpendicular AD on C M, and
put AM=f, we shall have, ^ 5 ^ -^
CD= CA.cos.A CM = u. cos.6j AD= CA.sin.A CM=u.sm.6,
MD= CM— CD=r — u. COS. 6,
and since AM^ = AD^ -{-MD% we shall get
p={u. sin. ^f-\-{r — u . cos. ^)2 = r^ — 2 r w . cos. ^ + u\
as in [471]. This is the formula 63 of the Introduction.
X (334) The mass of the particle u^du.dzs.dd. sin. 6, [470'^»'], being multiplied
by the force (p (/), gives the attraction of the particle in the direction MA. This multiplied
MD r-u. COS. 6 ^ ^.jj ^.^^ .^^ ^^^^^^ [1 1], in the direction M C, as in [472].
by
AM
f
11. ii. § 12.] ATTRACTION OF SPHERES. 293
direction parallel to r, and tending towards the centre of the stratum,
will be
w'du.dvi, d6 . sm.d,^^-^^^-^.ip(f) ; [472]
but we have*
therefore the preceding quantity may be put under the form
w'du.dvi.dd.sin.d. (^) • ? (/) 5 [474]
and if we put
fdf. f (/) = <,, (/), im
we shall obtain the whole action of the spherical stratum, upon the attracted
point, by means of the integral u^ du .fd -a .d 6 . sin. 6. (p^(^f')^ taking its [^74"]
differential relative to r, and dividing it by d r.f
This integral ought to be taken relative to w, from « = 0 to th = the
circumference of the circle ; after this integration it becomes
2'^.uKdu./dd.sin.6.(p^(f) ; [475]
ff being the ratio of the semi-circumference of a circle to its radius. If we [475*]
take the differential of the value of /, with respect to 5, we shall havej
* (335) Since /=v/'^2_2r«. COS. ^+m2, [471], we have
/df\ r — «.cos. ^ r — u.cos.^
\dr J \/r^ — 2ru. cos. ^ -f- "^ /
f (336) By [474'], 9^ (/) =fdf. 9 (/), its partial differential relative to r, is
\-~-i — ) ^^\l~j'^ (/)' hence the element of attraction [474], becomes,
Taking its integral relative to the whole surface of the stratum, it becomes
u^du.fdzi . d ^ .sin. 6 . (^^^^\
and since -m, 6, are independent of r, if we put f dTS . d6 .sin. 6 .cp^(f) = F, this [474o]
expression will become u^ . d u . (—\ as in [474"].
X (337) In the integral fd 6 . sin. d . cp^ (/), [475], the quantities /, 6, are considered as
variable, r, u, as constant, because r, u are the same for all the particles AB ab, of the
74
294 ATTRACTION OF SPHERES. [Mec. Cel.
[476] d6. sin. 6 = ^i- ;
ru
consequently
[476'J 2^.u'dujd6.sm.d.cp^(f)=2^,~.ffdf.<p,(f),
[476"]
The integral relative to 6 being taken from ^ = 0, to ^ = * ; and at these
two limits, we have* f=^ — w, and f=r-\-u; therefore the integral
relative to jT, ought to be taken from f=r — u, to f==r->ru; suppose
therefore
m'l ffdf.%(f) = ^(f);
we shall have
[477] -—ffdf, 9, (/) == ~- . [^ (r + l^) — ^(r — w)}.
The coefficient of d r, in the differential of the second member of this
[477^ equation, taken with respect to r, will give the attraction of the spherical
stratum upon the attracted point [474"] ; hence it is easy to perceive that in
[477'] the case of nature, where (p (/) = -^ , this attraction is equal tof — '— ;
spherical surface. Now the differential of /^, [471], taken in this hypothesis, is
2 fdf= 2ru.dd . sin. 6, hence d 6 . sin. 6 = —,
•' •' ru
which being substituted in fd6. sin. 6 . cp^ (/), it becomes — 'ffdf- (p, (/), the constant
quantities r, u, being brought from under the sign/. This being substituted in the expression
of the attraction [475], it becomes as in [476'].
* (338) The integral relative to xs, being taken in [476'], from '5J = 0 to t^sTrthe
circumference 2 ff, it will represent the attraction of an annulus of the stratum, formed by
the revolution of the arch A a, about the diameter Pp. In order therefore to embrace the
whole surface of the spherical stratum, it will be necessary that the point A should move
along the semi-circle P Ap, from P to the opposite point of the axisp, or from ^ = 0 to
d = the semi-circumference ir ; at the first limit, the point A falls in P, and / becomes
MP = r — w, and at the last limit the point A falls in p, where d = *, and the value of /
becomes Mp = r-\-u.
t(339) <?(/)=-^ gives [474'], %{f)^fdf.^{f)=^f^-^ = -y,
and by [476'"], 4^ (/) =//^/. % (/) = -/^/= -/j ^ence,
[479^
If. ii. <^12.] ATTRACTION OF SPHERES. 295
that is, it is the same as if all the mass of the spherical stratum was united at
its centre, which furnishes another demonstration of the property we have [477'"]
before spoken of [470"] relative to the attraction of spheres.
Let us now ascertain the form of 9 (/), upon the supposition that the
attraction of the stratum is the same as if its mass was united at its centre.
This mass is equal to At-Jt .u^du, [2756], and if it was united at its centre, [477»'']
its action upon the attracted point, would be 4ir . w^ Jm . 9 (r), we shall
therefore have [477', 477'^]
o J W' s - • (4^ r^ + wl — ^^[r — ul) } f L oj .^ ,r\\
2'>f.udu.< ^yV^Lij ^L JJ ^ >=4iif,u^du,(p(r) ; (D) [478]
( dr )
By taking the integral relative to d r, we shall have*
■^(r-{-u)—-^(r^u) = 2ru.fdr.cp(r)'i-rU; [479]
U being a function of u, and constant quantities, added to the integral
2u.fdr.(p(r). If we represent ^(r-\-u) — 4(r — w), by i?,t we shall have,
'^{r-{-u) — -^{r — m) = — (r-fw) + (r — u)== — 2 m, therefore
-—'{■^{r + u) — 4.{r — u)l= ,
the differential taken relative to r, and divided by d r, expresses the whole attraction of
the stratum [477'], — '—- — , as in [477"] ; but, by [2756], the mass d! m of a spherical
stratum, writing u for R, is 4it .u^duj and if this mass was collected m the centre of
the stratum, its attraction on the proposed point would be ' ' — , which is equal to the
preceding expression.
* (340) Dividing the equation [478] by — '- , we get
\-±I j-^ > ^ .dr = 2u.dr.cp{r),
which, by integration, relative to r, gives - . (-^^ [^ 4" w] — 4^ [^ — m] ) = 2 ufd r.(?{r)-{-U.
This multiplied by r gives [479].
f (341) The equation [479], by puttmg jR for its first member, becomes
R
= 2ru.fdr.(f)r-\-rU; hence (—j = 2u.fdr.(pr-{-2ur(p{r)-\-Ut
296 ATTRACTION OF SPHERES. [Mec. Cel.
by taking the differential of the preceding equation,
i/ddR\ . /• N , ^ d.(p(r)
fddR\_ fddU'
[480]
w' ^
»*
but by the nature of the function i?, we have^
thereforef
or
[482'] ^iW + ^•'pCO =, _L . (dduy
r dr 2u' \ du^ J'
Now as the first member of this equation is independent of u, and the second
[482"] member is independent of r, each member ought to be equal to an arbitrary
constant quantity, which we shall denote by 3 ^ ; therefore we shall have
[483] ^liW + ^^i^SJ;
r dr
* (342) Since R=-].{r-{-u) — -^{r — u), we have
[480a] (^^^ = ^'{r-u)-V{r-u), and (^) = 4." (r - ^/,) - 4." (r + 1.),
denoting, for brevity, (-^), {^f-} ^7 V if), and 4^" (/), respectively.
In a similar manner we have,
(^)=4.'(r + «) + +'(r-n), and (i^-^'j =. ^" {r + u) -^ V {r - u),
consequently (7;i-)= (7^). ^s in [481].
f (343) Substituting the values of (-^\ ("^■^j' [480], in [481], we get
4 M . 9 (/■)]+ 2 r M . ' =r .( -r-^ j, as in [482]. Dividing this by 2 ru, it becomes as
in [482'].
n. ii. § 12.] ATTRACTION OF SPHERES. 297
whence by integration^
m Cr'\ = 4 r -X-
7?
^{f) = Ar-\- — ', [484]
B being another arbitrary constant quantity. All the laws of attraction in AttTac°tL
.-,,.., . ii which
which a sphere acts upon a point placed without its surface, at the distance r ^Js^aT
from its centre, as if all its mass was united at its centre, are therefore ma^ssww
collected
comprised in this general formula ^^^»';^
v4 r + -^ ; [485]
and it is easy to prove that in fact this expression satisfies the equation (D)
[478], whatever values are taken for A and ^.f
If we suppose ^ = 0, we shall have the law of nature ; and it appears,
in the infinite number of laws, which render the attraction very small at great [^SS]
distances, that the law of nature only possesses the property of making
the attraction of spheres the same as if their masses were united at their
centres.
* (344) Multiplying [483] by r^ ti r, it becomes
whose mtegral is r^ . 9 (r) = w2 r^ + -^ 5 dividing this by r^ we get, (^{r)-=Ar-\- ——,
[484].
f (345) If we put (p{r) = Ar-{- — , we shall have [474'],
<PAf)=fdf-^{f)=^fdf. {^f-\-~) =i^/^-y-+ C, [486«]
C being a constant quantity. Hence by [476'"],
W)=ffdf-9,{f)=f{iApdf—Bdf-\- Cfdf]^^Ap-Bf+iCp + D, [4866]
D being another constant quantity. This gives
^ {r -\- u) — 4. {r — u) =i A . {(r + «)''— (r — w)^} _ jB . {(r + m) — (r — m){
which by reduction becomes A . {r^ u -\- r u^) — 2Bu-\-2 Cru, hence
7-(4^[^+"]-4'[r-i/])=^.(r2" + w3)-.^-}-2C«,
2 J? u
and its differential relative to r, being taken, and divided by d r, becomes 2Aru-{- —-^,
Substitute this in the first member of [478], also cpr, [484], in its second member, and the
equation will become identical, leaving A and B indeterminate.
75
[486c]
298
ATTRACTION OF SPHERES.
[Mec. Cel.
This is also the only law, in which a body situated within a spherical
stratum, of uniform thickness, is equally attracted in every direction. It
follows, from the preceding analysis, that the attraction of a spherical
stratum whose thickness is du, on a point situated within it, is expressed by*
' dr J
To make this function nothing, we must have
■^ (u-\-r) — -^^{11 — r) ^ r . U ;
U being a function of u independent of r, and it is easy to show that this
[487'] equation is satisfied in the law of nature, in which we have 9 (/) = — . f
But to prove that it is satisfied by no other law, we shall denote by -^l (/),
[485"]
[480]
[487]
[487^
* (346) When the point M falls within the stratum,
as at M\ the limits of the integral ■\>{f) in the equation
[477], will be
f=PM' = u — r, and f==pM' = u-{-r,
and the integral of the first member of [477] becomes, in
this case, — '- . l-\y{u-\-r) — -^(u — r)l. To
render the force nothing, we must put the differential of
this expression, taken relative to dr, equal to nothing
[474"], which will give, by neglecting the constant factor,
2 If .udn.
-AX
E
<Z . — . ^ 4^ (m + r) -— 4- (m — r)l
i.
M
P
X> • \
dr
'=0.
V
Pv
Multiplying by c? r and integrating, we get - • J -4 (« + 0 — \'{u-^r)\=^TJ, hence
4, (if -)- r) — \{u — r),=^rU, as in [487].
f (347) Put A=0, and r==f, in [484], and it becomes 9 (/)=—, as in the law of
nature [487"], and then -^ (/), [486i], becomes 4. (/) = — Bf+ (cp + D-, hence,
^(^u-^r)—^.{u—r)=—B.\{u+r)—{u—r)}-^^C.{{u-\-rf—{u-^-f\==—^Br-\-2Cur,
hence by [487], U= — 2B-\-2 Cu, and since this value of U is independent of r, as is
required in [487'], the assumed value of 9 (/) = — , must satisfy the proposed equation
[487].
I
[4881
II. ii. <^ 13.] ATTRACTION OF SPHERES. 299
the differential of 4^ (/), divided by df ; and by 4." (/), the differential of
■\! (/), divided by df, and in the same manner for others ; vre shall then [487'"]
have, by taking, twice in succession, the differential of the preceding equation
With respect to r,*
4." (w + r) •— 4^" (w — r) = 0. [488]
As this equation exists for all values of u and r, it follows that 4^" (/) must
be equal to a constant quantity, whatever be the value of /, consequently
4,'" (y* ) := 0 ; now by what precedes we havef
+'(/)=/•<?.(/); [489]
hence we deduce
4.'" (/)= 2. 9 (/)+/. ?'(/); [490]
therefore we shall have
0 = 2. ?(/)+/. ?'(/); [491]
which gives by integration, t (p(/) = — , corresponding to the law of [49]/]
nature.
13. Let us now resume the equation (C) of ^ 11, [466]. If we could
* (348) The first difTerential of [487], gives 4.' {u + r) + 4.' (m — r) = C7, its second
differential is as in [488], from which we get 4^" {u-\-r)=. 4*" (m — r), whence it would
follow, as in [482"], that we must have generally ■\f'{u-\-r), or 4-" (/), equal to a
constant quantity.
f (349) By[476"'], we have ■^{f)=ffdf.(p,{f), its differential divided by <Z/,
gives 4' (/) =/• 9/ (/)• Again, taking the differential and dividing by df, we get 4-" (/)>
and by writing, for brevity, <?/(/), %" if), for the differential of (p,{f), divided by
df, and that of 9; (/), divided by df we shall have, 4-" (/) = 9. (/)+/•<?/(/) ;
again taking its differential and dividing by df, ■^"' (/) = 2 9/ (/) +/. 9/' (/) ; but
9^/) =fdf.^{f), [474^, gives 9/ (/) = <? (/), and 9/' (/) =■ 9' (/), hence,
4.'" (/) = 2 9 . (/) +/. <P' (/), as in [490], and as V (/) = 0, [488'], we shall
have 0 = 29(/)+/.9'(/), as in [491].
% (350) IVIultiplying [491] by fdf it becomes 0 = 2fdf. 9 (/) +/2 df. 9' (/) or
0=2/t?/.9(/)+/^.<^.(9(/)), whose integral is 5=/«. 9 (A whence 9(/) = -^,
as above.
300
ATTRACTION OF SPHERES.
[Mec. Cel.
[491"J
[491'"]
[492]
obtain the general integral of this equation, we should have an expression of
V which would contain two arbitrary functions, which might be determined
by seeking the attraction of the spheroid upon some point selected, so as to
simplify the calculation, and then comparing this attraction with the general
expression. But the integration of the equation (C) [466] is impossible,
except in some particular cases, such as that in which the attracting spheroid
is a sphere, which reduces the equation to common differentials ; it is also
possible, when the spheroid is a cylinder, whose base is an oval, or re-entering
curve, and whose length is infinite ; we shall see in the third book [2075],
that this particular case includes the theory of Saturn's rings.
Let us take the origin of r on the axis of the cylinder, which we shall
suppose to be infinitely extended on both sides of this origin. Putting r' for
the distance of the attracted point from the axis ; we shall have*
r' = rv 1 — iJ?.
It is evident that V depends solely on r' and «, since it is the same for all
points in which these two variable quantities are the same ; it does not
[493a]
[4936]
[493c]
* (351 ) In the annexed figure C AE e is the axis of the
cylinder, taken as the axis oi x ; D the attracted point, C the
origin of the co-ordinates ; C A = x, AB :^y, B D = z,
CD = r, angle D CA = 6, DAB^-m, and
D^ =ir' = r. sin. ^ = r.v/.r=^, [465'],
as in [492]. Hence we get
m=^
.fx2,
Vm-/
rfA
which are used in [493, 493c].
As V does not contain r,
shall have
except through r', [492], we
and in a similar manner (-jy ) = ("T7^ ) • (^ — V'^)' ^^"^
the equation [466] becomes by developing the terms affected
by the sign d,
/ddV\
11. ii. §13.] ATTRACTION OF A CYLINDER. 301
therefore contain fx, except by means of r', considered as a function of that
quantity ; this gives [4936]
dV\ fdV\ /dr^ rfi, /"dV
d^J \di^ J \diij w'j 2 V^^'
ddV\ rV^ /ddV\ r fdV
[495]
[493]
li\^^ J ~ 1 — H.2 * \^d7^J ~ (1—72)^ * \d7) '
therefore the equation (C) [466] becomes
hence by integration*
V=(p . [r' . cos. iys-{-r' .V^ — 1 . sin. -sjJ -{--^[r' , cos. -a — r' .\/ — 1 . sin. wj ;
Substituting the values [493, 493c], we get
Reducing and multiplying by 1 — fji-^, we shall get
o /, ox fddV\ , /ddV\ . ^ /dV\
which, by substituting the value of /, [492], becomes as in [494].
* (353) It is easy to prove that the value here assumed for V satisfies the proposed
equation. For, if we notice only the function 9, which may be done, because the
demonstration is the same for 9 as for 4', and put for brevity, <p instead of
(p . I / . cos. zi -{-r' \/—l. sin. ■zff | , we shall have ( T/ ) "^ (^°^* '^ ~i~ S/—^' ^^°* '^)'^' 9
i— )==( — r'.sin. ts-f-r'.\/IirT. cos.'w) .9'; f — — - j = (cos. ■2* + \/— 1 . sin. -si)^ . cp" ;
( j = — / (cos. Ttf + V--^ • sin* ■^) • 9' + ( — f^ • sin. zJ + / . \/^^. cos. •rt)^ . 9".
these values being substituted in [494], it becomes,
' ^ . (cos. •« -|- \/— 1 • sin. ■rt)^ . 9" — r' . (cos. ■cs + v/^^^^ sin. -zs) . 9'
+ ( — / . sin. 'ui-\-r' . \/—l . cos. •bj)^ . 9" -f- ^ • (cos. vi -f- ^— 1. sin. 'zrf) . 9' ;
which is identically nothing, the first term being balanced by the tliird, and the second by the
fourth. The same thing takes place with the function -v^, by changing 9 into 4/, and writing
— \/^-\ for \/—\. Therefore the value F, [495], satisfies the proposed equation, and
as it contains two arbitrary functions, it must be the complete integral.
The equation [494], is of the second order of partial differentials, and is a simple case
76
302 ATTRACTION OF A CYLINDER. [Mec. Cel.
(p(r') and ^^(r') being arbitrary functions of r', which may be determined, by
seeking the attraction of the cylinder, when * is nothing, and when it is a
right angle.
If the base of the cylinder is a circle, V will evidently be a function of r',
independent of « ; the preceding equation of partial differentials [494] will
then become
which gives by integration*
dV\ H
H being a constant quantity. To determine it, we shall suppose r' to be
extremely great with respect to the radius of the base of the cylinder, which
permits us to consider the cylinder as a right line infinitely long. Let the
[497] base be A, and put z for the distance from any point of the axis of the
cylinder to the point where this axis is intersected by r' ; the action of the
cylinder considered as concentrated on its axis, will be in a direction parallel
to r', equal tof
/Ar' .dz
of a much more extensive class of equations, which has been treated of by several
mathematicians. It comes under the form of that given by La Croix, in § 750, edit. 1, or
§ 756, edit. 2, of his " Traite du calcul differentiel, &:c." For by putting V=z, / = a?,
■S3f = y, in [494], it becomes ^ = *^^ ' ("7^) "^ ("^~t) ^~^' (t")* That treated of by
La Croix being
which corresponds to the present example by putting R=x^, S = Oj r=l, P=x,
Q = 0, JV= 0, M= 0 ; and by following the method he has given, we should obtain
for z or F", the expression [495], which we have demonstrated synthetically ; this method
being used for brevity.
* (354) Multiplying [496] by --^, it becomes 0 — —r'dr' -(77^)—^^' • (^ )»
whose integral is — / . ( — j = H, whence — f — j = — , as in [497].
f (355) Suppose the whole mass of the cylinder to be collected in the axis C A E e,
and put AE = z'j. Ee = dz'. Then the mass of matter in the space E e will be
n. ii. § 13.]
ATTRACTION OF A CYLINDER.
CO to 2: ^ CO, which reduces it to*
the integral being taken from z = —
this is the expression of — ("3~r)» when r' is very great. By
r - \dr'
comparing it with the preceding, we have H=2A, and we find that
whatever be r', the action of the cylinder on a point placed without its surface,
IS
If the attracted point is placed within a circular cylindrical stratum, of
uniform thickness, and infinitely long, we shall also have — \~hv)'^~^^
[497] ; and as the attraction is nothing when the attracted point is on the
axis of the stratum, we shall have H=0 ; consequently, a point placed
within such a circular stratum, is equally attracted in every direction.
303
[4981
[498"]
represented by A.dz'. Dividing this by D E^, we get the attraction in the direction
D E. Multiplying this by -^ttt, gives the attraction in the direction D A equal to '3,
whose integral gives the whole attraction as in [498] ; the letter sf being accented, to
distinguish it from the co-ordinate BD = z. This integral is to be taken through the
whole length of the cylinder from z' = — 00, to «'=co.
[498a]
* (356) Putting the angle A D E = s, we shall get
2/ = r' . tana;. £, hence r^^ -\-2^^ = — and
d z' = r— , / being constant. Hence
COS.2 £ "
Ar'.dzf /^ A. ds. COS. s ^.sin. f
/jir .a z p,
(r'2 + 72)t~J '
-}- constant.
(r'2+z'2)t J 1' /
Now at the first limit of this integral, where z' = — co,
s = — J -n-, * being the semi-circumference of a circle whose
radius is 1, this becomes 0= 1- constant, hence the
- . , . w3.sin. £ , A
corrected mtegral is — 1-
and s=^')r, becomes
K
This, when «' = 00,
2w3
as above.
Putting this equal to — , [497], we shall get, as above,
^04 DIFFERENTIAL EQUATIONS OF THE [Mec. Cel.
14. The equations (A), (B), (C), of § 11, [459, 465, 466], may also be
applied to the motion of a body ; and an equation of condition may be
obtained from them, which will be very useful in proving the calculations
made by the theory, or in verifying the theory of universal gravity itself.
The equations (1), (2), (3), of § 9 [416—418] by which the relative motion
of m about Mis determined, may be put under this form*
f499j ddx _ fdq^ , d^y _ ^dq\ , ddz ^ /dq\ ,
[499']
[4986]
[499a]
ddx
dt^ ~
/JQ\ ddy fdq\
'\dx J ' di^ \dy J
in which
^ M-\-m ^ m[xx'-{-
■yy-\-z^) X
73 "^ m
* (357) The assumed value of Q, [499'], gives
/d G\ Mx mx m' xf . 1 fd'kX
' ' {M-{-m) /dr\ — (3f+m).r m'xx'
\dxj
because \ / = . ( ■— - ) = , and the terms — 2 .
dx y r^ \dx/ r3 ^3
mx
7)1/ X: 7Th OC
produce — 2 . — j— , in which — is not included. If we therefore include the terms
mx , , . , 11 1 /dQ\ Mx mx , 1 /dX\ ...
under the sign 2, we shall have, -— = — 2 • -r- H • "7~ > which
r3 o ' \d x/ r3 r^ m \d X /
being substituted in [416], it becomes -— - == (l~ )) ^^^ the equations [417, 418], give
d t'* \(t X J
. iM dd%j /dQ\ ddz /dQ\ ^^ , - r^ d d z . ^
m hke manner -— = (^— j, -— = (^— j. If we compare the equation 0=-— +P,
ddx /d 0\
[364], with 0 = — - — ( ;^ jj [499], it will be evident that the force Py acting on the body,
in a direction parallel to the ordinate x, and tending towards the origin of the force, (363''*] is
equivalent to — (t~ )' ^^ ^lie motion of m about M; therefore (t~" ) will express the
force acting on the body m in its relative orbit about M, in the direction parallel to x, and
tending to increase the co-ordinate x. And as the ordinate x is arbitrary, we may say
generally that [~r~)i (t^)» (t^)' represent tlie forces acting on the body m in its
relative motion about M, resolved in the direction of die lines y, z, r, respectively, and
tending to increase those lines. This agrees with tlie remarks made in tlie note page 253.
11. ii. § 14.] RELATIVE MOTIONS OF A SYSTEM OF BODIES. 305
and it is easy to prove that*
supposing the variable quantities x', t/, zf, x", &c., contained in Q, to be [soo]
independent of x, y, z.
We shall transform the variable quantities x, ?/, z, into others, more
convenient for astronomical purposes. Putting r for the radius drawn from
the centre of Mto that of m, v the angle vrhich the projection of this radius [500"]
* [358) Taking the partial differential of (^)j [498a'], relative to x, we get
/ddq\ {M+m) 3.{M-\-m).x /dr\ J^ /d dx\
because 2.-—— does not contain a?, [500']. Substituting for (~) its value -
it becomes
fddq\^
wx2 y
(./If+m)
r3
[463«],
3.(Jlf+TO).x2 1 /ddX\
+ -^'{-1^)'' and by changing ^^,
r5
• , . , ^ . /ddq\
successively x into y and z, we find ( -— 1
/ddq\_
\dz^ )
(J»f+m) 3.(J»f4-m).i/2
(Jtf+m) 3.(./Jf+m).22
m ' Vrfj/2/ '
"1 . [~~r^)' Adding these three equations
f3 f5
togedier and putting for x^ -{- y^ -{- z^, its value r^, the terras multiplied by {M -\- m) will
destroy each other, and the sura will become,
/ddq\ , /ddq\ , /ddq\ 1 ( /dd^A , /ddX\ , /ddX\y
Now each of the terms of which X, [412], is composed, being substituted in the second
raember of this equation, renders it equal to nothing. For example, the first term of X,
[412], by using the value of p, [455*''], becomes mm ^, which produces, in the second
member of the preceding equation, the terms m' . < { , ^) + [~Tt) ~\~ ["TTj ( »
which by [458] is nothing, and the same would take place if we put
Hence in general, (-j^^ + (-—^ + (^) = 0 ; consequently
/ddq\ /ddq\ /ddq\
77
[500r]
306
[501]
[502]
DIFFERENTIAL EQUATIONS OF THE [Mec. Cel.
upon the plane of x, y, makes with the axis of x ; and ^ the inclination of r
above the same plane ; we shall have*
X = r . COS. & . COS. V ;
y = r . COS. (5 . sin. v ;
z = r . sin. 6.
The equation (E) [500], referred to these new variable quantities, will be,
by§ll,t
fddq\
'^•'^)+--(4?)+'tS/
, fddq
COS. ^
d6
* (359) Let C be the place of the body M,
D that of m, CA the axis of x, AB, B D,
lines drawn parallel to the axes o( y,z; making
CA = x, AB = y, BD = z, CD = r,
angle A C B = v, angle BCD = L Then
CB=^ CD. COS. B CD = r. COS. 6, this
being substituted in CA=CB. cos. A C B,
AB=CB .sm. A CB, we get
[501a] x = r . cos. & cos. v, y = r . cos. & . sin. v.
Lastly, BD=CD. sin. J5 C D, hence,
z = r . sin. ^, agreeing with [501].
f (360) It is proved in [465^ — w], that by putting x= r . cos. 6, y= r .sin. d. cos. «,
z = r . sin. 5 . sin. raf, [460], the equation [459] would change into [465]. And as the
equation [4 59] would not vary, by writing z for x, x for y, and y for z, it will follow that if
we had put z = r . cos. ^, a: = r . sin. ^ . cos. zs, y=r. sin. ^ . sin. •ra, the equation
[459] would change into [465]. If in these values of a?, y, z, we write v for w, and 90 — 6
for ^, they will become x=r . cos. & . cos. i;, y = r. cos. 5 . sin. «, z = r . sin. d, which
agree with those in [501], and the result from substituting these last values in the equation
0 = (--—) + ( -, 2 ) + ("tJ^)' W'J^ ^^ obtained by writing, in the equation [465], Q for
V, V for 73, 90 — ^ for 6, therefore -^d6 for c? ^, and it will then become
/dd_q\
\d6^J
sin. &
cos.^
d6/
and by substituting, as in [465 m],
it becomes as in [502 J.
2r
;^)+.^.Q, fo.*e.e™ r.(i±^)
II. ii. § 14.]
RELATIVE MOTIONS OF A SYSTEM OF BODIES.
307
If we multiply the first of the equations (i) [499], by cos. d . cos. ?? ; the
second by cos. 6 . sin. v ; the third by sin. 6 ; and for brevity put*
M =
ddr r.drr^
. COS.^ 6
r. di
dt^
dt^ dt^
we shall have, by adding these products.
Likewise, if we multiply the first of the equations (i) [499], by
— r . cos. ^ . sin. v ; the second by r . cos. ^ . cos. v
[.503]
[504]
* (361) Considering in the first place, Q as a function of r, ^, «, and then of a?, y, z, we
Shan have (^^) = 0 . (^) + (|) . (^) + (lf) . (0 Now fro» [5«],
we get ( — j = COS. d . COS. v, f — j = cos. ^ . sin. v, f — j = sin. 5. Hence
/dQ\ /dQ\ , . /<iQ\ , . . /dQ\
[17) = ''o'- ' ■ "^- " • W) + <=°^- ' ■ ^'"- " • W j + '""■ '■{Ji}'
and by means of the equations [499], this becomes
(^)=-^''-^'{ddx.cos.v-\-ddy,sin.v^+s^'^'-a^-
In finding ddx, ddy, we shall, for brevity, put r. cos. ^ = 72, which gives, by [501],
X = R . cos. V, y = R. sin v. whence dx = dR . cos. v — Rdv . sin. v ;
ddx= [dd R — R. dv^) . cos. v — {2dR .dv-\- Rddv) .sin.v,
and in a similar way,
dd y= [ddR — Rdv^) . sin. v-\-{2dR.dv -\-Rddv) . cos. v.
The former being multiplied by cos. r, and the latter by sin. r, and the products added, we
shall get
ddx . COS. V -{-ddy . sin. v =-dd R — Rdv^.
Substitute this in [503a], and it becomes ( — ) = ( — ^; ;?- ) . cos. 6 -J ~ . sin. 6.
■- -^ \drj \ dfi dfi J ^ d^
Again fi:om i2 = r.cos.^, we get ddR={ddr — rd(r^). cosJ — {2dr.d&-{-rddd).sm.d,
also, ddz = {ddr — rd6^) .sin. d + {2dr.d6-{-rdd6) . cos. ^. These values being
easily deduced from those of rf<?cc, ddy, [503c, (fj, by writing r,6, for 72, v, respectively.
Hence, ddR. cos. 6-\-ddz .sm.6 = ddr — rdd^, this being substituted in [503/*],
, ., /dq\ ddr d^ „ rft;2
we shall get, ^^— l = -^_r . — — jR.cos.5.— , and by putting for R its value
r . cos. 6, it becomes as in [504].
[502a]
[5026]
[503a]
[5036]
[503c]
[503rfJ
[503€]
[503/]
[503g-1
308 DIFFERENTIAL EQUATIONS OF THE [Mec. Cel.
and add these products, supposing
d.( r^ '-r ' cos.^0
[505]
[506a]
dt
we shall have*
[506] N' = f—^].
\ dv J
Lastlj, if we multiply the first of the equations (i) [499], by
— r . sin. & . COS. v ; the second by — r . sin. & . sin. v ; the third by r . cos. & ;
and add the products, putting
[507] p'_^ i^4_^ if. sin & cos &4-^-l^~'
we shall findf
[508] p' _
dq
dd
* (362) In the same manner as (t^) was found in the preceding note we shall find
(£}=-r-<^os.i.An.v; (^) = r . cos. « .cos. . ; (^)=0;
Hence, ( ) == — 't - cos. ^ . sin. v . ( l~ ) + ^ • cos. & . cos. '^ '\i~\ and by substituting
^ -, . , . d d X . ^ d dy .
[499 J, It becomes — r . cos. & . sin. v , -— -f- r . cos. 6 . cos. v . -7^, or as it may be written
[ — j = ' -— . \ — sin. V . dd x-\- cos. v .d dy {. Substituting the values ddx^ ddy,
[503c, <Z], in the expression — sm.v.ddx-\-cos.v.ddy, itbecomes 2dR.dv-\-Rddv,
dJB^dv) ,. 1 . , ,. d.{r^.dv.cos.^6) , . n rcnozn tr
or — ^ — - , which IS changed into -, by using R, 1 5036 J. Hence we
have
,iq:Xi.i^.dv.oon.^_''-{^'-/t-'°'"')_ ^, [505], as in [506].
\dv J dt^ dt
[507a]
f (363) As in the two preceding notes, we find,
{dq\^/dq\/dx\ /dq\ /dy\,(dq\ /^y
\d6j \dxj \dj'^\dyj '\ddj~^\dzj '\ddj'
{^)—r.sin.&.oos.v; (^) = -r . sin.^ . sin. .; (^)=r.cos.5.
n. ii. -§> 14.]
RELATIVE MOTIONS OF A SYSTEM OF BODIES.
309
The values of r, v, &, contain six arbitrary quantities, introduced by the
integrations.* Let us consider any three of these arbitrary quantities,
which we shall denote by a, 6, c ; the equation M' = (-^\ will give the
three following equations if
(M)
fdv\ (ddq:
fddq\ }dv\ }ddq^
\drdv)' \dc)^\drd&^
ddq
J?
ddq
~d?~
ddq
dr"
d r
da
dr
lb
d r
d c
, (ddq
\dr dv
\dr dv
+
,(ddq
~^\drd^
(d^ _ fdM\ ^
\da)~\ da ) '
dd\_?dM'
dbj ~ \~db'
d6\
d^J
/dM'\
V do )'
We may obtain from these equations, the value of (— r^)* and if we make
m =
n =
P =
d V
Jb
dv
dc
d V
da
dc) \d~c)'\dbj
d6
da
d6
Jb
dv
da
dd
dc
_fdv\ /df\
\db) ' \d^)
[509]
[510]
These and the equations [499], give
^dQ\ . ^ dd X ... ddy ,
-— ^ = — r . sin. 6 . COS. v . -— r . sin. 6 . sin. v . -— ^ + r . cos.
^ddj dfi rff2 '
d dz
~dW
r.sin.^
rf<2
< dd X . cos. V -\- ddy . sin. © S -|~ »" • cos. 6 . -— -
Substituting the value of ddx . cos. v-\-ddy . sin. v, [503e], it becomes
/1?W^. \—ddR.sm.6 + ddz.cos.d] + '''^.RdvK
Now from ddR, ddz, [503^], we get — ddR.sin.6-\-ddz.cos.6=2dr.d6^rdd6,
which being substituted, and also r . cos. 6, for R, we shall find
/dq\ dd6 . ^ di^ 2r.dr.d&
Vrfi j = ^ • -:715- + ^ • 7^ • sm. « . cos. 6 +
being the same as P', [507], which agrees with [508]
rf(2
* (364) Each of the three equations [504, 506, 508], is a differential of the second order
in r, V, 6'j their integrals must therefore contain six arbitrary constant quantities.
t (365) The first equation is found by taking the differential of M' = (—\ [504],
relative to a, considering Q as a function of r, v, 6, and these quantities as functions of a, 6, c.
The other equations are found by changing a into b and c successively.
78
^^^ DIFFERENTIAL EQUATIONS OF THE [Mec, Gel.
[5101
[511o]
/dr\ /rft;\ /dd\
~\daj'\dbj'\dc)
\daj'\dcj' \dbj ~^\dbj' \d cj * \d a)
/dr\ /dv\ fd6\ /dr\ /dv\ /d6\ /dr\ /dv\ /d6\
\dbj \daj \dcj~^\dcj'\daj'\dbj \dcj'\db)'\da)'
we shall have*
''''' ''{-d^)=''\-jr)+'''{-dT)+P'{:
In like manner, if we put
''d r\ /'d6\ /'dr\ /dd
:J'\d
dM'
dc
^'=©•6
' dcj ' \db J \dbj'\dc
cJ \dcj ' \da
dr\ /'d6\ /dr\ /dd
p'
dbj \daj \daJ \db
* (366) If we multiply the three equations [509] by m, n, p, respectively and add the
products together, we shall find,
Substituting the values of m, n,p, [510], in the coefficient of [~r~^)i it becomes equal to
the quantity denoted by ^, [510']. The coefficient of [~r~T~) becomes
\da)'i\db) '\dc) \dc) '\db)l~^\dbj'l\dc) '{daj \daj'\dc)l
'^\dcji\da)'\db) \db)'\da)\'
and of the six terms of which this is composed, the first is destroyed by the fourth, the
second by the fifth, and the third by the sixth, and thus the coefficient is reduced to nothing.
The coefficient of { - — ;-- ) becomes
\drd6j
/dj\ ( /dv\ /d^\ _ /dv\ /d^S •) /d_6\ C /dv\ /dd\ _ /dv\ /dj\ ^
\da)'i\db) '\dc) \dc) '\db)l~^\db)' l\dcj '\da) \da/'\dcjl
In which the first term is destroyed by the sixth ; the second by the third, and the fourth by
the fifth, thus reducing it to nothing. Consequently the equation [511a] becomes as in [51 1].
TI. ii. §14.] RELATIVE MOTION OF A SYSTEM OF BODIES. 311
the equation -^'=(3^)' will give*
Lastly, if we put
P ~^J~aJ'\dh) \dbj\da
the equation F = (—^ \ will givef
The equation (F) [502] will thus becomej
* (367) Taking the differential of the equation [506], relative to a, b, c, we shall obtain
three equations similar to [509], and which may be deduced from them, by writing v for r,
r for V, JV' for M'. This change being made in m, n,p, [510], they will become
respectively, — mf, — n', — p', [512]; also p, [510'] will change into — p. These
quantities being substituted in [511] it will become
/ddq\ , /dJV'\ , /djsr'\ , /dj\r\
and by changing the signs of all the terms, we shall obtain the equation [513].
f (368) The equation [508] gives three equations similar to [509], by taking the
differentials relatively to a,b,c; and these equations may be deduced from [509], by
changing 1; 6, M' into ^, r, P\ respectively. By these changes the values of m, w, p, [510],
become — w", — »", — p', [514], respectively, and p, [510'], becomes — ^. These
changes being made in [511], it becomes
and by changing the signs of the terms we obtain [515].
J (369) The equation [502], being multiplied by p . cos.^ 6, becomes, by arrangmg the
terms in a different order,
0 = ..cos.= ..,.(^)+.Q+co..M.(^^)
312 DIFFERENTIAL EQUATIONS OF THE [Mec. Cel.
t«ei + "•' • (^) + "' • (^) +P' ■ (^) (G)
+ «.".cos.^,.(^) + «".cos.^..(!^)+y'.cos.»..(^)
+ P.52ri)f'. cos.^^ — P'.sin. ^.cos. ^}.
In the theory of the moon, we neglect the perturbations produced by the
moon's action upon the relative motion of the sun about the earth, which
amounts to the same thing as to suppose the moon's mass to be infinitely
small. In this case the variable quantities a;', ?/, 2!, corresponding to the sun,
would be independent of a:, y, z, which correspond to the moon ; and the
[516] equation (G) [516] would take place in this theory [500'] ; therefore the
values found for r, v, 6, ought to satisfy this equation ; which furnishes a
method of verifying these values. If the observed equations in the motion
of the moon, result from the mutual attractions of the three bodies, the sun,
earth, and moon, it must necessarily follow, that the values of r, v, and 6,
deduced from observations, would satisfy the equation (G) [516] ; which
furnishes a method of verifying the theory of universal gravitation ; for the
[516"] mean longitudes of the moon, the perigee, and the ascending node, enter into
these values, and we may take a, b, c, for these longitudes.
In like manner, in the theory of the planets, if we neglect the square of
the disturbing forces, which may almost always be done, and then put x, y, z,
for the co-ordinates of the planet whose orbit is to be computed, we may
[516'"] suppose the co-ordinates x', y', 2f, x", &c., of the other planets to correspond
to their elliptical motions,* and they will therefore be independent of x, y, z;
and by substituting the values of the terms computed in [511, 513, 515, 504, 508], it
becomes as in [516].
* (370) By neglecting the terms multiplied by m', m", he, in Q, [499'], the equations
[517] will give the elliptical motion of the body m. The neglected terms will be of the
order mf a/, m' 1/, he. Now any one of the co-ordinates x', 3/, &;c. may be supposed to be
divided into two parts, the one depending on the elliptical motion, the other on the
disturbing forces of m, m", Stc. This last part being of the order m x', m y', Sic. ; and it
must evidently produce in Q terms of the order m m', Stc, or of the square of the
disturbing forces.
n. ii. -§15.] RELATIVE MOTIONS OF A SYSTEM OF BODIES. ^^^
hence it would follow from [500'] that the equation (G) [516] would also
take place in this theory of the planets.
15. The differential equations of the preceding article*
ddr r.dv^ o, d6^ f dO
. COS.^ — T.—r-^='
I
d V
as in [518].
79
dt^ dt^ ' dt^ \d
c^.f r^.-— . cos.^d ) ,7^v 1
V dt J _ fdq\ ^ \ rm [517]
Tt \dv)^
^ dd6 , ^ dv^ . , , 2rdr.d6 /d Q
r . -r-s- + r^ • -r^ • Sin. 6 . cos. 6 -\ -— — = -—^
dt^ dt^ dt^ \dd
are merely combinations of the differential equations (i) [499] of the same
article ; but they are more convenient, and better adapted to the use of
astronomers. We may put them under other forms, which may be useful
on several occasions.
Instead of the variable quantities r and 5, let us use u and s, putting
1
r . COS. 6
or u equal to unity divided by the projection of the radius vector upon the
plane of a:, y, and
s = tang. 6, t^^^"^
or 5 equal to the tangent of the latitude of m above that plane. If we
multiply the second of the equations (H) [517] by r^ dv . cos.^6, and then
take the integral,! we shall have
* (370a) The values of M', [503, 504], being put equal to each other, we get the first
of the equations [517]. In like manner the values of JV*', [505, 506] give the second, and
those of P', [507, 508], give the third of the equations [517].
f (371) The product thus formed is
(r2.^.cos.2d).rf.(r».^.cos.2^) = (^).r2.rfr.cos.2^
orbyputtmg,asm[517'], ..cos.^ = -, {^TtJ ' ^ ' [^t) = [j^J ' ^ '
which, being multiplied by 2, and integrated, gives i^Tt) =^^+2./ {~r~) •
314 DIFFERENTIAL EQUATIONS OF THE [Mec. Cel.
h being an arbitrary constant quantity, hence we have
dv
[519]
dt
\/^'^--fm'i
w
If we add the first of the equations (H) [517] multiplied by — cos. 6, to
the third multiplied by — '—, we shall find*
r
d^ u dt^ \du J ^
* (372) The products, being added together, make the following sum
[520o] — . ] — (^<Zr.cosJ+2c^^'^^'Sin.^+^'^<^^-sin-^+^'<^^^.cos.^-}"^.<?v^.cos.^.(cos.^4+sin.^^) [
^ /dq\ . sin. ^ /dq\
Now — = r.cos. d, [517'], gives
— d^ . (-)= — <?^.(r.cos.^)= — ddr.cos.6-{-2dr.d^.sa\.(i^r.ddd.smA-\-r.d&^.cos.&,
the second member of this expression contains the four first terms of [520a], and by
substituting — d^ .(—\ for those terms and — for r . cos. ^, in the last term, that equation
d'-- 1 dv^ d sin
[5906] becomes ^f +:^ ' ^'=- ^°^- ^- (^) +^^ ' (^)- Now if we consider Q
as a function of r, 6, and then as a function of «, s, we shall have
\drj \duj '\drj~^\dsj ' \drj' \d & J \duj'\ddj~^\ds/'\dd/
_, 1 , ^ . /du\ 1 /du\ sin. ^
But w = -, and 5 = tang.fl give (7- = 5 ^j (77) = 515
r.cos.^ \dr/, r^.cos.6 \d 6 / r.cos.^6
raOl /rfQ\_ sin.^ /^N , _L.. /^N
These being substituted in the second member of [5206], it becomes
\du)' i r2 "'"^rS.cos.s^ l~^ r.cos.^d ' \d s J^
and by reduction it is equal to
m\ __1 _, tang.^ (d_q\ ^^ /d_q\ ^ /d_q\
\du J ' {r. cos.dJ^~ r.coa.6 ' \ds J' \duj ^ ' \d s /'
II. ii. §15.] RELATIVE MOTIONS OF A SYSTEM OF BODIES. ^15
hence we deduce
Substituting for dt its value [519], supposing dv to be constant, we shall [Siii]
have*
/^\ ^ _du_ _ /dq\ s^ /dq\
\dv J ' u^dv \du J u ' \ ds J
ddu , ^ \dv J vrdv \du J u \dsj [522]
The third of the equations {H) [517] becomes, in the same manner,! [523]
supposing dv to be constant.
-^ (ids UU V \ VU U / \ U/ O / \ Ul U/ /
0 = 7^ + » + 7 --TrR-T--. [524]
f Hence tlie expression [520J] becomes as in [520] ; and as — d . — = — , if we
substitute this in [520], multiplied by d t^ we shall obtain [521].
* (373) Put for brevity, \/ h^ +^*/'(^) * ^ ~ '^* "^^ ^^^^ squared and
its differential taken gives dA=ij^\ --r-^- Substitute this value of w2 in [519]
[522al
[5226]
and we shall get dt = -— r- , hence,
\u^dtj \ dv J dv ~ dv dv ' \dv/ Aifi^
d V being constant. This and the value of d t, bemg substituted in [521] it becomes
dv ' \dv/ Au^ ' A l\du/ ' u \ds/)
Dividing by Adv, transposing the two last terms, and substituting for A^ its value, we shall
obtain the equation [522].
t (374) The third of the equation [517] may be thus written,
d (r^ —\
•y ;^^y-|.^.^.cos.^a.tang.^ = f^l [522c]
dt ' dfi ° \dd/
Now s = tang. 5, [517"], gives ds = — ^, or dd^ds .cos.^ 6, hence
316 DIFFERENTIAL EQUATIONS OF THE [Mec. Cel.
We shall therefore have, instead of the three differential equations [H) [517],
the following expressions in which dv \s constant :
dv
dt =
u^
dq\ du /'dQ,\ s fdq
riv)K\ r^ ddu , , \ dv J u^dv \ du J u \ ds
i^'^i 0 = —^ -\-u-f-
•^ \ dv J w*
ds /dO'
' ^ ' US.
^ d d s . , d V \ d V
( •/ \dv y w'
If we wish to avoid fractions and radicals, we may put these equations under
the following forms :*
9 d & „ . d s 1 ^5
rKcos.^6. - = -.-, [517'],
dt dt u^ dt
\ A Ads
and as -— - = — - , [52261, it becomes, , whose differential, considering c? v as
v/^dt dv *- ■' dv
. Adds-\-dAds . dds , /dQ\ ds ^^^^ ^ ^, . ,. ., ,, j^
constant, IS 7 , or A. hl— ^). , r522al. This divided by rf i,
dv ' dv ' \dvj Au"^' ^ -^ •'
dv . , - ^ , . ^ _ . ^. dds , /dQ\ ds
or — - , gives the first ternoi of the preceding equation, [522c], u^ A^ . —- + ( "3~ ) • ^ •
Ai 2 9 /I . . dv'^ s dv^ sdv^ /u^A\^ „ .„ a • u tkoa t
Also H . cos.'^ 6 . tane. 6 . — = — . — = . ( ] =u^s . A^. Again by r520c
^ dt^ u^ d^ v^ \dv J ^ -^ •- -*
, /dq\ sin.a /dQ\ , 1 /dq\ . ,
we have ~ ) = . ( — ^ 1 -] . { — ^ I, or as it may be written
\dd/ r.cos.24 V<^M/ cos.s^ \dsj' •'
These values being substituted in [522c], it becomes
dv^ ' \dv/ dv ' \du/ ' ^ ^ \d s /
Transposing the two last terms, and dividing by u^ A^, we obtain the expression [524.]
Collecting together [519, 522, 524], we obtain the equations [525].
* (375) Using the value A, [522a], as in the two preceding notes, the first equation
[525] will give u^dt.A = dv, and its differential, considering dv as constant, and
substituting the value of dA, [522a], will be
2u.du.di.A-\-u^.ddt.A-\-u^.dt.(^) .-^ = 0,
II. ii. § 15.]
0 = ^ +
RELATIVE MOTIONS OF A SYSTEM OF BODIES.
2 du.dt
udv^
^^ \dvj' dv^'
/ddu , \ ( 1 , 2 /-/f/QX dv )
'^ h^'\\dvj' u^dv \duj u'\dsjy
^du
d^
dv
(L)
dv
dq
)\-
317
[526]
By using other co-ordinates, we may form new systems of differential
equations. Suppose, for example, that we change the co-ordinates x and y
of the equations (i), § 14 [499], into others, relative to two moveable axes,
situated in the plane of the co-ordinates a:, y ; so that the first of these new
axes may correspond to the mean longitude of the body m, whilst the other is
perpendicular to it. Let x^ and y^ be the co-ordinates of m, referred to these
axes, and nt-j-s, the mean longitude of m, or the angle which the moveable '■^'^^ ^
axis of x^ makes with the axis of ar ; we shall have*
[526']
and, by multiplying by ^, {u^.ddt-{-2u.du.dt).^-\-dt.dv. (-^\=0.
Substituting the value of A
dv
[522&], it becomes
v^dt
(u^.ddt4-2u. du.dt). , , ^
...<i..0=o.
This multiplied by " "'^ , and reduced, corresponds to the first of the equations [526].
The second and third of these equations may be deduced from the second and third of the
equations [525], respectively, by multiplying by — , or 1 + — . / (t^) • -j, [522a],
and reducing.
* (376) In the adjoined figure let the rectangular
co-ordinates of the point i^be either AB=x, BF=y,
or AD = x^, and D F= y^. Draw D C perpen-
dicular to A B, and D E parallel to A B. The angle
CAD = D FE = nt-{-s, and in the right angled
triangles DEF, A C D, we have
80
y
\
D
^""^'^ X
BC X
318 DIFFERENTIAL EQUATIONS OF THE [Mec. C61.
^527j x = x^.cos.(nt + s) — y^. sin. (nt + e);
y = x^. sin. (nt-\-s) -\-y^. cos. (nt + s) ;
whence we deduce, by supposing d t constant,*
d dx. cos. (nt-\-s) -{- ddy , sin. (n t -\- s) = d d x^ — n^x^.df — 2ndy^ . dt ;
ddy . COS. (nt-\-£) — ddx . sin. (nt-\-s) = d dy^ — n^y^.df-\-2ndx^.dt.
By substituting the preceding values of x and y in Q, we shall havef
[5-28]
[529]
[528a]
A C=AD. COS. CAD=x,.cos.{nt-\-s)', CD{=BE)= AD. sin. C A D=x,.sm.{nt-i[-s)',
FE=DF.cos. D FE=y,.cos.{nt-\-s); DE{=B C)=DF.sm.DFE=y,.sm.{nt-\-s).
Hence x = A C — B C = x^. cos. {nt -\-s) — y^ . sin. {nt-\-s)'j
y = B E -\- F E =. X, . sm. {nt -{- e) -\- y, . cos. {nt-\-t)',
as in [527].
* (377) The differential of x, [527], is
dx=dx^. COS. [nt-^s] — nx^.dt,sm.{nt-\-£) — dyi. sin. {nt-\-s) — ny^.dt . cos. [ni-\-s).
Its second differential, supposing d t constant, is
ddx=\ddXi — n^ Xi.dt^ — 2n .dy^.dt\ . cos. {nt-\-B)
— \d dy, — 'n^y^.dt^-\-2n .dx^.dt] . sin. (« < + s).
Now by writing s — J ir for e, the expression of a?, [527], becomes that of y, the same
change being made in ddx, gives
[5286] ; ddy=^ddxj — n^x,.dt^ — 2ndyi.dt\.sm.{nt-\-s)-\-\ddy, — n^ yi.dt^-\-2ndXi.dt\, cos. {nt-\-s).
]VIultiplying [528aj by cos. (n ^ + s), and [5286] by sin. (w^+s), and adding the
products we shall obtain the first of the equations [528]. Also, multiplying [528a] by
— sin. {nt-{- e), and [528&] by cos. {nt-\- s), and taking their sum, we shall get the
second of the equations [528].
f (378) Considering Q as a function of x, y, and then as a function of x^, y„ we have
JMultiplying the first of the equations [527] by cos. {n t + s)> the second by sin. (n t + 0'
and adding these products we shall get x^ = x. cos. {ni -\- s) -\-y . sin. {ni -\- s), hence
[5296] ( — ) == COS. {nt-{-B)'f (-r-J == sin. {nt-\-s). If we had multiplied the first of the
equations [627] by — sin. (n t -\- e), the second by cos. (n t -\- s), the sum would have
n.ii.§15.] RELATIVE MOTIONS OF A SYSTEM OF BODIES. ^19
This being premised, the differential equations (i) [499] will give the three
following equations*
0 = 77-"^
[530]
dt^ \dz)
been y^ = — x .sm. {nt-\-z)-\-y .cos. {nt -{•&), hence f — ' j = — sin. (n^ + e), [529c]
(— ' j = COS. {nt-\- s). These values being substituted in [529a], they will become as
in [529].
d dy
dt^
In the equations thus obtained, we must substitute the values of the first members [528],
and they will become like the two first of the equations [530]. The thu-d of these equations
is like the third of the equations [499].
The calculation in this part of the work, and in the two preceding notes might have been
done in rather a more simple manner as follows. First, we evidently have
\dxj \dx) ' \dxj'^\dy) ' \dxj ' \dyj \dx) ' \dyj "^ \dy) ' \dyj '
and from [527] we get (d/ "^ ^°^* (** ^ + ^) » f -p J = sin. (n < + 0 >
f — j = — sin. {nt-\-s); (-^\=r. cos. {nt-\- s). Substituting these values, and
(t-)) (t- )> given by the equations [499], we shall have
[529rfJ
* (379) Substitute the values [529] in the equations [499], and we shall get
ddx /dq\ , , , /dq\ . , , .
Multiplying the first of these equations by cos. {nt-\- e), and adding it to the second,
multipUed by sin. (» i + s), we get ~ . cos. (n ^ + e) + -^ . sm. (w < + s) = ^— ^. [sgge]
Again, the first of the preceding equations, multiplied by — sin. (ni+s), added to the
second multiplied by cos. (n < -f- s), gives
. COS. (n ^ + s) — ^ . sin. (» ^ + s) = (^). [529/]
320 RELATIVE MOTIONS OF A SYSTEM OF BODIES. [Mec Cel.
After having given the differential equations of the motions of a system of
bodies mutually attracting each other, and having deduced from them all the
complete integrals vi^hich have yet been discovered ; it now remains to
integrate these equations by successive approximations. In the solar system,
the heavenly bodies move in nearly the same manner as if they strictly
obeyed the principal force which acts on them, and the disturbing forces are
[53(y] very small ; we may therefore, in the first approximation, consider only the
mutual action of two bodies, namely, that of a planet or a comet and the
sun, in the theory of the planets and comets ; and the mutual action of a
satellite and its planet, in the theory of the satellites. We shall therefore
begin with an exact computation of the motions of two bodies, which attract
each other : this first approximation will lead to a second, in which we shall
notice the first power of the disturbing forces ; then we shall consider the
[530"] squares and products of these forces ; and by continuing in this manner, we
shall determine the motions of the heavenly bodies with all the precision
required by observation.
[529^1
(dQ\ d dx , , s , d d y . , ^ , .
(dQ\ ddx . , , . , d dy /..v
Substituting, in the second members of these equations, the values [528], we shall obtain, by
a small reduction, the two first of the equations [530].
n. iii. § 16.] ELLIPTICAL MOTION OF TWO BODIES. 321
CHAPTER III.
FIRST APPROXIMATION OF THE MOTIONS OF THE HEAVENLY BODIES, OR THEORY OP THE ELLIPTICAL
MOTION.
16. We have already shown in the first chapter [380'''], that a body
attracted towards a fixed point, by a force in the inverse duplicate ratio of
the distance, describes a conic section. Now in the relative motion of m
about M, this last body being supposed at rest, we must transfer to w, in a [530'"]
contrary direction, the action which m exerts on M; so that in this relative
motion, m is attracted towards M, by a force equal to the sum of the masses
M and m, divided by the square of their distance ; the body m will therefore,
upon the same principle, [380'''], describe a conic section about M. But the
importance of this subject in the theory of the system of the world, requires
that we should resume the investigation, in order to place it in a new point
of view.
For this purpose, let us consider the equations (JST) § 15 [525]. If
we put
it is evident by § 14 [499'], that if we notice only the reciprocal action of M
and m, we shall have*
Q = ^=:-=r; [530V]
I
*
(380) Putting m', m", he. equal to nothing, it makes X [412] vanish, and Q, [499'], becomes
simply Q = -^=-. But - = M . COS. d, [517'], and tang.fl = «, [517"], hence, [530a]
cos. 6 = = /-— i — = -y, , , consequently — = >■■ , , and
81
322 ELLIPTICAL THEORY OF THE [Mec. Cel.
[53Gv:] the equations {K) [526] will then become, dv being constant,
Difteren- J 4 ^^
tial equa- U t = -; — r" I
tionsofthe fi y^
motion of
one body rl f1 ii a
about Q _ ^_^ I f*
another '-' 7 q T^ •*
considerert
as at rost.
^^^ A^(l+55)t'
[531] "=rf^+^!
The area described during the element of time dt, by the projection of the
d V
[531'] radius vector, being equal to* ^ • — ^ » the first of these equations shows that
this area is proportional to that element, consequently in a finite time, the
area is proportional to the time. The last equation gives by integration,!
[532] 5 = 7 . sin. (V — 6),
y and ^ being two arbitrary constant quantities. The second equation [531]
gives by integration,!
t^^ i.=^q-^.{v/TT?^ + e.cos.(^-^)} = ^±^;
These bemg substituted in [525] they will become as in [531].
* (3S1) By [372a] this area is equal to the square of the projection of the radius vector,
multiplied by half the differential of the arch, or by referring to the figure in page 306,
1 dv
[530c] ^ . CB^ Xdv, and since C B = r . cos. 6 = -, [517'], it becomes i .
tt2
as
m
[531'].
f (382) This equation is obtained as in [864a], putting y = s, t = v, a = 1, 6 = 7,
9 = — d. And it is easily proved, for 5 = 7. sin. {v — &) gives ds=^jdv . cos. {v — ^),
and as rf « is constant [530"], dds = — 'ydv'^. sin. {v — 6), hence -— + « = 0-
{ (383) That the assumed value of u, [533] satisfies the second equation [531] is easily
proved by substitution and reduction, and as it contains two arbitrary constant quantities, it
must be the complete integral. For, by putting e = /ii^ax ' ^® terms of [533],
depending on the angle v — « will become e . cos. {v — th)', substituting this for u in the
second of the equations [531], it produces the terms — e . cos. {v — -a) -\-s. cos. {v — -sy),
n. iii. <^ 16.] MOTIONS OF TWO BODIES. 323
e and -a being two new arbitrary constant quantities. Substitute in this
expression of u the value of 5 in terms of v [532] and then this value of u in
d V
the expression dt = - — ^ ; the integral of this equation will give t in a. [533']
function of v ; we shall then have v, u, s, in functions of the time.
which mutually destroy each other, so that it will be only necessary to notice the other term
dependmg on ^l-\-ss, and if for brevity we put —6, this term will become
6.(14-* *) } which, being substituted in the second of the equations [531], produces the
following terms, observing that fjt = 6 A^ . (1 + 7^) in the last term,
The 6rst of these terms b — —f^ — being developed becomes
and since ~TY^^ — *' [^^1]> ^^ changes into 6 . (1 + ««)~ ] T"^ — ^ — *"* ( >
connecting this with the second term of [532a], b . {I -\-s 5)*, which may be put under
the form b . {l +ss)~Kl\ -\-2s^-{-s^], the sum becomes b.{l-{-ss)'~^.\l-^^-^~\,
and if we substitute the value of s, [532], it becomes
b.{l+ss)~^Al+f.sm.^{v—d)-{-y^.cosJ^{v — d)l =6.(l+«2r^.(l + 72),
which being equal, and of an opposite sign to the third term of [532a] renders the whole
equal to nothing, therefore the assumed value [533], satisfies the second of the equations
[531]. Fin%, by [517'] we have u = — — =- .v/l + taDg.2d=^^:^SIi, as [5326]
in [533].
We shall now show how the same equation may be solved directly by the method given
in [865a, J] . Putting in [865a], y = u, t=zv — L a=l, a=— . Q= — ^3,
^^ ^ ~ /[TTiIlIT^ ' *? = ^ — ■*' \ivi\aa become like the second of the equations [531],
and the general solution [8656], will give the following value of m,
f** / \ I . /*dt.cos.t /*dt.s\n.t
324 ELLIPTICAL THEORY OF THE [M^c. Cel,
The calculation may be considerably abridged by observing that the value
of s [532] indicates that the orbit is vs^holly in a plane inclined to the fixed
plane by an angle w^hose tangent is 7, and the longitude of the node 6,
In which, for brevity, t is retained instead of v — ^, and this makes [532] become
s / 5^
s = y . sin. t, hence sin. ^ = - ; cos. t = 1/ 1 ^^ whose differentials are
^5
d! % . COS. < = — ; it. sin. t = — "^" , these being substituted we shall get
v^-'i
a .1 / 1 ^
fie as p ds Y 7 /»
ds 'k 7^ y» sc^«
/d s s
— — — -3 = —,===, as is easily proved by differentiation ; also
5<?5 y . y/72 — 52
(1+7') ViT^'
-[-S«)2
>t7
i-P-('+")^
for the differential of the second member is
ysds y.^yH—.s^'Sds
ysds sds
Hence,
«.^i
The two last terms of which may be thus written,
a ( 2, (2^:zii)> g 7!i(i±fi) _ fLi\/iTii_
/./rqr^-|* -r i_|-y2 5 — ;^^V/1+^' 1+7^ 1 + 7^ '
and by substituting the value of «, it becomes ^'V, ,' f, . Hence
" == zXTiT^ * 1 ' * ''°'' ^"^ "" ""-^ "^ ^^*^^^ 1 '
as in [533].
II. Hi. § 16.]
MOTION OF TWO BODIES.
325
counted from the origin of the angle u.* Referring therefore to this plane ^533///^
the motion of m, we shall have 5 = 0, and y = 0, which gives [5SS']
u = -= -^.{l +e . COS. (v — -:^)} ;
This equation corresponds to an ellipsis in which the origin of r is at the
focus ; — ^ is the semi-transverse axis,t and we shall put
[534]
f^.(l-e')
A2
a =
= the semi-transverse axis,
e = the ratio of the excentricity to the semi-transverse axis, [534']
CT = the longitude of the perihelion.
* (384) Suppose a spherical surface AB C to
be drawn about the origin of r, with a radius equal to
unity, to intersect the plane of x, y, in the great circle
A B, and the plane of the orbit in its ascending node
A. Through this point draw, on the spherical
surface, a great circle A C, such that the tangent of
the angle BAG may be equal to y. IVIake the arch
AB =-v — ^, and draw, perpendicular to it, the arch B C cutting A C in C. Then by
spherics, tang. B C = tang. B A C . sin. A B = y . sin. [v — 6), or, by [532], 1-533--]
tang. B C = s, and since s [517"], represents the tangent of the latitude, it follows that the
planet in its motion must have the same latitude as if it moved in the plane of the great
circle A C.
t (385) From [378] we get ^ = a.{l-e^) * ^ ^ + ' " ^^^- (« — *)}'
/l2
equal to -, [534] we shall get -^j--^ = — , or ^ (i_^,)
semi-transverse axis, [377"], a e the excentricity, &;c. This gives
A = vV.a^.(l— e^)^,
which being substituted in the first of the equations [531], it becomes
dv
a:
dt^
\/^.a^.{\—e^f.y
putting this
a being the ^534^]
[5346]
[534c]
1 1 ,.| J c COS i'O '^\
and since m = - = — I— ^- — '-- [534,378], it will become, by substitution, as
r 0.(1 — f^) *-
in [535].
82
^^^ ELLIPTICAL THEORY OF THE [Mec. Cel.
The equation dt= — ^ [^31] will by this means become
[535] ^ ^ aK(l--e')^.dv ^
V^.{l+e.cos.(i; — «)f
We shall develop the second member of this equation, in a series of cosines
of the angle v — w and of its multiples. For this purpose, we shall
commence with the development of the quantity ; r, in a
^ ^ -^ 1 -f e . COS. {v — Hi)
similar series. If we put*
[536] X = "
l_|-\/l_e2
we shall havef
1 !___ ^ 1_ X.C-^^-^^-^
[537] 1 -f- e . COS. (v — -cr) """ \/Y^^ ') J .ix.c^^""*^'^^"^ 1 +X.c~^^~'^^'*^~^
* (386) The expression [536], may be put under other forms which it will be useful
to notice. First, by multiplying it by 1 + \/l — e^, and transposing X, we get
X . v/l — cc =e — X; squaring, rejecting X^ from each side of the equation, and dividing
by e, we get — x^ e = e — 2 X, whence
2X
[536al ^-I+TX-
This gives
[5366] l-e = :j-^-; l+e-^-j^^^-, — -_^^_^^^, ^___^_^^
^^{i-x)2, , , (i+x)2_ l-e (i-x)2^ ^ yr=T"_ir-^
Also,
4X2 i_2x2+X4 /1_XX\2 ^ 1— XX
Again, ^^^-l = — -l = j^, and y/^^+ 1=^_^ + 1 =j-^.
v/^
i— 1
[536rf] consequently ■ = X
v/^
^+1
(«— '5j).v/:ri _(v — •z3).v/iri
- c
[5376] putting for brevity «=c ~\ becomes cos. (v — «)=:^a? + Ja?~S and since
[537a] I (387) By [12] Int. cos. (v — vs) = - i^ '- This, by
II. iii. <^ 16.] MOTION OF TWO BODIES. 327
c being the number whose hyperbolic logarithm is unity. By developing ^537/1
the second member of this expression in a series, of which the first part is
arranged according to the powers of c )-v-i^ and the second according
to the powers of c~"^^~"^' , and then substituting, instead of the
imaginary exponential quantities, their corresponding sines and cosines ;* we
shall find
1 _ 1
1 + e . COS. {v — -sj) \/l — e2 [538]
.{1— 2x. COS. («?—«) + 2x2. cos.2(?j—7tf)— 2x3. cos.3(v-—n)+&c.}
Put the second member of this equation equal to 9, and q = — ; we shall [538']
have in generalf
t == ± Vg/ . [539]
{1+e.cos. (zj — zj)}"*+^ 1.2.3....OT.(?9«'
, [536a], we shall get
1+XX
1 1 1 + X2
l+c.cos.{«— «) 1 I ^ .{x \ j;-^) l + XS-fXx+Xx-i
1-j-X A
1 + X2
1 + X2 ^ 1 Xa:-1 )
;7=T) — nZxa* (l-|-Xx"~l-fXx-i> '
I
{l-|-Xa:).(l-j-Xi
substituting = —;===, [536c], it becomes as in [537].
* (388) Using the symbol x [5376], and developing the terms of [537], according to
the powers of x we shall find = 1 — Xa^ + X^a;^ — X^a^-j- &^' and in like
Xx-i
manner — -—— r = — 'kx~'^-\-y<^ x~'^ — X3a?~3 + &c. The sum of these two
l-|-Xx-i '
seriesis \ — \.{x-\-x-^)-{-\^.{a^^x-^) — \^.{!i^-\-x-^)-^kjc,', and by [537a, 6],
a;4-a;~^=2cos. (t; — -m); a;^+*~^==2cos.2.(t; — -a); a;3-j-*~^=2cos.3.(« — «),&«.
therefore the preceding series is equal to
1 — 2X . cos. (v — «) +2 X* . cos. 2 . (t> — 1*)— 2 X^ . cos. 3 . (t> — «) -f See.
consequently, the formula [537] becomes as in [538].
+ (389) This value of (p=-- ; rives i = e<Pf or
^-\-'^'^^'{^-^) ^ ^ + cos.(t;-«) [538«]
328 ELLIPTICAL THEORY OF THE [Mec. Cel.
d q being supposed constant, and the sign + or — taking place, according as
m is an even or odd number. Hence it is evident that if w^e suppose
[540]
[541]
1
= (1~.^)
.{l+-£^^'^cos.(«5— •ts) + £(2)^cos.2(D— ^)-}--E^'^cos.3(2;— 7.)+&c.| ;
we shall have, whatever i may be*
2e'.?l+»V-l-^j.
= — , and by putting for brevity g' + cos. (u — zs) = Jl^ A~^=—, now
the assumed value of A gives f — j = 1 , and if we take successively the differentials of
the equation A~^ = ( — ], considering g' or e only as variable, and substitute f — j=l,
we shall have
dq d(^ ag3
2.^A.Jl-^=^ — A^; &c. ±1.2.3.4....m.^-'«-^= — SlL ,
dq'^ ' dq^
Hence ^-»-. = ± j^^^_W_, b«,
l^'-f cos.(u — zi)l '» + ! n-}-e.cos.(v — zJ) | ™ + i '
putting these two expressions equal to each other, we shall get
= ± '- '-^ ; [539].
51 + e.cos.(t7 — trf)|'»+i 1.2.3. ...m.dq'>
-^e-Kd.n.)
[541a] *[390) Putting m= 1, in [539], it becomes ___^--^ = dr~
Now from the assumed values of 9, q, [538'], we shall get,
'^ ^— - . $1— 2X.C0S. (v— «) + 2x2.cos.2(v — -cj)— 2x3.cos.3(«— ttf) + &;c.?.
9 \/9 9
Hence
iL.iipoatii
-AiZ = — --^n . W — 2 X . cos. (« — -51) + 2 X2 . COS. 2 (v — •«) — &c. >
11. iii. § 16.] MOTION OF TWO BODIES. 329
the sign + taking place if i is even ; and the sign — , if i is odd ; supposing
therefore
we shall have*
ndt=dv.{l + E^'\cos.(v^vs)J^E^^Kcos,2(v—zi)JrE^\cos,3(v—^)-\-&Lc.}; [542]
and by integration
nt+e = v+E^'Ksm.(v—'^) + i.E^'^.sm.2(v—:^)+^.E^'\S(v—zs)-\-kc.; [543]
£ being an arbitrary constant quantity. This expression of nt-\-s is very
converging when the orbits are nearly circular, as is the case with the orbits [543^
of the planets and satellites ; and we may, by inverting the series, find the
value of «; in i ; we shall attend to this subject in the following articles.
this being multiplied by —e-^, or —q% gives -r— —-,[5410], equal to
Jl-}-c.cos.(v — '^)\^ J' ^l
r~^l)i • ] 1 — 2 X . COS. (» — -is) + 2 X^ . COS. 2 (v — xrf) — &c. I
— ^ 2_1 • ] — ^ ^°^- (^ — -5^) + 4 X . COS. 2 . {v — -Si) — &c. ^ . — ,
and if we put, as in [540], the terra of this series corresponding to cos. i.{v — -a), equal
to (1 — e^) ^.E^^, we shall have
(l-e2)"lE»=±r-^A.2X'=F7^.2t.X'-i.^,
' [qq—W ^^q^—l dq'
or by substituting q = -, and multiplying by (1 — ee)^ ;
e dq
Now X=:
rfX — 1 — X — Xe ^
hence E^^ = ±:2>} ,[\ -\- i . \/l — c2 1 , and by substitutmg for X its value y ,
[536], it becomes as in [541].
* (391) IMultiply the first member of [535] by n, its second member by the value of n
_ 3
[541'], a ^ . v//A , and substitute the expression [540], it will become as in [542],
83
330
[543']
ELLIPTICAL THEORY OF THE [Mec. Cel.
When the planet returns to the same point of its orbit, v is increased by
the circumference of the circle, which we shall always denote by 2'j( ; putting
T for the time of a revolution, we shall have,*
[544] rr ^-ff 2ir.a^
[544^
This expression of T may be deduced directly from the differential expression
of dt without having recourse to series. For by resuming the equation
[631] dt = -^, or J< = ^, [534], it gives T =J^ .
fr^dv is double the surface of the ellipsis [372«], consequently it is equal to
[544'] 2*.a^\/rir^ [378v] ; also /t' is equal to m- a . (1 — e^) [534a] ; hence
we deduce the same expression of T as that above given [544].
If we neglect the mass of the planets with respect to that of the sun, we
[544'"] sJiall have \/jr = \/~m [530'"] ; the value of ^'iL will then be the same for
all the planets ; T is therefore proportional to a^ , consequently the squares
of the times of revolution are as the cubes of the transverse axes of the
[544i^] orbits. We see also, that the same law takes place in the motion of the
satellites about their primary planet, neglecting their masses in comparison
with that of the planet.
17. We may also integrate the differential equations of the motions of
The
mution of
about' two bodies Mand m, which attract each other in the inverse dui^licate ratio
another,
computed
in a
(lilferent
manner.
of the distances, in the following manner. Resuming the equations (1), (2)
and (3), ^ 9 [416 — 418], they will become, by considering only the action
[544 V] of two bodiesf M and m, and putting M+ wi = m- [530'"],
* (392) Since nt-\- s = v -\- E^^'> . sm. (v — 'a) -{- he. [543]. If we increase t by T,
and V by 2 -r, we shall have n . {t-{-T)-\-s = (v -{-2 'ir) -\- E^^K sin. {v — w)-f-&;c.
Subtracting the former from the latter we get n T=2ir, or T= — ; substituting n,
[541'], it becomes T=-y=r , as in [544].
f (393) In this case X, [412], vanishes, and the equations [416 — 418] become as
in [545]. 1 Ubf ll..
n. iii. <^17.] MOTION OF TWO BODIES. 331
„ ddx (x.a:
0 = ^ + ^ ). (O)
^ ddz [t'.z
The integrals of these equations will give the three co-ordinates a:, y, z, of
the body m, referred to the centre of M, in functions of the time t ; we shall
then have, by § 9, the co-ordinates ^, n and / of the body M, referred to a
fixed point, by means of the equations [421, 422],
|=ffl+6<— ^; n = «'+6'«_-|^; y=«"+6"«_^^; [546]
Jvl-Ym M-\-m M-\-m
Lastly, we shall have the co-ordinates of m with respect to the same fixed
point, by adding a; to <^, ?/ to n, and z to y ; we shall thus have the relative
motions of M and ?w, and their absolute motions in space. All that is now
required is to integrate the differential equations (0) [545].
For this purpose, we shall observe, that if we have between the n variable
quantities sP''^, x^^\ x^^ a;^"\ and the variable quantity t, whose differential
is supposed constant, a number of differential equations denoted by n, of the
following form.
in which we suppose s to be successively equal to 1, 2, 3, n\
A, B, H, being functions of the variable quantities x^^\ 3f^\ xP\ &c., and
t ; A, B, H, being symmetrical* with respect to the variable quantities
xf^\ x^^\ x^*^ ; or, in other words, they will remain the same, when we
change any one of these quantities x^^\ aP, a;^"), into any other of them,
and the contrary ; we may suppose
a;W = flW . a:(»-* + '> + 6^') . a:(»-'+^ + h^"-^ . a;<»> ;
aj^*-*") = «(*—■> . ar^"- *+ ^> + 6^"— "^ . a:("-*+^ + ^("-'\ a:^ ;
* (393a) The only condition necessary to be observed relative to the quantities
Ji,B H is that they must be the same for all the differential equations of the form
[547], it is not generally necessary that they should be synmietrical.
[547'J
[548]
332 ELLIPTICAL THEORY OF THE [Mec. Cel.
a^^\ b^^\ h^^^ ; a^\ b^^\ &c., being arbitrary constant quantities, the
number of which is i.(n — i). It is evident that these values satisfy the
proposed system of differential equations ;* moreover they reduce these
equations to i differential equations, between the i variable quantities
^(n-i+i)^ ^(n-i+s)^ ^(n)^ Thclr Intcgrals will introduce i^ new constant
quantities, which being united with the i . (n — i), just mentioned, will
* (393) To illustrate this we shall take the case where i = 3, and it will be easy to
extend the demonstration to i = 4, 5, &tc. In this case the n differential equations of tlie
order i are
^ <Z3^(3) ^2^(3) dx^'^ .rr (3)
[548a] 0=^ + ^il^ + 5.i^-fff.«:(.);
dt'^ at^ at
dt3 ^ dt^ ^ Tt ^^'^ '
In this case the expressions [548] will give the quantities xf'^\ ccP^ aP^ ocf^^ 3fi*~^\ in
terms of x^^~^\ a;^"~^\ «("^, any one of these quantities, as x^^\ will be expressed in the
[5486] following manner, af^^ = d«'> . x^""'^^ ~\~ ¥^K x^''-^'^ -{- c^^K af""^ ; d^\ h^\ 6''\ being
arbitrary constant quantities. For this value of 0!^^ gives
d xf«^ = afe> . <? a:(«-2) -f- 6^) . rf a;^"- " + c(«> . df a; « ;
d^ a^*) = a^^^ . d^ a?("-2>+ A^*^) . d^ «("-i> -{-(^^Kd^ a^»> ;
ft jj 1
multiplying these values respectively by H, — -, — , — , and adding the products
together, the sum will be equal to ^1^ + A . ^^ + JB . ^^ + H . x^\ and this
ought to be equal to nothing, if the assumed value of x^^^ is correct. Now this sum is equal
to the following expi:ession.
n-iii. §17.] MOTION OF TWO BODIES. ^33
make the i n constant quantities necessary to complete the integrals of the
proposed differential equations.
If we apply this theorem to the equations (0) [545] ; we shall find that
z = ax-\-hy<, [548']
a and h being two arbitrary constant quantities.* This equation is that of a
plane passing through the origin of the co-ordinates [19c] ; therefore the [548"]
orbit of m is wholly in the same plane.
The equations (0) [545] givef
+ 6(.).^!^"_!i + ^.60r).^!^l! + 5.6(.).^£^ [548rfJ
di^ dv' at
which is evidently equal to the sum of the three last of the differential equations [548a],
multiplied respectively by a^^\ b^\ c^^\ and added together ; therefore this sum is equal to
nothing, and the assumed value of x^«\ [5486], containing the three constant quantities a^*\
U^\ c^^\ is correct, and each of the n — 3, or n — i, of the first of the proposed equations [548a],
furnishes 3 or i constant quantities, making in all (n — i) .i quantities. Again, the values [548e]
jP^^aF^ a;^"~^^, [548] being substituted in the three, or i, last equations [548a], they
will contain only the quantities o;^""^^, a;^"~^^, a;^"^, and their differentials of the order 3 or i.
These three, or i equations of the order 3, or t, being integrated will introduce 3X3, or r*,
new arbitrary constant quantities, adding these to the in — v^ quantities [548e], the sum
becomes i n, which is the whole number required to complete the integrals of the n proposed
equations of the order i.
* (395) The equations [545] being compared with the general form of the expression
[547] give n = 3, i=2, A = 0, H=-, r and H being symmetrical in x, y, z.
In this case n — i becomes 1, and the series of equations [548] will be reduced to the first
a/i> = a(« . a^^^ + 6"> . aP\ and by putting a^" = z, x^^^ = y, x^^^ == x, ¥^^ = a,
d^'>=b, it becomes as in [548'].
f (397) IVIultiplying die equations [545] by r^ and taking their differentials we get [549].
The differential of r^ = x^ -\- 7^ -\- z^, [411], gives rdr = xdx-{-ydy-{-zdZj [549']. [54903
84
^^^ ELLIPTICAL THEORY OF THE [Mec. Cel.
[549] 0 = d.(r'.^-^^ + ^.dij \. (ff)
Now by taking the differential of the equation [411]
[549^ rdr=^xdx-^ydy-\-zdz
twice in succession, we shall have
[550] r.d^r+Sdr.ddr=x.<Px+y.d^y+z,d^z-{-3.{dx.ddx-]-dy.ddy+dz.ddz};
consequently*
ddzi
r«;^n d ( r" ~Vr^ \x ^4-v ^-^-A-z — Usr^ \dT —''-\-dij ^^^ 4-d2
[551] a.yr. ^^^j-r.jx. ^^, -ty-^^^ -\-z. ^^^ ^+^r^.jdx. ^--\-dy.-^-\-dz.
dfi
Substituting, in the second member of this equation, for ^x, ^y, ^z, their
values given by the equations (0') [549], and then, instead of ddXy ddy,
dd Zj their values deduced from the equations (0) [545], we shall findf
[552] o==d,(t^.~^-hi^dr.
r2
* (398) IVIultiplying the equation [550] by -— , the first member of the resulting
//3 ^ d d T
equation becomes r^ . — -[- 3 r^ . <? r . -— , which is evidently equal to the diiFerential
d d T
of r^. -— - . The second member, without any reduction, is of the form [551].
f (399) The terms of the second member of the equation [551], depending explicitly
d^x d d X _
on X, are t^ .x . — -j- 3 r^ . rf a? . -— . The first of the equations [549] being developed
X d^x n d 3* Hi X d X
and multiplied by - , gives ^•^•TTi^ — ^rdr.x. — — j substituting this
. , J. . _, ddxu,xdx.^„^ddx .
m the precedmg expression we get — 3 r a r . a? . -r-^ f- 3 H . a a? . -— - , and
ddx iLX r^.^-i .1 ^ ■, 1*3: iLxdx ^ „ , M<x
smce -7— = -, [545 1, It becomes 3rrfr.a;.'--r ^r^ .dx . — , or
dv^ ir^ 1^ r r3
4 Ml 3 It c? r
by reduction .xdx-\ — .x^. In a similar manner the terms depending on y
I
n. iii. § 17.] MOTION OF TWO BODIES. ^^^
If we compare this equation with the equations (O') [549] ; we shall have,
d cc d 'u d z d T
by means of the theorem above given [548], supposing — -, -r^, — , — , [552'j
CL z ax ct z Cv z
to be the variable quantities aP, xf-'^\ xf^, a^^^ ; and r to be a function of the
time t ;*
dr^\.dx-i-y,dy ; [553]
4 y< 3 M" rf r
and z, found by changing successively x into y and z, are .ydy-] — . ^ ;
.zdz-\ — . z^. The sum of these three expressions, putting
xdx-\-ydy-\-zdz = rdrj ocr^ -{-i^-\- z^ = r^, [549a],
4 ifc 3 fJi'dt 7*
is r dr-\ — . r^ = — i^'dr. This being substituted for the second member
of [551] gives d .(r^ . -f^j = — l^dr, as in [552].
dr
* (400) Divide [552] by d t, and put — = a/^^ it will become
Dividing this by r^ and putting A = —r- , H=:-, we shall get
d d t(1> d r('>
rff'^ at
dz
Putting —=ixf^\ in the third of the equations {(y), divided by dt, we shall get
0 = d .(r^. ■ - j-\-it.aP^ , which bemg developed and divided by r^ becomes
0 = ^~ + A . ^:^ + H. a;(2> . The first of the equations (O'), developed in the
0/ Z CL Z
^ ^^ .JVi ■ n d doP^ , a d X^^^ , rr ^-^^ j
same manner, puttmg —-^ar^ , gives 0 = — — - — -\-A . -■ + ^ • ^ > ^^^
at dv^ dt
d v
the second of the equations (C), by putting — = a;(^> , becomes
In these four equations in ccf^\ «^'^^, nP^, x^^\ the terms A, H, may be considered as
functions of f, being all similar to the equation [547], making t = 2, n = 4, and they will
furnish two equations of the series [548],
a^C) = a^') . aP> + 5<'> . a^^> ; z^^ = o^^ . a<3) _^ j® . a:(4).
^6 ELLIPTICAL THEORY OF THE [Mec. Cel.
X and y being constant quantities ; and by integration,
[554] 7- — ^xa: + 7«/,
— being a constant quantity. This equation, combined with the following
[548', 549«],
[555] z=^ax-\-hy \ r^ z= '3^ -\- y^ -^ z^ \
gives an equation of the second degree, in x and y^ in x and z^ or in y and z ;*
whence it follows that the three projections of the curve described by m about
ilf, are of the second order ; and as this curve is wholly in the same plane
[555'] [548"], it is itself a curve of the second order, or a conic section. It is easy
to prove, from the properties of curves of this kind, that as the radius vector
[555"J r is expressed by a linear function of the co-ordinates a:, y ; the origin of
these co-ordinates must be at the focus of the section.!
3t T d oc d li
Substituting in the first, for a;(^\ a;^^^ a;^'*\ their values —5 T~) jfj [552'], it becomes
Of V d Z (t V
dr = d^'> .dx-{- ¥^^ . dy, which agrees with [553], putting d^'> = X, Z>^^> = 7. The
second becomes dz = a^^^ .d x-\-b^^Kdy, and agrees with z = ax-{-by, [548'],
putting d^^ = a, U^^ = b.
* (401) Substituting the value of r [554], in the second of the equations [555], it
[553a] becomes f [-"Kx -\-yy\ = x^ -\- y^ -\- z^ ', and if in this we substitute either the
value of z, x, or y, deduced from the first equation, [555], z = ax-{-by, we shall obtain
an equation of the second degree. Thus jf the value of z is substituted, the result will be
an equation of the second degree in x and y, representing the projection of the curve upon
the plane of x, y, as in [555']. The projections of the curve being of the second degree,
the curve itself must be of the same order. This result may also be obtained by observing
that since the curve described by the body is on a plane, [548"], we may take, on that
plane, the rectangular co-ordinates x^^^, y^^,, to denote the point corresponding to the
co-ordinates x, y, z. Then by [172rt],
[5536] « = '^o^/;/ + -So2//«; y==AXn,-\-^iyn,y Z = A^X,,-\-B^y„,',
because z^,, = 0, the body being supposed to move on the plane of a?,,^, y,^,. These values
of a?, y, z, being substituted in the equation [553a], will produce the equation of the curve
described by m about M, expressed in terms of Xm, y^,,, and this will evidently be of the
second degree, or a conic section.
•j- (402) The equation [554] may be reduced to an expression of r in terms of the
co-ordinates x^,^, y^^^, .taken on the plane of the apparent path of the body m about M, by
II. ui. § 17.] MOTION OF TWO BODIES. 337
Now the equation r — \.\x-\-yy [554] gives, by means of the
equations (0) [545],*
r._ddr \ ^ [556]
substituting for x and y their values [5536], which give
^ = ir + (^-^o + y-^i)-^'"+('^-^o+r^i).y///. [554o]
In the plane of x,^,^ y^^^ take two other rectangular co-ordinates, x", y", so that the axis of x"
may make, with the axis of x^i, an angle denoted by s ; then the co-ordinates x'\ y", and
*///» Villi being supposed to correspond to the same point of the curve, we shall have, as in
[252], by writing x,,,, y,,„ for a/, y',
x,,^ = od' . COS. s + y" . sin. s ; y,,, = y" . cos. s — a/' . sin. s.
These being substituted in the preceding expression of r, it will become
+ f . [(X^o + r ^i) . sin. s-j- (X So + r-Bi) . COS. s\.
Now as s is arbitrary, we can take it so that the coefficient of y" may be nothing ; this value
of 6 being substituted in the coefficient of a/', let its result be — e, and we shall have
r = ex" = e.< a/'V. Which is the noted theorem used in page 243 to
r" (. r" ^ J
demonstrate the properties of the conic sections. For by referring to the figure in that page, and
putting SD = — , SF=x", SP=r, the preceding equation will become [5546]
SP = e.{SD — SF) = e.PE, being the same as in [378a], where the origin of the
co-ordinates is taken at the focus S.
* (403) The second differential of [554] divided by dfi, gives
ddr ddx . ddy
Ifi' ~ '~dfi'~^'^'~d^'
and from [545] we get -^^=_— ; -_=_—; hence -^=-_ . (Xa:-f-yy) ;
now from [554] we get (X a? -j- 7 y) = r , whence = . (r ),
as in [556]. This being multiplied hy 2 dr becomes 0 = 2. ^^^^ _j- 2 ^ . ^ — ^^^ ,
whose mtegral is — -f - -|- — - = 0, and this multiplied by r^ gives
'•'•S~2^r + ^ + A^=0, [557],
whence we easily obtain d t, [558].
85
338 ELLIPTICAL THEORY OF THE [Mec. Cel.
Multiplying this equation by d r, and taking the integrals, we shall have
[557] ^.iz!_2^r + ^-^ + /i^ = 0;
at 0,
a' being an arbitrary constant quantity. Hence we deduce
rdr
[558] dt =
v/2.-4-fv-;
this equation will give r in a function of t ; and as x, y, z, are given, by what
[^ J precedes [554, 555], in functions of r ; we shall have the co-ordinates of m,
in functions of the time.
18. We may obtain these equations by the following method, which has
the advantage of giving the arbitrary constant quantities, in functions of the
co-ordinates rr, y, 2, and of their first differentials ; which will be useful in
the course of this work.
Suppose that V= constant is an integral of the first order of the equations
d CG d ^ d z
[558"] (O) [545], V being a function of x, y, z, ~J~ -> -f- •> ~i~ 'i ^"^^ if we put
, dx , dy , dz
[558'"] X =-r-, 1/ = -^-, 2f — -r,
the equation V= constant will give, by taking its differential,
^ ^ \dxj dt~\dy) dt~\dz) dt~\d3fj dt^Xdy'J dt^\d2fj dt '
but the equations (0) [545] give
[560]
docf ii'X di/ l^y . ds/ 1*2:^
77 "^ "~ 73" ' ~dT "^ "" "^ ' ~dT^'~'^'
therefore we shall have this identical equation of partial differentials,*
d 3/ d d cc d u d d 11
[560a] * (404) The differentials of [558'"], divided by d t, are — = — — , J7 = -r# »
(It (t t CL Z eft*
rf z' ddz
-—==——, hence the equations [545] become as in [560]. Substituting these in the
at d^
differential equation [559], and putting for — , -f- , — , their values a?', y', 2', [558'"],
CL Z Ct Z Gr Z
we shall obtain [561], which is a differential equation of the first order, without a constant
quantity, and is evidently identical.
[562^
[56^']
n. ili. '^ 18.] MOTION OF TWO BODIES. 339
It is evident that every function of x, y, z, a/, y', z', which substituted for
V in this equation renders it identically nothing, becomes, by putting it equal [56iT
to an arbitrary constant quantity, an integral of the first order of the
equations (0) [545].
Suppose
V=U+U'+U" + kc.; [562]
U being a function of the three variable quantities x,y,z; U' 3. function of
the six variable quantities x, y, z, x', y', zf, but of the first order* relative to
x\ y, z' ; U" being a function of the same quantities, but of the second order
with respect to a/, y, z' ; and so on for the vest. Substitute this value in
the equation (/) [561], and compare separately. First, the terms without
a/, y', z' ; Second, those containing the first power of these quantities ; Third,
those containing their squares and products ; and so on for others ; we shall
have
^■•ci^)+^'•(-.y)+^•(^)=s•^•(f->^(f->-c^)}
&c.
The integral of the first of these equations is, by the theory of partial
differentials,!
C7' = fiinction{a:y — yxf, xz' — 2 a/, y zf — zy', x, y, z,] ; [564]
* (405) By the first order is to be understood terms of the first degree in a/, y', z',
excluding their powers and products. By the second order, terms of the second degree in
a?', 2/, z', excluding their powers and products of the third degree, or above, &;c. ; and
since by substituting the value of V = U -{- U' -\- he. in the equation [561], it ought to
be identically nothing, the terms of the first, second, tliird, he. orders, must be separately
equal to nothing, which will give the equations [563].
f (406) In the equation [563]
fdU'\ , /dU'\ , /dU'\
^^0 ELLIPTICAL THEORY OF THE [Mec. Cel.
as the value of U' ought to be linear in x\ 3/, z!^ [662'], we shall suppose it
to be of this form,
[565] U'=.A.(xiJ^y3^)-\-B.{xz' — zx') + C.{yz!^z'i/) ;
the partial differentials relative to x\ y', z!, exist, but those relative to «, y, z, do not occur,
so that it is in the same situation as if U' was a function of only the three variable quantities
a/, y', z'. In this supposition the general value of d U' will be
and if to this we add the preceding equation, multiplied by , we shall eliminate
{^) ana Shan find . W = (ijf) . f-^-i^^ + (i^) . {^-^).
j-gg.,. If we now put xy^ — yoc/=p, xz' — z x' = q, y sf — zy' = s, we shall get, by
taking their differentials, xdy' — ydx' = dp, xdsi — zdx' =^dq', therefore
d U'=— . (-^-7) 'dp-\ — . ('T~r) • ^ ? J in which the second member must, like the
[564c] first, be an exact differential. This condition is evidently satisfied by putting U'=(p. (p, q),
(p being a function of p, q, and we may include in it the terms x, y, z, considered as constant,
putting U' = (p.{p,q, X, y, z). But - .q .p, is a function of the five quantities
Pi q^ '3?, y, z, included under the sign 9, we might therefore introduce this quantity also, or its
equal, - .{xz' — zx') • (^ y' — y^)=y^' — ^if =^s, [5Q45]. Hence a more
[564(Z] symmetrical expression is U' = cp . {p, q, s, x, y, z), which is the same as that assumed in
[564]. To prove its correctness a posteriori, we may substitute it in the proposed equation
[564a], writing for brevity, 9 instead of cp . {p, q, s, x, y, z), we shall have
and since by [564J], (^) = -y, (^) = _z, (^) = 0, it becomes
/dU'\ /d(p\ /d(p\ T • -1
X . {-r-r )== — ^^'V^j — ^^*\^]' similar manner
/'dU'\ /d(p\ /dcp\ /dU'\ /d(r>\ , /d(p\
The sum of all these is
and as the terras of the second member mutually destroy each other, it becomes as in [564a] ;
therefore the assumed value of U' [564], must satisfy the proposed equation [564a].
II. iii. § 18.] MOTION OF TWO BODIES. 341
A, B and C being constant quantities. Suppose the value of V to terminate
at U% making U'\ U", &c., nothing ; the third of the equations (/') [563]
will become
The preceding value of U' satisfies also this equation.* The fourth of the
equations (/') [563] will become
J /dU"\ , J fdV"\ , , /dU"\
the integral of which isf
C7" = function [a; i/ — ya/, X2! — zod, y 2! — z'lf, a/, y', 2'}. [568]
This function ought to satisfy the second of the equations (/') [563], and
the first member of this equation multiplied by dt is evidently equal to
dU \X the second member ought therefore to be the exact differential of a
*(407) From [565] we get (17-)=='^^' + ^^' (-^) = — ^ a/ + C «',
(i£-\= — Boc/—Cil, Hence
in which the terms of the second member mutually destroy each other, and it becomes as
in [566].
f (408) The integral of the equation [567] may be easily deduced from that of [564a],
since the former may be derived from the latter by changing U\ x, y, z, a?', y', 2', into
C/", x', y', z', X, y, 2, respectively. •*
Now these changes being made in p, q, s, [5645], they would become respectively
— p^ — q^ — 5j and the expression of U', [564crj would be changed into
U"=cp. {—p, — q, — s, X, y, z').
or by neglecting the signs of p, q, s, U" = function (p, q, s, x\ y, sr'), as in [568].
X (409) The first member of this equation being multiplied by d <, gives
and by substituting x^ dt=:dx, y' dt=i dy, z' dt=^dz, [558'"], it becomes
d X . {-— \-\-dy .i—\-^dz. (— \ which is evidently equal to d (7, because by
86
342
[569]
ELLIPTICAL THEORY OF THE [Mec. Cel.
function of x, y, z. Now it is easy to see that we may satisfy this condition,
and at the same time conform to the nature of the function f/", and to
the supposition that this function is of the second order in a;', ^, 2! ; by
making*
U"=={D^ — Ex').{x'{/^yx') + {Dz' — Fx'),(xz'^zx')
-{-(E z' -^Fy') .(y z'-zy') + G , (of' + i/' + z'') ;
hypothesis [562'], U does not contain a/, 1/, z'. Therefore we shall have
and the second member, like the first, must be an exact differential in x, y, z.
* (410) This assumed value of Z7", [569], is of the form required by [568]. It is also
of the second degree in a?', 3/', sr', as it ought to be by [562']. Moreover it is necessary that
it should render the value of d U, [568a], integrable. Now as D, E, F, G, are supposed
to be arbitrary constant quantities, wholly independent of each other, it will follow that the
terms multiplied by each of these coefficients must be separately integrable. We shall
therefore examine each of them in succession. The term of Z7", [569], depending on D,
is J), ja??/'^ — y x' 1/ -\- X z' '^ — zx'2f\. If this be substituted for U", in [568a],
and then reduced by means of [549a], it will become
dU=-^. Ix .{—yyf — Z2!)-\-y .{2xy^—y(i(!)-\-z .{2xz' — zx')\
^^.[x.{yy' + z^)-^xf.{y^^z^)^^
[569a] =.^Ax,{ydy-{-zdz) — dx.{y^-\-z^)\
= — .\ X .{r dr — X dx) — dx .{r^ — c^) >
The terms depending on E, F, may be found in the same manner, or much more simply, by
the consideration that the function U", [569] is symmetrical, as it respects the three series of
quantities £>, E, F, x, y, z, x', y', z'. So that the expression [569] will not be altered
by changing each of these quantities into the following one of the same series, commencing
each series again when we arrive at the last terms F, z, and z' ; this would not affect the
value of r = \/x2-\^f-\-z^ , or the coefficient of G', [569], If we make these changes in
the term — Dy^.d . (-\ [569a], we shall obtain the terms depending on E and F, which
[5696] will be respectively —Eii.d.f-j, —Fi)..d.(^\ Lastly the term depending on
n. iii. §18.] MOTION OF TWO BODIES. ^^
D, E, -P, G, being arbitrary constant quantities ; and then r being equal to
^a;2 -1-1/2+22, we shall find
U= — ^.{Dx + Ey + Fz+2G}; [-'^70]
we shall thus have the values of U, U', U" ; and the equation V= constant
[558", 562], will become [570, 565, 569]
constant = — ^.{Dx-{-Ey + Fz-{-2G}+(A-{- Di/ — Ex').(xi/--yxf)
+ (B + D2f — Fa/).(xz' — zx')-{'(C-{-E2f-^Fy').(y2' — zt/)
+ G.(x'' + ^/' + n'
This equation satisfies the equation (/) [561 ] , consequently also the diflferential ,gyj„
equations (0) [545, 561'], whatever be the arbitrary quantities J, B, C, D,
E, F, G. If we suppose. First, that all except A are nothing ; Second, that
all except B are nothing ; Third, that all except C are nothing ; &c., and
then resubstitute -^, -r, -r^ for a/, V, 2^, we shall obtain these
at at at
integrals :
G, [569], being taken for U", and substituted in d U, [568a], will produce the quantity
G.^-A2xa/-j-2yy'-^2zz'\ = G.^A2xdx-^2ydyJr^!sdz\
= GA^.2rdr = 2Gii.^ = — 2Gii^.d.(-\
Connecting together all these terms [569a, &, c], we shall get the complete value of d U,
dU=^D^.d.(^ — Eiu.d.(^--Fis..d.(A — ^G^.d.(^,
the integral of which gives U, as in [570].
In order to abridge the demonstration, it has been supposed that the form of the function
U" is given as in [569]. If this form were unknown, it might be investigated, by the
consideration that U", [562'] is of the second degree in p, q, s, xf, y', z' ; and the most
general form of a function of this kind, connected with constant coefficients a, a', a", he. is
to be substituted for U", in [568a], and the constant quantities a, of, &ic. are to be taken, so
as to make the second member of this equation to be, like the first, a complete difierential.
In this way we might obtain the function [569], connected with a few other terras, which
were neglected, not being of any use in the subsequent calculations. It was not thought
necessary to explain this calculation more fuUy ; it may however be proper to remark that in
making these substitutions, we may consider p, q, s, as constant, since the terms arising in
[568a], from <i/>, dq, ds, mutually destroy each other.
[571]
[569c]
^^ ELLIPTICAL THEORY OF THE
Important^ xdy~-ydx ^ , _ xdz — z d X „ ydz — zdy
integrals t. , C •-- ; C = —
correspon- ul ut dt
ding to the
[Mec. Cel.
relative
motion of
one body
about an-
other in
a conic
section.
(\ — fA.<^ S ^ f^y^-\-^^^\ \ , ydy.dx zdz.dx
0 = f'4-v ^ ^- fdx^ + dz^\ ) xdx.dy zdz.dy ^ \
-^ ~^^'\r \ dt^ 7 5"^ dt^ '^ dt^ ' / ^^
HA 2 \ju d x^ -\- d y^ -{- d z^
a "" T ^ dt^ '
xdx.dz ydy.dz
dt^ "^ 'd¥~
c, c', c", f, /', /", and a being arbitrary constant quantities.
The differential equations (O) [545], can have but six distinct integrals of
the first order,* and if from these we eliminate the differentials dx, dy, dz,
we shall obtain the three variable quantities x, y, z, in functions of the time
t ; therefore at least one of the seven preceding integrals is comprised in the
[57^ ] six others. In fact it is easy to perceive a priori that two of these integrals
ought to be contained in the remaining five. For these integrals do not
contain the time t explicitly, but merely its differential d t, therefore they
cannot give the variable quantities x, y, z, in functions of the time,t
consequently they are not sufficient to determine completely the motion of m
about M. We shall now examine in what manner these integrals are
equivalent only to five distinct integrals.
z d ^ ' ' "U d z
If we multiply the fourth of the equations (P) [572] by — — , and
* (411) A differential equation of the second order, has generally in its complete finite
integral only two distinct arbitrary constant quantities, and if between this integral and
its first differential, we eliminate first the one, then the other of these constant quantities, we
[571c] shall obtain two equations of the first order, each containing a different constant quantity.
In this way the three equations [545] may have six distinct integrals of the first order,
containing a:, y, z, d x, dy, d z, and by eliminating dx, dy, d z, there would remain
three equations containing x, y, z, in functions of t, and of the six arbitrary constant
quantities.
f (412) All the equations [572] contain d t, but none of them contain t explicitly,
therefore t cannot be obtained from them without another integration.
n. iii. §18.] MOTION OF TWO BODIES. ^5
00 d z —^^ z d oc
add to it the fifth multiplied by ; we shall have* ■»
^ ^ fzdii — ydz\ , ^, fxdz — zdoo\ , ^xdy — ydxX ( w- /dx^-^-diP'
0=/. ^-^- +/. — jz — +^- J y \ ^ I -T y
dt J \ dt J V dt J f r \ di
fxdy — y dx\ ( X dx . d z y dy . d z ")
"^ V d} ) ' I df ' J¥~ \ '
[573]
-r, , . . ^ xdy — ydx xdz — zdx ydz — zdy , . ,
13 y substituting for — ~t — > j > r i their values
at dt dt
given by the three first of the equations (P) [572] ; we shall have
n /'c'— /<^"i^ ^ f^ /do^dy^W xdx.dz ydy.dz ^ ,--.
^ = — i — + ^' ir-[r~~d¥~~j \ ^—dr--^—d^ ' ^ ^
* (413) In performing these reductions it will be convenient to put the fourth, fifth and
sixth of the equations [572], under a different form, by substituting d(,?^da^-\-dy^-\-dz^,
rdr = xdx-\-ydy-{-zdz, [549']. For by this means we shall have in the fourth, '■ ^
dr^-\-dz^^du^ — dx^ and ydy.dx-\-zdz.dx^dx.(ydy-\-zdz)=dx.{rdr — xdx),
Cf* {diJ^-dx'i\1 dx.{rdr-xdx) . . .
hence 0 =/ + a: . | - — (^ — -- — ) ^ -\ — , or by reduction
„ , C f* duf^") , rdr.dx
And by changing successively x into y, z, and /into /', /", we shall obtain the fifth and
sixth equations [572],
/./ ■ ( M- d(J^ ) , rdr.dy ^ ... , C |x ^^2 ) rdr.dz
Multiplying the fourth by — , the fifth by — , and adding the products
we shall get
- /zdy — ydz\ . -, /xdz — zdx\ Cfi rfw^^ <ix.{zdy — ydz)'\-y .{xdz — zdx)"^
^=f'[ — jr-J^-f'K Jt )-^lV~~^S'l dV 5
rdr ( - /zdy — y dz\ , . /xdz — zdx\')
but X . {zdy — y dz) -{-y .{x d z — zdx) = z .{xdy — yd x), also
dx .{z dy — ydz)-\-dy.{xdz — zdx) = dz.{xdy — ydx), hence
- /zdy — ydz\ , ., /xdz — zdx\ , C //Ji. d(^\ , rdr dz ) /xdy — ydx\
resubstituting for d (J^ its value d a^ -\- d i/^ -{- d zr^, and r dr = xdx-{-y dy -{- z dz,
[5716], and neglecting the terms multiplied by zdz^, which mutually destroy each other,
we obtain the equation [573]. Substituting in this the values c, d, c", [572], and dividing
by c we get [574].
87
346 ELLIPTICAL THEORY OF THE [Mec. Cel.
which becomes the same as the sixth of the equations (P) [572], by making
[574'] 0 =/c" — /' e +/" C.
Hence the sixth of the equations (P) [672], results from the first five, and
the six arbitrary quantities c, c', c", /,/'?/", are connected by the preceding
equation [574'].
If we take the squares of the values of /, /', /", given by the equations
(P) [572], and add them together, putting for brevity
[574"] /2_|_y/2_|_y.2_^.
we shall have*
[575] ^^-.^ = |r^(^ J^ )-\-lT) ]■[ — -JF- -7]'
but if we add the squares of the values of c, c', c", given by the same
equations [572], putting
[575'] c2_|-c'2 + c"^ = A2;
we shall havef
* (414) Putting for brevity —- = d, —-=: B, we shall have by [572a, 6],
—f=Ax + Bdx; —f = Ay-\-Bdy; —f" = Az-^Bdz. The squares of
these added together, using P, [574"], make
P = A^.{x^+y^ + z^)-^2AB.{xdx-\-ydy-Jrzdz)+B^.{da^-{-dy^ + dz%
or by [5716], l^=^A^.r^ + 2AB.rdr+B^.d(J^. Substituting the values of A, 5, it
beeo.e. l^-r^^-'-^J+^.m' ■ {^-^) + (-^^^J- The .s.
term of the second member being developed becomes f/-^ — 2 (x r . -— -{- ^ • -TTf
Substituting this, and connecting together the terms depending on the different powers of f*,
— M- —[-^ J^J • "^ ~ • V • rff2"~ dfi )
~l~Jt^ \dt) yidfi r y
and by resubstituting dcJ^^^dx^-^- df -\-d z% [5716], it becomes as in [575].
f (415) The values of c, </, c", [572], multiplied by d t, and squared, give
c^.dt^ = x^.dy^ — 2xy.dxdy-\-f.dx^', d^ .dt^==x^.dz^—2xz.dxdz-{-z^dx^',
_c"^.dt^ = y^.dz^ — 2yz.dydz + z^.df;
n.iii. §19.] MOTION OF TWO BODIES. 347
\ dt^ J \dt J ~' '
therefore the preceding equation will become
dar^-\-dy^-\-dz^ 2 f* , \>^ — J^
By comparing this with the last of the equations (P) [572], we shall obtain
this equation of condition,
''-^=-. 1578,
The last of the equations (P) [572] is therefore included in the six others,
which are equivalent to five distinct integrals only, the seven arbitrary
quantities c, c', c", /, /', /", a, being connected by the two preceding
equations of condition [574', 578]. Hence it follows, that we shall find the
most general expression of F, which satisfies the equation (7) [561], by [578']
taking for this expression, an arbitrary function of the values of c, c', c", /,
and /', given by the five first of the equations (P) [572].*
19. Although these integrals are not sufficient to compute a;, y, 2:, in
functions of the time, they determine however the nature of the curve
and if to the sum of these we add [r drY = {x dx-\-ydy-\-z d zf, or by developing
{rdrY = o^dx^-\-y^dy^-\-z^d3?-\-2xy.dxdy-\-'2.xz.dxdz-\-2yz.dydz,
we shall have
(c2-f c'2 + c"2) .d<2_|_(^^^)2^(34J_^y5S_j_^). d-c2_^(^2_|_y2_j_ ^2), df-\-{a^-\-y^-\-Z^).dz\
and by substituting c^ -\- c'^ -{- c"^ = 1? ', (c^ -\- y^ -\- z^ ^ r^, it becomes
h"" .df-\-{rdrf=r^ .{dx'^^dy^-Ydz'^).
Dividing this by dt^, we get [576 j. Substituting this value of h^ in [575], divided by A^,
we shall get [577].
* (416) Using for brevity, the letters c, c', c", /,/', to represent the quantities to which
they are respectively equal in the equations [572] ; then it is stated, [578'], that the most
general value of F'will be expressed by ^==9 • (c, c', c",^/'). To find whether this
will satisfy the equation [561], we shall suppose
denoting, as usual, by 9', 9", &£c., the coefficients of dc, dd, &;c. in the general differential
of V. If we put successively the values of c, c', c",/,/', for V'm the second member of
3^8 ELLIPTICAL THEORY OF THE [Mec. Gel.
described by m about M. For if we multiply the first of the equations (P)
[572] by z, the second by — y, and the third by a: ; we shall have, by
adding these products,
[579] 0=cz — dy^c"x;
which is the equation of a plane* whose position depends on the constant
quantities c, c', c".
If we multiply the fourth of the equations (P) [572] by a:, the fifth by y,
the sixth by 2, we shall have, by adding these products,t
7^ dr^
[580]
[5786]
0 =/;.+/' 2, +/" z +. r - ^ . (1^4^±^) +
dt^
the equation [561], and represent the resulting quantities by C, C, C", F, F', we shall
have
&z;c.
[578c] Then we shall have [571', 572], 0 = C, 0 = C", 0 = C", 0 = F, 0 = F'.
Now if we substitute the above assumed general value of V = cp . {c, c', c",f,f'), in the
second member of [561] it will be cp' . C + 9" . C + 9'" . C" + 9'^ P+ <p- . F', and
this becomes nothing, by means of the equations [578c] ; therefore V= 9 . (c, </, c",f,f'),
satisfies the equation [561], and as it contains an arbitrary function 9, it will be the complete
integral.
* (417) The first member of this sum is cz — <^ y -{- c" x, its second member,
omitting for brevity the divisor d t, is
z . {x dy — y d x)-\- y . (z dx — x dz) -\- x . (y dz — z d y),
the terms of which mutually destroy each other, therefore we shall have 0=cz — c'y-}-cf'x,
as in [579]. If we put c^^Bc, c" = — Ac, and divide by c, we shall get
z=:Ax-\-By, which is the equation of a plane, [1 9c] passing through the origin of the
co-ordinates.
f (417a) The equations being put under the same forms as in [572a, J], and multiplied
respectively by x, y, z, the sum of the products will be
0==f^+f'y+rz + (^^~^^y{aP + y^ + z^)V^,{xdx-^ydy+^dz).
[579a]
[582"]
U. iii. § 19.] MOTION OF TWO BODIES. ^9
but by the preceding article [576]
^='^-(, -^ )~^7^' t^l^
therefore
0 = i,r — h'+fx+fy+f"z, [582]
This equation being combined with the following [579, 555],
0 = c"x — c'y-\-cz; r^ == x^ -{- if -^ :^ ; [582^
gives the equation of the conic sections,* in which the origin of r is in the
focus. The planets and comets describe therefore, about the sun, nearly
conic sections, the sun being placed in one of the foci ; and the motion of
any planet is such that the areas described by the radius vector are
proportional to the times of description. For if we put dv for the angle
included between the infinitely near radii r and r -}-dr, we shall havef
dx' + df + dz' = r'dv^-{-dr^; [583]
Which by substituting [549a], rr = xx-\-yy-\-zz, and rdr=xdx -\-ydy-\-z d z,
becomes 0 =:fx +/' y +/" z-\-iir — r'.^-\- (j^J, as in [580]. This being
added to [581] gives [582].
^2 f fi f"
* (418) The equation [582] gives r =: -.x — -.y — — .2:, this being [582o]
(^2 f f f" \2
.X — .y — •^) =-^-\-y^-\-^^i
which is of the second degree in a?, y, z. The equation [579], 0 = cz — c y -{-(!' x^
may, as in [579a], be put under the form z=:^ Ax-\- By. From this and the preceding
equation, we find the equation of the conic section, as in note 401, page 336. Again, if we
substitute in the above value of r, [582a], the expressions of x, y, z, [5536], it will become
A2
of the form r = — \-Dx„^-\- E y^^, which is similar to that in [554a], from which we
have proved in [5546], that the origin of r is at the focus.
f (419) By [372], r^dv^-\-dr^ expresses the square of the space passed over by
the body in the time d t, being limited by the two radii r, r -f- rf r, and the included angle
dv, and by [40a], the square of the same space is also expressed by d a^ -\- d if -{- d z^,
according to the common principles of orthographic projection, with three rectangular
co-ordinates, x, y, z. Putting these two expressions equal to each other, we get, [583].
88
350 ELLIPTICAL THEORY OF THE [Mec. Gel.
and the equation [576]
becomes r'^di^ = h^dt^ ; therefore
7 h dt
[585] dv = -^.
Hence we find that the elementary area ir^dv, described by the radius
vector r, is proportional to the element of the time d t ; the area described
[585'] in a finite time is therefore proportional to the time. We also see that the
angular motion of m about M is, at each point of the orbit, inversely as the
[585"] square of the radius vector ; and as we may, w^ithout any sensible error, take
very small intervals of time, instead of an infinitely small instant ; we shall
have, by means of the preceding equation, the horary motions of the planets
and comets in difierent parts of their orbits.
The elements of the conic section described by m, are the arbitrary
constant quantities of its motion ; consequently these elements are functions
[585'"] of the quantities c, c', c",f, /', /", -. To determine these functions, let
6 be the angle which the axis of x makes with the line of intersection of the
plane of the orbit with the plane of x, y, which line we shall call the line of
nodes ; let (p be the inclination of the planes to each other. If we call a/ and
y the co-ordinates of m referred to the line of nodes as the axis of the
abscisses ; we shall find*
j-^gg, xf = X . COS. ^-\-y ' sin. & ;
y' =:y . COS. 6 — X , sin. 6 ;
we shall have also
[587] z = y' , tang. 9 ;
(j-2 (^ 1,2 _L_ ^ 7.2 \ r^dr^
~ J T^ = h^' Multiplying by d t^, and
reducing we obtain r^diP=ih^d t% whose square root is r^ dv = hd t, or
rfv=— — -, as in [585]. Now by [372a], r^dv is double the area included by the radii r
and r-\-dr, therefore this area is proportional to the time.
* (420) In the annexed figure let C X, C Y, be the rectangular axes of x, y;
CX', CY', those of a/, 3/; the line C X' being the line of intersection of the
11. iii. «^ 19.]
MOTION OF TWO BODIES.
351
therefore we shall have
z^=y , COS. & . tang. 9 — x, sin. 6 . tang. 9.
By comparing this with the equation [579],
0 = c" a: — c' 2/ + c 2 ;
we shall get*
c' = c . cos. & . tang. <p ;
c!'z=c . sin. ^ . tang. 9 ;
whence we deduce
c"
tang. ^ = — ;
tang.
_{/&^-^c"2 ^
[588]
[589]
[590]
[591]
plane of the orbit with the plane of a?, y ;
and the axis of z bemg perpendicular to
the plane of the figure ; so that the co-
ordinates of any point D, may be expressed
either by CH=x, HB^y, BD==z,
or by CA^od, AB = ^, BD = z.
The angle HCF=HBE=6, angle
BAD = (p, FH, AE, being parallel
to the axis of /, and HE parallel to the
axis of x'. Then we evidently have
CA=CF-{'HE = x.cos.&^y.sa\.b.
J1B = BE — FH=y. COS. d — x.sm. 6.
which agree with [586]. Again, in the
rectangular triangle DBAjWe have
BD=r=.AB. tang. BAD — if . tang. 9,
as in [587]. Substituting in this the value
of y', [586], it becomes as in [588].
1
/
A
X
B
9'
A-
y^
V
/«- ■'.
A.
K
^
*^>^x
/
v.
JT
:x:
* (421) Multiplying [588] by — c, we get
0 = (c . sin. d . tang, (p) . a; — (c . cos. 5 . tang. 9) . y -j- ^ • «;,
which being compared with [589] gives the values of c', c", [590]. The latter divided by
the former gives tang, fl, [591]. The sum of the squares of [590] is c'^-|-c''^=c^'tang.^9,
whence we deduce tang. 9, [591].
If the plane of the orbit be taken for the plane of a;, y, we shall have tang. 9 = 0, and
the last equation [591] will give c'^ -|- c"^ = 0, which requires that (/ = 0, c" = 0,
and then the expression of h, [575'], will become h. = y/c2-|-c'2+c"2 = c.
[591a]
352 ELLIPTICAL THEORY OF THE [Mec. Cel.
S^f Thus the positions of the nodes and the inclination of the orbit will be
and"inc|- determined in functions of the arbitrary constant quantities c, c', c".
At the perihelion, we have*
[592] reZr = 0; or xdx-]-ydy-\-zdz==^0;
[592'] let X, Y, Z, be the co-ordinates of the planet at this point ; the fourth and
fifth of the equations (P) [572] of the preceding article, will givef
[593] - = ^-.
[593'] ^ut if we put / for the longitude of the projection of the perihelion upon the
plane of x and y, this longitude being counted from the axis of a:, we shall
havej
Y r
[594] -^ = tang. 1 ;
therefore
[594'] tang. /==-^;
Uetcrmi-
thepfaw which determines the position of the transverse axis of the conic section.
of the
perihelion. _ ^ ^ . _^^_ _ /d x^ + dv'' -\- d Z^\ r^dr" ,„
If from the equation [576] , r^. I ^^ ) 1-^ = ^ ' ^^
* (422) At the perihelion the radius r is a minimum, at the aphelion it is a maximum,
consequently at those points the differential dr must be nothing, hence r £Zr= 0, or [549'],
xdx-\-ydy-\-zdz=0, and by using the capital letters, in conformity to the notation
[592a] [592'], it becomes Xd X-\- YdY-\- Z d Z = 0.
f (423) Putting rdr = 0, in the values of /, /, deduced from [572a, 6], we shall
find, f=X.(— — '^): f=Y.(^~^). The latter divided by the former
gives ^ = y. [593].
{ (424) Suppose, in the preceding figure, D to be the place of the perihelion, so that
CH—X, HB = Y, BD = Z, the angle HCB=I, we shall evidendy have
HB Y
tang.flCjB=—- -, or tang./=-— -, as in [594], and this, by means of [593],
f
becomes tang. /= — r , as in [594'].
II. iii. § 19.] MOTION OF TWO BODIES. 353
eliminate ~ , by means of the last of the equations (P) [672] ;
we shall find
'Z^r-^-'^^h^- [595]
a dr
but d r is nothing at the extremities of the transverse axis ; therefore we
shall have at those points
0 = r^ — 2ar-\ '■ — . [596]
The sum of the values of r in this equation is the transverse axis of the conic
section, and their difference is the double of the excentricity ;* therefore a E«entri-
is the semi-transverse axis of the orbit, or the mean distance of m from M ; transverse
' ' axis.
and 1 / 1 is the ratio of the excentricity to the semi-transverse [596']
axis. Let e be this ratio ; we shall have by the preceding articlef [596"]
J
[597]
* (425) The equation [596], gives r=^a±a. | X I — The greatest value
being a-fa.t Xl — -i_ , the least a — a.\ /\ — — , whose sura is the
transverse diameter 2 a, and difference 2 a . t X 1 — _ is double the excentricity.
Hence l / 1 expresses the ratio of the excentricity to the semi-transverse axis, [596a]
and as this ratio is put equal to e [596"], the excentricity will be represented by a e.
f (426) Since e==t/ 1 — —., [596a], we get, by squaring and reducing
A2 = a^.(l_e3). iMuitiplying by ;rf^,wefind ^ ^ f^^f^) ^ M^ ^ ^^^^^ [5965]
Hence we have y? .{\ —e^) — \i?— F, therefore |j-2 ^ = ^2^ and m- e = I, [597'] .
The preceding value of h\ by substituting e = -, [597'], becomes A^j = - . (^2 _ P),
[599]. Wlien the plane of the orbit is taken for the plane of a?, y, we shall have as in
[591a], A = c, hence, in this case A = c = v/fjLa.(l — c2) , [599]. t^^
89
354 ELLIPTICAL THEORY OF THE [Mec. Cel.
[578] ; therefore we shall have
[597] {J^e = 1.
We shall thus know all the elements which determine the nature of the
conic section, and its position in space.
20. The three finite equations, found in the preceding article [582, 582'],
between x, y, z, and r, give x, y, z, in functions of r ; therefore to obtain the
co-ordinates in functions of the time, it will be sufficient to ascertain the
value of r, by a similar function, which requires another integration. To
obtain this, let us resume the equation [595]
P981 2^r ~ J^ = h';
we have by the preceding article [597, 597'],
[599] hJ' = -.(ii.'' — P)=ai^.(l^e');
hence we shall have
rdr
[600]
dt =
V/fJ^-l/ 2r— ^— a.(l— e^)
[600] To obtain the integral of this equation, put r = a. (I — e . cos. u) ; we
shall have*
[601] di = —i=- ,(\—e, cos. u) ;
V ^
hence by integration,
[602] t^T = -— .(u — e. sin. u) ; (S)
* (427) This value of r gives
2r = a.(2 — 2e. cos. u), = a.( — 1 + 2 e . cos. u — e^ . cos.* u}.
a
Hence the term of the denominator,
I y^ 2r — — — a.(l — e^)= \/ae^.{l—co3.^.u) = ^ae^.Bm.^u = a^ e . sin. m.
The same value of r gives dr = ae.du.sm.n, hence
rdr = a^e.du. sin. w . ( 1 — e . cos. m).
These being substituted in [600], we shall get dt^"' \-^ . (1 -— e . cos. u), as in [601].
11. iii. § 20.]
MOTION OF TWO BODIES.
355
T being an arbitrary constant quantity. This equation gives w, and thence
r, in functions of t ; and as x, y, z, are given in functions of r, [558'] ; we
shall have the values of these co-ordinates, at any instant.
Thus we have completely integrated the differential equations (0) § 17
[545], w^hich has introduced the six arbitrary quantities a, e, /, 5, ?, and T :
the two first depend on the nature of the orbit ; the three following, on its
position in space ; and the last, on the position of the body at a given
epoch, or, which is the same thing, it depends on the time of passing the
perihelion.
We shall refer the co-ordinates of the body m to others more convenient
for astronomical uses ; for this purpose, let us put v for the angle which the
radius vector r makes with the transverse axis, counted from the perihelion ;
the equation of the ellipsis [378] will be*
[602']
[602"]
r =
1 -f- e . COS. V
The equation r =: a . (1 — e.cos.u), of the preceding article [600'],
indicates that u is nothing at the perihelion,! consequently this point is the
origin of both the angles u and v ; and it is easy to perceive that the angle u
is formed by the transverse axis of the orbit, and the radius drawn from its
centre to the point where the circumference of the circle described about
the transverse axis as a diameter, is intersected by the ordinates drawn from
the body perpendicular to the transverse axis.f This angle is what is called
[602"']
Polar
equation
of an
ellipsis.
[603]
[603']
[603"J
Excentric
anomaly.
* (428) This is the same as [378], writing « for r — •«, or counting the angle v from the
perihelion.
t (429) At the perihelion r = a . (1 — e), putting this = a . (1 — e . cos. u), [600'],
we get COS. m = 1, hence m = 0.
{ (430) In the annexed figure, which is similar to that in
page 243, AP B is a semi-ellipsis, whose transverse axis is A B,
foci S, s, centre C; Pa point of the orbit corresponding to
SP = r, ASP = Vj the angle ■&, [377"], being nothing.
Draw the ordinate FP perpendicular to A B, and continue it to
meet the semi-circle AP'B, in the point P' join CP'. Then
CS = ae, [378e], SF=SP. cos.ASP=r. cos. v
C F= C S -\- S F= ae-^r . cos. v; in the triangle
H
F
'<""""
'v^
f'p.
5^
N\
A F S C
B
, consequently
CFP' we have
^6 ELLIPTICAL THEORY OF THE [Mec. Cel,
the excentric anomaly, and the angle v is called the true anomaly. Comparing
these two expressions of r [600', 603], we find
True
anomaly.
[604] 1 — e . cos. u =
[605]
1 -)- e . COS. © '
hence we deduce*
tang, iv = \ / 1+! . tang. J u.
V 1 — c
If we fix the origin of the time t at the moment the body m passes the
perihelion, T will be nothing,t and putting for brevity [530'"]
[605'] n =
a^ ai
[605"] we shall have [602] nt = u — e . sin. u. Collecting these equations of the
motion of m about M [605', 600', 605], we shall have
^^'^i^°^ nt = u — e . sin. u
elliptical
motion.
r = a.(l — e.cos. w) i , /•>,
[606] >— — ' ' ^-^^
tang, i V = \X IX— . tang, i u
Y 1 — e
[603a] CF=CF.cos.ACP'=a.cos.ACP', hence a.cos.ACP'=ae + r.cos.v,
the angle A C P' being the excentric anomaly. Now the expressions of r, [600', 603],
«.(! — e2) a.(e.cos.v+e2)
being put equal to each other, we get a e . cos. u= a — — = — — -; .
o sr ^ JO l-j-e.cos.i' l-^-e-cos.v
■n.. .,. ,. , ^ , a.{cos.v-{-e) , a.{l — e^).cos.v
[6036] Dividing this by e we find a . cos. u = -—, — = ae-\ — , and by
•- ■■ o J l-j-e.cos.v l-|-e.cos.« "^
using the value of r, [603] it becomes a . cos. u=ae-\-r . cos. v = a . cos. A C P",
[603rt], hence cos. u = cos. A CF, and u=.A O P' = the excentric anomaly.
COS. V 4- e
* (431) Dividing a . cos. u, [603&], by a, we get cos. u = ]X7~-~ » "^nce
(1— e).(l — cos.r) , . , (14-c).(l + cos.v) -.. ... ,
1 — cos. u = ';^ , and 1 + cos. u = , / ^ • Dividing the
1 + 6.003.1; 1 + e.cos.v
, , , 1— COS.W 1 — C03.V 1 — e -, 1 — COS. M 2 .
former by the latter gives — ; = —; • TT"- ^^^ , ■ ^^„ „ = tang. ^ u,
^ ° 1-|-C0S.« l+COS.V l + « 1 + cos.tt
1 I COS W • -1
[401 Int. and — ; — = tang.^ i v. Substituting these and extracting the square root
*- ■' I -f- cos. V
we shall get tang. ^ v, [605].
f (431a) If ^=0, when m = 0, we shall get also r=0, [602], and this by means of
[605'], gives n^, [606].
n. iii. ^21.] MOTION OF TWO BODIES. 357
the angle n t being what is called the mean anomaly. The first of these Mean
equations gives w in a function of the time t, the two others give r and v,
after u is ascertained. The equation between u and t is transcendental, and ^ ^
can only be solved by approximation. Fortunately the circumstances of the
celestial motions render these approximations very rapid. For the orbits of
the heavenly bodies are either nearly circular or very excentric, and in both
cases, we can determine u in terms of t, by very converging formulas, which
we shall now investigate. For this purpose, we shall give some general [606"]
theorems on the reduction of functions into series, which will be useful in
the rest of this work.
21. Let u be any function of «, which is to be developed according to the [606"']
powers of «. By representing this series in the following manner,
W = U + a.^i + a^g'2+a3.^3 _j- „« . gr^ + a "+i . 9 „^l + &C. ; [607]
u> ^11 5'2? ^c., being quantities independent of a ; it is evident that u is the
value of w, when a is nothing, and that whatever be w, we shall have*
(^) = 1.2.3....7i.9„ + 2.3....(n+l).«.g„+i + &c. ; [608]
* (432) Taking the differential of u [607] relative to a, n times, we shall get the expression
[608] ; then putting a = 0, and dividing by 1 . 2 . 3 .... n, we shall obtain the value of q^,
[609]. Therefore the values of g^, q^, q^, he. may be found from the first, second,
third, &c. differentials of u, by putting a=0 after taking the differentials, and dividing by the
factors 1 , 1.2, 1.2.3, he. respectively.
Hence we shall obtain the development of the function u =/(a?), according to the powers Maciau-
of X, by changing a into x in [607, 609], therefore Theorem.
„=/(ap) = u + (-).^ + (— ) . U+{j^)- 1.2.3 + ^- t«"<")
observing to put a? = 0, in u and its differential coefficients. This formula is generally known
by the name of JVIaclaurin's Theorem. The use of it may be shown by a few examples.
Thus if M = a^, we shall have, by using hyperbolic logarithms, — = a^. log. a,
-— - = «^ . (log. of, &;c. and when x = 0, we get u = 1 , ( — - j = log. a,
f — -^ j = (log. aY, &;c. thus the expression [607a] becomes
a^=l J^x . log. « + 5^- (log.a)^+ 5^ . (log. «)» + &c. ^6^^^
90
358
[609]
[611]
DEVELOPMENT OF FUNCTIONS. [Mec. Cel.
the differentials i -— - j being taken upon the supposition that every thing
which varies vrith a must vary in u. Therefore if we suppose, after taking
^d " M \
the differentials, that a = 0, in the expression of ( -— j, we shall have
'd^'u'
Let u be a function of two quantities « and «', which is proposed to be
developed in a series according to the powers and products of a and a. If
we represent this series in the following manner,
[610] +«'-9'o,i+««'-9i,i+ ^C.
the coefficient q^^^' of the product «". « "', will in like manner be
^n,n'
\,2.S..,.n.\.2.S....n' '
and if a=:c=i number whose hyp. log. is 1, we shall find
[607c] '^=^ + ^+]S + r:f:3 + ^''*
It M == sm. X, we shall have -— = cos. x, -r~^ = — sm. x, -r-^ = — cos. x,
dx d x^ d x^
d'^u . /d^\ /rf2u\
— =sin.a;,&c. and when x = 0, we get u = 0, (^^j = ^' 1,7^;=:^'
f T-5) = — I J &«. then the expression [607a] becomes
1 = — 1, &c. then the (
z3 , x5 a:7
&;c.
[607rfJ sin. x=:x-jj^ + T^J:^ - 1X3X^X7
T^ , „ . du . d^u d3u .
It u = cos. a?, we shall have -7— = — sin. x. —- = — cos. x, — - = sm. a?, oic.
dx d x^ dx3
and when x = 0, u=l, /~-^j = — 1, ^--^j = l, &ic., the other
coefficients depending on du, d^u, d^u, &c. being nothing, hence [607a], becomes
[607eJ ^°^-^—^~" 1.2"^ 1.2.3.4 1.2.3.4.5.6"^
II. iii. §21.] DEVELOPMENT OF FUNCTIONS. ^9
a and a being supposed nothing after taking the differentials.*
In general, if w is a function of «, «', «", &c., which is to be developed in a
series arranged according to the powers and products of «, a, a", &c., the
coefficient of a» .«'"'. a""" . &c., being represented by 9n,n',n", &o.» we shall
have
qn,n',n",S.c. — J . 2 . 3 . . . . % . 1 . 2 . 3 . . . . Ij' . 1 . 2 . 3 . . . . W" . &C. '
provided that we suppose «, a, a", &:c., to be nothing after taking the
differentials.
Supposing now u to be a function of «, a', a", &c., and of the variable
quantities t, if, f, &c. ; and that by the nature of this function, or by an
equation of partial differentials which represents it, we can obtain
^n + n'+n"+&c. ^
(-
<^a».^a'»'.&C. y'
in a function of u, and of its differentials taken with respect to t, t, &c.
Then if we call this function F, after u is changed into u, u being the value
of u, when «, a , a", &c., are supposed equal to nothing ; it is evident that we
shall obtain qn.,n',acc.i by dividing Fhy the product
1 .2. 3... .71. 1.2.3....w'.&c. ;
we shall therefore have the law of the series in which u is developed.
* (433) Taking tlie differential of [610] n times and dividing by d a", considering d a
as constant, all the terms of u, depending on a""^, and lower powers of a will vanish
in eZ" M ; the terms depending on a", will obtain the factor 1 . 2 . 3 . . . . w, multiplied by a° or 1 ;
the terms multiplied by a^+^j a"+^, he. will produce terms multiplied by a, a^, &£c. If we
now take in the same manner the differential of d" u, n' times relative to a', and divide by
c?a"', all the terms, which in u are multiplied by a'"'~^, and lower powers of a! will vanish,
and the term of u multiplied by a^ . a'"', will produce in the differential the quantity
1.2.3 — n . 1.2.2....n' .q„^J^,, the other terms of the differential being multiplied by a,
or a' and their powers and products will become nothing, when a = 0, a' = 0, and then
{da^.da'^')^^'^'^""'^'^'^'^""'^'-^^'^'' which gives 5^,^,, as in [611]. In the
same manner we shall obtam the general expression of g'n,n',w",&c, [612].
[612]
[613]
[613^
mo -JDEVELOPMENT OF FUNCTIONS. [M6c. C61.
Suppose in the first place that u is equal to any function of i + «, f -\-a',
if' -\- fit!'., MQ'.', wjiich we shall denote by
[613"] u=^.(t + u, f-i-a', f + a", kc.) ;
in this case the differential of w, relative to a, being taken a number of times
denoted by i, and then divided by d a\ will be evidently equal to the like
differentials taken with respect to t, and divided by dt\* The same
equality exists between the differentials taken with respect to « and ^, or
with respect to a" and f, &c. ; hence it follows that we shall have in
general
da-.da'-'.da"-''.kcj ^ \ d t . d tf^' . d f''" . &LC. ) '
By changing in the second member of this equation, u into u, that is into
<p (tf t, f, &c.) ; we shall have, by what precedes, [612, 614].
/J''+"'+""+^°- .9 (^, t, f, &C.)\
[615] V dt.dt''' .df^'^kc. y .
9«.n'.«".&c. - 1^2. 3. ...n. 1,2. 3.... n'. 1 .2 .3 ....riV&^ '
If M is a function of t-{-a only, we shall have [615]
'S^. thereforet
[6,7] K<+«)=»(0+«--^^+r:^--^+r:^.3-^t^ + &c. o)
rgjyn Suppose now that u, instead of being given explicitly in « and t, as in the
preceding case, is a function of x ; x being given by the equation of partial
differentialsf
^d oo\ /d x^
lfil7"] KdaJ-'^'KdtJ'
* (433a) Because t and a occur in 9, only under the form of ^ + a, and its functions,
a„dwehave(l:i^i^) = (i±±^).
f (434) Substituting in [607] the values deduced from [616], we shall get the
formula [617].
J (434a) It will be proved in [632'] that the integral of this equation is a; =?= (p (f+ a z),
(p denoting an arbltrar)"- function.
n. iii. §21.] DEVELOPMENT OF FUNCTIONS. 361
in which z is any function of x. To reduce w to a series arranged according [617"]
-T-n] [609, 607], in the
case of a = 0 ; now we have, by means of the proposed equation of partial
differentials,
XTa.) ^ \d^) ' \J^J ^ ^ ' [d^J ' \dtj '
therefore we shall have*
•^^'^''^ ; (k) [619]
[618]
© = (^
dt
taking the differential of this equation with respect to «, we shall have
ddu\ / dd .fz d u
dc^ J \ d adt
now the equation {k) [619] gives, by changing n into/zc?w,t
[620]
therefore
/^\ _ /dd.fz^du\ ^^^
* (435) u being any function of x, [617'], and x a function of t, a, we evidently have
(i^) = (jf ) • (sf )• =""* *''' •'5' ™"^' °f ["'"J' ''^~™= (^) = ^ • (jl) • (^)'
as in [618] ; and as z, u, are both functions of x [617', 617'"], vee may find the integral
fzdu = z\ z being also a function of x; hence z .(-;—) = (-;—), and the preceding [617aj
\d xj \dx /
expression will become (t— ) = (t~) ' {'T~)> ^^ second member of which is evidently
equal to (tt)» because 2/ is afunction of x, and x a function of t, a, hence (— ) = f;?— )j [GlSa]
which becomes the same as in [619] by substituting d.fzdu for rf«', [617a].
f (436) The equation [619] was derived from [617"], and in it we may take for u any
function of x whatever, as fz d u, fz^ d u, fz^ d u, he. and the equation will still exist,
or in other words, we may, instead of d u, put zdu, s?du, z^ du, &ic. Thus by
changing d u into zdu, in the equation [619] we shall obtain ( v^ ^\ __ /_j/i — ^\
as in [621], the differential of this relative to t is ( i , ) = ( — '"{^ -), which being
substituted in [620] gives [622].
91
^62 DEVELOPMENT OF FUNCTIONS. [Mec. Gel.
Taking the differential again with respect to «, we shall have
da^ J \ dadt^
now the equation {k) [619] gives, by changing u into fz^du^*
/d.fz^du\ _ /d.fz^du\ ^
\ da. )~\ Tt
therefore
'^d^u\ fd^.fz^du
[623]
[624]
[625]
[626] (d^u\^/d-.fz^du _
[626'] Suppose now that by making a = 0, we should have x = T^ T being a
function of t ; and that this value of x, being substituted in z and w, makes
[626"] those quantities become Z and u ; we shall have, by supposing a = 0,
da? J \ dfi
Proceeding in this manner, it will be easy to conclude that in general
[627]
d''u\ V dt
da!' J dt""-'
therefore by what precedes
[628]
9.n
\ dt
l.2.3....n.dr-' '
which gives [607, 628]t
d 2 ^
du . a^ V dt J . a^ V dt
* (437) This change of u into fz^ du, ox du into z^ d u, may be made for the reasons
stated in the preceding note, and [619] will be changed into [624] ; taking its diiEFerential
, . /d3.fz^du\ /d?.fz^du\ ,., , . , . ,. r^^^-.
twice relative to t we get ( ) ^^ i — Jh — J' ^"*^" being substituted m [623],
it becomes as in [625]. Proceeding in this manner we shall evidently obtain the general
expression [626].
f (438) The formula [629], given by La Place, is of great use in the inversion of series
and is frequently referred to ; it may not therefore be amiss to collect in one point of view
TI. iii. §21.] DEVELOPMENT OF FUNCTIONS. ^QS
It now remains to determine the function of t and «, which is represented
by X. This is done by taking the integral of the equation of partial
differentials [617"] {'J~) = ^ ' \~j~:)' For this purpose we shall observe
that*
the forms of the different functions used. Supposing therefore 9, ■^'j jP, to denote
the characteristic of functions, we shall have [632', 632", 617'", 626", eiTQ,
X = (p{t -\-a.z); T=cp{t) ;
z = F{x)', Z=F{T) = F.\cp{t)\; [629a]
u = ^{x) = 4^.\<p{t-i-az)]; u = 4.(T) = 4..J9(0!;
T, Z, u, being the values of x, z, u, respectively when a = 0.
If we take, as a simple case of the formula [629], x = t-{-az=t-]-a .F{x), we
shall have u = ^{t-{-az), and that formula will become, by putting -^ — = -s^' {t),
« = 4.(0 + a.F(0.4'(0 + j:i- J, + 1X5' — d^ — + ^^- ^^^^^
If in this we put a = 1 , we shall obtain the celebrated Theorem of La Grange, which has
been of such great use in analysis. In this theorem we have Theorem
X = t-\-F{x), Grange.
For an example of the use of this last formula we shall suppose x = t-\-a.x^, and that
it is required to find a; in a series arranged according to the powers of t. This value being
compared witli x=^t-\-F [x), [629c], gives F (j;) = a a;", -^^ {x) = x, hence
F{t) = oL V\ ^ [t) = t, and -4.' (0 = 1, and the formula [629c] becomes . . .p
~ '1.2 dt '1.2.3 dpi '
or by development
a; = ^+af' + -^.2w.^«-i + --^.3n.(3n — l).i3n-2_^^c.
in which the law of continuation is very manifest, and it is one of the great advantages of
this beautiful formula, which is much used in the course of this work, as in [652, 657, 658,
666], which may be referred to, as striking examples of the importance of this method of
development.
* (438«) In the equation [617"] the partial differentials of x relative to a and t only
occur, or in other words x is considered as a function of t and a only, and then its complete
differential dx will be as in [630]. i v. . ; ,; ^^ ^^ u ^ _|;. ^\ ,._-- ,; ;> ^^j
^^ DEVELOPMENT OF FUNCTIONS. [Mec. Cel.
by substituting for (j^\ its value z . (-^\ we shall have*
therefore we shall have
•+-0-(t:)
which gives by integration
[632'] X — (p(t-}-az) ;
(p(t-\-az) being an arbitrary function of (t-\-az) ; hence it follows that the
[632"] quantity we have denoted by T [626'] is equal to <p (t). Therefore whenever
there is given between x and a an equation which may be reduced to the
form x = (p(t-\-az) ; the value of u will be given by the formula (p) [629],
in a series arranged according to the powers of a.
Suppose now that t< is a function of two variable quantities x and oif, these
[632"] quantities being given by the equations of partial differentials,
* (439) The last member of [631] is easily deduced from the preceding, by substituting
for dt-\-zda, its value d.(t-\-az) — a.dz, and as « is a function of a?, [617'"],
we may write (~—j,dx for fZz, we shall thus obtain
Transposing the last term and dividing by ^ ~f" " • (j") • (ttJj ^^ ^^^^^ obtain
dx
'+«-©-(?7)
/d_x\
Putting now for brevity i-]-az = u, j^^ — 7dJ\~ "^ ^' " becomes a
common differential equation dx = Vd w, and as the first member is a complete
differential of a?, the second member must also be complete, which cannot be, in general,
unless we take V such that x may be a function of w or t-\-az, which may be represented
by x-=(p{t-\-a.z), as in [632'].
II, iii. ^S21.] DEVELOPMENT OF FUNCTIONS. ^^^
/'dxS /dx\ / dCC'\ , /d3t/\ r^gg-,
V^y^^'W' \d7j'^^'\df'j'
in which z and z' are any functions whatever of x and a/. It is easy to [^^3^
prove* that the integrals of these equations are
x = ^(t + az); af=4^(if + a'zf); [634]
(p(t-\-az) and ^^ (i' + a 2f) being arbitrary functions, the one of t-{-az, and
the other of tf -\- a z'. We shall also havef
This being premised, if we suppose a/ to be eliminated from u and z, by
means of the equation x' = -^(tf-{-a2f) u and z will become functions of
a:, a, ^, without a or ^ ;t therefore we shall have, by what precedes [627]
/d^ju\ _ [ \dtj J [636]
If we suppose a = 0, after taking the differentials, and substitute also in the
second member of the equation 16361 x = c? (t-^-a z^), consequently^
( — j = 2" . f — j ; we shall have, by means of these suppositions, [636']
dp-^
* (440) This demonstration may be made as in the preceding note.
f (441 ) Put t -{- az = u, t' -{-a z = w', then u may be considered as a function of
.,.,[63^034], and we Shan have {^) = {^) . i^); (^) = (^) • (^>
but from .=:t + az, we get (^) = 1, (^) = ^> therefore (^) = (^) ;
(^) = (^)-^=(^)-^' ^si°[635]. Inlikemamierwefind (^,)=^.(^).
J (442) 2r, w, do not contain a, ^, explicitly, but implicitly only, by means of a?, and they
then come under the form supposed in [617', 617'"], and the result found in [627], from the
suppositions made in [617 — 617'"], may be applied to this case, as is done in [636].
§ (443) This calculation is made in the same manner as in note 441, merely changing
z into 2", so that w = ^ + a z", by which means we shall find that (-—) = z'^ . ( j-)-
Substituting this in [636] it becomes as in [637].
92
[637]
366 DEVELOPMENT OF FUNCTIONS. [Mcc. Cel.
consequently*
^^^ / d^+^.U \ V da J
\ du^.da!''' )~\ dr^' J'
in like manner we shall havef
[639] /d-'u\ f ^"'~'*(^
[640]
supposing a'=0, after taking the differentials, and putting also in the second
member of this equation a;'= -j- (^ + «' 2;' "') ; therefore we shall havej
/ tZ^ + ^'M \ __ ( '\dada'J J ^
yda^.dct'^' ) ~~\ dt^'-Kdf^'-^ / '
provided we put a and a equal to nothing after taking the differentials, and
also, in the second member of this equation, make
[641] x=^(t + az''); a;' = 9(^ + «2^"') ;
which amounts to the same thing as to suppose in both members
[642] x=:^(t+az); x' =(p(t' + u'z'),
changing at the same time in the partial differential I -j—j-f ) ^^ ^^® second
[642'] member, z into z™, and z! into z!""'. These suppositions being made, and
withal changing z into Z, z! into Z\ u into u [626"], we shall find§
J ''--•G-^^) \
9n,n' — y J 2 . 3 .. . . W . 1 . 2 . 3 . . . . n' . <? P-^ <^ ^"'- V *
[643]
* (444) Taking the differential of [637] n times relative to a', we shall get [638].
f (445) The expression [639] was found in the same manner as [637] and it may be
deduced from it by changing w, a, t, into w', a, ^, respectively.
X (446) Substituting the value of (^), [^39] in [638] it changes into [640]
§ (447)
get [643].
§ (447) Substituting in q^,^., [611] the value of (dSriT^)' ^^40], we shall
therefore*
"n
(448) Putting n' = 0, n" = 0, &£c. and r=l, the general expression [645]
become y. = __^?^. Sub^
z mto 2", as in [645'] we shall get [647].
will become gr„ = _^__J^|^ . Substituting the value of (-^Y [646], and changing
[644]
U. iii. §21.] DEVELOPMENT OF FUNCTIONS. 367
By following this method of reasoning, it is easy to perceive that if we have
r equations
a/ = 4'(^ + «V);
a/'=n(f + a"z") ;
&c. ;
z, 2f, 2", &c., being any functions whatever of x, xf, a/', &c. ; and if we
suppose w to be a function of the same variable quantities, we shall have in
general
\da.da'.da".&.c.J J
(d^u \
^ ), 2 into rg^gr,
2", 2f into 2'*^, &c. ; afterwards 2 into Z, 2' into Z', 2" into Z", &c. ; and then
u into u.
If there is but one variable quantity x, we shall have [OSS']
(d u\ fd u\
[645]
[647]
If there are two variable quantities x and a/, we shall have [635]
fd u\ fd u\
taking the differential of this with respect to a , we shall find
(ddu \ f^^\ /^^"\i_ f ddu \
368 DEVELOPMENT OF FUNCTIONS. [Mec. Cel.
[f)49T now we have* ('T~f)=^^''\'JZ}) [^^^] 5 ^^^ ^J changing in this equation
[649"] u into 2, we shall get ( ji ) = 2:' . ( -r^ ) ; thereforef
Ja'J \dif
/ J A ., \
[650]
/ddu\_ ( '^ \dt') ] , /</A /Jm\
V7^o7y~^'\ dt J^^ '\d7)'\di)'
Supposing a and «' to be nothing in the second member of this equation,
and changing z into Z", z' into Z'"', and u into u [642'], we shall find the
value of ( - I ) corresponding to these conditions ; hence we getf
^"•^-•0^)+^'--(^)-(§-:)
[651] / + Z" .
rfi y \di
1 ,2.S....n.dr--K 1 .2.3.... 71. ^^''-^
* (449) By [634] we have a/ equal to a function of tf -\-a' zf, and from this we have
deduced in [635], ( — ]== z' . ( — ] ; m being any function of «, a?', [632'"]; and as z
is also a function of x, x', [633'], we may also in this equation change u into z," and we shall
obtain ( — ; ) = ^' • (77)' [649"]. In this calculation we have neglected the consideration
that z depends also on a; or t-j-az, because the partial differentials relative to a, t, do not
occur in the equations treated of in this note.
f (450) Talcing the differential of the equation (—\=z^. (ttj), [649'], relative to t,
"^ *^' S<=' (l^) = Qf) • © + ^ • {if^} S">'^M*ng this, and the value of
{—) = z" . (^\ [«49"], in [649], it becomes
in which the terms between the braces, or the factor of z, may be put under this form,
'^MW
' , by which means it becomes as in [650].
.-,,.. / ddu \ ^^_^ ^ . ' \d ada'J
t(451) Substituting (^^~^,j, [650a], m gn..'=^ j.a.3....n.rf!^>-i.l.2.3....n.rf^>'-i>
[645] ; changing also z into Z", z' into Z'", [645'], it becomes as in [651].
[652]
n. iii. § 22.] ELLIPTICAL ORBIT OF A PLANET. 369
By continuing in this manner, we shall obtain the values of qn,n',n",ttc.i ^oi
any number of variable quantities.
Although we have supposed w, z, z', 2f', &c., to be functions of a:, a/, a/',
&c., without t, t, f, &c., we may however suppose them to contain these
last variable quantities, but we must then denote these quantities by t^, t^, t", [6511
&c., and we must suppose t^, ^/, t", &c., to be constant in the differentiations,
and after taking these differentials we must resubstitute t, if, &c., for
t,i ^;, &c.
22. Let us apply these results to the elliptical motion of the planets.
For this purpose we shall resume the equations (f) ^ 20 [606]. If we
compare the equation nt = u — e . sin. u, or u =nt-{-e . sin. w, with [651"]
x = (p(t-\-az) [632'], X will change into u ; t into nt ; a into e ; z into sin.M ;
and 9 (^ + a z) into nt-\- e . sin. u ; hence the formula (p) [629] of the
preceding article will become*
+ TT2:3- ^<fF +^^-'
■^' (n i) being equal to ' ^ . To develop this formula we shall
observe that c being the number whose hyperbolic logarithm is unity, we
shall havef
* (452) The symbol u is used in a different sense in [651"], from what it is [629]. To
prevent any confusion, it was thought best to accent the letters m, <, in this last formula, which
will then become,
W_u -t-a^.^^-t-j g. ^^ ^1.2.3 572 ^^'
In which as m [629a], a: = (?(<' + a ar), T=<p(f'), u' =z -^^ (x) = 4. . \ cp {t' -\- a z) \ ; [6526]
u'=-\^. \(p{i')\. And by comparing x = <p(^-{-az) with u = nt -}- e .sin. u,
we shall have as above x^^u, a=e, 2;=sin.M, t!=nt, <»)(<') = n^, u'=-^(nt)j
u __ -S-i^^ _. ^/ ^^ ^^ . ^__ gjjj^ jj ^^ Substituting these in [655a], we shall get [652],
f (453) Putting z=nt, in [11, 12] Int. and involving to the power i, we shall get [653].
The formulas [653^] are derived from [15, 16] Int. by putting zsssrt, '--'i SMu''^ uv/
93
370 ELUPTICAL ORBIT OF A PLANET. [Mec. Cel.
i being any number whatever. By developing the second members of these
equations, and substituting instead of c^^''^~\ and c~^^ i their
[653'] values cos.r 71^+^111. sin. rw ^, and cos.rni — [/—i • sin. rnt; r being
any quantity vs^hatever ; we shall have the powers i of sin. ni and cos. 7i^,
developed in a series of sines and cosines of the angle n t and its multiples ;
this being supposed, we shall find, that if the following function be put
equal to P
[653"] sin.n^ + --^.sin.'n^4---|— .sin.'yi^+ % .sin.'n^+fcc. ;
we shall have [1 — 5, Int.]
P=sin.n^ — -4-7;' \ cos.2nt — 1 i
1.2.2 ( 3
. < sin. 3ni — 3 . sin. nt i
[<j^\
1.2.3.22
[654]
e^ C 4.3 >
^ . < cos. Aitit — 4 . cos. 2nt-\-i . — - >
1.2.3.4.23 ( ' ^ 1.2 3
— . < sm. ont — o.sm.on t4-- — -,sm.nt >
.5.2^ ( '1.2 >
1.2.3.4,
6.5.4
— &C.
— -—; 7r-7:r ' \ cos.6nt — 6.cgs.4w^-| — ^.cos.2w^ — i--^—^
1.2.3.4.5.6.2-^ ^ ' 1.2 1.2.:
[654'] Multiply this value of P by ■^' (n t), and then take the differential of each
of its terms, relative to t, a number of times equal to the exponent of the
power of e by which it is multiplied, dt being supposed constant ; afterwards
[654"] divide these differentials by the corresponding power of ndt. Let P' be
the sum of these differentials thus divided ; the formula (q) [652] will
become
[655] ■^(u)==-^(nt) + e F.
It will be easy to obtain, by this method, the values of the angle w, and of
the sines and cosines of its multiples. Supposing, for example, •|(M)=sin.^^«,
[655'] we shall have ■\l(nt) = i . cos. i n t. We must multiply the preceding value
n. Hi. § 22.] ELLIPTICAL ORBIT OF A PLANET. 371
of P by i . COS. i n t^ and develop the product,* in sines and cosines of the
angle n t and its multiples. The terms multiplied by the even powers of e
will be sines, and those multiplied by the odd powers of e will be cosines. [655']
Then we must change any term of the form K.e^'' .sin. snt into [655"]
dzK. e^"" . s^"" . sin. snt ; the sign + is to be used if r is even, and the
sign — , if r is odd.f We must likewise change any term of the form
K.e^""^^ .COS. snt, into ^ K. e^''+^ .8^''+^ . sin. snt ; the sign — is to ressivj
be used if r is even, and the sign +> if ^ is odd. The sum of all these terms
will be the value of P', and we shall have|
sin. iu = sin. int-{-eP'. [656]
* (454) If we multiply the above expression of P, [654] by i. cos. int, and change
the products like i .sinmt . cos. i n t, into ^i • \ sin. [m -{- i) . n t -\- sin. (m — i) .nt\,
[18] Int., or as it may be written ^i .sin. {mdtzi) .nt ; also products like i.cos.mt.cos.int,
into i z . I cos. {m-\-i) .nt-\- cos. (m — i) .ntL or J i . cos. (m ± »') • n t, [20] Int., it
will become
i . P . COS. int = —- .sin. (1 rb*) .nt — , _ ^. jcos. (2 ± *) • n < — 2 cos. in ^1
1.2 ^ ' 1.2.22 ( ^ ' >
""l.a's.Sa '{sin.CSdrij.n^ — 3sin.(l±z).n^}
+ 1.2^4,24 • jcos.(4zht).w< — 4.cos.(2rbt).?i<4-j^.cos.tn^|
g4 j _ 5.4
} sin. (5 ± *) . n < — 5 . sin. (3 ± i) • w < + -^ . sin. (1 -^i) .nt\
JL • <«
' 1.2.3.4.5.25
g5 ^ 6 5 6 5 4
~ 1.2.3.4.5.6.26 • ? cos.(6±i}.n<--6.cos.(4±i).ni+— .cos.(2±i).n<--j-^.cos.iW } + &c
which is like the form mentioned above.
f (455) These values for the signs, and the factor s^*", necessarily follow from taking the
differential a number of limes denoted by the exponent of e ; that is 2 r times. For if r
is even as 0, 2, 4, &xj. rf^'' . sin. snt wiU evidently have a positive sign ; but if r is odd,
as 1,3, 5, &£c. it will be negative. It being easy to perceive that the sign of the coefficient
of Jf e^*" is the same as that of Ke'^'"^'^. Similar remarks may be made on the term
^e^''~*"^.cos. sn ^
J (456) Taking the several terms of the expression of iP .cos. int, [654«], for
Kf?'' .sin. snt, or Ke^'"^^ .cos. snt, and deducing from them ±^e^'" .5^'' .sin.^n^
=F Ke^""^^ . s^r+i . sin. gnt, by the above rules, we shall obtain the value of P, which
being substituted in [656], it becomes
372 ELLIPTICAL ORBIT OF A PLANET. [Mec. Cel.
[656'] If we suppose 4- (yd = u, we shall have ■^' (nt) = 1, and we shall find*
ic3
1.2.3.23
ie4
\.iu=sm.ini-\---Z' sin. (I ±i) .nt-\- ■.{(2±t') .sin. (2±'i) . w^ — 2*.sin. iw<^
. { (3 dr i)^ . sin. (3 d= *) . w ^ — 3 . (1 zhif. sin. (1 ± i) . w ^}
4 3 >
[656a] + ^ g g ^ g4 . {(4 ± i)^ . sin. {4±i).nt-—4. (2±i7.sin. (2 ± i) • ^^ + J^ • ^ • ^^' *^^ 5
+ ^ ^ g'"^ g ^^ .{(5±t)^sin.(5d=0-^^~5.(3±i)^.sin.(3±i>n^ + ^.(l±:i)4.sin.(l±i).n^|
^.g6 ((6 db i)^ . sin. (6 ±t) . w ^— 6 . (4 ± i)^ . sin. {4±i).nt
^i^-^^-"^ (+'^.(2±^)^sin.(2±^).n.--^.^^sin.^•n^
By putting successively i= 1, 2, 3, 4, 5, 6, fee. and making the deductions arising from
sin. ( — mnt) = — sin. mnt, we shall get
e e^ e^
sin.M=sin.w t+r-z- sin.2n t-\- . bin. 3 nt — sin.n t \ + . j 42.sin. 4nt — 8 sin. 2 nt \
+ nrrH-Tn: • I ^^ • sin. 5 n ^— 3 . 3^ .sin. 3n < + 2 . sin. w ^ j
[6566] 1.^.3^.4.24
+ , o o . . ^,.16^sin.6n<— 4.4^sin.4?i^4-5.2^sin.2w^| + &c.
1.2.3.4.5.25 ( '
2e 2c2
sin . 2 M = sin. 2 w ^ + — . j sin. 3 w < — sin. n ^ j + ^-y^ . 1 4 . sin. 4 w ^ — 4 . sin. 2 » ^ |
4- -^ . J 52 . sin. 5 n ^ — 3 . 32 . sin. 3 n < + 4 . sin. n ^ I
[656c] ~ 1.2.3.23 I ° '5
-4 ^^^^^ .\ 63 . sin. 6 w < — 4 . 43 . sin. 4nt4~^ .2^. sin.2 n < | + &tc.
^ 1.2.3.4.24 < ' M
3 g 3e2
sin.3w=sin.3n^+-^.{sin. 4w< — sin.2n<j + r-^-^-|5.sin.5n< — 6.sin.3»r+sin.»^}
[656rf] -1 ^^^.{62.sin.6n^— 3.42.sin.4n<+3.22.sin.2n<}+&;c.
4e
sm.4M=sin.4n< + 7i;-isin.5n< — sin. 3wq
[656c] 42 J
4. ^— A6, sin. 6 n i — 8 . sin. 4 n < + 2 . sin. 2 » < 1 + &MJ.
1.2.22 t
[656/] sin.5M=sin. 5n < + — .^sin.6ni — sin.47Uj+^c.
[656g] sin. 6 tt = sin. 6 n < -}- Sic.
*(457) 4,(w)==«, [656'], hence 4.(71 0 = »^ and 4-'(wO= ndt '
consequently the factor of P, [654'], is equal to unity. In order therefore to find F, [654"],
n. iii. § 22.] ELLIPTICAL ^ ORBIT OF A PLANET. 373
u — nt+e.sm.nt-^ .2.sin.2n^+ . ] 3^,sin.Snt — S.sin.nt [
1.2.3.4.2"* ( ^
( 5 4 )
[657]
' 1.2.3.4.5.2*
+ &C.
This series is very convei^ing for the planets. Having thus determined u
for any instant of time, we may deduce from it, by means of the equations
[606], the corresponding values of r and v. We may also compute r and v
by converging series in the following direct manner.
For this purpose, we shall observe that by § 20 [606] we have
r = a . (1 — e, cos. u) ; [657^
now if in the formula (q) [652], we suppose 4^ (w) = 1 — e. cos. w, we shall
get* ^' (nt) = e . sin. n t ; consequently
l-e.cos.u=l-e.cos.nt+e'.sm.^nt+j-^ . --^^ + j^ . ~^^ + &c. ; [658]
we shall thereforielind by the preceding analysisf
we must take the differential of each of the terras of P, [654], a number of times denoted
by the exponent of e, in that term, and must divide by the corresponding powers of nd t.
The general expression [655], -^^ (u) =■- -^ [n t) -{- e P', becomes, in tliis particular case,
u = nt-{-eP', and by substituting the value of P', found in the preceding manner, we
shall obtain the value of u, [657].
* (458) Having 4. m = 1 — e . cos. w, it gives ■\,(nt) = 1 — e. cos. n t, hence
^'{ni)=-^-— - = e. sin. nt, these being substituted in [652], give the vadue of
1 — e.cos. M, [658].
f (459) Substituting for sin.^n^, sin.^nf, &c. their values [1 — 5] Int., then taking
the differentials as in [658] we shall obtain the expression of 1 — e . cos. m = — , [606],
as in [659].
94
374 ELUPTICAL ORBIT OF A PLANET. [Mec. Cel.
- = H e.cos. n^ — -.cos. 2n<
a 2 2
— r-4-^o- \ 3.cos.3w^ — 3.C0S. n< \
1.2.22 ^ J
— ^^ — r.^4^cos.47i« — 4.2^cos.2n« J
1.2.3.23 ^ ^
gS ( 5.4 )
r- < 5^cos.5?^^ — 6, S^, cos.Snt -{--—. cos. nt >
1.2.3.4.2'' ( ' 1.2 >
^ 6^cos.6?l«— 6.4*.cos.4?i<+T^.2^cos.27i^ J
C 1'2 )
[659]
[660]
1.2.3.4.5.2^ *
— &C.
We shall now consider the third equation (/) ^ 20 [606] ; it gives
sin. J« yi-l-e sin. ^M
COS. ^ V y^ 1 — c * COS. i M *
Substituting in this equation, the exponential values of the sines and cosines,
we shall find*
* (460) Substituting the exponential values of sin. ^ v, cos. J v, sin. i u, cos. ^ m,
[11, 12] Int. in — ^ r-— = i / r^ • — ^k ; ' deduced from [660], it
^ ' -" S.cos.iv ^^ 1 — e 2.cos.iM
becomes ^j^^.^:rT_^^-i«.v/=r[ = ^ i^e ' ^iw./— i_^^_i«./:zi '
Multiplying the numerator and denominator of the first member by c^^'*^~\ those of the
second by c*"**^~^, we shall obtain [661]. Now the value of X, [662], being the same
as in [536] we get by [5365], i y^^i! = -^— , which being substituted in [661]
we get =z = ==1 ' 7=^' From each member of this
^ c^V-i_^] c".v/-i__x + l— Xc^-^-i
equation, we may deduce new fractional expressions, by adding the numerator and
denominator for a new numerator, and subtracting the numerator from the denominator, for a
2c^-^^ 2c~-»^^— 2X
new denominator ; by this means the expression becomes x = — .— ^ ,
2^— 2 A.c
«.V/^ , ^ ( 1 V -— w./^ )
or c''-v^=-^ -A==c^-^-^< ^ ->, asin[663]. Taking
the logarithms of both sides, and dividing by y/— 1, we shall obtain [664].
n. iii. §22.] ELLIPTICAL ORBIT OF A PLANET. 375
1—- i = | /i+i ) f i ( .
c
supposing therefore
x = — ^— •
we shall have
[661]
[662]
^..v/-i _ ^nV-i ^ > ^^^£ V . ^6^3^
1 — x.c
u.y/_
consequently
log. (1— x.c-"-^~)— log.(l— x.c^-v^^)
t, = >j^ -f- ^ A—- 1 L , [6641
V/-l
hence, by reducing the logarithms into series, we shall obtain*
2 X^ 2 X^ 2 X^
v = u-{-2'K.sin.u-i — —.sin.2wH — — .sin.3w + — — .sin. 4m + &c. [665]
^ o 4
We shall have, by what precedes, [657, 656a — -g], u, sin. w, sin. 2w, &c.,
in series arranged according to the powers of e, and developed in sines and
cosines of the angle n t and its multiples ; all that is now required to obtain
t> by a similar series, is to arrange the successive powers of x, in a series
proceeding according to the powers of e.
The equation! u = 2 , will give, by the formula (p) [629] of the [6651
* (461) By [58] Int. log. {I + x) — x — ^ aP -\- ^ a^ — i x*, kc. hence
log.(i_x.c-«-^^)=-X.c-«-^^-J.x2.c-2«V::^_j.x3.c-3"V-i_&e.
-log.(l— x.c"-^=^)=X.c"-^~i+J.x2.c^«-v/^+J.x3.c3"-v^i + &c.
Their sum is
X.(c"V:^-_c-"-^+i.x2.(c2"-v^^— c-^"-V^=^)
+J.x3.(c3«-^^— c-3"-v^) + &c.
and by [11] Int. this becomes 2 . y^— l . (x . sin. « + i • ^^ • sin. 2 « -j- i • ^^ • sin. 2u-\- &c. )
hence [664] changes into [665].
f (462) This equation multiplied by «, is m^ = 2m — ee, hence u = l-\-\/l — e^ [665a]
as in [666']. Substituting this in [662] we get X=— , hence X'=-^. Therefore to [666o]
376 ELLIPTICAL ORBIT OF A PLANET. [Mec. Gel.
preceding article,
[666] J_==±4.ii^4.i:(!±^ _!!_-Liili±iM^±l) __i!_4.&p .
ui 2»~2»+2^ 1.2 '2»+4~ 1.2.3 • 2^+6 "^ '
obtain X* in a series arranged according to the powers of e, it is necessary to find — r ,
arranged according to the same powers of e, which may be done by means of the formula
[652a]. For by comparing w = 2 , [665'], with x= cp{t' -{-a,z), [6525], we
find a? = M, /' = 2, a = e^, z=i , u = 2, and (p{t'-\-az), becomes
t'-\-az=2 = w. Now putting in [652a] w' ::=«"', it will become
"■i'^-'-'-^) ee '^^•(-=-^')
— ) — i S2 'K/.U""' 1 ^ V, ai I ^ \ ai I \ 8,^
u '=u «-e^-.-^^ + j^.^-^p ^-17273 •-- ^-^?^ ^ + ^^-
and since w = ?' + a z, we shall have __ == 1 , therefore '.^ = — * . u * ^ ,
ar at
consequently m '=u *-j-ie^.u ^ — J^ ^-^^; " "^ iTO fe ^ — &c.
Developing the difierentials indicated in the formula, it changes into
Which, by putting u = 2, becomes as in [666]. This value of w~* being substituted in
[666a], X^ = — , it becomes as in [667]. It may be observed that the factor (i-|-4),
in [666, 667], was by a small mistake in the original written (i -|- 3). The correctness of
the present form is easily verified by examining a simple case, as for example, when
i = l, corresponding to X= ■■ _i_ /, Developing the denominator by the usual
rule for extracting the square root, it becomes 2 — \^ — | e^ — -^-^ e^. Dividing e by
this, by the usual method of division, we get X = ^ e + i e^ + tV ^ ~\~ t^ ^t ^ which
the coefficient of eJ is y|^. This agrees with the corresponding term of the formula [667],
^3—^ . {^y by putting z= 1, whereas -—^ . (-) , would
be tI^^^' If in the expression of X^, [667], we put successively : =1, 2, 3, 4, 5, 6, we
shall get
n. iii. § 22.] ELLIPTICAL ORBIT OF A PLANET. 377
and as u=l -\- ^/T^^ [GSBa'], we shall have [^^1
This being premised, we shall find, by continuing the approximation to terms
of the order e^ inclusively,*
1097 5 . r ^ , 1223 6 . ^ ^
. e^ . sm. 5r^^+ — —— . e** . sm. bn^.
-=\-\-ie^—(e—ie^).cos.(nt+s—':s) — (ie^—ie').cos.2(nt + s—z,)—8ic.;
* (463) If in the expression of «, [665], we substitute the value of u, [657], those of
sin. w, sin. 2 u, he. [6566 — -o-], and X, X^, x^, &tc. [667a], it becomes, by placing the terms
in the same order as they occur in these formulas, • '> '
95
[668"']
' 960 ' 960
The angles v and n t are here counted from the perihelion ; but if we
wish to count them from the aphelion, we must evidently make e negative in [6681
the preceding expressions of r and v. The same result might also be
obtained by increasing the angle ?i ^ by two right angles, which would render
the sines and cosines of the odd multiples of n t negative ; now. since the [668"]
results of both methods ought to be identical in the values of r and v, it is
necessary that the sines and cosines of the odd multiples of n t, should be
multiplied by odd powers of e, and that the sines and cosines of the even
multiples of the same angle, should be multiplied by even powers of that
quantity. Which is confirmed by calculation a posteriori.
Suppose that instead of counting the angle v from ^the perihelion, we fix
its origin at any other point whatever ; it is evident that this angle would be
increased by a constant quantity, which we shall denote by w, and this will [668*^]
express the longitude of the perihelion. Instead of fixing the origin of f, at
the instant of passing the perihelion, if we fix it at any other instant, the
angle n t will be increased by a constant quantity, which we shall denote by
£ — w; the preceding expressions of -, and t>, will thus become [668 »]
[669]
378
ELLIPTICAL ORBIT OF A PLANET.
[Mec. Cel.
[669'] ^ ^^ t^® *^"® longitude of the planet, and nt-\-s its mean longitude, these
longitudes being counted upon the plane of the orbit.
[688aJ
C3
nt-{-e. sin. w ^ + ro~o' * ^^"- ^ " ^~i~i o o oa • {^^ • sin. 3 w i — 3 sin. w i}
C4
1.2.3.4.23
f 4^ . sin. 4:nt — 4.2^. sin. 2nt\
5.4
1.2.3.4.5.24
. 1 5* . sin. bnt — 5.3^. sin. ^nt-\- — . sin. nt\
6.5
1.2.3.4.5.6.25
+2.{^e+^e3+^V«'l'
. ^6^. sm. ^nt — 6.4^. sin. An t-\--r-i.2^ .sa\.2nt\
JL .<«
e . e^ .
sm. n^ + r-r .sin.2n< + r-?r-?S' 13 sin. 3w^ — sin. w t\
' 1.2 1.2.22 * >
e3
1.2.3.23
1.2.3.4.24
e5
1.2.3.4.5.25
. {4^ . sin. 4 n ^ — 8 sin. 2nt\
.{53.sin.5w^— 3.33.sin.3n<4-2sin.w^}
. { 64.sin.6ni—4.4''.sin.4n^-f 5.24.sin.2ni|
2e
sin
in. 2nt-\- — . \ sin. 3 » < — sin. n t\
. 2e2
+ |.He^ + *e'' + ^«*'}-'
1.2.22
2e3
^1.2.3.23
2e4
1.2.3.4.24
. 1 4 . sin. Ant — 4 . sin. 2nt\
.f5^sin.5n^— 3.3^.sin.3n^4-4.sin.»f}
. \ 6^.sin.6n^ — 4.43.sin.4w<+7.23.sin.2n< \
+§.l*e3+^e5|.^
3e
sin. '^nt-\-—z. {sin. Ant — sin. 2nt\
3e2
4" riT-^ • { ^ • sin. 5 » ? — 6 . sin. 3nt -\- sin. n t]
+ — ||— .{62.sin. 6«^— 3.42.sin.4n< + 3.22.sin.2n<}
4e
ksin. Ant -\- j— g-.fsin. 5 ni — sin. Snt]
1.2.22
.{sin. 6 w < — 8 . sin. 4 w < + 2 . sin. 2 n <^
+ f • {i^7 «^l • 5 sin. 5nt-\- — .{sin. 6nt — sin. 4 n ^} i
+ %^\^e^.{sm.6n_t]^hc.
n. iu. ^22.] ELLIPTICAL ORBIT OF A PLANET. 379
We shall now refer the motion of the planet to a fixed plane, which is
inclined by a small angle to the plane of the orbit. Put*
(p = the angle of inclination of the two planes ;
6 = the longitude of the ascending node of the orbit, counted upon the
fixed plane ; [669^/]
3 = the longitude of the ascending node of the orbit, counted upon the
plane of the orbit, so that & may be the projection of ^ ;
v^ = the projection of v upon the fixed plane ;
then we shall have
tang, (v^ — 6) = cos. ? . tang, (v — ^). [670]
This equation will give v^ in v and the contrary ; we may also obtain these
angles by very converging series, as follows.
We have before deduced the series [665] f
connecting together the terras depending on sin.n^, sm. 2 nt, &c. and making the
necessary reductions we shall obtain the expression of v, [668]. This method appears long,
but it is incomparably more simple and easy than the method formerly used by astronomers, as
explained in La Lande's Astronomy. Several astronomers have calculated these series as
far as e^^ or e^^, and Schubert has shown how to calculate any one term of the series,
independent of the rest.
* (464) Suppose a spherical surface to be
described with the radius 1, about the focus, which
is the origin of r, as its centre, and let the intersec-
tion of this surface by the plane of the orbit, and
the fixed plane, be represented respectively by the
great circles FDBA, E D C A. B being die
place of the planet, C its projection on the fixed plane, D the place of the node, F the point
fi-om which the longitudes are counted, and E its projection on the fixed plane ; the angle
BDC=EDF=((>; FD = ^', ED = d; FB = v', EC = v,;
DB = V — p; D C = Vj — 6. Then in the right-angled spherical triangles D EF,
BCD, we shall have tang. D C = cos. B D C . tang. DB; and
tang. D E = cos. ED F . tang. FD, which in symbols become
tang, {v^ — d) := COS. <p . tang, {v — (s), and tang. 6 = cos. 9 . tang. ^, [670ol
as above.
f (465) This is the equation [665] divided by 2.
380 ELLIPTICAL ORBIT OF A PLANET.. [Mec. Cel.
[671] iv = ^u-\->^. sin. u-\- — . sin. 2u + ~ . sin. 3u-\- &c.,
from the equation [606]
[672] tang. iv = l/^±^ . tang. i w,
by putting*
[673] X = '^ '-' .
[673Q If we change ^v into t?^ — 6; ^u into iJ — ^; and tylltl. into cos. 9;
we shall havef
[674] X
COS. 9
-— - = — tang.H? ;
COS. 9+1
[675]
[675']
the equation between ^ ?? and J ?*, will change into an equation between
v^ — d and V — ^, and the preceding series [671] will become
v^ — 6 ^v — ^ — tang.^ i 9 . sin. 2(v — ^) + ^ . tang.'* i 9 . sin. 4 (v — p)
— ^ . tang.^ i 9 . sin. 6 (v — p) + &c.
If in the equation between ^ v and i u, we change ^ 2? into v — p, i w into
V, — 5, and I / J-ii into ,t we shall find
' Y l—e COS. 9
[676] X = tang.^ i 9,
* (466) The value of X, [536, 662], is in [536<Z] reduced to the form [673].
f (467) These changes of J w into v^ — 6, ^u mlo v — p, and 1 y^ -i-^ into cos. 9,
being made in [672], it will become as in [670] ; the same changes being made in X [673],
will produce the first value of X, [674] ; its second value being deduced from the first by
means of [40] Int. By the same process [671] will produce [675].
f (468) Dividing [670] by cos. 9, we get tang, {v — ^) = — ^-^- ^, which might
be derived from [672] by changing in this last expression ^ v into v — p, iu into v, — 0,
and I /p^ into ; and then [673] would become
1/ 1 — e cps.9 ' I- -•
n.iii. §22.] ELLIPTICAL ORBIT OF A PLANET. 381
and
V — (3 = 2)^ — ^-f- tang.^ f <p . sin. 2{v^ — 0 + ^ tang.'* i 9 . sin.'* (y^ — d)
+ ^ . tang.H <p . sin. 6 (i;^ — ^) + &c.
We thus see that the two preceding series [675, 676'], mutually change into
each other, by altering the sign of tang.^ ^ ?, and writing v^ — ^, for v — 0,
and the contrary. We shall have v^ — ^, in a function of sines and cosines {^'^^'^
oi nt and its multiples, observing that by what precedes*
''.'■'■ ' ' ' ^ = nt + ^.+ eQ, [677]
Q being a function of sines of the angle nt-\-s — zs and its multiples ;
and the formula (i) § 21 [617] gives, for any value of z,t
sin. i (v — p) = sin. i(nt-{-s — ^ + eQ)
= 1 1 ni^+IT^ih — ^^•5-S^»-*(**^ + ' — ^) [678]
Lastly, 5 being the tangent of the latitude of the planet, above the fixed [678']
COS. (P 1 COS. <P at
1 , 1 + COS. 9 & -s T>
COS. 9 '
[40] Int., as in [676]. These changes bemg made in [671] it becomes as in [676*].
* (469) This is evident from the series [668] altered as in [669] and substituted
in [675].
f (470) Writing r for t in [617], to distinguish it from the time tj used in this article, and
for (p (t -f- a) putting sin. (r -f- «)) vve shall get
, , . . , d.sin.T , a^ d^.sm.T , a3 rf3.sin.T , „
= l'-]r2+ri¥.4-'^-l-='"-^+l''-il3+i:oz5-*^- !•'='''•■'•
which by substituting r = i{nt-\-s — p), a = ieQ, becomes as in [678]. This may [678a]
also be obtained from [21] Int. sin. {r-\-a) = cos. a . sin. r -\- sin. a . cos. r, for by substituting
the values of sin. a, cos. a, deduced from [43, 44] Int. it becomes as in [678].
96
382 ELLIPTICAL ORBIT OF A COMET. [Mec. Cel.
plane, we shall have*
[679] s = tang, (p . sin. (v^ — ^) ;
[679] and if we put r^ for the projection of the radius vector r upon the fixed plane,
we shall havef
[680] r^ = r.(l +5^)-!i = r.{l— 1.5^ + f .5^ — &c.] ;
we may thus determine v^, s, and r , by converging series of sines and
cosines of the angle n t and its multiples.
23. Let us now consider very excentrical orbits, like those of the comets.
For this purpose we shall resume the equations of § 20 [603, 606].
a.(l — e2)
I -f- e . COS. V
[681] nt = u — e . sin. u ;
tang, 1^ = 1 / i+_l . tang. ^ w. 4 i
Y 1 — e
In the case of very excentrical orbits, e differs but very little from unity ;
we shall therefore suppose
[681'] 1 — e = «,
a being very small. If we put D for the perihelion distance of the comet,
we shall have
[681"] D=a.(l —e) = aa ;
the expression of r will therefore becomef
(2 — a).D D
[682]
2.cosU^;-«.cos..; cos.^^z; .jl +^^ . tangf ^ t j '
* (471) Referring to the figure in page 379 we shall have in the right angled spherical
triangle BCD, tang. B C = tang. B D C .mi. D C, or by the symbols used in that
[679a] note s = tang, cp . sin. {v, — 6 ).
f (472) In the figure page 35 1, we have C D = r, C B = r^, tang. B C D = s,
or cos. B CD = —^=-= ( l -f s^)"^, and it is evident that CB=CD. cos. B CD,
hence r,==r . {l-\- ^) , as in [680]. Developing (1+5^) » by the binomial theorem
we shall obtain the second formula [680].
t (473) Substitute e= 1 — a, [681'] in r [681] and it becomes r== "«-^ ~^ —
+ \ / 5 L J L J 14-cos.v — a.cos.v
cos.^ i V + TT- sin.^ J V cos.^ i V . \ 1 + r-^ .tane.^ i v\
2 — a ^ 2 — a o «< 3
as in [682].
* (474) Put z = ^u, and tz=: tang. ^ m, in [48] Int. and we shall get [684].
Again tang. ^ w = I / ^ • tang, i v, [68 1], and — ^ = --^— , hence we obtain
[685]. Substitute this in [684], it becomes as in [686].
f (475) The first expression of sin. u, [687], is easily deduced from [30"] Int. This
being developed in series becomes like the second formula [687].
J (476) Multiplying the first member of the expression [687] by c, and the last member
by its equal 1 — a, then substituting for tang. J w, its value [685] we shall get [688].
n. iii. § 23.] ELLIPTICAL ORBIT OF A COMET. 383
which reduced to a series, gives
r = — ^—.\ 1 — — ^.tang.^ii; + (— ^) . tang.^i?; — &c. > . [683]
cos.^l?; i 2 — a ° \2 — a/ ^ ^ ) '--'
To obtain the ratio of v to the time ^, we shall observe, that the expression
of an arch by its tangent gives*
u = 2. tang. ^ w . { 1 — i . tang.^ ^ w + i tang.'' ^u — &c. } ; [684]
now we have
tang, i w = y/^-^ . tang. i?J ;
[685]
[686]
[687]
therefore we shall have
M=2.^_£_.tang.i|^.|l— ^(^^).tang.^ii;+i.(^^J.tang.^^^
moreoverf
sin. u = — i- — --ir— = 2 . tan? . huAl — Um.^ hu-\- tang.^ ^u — &c. \ :
1 + tang.2 1m to 2 I &2ib2 j?
hence we deducet
e.sin.w=2.(l— a).i^/^.tang.i«>.|l — (^^.tang-^lr+^^^ltang.'^it^ — &c.| [688]
but aa = I>, [G81"], cos. « = cos.^ i V — sin.^^v, [32] Int. and 1 4-cos.«>=2 .cos.^^t?,
[6] Int. ; hence by substitution
D.{2 — a) Z?.(2 — «)
2 C0S.2 i » — a . (cos.2 i u — sin.2 it?) (2 — a), cos.2 iiv-\-a. sin.2 ^ v
D D
384 PARABOLIC ORBIT OF A COMET. [Mec. Cel.
Substituting these values of u and e.sin. m, in the equation [681]
nt = u — e . sin. u ; we shall have the time t, in a function of the anomaly
V, by a very converging series ; but before making these substitutions, vee
shall observe that we have, by § 20 [605'], n = a~^ .s/'^, and as D=aa
[681"], we shall have
1 D^
[689] i = .
This being premised, we shall find*
[690] t = -4^z£L: . tang.i 2? .h + 2z^ . t^rls.^v—^4p^' tang.^i|^ + &c.^
If the orbit is parabolical, we shall have « = 0 [681', 3786], consequently,
[682, 690],
D
[691]
r =
cos.^lw '
t = Z . {tang. ^v + i. tang.^ ^ v}.
■'^ (477) From [605'] we get - = —7=, and by means of a = — , [681"], it
u ~^~ c Sin u
becomes as in [689]. Substitute this in t = ' — '— , [681], and we shall get
t = . \ u — e . sin. u v . From u [686], and — e . sin. u, [688], put under the
following form
y—- C-(l-a) + (l-a).(^).tang.Ht' ;
— e.sin.M=2.l/^-.tang.it;.; ^; \^
^ ( -(l-a).(^).tang.H. + &c.|
we shall get
(«+(§-«).(— :-).tang.Ht'
w — e.sin.M=2.1 /— ^.tang.^v. / 2
^ " ^-(l-a).(^) .tang.^it; + &c.
Substituting this in the preceding expression of t, it will become as in [690].
TI. iii. §23.] PARABOLIC ORBIT OF A COMET. 385
The time t, the distance /), and the sum of the masses of the sun and comet vQ^r]
fjL, are heterogeneous quantities, and to compare them with each other they
ought to be divided by the unity of measure of each species. We shall
therefore suppose that the mean distance of the sun from the earth is the
unity of distance, and that D is expressed in parts of that distance. Then if [691"]
we put T for the time of a sidereal revolution of the earth, supposing it to [691'"]
commence at the perihelion, we shall have, in the equation nt=u — e.sin.w
[681] w = 0 at the commencement of the revolution, and w = 2^^ at the
end ; < being the semi-circumference of a circle whose radius is unity. We
shall therefore have nT^ 2* ; but we have n = a ^.^/-—.^ [605']; [e9i'']
because a = 1 [691"] ; therefore
2 *
\/l^ = -y- [692]
The value of [t' is not exactly the same for the earth as for the comet ; since
in the first case, it expresses the sum of the masses of the sun and earth ; [692']
whereas in the second case it expresses the sum of the masses of the sun
and comet ; but the masses of the earth and comet being much less than that
of the sun, we may neglect them, and suppose that fx is the same for all these
bodies, and that it expresses the mass of the sun. Substituting therefore^
_ 2*
instead of \/(jl, its value -=7 1|692], m the preceding expression of < [691]; [692"]
we shall have
^ ^ '^^Y^ ' ^^^°^' ^ ^ + * • tang-H 1;}. [693]
This equation now contains only such quantities as are comparable with each
other ; and by it t may easily be obtained from v ; but to find v when t is
given, it will be necessary to solve an equation of the third degree, which
has but one real root. We may dispense with this solution, by making a [693']
table of the values of v, corresponding to those of t, in a parabola whose
perihelion distance is equal to the mean distance of the earth from the sun,
represented by unity. This table will give the time corresponding to the
anomaly v, in any parabola whose perihelion distance is Z), by multiplying by
Z)^, the time which corresponds to the same anomaly in the table. We [693"]
shall have the anomaly u, corresponding to the time t, by dividing t by
97
386 ELLIPTICAL ORBIT OF A COMET. [Mec. Cel.
[693'"] D^, and seeking in the table the anomaly corresponding to the quotient of
this division.*
Suppose now that the anomaly v, corresponding to the time ^, is required
in a very excentrical ellipsis. If we neglect quantities of the order a^, and
[693''] resubstitute 1 — e for a [681'] the preceding expression [690], of t in v,
in the ellipsis, will givef
[694] j^^-P^Va ^tang.itJ + i.tang.^ii; ^
^^ ( + (1— c). tang. |«J.{J — J.tang.^i?? — i.tang.^'lt;} y
* (478) Let f be the time corresponding to the anomaly v, in a parabola, whose
perihelion distance D is unity, t being the time corresponding to the same angle v and the
perihelion distance D. In this case we shall get from [693],
[693a] Comparing this with [693] we shall get t = D^.t', and ^=-3:. Now if «; be given,
3
[6935] we may find «' from the table, and then t=D^.tf; but if t be given we must find
if = —r. and then find in the table, v from this value of f.
Delambre computed a table of the values of v corresponding to the argument t, from 0 to
200,000 days, which has been republished in several works on astronomy. Burldiardt has
lately made a very useful change in this form of the table, by taking for the argument the
logarithm of t. This table was printed by him, in 1814, in an octavo form and is very
convenient for use.
f (479) In the expression [690] if we neglect terms of the order a^ we may put
< = :^-^.(l+ia).fl+(f — a).(J + Ja).tang.2it; — ia.tang.'*!?^! .tang.i»,
or by reduction,
<=::^-^.|(l+^a)+(i — ia).tang.2ii;— |a.tang.4.J«}.tang.it;
==:5-^. {tang. i« + J. tang.3ii;4-a. tang.it;. (i—i.tang.2ii; — i.tang.''it;)|,
as in [694].
*
II. iii. §23.] ELLIPTICAL ORBIT OF A COMET. 387
We must find, by means of the table of the motion of a comet, the anomaly
corresponding to the time t, in a parabola in which D is the perihelion
distance ; let Uhe, this anomaly, and U-{-x the true anomaly in the ellipsis, [694']
corresponding to the same time, x being a very small angle. If we substitute,
in the preceding equation, U-\-x instead of -y, and reduce the second member
into a series, arranged according to the powers of ar, we shall have, by [694"]
neglecting the square of a:, and the product of a; by 1 — e,
Sec
ftane;. \U-\-l. tans;.^ hU\+ -rr-Tr
i_e } '-> [695]
+ i—' . tang. IU,{\— tang.^ ^ £/— t tang.^ 4 U\
* (480) Putting «= U'+a?, we obtain from [29] Int.
tanff. i t> = tana;. i\U4-\x)=^ — — ~— — ^r— -^--— ,
^ ^ & V2 T 2 y 1— tang. i 17. tang, ix '
and by developing the denominator in a series, neglecting a?^, we get,
tang. \v = tang. \ U+ tang, i a; . (1 + tang.^ ^ C7) = tang. \U-\- ^^~ •
The cube of this divided by 3 is
hence
tang.it; + ^tang.Ht^=tang.iC7+J.tang.4 C7+*£|i^. ^i+tang.^^ u]^
tang, i X
=tang.i [7-1-J.tang.H U-
. i 0?, the arch itself | a?, it w
tang, i CZ+J.tang.^i U-
cosA hU'
And by puttbg for tang. \ a?, the arch itself | a?, it will become
X
2.cos.4it7
Substituting this in [694], and in the terms multiplied by a or 1 — e, putting 27 for r, it will
become as in [695]. Making tiiis equal to the value of t, [696] deduced from t [691] by
D^ 1/2"
changing v into U, according to the hypothesis [694^], and dividing by — jr=—i we shall
S^* ^==2:^^Iin7"^"l^"^^"^-^^-^^~'*^S*'^^~^-^S-Hf^i- Multiplybgby
2 COS.'* ^ Uj putting for cos. ^ U . tang. | t7, its value sin. \, Uj and sin. x for a;, we get
sin. a; = -Iz:! . tang, i [7. { — COS.H Z7+ COS.4 C7 . sin.4 C7+ 1 . sm.4 C7| ,
substituting sin.2 i U= 1 — cos.^ | U,
sin.4 i C7= (1 •— cos.2 J C/)2 = I — 2 cos." J t/+ cos.'' i C/,
and reducing we obtain [697].
388 HYPERBOLIC ORBIT OF A COMET. [Mec. Cel.
but by hypothesis [694', 691] we have
[®*1 .,,„ , «=^^.Stang.iC/+i.tanff.nf/i;
we shall therefore hare, by substituting for the small arch x its sine,
[697] sin. a;= tV . (1 — e) . tang, i C7.[4 — 3. cos.^ ^ U~ 6 . cos.^i U] ;
and if we compute a table of the logarithms of the quantity
[698] T^. tang. iC7.{4 — S.cos.^if/— 6. cos.^U],
it will be only necessary to add this tabular logarithm to the logarithm of
1 — fe, to obtain log. sin. x ; in this manner we may find the correction x to
[698'] be made in the anomaly U, computed for a parabola, to obtain the corresponding
anomaly in a very excentrical ellipsis.
24. It now remains to consider the motion in an hyperbolical orbit. For
this purpose we shall observe, that in an hyperbola, the semi-axis a becomes
negative^ and the excentricity e exceeds unity [378']. Putting therefore, in
u'
[698"] the equations (/) § 20 [606], a^ — a', and u = , and substituting
for sines and cosines their imaginary exponential values, the first of these
equations will give*
The second will becomef
* (481) Substituting in the first of the equations [606] the value of n [605'], also the
exponential value of sin. m, [11] Int. it will become
and if we put a = — a', u = - — ^ as in [698"], it will change into
\/fx.^
Multiplying this by — v/'--Jj ^'^ shall obtain [699].
f (482) The second equation [606], r=a.{\ — e . ccs. u), by substituting — a'
for a, and -^ , or -—^ — for cos. m becomes
r = a .\\e.{e' -\-c-''') — l\, , . gnioul>o-i bn
as in [700].
II. iii. <5 24.] HYPERBOLIC ORBIT OF A COMET. 389
r = a' .{i e . (C^ + c-"') — 1 }. [700]
Lastly, by taking the sign of the radical of the third equation, so that v and
u may increase with t, we shall have*
tang. hv = \ y/t±l. . \'A^l . [701]
Suppose in these formulas, w':= log. tang. (J* + ^to), * being the [7011
semi-circumference of a circle whose radius is unity. The preceding
logarithm being hyperbolical, we shall havef
,3 = c . tang. « — log. tang, (i * + 4 ^) ;
a ^
r = a! ,1 — : IS; [702]
( COS. ^ 3
tang. \v = y/'L±L . tang. 4 «.
*
(483) From [11, 12] Int. we get
3io.i«= ^_^^= _______ and
cos.iM= -^ = ,
, , sm. iM c -• — c " ,. , . ,
nence tang. ^ m = — = , — — - , or by multipljong the numerator
COS.!iW ^-—J^^)^u_^^ — hU^
and denominator by c^", tang. ^ m = ——-;—-. Substituting this in the third equation
[606], it will become as in [701].
f (484) Putting for brevity J at -|- i ts = 5, we shall have u' = log. tang, b, [701'], [702a]
, „/ , sin. b , , COS. ft
hence r=tang.6 = -, and 0""^ = ^— r, therefore [7026]
„' _^, sin. 6 COS. 6 sin.2 6 — cos.2 6
COS. 6 sin. b sin. b . cos. b '
butsin.^S — cos.^b= — cos. 2 J, and sin. 6 . cos. 6 = ^ sin. 2 J, [31, 32] Int. hence
, , cos. 2 b
e — c " = — 7^1^= — 2 . cotang. 2 6^ — 2 cotang. (^ *-}-«) =2 tang. w.
Substituting this and u' [701'] in [699] we get the first of the equations [702].
98
390 HYPERBOLIC ORBIT OF A COMET. [Mec. Ct:l.
[702'] The arch Z' , is the mean angular motion of the body m in the time t.
/^
a^
supposing it to move in a circular orbit about M, at the distance a'.^ This
arch may easily be found in parts of the radius ; the first of these equations
[702] will give, by a few trials, the value of the angle * corresponding to
the time t ; the other two equations will then give the corresponding values
of r and v.
25. Since T [691'"] expresses the time of the sidereal revolution of a
planet whose mean distance from the sun is a, the first of the equations (f)
[702"] ^ 20 [606], will give w r= 2 « [691'^] ; but we have by the same article,
[605'], ^-~ = n ; we shall therefore find
[703] T= ^"^-^"^
V^
If we neglect the masses of the planets, with respect to that of the sun, fx
will denote the mass of the sun, and this quantity will be the same for all
[703'] the planets ; hence for a second planet, in which a' and T' represent the
mean distance from the sun, and the time of a sidereal revolution, we shall
also have
[704] T' = — --- ;
Again the same values of c"', c""'*', give
, , , sin. 6 , COS. 6 sin.2 6-|-cos.2 6
COS. 6 sin. 6 cos. ft. sin. 6 cos. 6. sin. 6 sin. 2 6 sm.{h<K-\-T^) cos.'zrf
which being substituted in [700] gives the second of the equations [702].
Lastly, since tang. :J*= 1, [701'], and c"' = tang. h, [7026], we shall get
c"' — 1 tang. & — tang. I ^ . // i \
, I .. = — - = tang. (6 — t*),
c"'-f-l tang.6.tang.i<7r-|-] ° ^ ^'
f-u' 1
[30] Int. ; and as & — J<n' = ^ij, [702a] this will become ^ = tang. \ -a. Substituting
this in [701], we shall obtain the third of the equations [702].
* (485) In a circle, e = 0, and [668] gives v==nt, and by [605'] ^LK=n, hence
V = -iXJt as above,
a'^
U. Hi. §25.] MASSES OF THE PLANETS. 391
therefore we shall have
T^iT^'iic^ia!^; [705]
that is, the squares of the times of revolution of different planets are as the
... Keplers
cubes of the transverse axes of their orbits ; which is one of the laws ^*'^-
discovered by Kepler. We see by the preceding analysis, that the law is [705']
not rigorous, and that it exists only in the supposition that the attraction of
the planets upon each other, and upon the sun, is neglected.
If we take the mean motion of the earth for the measure of time, and its
mean distance from the sun for the unity of distance, T in this case will be [''^^1
equal to 2 *, and we shall have a == 1 ; the preceding expression of T [705'"]
[703] will therefore give* (x = 1 ; hence it follows that the mass of the sun
ought to be taken for the unity of mass. We may therefore, in the theory
of the planets and comets, suppose fx = I , and take for unity of distance, [705>''J
the mean distance of the earth from the sun ; but then the time t will
be measured by the corresponding arch of the mean sidereal motion of the [705 »]
earth.
The equation [703]
T=^, [706]
furnishes a very simple method of determining the ratio of the mass of a
planet to that of the sun, in case the planet is accompanied by a satellite.
For by representing the sun's mass by M, if we neglect the mass m of the [706']
planet in comparison with M, we shall have
T = ,_ . [707]
If we then consider the satellite of any planet m' ; and put p for the mass of
the satellite, h its mean distance from the centre of m', and T the time of its [707']
sidereal revolution ; we shall have
T = 4^ ; [708]
V/m'-fp
therefore
i'-\-p P /TV
* (486) T='2if, and a= 1, substituted in the equation [703] evidently gives (* = !.
392 MASSES OF THE PLANETS. [Mec. Cel.
This equation gives the ratio of the sum of the masses of the planet rn! and
its satellite, to the mass M of the sun ; by neglecting therefore the mass of
the satellite with respect to that of the planet ; or by supposing the ratio of
[709'] these masses known ; we shall have the ratio of the mass of the planet
to that of the sun. We shall give, when treating of the theory of the
planets, the values of the masses of those planets about which satellites have
been observed.
Il.iv. §26.] ELEMENTS OF THE ELLIPTICAL MOTION. 393
CHAPTER IV.
1»Xi»MINA'EION OF THK ELEMENTS OF TB£ ELLIFTICAii MOTION.
26. After having explained the general theory of the elliptical motion,
and the manner of computing it, by converging series, in the two cases of
nature, namely, that of orbits nearly circular, and that where they are very
excentrical ; it now remains to determine the elements of these orbits. If
the circumstances of the primitive motions of the heavenly bodies were
given, we might from them easily deduce these elements. For, if we [709"]
put V for the velocity of w, in its relative motion about M, we shall
have [40«]
V= j^, ; [710]
and the last of the equations (P) ^18 [672], will give
To eliminate ps- from this expression, we shall denote by U the velocity which
m would have, if it described about M a circle whose radius is equal to the [7lY\
unity of distance. In this hypothesis, we shall have r=a:=l, consequently*
U'^ = ii ; therefore
V'=Uk\-—-\. [712]
I r a }
This equation will give the semi-transverse axis a of the orbit, by means of [712*]
*(487) Substituting r = fl=l, and V= U, in [711] gives U^z=ii, hence [711]
becomes as in [712],
99
^^^ DETERMINATION OF THE ELEMENTS [Mec. Cel.
the primitive velocity of m, and its primitive distance from M.* a is positive
in the ellipsis, infinite in the parabola, and negative in the hyperbola ;
therefore the orbit described by m is an ellipsis if F< Ua/^-, a parabola
[712"] *'^ ** •
if F=C/.l/_, and an hyperbola if F>C/.l/^. It is remarkable
[712'"] that the direction of the primitive motion has no influence on the species of
the conic section described.
To determine the excentricity of the orbit, we shall observe, that if we
put s for the angle which the direction of the relative motion of m makes
[7i2iv] with the radius r, we shall havef -7-3- = V^ . cos.^ s. Substituting for
V^ its value f* . ( j [711], we shall have
[^13] — = f.Y--iycos.s^;
* (488) Putting F" for the primitive velocity, and r for the primitive distance, in [712],
1 2 rs 1 . .
we shall set — = — — , from which we may compute the value of — or a ; and it is
a r U^ a
evident that - is positive if F'<Z7.1y^-, - = 0 if F= ?7.1 y^- , and- is
negative if T^^U.t / -•> ^<1 since a is infinite when -= 0, it will follow from [378']
that the curve will be an ellipsis if V <^V .\ / - , a parabola if V= ^-X/ - '
and an hyperbola if V^U.t / ~-
f (489) In the adjoined figure let M be the place of the
bodyJkf, m that of the body m, m mi the primitive direction
of the body m in its relative motion about M; mm! being the
space described in that relative orbit in the time d t. Then
Mm^r, Mm'==r-\-drj and taking on M m, continued, Ma = Mm', ma=:dr,
the angle Mmm'—s, and mm'=Vdt. Then in the triangle mam' we have
ma = mm' .COS. amm'=^-^ mm' .COS. s, or dr = — Vdt.coss, squaring we find
~=:V^. cos. s2, hence from F^ [711], we get [713]. The value of h^, [599], being
n.iv. §26.] OF THE ELLIPTICAL MOTION. 396
but we have by § 19 [598, 599]
2M.r -^ ^- = ,xa.(l— e^) ; [714]
therefore we shall have
a . (1 — e^) = r^ . sin. s^ . (^^ — 1^ ; ^7,53
which will give the excentricity of the orbit a c. [7151
The equation of conic sections [378]
a.(l — e2)
7* =
1 + e . COS. V '
gives
a.(\ — e^) — r
cos. V = — ^^
er
[716]
[716^
Hence we shall find the angle v^ which the radius vector makes with the
perihelion, consequently the position of the perihelion will be obtained. The
equations (/) § 20 [606], will then give the angle w,* and by this means
the time of the passage by the perihelion may be found. [716"]
To obtain the position of the orbit with respect to a fixed plane passing
through the centre of M, supposed to be at rest ; let 9 be the inclination of
the orbit to this plane, and |3 the angle which the radius r makes with the
line of nodes ; also, let z be the given elevation of the body m above the [716'"]
used in [598] gives [714]. Substituting in [714] this value of — , we find
2fxr ^^fA'f ) -COS. s^ = ^a . (1 — e^).
a \r a /
Dividmg by fx and reducing the first member, it becomes
(^— ^y r2 . (1 — COS. 62) = a . (1 — e2),
and as 1 — cos. r* = sin. s% it changes into [715]. Now as the primitive value of r is
given, and a is known by [712], we shall easily obtain e from [716].
* (490) The last of the equations [606] gives « by means of e, v, which had been
previously computed [715', 716']. Having «, we may obtain t by means of the first
equation [606].
396
[717']
[717"]
[718]
[718']
[719]
DETER]VIINATION OF THE ELEMENTS [Mec. Cel-
fixed plane, at the commencement of the motion ; we shall have*
r . sin. |3 . sin. <? = z ;
hence the inclination of the orbit ? will be known when /3 shall be determined.
For this purpose let x be the angle which the primitive direction of the
relative motion of m makes with the fixed plane, this angle being supposed
to be given ; if we refer to the triangle formed by the line of nodes, the
radius r, and the line of the primitive direction of the motion continued till
it meets the line of nodes, and put I for the side of the triangle opposite to
the angle p, we shall havef
r . sin. p
also J = sin. x
sln.(^4-£)
hence we shall get
tang^»|3=:
z . sin. s
r . sm.X — z.cos.s
* (491) Let M, m, be tlie places of the bodies M, m,
MJV the line of nodes ; m m', the line of the primitive
direction of the relative motion of the body m about M, which
line being continued meets JkfJV in JV. Draw m.^ perpen-
dicular to MJV, and m B perpendicular to the fixed plane,
to meet it in 5. Then Mm = r, TnB = z, JYm = l,
angle JYMm = p, angle MmJV=s, consequently the angle MJVm = * — p — e,
* being equal to two right angles, angle B Am = cp, angle B JVm='k. Then in the
right angled plane triangles MAm, ABm, we have Am = Mm . sin. JVMm = r . sin. ^,
and mB or z= Am .sm. BAm = Am . sin. <p=^ r .sm.^ . sin.cp, [717].
f (492) In the plane triangle MJVm, we have sin. MJVm :Mm:: sin. JVM miJVm,
or in symbols sin. (-tt — j3 — s) : r : : sin. ^ : Z, hence I =
r . sin, |3
as m
[718].
6in.((3 + £)'
Again, in the right angled plane triangle JVB m, we have mB = JVm. sin. B JVm,
which in symbols is z = Z . sin. X, or Z= —
sin. X
Substituting this in [T18] we get
sin. X
2.sin.(p + £) 2.(sin. |3.cos. ?-[~cos. /B.sin. s)
r . sin. j3 r . sin. ^
numerEiior and, denominator by cos. (3 it becomes sin. X :
we easily deduce tang./s, [719].
[21] Int., and if we divide the
2 . (tang. ^ . COS. £+ sin. s)
r.tang. ^
hence
II. iv. § 26.] OF THE ELLIPTICAL MOTION. ^97
The elements of the orbit of the planet being determined, by these
formulas, in functions of the radius r, the elevation z, the velocity of the
planet, and the direction of its motion ; we can find the variations of these
elements, corresponding to any supposed variation in the velocity or in the
direction of the motion ; and it will be easy, by the method we shall hereafter
give, to deduce therefrom the differential variations of these elements, arising
from the actions of the disturbing forces.
We shall now resume the equation [712]
F==?7\J^-i-j. [720]
In the circle a = r, consequently V= U. \/- \ hence the velocities [72(y]
of planets in different circles are* inversely proportional to the square roots of
their radii.
In the parabola a ^ co, hence F=C7.I/ _ [720] ; therefore the velocity [720"]
in any point whatever of the orbit is inversely proportional to the square root
of the corresponding radius vector r ; and the velocity of the comet will be
to that of a planet, which should revolve about the sun in a circular orbit at
the same distance r, as ^2 to l.f [720"/]
An ellipsis infinitely flattened becomes a right line ; and in this case V
would express the velocity of m, if it should fall in a right line directly
towards M. Suppose that tn should fall from a state of rest, and that its [720i»]
distance from M at the commencement of motion was r, and when it has
arrived at the distance r', it should have acquired the velocity V ; the [720^]
preceding expression of the velocity will give the two following equations :%
* (493) In the original the word inversely was accidentally omitted.
f (494) The velocity in the parabola having been found to be Z7 . | y^- , [720"], and
the circle ^'\/ -•> [720'], these are evidently to each other as \/2 : 1.
in
J (495) The first of these equations is found by putting F= 0, [720], at the
his gi
100
2 1 12
commencement of the motion, this gives =0j hence - = -, which being
T ft. n. 7* O
[721] 0 = -
r
hence we deduce
^^^ DETERMINATION OF THE ELEMENTS [Mec. Cel.
[722] V'=U.\ /2.(r-"7j .
which is the expression of the relative velocity acquired by m, in falling from
[722'] the height r, towards M, through the space r — /. We can determine
easily, by means of this formula, from what height the body m, moving in
a conic section, ought to fall towards M, to acquire in falling from the
extremity of the radius vector r, a relative velocity equal to that which it
has at that extremity ; for V being this last velocity, we shall have
[723] V'==UKi^ — -l ;
but the square of the velocity acquired by falling from the height r — /, is
[723'] 2 U^ . — -— [722] ; putting these two expressions equal to each other, we
[724]
shall find*
, r .(2 a — r)
r — r = — 7
4 a — r
[724'] In the circle, a = r, and then r — r'=^r [724] ; in the ellipsis, we havef
r — r' <i\r \ in the parabola a is infinite, and we have r — r' = \r ; and
in the hyperbola, where a is negative, we have r — r' ^ \r.
substituted jn, the second equation [721] V'^= V^ .\- > , it becomes
r'^^m_(l_^\^U.,l±=n, hence V'^U.\/IS=n,
Vr' r/ r/ 1/ rr
as in [722].
* (496) This equation, by rejecting the common factor U^, becomes ^ — = ,
hence / = - , and r — / = — ; , as m [724 J.
4a— r 4o — r
7-2
f (497) The expression [724] may be put under the form r — r' = ^r — ^ i^a—r\ '
and in the ellipsis where A a — r is always positive, the last term must be negative,
consequently r — r' <^^r. In the parabola, where a = co, that term vanishes, and we
get r — r'=-^r. In the hyperbola a becomes negative, and then by putting a = — a',
,•2
it becomes r — r'=^^r-\- , which evidently exceeds J r.
n. iv. § 27.] '■-'■ GFTHE EEOPTICAL MOTItaNr.^. . ^^
27. The equation [572] 5Y£ii IIm*';^ e> >/
is remarkable, because it gives the velocity independently of the excentricity
of the orbit. It is comprised in a more general equation, between the [725']
transverse axis of the orbit, the chord of an elliptical arch, the sum of the
extreme radii vectores of the arch, and the time of describing the same arch.
To obtain this last equation, we shall resume the equations of the elliptical
motion, given in § 20 [603 — 606], supposing for greater simplicity (*= 1. [725"]
These equations thus become*' ^"^'*^ ''''^^'^*''
:i i;iiiyu "5
2\. - <:?
a. (I — e2) J
y i £_ •
1 -{- e . COS. V '
r = a.(l — e. cos. u) ; [726]
t = a^ . (u — e. sin. u).
Suppose that r, v, u, U correspond to the first extremity of the elliptical [726']
arch, and r', v\ u\ tf, to the other extremity ; we shall have
i-\-e . COS. V
r' = a.(\—e. cos. u') ; [''^^l
tf = a^ . (u' — e . sin. m').
Put
r-t=T; ^ = /3; ^ = ^' ; f' + r^R. [728]
If we subtract the expression of t [726], from that of if [727], observing
thatf
sin. u' — sin. % = 2 . sin. |3 . cos. ^' ; [729]
* (499) The first of these equations is as in [603]. The second is like the second
of [606]. The last is the same as the first of [606], substituting w = i-ii, [605'], putting
(* = 1, [725"], and multiplying by az. Accentmg the letters r, v, m, t, [726] we obtain
[727].
t (500) By [26, 27] Int. we have sin. w' — sin. « = 2 . sin. ^^^V cos. 0^^\
and cos. u' + cos. w =2 . cos./ — - — j .cos. ( — r — j, which, by u^g,j.thei valines, of
400 DETERMINATION OF THE ELEMENTS [Mec. Cel.
we shall have
[730] T = 2«^.{^ — e.sin.|3.cos.|3'}.
If we add together the two expressions of r and r' [726, 727], in u and u',
observing that
[731] cos. u' + COS. w = 2 . COS. |3 . cos /3',
we shall have
[732] R=2a.{\ — e . cos. p . cos. |3').
Now c being the chord of the elliptical arch, we shall have*
[733] c^=r^-|-/^ — 2rr'.cos. (u — v') \
.,. _, but the two equations [726]
«.(1 — e2) .^ .
[734] r = — -^ ; r = a.(\ — e . cos. u) ;
1 + e . COS. V ^ ^
givet
\ COS. u — e I . a. \/l — e^ . sin. u
[735] COS. v = a. ; sm. v = — .
r r
And in like manner
_o.ji / {cos.m' — e} . , a-v/l — e2.sin.w'
[736] COS. v' = a. -J ; sm. v == — ^ j ;
r r
3, |3', [728] become as in [729,731]. These being substituted in f — t, and Z + r,
deduced from [726, 727] give [730, 732].
* (501) As in [471] or [63] Int., putting, in the second figure, page 292, A C = r,
C M= /, angle A C JW= v — v'.
(lA\ — ^\ ,•
f (502) The first value of r, [734], gives cos. ■« = -^ , and by substituting
J COS W II I C t
in the numerator the second value of r, [734] it becomes cos. t> = a . '■ j hence
sin
«=\/l — cos.2v = | y^ 1 2-* (cos. M — e)^=-. \/r2— a2.(cos.w— c)2
Substituting r = a . (1 — e . cos. m), [726], in the radical, it becomes
o a . v'l— e2 . sin. u
sm. « = - .\/(I— cos.2«).(l — ee)= ,
as in [735] ; and by accenting the letters r, w, v, [735], we get [736].
[739]
II. iv. § 27.] OF THE ELLIPTICAL MOTION. ^1
therefore we shall have* ., ::r4,M
r / . COS. (y — v') = a^.(e — cos. u).(e — cos. u') + (f .(\ — e^) . sin. u . sin. u' ; [737]
consequently
c^ = 2a^ (1 — e^). { 1 — sin. M. sin. m' — cos.w . cos. w'} -\-a^e^. (cos. w — cos. w'/ ; [738]
now we have
sin. u . sin. u' + cos. u . cos. i*' = 2 . cos. ^^ — 1 ;
COS. u — cos. u' = 2 . sin. |3 . sin. ^' ;
therefore
c" = 4 a^ sin.^ p,(l—e\ cos.^ |3') ; [740]
hence we have the three following equations [732, 730, 740] :
R = 2a.{l — e. cos. ^ . cos. ^} ;
T = 2a^.{^ — e . sin. |3 . COS. |3'} ; [741]
c^ = 4 a^ sin.^ ^ . { 1 — c^ cos." /3'}.
The first of these equations gives
_, 2a — R
e . cos. p = ; [7421
' 2 a. COS. p* L^^^J
* (503) Since cos. (» — t/) = cos. v . cos. i/ + sin. v . sin. v', [24] Int., by using the
values [735, 736], we shall get,
r / . cos. {v — v') = a^ . (e — cos. m) . (e — cos. m') -|- a^ . (1 — e^) . sin. u . sin. u',
[737]; substituting this, and r = «.(l — e.cos. m), / = «.(! — e.cos. m'), [734],
in [733], it becomes
c2=a^.(l — e.cos.M)^+a^.(l — e.cos.w')^ — 2a^.(e — cos.M).(e — cos.m') — 2a^.(l — e^).sin.M.sin.w'
=2a^ — 2a^e^ — 2a^.(l — e^).sin.M.sin.tt'-|-a^e^.cos.^M — ^2a^.cos.M.cos.tt'-}-a^e^.cos.^M',
in which the three last terms
aV.cos.^M — ^2a^.cos.M.cos.M'-f-aV.cos.V=aV.(cos.!< — cos.m')^ — 2 a^.(l — ee) . cos.w.cos.m',
being substituted we get ^
<pz=2a^.{l — e^). {I — sin. M. sin. m' — cos.M.cos.tt'}+a^e^.(cos.M — cos.m')*,
as in [738], but sin. u . sin. u' -\- cos. u . cos. u = cos. {u' — w), [24] Int., and this by
using [728], is = cos. 2^ = 2. cos.^ ^ — 1, [6] Int. Also by [17] Int.
2 . sin. 3 . sin. p' = cos. ((S' — p) — cos. (^' -|- ^) = cos. u — cos. m', [739].
These being substituted in [738], we get
c2= 2 a^ . (1 — c2) . |2 — 2 . cos.2^} -{-0^6^.(2. sin. fi . sin. p^,
and by putting 2 — 2 . cos.^ ^ = 2 . sin.^ ^, it becomes
c3 = 4 a2 . sin.2 ^ . { i __ e^ + e^ . sin.^ ^'\ = 4 a^ . gin.^ ^ . 1 1 — e« . cos." 0'}, [740].
101
4^02 DETERMINATION OF THE ELEMENTS [Mec. Cel.
substituting this value of e . cos. f3' in the other two equations, we shall
have*
[743]
T = 2«- .| f3 + (^-^— J . tang. |3 ^ ;
= 4 a^ tang.^^ . ^ cos.^ |3 — f^^^—)' \ -
(? =
These two equations do not contain the excentricity e ; and if in the first,
we substitute for (3 its value given by the second, we shall have T in a
[743'] function of c, i?, a. Hence we see that the time T depends only on
the semi-transverse axis, the chord c, and the sum R of the extreme radii
vectores.
If we put
2a — R + c J 2a — R^c
the last of the preceding equations will givef
[745] cos. 2 ^ =ZZ'+V/(1— s2).(l— ^2) ;
* (504) Substituting the value of e . cos. ^', [742], in c?, [741], we shall find
(^=4a^.] sin.2 ^ — ( -— ) . — f- [ ,
( ^ \ 2a J cos 2 ^ y
11 sin. ^
and by putting in the first term sin. ^ = cos. p . tang, p, and m the last - — -=tang. p,
it becomes as in [743].
[745a] t (505) From [744] we get 2!—z' = ^, i{z + z')= ^^ , which, bemg
substituted in -, [743], give
{z — z'f = 4 .tw^.^ ^ . {cos.'' ^ — i{z + z'f I =4.sin.2^ — (2: + z')2.tang.2^.
Now by putting cos. 2 ^ = v, we shall have sin.^ p =i — i • cos-2|3 = ^ — | v, [1] Int.
ta„g.= g = i=^-||f=i=^, [40] In..; hence (.-z')^=4.-(4_i«)^(.+z')=.tr.
Multiplying by ^ (1 + v) we shall get
l^v''=:^:^{l^v).{z^z'f + i{l^v).{z-{-t'f = z^-\-z'^ — 2vzz', or
v^-.2vzz' + z^z'^=l—z^ — zf^ + z^gf^ = {l—z^).{l—z'^),
extracting the square root, we shall find v — zz' = v/(l— z!2).(l— z'^), and by resubstituting
11. iv. § 27.] OF THE ELLIPTICAL MOTION. ^
heiice we deduce
2 (3 = arc. COS. 2f — arc. cos. z ; [746]
arc. COS. z denoting the arch which has z for its cosine ; hence we shall
have*
sin. (arc. cos. z') — sin. (arc. cos. z) , .
tang. f3 = ^ z^fT' ' ' ^ ^
we shall have also z -{- z' = [744] ; the expression of T will
therefore become, by observing that if T be the time of a sidereal revolution
of the earth, and the mean distance of the earth from the sun be taken for [747']
unity, we shall have by § 16, r = 2* [705"],t
T= — ^.{ arc. COS. 2:' — arc. cos. 2: — sin. (arc. cos. 2:')+ sin. (arc. cos. 2:)}. (a) [748]
As the same cosine may appertain to several arcs, this expression of T is
ambiguous, and we must carefully distinguish the arcs to which the cosines
z and zf correspond.
V = COS. 2 p, we shall obtain cos. 2^=zz' -{- v/{l — z2).(l— z'2), as in [745]. Now
if we put z = cos. Aj z = cos. B, this will become [7456]
cos. 2 p = COS. ./2 . cos. jB -f- sin. v2 . sin. 5 = cos. (jB — A)^ [24] Int.
Hence cos. 2 (3 = cos. (.B — A), therefore we may put
2 p = i5 — A= arc. cos. z' — arc. cos. z, as in [746]. [745o]
*(506) We have tang.i.(5— .^) =^^^^^, [36] Int., also, ^.(5— ^)==p,
COS. B = z', COS. A = z, [745c, 6], hence we find
sin. (arc. cos. 2/) — sin. (arc. cos. z)
tang, p == j:^ ,
as in [747].
f (507) Having by [705"], T=2'r, the expression of T, [743], becomes by
multiplying by — , which is equal to unity, T = — — . < 2 p + ( j . tang, ^i;
j = — {z-{-g/); using
these and tang. ^, [747] we obtain [748].
404
[749]
[750]
DETERMINATION OF THE ELEMENTS [Mec. Cel.
In the parabola the semi-transverse axis a is infinite, and we shall have*
arc. COS. z' — sin. (arc. cos. 2') = ^ . ( ) .
By making c negative, we shall have the value of arc. cos. 2 — sin. (arc.cos.^) ;
the formula (a) [748] will therefore give, for the time T employed in
describing the arch subtended by the chord c,
the sign — taking place when the two extremities of the parabolic arch are
situated on the same side of the axis of the parabola, or when one of them
being below,t the angle formed by the two radii vectores is turned towards
* (508) The transverse axis 2 a being very great in comparison with R,c, the values of
z, z', [744], must be very nearly equal to unity; and A,B, [745b'] may be considered as
very small, therefore if we neglect B^ and its higher powers, we shall have
sm.B = B — ^B\ cos. 5 = 1 — i J5^ [43, 44] Int.
but by [744, 7456], cos. B = 2^ = , hence
2a
2a
{R±c)
2a
J2 + C
therefore B^ = nearly. This gives ^ B^
=^m
in the preceding value of sin. B, gives B — sin. B=-l .( 1 '
which being substituted
as in [749]. Again,
since z may be derived from z', [744], and therefore A from B, [7456], by changing the
sign of c, we may, from the preceding expression of B — sin. B, obtain the value of
A — sin. A = ^ . ( ) ^ , corresponding to z. These being substituted in [748],
putting also R=zr-{-r', [728], it will become as in [750].
f (509) Let ABDE F be a parabola, whose axis is D C O,
vertex D, focus C, A B the proposed parabolic arch, whose
chord AB=c, CA = r, C B = r'j and suppose A C to he
continued to E; A being always taken for the point most distant
from the vertex. Now to ascertain the sign of the terms
^^r _j_ / — cf in T, we shall observe that when the time is very
small, it must be nearly proportional to the chord A B, which will
be small in comparison with r, /, and if we develop T, [750],
n. iv. §27.] OF THE PARABOLIC MOTION. ^05
the perihelion ; in other cases we must use the sign -f • ^ being equal to
365^25638, we have -|^ = 9^,688724. [75(r\
12 If
according to the powers of c, neglecting c^, (?, he. we shall have, by using Rz=r-\-r',
for brevity, T=r^ • ] R^ +| il*c=F-R'^zh|ii c > , which cannot be proportional to [750a]
0 3.
c, unless the two terms R- =f:/J^ destroy each other, which is the case when the upper sign
takes place ; therefore when c is small we shall have
'^^i--[i^+''+'f-i'+''-'f]-
The most distant point A from the vertex being fixed, suppose the other point B, to move
from A towards E, the terms r -j-r^ -j-e, and r-\-r'-r-^c, will always be positive,
because in the triangle B C A we have A C+B C^AB, or A C+B C—AB^O,
that js r -j- r' — c ]> 0. At the point E we shall have r-^r' — c = 0. In proceeding
from E towards F, and beyond F, r -{-r' — c always exceeds 0, so that there caij be np
change of sign except when the point B passes through E ; and if a change of sign then
take place, we shall have in the branch EF, T = r^. \ {r -\- r' -\- c)- -\- {r-\-r' — c)^ > ;
and the rules for applying the signs will agree with the above. It only remains therefore to
examine whether this formula is exact for any one point of the branch E F. Now putting
C D=-e, we shall take the points A, JP, so that the absciss D G = 4 e, and
CA = CF=r=r^. Then by the nature of the parabola the ordinate CH=CI=2e,
the ordinate FG = AG = 4e; and C F= 5e. Hence r = r = 5 e,
c = 2.FG = 8e; r + / + c= 18 e, r-{-r' — c = 2e, and [750] becomes
In a similar manner we may find the time of describing ID H, by putting C 1= r=2e,
CH=r' = 2e, Jf/=c = 4e, hence r-{-r'-\-c = 8e, r-\-r' — c = 0, and
from [750] t = r^ . ^ (8 e) ' | = rf^ . (2 e)^ . 8. This value is to the former [750<r|,
as 8 : 27 =F 1 , which ought to represent the ratio of the areas ID H, A D F C A,
[3G5'] ; the former of which is =§.jff/. DC = §.4e.e = fe2. The latter is equal
to the parabolic space AD F G less the triangle A C F. This parabolic space is
=%.DG.AF=%.4e.8e=-%^-e', the triangle ^ CF=^CG.^F=i.3e. 8e=12e3,
and -6^ e^—12e^=^'-. Hence the space ID H : space A DEC A ::^e^:^fi::8:28,
and as this ratio ought to be the same as 8:27=Fl) the lower sign must take place at the
point F, which was to be proved.
102
[7506]
[750c]
[750rf]
[750e]
^^ ELEMENTS OF THE HYPERBOLIC MOTION. [M6c. C61.
In the hyperbola, a is negative, [378'] ; z and z! become greater than
unity ;* the arches arc. cos. z, and arc. cos. 2', are imaginary ; and we shall
have, by using hyperbolic logarithms,t
arc. COS. z = -—^ . log. (z -f \/z^—\) ;
[751] v—i o V V
arc. COS. z! = -—= . log. {2! + ^/T^I^T) ;
the formula (a) [748] thus becomes, by changing a into — fl,
J T
[752] T = — . {v/«'2_ 1 =p ^z^^i — log. (z! + v/;?'2 — 1) d= log. (z + v/^^—l)}.
The formula (a) [748] gives the time employed by a body in descending
in a right line towards the focus, setting out from a given point with a given
velocity; to obtain this, we must suppose the ellipsis to be infinitely flattened.
If we suppose, for example, that the body sets out from a point at rest at
the distance 2 a from the focus, and the time T be required, in which it
would fall through the distance c ; we shall have, in this case, i2 = 2«+r;
r=2a — c ; hencej zf = — 1 ; z — ^^^ ; the formula (a) [748] will
therefore give
[753]
-^=^"^•1— •-•(^)+v/^^l-
* (510) Putting a negative in the values of z, z'^ [744], they become
— 2a — JR + c 1 I ^ — <^ t — 2a — -R — c , i ■R+<'
and as R= CA-j- C B, [728], always excels A Bore, the term R — c must be
positive, consequently z, z', exceed unity.
f (511) We have c =cos. ^ +V^— T* sin* A, [13] Int. whose logarithm divided
by \/^^ gives A = — =r . log. < cos. A + V^^^ • sin. A > ; and as « = cos. A, [7456],
this becomes arc. cos. z = -— r . log. <z -\-\/z^—l \ . The expression of arc. cos. a^, is
found in a similar manner ; these agree with [751].
f (512) These values of jR, r give 72= 4 a — c, which being substituted in ar, «',
[744], they become a^ =■ — 1, « = -^- , and as cos. A = z, cos. B=s!, [745J],
n. iv. § 28.] ELEMENTS OF THE ELLIPTICAL MOTION. 407
There is however an essential difference between the direct motion
towards the focus, and the motion in an infinitely flattened ellipsis. In the
first case the body having arrived at the focus, passes through it, and ascends
to the same distance on the opposite side ; in the second case, the body [753']
having arrived at the focus, returns back to the point from which it set out.
A tangential velocity at the aphelion, however small it might be, would be
sufficient to produce this diflerence ; and such a change in the velocity would
have no effect in altering the time of descent to the focus.
28. As the circumstances of the original motions of the heavenly bodies
are not known from observations, we cannot determine, by the formulas of
^ 26, the elements of their orbits. It is necessary for this purpose to
compare their respective positions, found by observations, at different epochs ;
this is rendered more difficult by the observations not being made from the
centre of their motions. With respect to the planets, we may, by means of
their oppositions or conjunctions, obtain their longitudes, as if they were
observed from the centre of the sun. This circumstance, taken in connexion [753"]
with the smallness of the excentricities, and the inclinations of the orbits
to the ecliptic, furnishes a very simple method of obtaining their elements.
In the present state of astronomy, the elements of these orbits require but
very small corrections ; and as the variations of the distances of the planets [753"']
from the earth, are never so great as to render them invisible, we may observe
them at all times, and by comparing a great number of observations, we may
rectify the elements of their orbits, and correct the eff*ect of small errors to which
the observations are liable. This is not the case with comets ; we see them
only when near the perihelioYi, and if the observations made at the time of
their appearance are not sufficient to determine their elements, we shall have
no method of tracing in our minds the paths of these bodies in the immensity
of space ; and when in the course of ages they shall approach again towards [753'*]
the sun, it will be impossible to recognise them. It is therefore important
we shall have B=:ir, whose cosine = — 1 , and sine = 0, also cos. A=z= —
a
hence sin. A = 1 /'l — (^-^) = \ X^^^— . These values being substituted in
[748] it becomes as in [753].
408 INTERPOLATION. [Mec. Cel.
to determine the elements of the orbit, by the observations made during the
appearance of the comet ; but this problem, taken rigorously, exceeds the
power of analysis, and we are obliged to have recourse to methods of
approximation, to obtain the first values of the elements, which may
afterwards be corrected with all the precision that the observations may
require.
If we use observations taken at distant intervals, the elimination of the
unknown quantities leads to impracticable calculations ; we must therefore
confine ourselves to observations made near to each other, and even with this
[753"] restriction, the problem is extremely difficult. After having reflected on the
subject, it has appeared to me that instead of using directly the observations,
it would be better to deduce from them certain quantities which would furnish
an exact and simple result ; and I am convinced that the quantities which
best fulfil this condition, are the geocentric longitude and latitude of the
comet, at a given time, and their first and second differentials divided by the
corresponding powers of the element of the time ; for by means of these
given quantities, we may determine the elements, rigorously and with
[752"'] facility, without any integration, using merely the differential equations of
the orbit. This manner of considering the problem allows us to use a
great number of observations taken near to each other, but comprising a
considerable interval between the extreme observations, which is very useful
[75.3vii] in diminishing the influence of the errors to which these observations are
always liable, on account of the nebulous appearance surrounding comets. I
shall, in the first place, give the necessary formulas to determine the first
diflerentials of the longitude and latitude, from any number of observations
taken at short intervals ; I shall then determine the elements of the orbit of
[753viii] a comet, by means of these first diflerentials ; lastly, I shall explain the
method which appears to me the most simple, to correct these elements, by
three observations taken at distant intervals.
r753'»l ^^- ^^ ^ given epoch, let « be the geocentric longitude of a comet, 6 its
northern geocentric latitude, the southern latitudes being supposed negative.
If we denote by 5, the number of days elapsed since the epoch, the
[753 ^'l geocentric longitude and latitude of the comet, after that interval, will be
expressed by means of the formula (^) of § 21 [617], by the two series
TI. iv. §29.] INTERPOLATION. ^^^
[754]
We shall determine the values of «, (t-)j ("T^)'^^'' ^» v^)' ^^* '
by means of several observed geocentric longitudes and latitudes. To obtain
these quantities in the most simple manner, vre shall consider the infinite
series expressing the geocentric longitude. The coefficients of the powders
of 5 in this series, ought to be determined by the condition that it will [754']
represent each observed longitude, by substituting for 5, the number of
days w^hich corresponds to it ; we shall thus have as many equations as
observations, and if the number of observations be n, we can determine, by
means of them, only n quantities of the infinite series «, ( t~ )? ^c. But [754*^
we ought to observe, that s being supposed very small, we may neglect the
terms multiplied by 5", 5"+^, &c. ; this will reduce the infinite series to its
n first terms, which may be determined by the n observations. These
values will be merely approximations towards the truth, and the degree of
correctness will depend on the smallness of the neglected terms. They will
become more correct by decreasing s, and by increasing the number of [754'"]
observations. The theory of interpolations is reduced by this means to
the finding of a rational and integral* function of s, of such form, that by
substituting for s the number of days corresponding to each observation, it [754iv]
will become equal to the observed longitude.
We shall represent by |3, p\ p", &c., the observed longitudes of the comet,
and by i, i', i", &c., the number of days they fall after the given epoch ; [754»]
these numbers being supposed negative for observations made before the
epoch. If we put
* (513) A rational and integral function of s, is of the form A-\-Bs-\-C^-\-Ds^-\-&ic.
depending only on integral positive powers of «, without surds, and without fractions containmg
s in the denominators.
103
410 INTERPOLATION. [M«c. Cel.
[758]
52p'_,52^
= S^^; &C. ;
&c. ;
the required function will be
[756] |3 4-(5 — t).5|34-(5 — Z).(5— «')j2|3 + (5 — Z).(5 — *').(5— i").^'^ + &C. ;
for it is easy to prove that if we put in succession s = i, s = i\ s = i", he. ;
it will become ^, ^', /3", &c.*
If we now compare the preceding function with the following, [754],
d a\ . s^ / d^ a\
we shall have, by putting the coefficients of like powers of 5 equal to each
other,
a = |3 — i . ^ p +i . i' . (52^ — z. z' . i" . .53 ^ + &c. ;
[756a]
i.(^) = ^^|3-(z + z' + 0.^^|3 + &c. ;
* (514) Thus if 5= i, this becomes p. If s=i', it becomes ^-\-{i' — i)'Sp, all the
Other terms vanishing; and by substituting Sfi=— — r, it changes into ^-{-.{^' — ^), or
simply p'. If 5 = i", it becomes p + {i" — i).8^-\- {i" — i) . {i" — i') . 6^ ^, and by
substituting for {i" — i) . ^ p its value [755], 6^' — 5 ^, it becomes
^4-(i"_i).5^4-(i"_i').(5^'__5|3) or p + (i' — t).5p + (*" — *') -^P'^
which by using the values of (J p, <5 p', [755], changes into (3 + (^' — ^) + (p" — ^'), or
simply p" ; and in the same manner the others may be proved. It is to be observed that
this is the usual rule of interpolation, as given by Newton, in page 129, Vol. Ill, of Horsley's
edition of his works. This appears by changing the symbols of Newton into those of the
present section in the following manner,
for a ; 5, 2 6 Sec. ; c, 2 c &;c. ; d, 2d he; e, 2 e &tc. ;
write ^; — (Jp, —(Jp'&cc.j 8^ ^, &^ ^' he. ; ^6^^, —P ^' he; S^^,6^^hc.;
n. iv. §29.] INTERPOLATION. 4,11
the differential coefficients of higher orders will not be of any use.* The
coefficients of these expressions are alternately positive and negative ; the [^^^8']
coefficient of <Z'^|3, neglecting its sign, is the product of r quantities z,*', [758"]
i" .... i^'"~^\ taken r by r, in the value of a ; it is the sum of the products of
the same quantities, taken r — 1 by r — 1, in the value of (y ) ; lastly,
it is the sum of the products of these quantities, taken r — 2 by r — 2, in [758'"]
the value of ^(^)-
If we put 7, /, y", &c., for the observed geocentric latitudes of the comet ; [758»»]
we shall have the values of 6, (t~)' ("TIs)' ^^•» ^J changing in the
preceding expressions of «, (j-J? ("T^)» ^c*' the quantities |3, ^', ^", [758^]
&c., into 7, 7', 7", &c.
These expressions are rendered more ^accurate by increasing the number
of observations, and decreasing the intervals between them ; we might
therefore use all the observations near the epoch, if they were accurate ; but [758^^]
the errors to which they are liable would lead to an inaccurate result ; to
diminish the influence of these errors, we must therefore increase the interval
of the extreme observations, when we augment their number. We may in
this way, with five observations, embrace an interval of thirty-five or forty [758vii]
degrees,! which ought to give with considerable exactness the geocentric
longitudes and latitudes, and their first and second differential coefficients.
If the epoch made choice of, is such that there is an equal number of r758Tiiii
observations before and after it, so that each longitude after the epoch, has a
[7566]
for p; q; r, &c. ;
write — {s — ») ; {s — i) . (s — i') ; — (s — i) . {s — i) . {s — t"), &c. ;
and then Newton's value of RS = a-\-bp-\-cq-{-dr-\- &£c. will become the same
as [756].
* (515) It will be seen in the final equations [806], that no differentials of a, 6, higher
than the second order occur, it will therefore be of no use to compute them.
f (516) The degrees here mentioned are of the centesimal division, they correspond to
31'' ^ and 36'^, in sexagesimals. To form a rough estimate of the degree of accuracy of this [7570]
412
INTERPOLATION.
[Mec. Cel.
corresponding one at an equal interval before the epoch ; this condition will
render the values of «? (t-)j 2 • \~TY]i more correct ; and it is easy to
prove that additional observations, taken at equal distances on each side of
[7586]
method, the formula [756] was applied to the values of p, p', p'', p"
following table, in sexagesimals.
p'", p"" , given in the
days
d in s
^ = 0 00 00,0
(3' = 8 20 01,3
(3" =16 29 42,5
(3'"=24 20 03,1
^iv=31 44 16,7
^v=38 38 11,5
d m s
8 20 01,3
8 09 41,2
7 50 20,6
7 24 13,6
6 53 54,8
— 10 20,1
— 19 20,6
26 07,0
— 30 18,8
3il diff.
— 9 00,5
— 6 46,4
— 4 11,8
4lhdif |5thdif.
2 14,1
2 34,6
20,5
which numbers were taken for intervals of six days, from Delambre's table of the
heliocentric motion of a comet in a parabola, whose perihelion distance is equal to the mean
distance of the earth from the sun j it being supposed that if these heliocentric values were
assumed for the geocentric longitudes, in the formula [756], the errors from the neglect of
any of the terms of that formula, would generally be of the same order in these heliocentric
longitudes as would occur in the corresponding geocentric places of the comet ; we shall
therefore, in the rest of this note, suppose ^, (3', &;c. to represent the observed geocentric
longitudes of the comet at intervals of six days. In this case the first observation is p = 0,
the fifth p'"^ 3 1'* 44'" 16^,7, the interval being greater than 31''^, one of the limits
mentioned in [757a] for five observations. The sixth term, or ^^ = 38*^ 38'" 11 ^S, was
added so as to include the differences of the fifth order. Then the intervals
i' —i=i" —i'=^i!" —i" =i"" —i'" = ^', i"—i=i"'—i'==hc. = 12 ;
i"'—i=^i""—i'=hc. = 18.
Therefore, by dividing the numbers in the column of first differences by 6, we shall obtain
^ Pj ^ ^'5 ^ ^"} &^c. [755]. Those of the column of second differences divided by 6.12
or 72 give (5^3, ^ ^', he. Those of the column of third differences divided by 6 . 12 . 18,
20^,5
[758a] or 1296 give 6^ p, 6^ js', he. ; and in the same manner 8^ ^ =
so that by
6.12.18.24.30 '
taking only the five first observations, and rejecting the sixth, we should neglect this value of
^5 3. We shall now compute the effect of this neglected term, in the values of a, (~),
and ^'(jli)- The terra produced in a, [758], is, — i.i'.i".i"
.i"".S^^, which
becomes 0, if we take the epoch at the time of one of the observations ; for then one of the
quantities i, i', i", he. will be nothing, and the expression will become 0. If we suppose the
epoch to be taken at the middle time between the two first observations we shall have
i = — 3, i' = 3, i" = 9, i"'=l^, i"" = 21, and the preceding expression will
last terms, of the same order as the ratio of 5^ . ( -r-j ) to «.* This [758«]
n. iv. § 29.] INTERPOLATION. ^^^
the epoch, will only add to these values, quantities which will be to their
3 3 9 15 21
become — ' ' — - . 20*,5 = 0",5. If we take the epoch at the middle time between
6 . 1^ . 18 . 2% . 30
the extreme observations it will be less, and in no case will this term amount to a second,
when the epoch is taken between the extreme observations. The coefficient of 6^ ^ in the
value of (^) is i . (Jf.{'.i:''-\-i/.i!'.i!''' -i-if.if''.t''' + ^'.if''.'i!''') +i!.if'.'if''.r. This,
in the case mentioned in [7586], where i = — 3, i' = 3, i'' = 9, i"'=15, i'"' = 21,
becomes — 5751, and the corresponding term of {-;—]is-——--————.20',5== — 0»,1.
\d s / o . 1^ . lo . -i4 . oO
Taking the epoch any where between the first and fifth observations, it is evident that no one
of the terms of this coefficient of (t~)> as i, i', i", i'", can exceed 6 . 12 . 18 .24, so that
the five terms, which compose this coefficient, cannot be so great, and in general must be
much less than 5.(6.12.18.24); therefore the term of (t~) produced by ^p
must be much less than this quantity multiplied by 6^ ^, that is, it must be much less than
5.6.12.18.24 ^^ , 20^5 „ , tt • • j i • , „ »
—————— . 20*, 5 = — ~. = 3*,4. Hence it is evident that it must be very small. In
like manner the coefficient of 6^ p in the expression of | . f — ] must, independent of its
sign, consist of the sum of the products of the five quantities i, i', i", i"\ i'''', taken three and
three, thus i.i' . i" -j- i . i' . i'" ~\- &;c. The number of terms of this series, by the doctrine
1.2.3.4.5
of combinations, is ~r~o~o~ ^^ ^^» ^"^ ^^ ^^^ greatest term, when the epoch is between
the first and fifth observations, cannot exceed 12 . 18 . 24, the whole sum must be much less
(cPa\
— j, independent of its sign,
, , , , , . . , . V , 1 « 20.12.18.24
must be much less than this quantity multiplied by o^ p, or — .20*,5^2',2;
consequently this term must be very small. From this rough essay we perceive that, with
the limits assigned by the author, in the length of the described arch, and in the number of
observations, the errors of the formula must be very small.
(517) This ratio is not generally correct for any one of the quantities a, (~—\
— j. It is however correct for the last of them, when the epoch is taken at the middle
*
104
^1^ INTERPOLATION. [Mec. Cel.
symmetrical form takes place when all the observations are equidistant, and
the epoch is placed at the middle of the interval comprised by the
[758-^] observations; it is therefore advantageous to use such observations. In
observation, and an equal number of equidistant observations are taken on each side of the
epoch. In general the neglected terms are rather greater than is stated by the author. This
may be proved in the following manner. The general expression of the longitude, virhich we
shall call I, is
[758c] l=::^J^{s—i).5^-\-{s — i).{s — i:)A^^-{-{s — i).{s'—i').{s — i").6^^-\-Uc.
[756}, which may also be put under this form
r..«.i Z=^' + (5-i') .5^' + (.-*') . {s-i")A^^'-^r{s-i') . {s-i") . {s-i"')A^^
75oa I
_|- {s — i') . {s — i").{s — i"') . {s—i"") . ^4^' + &c.
The only difference in these two expressions consists in commencing the series of longitudes
fi, p', &,c. and times i, i', he. at |3', i', instead of p, i. Now supposing in the first place, that
there were five observations, ^, ^', p", ^"', p"", and that if the series were extended on
either side, their fourth differences would be constant, the expression of Z, [TSScZ] would give
the true longitude, neglecting 6^ ^', he. which would vanish, because the fourth differences
[758e] are constant. If we suppose only the three middle observations, p', p", p"', to have been
made, the formula [758c] would, according to this method of calculation, give the value of I,
by putting another accent on ^ and i, because the first terms of ^', ^", ^"', would commence
with p' and i', instead of |3 and i ; this value of I would therefore be
[758/*] Hlv/^^lvyv /^-
Hence we see that by taking three observations ^', js", p'", and afterwards adding another
observation at each extreme, as ^, p"", the value of I will be increased by the terms depending
on b^ ^', 5"* |3' [758^], which we shall denote by L, and we shall have
L^{s — i').{s — i").{s — i!").S'^'+{s — i').{s — i").{s — i"').{s—i"").6''^
= \—i!. i" . i'" .6^^' + i'. i" . i'" . i"" . S^ p'}
^'^^^^ j^s.{{i!.i"-{-i'. i"'+i" .i"').6^^'—{i: . i" . i"' + i'.i".i""-{-i'.i"'.i""-\-i".i"'.i"").S'^'\
+ ^2 . 1 — (i' + i" 4- i'") . <53 p'+ (*'. i"+ i'. i"'-{- i' . i"" + i". r+ r, i"" + r . i"") .s^^'i + hc.
If we compare the value of /, [758/], with the general formula [757],
[758/t] weshallget « = p'-i'.5p' + i' . i" . 6=^', (^^^ = 6 ^' - {if + i") . ^^\
{—\ = 2 62 13'. And if we denote by a', (^Y and Cj^\ the increments of the
n. iv. § 29.] INTERPOLATION. 4,1 5
general, it will be useful to fix the epoch nearly in the middle of this interval ;
because the number of days from that time to the extreme observations vrill
be less, w^hich will render the series more converging. The calculation may [758*^]
preceding terms respectively, arising from the introduction of the terms of L, ['^Sg"], we
shaUhave «' = — i' . i" . i'" . 63p' + i' . i" .T .i"" . 5^p';
Putting also a", {-j-\ (tt j' ^^^ ^® ^^^^ ^^^"^^ °^ "' id)' (d "^)' C'''^^^]'^® ^^^
have a" = i'.i".<52^'; (j^\ :=. — {i! J^ H') . ^ ^ , (^^^2^^. Now if the
epoch be taken at the time of the second, third, or fourth observation, we shall have one of
the quantities i', i", i", equal to nothing, consequentiy a' = 0. In general we shall have the
ratio of a to a" expressed by ( — i!" .b^^-\- %' . %'" . 6^ ^') : ^ ^, and as 5'* ^' is of a less
order than b^ ^ it may be neglected, this ratio will become — il" .^^':S^^, and since H" is
of the same order as s, it will be of the same order as s8^ ^' :^ fl. This may be expressed
in a different manner, by observing that from [758A], a is of the finite order p', ("r-j is of
— - ) is of the second order ^ ^', and as 5^ p' is
a s^/
of the next higher order to 6^ p', we may, in counting the order of the terms, consider the
ratio of ^ p' to (5^ p', to be of the same order as ( T^ ) to a, consequently o' : a" is of the
same order as ^•(t~)=«> instead of ^•\1~^'"'^ ^ ^^^ stated by the author in
[758'*]. In like manner
— {i'.i".i"'-{-i'.i".i""+i.i'".i""-\-i".i"'.i"").S^fi': — {'i:-\-i'').6^^',
and by neglecting the term S'^ p', as of a less order than 6^ p', and observing that t', t", t'", are
each of the order s, and the ratio of P^ to ^^IS* of the same order as ( -^ j to a, this ratio
of f ^] '• ( T~) wil^ become of the same order as * . (t~) * «> which also differs from
[7 58"]. Lastly,
K in this we take the epoch at the middle observation, making i"=0, and take i' = — t'".
416 COMPUTATION OF THE [Mec. Cel.
also be simplified by fixing the epoch at the instant of one of the observations ;
which will give directly the values of « and 6.
When we shall have found, in the preceding manner, ( ;r- ) » ( -rj ) '
— j, and (-y-^jj we may deduce from them the first and second
differentials of « and 6, divided by the corresponding powers of the element
of the time, in the following manner. If we neglect the masses of the
[758xiii] planets and comets, in comparison with that of the sun, taken as the unity
of mass ; and take also for the unity of distance, the mean distance of the
earth from the sun ; the mean motion of the earth about the sun will be by
[758«'] § 25 [705"], the measure of the time t. Therefore let x be the number of
seconds which the earth describes in a day, by means of its mean sidereal
motion ; the time t, corresponding to the number of days s, will be x 5 ; we
shall therefore have*
,«^„, /da\ 1 /'da\ /d^a.\ 1 ^ d'^ a
[759]
dij X \ds)' \dt^J X2 \ds
Using common logarithms, we have by observation, f log. x = 4,0394622,
[759'] X
[or 3,5500072 sex.] ; also log. x^ = log. x + log. — , R being the radius
it will become simply i . i" . 5* ^' : 2 6^ ^', and by putting i' . i"\ of the order ^, and the
ratio of 5"* p' to 6^ (3', of the same order as that of ( — - ) to a, it will become of the same
order as ^ . (t^) :«) as is stated in [TSS''^]; but this takes place only when i'-\-i"-{-i"'=0,
for if this quantity is finite, and of the order s, the term S^ ^' will not vanish, and we shall
have {-TY) ' \TYl °^ ^^ sdiVae order as s . (t^) : «• Thus the ratio of s^ . [t~^] to a,
given in [758'*], can hardly be said to be correct, in any point of view, in the example we
have now computed, for three and five observations ; and it is evident that the same reasoning
will apply with scarcely any alteration to a greater number of observations.
* (517a) The second members of the equations [759] are deduced from the first, by
changing dt into Xd s, as in [758'''].
f (518) Using the centesimal division of the circle and day, the number of seconds in
the whole circumference is 4000000", the number of days in a sidereal year, 365,25638,
ll.iv. §30.] ORBIT OF A COMET. ^17
of the circle reduced to seconds ; hence we have log. x^ = 2,2750444, [or [759"]
1,7855894 sex.] ; therefore if we reduce into seconds the values of ( ^)
and (-Jj\ we shall have the logarithms of (-^j and (-j^\ ^7
subtracting from the logarithms of these values, the logarithms 4,0394622, [759'"]
and 2,2750444 respectively, [3,5500072, and 1,7855894 sex.] In like
/d6\ / d^ 6\
manner we shall have the logarithms of f — j and (-r^j? by subtracting [759i»]
the same logarithms respectively from the logarithms of their values reduced
to seconds.
It is on the precision of the values of «, (-tt)> ("Jt)' ^' ("Tt)' ^"^ ^759^]
(— -— j, that the exactness of the following rules depends, and as their
computation is very simple, we must select and augment the number of
observations, so as to obtain these quantities with the greatest correctness.
We shall now determine, by means of these values, the elements of the orbit
of the comet ; and to generalize the results, we shall consider the motion
of a system of bodies acted upon by any forces whatever.
30. Put X, y, z, for the rectangular co-ordinates of the first body ; a/, ?/, z'j [759^']
for those of the second body, and so on for the rest. Suppose that the first
body is urged in directions parallel to the axes a:, y, z, by the forces X, Y, Z, [759^^1]
respectively, tending to decrease these co-ordinates ; that the second body is [759viii]
urged in directions parallel to the same axes, by the forces X', Y', Z' ; and
hence log. X = log. 4000000 — log. 366,25638 = 6,6020600 — 2,5625978 = 4,0394622.
The radius being 1, the semi-circumference is *= 3,1459, &c. whose log. is 0,4971499,
and as the number of seconds in the semi-circumference is 2000000 whose log. is 6,3010300,
we have log. radius in seconds =6,3010300— 0,4971499 = 5,8038801 = log. R. Hence [759a]
log.x2=21og.X—log.i2=2X4,0394622 — 5,8038801=2,2750443, nearly. If we wish to
use the common division of the circle into 360"*, or 1296000* instead of 4000000", we must add
1296000
to the preceding logarithms the logarithm of - = log. of 0,324 = 9,5105450 ; adding [7595]
this to 4,0394622, 2,2750444, and neglecting 10 in the index, they become respectively [759c]
3,5500072 and 1,7855894.
105
418 COMPUTATION OF THE [Mec. Cel.
so on for the others. The motions of all these bodies will be given by the
differential equations of the second order,*
[760]
ddx ddy ddz
^J^ rff^y' ddz'
&c.
If the number of these bodies is w, the number of equations will be Sn, and
[7601 their finite integrals will contain 6 n arbitrary terms, which will be the
elements of the orbits of the different bodies.
To determine these elements by observation, we shall transform the
[760"] co-ordinates of each body into others whose origin is at the place of the
observer. Suppose therefore a plane to pass through the eye of the observer,
[760'"] and to maintain a situation parallel to itself, while the observer moves on a
given curve ; we shall call p, p', p", &c., the distances from the observer to
the different bodies, projected on this plane ; «, « , a", &c., the apparent
[760iv] longitudes of these bodies, referred to the same plane, and ^, ^', ^", &c., their
apparent latitudes. The variable quantities x, y, z, will be given in functions
of p, a, 6, and of the co-ordinates of the observer. In like manner a/, i/, z',
[760 ▼] will be given in functions of p', a, 6', and of the co-ordinates of the observer,
and so on for the rest. Also if we suppose the forces X, Y, Z, X', Y', Z',
&c., to depend on the reciprocal action of the bodies of the system, and upon
external attractions, they will be given in functions of p, p', p", &c., «, «', a",
[760»i] ^c.^ 6^ e'^ 6"^ &c., and of known quantities ; the preceding differential
equations will thus correspond to these new variable quantities, and their
first and second differentials ; now by observations we can find, for any given
instant, the values of a, (^^) , (^^) [759] ; «, (^), (•^) ; »',
(-~\ &c. ; there will therefore remain unknown only the quantities
* (519) These are similar to the equations [38] or to those deduced from [142], by
changing — P, — Q, — R, he. into X, Y, Z", &c. because the forces P, Q, R, in
note 66, page 96, are supposed to increase the co-ordinates, but the forces X, Y, &;c. [759"'"],
are supposed to decrease them, consequently P, Q, &c. ought to have different signs
from X, Y, See.
£760^"]
n. iv. <^31.] ORBIT OF A COMET. 419
P, p', p", &c., and their first and second differentials. The number of these
unknown quantities will be 3 n, and as we have 3 n differential equations, [760^«]
we shall be able to determine them. We shall have even this advantage,
that the first and second differentials of p, p', p", &c., will appear only under
a linear form* in these equations.
The quantities a, ^, p, a', d, p', &c., and their first differentials divided by d t,
being known ; we shall have for a given instant, the values of x, y, z^ xf, y', z', [760«]
Sic, and their first differentials divided by d t. If we substitute these
values in the 3 n finite integrals of the preceding equations, and in the first
differentials of these integrals, we shall have 6 n equations, by means of [760 »]
which we can determine the 6n arbitrary constant quantities of these
integrals, or the elements of the orbits of the different bodies.
31. We shall now apply this method to the motion of comets. For this
purpose we shall observe that the principal force which acts on them is the
sun's attraction ; we may therefore neglect the other forces. However, if a
comet should pass so near to a great planet as to be sensibly disturbed by it, [760«»]
the preceding method would give the velocity of the comet, and its distance
from the earth ; but as this case very rarely occurs, we shall in what follows,
only take notice of the action of the sun.
If we take the mass of the sun for the unity of mass ; the mean distance [760"i]
of the earth from the sun for the unity of distance ; and fix at the centre
of the sun, the origin of the co-ordinates a:, y, z, of a comet, whose radius
vector is r; the differential equations (0) ^17 [545] will become, by [760"«]
neglecting the mass of the comet in comparison with that of the sun,t
^ ddx X
0 =
d dz
dt^
* (520) This is evident from the form of a?, y, «, &c. [762], which give dx, ddx,
dy, ddy, dz, ddz, he. in terms of p, rf p, ddp, in a linear form [125a], which being
substituted in [760] produce the linear equations mentioned above.
f (520a) In this case M-^m=(x, [544'], and by putting M=l, [760"'], and
neglecting the mass m of the comet, we get fii. = 1.
I
420
COMPUTATION OF THE
[Mec. Cel.
[761'] Suppose the plane of x and y to be the plane of the ecliptic ; the axis of x to
be the line drawn from the centre of the sun to the first point of aries, at a
[761"] given epoch ; the axis of 7j to be the line drawn from the centre of the sun to
the first point of cancer, at the same epoch, and the axis of z to be directed
[761'"] towards the north pole of the ecliptic. Then put x' and i/ for the co-ordinates
of the earth, and R for its radius vector ; this being premised,
We shall transform the co-ordinates x, y, z, into others referred to the
place of the observer. For this purpose put
« = the geocentric longitude of the comet ;
[761 »^1 ^ ^^ ^^® geocentric latitude of the comet ;
P = the distance of the comet from the earth, projected upon the plane
of the ecliptic ;
we shall have*
[762] a; = a;' + p . cos. « ; y = y' -\-p , sin. a ; z = p . tang. 6.
If we multiply the first of the equations (k) [761] by sin. «, and subtract
from it the second multiplied by cos. «, we shall have
[763]
0 = sin. a .
dd X
dt^
COS. « .
ddy X . sin. a — y . cos. a
* (521) Let A be the sun supposed at rest, FD G the
orbit of the earth, A F X ^e axis of x, drawn through the
first point of aries ', A GY the axis of y, drawn through the
first point of cancer; EH the orbit of the comet projected
upon the plane of the ecliptic, the earth being at D when the
projected place of the comet is at i^ ; we shall have
[761'— 761'^], the co-ordinates AB = x', BD = y',
AC = x, CE = y, AD = R, DE=p, the angle
XAD = A; then if we draw D /parallel to ^ X to meet CE in I; and put the angle
EDI=a, we shall have in the rectangular triangle DIE,
DI=DE.cos.EDI==p.cos.a, EI= D E . sm. ED I=p , sin. a.
Now ^ AC=AB-{-B C = AB-\-DI, hence x = xf -\- p . cos. a., [762], also
C E= CI-\-EI=BD-{-EI becomes y = ?/' + p . sin. «, [762] ; and as the comet
is elevated above the plane by an angle which, viewed from D, is equal to 6, the distance
DE being equal to p, the actual elevation will be equal to p . tang 6, hence z= p . tang. 6,
as in [762j.
ddx' a/
dt^ ' E3 '
<""4^
y' .
which give
d d ccf
Sin. a . — — COS. « .
dt^
ddi/
1/ . COS. a — ocf .
. sin. a
>
we shall therefore have
II. iv. §31.] ORBIT OF A COMET. ^21
by substituting in it the values of ar, y, given by the preceding equations, we
obtain*
ddo[f ddi/ , a/, sin. a — V-cos-a „ ^d p\ /'daS /dda\
O = sm.«.^^-cos.<..^-+ ^ 2.(;^). (^^J-P.(^^j. [764,
The earth being retained in its orbit, like the comet, by the attraction of the
sun, we shall havef
[765]
[766]
0=(y.cos..-..sin.).;±-±]-2.@).@_,(^«). ,^
* (522) The differentials of the equations [762] give
dx = d a/ -\-dp . COS. a — pda . sin. a; dy=di/-\-dp. sin. a -{- p <i a . cos. a ;
dz = dp. tang. 5 + P ^ ^ • (cos. d)~^.
Again taking the differentials, and putting for brevity
B = 2 d p . d a -\- p . d d a, and C=ddp — pda% [764a]
we obtain
ddx = dd j/ -\- C . cos. a — B . sin. a; ddy = d di/ -\- C . sin. a -f- -5 • cos. a ; [7646]
ddz = d^ p .tang. d-f"2<?p.<Z^ .(cos. ^)~^ + 2 p .<i^^.sin.4.(cos.d)~^-f-p'^<^^*(cos. ^)~^. [764c]
These values oi ddx, ddy, being substituted in sin. a . d d x — cos. a.d dy, it becomes
equal to sm.a.ddx' — cos. a.d dy' — B. Dividing this by </ i^ we get
ddx ddy . ddi! ddi/ r^ /d p\ /d a\ /d d a\
Sin. a .-— COS. a.—-— =sin. a.——- cos. a.—-- 2. — ) . I — ) — p . ( 1. r764/il
rf(2 dt^ dpi rf<2 \dt J \dt J ^ \ dfi ) y^>*a\
Again, the values of a;, y, [7G2] give x . sin. a — y . cos. a = x' . sin. a — r/ . cos. a, which
being divided by r^, and added to the equation [764<?], the first member of the sum is
nothing, by means of [763], and the second member becomes as in [764].
f (523) The earth being attracted by the sun in like manner as the comet, we shall have
the equations of the earth's motion by changing in the two first equations [761], x, y, r, into
x', y', Bj respectively. The third equation is not used, because by hypothesis, [761'], the
earth moves in the plane of x, y. Hence we obtain the equations [765]. Multiplying the
first by sin. a, the second by — cos. a, and taking the sum of the products, we get
[766]. Substituting this in [764] it becomes as in [767].
106
^^ COMPUTATION OF THE [Mec. Cel.
[767] Let A be the longitude of the earth seen from the sun ; we shall have*
t768] x' = R.cos.A; y' = R. sin. A;
therefore
[''69] y • COS. « — x' . sin. a = jR . sin. (A — «) ;
the preceding equation will thus become
[770]
^d^\_R.S\W.(A^oi) CI 1 \ ^'\d¥)
\dt) --Jd^'lR^ r^l 2.(—) ' ^^^
We shall now investigate another valuef of ( -p ) • For this purpose, we
shall multiply the first of the equations (A;) [761], by tang. ^. cos. « ; the
second by tang. 6 . sin. « ; and from the sum of these two products shall
* (524) In the preceding figure, page 420, we have
AB = AD.cos.XAD; B D=AD .sm. XAD ;
which in symbols are x' =R . cos. A] y' = R. sin. A, [768]. The first multiplied by
— sin. a, and added to the second, multiplied by cos. a, gives
i/ . COS. a — x' . sin. a = R . (sin. A . cos. a — cos. A . sin. a) = i2 . sin. (A — a),
[22] Int. as in [769]. Substituting this in [767], we get
o = «..„.(^-«).(^-i)_..(if).(^«)-..(^,f).
Transposing —^'Cjj) • (jf\ and dividing by 2 . ^^Y we get [770].
f (525) It may be observed relative to the calculation here used, that by substituting the
values of x, y, z, [762], and their differentials computed in note 522, in the equations [761],
we shall obtain three differential equations, linear in p, — -, --— . Deducing from these, in
'■ a t dt'^
any manner three values of ~—~ and putting them equal to each other, we shall obtain two
independent equations containing p and — -; from each of which if we can find a value of
(—V like that in [770] or [774], and put them equal to each other, we shall obtain an
equation containing p,- without its differentials, as in [776].
II. iv. §31.] ORBIT OF A COMET. 423
subtract the third equation ; we shall thus have
^ ( ddx , . ddy) , {a?.cos.a+V'Sin.a| ddz z
O = tang.«.jcos.«.^ + s.n.a.^|+tang.«. / <_____. p,j]
This equation will become, by substituting the values of a:, y, z*
^ ( /dda/ , a/\ , /ddy' , VX . ) V^V V
2.1-;-) sin.^
^ cos.^d ^ cos.^a \dtj ^
now we havef
(^^ + ^J • "^'- "+ V^ +-;^; .sm.a=(a/.cos.a+y'.sm.«). (^-^ - ^j
= i2.C0S.(J— a).|-l — i|;
* (526) Substituting the values of a?, y, z, [762], in [771], the terms depending on
a/, y, which for brevity we shall call X, will agree with those in [772], and as «, y, z, enter
the equation [771] under a linear form, we may find the parts of [772], independent of a;', y',
by substituting in [771] the parts of x, y, z, [762], independent of a/, i/, namely
x = p . cos. a ; y = p. sin. a ; z = p . tang. 6. These give x . cos. a-\-y . sin. a = p ;
and by [7646], neglecting x', y, we have ddx= C . cos. a — B . sin. a,
ddy = C . sin. a -|- -B . cos. a, hence cos. a. . ddx-\- sin. a .ddy= C=dd p — prfa^,
[764a]. Substituting these in [771], and then in the last term putting z=p . tang 6, [762],
we shall get
and by reduction,
ddz p. tang, d
ddz
d^ '
but from [764c] we have
_rfrfz_ fddj\ /dp\ /d_6\ _1 /HV
hence by substitution we get [772].
^dd\^ sin.^ /ddei\
f (527) From [765] we have -r^——^ ', -/^——^ • Substituting these in
[772]
[773]
dt^ ijs ' d<2 iJ3
[774]
[776]
[776"]
4^24 COIMPUTATION OF THE [Mec. Cel.
therefore
( /dd6\ /day .
{, \dtj \dt
R . sin. 6 . COS. 6 . cos. (A — a) C 1 1 ^ (2)
If we subtract this value of — from the first [770], supposing
da\ /dd(i\ /d&\ fddoL\ , _ fdo\ /dsy , /day .
.-t7V(-t:f) — ( — )-(-t:f)+2.( — ).(—). tang.^ + ( — ].sm.^.cos.d
[775]
)\ /dda\ , _ /da\ /dSy
_ dtJ\dt-J \di)\i:¥r^\Tt)\Tt)'^^''^-'+\dt
f-T^ j.sinJ.cosJ.cos.(^ — «) + (—]. sin.(^ — «)
we shall have
[776'] The projected distance of the comet from the earth p, being always positive,
this equation shows that the distance r of the comet from the sun is less than
the distance R of the sun from the earth, if n*' is positive, but r is greater than
J?, if M-' is negative ;* these two distances are equal, if \>! = 0.
{~dW~^i) • ^°^ « + {-^ + ^) • ^^"- «' ^^ becomes (a/ . cos. a + 3/ . sin. a) . (^-^3),
as in [773] ; and by using the values of a/, y', [768], we find
[774al ^' • COS. « 4" 2/ • sin. a^R . { cos. w3 . cos. a -\- sin. .4? . sin. a^ =R . cos. (./2 — a),
[24] Int. ; hence we obtain the last expression [773]. Substituting this in [772], then
COS ^ A
multiplying by — tJ-jt- , putting also cos. 6 . tang. 6 = sin. 6, and reducing, we shall
get [774].
* (528) Subtracting tbe equation [774] from [770], we get
\\dt) \dt) \dt) -/
CI ^ ) C sin. 6 . COS. 6 . cos. {A — g) ^^ sin.(wj — «) ^
"^^'"^'^1 -(^) -(77)!'
II. iv. §31.] ORBIT OF A COMET. 4,25
We may, by the inspection of a celestial globe, ascertain the sign of m-', [776'"]
and by that means determine whether the comet or the earth is most distant
from the sun. For this purpose, suppose a great circle to pass through two
geocentric places of the comet, infinitely near to each other. Let the
inclination of this circle to the ecliptic be 7, and x the longitude of its [776'»]
ascending node ; we shall have*
tang 7 . sin. (a — x) = tang. ^ ; [777]
hence we deduce
d d . sin. (a — x) = <? a . sin. ^ . cos. ^ . cos. (« — x) ; [778]
which being multiplied by ^ ■\J7) ' vfrp and then divided by the coefficient of p, gives
0 = P ; . ] - — ^ [ , hence we get p, [776]. Now when r <^ i?,
1 1 JJ3 ,-3
- = — — — will be positive, therefore |x' must then be positive to render p positive.
On the contrary when r'^ R, - — — will be negative, and then (jf must be negative to
render p positive. When r= R, the numerator of the value of p becomes 0, consequently,
fx = 0, because p must have a real positive value.
* (529) het AB C be the ecliptic, D, E, two observed
geocentric places of the comet, infinitely near to each other,
D C the circle of latitude. Continue JE X) to meet the ecliptic
in the ascending node B. A being the first point of aries, _^ j^
we shall have AB = \ AC = a, BC = a — \, "^ B C C'
D C=^&, the angle D B C = y', then by spherics, tang. D C = tang. DB C . sm. B C,
or tang. ^ = tang. 7 . sin. (a — x), as in [777]. The differential of this, supposing d, a,
variable, is
da.tang.7.cos. (a — X) = -— ^, f777„j
... tang. ^ r T . 1
and by substitutme; tang. 7 = -: — ; — , 777 , it becomes
•^ 00/ sm. (a — X) ■- -^
tang. 6 . . dd
a a . -; — ; . COS. (a — X) =
sin. (a — X)' *^ ^ cos^^'
this being multiplied by cos.^ 6 .sin. (a — X), becomes as in [778]. Taking the differential
of [777fl], we find,
ddaAnns^.y . cos. (a — X) — <? a^ . tang. 7 . sin. (a — X) = — irr + S d6^ .^^^, .
107
-^26 COMPUTATION OF THE [Mec. Cel.
and by taking the differential of this, we shall have
[779'] dds^ being the value of dd&, which would take place if the apparent motion
of the comet continued in the great circle. Therefore by substituting for
J, . 1 c/ a . sin. ^ . COS. ^ . COS. f a — '^) ^„„nt i ti i • i /. i,
[779"] d6 Its value ~J~ZZ~\ — ^ U^^l^ we shall obtam the followmg
value of M-' :*
[780]
C /dd6\ /dd6\ ) . ,
sin. 6 . cos. 6 . sin. (A — x) '
Substituting in this for tang. 7 . cos. (a — X), and tang. 7 . sin. (a — X), their vakies
deduced from [777a, 777], namely, ~, and tang. 5, we get
ddu.-^- cZa^tang.^=-^+2^^^ ^"'^
da. cos.^ 6 cos.2^ cos.3^
which being multiplied by cos.^ 6 . — , putting — '— = tang. 6, and transposing all the
(t V COSa (J
terms to the second member, becomes as in [779].
* (530) Substituting d 6, [779"], in the denominator of /x', [775], it becomes
/d a\ . . . ,a \ I /^ «\ • A A sin.(./3— a)-cos. (a — X)
I -r- ) . sm. & . cos. ^ . cos. (A — a) + ( -7- I • sm. ^ . COS. & . -. — 7 —
\dt/ ^ ' \dt/ sin. (a — X)
= (——). . ' ' — '— . \ sin. (a — X) . cos. [A — «) -[- cos. (a — X) . sin. [A — «) [ 5 .
of which the part between the braces is, by [21] Int. equal to
sin. f(^ — «) + (« — X)} =sin. (.4 — X),
therefore the denominator of U-' is (-7^). — . ', ' — ^^ -. Again, by subtracting
\dtj sm.(a — X) o ^ j o
from the numerator of y!, [775], the expression [779] which is equal to nothing, it will
become (~j~)- (~7~^) — ("J") • i~y^)' This being divided by the preceding expression
of the denominator, gives, by rejecting the term ( — j, common to the numerator and
denominator, the value. of fj^', [780].
11. iv. § 31.] ORBIT OF A COMET. 427
The function ^^- is always positive ;* the value of |x' is therefore [780']
sin. 6 . COS. 6 J r
,. /dd(i\ /ddd\ . ^ , ^
positive or negative, according as ( -^ J — ( --y j is oi the same or or a [780"]
different sign from sin. (A — x) ; now (A — x) is equal to the distance of
the sun from the ascending node of the great circle increased by two right
angles [767', 776'"] ; hence it is easy to perceive that f^' will be positive or
negative, according as the comet shall be found on the same side of the great
circle on which the sun is, or on the opposite side, at the time of a third
observation, taken immediately after the two preceding observations, and [780'"]
infinitely near to them.f Suppose therefore, through two very near geocentric
* (531) As ^ never exceeds a right angle, its cosine is always positive; and in the
figure page 425, it is evident that ^ remains positive, while the arch B C increases from 0 to
two right angles, during which the signs of sin. ^, sin. (a — x) are always positive. In
the other semi-circle, where a — X is between two and four right angles, 6 is negative, and
11 1 • /• ^ . . , sin. (a — X) . ,
Its sme, as well as the sme ot a — X, is negative, consequently -: is always positive.
SXIi* tj • COS* Q
The same result may be obtained from [777], which gives
, . tang. 6 sin. &
sin. (a — X) = = -
tang. 7 tang, y.cos.d'
and this being divided by sin. 6 . cos. 6, becomes -r-^ -== , the second
sm.^.cos^ tang.y.cos^^ '
member of which is evidently positive.
f (532) Supposing ir to be equal to two right angles, the distance of the comet from
the south pole of the ecliptic will be J * + ^, which we shall put = 6', and write d d 6/
for the value of ddd^, [779'], also d d 6f for d d 6, observing that ^' is always positive. By
this means the expression [780] becomes
^,^sin.(a— X) \\di^)~\d^) \
sin. 4 . COS. d ' sin. [A — x)
Now it is evident that if (7^) — (7^) is positive, the south polar distance of the comet
at the third observation, will be greater than it would if it continued to move in the great
circle ; therefore, instead of moving on the arch E F, on the continuation of the great circle
BE,^g. page 425, it will fall to the northward of itonE/; but if f—f\^f-^\ is negative,
the comet will fall to the southward of the great circle, towards g. But C being the place of
428 COMPUTATION OF THE [M.'c. Cel.
places of the comet, a great circle to he drawn. Then if at another third
observation, taken very soon after the two others, the comet deviate from the
great circle towards the part of the heavens where the sun is, the comet will
be situated ivithin the earth^s orbit, or nearer to the sun than the earth is : but
reTfisu .1 if the deviation be to the opposite side of the great circle to that in which the
iffhf ^ww is placed, the comet will be without the earth's orbit, or farther from the
ofa Comet sun thttu thc carth is. If the comet continue to move in the srreat circle, the
comet and the earth would, both be equally distant from the sun. Thus the
[780 V] various inflections of the apparent path, will enable us to estimate the
variations of the distance of the comet from the sun.
To eliminate r from the equation (3) [776], so that this equation may
contain only the unknown quantity p, we shall observe that we have [555]
[780vi] r'' = x^- + y'' + z' ;
and by substituting for x, y, z, their values in p, a and 6 [762], we shall
have*
[780'^]
Use of a
<"elfi.sti.ll
Gloiio in
juflgins
of the
distance
ofa Com
from the
Sun
[781] r^ = x'^ + 2/' + 2 p . {:r' . cos. « + 2/' • sin. «} +
cos.^r
the earth in the ecliptic, we have ./2 C'=./4, [767'], hence BC' = A — X, is the
distance of the earth from the node B, and A — >. -[- '^, is the distance of the sun from the
same node. Now it is evident that when A — X is between 0 and *, the sun will fall to the
northward of the great circle B E, and when A — X is between -r and 2 -r it will fall to the
southward. In the first case sin. (A — X) is positive, in the second negative. Hence it
evidently follows that when the comet and sun fall both on the same side of the great circle,
/ddd'\ /dd6\
W<2 )~\~d~^)
that is, both to the northward or both to the southward, the sign of .^^ / , ^ ^-— — ^ , and
° sin.(./3 — X)
therefore that of fj^', will be positive ; but if the sun and comet fall on different sides of the
great circle, that quantity will be negative.
* (533) The values [762] being substituted in [780"], we get
r^ = (x' -j- p • cos. a)2 + (/ + p • sin. a)^ -f- p^ . tang.^ 6
= *' ^ + y ^ + 2 p . (a;' . COS. a 4" 2/ • sin. a) -}- p^ . (cos.^ a -f- sin.^ a -\- tang.^ 6).
1
C0S.2 d
[781a]r Now cos.^ a -\- sin.^ a -f- tang.^ ^ = 1 -|- tang.^ 6 = g- , hence the preceding expression
becomes as in [7S1].
II. iv. §31.] ORBIT OF A COMET. 429
but we have [768] af = R. cos. A; iJ = R> sin. A ; therefore* [781T
r' = -^ + 272p.cos.M — «)+i2^. [782]
If we square both sides of the equation (3) [776], after having put it under
the following form,t
r^{fx'i^P+l} = i2^ [783]
we shall have, by substituting for r^ its value [782] ^
|-^ + 2i2p.cos.(J-«) + i2^|'.{/.'i?^P + l}^=7^; (4)
[784]
in which equation there is only one unknown quantity p, and it is of the
seventh degree, because the known term of the first member being i^, as in [784^
the second member, the whole equation becomes divisible by p. Having thus
d p
found p, we shall obtain (j-j by means of the equations (1) and (2),
[770, 774]. By substituting, for example, in the equation (1) [770], for
-^ — ^, its value ~, given by the equation (3) [776], we shall [784"]
have
'd^\ p
-.{(^)+.'.si„.(^-«)J.
dtj ~ /da\ ' I \dfij ' '^ • ^ ^ 5 • [785]
The equation (4) [784] is often susceptible of several real positive roots.
For by transposing the second member, and dividing by p, its last term [785^
will bej
2.7^^cos.«^.{M-'i23 + 3.cos.(^~a)} ; [786]
* (534) From [774a] we have x' . cos. a-\-y' . sin. a = R. cos. (A — a) ; also
the sura of the squares of x', y\ [768], gives a;'^ _j_ y/2 __ jf^^ ^j^ggg being substituted in
[781], we obtain [782].
f (535) This is obtained by multiplying [776] by R? r^ /, which makes it
li.'pR^r^ = R^ — r^, lience R^ = r^ + ii! p R^ 7^ = rK{i.' B^ ?+ I],
as in [783]. The square of this is r^ . {/x'\R2 p _|_ j p __ /je^ g^j ^y substituting for r^
the cube of r^, [782], we obtain [784].
J (536) The equation [784] is supposed to be multiplied also by cos.^ 6, to avoid
fractions.
108
430 COMPUTATION OF THE
[Mec. Cel.
[786']
[786"]
therefore, as the equation in p is of the seventh degree, or of an uneven
degree, it will have at least two real positive roots, if i^'R^-\- 3. cos. (A — a)
is positive ; for it ought always, by the nature of the problem, to have one
positive root, and it cannot then have an uneven number of positive roots.*
Each real and positive value of p, gives a different conic section, for the orbit
of the comet ; we shall therefore have as many curves which satisfy the three
^ observations, as p has real and positive values ; and to determine the orbit of
the comet, we must then have recourse to another observation.
32. The value of p, deduced from the equation (4) [784] would be
[?86"] rigorously correct, if «, (^), (^), t, (^), (^^), were accurately
known ; but the approximate values of these quantities only have been found.
It is true we may approach more and more towards the exact values, by using
the method before explained, and taking a greater number of observations,
which has the advantage of embracing longer intervals, and compensating
the one by the other for the errors of observations. But this method has
the analytical inconvenience of using more than three observations, where no
more than three are absolutely necessary. We may obviate this inconvenience
[786iv] in the following manner, and render the solutions as exact as may be required,
using only three observations.
For this purpose, suppose that « and & represent the geocentric longitude
and latitude of the middle observation ; if we substitute in the equations (k)
[78C>y] [761] of the preceding article, for x, y, z, their values [762] a:' + p . cos. a.
* (537) Supposing a, a", a'", a"", he. to be real positive quantities. The factors of
this equation depending on imaginary roots, which always enter by pairs, are of the forms
p rt: a' + a" . v/— T= 0, p±a' — a" . \/— 1= 0' whose product is
p2±2pa'+(a'2 + a"2)^0.
Negative roots depend on factors of the form p + «' " = 0, and positive roots depend on
factors of the form p — a"" = 0 ; and the constant term of the proposed equation of the
seventh degree, must be formed by products of the constant terms of these factors, that is
by terms of the forms a'^-{- a"^, + a'", — a"". Now in order that this final product
may be equal to the positive quantity 2 R^ . cos.*^ ^ . ffx' jR^ -j- 3 . cos. {A — a)|, it is
necessary that the number of the factors of the form — a"" should be even, or in other
words, that the number of positive roots should be even.
n. iv. § 32.] ORBIT OF A COMET. 451
y + p.sin.«, and p. tang. ^, they will give (-jj\ (:^)' ^^^ (j^/ ^^^'•'
in functions of p, a and 6, and of their first differentials with known quantities.
and (-jt)' ^^ functions of p, « and ^, and of their first and second
differentials. We may eliminate the second differential of p, by means of its
value, and the first differential by means of the equation (2) [774] of the
preceding article. By continuing to take the differentials of {-fYJi ( T~3" )' [786™i]
successively, and eliminating the differentials of a and 6 above the second,
and all the differentials of p, we shall have the values of [-ty\ \~J^n
^''■' (-dj)' {tf)' ^'■' i" fon^ions of P. «' (^) - (jf)' «-
(d d\ /^ d^ 6\
Ttj' KdW)''' t^is being premised.
Let a , a, «', be the three observed geocentric longitudes of the comet ; [786«]
fl^, ^, ^', the geocentric latitudes corresponding ; i the number of days [786 «]
between the first and second observation, i' the number of days between the
second and third observation ; x the arch in seconds which the earth describes [786«i]
in a day, by its mean sidereal motion ; we shall have, by ^ 29 [754, 759],*
^ \dtj^ 1.2 V^^V 1.2.3 V^^V
'^V!!i^^ f^\_il^ r^^4-&c •
JtJ^ 1.2 • V^^V 1.2.3-VrfiV '
, i-'2.X2 /d^6\ , i'3.X3 /d^d\ , e,
+ T72-UT;+r2r3-U^) + ^^-
fl^ = d — i . X .
9' = 5-[-i'.X.
[787]
* (538) The two first of these equations were deduced from the first of the equations
[754], putting successively s = — i, s = i', and for (t~")j (tIj)' ^^* their values
^'\~dt)' ^^'\Tiir ^* C^^^]' ^^^ t^o l^st equations were deduced in a similar
manner from the second of the equations [754], or by changing a into 6,
''^^ COMPUTATION OF THE [Mec. Cel.
If we substitute in this series, for (^), (^), &c. ; g^), g^),
&€., their values obtained by the preceding method ; we shall have four
equations betweea the five unknown quantities p, ( "T" ) ' ( T^ n ( 7~ ) »
( -7-3- J . These equations will become the more correct by using a greater
[787] number of terms of the series. We shall thus have (-7^)5 (~7~y )' \J~\
I -v-y ], in functions of p and of known quantities ; and by substituting these
values in the equation (4) [784] of the preceding article, it will contain only
the unknown quantity p. However, this method, which I have given only
to show in what manner we can obtain, by approximation, the value of p,
[787"] by using only three observations, will require in practice very laborious
calculations ; and it will be more accurate, as well as more simple, to use a
greater number of observations, by the method of § 29.
33. When the values of p and ( --] shall be determined, we shall have
\dtj
those of X, y, z, (-r\ (7/)' ^"^ (tj)' by means of the equations*
[788] X =R. COS. A-\- p. cos. a ; y = R . sin. A-\- p . sin. « ; z = p. tang. 5 ;
and of their differentials divided by J ^
d(>
dz\ ^dp\ ^ , ^\dl
The values of (-7—), and [— — ), are given by the theory of the earth's
[789'] motion : to facilitate the calculation, let E be the excentricity of the earth's
* (539) The values of x, y, z, were derived from [762], by substituting x', y', [768].
Their differentials, divided by d t, give the equations [789]^ without any reduction.
II. iv. § 33.] ORBIT OF A COMET. ^^
orbit, H the longitude of its perihelion ; we have by the nature of the
elliptical motion,*
/dA\ _ ^^^W l-E^
These two equations givef
fdR\E^n^-H)
\dt J y'T^rm
Let R be the radius vector of the earth, corresponding to the longitude A [791']
of that planet, increased by a right angle ; we shall havef
1 ^3
hence we deduce
n • /-A TT\ ^ — I -\- E^
E. sin. (A — H) = —^ ;
*(540) Wehave cZ^=^^, [531], m = ^ = -^ . ^ 1+ e .cos. (« — -n)}, [534],
— ^=«> [534'], and if in these we put a=l, m.= 1, e=E, v=A, -^=11, r=R,
they wiU become dt = j-, w = -=-^ . ^ 1 +E .cos. (^ — If) | , ^^=1,
the last gives h = y/l — IP, which being substituted in the second, gives
i-=fi: '-^
tt 1 -l[-E. COS. {A — H)'
as in [790]. These values of h, and - being substituted in the expression of dt, it
- dA.m , /dA\ i/JITW . n .
becomes dt^=-—=^, hence (-—-j==*-—2 — , asm [790].
f (541) Taking the differential of the second of the equations [790], R and A being
variable, we shall find
\dt)~\di) ' n-\-E. CO
8m.{A—H) \/\Trw (1— JE2).E.sin.(^— H)
|14-£.cos.(^— H)p iJ2 • |14-£.cos.(^ — H)|2 '
[790], and by substituting for the denominator its value (1 — E^Y, deduced from the
second of the equations [790], it becomes as in [791].
X (542) By writing R for R, and J * + ^, for A, [791'], in the second equation [790].
109
[793]
434 COMPUTATION OF THE [Mec. Cel.
therefore*
[794]
dR\ R + E^ — l
[795]
dt J R.s/l—E^ '
If we neglect the square of the excentricity of the earth's orbit, which is very
small, we shall have
'dA\ 1
(4?)=^'-^ =
[796]
dt J m'
the preceding values of i-nX \-^\ ['789], will by this means become
[796^ R, R, and A, being given directly by the tables of the sun, the calculation
of the six quantities x, y, z, [-f)i ('^f)' (77)' ^^^^ ^® easy, when p and
( —^ j shall be known. We may thence deduce the elements of the orbit of
m?nftto the comet, in the following manner.
direct or
retrogiade ~.-, 11 t*ii •• r 1 i«
thec^mlt ^^® mfinitely small sector which the projection of the radius vector of
the comet describes on the plane of the ecliptic, during the time d t, is
X d n ^— 1/ d 00
[796"] — ^ — ^ — [167'], and it is evident that this sector is positive if the motion
of the comet is direct, but negative if the motion is retrograde ; therefore by
[796'"] computing the quantity ^•(■^) — V ' {~J~)^ ^^ ^^^^ indicate, by its sign,
the direction of the motion of the comet.
*(543) Substituting the value of E.sm.{A — H), [793], in (~\ [791], it
becomes as in [794], and by neglecting E^, it changes into , and as the numerator
is of the order E, we may, by neglecting IP, put the denominator = 1, making
f--\ ^R—l, as in [795]. Neglecting E" in (^\ [790], it becomes as in [795].
Substituting these in [789] we obtain [796].
n. iv. <^33.] ORBIT OF A COMET. 4-35
To determine the position of the orbit, let 9 be its inclination to the
ecliptic, and / the longitude of the node which would be ascending, if the [796i']
motion of the comet was direct, we shall have*
z = y . COS. / . tang. 9 — x . sin. / . tang. 9 ;
© = O ■ '=<''• ^- t'-s- * - (57) • ''"• ' ■ ^'"g- *•
These two equations givef
tang. /=
tang. 9 =
^ \dtj \dt
\dtj \dtj
sin./.j.:.(^)-y.(^
[797]
[798]
* (545) In the figure page 351, let C be the place of the sun, D that of the comet,
B its projection on the plane of the ecliptic, C X the axis of x, in the direction of the first
point of aries, C Y the axis of T, C X' the line of the node, which would be the ascending
node if the comet's motion be direct ; then the angle XC X' = 1, [796'''], is called d in
[585'''] ; 9 being the same in [5S5'''] as in [796'''] j therefore, to conform to the present
notation, we must change 5 into /, to obtain from z, [588], its value [797]. The differential
of z being taken, and divided by d t, considering x, y, z, only as variable, gives the second
of the equations [797], which was accidentally omitted in the original work.
f (546) Multiplying the first of the equations [797] by — ( J^)> die second by y, and
adding the products we get
y ■ (^)-*-(57)=="-^-'™s-»- ^' (?i)-y-(jf) ] ■ [w.]
Again, multiplying the first of the equations [797] by — i'J~\ ^^^ die second by a?, and
"taking the sum of the products, we get
^•(7r)-^-(jf)=^°^-f-'»»s-'- ^•(?f)-y-(^)l •
Dividing the former by the latter, and putting tang. / for ^^,, we obtain the first of the
COS. I
equations [798] ; the second of these equations is the same as [797a] divided by the coefficient
of tang. 9.
-I^i
4§P COMPUTATION OF THE [Mec. Cel.
[798'] As cp ought always to be positive, and less than a right angle, this condition
will determine t^e sign of sin. / ; now thp tangent of /, and the sign of its
sine being determined, the angle / will be wholly determined.* This angle
[798"] is the longitude of the ascending node of the orbit, if the motion be direct ;
but we must add to it two right angles to obtain the longitude of this node,
if the motion be retrograde. f It would be more simple to consider the motion
[798'"] always as direct, making the inclination 9 to vary from 0 to two right angles ;
for it is evident that then the retrograde motion corresponds to an inclination
[798^''] greater than a right angle. In this case, tang. 9 is of the same sign asf
x.(-^) — y.l—j; which determines sin./, consequently the angle /,
which always expresses the longitude of the ascending node.
* (547) When / does not exceed a right angle, sin. I and tang. / are both positive ;
between one and two right angles, sin. / is positive tang./ negative ; between two and three
right angles, sin. / is negative and tang. / positive ; between three and four right angles,
[7976] sin. / and. tang. / are both negative. Therefore by knowing the signs of tang. / and sin. /,
we can determine the affection of /; now the first of these is determined by "the first of the
equations [798], and the second by the second of these equations.
f (548), If the motion be supposed direct, the values oi d x, d y, d z, must be considered
as positive, and if it be retrograde thesg. ^i^grentials would be negative. Tliis would make
the second equation [797] become
which by changing the signs of all the terms would become identical with the equation [797],
from which it Was derived y therefore the two equations [797] would be the same whether
the motion be supposed direct or retrograde. The same must take place in the equations
[798] deduced from [797]. The angle /, determined by these equations would give the
place of that node, which would be ascending if the motion be direct, and if the motion be
retrograde the numerical value of / would remain the same, because the terms of [798] would
remain unaltered, but in this last case the value of / would correspond to the descending
node, and we must add to it two right angles to obtain the longitude of the ascending node.
f (549) It was observed in [796"], that cc . (tt) — y '{tj) ^^ positive when the motion
is direct, in which case 9 is supposed to be less than a right angle and tang. 9 positive.
When the motion is retrograde, '^•(77) — y-ill) becomes negative, and as 9 then
II. iv. § 33.] ORBIT OF A COMET, 437
a and ae being the semi- transverse axis and the excentricity of the orbit [798 ^j
[596'], we have by ^ 18, 19, supposing fj^=l,*
a ~ r \dtj \di) \cUj '
The first of these equations determines the semi-transverse axis of the orbit,
the second its excentricity. The sign of the function
^•(^)+2'-(^) + ^-0' t^
shows whether the comet has passed the perihelion ;t for if it is approaching whether
towards the perihelion, this function is negative ; and in the contrary case, the ^^^^^'^
comet is receding from the perihelion.
exceeds a right angle, its tangent becomes negative. In both. cases we have tang, (p of the
same sign as ^•(^)-y-(^)- The product tang. <p. { a:.(^)-y .(^) | ig
therefore always positive, and as the second of the equations [798] gives
,a„g.,.^..(ll)-y.(^)|
we shall have sin./ of the same sign as y-[~7~) — ^•(j^)' with this sign, and the first
equation [798] we find / by the directions in note 547.
* (550) The first of these equations is the same as the last of [572]. The second is
deduced from the equation [598], putting fA = l, h^ = a.{l — e^), [599], and [799a]
rdr = xdx-{-ydy-^zdz, [549'].
f (551) This function '^ • (3~)4~y •(t^)4"^-(t~) i^ by the last note equal to
r .( — j, and as r is positive it must have the same sign as (l~)) which must, from the
nature of the perihelion, where r is a minimum, be negative before passing the perihelion,
positive after passing it. The value of the function ^ '{~r~)~\~y '\~r') '{'''' '[tt] ^^ ^^^^^
by using the values of x, y, z, [788] ; (~y (^), [796], and (^), [789].
110
sin.J= liiZ IfLL^ , [798«]
438 COMPUTATION OF THE [Mec. Cel.
Let T be the interval of time between the epoch and the passage of the
[799"] comet through the perihelion ; the two first of the equations (/) ^ 20 [606],
will give, by observing that m- having been put equal to unity [798''], makes
n = «~^[605'],*
[800] r = a . (\ — e . cos. w) ; T ==a^ . (u — e . sin. u).
The first of these equations gives the angle w, the second the time T. This
time added to the epoch, if the comet is approaching towards the perihelion,
but subtracted from the epoch if the comet is receding from the perihelion,
[800] will give the instant of its passage through this point. The values of x and
y will determine the angle which the projection of the radius vector r makes
with the axis of x ; and since we know the angle / made by this axis and
[800"] the line of nodes, we shall have the angle which this last line makes with
the projection of r ; hence we may deduce, by means of the inclination of
the orbit <p, the angle formed by the line of nodes and the radius r.f But
the angle u being known, we shall have, by means of the third of the
[800^"] equations {f) § 20 [606], the angle v which this radius makes with the
line of apsides ; hence we shall have the angle included between the lines of
apsides and nodes ; consequently the position of the perihelion, and all the
elements of the orbit, will be determined.
34. These elements are given by what precedes, in functions of p, i-j-
[sooiv] and known quantities ; and as (t-) is given in p by § 31 [770], the
elements of the orbit will be functions of p and known quantities. If one
* (552) The first of the equations [606], putting m^ = 1, n=a ^, [605', 798''], and
3 2l , ...
multiplying by a^, gives t or T= a'-^ .{u — e. sin. u), as in [800]. In the original, sin. u
was printed cos. u, by a typographical error.
f (553) The sngle formed by the line of nodes and the projection of the radius vector,
may be considered as measured by the arch D C in the figure page 379, the angle B D C
being cp, the arch BD will be the measure of the angle formed by the radius and the line of
nodes, and by spherics we shall have cotang. B D= cos. 9 . cotang. D C.
n. iv. § 34.] ORBIT OF A COMET. 439
of them be given,* we should have another equation, by means of which
we might determine p ; this equation would have a common divisor with the
equation (4) ^ 31 [784], and if we seek this divisor by the usual methods, we
should obtain an equation of the first degree in p ; we should also have an [800 »]
equation of condition between the quantities given by the observations, and
this equation would be that which ought to take place, in order that the given
element may appertain to the orbit of the comet.
We shall now apply this principle to the case of nature. For this purpose
we shall observe that the orbits of comets are very excentric ellipses, which [800^*]
nearly coincide with a parabola, in the part in which these bodies are
visible ; we may therefore suppose, without sensible error, a = co , [SOOvii]
consequently - = 0 ; the expression of - of the preceding article [799],
will in this case become
r dt^
If we substitute for, (-7-)? (3^)' ^^^ (t~)' ^^^^^ values found in the
same article [796, 789], we shall have, after making the necessary reductions, [800**]
and neglecting the square of R — 1 ,t
+ ^-(~^)-\(R'-1)-cos.(A-u)-
fda\ C . COS. (A — a) ^ 1 2
* (553rt) That is, if one of the elements be given. This is supposed to be the case in
calculating the orbit of a comet, in which a is supposed to be infinite, as in [800'"], from
which is deduced the equation in p, [805].
f (554) Changing the signs of [800""], and substituting for (-^j its value in [789],
we shall have, 0= </dp\ , ,^'\dt)> , /dx\^ , /dy\^ 2
^ '-^ ms.i + ^^ ^ +(-) +[^) --, ■„ which are [801.)
440 COMPUTATION OF THE [Mec. Cel.
by substituting in this equation, for (-r~ii its value [785],
— j , [jTJi [796]. In making this substitution we shall, for brevity,
[8015] piit (B'— l).-cos.^— -^=A {R'—l).sm.A + ^=iy,
(ff)-^^'-"-P-(^) •^in-'« = ^' (^) •^^"- «+P.(^) .cos.a = E', by
whifch mearis 'the formulas [796] wiU become, {^\ = D + E, (^) = iy + E\
and the sum of their squares is
[801c] (^y+ i^y^ ^^' + ■^') + (E^+E'^) + 2 D E + 2 jy E'.
Now the values of D, U, E, E', evidently give jD^+D's^ (^R' _ i)2 _|_ A=^ ^
neglecting the square of R'—l, [800"^], and ^ + £'2 ^ (7^")^+ P^ ' (jlf' ^"
finding the value of 2 D E-{-2 U E'^ we shall connect together the terms multiplied by
2 . ( — ), and in another group those multiplied by 2 p . f — j, and we shall have
. J . V (i?' — 1 ) . (cos. A . cos. a -|- sin. A. sin. a) ,
2DE + 2irE'=2.(-^).^ 1
\at/ i — — . (sin. ./2 . cos. a — cos. .4! . sin. a) '
-|-2p. ( — ) . < {R' — 1 ) .(sin. A . cos. « — cos. A . sin. «) + ^^ • (cos. A . cos. a -f- sin. A . sin. a) >
:=:2.(^^y[{R-l).cos.{A-a)-^.sm.{A-a)^
+2p.(^).^(i2'-l).sin.(^-«)+^.cos.(^-«)],
[24, 22] Int. These being substituted in [801c] we get
+ .p.(^). [(«'_.).. in. (^-«)+^^'|,
hence [801a] becomes as in [801].
n. iv. § 34.] ORBIT OF A COMET. 441
found in § 31 ; and then putting
+ ) tang. d.f^.-Vf^'. tang. ^. sin. M-«) '\dtj\dtj\ ;
( TT ) + f* • sm. M — ^)c-(a\ X
^ Vc^^V ^ ^ Csin.(.^— a) 1\ ^^WJ ^t
C= ^— I ^ (i2^1).C0S.(J^a)j
we shall have*
0 = B.p'+C.p+^-^ [804]
* (555) Put for brevity f -— - j -f ju,' . sin. (./^ — a)==-F, and we shall get from
[802], [J\^—^jj^ . Substituting this in [801], we find
CPS.2a
P-^ Sfvti i\ /^ \ sin. (.4 — a))
— 7rf^\ • ^(^— 1).C0S.(^ — a) ^ ^^
and by arranging according to the powers of P,
/dS\ /da\\ 2^
'\dt) ' \d t)
^da\^ , F2 ^ ^ 2
COS.24
^ !_(/,_!). e„,(^_„j + !iM^-)| 1 ^
+f-<Vrf<;... . ._ ' >+:^-7.
111
44-2 COMPUTATION OF THE [Mec. Gel.
consequently
[805] y.|5.p^+C.p + i| =4;
this equation is only of the sixth degree, and in this respect it is more simple
[805'] than the equation (4) ^31 [784] ; but it is restricted to the parabola,
whereas the equation (4) [784] extends to every kind of conic section.
35. We find from the preceding analysis, that the determination of the
parabolic orbit of a comet leads to more equations than there are unknown
[805"] quantities ;* we may, by combining these equations in different manners,
form several different methods of computing these orbits. We shall examine
those from which we ought to expect the greatest precision in the results,
and which are the least affected by the errors of the observations.
It is chiefly in the values of the second differentials (-7-^)5 and
^dt^J' \dt^
that these errors have a sensible influence. In fact, to determine them, it is
[805'"] necessary to take the second differences! of the geocentric longitudes and
latitudes of the comet, observed in a short interval of time ; now these
differences being less than the first differences, the errors of observation will
be a greater aliquot part of these second differences ; moreover, the formulas
of § 29 [758], which determine, by combining the observations, the values
°^ "' '' U/ \dl} [j^} Wj' Si^e with greater precision the
[805>'] four first of these quantities, than the two last ; it is therefore advantageous
in which the coefficient of p^ is equal to B, [803], and that of p is equal to C, [803], hence
12,
the preceding equation becomes 0==Bp^-{-Cp-\-— , as in [804]. Transposing
2
-, multiplying by r, and squaring both sides, we obtain [805].
r
* (556) Thus the /our independent equations [782, 770, 774, 801], which compose the
equations [806], are given to find the three unknown quantities p, i~rX and r, being one
more than is absolutely requisite.
f (557) The word here translated " second" was in the original printed " finies" instead
of "secondes."
n. iv. §35.] ORBIT OF A COMET. 443
to depend as little as possible on the second differences of « and & ; and as
we cannot reject them both, at the same time, the method which uses only
the greatest, must give the most accurate results ; this being premised,
We shall resume the equations of % 31 and 34, [782, 770, 774, 801]
r^ = -^^ + 2i2.p.cos.(^ — «)+i2^
(ddd
~dl
'\h--^\- ^ fL\ ' (^)
dt J \dt
FuBcIa-
mental
_P
^^^ C) ( \ '-■"' ' } <^ { ^ '^ \ equations
•^•1,1 /6( . I — r~ J for com-
puting the
orbit of a
dd^\ fdo\^ . \ •=•»""'•
-^ ) . sm. ^ . cos. ^
( \dtj \dtj
R.sm.6.cos.d .cos.{A — a) C _1 l_} [806]
0 =
dt J i [ -r:] ' tang.
•©
^* C0S.2 d
+ 2 . g) . J (i?- 1) . COS. (^-»)_!!^^^
COS. (./3 — a) ) 1 2
+2p.(^).[(i2'-l).sin.(J-a) +
If we would reject ("Tt)' "^^ must use the first, second and fourth
of these equations ;* by eliminating ( j^ ) from the last, by means of the
second, we shall obtain an equation, which being cleared from fractions, will
* (558) It is to be observed that by neglecting one of the equations [806], the resulting
equation in p or r is of a higher order ; for instead of being of the sixth degree in p, as in
[805], it becomes of the sixteenth, as in [606']. Upon further consideration of the subject
the author finally concluded, not to reject wholly either of these equations, but to combine
two of them together, in a manner which he supposed would probably lead to the most
accurate result. We shall hereafter, [815a, Sic], speak of this method, which is particularly [SOGa]
treated of by the author in Book XV, <§,5,
444 COMPUTATION OF THE [Mec. Cel.
[806] contain a term multiplied by r^ p^, and other terms affected with even and
odd powers of p and r. If we place on one side of the equation all the terms
containing even powers of r, and on the other all the terms containing the
odd powers, and then square both sides to obtain only even powers of r,
the term multiplied by r^ p^, will produce one multiplied by r^^ p"* ; and by
[806"] substituting the value of r^ given by the first of the equations (L) [806],
we shall finally obtain an equation of the sixteenth degree in p. But instead
of forming this equation, to resolve it afterwards, it will be more simple to
satisfy the three preceding equations by trials.
If we would reject (-ty )' ^^ must use the first, third and fourth of the
[806"] equations (L) [806]. These three equations lead to a final equation of the
sixteenth degree in p ; which equations may be easily satisfied by trials.
The two preceding methods appear to me to be the most accurate that
we can use in finding the parabolic orbits of comets ; it is even absolutely
necessary to have recourse to them if the motion of the comet in longitude
[806i^] or in latitude is insensible or very small, in order that the errors of the
observations may not alter sensibly the second differential ; in this case we
must reject that one of the equations (L) [806] which contains that second
differential. But although in these methods we use only three of the
equations, the fourth will be useful to determine, among all the real and
[806^] positive values of p which satisfy the system of the three other equations,
that value which ought to be assumed.
36. The elements of the orbit of a comet, determined in the preceding
manner, would be exact if the values of «, 6, and of their first and second
[806"] differentials, were rigorously correct ; for we have taken into consideration,
in a very simple manner, the excentricity of the earth's orbit, by means of
the radius vector R of the earth, corresponding to its true anomaly increased
by a right angle ; we have only neglected the square of this excentricity, as
being so small a fraction that its neglect could not sensibly affect the result.
[8C6^''] But ^, «, and their differentials, are always liable to some error, on account
of the imperfection of the observations, and also by reason of the errors
arising from the approximate method of computing their differentials. It is
therefore necessary to correct the elements, by means of three distant
observations, whi.ch may be done by a very great variety of methods ; for if
II. iv. <§36.] ORBIT OF A COMET. ^^
we know very nearly two quantities relative to the motion of a comet, as,
for example, the radius vector at each of two observations, or the position of [806"«]
the node and the inclination of the orbit ; by calculating the observations first
with these quantities, then with other quantities which vary a little from
them ; the law of the differences between the results, will easily give the
corrections to be applied to those quantities. But among all the combinations,
two by two, of the quantities relative to the motion of comets, there is one [806"]
which furnishes the most simple calculation, and which, for that reason,
deserves particular attention ; it being of importance, in so complicated a
problem, to spare the calculator all unnecessary labor. The two elements
which appear to me to have this advantage, are the perihelion distance, and
the time of passing the perihelion ; they are not only easily found from the
values of p and ( t^ ) ; but may be very easily corrected by other observations, [806 »]
without being obliged, at each variation which is made in these two elements,
to determine all the other corresponding elements of the orbit. [806"]
We shall resume the equation found in ^ 19 [598, 599]*
a.(l— e^) = 2r — — — ^^!-^; [807]
a. (1 — e^) is the semi-parameter [383', 377"] of the conic section of which [807^]
a is the semi-transverse axis, and ae the excentricity ; in the parabola, where
a is infinite, and e equal to unity, a. (I — e") is the double of the perihelion [807"]
distance ; naming this distance D, the preceding equation becomes,
relatively to this curve,t
* (559) This is like the second of the equations [799], putting
xdx-\rydy-{-zdz = rdrf [799a].
f (5G0) Since D = a . (1 •— e), [681"], we have
a.{l—'e^)=a.{l—e).{l-^e)=D.{l-{-e),
and in a parabola, where e= 1, [378^], it becomes 2 D. This being substituted in [807],
observing that when a = (x; -=0, it becomes 2D = 2r ——. Dividing this by
112
446 COMPUTATION OF THE [Mec. Gel.
7* /Z 7* Tt n f^
is equal to — -^ — . Substituting for r^, its value
dt ^ dt
cos
2 4
+ 2i2p.COS.(^ — a) + i2',
[809]
and instead of (-7—)? and (-t-)j their values found in § 33, we shall
have, by putting for brevity P equal to the last member of the following
f. r dr
expression or -~ ,
+ P. { (iJ'-l).cos.(^-«)-?i:ili|=^ J
+ f.R.(^-^.sm.{A — <^) + R.{K — \);
[809^] if p be negative, the radius vector r would be decreasing, consequently the
comet would tend towards the perihelion ; but it would be receding from the
perihelion if P be positive.* We thus have
[810] D = T—IP^;
the angular distance v of the comet from its perihelion, is found by the polar
equation of the parabola [691]
[811] cos.^ 1 V = — ;
r
J811'] lastly, we shall have the time employed in describing the angle v, by the
table of the motion of comets. This time, added to the time of the epoch,
2 we get D, [808], which may be written D =r — \ . ( — — ) . Now half the differential
of the first of the equations [806], is
\d. r'=-^^-\-^''-^-.d&-{-Rdo. COS. (A — a)-\-o.dR. cos.{A--a)
" cos.2^ ' COS.34 I r \ J \ V \ /
-{- R p . {d a — d A) . s'm. (A — a) -]- R d R.
Dividing this by dt and substituting the values of ( — ), ("T~)> ["795], we obtain [809]
* (561) This is conformable to what is shown in note 551. The expression P [809],
substituted in [808], gives [810].
[8ir]
II. iv. §37.] ORBIT OF A COMET. 447
if P be negative, or subtracted from the time of the epoch, if P be positive,
will give the instant that the comet passes the perihelion.
37. Collecting together these various results, we shall have the following
method of computing the parabolic orbit of a comet.
GENERAL METHOD FOR COMPUTING THE ORBIT OF A COMET.
This method will be divided into two parts ; in the first we shall give the
method of obtaining very nearly the perihelion distance of the comet, and the
instant of passing the perihelion ; in the second we shall determine accurately
all the other elements of the orbit, supposing the former to be known very
nearly.
APPROXIMATE COMPUTATION OF THE PERIHELION DISTANCE OF A COMET, AND THE INSTANT
OF ITS PASSING THE PERIHELION-
We must select three, four, or five, &c., observations of the comet, as
nearly equidistant from each other as possible. With three observations we
may embrace an interval of 30° [27*^ of the sexagesimal division] ; with
five observations, an interval of 36' or 40' [32' 24™ to 36'' of the sexagesimal ^
division], and in like manner for a greater number ; but it is always !!,m*puung
^ ^ the orbit of
necessary that the mterval should be increased with the increase of the acoinet.
number of observations, in order to diminish the effect of the errors of the
observations. This being premised.
Let |3, /3', /3", &c., be the successive geocentric longitudes of the comet ; 7, 7',
7", &c., the corresponding latitudes, these latitudes being supposed positive .0111,1
if norths but negative if south. We must divide the difference ^' — 13, by the
number of days elapsed between the first and second observation ; in like
manner we must divide the difference f3" — (3', by the number of days elapsed
between the second and third observation ; we must also divide the difference
|3"' — p", by the number of days elapsed between the third and fourth
observation ; and so on for the others. Let these quotients be (J|3, 5/3', 5/3", &c.*
* (563) This and the following part of the article correspond to §29, [755,756,758,759].
The intervals between the observations are to be expressed in days and decimal parts of
a day.
^^ COMPUTATION OF THE [Mec. Cel.
We must divide the diiference S ^' — 6 /3, by the number of days interval
[811''] between the first and third observation ; in like manner we must divide the
difference 5 f3" — 5^' by the number of days between the second and fourth
observation ; the difference 6 ^"' — .5 13" by the number of days between the
third and fifth observation ; and so on for the rest. Let these quotients be
62^, 6^^', 6^^", &c.
We must divide the difference ^3^ ^' — s^^ by the number of days between
[Sll'^"] the first and fourth observation ; in like manner we must divide 5^^" — S^ ^'
by the number of days between the second and fifth ; and so on. Let these
quotients be <5^ |3, ^^ ^', &c. We must proceed in the same manner till we
[Sllviii] obtain ^""^ |3, n being the number of observations used.
This being done, we must take an epoch, which is equidistant, or nearly
[81l»'^] so, from the two extreme observations, and putting i, i', i", i'", &;c., for the
number of days it precedes each observation, i, i', i", &c., being supposed
negative in observations preceding the epoch ; the longitude of the comet,
after a small number of days, denoted by z, counted from the epoch, will be
expressed by the fallowing formula :
^-^i.5^-^ii'.6^^—ii'i"J^^-{-kc. (p)
[812] i-z.{l^—(i+iy^^^+(ii'-^ii''-i-iT).^^^—(iirWi'''+ii"^^^^
The coefficients of — (5|3, +'5^/3, — 6^^, &c., in the part independent of
[812] z, are, First, the number i ; Second, the product of the two numbers i and
i' ; Third, the product of the three numbers i, i', i", &c.
The coefficients of — 3^^, +'^^/3, — ^^'^ f3, &c., in the part multiplied by
[8i2"j z, are. First, the sum of the two numbers i and i' ; Second, the sum of the
products, two by two, of the three numbers i, i', i" ; Third, the sum of
the products, three by three, of the four numbers i, i', i", i'", &c.
The coefficients of — ^^ ^, -\-^'^^, — ^^ ^, &c., in the part multiplied
by 2% are. First, the sum of the three numbers, ^, ^', i" ; Second, the sum
[812"'] of the products, two by two, of the four numbers i, i' i", i'", ; Third,
the sum of the products, three by three, of the five numbers i, i', i", i'",
i"", &c.
Instead of forming these products, it is as easy to develop the function
[813] ^ + (z — i).5(3 + (z^i).(z — i').5''^ + (z—i).(z — i').(z—i").6^^-^kc.,.
U. iv.§37.] ORBIT OF A COMET' 449
rejecting the powers of z, above the square, which will give the preceding
formula [812].
If we perform a similar operation upon' the observed geocentric latitudes of
the comet, its geocentric latitude in z days after the epoch, will be expressed'
by the formula (^) [812], changing in it ^ into 7. Let us call the formula
thus changed (^) [813']. This being premised, [SIS']
a will be the part independent of 2; in the formula (j?) [812],^ will be the [813"]
part independent of z in the formula {cj) [813'].
Reducing into seconds the coefficient of z in the formula (^) [812], and
subtracting from the tabular logarithm of this number of seconds, the
logarithm* 4,0394622 [or 3,5500072 sex.], we shall have the logarithm of a [813^
number' that we shall denote by a.
Reducing into seconds the coefficient of 2*' in the same formula, and
subtracting from the logarithm of that number of seconds the logarithmf [8l3i']
1,9740144 [or 1,4845594 sex.], we shall have the logarithm of a number
that we shall denote by h.
By reducing in like manner into seconds the coefficients of rand 2*, in the
formula (5) [813'], and subtracting from the logarithms of these numbers the [813']
logarithms 4,0394622,* and l,9740144t respectively [or 3,5500072 and
* (564) If we use the common sexagesimal division of the quadrant into 90**, or
324000^ the logarithm must be 3,5500072 as is observed in [759c]. The values a, h
being respectively equal to (i~)> VT)'
f (565) The coefficient of r^ in the function [812] is the same as ^ • (— ^j, [758],
and it is shown, in [759c], that by subtracting from the logarithm of J . { — ), in seconds,
the quantity 2,2750444, for the centesimal division, or 1,7855894, for the sexagesimal
— j = log. J 6, [813']. Adding to this the log. of
2 or 0,3010300, we shall get the log. of h. JVIoreover, since
2,2750444 — 0,3010300= 1,9740144,
and 1,7855894—0,3010300=1,4845594,
we may obtain the log. of 6, by subtracting from log. \ . (t^) in seconds, the number
1,9740144, if centesimal seconds are used, or 1,4845594, if sexagesimal seconds are used.
113
^^^ COMPUTATION OF THE [Mec. Cel.
1,4845594 in sexagesimals], we shall have the logarithms of two numbers
which we shall call h and /.
The accuracy of this method depends on the precision of the values of «, 6,
[Si^'"] ]i^ I . and as the computation of these quantities is very simple, we must
select and increase the number of observations, so as to ascertain them with
all the exactness that the observations will allow of. It is evident that «, 6,
[8 i3vii] A, /, represent the quantities T^Y \-jj\ (j^\ \~d^J'' ^^i^^' ^^^
greater simplicity, have been expressed by the preceding letters.
If the number of observations be odd, we may fix the epoch at the instant
[813""] of the middle observation ; and then we may dispense with the calculation of
the parts independent of z, in the two preceding formulas ; for it is evident
[813«] that these parts would then be equal to the longitude and latitude of the
middle observation respectively.
Having thus determined the values of a, a, b, 6, h, and Z, we must find the
[813*] longitude of the sun, at the time of the epoch; let E be this longitude, R
the corresponding distance of the earth from the sun, and R the distance
corresponding to E increased by a right angle ; we must then form the
following equations :*
[814] 7^=—^ 2Rx.cos.(E — a)-\-R^', (1)
cos/ 4 ^
[815] y= i -^ I ___!__; (2)
C , ^ , I , a^ . sin. ^ . cos. & )
3, = -a;.|A.tang.« + ^ + -^ |
, i2 . sin. d . CDS. 4 ,ri \ ( 1 1 )
+ ^7i -COS. (£-«). I j^-^j;
tiontothe oQ,A ^ ., "'^Vf5> VSin. {jC< ttj „, . _ ./
'o'fThr 0 = 2/^ + « ^ +( y.tang.^+^^^J +2y. ^^^ L^(^R--\),cos.{E—a){
orbit of ft ^ • / V. J
'°""'" c^ S.-n, •i\ ' rrr s ^ COS. (E — a) } ,12 (4,)
[817] C ^ ) R^ r
* (566) These are the same as the equations [806]. Putting p = a?, f— j = y,
[810]
Equations
for com-
puting the
fir»t ap-
proxima-
(3)
U. iv. §37.] ORBIT OF A COMET. ^^1
To deduce from these equations the values of the unknown quantities x, y,
and r, we must consider whether 6, independent of its sign, be greater or less
than /. In the former case, we must use the equations [814, 815, 817], and [817']
two right angles. The quandty x being the distance of the comet from the earth, projected
on the plane of the ecliptic, r the distance of tlie comet from the sun. The remarks above
given relative to the equations to be used, are conformable to what was observed immediately
following the formulas [806].
We have already observed [806a], that the author modiSed this calculation, in Book XV,
§ 5, by changing the manner of computing the quantities a, h, h, I, and connecting together
the two equations [815, 816]. In this new method, the quantities a, b, h, I, are computed
in the following manner, by combining only three observations, instead of using a greater
number, as in the formulas [754 — 758]. Let the geocentric longitudes of the comet
corresponding to these three observations be a^ , a, a ; the geocentric latitudes 6^ , 6, 6', ^P^^*^f
respectively. Then fixing the epoch at the time of the middle observation, and putting i for t|'o™f"'*"
the interval in days and decimals of a day, between the first and second observations, also i'
for the interval between the second and third observations ; the general expression of the
longitude corresponding to s days, after the epoch, will be of the form a-\-sa-\-li^.b, [7 57],
and that of the latitude will be 6-{- s .h-{-^s^ .1. If we now put s = — i, they will [815a]
become a^, 6^, respectively; and if s = i\ they will become a, and d', respectively;
hence we shall obtain these four equations, in which a, 6, A, Z, a, ^, &£c., are expressed in seconds,
a — a^ia — ^v^.b; A — d^ = ih — Ji^.?; mSb}
u' — a = i'a-{-^i'^.b; ^ — fl = i'A + ii'2j;
The values of a, b, h, I, being found from these equations, in sexagesimal seconds, we
must from the logarithms of a, h, subtract the logarithm 3,5500072, [759'", 814a], and from the \si5c\
logarithms of b,l, in seconds, subtract the logarithm 1,7855894, and we shall obtain the
logarithms of the values of a, b, h, I, to be used in the formulas [815Z,'m, ri].
With the same epoch and the same middle observation a, 6, we may use another extreme
observation, a^, 6^^, made before the epoch, and another a", d", after the epoch, and by rglM]
means of the intervals corresponding to these observations we can compute other equations
similar to [8156], which may also be used in finding a, b, h, Z, so that it is not necessary to
confine the calculation to three observations, since the triple combinations of observations
may be augmented at pleasure. Any number of these equations may then be connected
together, to determine the values of a, 6, h, Z, in such manner as shall be judged most
advantageous.
The method recommended by the author for the combination of such equations, is
derived from the principle of making the sum of the squares of the errors a minimum, which [815c]
principle will hereafter be more fully explained. In the present case all the equations
^52 COMPUTATION OF THE [Mec. Cel.
form a first hypothesis for x, by supposing it, for example, to be equal to
unity; and then compute, by means of the equations [814, 815], the values
[8l7"j of r and y. Substituting these in the equation [817], if it become nothing,
containing a, h, are to be combined together. Firsts by multiplying each of the equations by
[815/*] the coefficient of a, in that equation, and taking the sum of these products for one of the
final equations, to be used in computing a, h. Second, by multiplying each of these equations
[815g-] by the coefficient of b, always noticing the sign of this coefficient, and taking the sum of the
products for the second final equation. From these two equations are to be computed the
values of a, h. In like manner from the equations in h and Z, two final equations are to be
found, for the determination of h and I. It may also be observed that if we denote, as in
[815A] [754^], by i, i', i", &;c., the number of days and parts of a day, which the several observations
follow the epoch, considering these numbers as negative if they precede the epoch, noticing
the signs and putting
[815i] A = l^-i-i'^+i"^ + hc.', 5=i3 + i'3 + i"3 + &C.; C=>'4 + t'4-fi"4_|_gjc.J
the terms depending on a, 6, in the two final equations, will be Aa-^-^Bb, and
[815k] ^ B.ar{-i C b, respectively, and the similar terms in tlie equations depending on h, I, will
be Ah-]-^Bl, and i Bh-{-^ CI, which maybe very expeditiously calculated,
when the numbers are large, by means of Barlow's excellent table of the powers of
numbers.
These final equations become very simple, when every positive term of the series i, i', &c.
[815A], is accompanied by a negative one of equal value, because in this case the quantity
B, [815i] will vanish, and the terms depending on a, b, in the final equations [81 5A;), will be
reduced to A a and ^ C b. As an example of this method, we shaU take the four following
equations, in which the series i, i', i", he. is represented by — 4, — 2, 2, 4,
respectively, the epoch being taken at the middle time between the extreme observations.
0 = 4a — 86 — 23,
0==2a — 2b — 15,
[8151] 0 = 2a + 2& — 23,
0 = 4a + 86 — 55.
Multiplying these equations by the coefficients of a, namely, 4, 2, 2, 4, and
adding the products, we get the first final equation 0 = 40 a — 388, hence a = 9, 7.
Again, multiplying the same equations by the coefficients of b, namely, — 8, — 2, 2, 8,
and adding the products, we get the second final equation, 0= 136.6 — 272, hence 6=2,
These values of a, b, being substituted in the second members of the equations [815Z], they
become — 0,2, 0,4, 0,4, — 0,2, instead of being nothing. The sum of the squares
of these errors is 0,40, and no values of a, b can be found which will make this sum. less,
as will be seen when we shall explain the method of the least squares.
n. iv. § 37.]
ORBIT OF A COMET.
453
it will prove that the value of x was rightly assumed ; if the result be
negative, we must increase the value of x ; but it must be diminished if the [817'"]
result be positive.* We shall thus obtain, by a few essays, the values of a:,
Instead of the four equations [814 — 817], the autlior finally adopted the three following
r2=
•2/2a;.cos. (E — a) + ii^
0 =
C03.2 ^
I a . sin. (-E — a) — ft . sin. ^ . cos. ^ . co3. (E — a)} -n (^ ^\
{ fe3 . tang. ^\\ah-\-\'hl-\-\a^'h. sin, h . cos. 6 j
h^x^ 2hyx.tAng.6
■a^ar^-
+ 2y
cos.^d COS.44
{E-a)
COS.2 d
{R—l).cos.{E — a)l
-2«..[(R'_.).sin.(£-„)+^^j+i-i,
C sin.
R
[8151']
[815m]
Final
equation
adopted by
the autlior.
[815n]
of which the first is the same as [814] ; the second is found by multiplying [815], by
cfi
, and [816] by
A2
and taking the sum of the products ; the third is the same
as [817], connecting together the terms depending on y^, and putting in its coefficient
— — for 1 -\- tang.^ 6. After substituting in these equations the values of a, b, h, I, we
may from them compute the values of x, y, r, and then the other elements in the manner
pointed out in [817' — 820']. We shall hereafter give a numerical example of this method.
* (567) Put Y equal to the second member of the
expression [817], and upon the line AD, taken as the axis
of X, erect the ordinates A B, F G, F' G', kc.
representing the values of Y, which correspond to x=0,
x = AF, x = AF', &tc. respectively; the positive
ordinates F' G\ &;c. being taken above the axis ; the
negative ones AB, F G, &,c. below. Through the
extremities of these co-ordinates draw the regular curve
B G C G' E, and it will intersect the axis at least once, at C, from the nature of the
question. This is also evident from the consideration that when x = 0, we shall have
r = /?, [814], y = 0, [815], hence F, [817], will become ^—^, as R, [790], is
nearly equal to unity, this will become F== — 1 , nearly ; so that when x = 0, T will
be negative. On the contrary, when a;= oo, we shall have r = 00, [814], y = db 00,
[815], hence Y=c3o, [817], because the three first terms of [817], depending on the
114
454
COMPUTATION OF THE [Mec Cel.
y and r. But as the unknown quantity may be susceptible of several real
t^^'^"] and positive values, we must select that which satisfies, either accurately, or
very nearly, the equation [816).
In the second case, that is when /> 6, we must use the equations [814,
[817^] 816, 817], and then the equation [815] will serve for verification.
Having thus the values of a:, y, r, w'e must compute the quantity [809]*
[818]
+ a: . I ^^^:=l^ — (i2'— 1) . COS. (^ — «) I — i2aa: . sin. (i: — «)
+ R.{E — \).
The perihelion distance of the comet Z), will be [810]
[819] i)=^—^p2.
the cosine of half the true anomaly v will be given by the equation [811]
[820] cos.' 1 ?; = -_ ;
squares of x, y, he. are infinitely greater than the others, and are all positive ; therefore,
when 00= CO, Twill be positive and infinite. Now, without examining into the nature of
this curve, we find that for every value of x, from 0 to oo, there is a real value of Y,
positive or negative, the negative value taking place when x = 0, the positive when
a?:^ 00, this could not be, unless the curve crossed the axis at some point C, between these
extreme values of x.
If the assumed value of x in the first hypothesis [817"] be A F, corresponding to the
negative ordinate F G, it is evident that by increasing the value of x, we shall finally obtain
a value A C, in which the ordinate Y is nothing, corresponding to the next following point C,
where the curve cuts the axis; but if the value of x, selected, should be AF', corresponding
to the positive ordinate F' G\ by decreasing the value of x, we should obtain the next
immediately preceding point C, where the curve crosses the axis. The same rule would
apply with a curve of this kind, which should cross the axis in more than one point ; it being
evident, from a little consideration, that if the assumed value of x corresponds to a negative
ordinate Y, we must increase x to obtain \\ie following point of crossing the axis ; but if the
assumed value of x corresponds to a positive ordinate Y, we must decrease x to obtain the
preceding point of crossing the axis.
* (568) The formulas [818, 819, 820], are the same as [809, 810, 811], substituting
the values [813''"]. "
n. iv. §37.] ORBIT OF A COMET. ^5
and we may deduce, from the table of the motion of comets [693"], the
time employed in describing the angle v. To obtain the time of passing the
perihelion, we must add this time to the epoch if P be negative, but subtract [820']
it if P be positive ; because in the first case, the comet approaches the
perihelion [809'], in the second case it recedes from it.
Having thus obtained, nearly, the perihelion distance of the comet, and
the time of its passing the perihelion, we may correct these elements by the ^^^'^
following method, which has the advantage of being independent of the
knowledge of the approximate values of the other elements of the orbit.
APCURATE DETERMINATION OP THE ELEMENTS OF THE ORBIT, WHEN WE KNOW NEARLY THE
PERIHELION DISTANCE OF THE COMET, AND THE TIME OF PASSING THE PERIHELION.
Select three distant observations of the comet, and by means of the
perihelion distance, and the time of passing the perihelion, obtained by the [820"']
preceding method compute three anomalies of the comet, and the three
radii vectores corresponding to the times of the three observations. Let v,
v', v", be these anomalies, those preceding the perihelion being supposed
negative, andr, r', r", being the corresponding radii vectores ; v' — v, v" — v, [820*^]
will be the angles contained between r', r, and r", r ; put U for the first of
these angles, and U' for the second, so that
U=v' — v; U' = v" — v. [820V].
Let «, «', a", be the three observed geocentric longitudes of the comet, referred
to di fixed equinox ; ^, ^, ^", the three geocentric latitudes, the southern latitudes
being supposed negative ; f3, |3', ^", the three corresponding heliocentric [820'']
longitudes, and ts, to, zs", the three heliocentric latitudes ; E, E', E", the
three corresponding longitudes of the sun, and J?, R\ R", its distances from [sso^ii]
the centre of the earth.
Suppose the letter S denotes the place of the centre of the sun, T that of
the earth, C the centre of the comet, and C its projection upon the plane of [820^!"]
the ecliptic. The angle ST C will be the difference of the geocentric
longitudes of the sun and comet ; and by adding the logarithm cosine of this
angle to the logarithm cosine of the geocentric latitude of the comet, we shall
456
COMPUTATION OF THE
[Mec. Cel.
[820«] have the logarithm cosine of the angle STC ;* therefore we shall have, in
the triangle STC, the side ST or jR ; the side S C or r, and the angle
STC: we shall then have, by trigonometry, the angle C S T.f We may
obtain the heliocentric latitude of the comet w, by means of the equation J
.«„,, . sin. d .sin. C S T
f""'' ="'•'' = -ir„7crs--
The angle TSC is the base of a rectangular spherical triangle, whose
hypotenuse is the angle TSC, and side t^ ; and from the two last, we may
easily compute the angle TSC, and then find the heliocentric longitude of
the comet |3.^
[820a]
* (569) In the annexed figure are marked the places of
the sun, earth and comet, as directed above ; the lines C P,
C P are drawn perpendicular to T S. Then in the rectan-
gular triangles T P C, TPC, we have for TP the
expression T C . cos. STC=TC', cos. STC'-, and
in the rectangular triangle T C C we have
TC'=TC. COS. CTC'=TC.cos.6,
substituting this in the preceding equation, and dividing by T C, we get
cos. STC = cos. STC cos. 6,
as in [820*^], which is also easily obtained by spherics.
t (570) For SC:ST::sm.STC:sm.SCT; then the angle
C S T=^ 180^— STC— SCT.
It may be observed that the angle S C T being found by its sine, has two values, as is
observed in [S26']. This might cause some embarrassment, when the angle SCT'is nearly
a right angle, and this is to be avoided as in note 575.
J (571) In the rectangular plane triangles T C C, S C C, we have
CC' = rC'.sin. CTC'==rC.sin.^, and C C' = S C .sm. C S C'= S C .sm.zs.
Hence T C . sin. 6=S C . sin. zi. Now in the triangle STC we have
TC'.SC:: sin. C ST: sin. C TS, which being substituted, we get
sin. C S r . sin. ^ = sin. C T S . sin. *,
hence we obtain sin. zs, as in [821].
§ (572) With the centre S, and radius unity, suppose a spherical surface to be described,
intersecting the lines ST, S C, S C, in the points t, c', c, respectively, and forming
the rectangular spherical triangle < c' c. Then the arches id, tc, cc', are of the same
n. iv. § 37.] ORBIT OF A COMET. 457
In the same manner we shall have -d^ |3', to", |3" ; and the values of |3, ^, j3", [821']
will show whether the motion be direct or retrograde.
If we suppose the two arcs of latitude ro, to', to be continued to meet in the
pole of the ecliptic, they will make there an angle equal to j3' — 13 ; and in [821"]
the spherical triangle formed by this angle, and the sides «*, z/,
nr being the semi-circumference of the circle, the side opposite to the angle
|3' — /3 will be the angle formed at the sun, by the two radii vectores r, /. [821'"]
This angle may be found by spherical trigonometry, or by the following
formula :*
sin.^ ^V= cos.® ^(^-{-z/) — cos.^ ^ (f3' — 13) . cos « . cos. ■o' ; [822]
in which V represents this angle. Now if we suppose A to be found from
the tables by means of the following equation,
sin.® A = cos.® ^ (|3' — (3) . cos. m . cos. ra', [822Q
we shall havef
sin.® ^ F=* cos. (^ zs -\- ^ 'a' -^ A) , cos. (| ra -f i w' — A). [823]
number of degrees as the angles T S C, C S T, C S C, respectively, of which the
two last are given, and we may obtain the first by the usual rule of spherics
, COS.fc m Ci y^i cos.CST
cos. tC = ;, or COS. 1 b C = .
COS.CCr COS. "Cy
* (573) If j1, B, C, be the sides of a spherical triangle, and c the angle opposite to the
side C, we shall have cos. C:= cos. A . cos. B -\- sin. A . sin. B. cos. c, [172t]. Putting
c = p' — p, v2 = J* — irf, B = iie — ■13', Jir being a right angle, and C=V, it
will become cos. V= sin. -a . sin^-s/ -\- cos. -a . cos. ta' . cos. {^ — ^). Now by [1, 6] Int.
COS. V^ 1 — 2 . sin.® \ V, cos. (^' — ^) = 2 . cos.® ^ (^' — p) — 1, hence by
substitution,
1 — 2 . sin.® ^ V= 2 . COS.* \{'^ — p) . cos. w . cos. -c/ — cos. th . cos. •5/ + sin. trf . sin. •ra'
= 2 . cos.^ i [fi' — p) . cos. w . COS. T^ — COS. {■a-j-'s/)
= 2 .cos.®i^(^' — p) . COS. -a. COS. •c/ — 2. cos.®^ (zi -\-v/) -f-1,
as appears by [23, 6] Int. ; hence, by rejecting the term 1 from each member, and dividing
by — 2, we obtain [822].
t (574) From [20] Int. cos. {B + A). cos. {B—A) = l cos. 2 B-{- ^ cos. 2 A,
and by [6, 1] Int. we get ^ cos. 2 5 = cos.® B — i, | cos. 2A = l — sin.® Aj
hence cos. {B -{- A) . cos. {B — .^) = cos.®5 — sm.^ A. Putting now B = ^zi-\-^z/f
and sin.® A, as in [822'], the second member will become equal to the value of sin.® ^ V^j
[822], and the first member will therefore represent sin.® ^ F", as in [823}.
115
458 COMPUTATION OF THE [Mec. Cel.
If we likewise put V for the angle comprised between the two radii vectores
r and r", we shall have
[824] sin.2 h'^' = COS. (Izi + ^z/' + A). COS. (1 « + ^ ^" — A').
[824'] A' being what A [823] becomes by changing -a' and |3' into «/' and |3".
Now if the perihelion distance of the comet, and the time of passing the
perihelion, were exactly known, and the observations were rigorously correct,
we should have [820"]
[895] V=U; V'=U'\
but as this very rarely happens, we shall suppose
[826] m=C7— F; 7n!=U'—V',
We may observe that the calculation of the triangle ST C, gives for the
angle CST, two different values [820a] ; in general the nature of the motion
[826'] of the comet will show which ought to be used, especially if the angles are
very different ;* for then the one will place the comet farther from the earth
than the other ; and it will be easy to judge, by the apparent motion of the
comet at the time of observation, which ought to be selected. But if there
is any uncertainty in this respect, we may avoid it, by choosing that value
[826"] which renders V and V nearly equal to U and U' respectively.
This method of finding F" has however no advantage over the
common method used in spherical trigonometry. To prove this, let
B, X), be the geocentric places of the comet at the first and second
observations ; A the pole of the ecliptic C F; ABC, AD F,
circles of latitude. Draw DP perpendicular to AB. Then
BC = T^, DF=zi', AB=:^r{ — ^i, AD = ^if-^z/,
BD=V, BAD = fi' — ^, CP = D, BP = E,
AP = ^i( — D. By spherics we have
tang. AP = tang. A D . cos. PAD, or, cotang. D = cotang. z/ . cos. (^' — (3).
Noticing the sign of D in the same manner as those of -a, z/. Then BP=^C P — B C,
gives E==D — -m, and by spherics
_. _, cos.AD .cos.BP T7- sin.tS'.cos.jE
COS. BD= T- , or, cos. V= r-j: — .
C03.AP sm.D
This requires less labour than the former method, but rather more attention to the signs.
* (575) If however the angle S C T should be very near a right angle, the observation
might be changed for another a day or two earlier or later.
n. iv. Ǥ37.] ORBIT OF A COMET. 459
We must then form another hypothesis, in which the time of passing the
perihelion is to be retained, while the perihelion distance is varied by a small [826'"]
quantity, as for example a fiftieth part ; and we must compute in this
hypothesis the values of C/ — F, and U' — V. Then put [826iv]
n=C7— F; n'=£7' — F'. [827]
Lastly we must form a third hypothesis, in which the same perihelion
distance is used as in the first hypothesis, while the time of passing the
perihelion is varied half a day, or a day, more or less. We must find, in this [827']
hypothesis, the values of U — F, and U' — V. Then put
p=U—V', p'=U'-^V', [828]
This being supposed, if we put u for the number by which we must multiply
the supposed variation in the perihelion distance, to obtain its true value ; [828']
and t the number by which we ought to multiply the supposed variation in
the time of passing the perihelion, to obtain the true time ; we shall have
the two following equations :*
{m — n).u + {m—p).t=m', ^^^
{m! — n') .u-\- (mf — p') .t = m';
* (576) Suppose the time of passing the perihelion, and the perihelion distance to be
respectively, in the 6rst hypothesis, T, D ; in the second T, D-\-5; in the third T-j- t, D ;
the true values being T-\-t'r, D-{-u5. By the equations [826, 827, 828], the angle
U — F", which ought to be nothing, was in the first, second and third hypotheses, m, n, and p,
respectively ; consequently the increment <5 in the perihelion distance produced an increase of this
angle from m to n, the variation being n — m, and if the variation of D, instead of being
5, were « S, the variation of the angle U — Fi or m would be nearly u . {n — m), because
these variations, when small, are proportional to the increments. Again, by increasing the time
of passmg the perihelion by t, the angle U — F^, or m is changed into p, increasing by
p — m, therefore if the time of passing the perihelion were increased by tr, the angle m
would be augmented by t .[p — m); hence it appears that by increasing the perihelion
distance by u 8, and the time of passing the perihelion by t t, the angle m will be increased
by the sura of the two quantities u . (n — m) -\-t . {p — m) ; consequendy when the
perihelion distance is D -^uS, and the time of passing the perihelion is T-{-tr, the
angle U — V will become m-\-u .{n — m)-\-t .{p — m) ; and since, by hypothesis,
this corresponds to the true orbit, the angle U — F" must then be nothing ; hence
m-{-u.{n — m)-\-t.{p — m) := 0, or (m — n).u-}-{m — p).i = m,
which is the first of the equations [829] ; the second is obtained in exactly the same manner
4^60 COMPUTATION OF THE [Mec. Cel.
from which we may find the values of u and t, and thence the corrected
perihelion distance ; also the true time of the comet's passing the perihelion.
In making the preceding corrections, it is supposed that the elements
found by the first approximation are so nearly exact that the errors may be
from the values of the angle C/' — V, and it may also be deduced from the first by
accenting w, n, p.
It may be observed that this method of correcting the assumed elements may be generally
used, in similar cases, making those alterations which the nature of the case may require.
Thus, if instead of the perihelion distance and the time of passing the perihelion, we assume,
as Newton has done, in Prop. 42, Lib. 3, Princip. the inclination of the orbit to the ecliptic
and the longitude of the node, the resulting equations for correcting these quantities ought to
be similar to those in [829]. It is however a fact that in all the editions of the Principia
which I have seen, these equations are given inaccurately ; and an attempt has been made by
Le Seur and Jacquier, in the commentary annexed to their edition, to prove these rules to be
correct ; and the same has also been done by Emerson in his " Short Commentary on Sir
Isaac Newton's Principia, Sic." Now if Newton's rules are correct, the equations [829],
must be erroneous, because they are both founded on the same principles. I have therefore
thought it necessary to enter into some explanation of the true rules which ought to be used
in Newton's method, to prevent any embarrassment from the incongruity of the two methods
as they now appear. Newton formed, in the same manner as above, three hypotheses. In
thembiuff the first, the inclination of the orbit was put = /, and the longitude of the node = K; in
the second, these quantities were put equal to J and K-\- P ; in the third, I-\- Q and K;
the true values being supposed /-f-wQ, and K-\-mP. In each of these three
hypotheses, he calculates the ratio of the areas described by the radius vector between the
first and second observation, and between the second and third, and denotes them by
G s y •
— , - , - , or simply by G, g, 7, respectively ; also the times of describing the areas from
the first to the third observation, which are denoted by T, t, r, respectively. Hence by
comparing the results of the first hypothesis with those of the second and third, the increment
P, in the longitude of the node, makes G increase by g — G, and T increase by t — T,
therefore the increment m P in the longitude will make these increments m . {g — G) and
m .{t — T) respectively ; and by comparing the first and third hypotheses, we find that the
increment Q in the inclination causes (rand T to increase by 7 — G and r — T,
respectively ; hence, by proportion, the increment n Q will cause the increments n . (7 — G),
w . (t — T), in these quantities. These increments applied to G and T, give the true
values of these quantities corresponding to the inclination I-\-nQ^, and longitude
K -{- m P, namely, the proportion of the areas will be G -\-m . (g — G) -{- n . {y — G),
and the time of description T-{-7n.{t — T)+w.(t — T). Now the areas are
Newton's
method of
correcting
U. iv. § 37.]
: ORBIT OF A COMET.
461
[829"]
[829'"]
considered as infinitely small. But if the second approximation do not appear
to be sufficient, we may have recourse to a third, using the corrected elements
like those of the first hypothesis, but making the variations less. It is even
sufficient to compute, by these corrected elements, the values of U — V and
U' — V ; for by denoting them by M and M' , vre may substitute them for m
and w', in the second members of the two preceding equations [829] ; we
shall thus have two other equations which will give the values of u and t,
corresponding to these last elements.
Having thus obtained the perihelion distance and the time of passing the
perihelion, we may thence compute the other elements, in the following
manner.
Lety be the longitude of the node which would be ascending if the motion [829*^]
of the comet was direct, and 9 the inclination of the orbit ; we shall have,
by comparing the first and last observation,*
tang, -a . sin. ^' — tang, -a" . sin. ^
tang.;==
* -«ui* - tang. <p =
tang, -a . cos. ^" — tang, -a/' . cos. ^ '
tang. -Bj"
sin.(^"-y)-
[830]
proportional to the times of description, which are known from observation, and by putting
the ratio of the time elapsed between the first and second observation, to that between the
first and third equal to C:l, and the whole observed time from the first to the third
observation = 5, we shall have C= G-\-m .{g — G)-{-n .{y — G), and
S = T-{- m. {t — T) -\-n. (r — T), which by tilinsposition become
G—C = m.{G—g)-\-n.{G — 'r); T—S = m,{T—t)-\-n.{T—'r),
hence m and n may be found. The equations given by Newton are
2G — 2C=m.(G— g)+n.(G — 7), 2 T— 2 S = m . (T— <) + n . (T— t),
which make m and n twice their real values.
* (577) Let rABDhe±e ecliptic, ACE the orbit of
the comet, T the first point of aries,./2 the ascending node, the
motion of the comet being supposed direct, C the place of the
comet at the first observation, and E its place at the last
observation, C B, ED, arcs of latitude. Then fA=j,
Y5 = p, '^D = ^", BC = vi, DE = v/', the
116
462 COMPUTATION OF THE [Mec. Cel.
As we can compare the three preceding equations two by two, it will be
most accurate to select those which give, to the formulas [830], the greatest
numerators and denominators.
Tang, y may either appertain to the angle j or I'+y, j being the least
[830'] positive angle corresponding to that expression ; to determine which of these
must be used, we shall observe that 9 is positive and less than a right angle ;
[830''] therefore sin. (|3" — j) ought to have the same sign as tang. ^'. * This
condition will determine the angle y, which will correspond to the ascending
node, if the motion of the comet be direct ; but if this motion be retrograde,
[830"'] we must increase the angle j by two right angles, to obtain the position of
this node.f
_, - ^ _, - _, ,, , . tang.jBC taxig.DE
r830a] angle BAC = DAE = (p, and by spherics tang. 9=—^ — — —== , hence
tang. B C . sin. w2D = tang. D E . sln.AB, or, tang. is. sin. (p" — j)=tang.zi''.sm.{^ — j).
Putting, for sin. ((s" — /), sin. (p — /), their values, [22] Int., we get
tang, zi . {sin. p". cos.j — cos. p" . sin./} = tang. z/'. {sin. j3 . cos.j — cos. ^ . sin.^j.
[8306]
,..,., . , . sin. J . .
dividing by cos.^, and putting ^. = tang.y, it becomes
cos.^
tang, zi . {sin. p" — cos. ^". tang.y} = tang, ■ci" . {sin. ^ — cos. ^ . tang.j},
hence we easily obtain tang.y, as in the first of the equations [830]. The second of
these equations is the same as the second of the expressions of tang, cp, [830a].
* (578) Having tang. 9 = -^ — — — -, [830], and 9 not exceeding a right angle, its
tangent must always be positive, consequently, tang, •ci" and sin. p" — j must always
have the same sign. Hence, if -s/' be positive ^" — j must be less than two right angles,
but if -a" be negative /s" — j must exceed two right angles, hence the affection of y may be
determined.
f (579) If the comet, instead of moving from C towards E, moved from E towards C,
the first observation would correspond to p", z/', the last to p, zi, and the expression of tang.y,
being found in the same manner as in note 577, from the equation [830&] would be identical
with the expression of tang.y, [830] ; therefore in both cases tang.y must be equal and of
the same sign ; but it is evident that when the motion is from E towards C, or retrograde,
A must be the descending node, and the angle y must correspond to that node, and to
obtain the longitude of the ascending node, we must increasey by two right angles.
n. iv. §37.] ORBIT OF A COMET. ^^S
The hypotenuse of the spherical triangle, whose sides are |3" — j and w", [830'^]
is the distance of the comet from its ascending node at the time of the third
observation ;* and the difference between v" and this hypotenuse, is the
interval between the node and perihelion, counted on the orbit.
If we wish to obtain the greatest degree of accuracy, in the theory of a
comet, we must combine together all the best observations, which may be
done in the following manner. Mark the letters m, n, p, with one accent [830^]
for the second observation, two accents for the third observation, Stc, all of
them being compared with the first observation, we shall have these
equations,!
[831]
(m — n) .u + (m — p) .t = m;
(rnl — n') .u-\- (ml — p') .t^m';
(^m"— n") . u + (m"—p") .t = m";
&c.
Combining tnese equations in the most advantageous manner to determine
u and t, we shall obtain the corrections of the perihelion distance, and the
time of passing the perihelion, resulting from the whole of these observations.
Hence we may deduce the values of f3, f3', |3", &c., w, is', «", &c., and we
* (580) In the spherical triangle A D E, of the figure, page 461, we have the base
AD = ^" — y, and the perpendicular D £ = ■bj", to find by spherics, the hj^otenuse
A E. Then P being the place of the perihelion, we have P E = v", the difference
between this and A E is equal to A P, the distance of the node from the perihelion.
f (581) These equations are exactly similar to those in [829], and require no farther
explanation. It may however be observed, that although this method is simple, it is attended
with the inconvenience, that any error in the first observation affects all the equations ; and if
the second, third, he. observations are very near to the first, and the described arcs very rgsia]
small, the resulting equations may be considerably affected by this circumstance ; moreover,
when the second, third, he. observations are very near to the first, there appears to be as much
propriety in combining them with the subsequent observations, as there is in using only the
first observation. This difficulty may be obviated by computing each observation separately,
with small changes in the elements of the orbit, in the manner which will be more fully
explained in note 591.
464 COMPUTATION OF THE [Mec. Gel.
shall have*
tang. «.{sin.p'-4-sin.^"+&;c.} — sin. p .[tang. •13' + tana;. -51" + Sic?
rggg-j ^' "^ tsug. •sf . j COS. ^'+ COS. ^"4" ^c. j — COS. ^ . | tBug. OT + tang. 'ss" "j" ^c. ! '
tang. ■5/ + tang, ts" + &;c.
tang. 9 = - — ;-, .—-. : ° ■■ „ , : — .
6 ^ Sin. (13' — ;) + sin. {^" —j) + he.
38. There is a case, which however very rarely occurs, in which the orbit
of a comet may be determined rigorously in a simple manner ; this happens
[832'] when the comet has been observed in both nodes. The right line drawn
through these two observed positions then passes through the sun's centre,
and coincides with the line of nodes. The length of this line is ascertained
[832"] by the time elapsed between the two observations ; putting T for this time
reduced to decimals of a day, and denoting by c the proposed right line, we
* (582) The equation [8306], for computing tang.^, using the second observation p', «',
instead of the third ^", -si", becomes
tang, -a . {sin. p' — cos. p' . tang./} = tang, z/ . {sin. p — cos. p . t^ng.y},
and in a similar way, by using $"', zi"\ instead of p", -a/', we obtain
tang, zs . { sin. p'" — cos. ^"' . tang.j } = tang, s/" . { sin. (3 — cos. ^ . tang./ ] ,
and other observations give similar expressions. By adding all these equations together we
obtain the following,
tang. Ttf . { sin. ^' + sin. ^" + &^c. } — tang, zi . tang.jf . { cos. ^' + cos. p" + Sic. }
= sin. (3. {tang.'!/ + tang.'cj"-|-&£c.| — cos. p .tang.y. {tang. ■5/+ tang. -n" + ^^c.},
which gives for tang.y the expression [832]. Again, the second equation [830], becomes
by using the second observation, (3', ^, instead of the third ^", t^',
tang, xs' = tang. 9 . sin. (p' — j),
and in a similar manner,
tang. Tz" = tang. <p . tang, {f — j) ; tang, vs"' = tang. 9 . tang. ((3'" — j), he.
These equations being added together, which makes each of them enter into the determination
of 9, we get,
(tang. -5/ + tang, ss" + Sic.) = tang. 9 . { sin. (^' — y) + sin. (^" — y) -|- &;c. | ,
hence the second of the equations [832] is easily obtained. This method of combining the
equations to find tang, j and tang. 9, is somewhat arbitrary, since the first observation is
connected with all the others, an arrangement which may sometimes not be conducive to the
attainment of the most accurate result.
[834]
[834']
[835]
fl. iv. § 38.] ORBIT OF A COMET. 465
shall have by ^ 27,*
Now let |3 be the heliocentric longitude of the comet at the time of the
first observation, r its radius vector, p its distance from the earth, and a its [833]
geocentric longitude ; also R the radius vector of the earth, and E the
corresponding longitude of the sun at the same instant, we shall havef
r . sin. ^ = p . sin. « — R. sin. E ;
r . cos. /3 = p . cos. a — R. cos. E.
flr-ffS will be the heliocentric longitude of the comet at the second observation;
and if we accent the quantities r, a, p, i2, E, corresponding to this time, we
shall have
r' . sin. ^ = R . sin. E' — p' . sin. «' ;
/ . COS. ^= R'. cos. E' p' . COS. a.
* (583) In this case r-{-r', [750] is evidently equal to c, because r, /, fall on the line
T i
of nodes, and that equation, [750], becomes T = -r— . (2 c)^ ; and by [750'] we have
T
= 9*^,688724, substituting this we get c [833].
f (584) These values are easily found by means of the
adjoined figure, in which S is the centre of the sun, B the
place of the comet situated in the ecliptic, and D that of the
earth at the first observation, C the place of the comet,
at the second observation, S H the line drawn from the sun
through the first point of aries, fi-om which the angles a, p, &£c.
are counted ; DF^ B E H Are perpendicular to S Hj and
DE parallelto SH. Then H S B = ^, SB=^r, SD = R, BD=:p, the angle
HSD= longitude of the earth = * -j- £, and the angle EBB equal to the geocentric
longitude of the comet a. Then m the triangles B S H, BDE, D S F, we have
B H= S B. s'm. H SB = r. sin. ^', S H= S B .cos. HS B = r .cos. ^;
BE = BD. sm. EDB= p . sin. a j DE=BD. cos. EDB=:z p . cos. a ;
DF=SD.sm.HSD = R.sm.{'^-{-E) = — R.sm.E;
SF=SD. COS. HSD = R. cos. (* + £) = — i? . cos. E.
Substituting these in BH=BE + DF, SH=DE-\-SF, they will become as
in [834]. When the comet is at C, its heliocentric longitude is evidendy * + p; substituting
this and changing p, r, &;c. into p', /, he. in [834], they will become, by changing the signs
of all the termsj the same as [835].
117
4*66 COMPUTATION OF THE [Mec. Cel.
These four equations give*
r-,„„^ ^ p. sin. a — R.sm.E o' . sin. a' — R' . sin. E'
[836] tang, p = = ;
° p.cos.a — R.cos.E p'.cos.a' — R' .cos.E'
hence we deduce
[■837] ,_ RR'. sin. {E — E') — R' p. sin, (a — E')
p . sin. (a — a) — jR . sin. (a' -r- E)
We then havef
(r + r') . sin. |3 = p . sin. « — p' . sin. «' — R. sin. E -\- R . sin. E' ;
(r + r') . cos. |3 = p . COS. « — p' . cos. a — i? . cos. jE + -R' • cos. E'.
By adding together the squares of these equations, and substituting c for
r + r', we shall have J
c'=:R — 2RR. COS. (E^E') + R'^
+ 2p.{R' . COS. (u — E') — R. COS. (a— E)]
+2p'.{R . COS. (c^'—E) — R . COS. (u' — E')]
+ p2 — 2pp'.C0S. (a' — «) + p'2.
[838]
[839]
* (585) These two values of tang. ^ are found by dividing the first equations by the
second in [834, 835], respectively. Putting these two expressions of tang, p equal to each
other, and multiplying by the denominators, we get
Ip.sin.a — jR. sin. £|.{ p'.cos.a' — ^72'.cos.jE'}=f p'.sin.a — 72'.sin.E'}.{p.cos.a — R.cos.El;
performing the multiplications, and connecting together the coefficients of p p', R p', jR R,
R' p, it becomes
RR . (sin. E . cos. E' — sin. E' . cos. E) — R' p . (sin. a . cos. E' — cos. a . sin. E'l
= p p' . (sin. a! . COS. a — cos. a' . sin. a) — R p' . (sin. a . COS. E — COS. a . sin. E),
which being reduced, by [22] Int. changes into
RR.s\n.{E — E')—Rp.sm.{u — E') = pp'.sm.{ct' — a) — Rp'.sm.{a' — E).
Dividing this by p . sin. (a — a) — R. sin. (a' — E), we obtain p' [837].
f (586) The first of the equations [834, 835] being added together, we get the first of
[838] ; and the second of the equations [834, 835] being added together, we get the
second of [838].
f (586a) Putting c for r + Z in [838] and taking the sum of the squares of both
equations, the first member will become c^ . (sin.^ p -|- cos.^ ^) = c^, as in [839]. In
squaring the second member of [838], there will be two species of terms, the one composed
of the squares of p, — p', — R, R, and the other of tlie double of the products of these
quantities two by two.- The former will, in the square of the first equation, produce terms of
n. iv. §38.] ORBIT OF A COMET. 467
Substituting in this equation the value of p' in terms of p given in [837], the
result will be an equation in p of the fourth degree, which may be solved
by the usual methods ; but it will be easier to assume for p any value at
pleasure, and to compute the corresponding value of p' [837] ; then we must [839']
substitute these values of p, p', in the preceding equation [839], and see if
they satisfy it. By a few trials, in this manner, we may determine p and p'
with accuracy.
By means of these quantities we shall obtain ^, r, /. Put D for the
perihelion distance, v for the angle included by the line D and the radius [839"]
vector r ; n — v will be the angle formed by the lines D and r' ; we shall
then have, by § 23,*
B^_ , _ B
which give
tang-H^ = 3; ^=7ih' [841]
r" r-\-r''
We shall therefore have the anomaly of the comet v, at the time of the first
observation, and the perihelion distance D ; thence it is easy to deduce the
the form (p . sin. a)^, which will be accompanied, in the square of the second equation, by
a term of the form (p . cos. a)^, the sum of these two terms will be p^, and the other
similar terms will produce p'^, jR^, R'^, [839]. In the second species of terms the double
product depending on p, — p', will produce in the sum of the squares of the two equations
[838], the quantity — 2 p p' . (cos. a' . cos. a -j- sin. a' . sin. a), which, by [24] Int.,
is = — 2 p p' . cos. (a' — a), as in [839]. In like manner we obtain the terms depending
on —2 pi?, 2pR, 2p'i2, — 2p'/2', -^2RR.
* (587) Let SP, in the figure page 465, be the perihelion distance, we shall have
BSP = Vj hence CSP = ir — v. The first equation [691] gives r= — iT" >
and / = -- — — — -= . „. , as in [840]. Dividing the value of r by that of r',
cos.2i.(^ — v) Bin.^hv L J O J J
V sin ^ n H
we get — = — YjF =^ tang.^ | v, [841]. The sum and product of r, /, are
, , _, sin.2Ji» + cos.2ir D . , Ifi
r-\-r^ = D. . .^^_ __^^^_ =.:_o.„ — oTTr» and rr' = -
sin.2 ^ V . C0S.2 i V sm2 ^ v . cos.2 i v ' sin.2 i v . cos.2 i v
dividing the latter by the former we get D, [841].
468 COMPUTATION OF THE [Mec. Cel.
, position of the perihelion, and the time of the comet's passing that point.
Thus, of the five elements of the orbit, four will be known ; namely, the
perihelion distance, the position of the perihelion, the time of passing the
perihelion, and the position of the node ; the only element which remains
to be investigated is the inclination of the orbit ; and for this purpose it will
[841"] be necessary to recur to a third observation, which will also serve to
determine which of the real and positive roots of the equation in p is to be
used.
39. The hypothesis of the parabolic motion of comets is not perfectly
correct ; the probability of it is even extremely small, considering the infinite
[841'"] number of cases producing an elliptical or hyperbolic motion, in comparison
with those producing a parabolic. Besides, a comet moving either in a
parabola or hyperbola, would be visible but once ; hence we may suppose,
with great probability, that the comets describing these curves, if there be
any, have disappeared a long time since ; so that those we now observe, are
such as move in returning or oval curves, which, at greater or less intervals
of time, come back to the regions of space near the sun. We may, by
[84iiv] the following method, determine within a few years the duration of the
revolution, when we have a great number of very accurate observations,
before and after passing the perihelion.
For this purpose, suppose we have four or a greater number of good
observations, including all that part of the orbit in which the comet was
visible, and that we have found by the preceding method the parabola
[841 V] which nearly satisfies these observations. Let v, v\ v", v'", &c., be the
corresponding anomalies ; and r, /, r", r'", &c., the radii vectores. Put also
[842] v'—v=U', v" — v=U'; v"' — v=U"; &c. ;
this being supposed, we must calculate by the preceding method, with the
parabola already found, the values of U, U', U% Stc, F, F, V% &c. ;
then put
[843] m = U—V', m'=C7' — P; w!'=U"—V"; nf'=U"'-^V'"; &c.
We must then vary by a very small quantity, the perihelion distance in the
parabola ; suppose in this hypothesis
[8443 n=U^V; n'=U' — V'; n"=U"—V"; n"'^U"'—V"'; &c.
We must then form a third hypothesis, in which we must preserve the same
n. iv. § 39.] ORBIT OF A COMET. 469
perihelion distance as in the first, and vary the time of passing the perihelion
by a very small quantity ; then putting
p=U—V; p'=U' — V'; p"=U" — V"; f'=U"'^V"'; &c. [845]
Lastly, with the perihelion distance and the time of passing the perihelion
of the first hypothesis, we must compute the angle v and the radius vector r,
supposing the orbit to be elliptical, and the difference 1 — e between its
excentricity and unity to be a very small quantity, for example g^. To obtain [845']
the angle v, in this hypothesis, it is sufficient, by § 23 [697], to add to the
anomaly v, computed in the parabola of the first hypothesis, a small angle,
whose sine is
■3^.(1 — e) .tang. ^2J.{4 — 3. cos.^ ^ « — 6 . cos.'*|i>}. [846]
Substituting, in the equation*
for V, the anomaly calculated in this ellipsis, we shall have the radius vector '
r corresponding. We must compute in the same manner, v', r', v'\ r", v"\ r'",
8tc. ; hence we may deduce the values of C/, U\ U", &c. ; and by § 37, [847^]
those of V, V, V", &c. Suppose in this case
q=U—V; (/=U'—V'; q"^U"—V"; ^"=U"'—V"'; &c. [848]
Lastly, let u be the number by which we ought to multiply the supposed
variation in the perihelion distance, to obtain its true value ; t the number by
which we ought to multiply the supposed variation in the time of passing the [848Q
perihelion to obtain the true time ; and s the number by which we ought to
multiply the supposed value of 1 — c, to have its true quantity ; we shall
form the equations!
* (589) This is the same as the equation [683], neglecting a^, which reduces it to
and substituting 1 — e for a, [681'].
t (590) The values of [843, 844, 845], are precisely like those in [826, 827, 828].
Those of q, 9', &c. [848] depend on the same principles, and it is evident that the equations
[849] are found like those in [829].
118
4^70 COMPUTATION OF THE [Mec. Cel.
[849]
(m — n) . u -\- (m — p) . t -\- (m — q) . s=m ;
(m' — n'). u + (m' — p').t + (m'—^),s==m';
(m"—n") . u + (m" —f) . t + (m" — q") .s = m";
(m!"—n"') . u + {m"'—p"') . t + (m'"— /') . s = m!" ;
&c.'
By means of these equations we may determine the values of u, t, s ; hence
we may deduce the perihelion distance, the true time of passing the perihelion,
and the correct value of 1 — e. Let D be the perihelion distance, a the
[849'] semi- transverse axis of the orbit ; we shall have a = [681""! ; the
1 — e '-
time of a sidereal revolution of the comet will be expressed by a number of
[849"] sidereal years equal to* a^, or (- Y, the mean distance of the sun
* (591) From [705] we have T^: T'^::a^ :a'^ and by putting T'^lyear,
■• a' = the unity of distance, or the mean distance of the earth from the sun, we find
T^ = a^, or T=a^.
The system of equations [849] is liable to the same objections that were made to the
system [83 J] in note 581. To obviate this the following metliod may be used.
Let the approximate elements of the orbit be the perihelion distance D, the time of
passing the perihelion T, the longitude of the perihelion counted upon the orbit of the comet
P, the longitude of the ascending node of the orbit JV, the inclination of the orbit of the
ecliptic /, the excentricity expressed in parts of the mean distance 9f the comet from the
sun E, this last element being omitted when the comet is supposed to move in a parabolic
orbit. With these elements, we must, for ajirst operation, calculate the geocentric longitude
and latitude of the comet at the time of any observation. The same calculation must be
repeated in six successive operations, varying one of the elements at each operation, by some
small quantity, while the others remain unaltered. In the second operation, the distance D
must be changed into D -{- d, d being a very small part o( D ; T must be changed into
[849a] T-\-t in the third operation, t being a fraction of a day ; P into P -{-p, in a fourth •, JV
into JV-f- n, in ^ fifth ; /into /+ h in the sixth ; p, n, i being small arcs or parts of a
degree. Lastly, if the ellipticity of the orbit be taken into consideration, we must for a
seventh operation, change E into E -{-e, e denoting a very small increment of the
r84961 excentricity E. Then representing the longitudes, or latitudes, computed in these successive
[849c] operations, by L', L", L'", L"", L^, L^'^, L"'", and the corresponding observed longitude or
latitude by Z/, and supposing the true elements to be D-\~d§, T -\- 1 r, P-^pir,
JV_j_,ir, I-{-ii, E-\-eSj each observed longitude or latitude will furnish an equation
471
[849d]
n. iv. <^39.] ORBIT OF A COMET.
from the earth being taken for unity. We shall then have, by ^ 37 [832],
the inclination of the orbit 9, and the position of the node j.
of this form, which was computed upon the same principles as those in [829, 849],
explained in note 576.
0 = {L — L')-ir{L'—L").S-\-{L'—L"').r + {L'^L"").',f
+ (L'— L^).v + {L— L-) . c + {L— V^) . £,
so that n observations of the comet will produce 2 n equations, each of which will be
independent of the others. Gauss, in his invaluable work, Theoria Motus Corporum
Coelestium, has given many differential formulas, by means of which the variations of the
geocentric longitudes and latitudes of the comet, corresponding to small variations in the
elements of the orbit, may be computed without the trouble of repeating the whole calculation
of the longitude and latitude, at every operation ; and, by this means, the equations of the
form [84 9<^], may be found with much less labour than by a direct operation. Bessel, in his
excellent work on the comet of 1807, entitled Untersuchungen iiber die scheinbare und
wahre Bahn des im Jahre 1807 erschienenen grossen Kometen, gives several of the formulas
of Gauss, with additional ones of his own, for the purpose of abridging such calculations.
Both these works deserve the careful perusal of any one who wishes for full information on
this subject. If there are only six of the equations of the form [849d], they will be just
sufficient to obtain the unknown quantities S, r, he, and thence the corrections of the
elements. If the observations of the comet were accurate, tlie orbit a perfect ellipsis, and
the variations of the elements infinitely small, all these equations, however great the number
might be, would be satisfied, by using these corrected elements ; but the imperfections of
the observations, and the finite nature of these variations, with other causes, generally prevent
this from taking place ; and the second member of any one of the equations, instead of [849e]
vanishing, becomes in general equal to a small quantity c', which may be considered as the
correction, or error of the particular observation, from which the equation was derived ; so that
by putting for brevity L—L'=A', L'— L"= B, L'—L"=C', L'—L""=U,
U— L' = E ', U— L"^ = F', L' — Z,'" = G^ ', the preceding equation will become,
c' = A'-i-B6-{-C'r-]-U'>r-\-E'v-{-F'i-i-G's. [84%]
If we have more than six of these equations, we must combine them together, so as to make
the sum of the squares of the errors c. a minimum. Before using this method it will often
be conducive to the accuracy of the result, to examine carefully the observations, and if any
of them are considered to be more imperfect than the rest, as might frequently be the case
with the observations made just before the time of the disappearance of the comet, when it is
very faint ; such observations may be made to have less influence, on the final result of the
calculation, by multiplying the equation, computed as above, by some fraction, less than unity? [849A]
as I, J, J, &«;., or by rejecting it wholly, if it shall be found to diiFer very much from the
rest. Moreover, we ought to multiply the equations derived from the observed longitude by
[849/]
472 COMPUTATION OF THE [Mec. Cel.
However great the accuracy of the observations may be, they will
always leave a degree of uncertainty on the time of revolution of a
the cosine of the corresponding latitude of the comet, in order to reduce the diifference of
longitude L — L', to the parallel of latitude of the star, so that it may correspond to the
actual arc, described by the comet in the heavens. This will appear, by referring to the
figure in page 216, supposing jE Q g' to represent the ecliptic. Pits pole, a the observed
place of the comet, and B its computed place, at the first operation ; then we shall have
L' — L equal to the arch Q q, and the arch ab or A B^ corresponding to the actual change of
place of the comet, in its parallel of latitude, will evidently be nearly equal to
Q q . COS. Q^A= (L' — L) . COS. lat.
In this way, by accenting the letters A', B', C, &tc. for the successive equations, we shall
obtain 2 n equations of the following form,
c' = ^' + 5' 5 + C ' T + D' * + E ' r + F' ^ + G ' f,
c"=^"+S"(5+ C" r-{-D"if + E" y -^ F" i-j- G"s,
[849i] cf"=A"'-\-B"6-]-C"''r-\-D"'ir+E"'v-{-F"'i-{-G"'s,
which must be combined together so as to make c'^ + c''^ + ^"^ . . . . + c^^"^^ a minimum
in the following manner. First, Multiply each of the equations [849«] by the coefficient of
6 in that equation, and take the sum of all these products for the first final equation. In
[849A;] other words, the first equation is to be multiplied by B*, the second by B", the third by
B'", he, always noticing the signs of these terms, so that if B' = — 3, the factor of the
first equation must be — 3. Second, in like manner multiply the same equations [849r] by
C, C", &c., the coefficients of T, corresponding to each equation; and take the sum of these
products, for the second final equation. Third, Muhiply the equations [849i] by jy, U', fee,
and take the sum of the products for a third equation. Proceeding in the same manner,
with the coefficients of "v, ', ?, we shall obtain three other equations, making in all six final
[849i] equations, from which the values of (5, t, *, r, <, s may be computed, by the usual methods.
In the parabolic orbit s must be neglected, and the number of final equations will be reduced
to five.
It is very easy to prove, that the method just given corresponds to the minimum value of
d^ -\- c"^ -\- &;c. For if we denote all the terms of the second members of the equations
[849i], independent of 5, by M', M", Sic. we shall have,
[849m] c' = B'6-irM', d' = B"8-\-M", c" = B'" 6 -^ M'" , he.
hence
[849n] c'2+c"^ + &c. = (B'^ + Jf')2 + (^"5 + J»/'7 + &c.
The minimum of this quantity, supposing 5 to be variable, is found in the usual manner, by
putting its differential relative to (5, equal to nothing. This differential being divided by 2 <? 5
Principle
of the least
squares.
ll.iv. §39.] ORBIT OF A COMET. ^73
comet. The most exact method of determining this time, is by comparing
the observations in two successive revolutions ; but this method is not [849"']
becomes
B'.{B'6-\-M') + B'.{B"S + M")+hc.=0. [849«>]
which is exactly the same as in the rule, given in [849^], for finding the first final equation.
The demonstration, relative to the other unknown quantities, is made in exactiy the same
manner, and it holds good, whatever may be the number of these quantities.
In case the number of observations is very great, this method would be too laborious, if
the calculation were made separately for each observation. This difficulty is avoided by
dividing the observations into five or six groups, comprising the observations of several [84S*p]
successive days, and using only the middle day of each group, correcting the observed
longitude on that day for the mean error of all the observations of the group, to which h
corresponds, as they were computed from the original elements, in the terms A', Ji\ &ic. of the
equations fS49i]. Bessel, in his work on the comet of ld07, combined 70 observations
in six different sets, each furnishing one equation for the longitude, and one for the latitude, [849?]
and the twelve equations thus obtained were reduced to six, by the method of the least
squares [849i — T\. The variations of the elements used by him, in his last calculations,
after he had obtained the elements to a great degree of accuracy, were c?:^ 0,0001,
t = 0''''",005, p = 1 0^, n = 1 0% i = 1 0% e = 0,000 1 . In the calculation of the
elements of the orbit of the comet of 1811, in the third volume of the memoirs of the
American Academy of Arts and Sciences, I used c?:^ 0,004, ^ = 0'^'',05, p = 10*,
n = — 10'", 7:=I0'».
In strictness, the observed longitudes and latitudes of the comet should be corrected, for
the perturbations caused by the attraction of the planets, before insertion in the equations
[849t]. If the comet should pass near to any one of the larger planets, it would be
absolutely necessary to notice this circumstance, and it is always conducive to accuracy to
doit. La Place has given a method for this purpose, in Book IX, § 1 — 13. The same [849rl
subject is also treated of by Bessel, in the above mentioned work, where he has given many
useful formulas, with their application to the comet of 1807. In vol. xxiv of the Memoirs
of the Royal Academy of Sciences of Turin, is an elaborate article, by Baron Damoiseau,
on Halley's comet of 1759, in which he computes the disturbing forces of the planets from
1759 to 1835, and fixes the time of passing the perihelion on November 16, 1835. Any
one who wishes to know, in detail, the methods of making such calculations, would do
well to refer to these works of Bessel and I)an)oiseau.
It has been observed, both by Gauss and Le Gendre, that the rule for taking the mean of
any number n of observations, follows as a simple result from this general method of the least
squares. For if a', a", a'", &;c. represent several observed values of an unknown quantity x,
119
474 COMPUTATION OF THE ORBIT OF A COMET. [Mec. Cel.
practicable, until the comet in the course of time, shall return back again
towards its perihelion.
the sura of the squares of the errors will be {x — a')^ -h (■^ — «")^ ~\~ {^ — «")^ + ^c.
Its minimum is found by taking the differential relative to x and putting it equal to nothing.
This differential being divided by 2dx, becomes
[849*] (x — a')-{-{x— a") + {x— a'") + Sic. = 0,
I. / I " I //' I 0 J a' + a''' + a'" + &c.
hence nx = a -\- a -f-a" -j- Sic. and x = . In like manner,
n '
if the rectangular co-ordinates of a point in space be x, y, z, and by one observation they be
a!, I/, c ; by another a", h", c", &;c. ; then the square of the distance of the point x, y, z,
from the point a', h', c', will be {x — a')^ -\- {y — 6')^ -\-{z — c')^, [12]. This represents
the square of the error of the first observation ; and by a similar calculation, that of the
second is {x — a"Y -\-{y — b"f -\-{z — c")^ + &;c. The sum of all these, using the
[849f] symbol 2 of finite integrals, as in page 9, is 2 . (a; — a')^ + ^ • {v — ^T ~\~^ - {^ — c')^,
which is to be a minimum ; therefore its differential, taken successively, relative to x, y, z, is
to be put equal to nothing. Its differential relative to a: is 2 c? a; . 2 . (a; — a')=0, hence
2 . (a? — a') = 0, and 2 a; = 2 a', and if the number of points be n, Sx will be the
same as n x, hence w a? = 2 a', and the differentials relative to y and z, will give similar
2o' 2 &' 2 c'
[849«] expressions, hence we get x = — , y = — , z = — ; these formulas are the
same as those in [154], for finding the common centre of gravity of n equal masses, situated
at given points, each mass being represented by m := 1 ; therefore we find that the common
of the centre of gravity of any body has this general property, pointed out by Le Gendre. ^ we
C6ntr6 of
gravity, divide the mass of a body into very small equal particles, considered as points, the sum of the
[M9v] squares of the distances of the particles from the centre of gravity, will be a minimum.
n.v. '^40.] MOTIONS OF THE HEAVENLY BODIES. ^75
CHAPTER V.
GENERAL METHODS FOR FINDING THE MOTIONS OF THE HEAVENLY BODIES, BY SUCCESSIVE
APPROXIMATIONS.
40. In the first approximation of the motions of the heavenly bodies, we
have only considered the principal forces which act on them, and have thence
deduced the laws of the elliptical motion. In the following researches we
shall notice the forces which disturb this motion. The effect of these forces
is to add some small terms to the differential equations of the elliptical [849iv]
motion, of which we have already given the finite integrals : we must now
determine, by successive approximations, the integrals of the same equations,
increased by the terms arising from the effect of these disturbing forces.
The following is a general method for obtaining such integrals by successive
approximations, whatever be the number and the degree of the differential
equations proposed to be integrated.
Suppose we have, between n variable quantities y, i/, if^ &c., and the [849^]
variable quantity t^ whose element dlvs, considered as constant, n differential
equations
0 = l^^f + i> + ».Q;
d'v t850]
0 = 7F + ^ + «-«''
&C.
P, Q, P', Q', &c., being functions of t, y, y, &c., and of their differentials
as far as the order i — 1 inclusively, and a being a very small constant [850']
coefficient, which, in the theory of the heavenly motions, is of the order of
the disturbing forces. Suppose also that we have the finite integrals of these
equations, when Q, Q', &c., are nothing. Then taking their differentials
416 MOTIONS OF THE HEAVENLY BODIES, [Mec. Cel.
i — 1 times in succession, they will form, with their differentials, in equations,
by means of which we may find, by elimination, the arbitrary quantities c, c',
c", &c., in functions of t, y, ij, y", &c., and of their differentials as far as the
[850"] order i — 1. Denoting therefore these functions by V, V, V", &c., we
shall have
[851] c = V; c' = y'; c"=V"; ' &c.
These equations are the in integrals of the order i — 1, which the differential
equations ought to have, and which, by the elimination of the differentials of
the variable quantities, give their finite integrals.
If we take the differentials of the preceding equations [851] of the order
i — 1 , we shall have
[852] 0 = dV; 0 = dV'; 0 = d V" ; &c.
Now it is evident, that these equations being differentials of the order i,
without arbitrary constant quantities, they must be the sums of the following
equations :
^^ 0=1^ + ^= ^-IF + P"' ^^-^
multiplied respectively by such factors as will render the sums exact
[853'] differentials;* putting therefore Fdtj F' dt, &c., for the factors to be
* (592) By hypothesis [850"], V, V, &;c., contain no differentials of a higher order
than i — 1. The differentials of these quantities cannot, therefore, contain any differentials
of a higher order than i, and the terms of this order must ap| ear of the first degree, or under
[852a] a. linear form. For if ^ contain a term of the form A . {d'~^y)"', its differential dV
would contain the term Am . [d^~^ y)^~^ • d'y, which is of the first degree, as it respects
d' y. Again, the integration of the equations [853] has produced the i n equations of the
form [851], containing the i n constant quantities c, d, c", &;c. If we now take the
differential of these last equations, we shall get the in equations [852], which are of the
same order i as the equations [853], from which they were derived ; but they will contain
none of the constant quantities c, c', &c., which were introduced by the integrations. The
equations [852] must therefore be deducible from the equations [853], by the common
rules of elimination in algebra. But we have just shown [852a], that d V, d V\ &ic., in
[852], contain c?'y, d y', &tc. only under a linear form, as they are in [853]. Therefore,
if we multiply the equations [853] by factors F, F', &xj., H, H', he, of an order not
exceeding i^l, and add them,, as directed above, the sums will be of the same fornis as
II. V. § 40.] FOUND BY SUCCESSIVE INTEGRATIONS. 477
used in forming the equation 0 = dV; also Hdtj H'dt, &c., for the
factors to be used in forming 0 = dV\ and in like manner for the rest ; we [853"]
shall have
dV'^H.dt,\^+p]+H'.dt.\^}^,+ F] + ^c.;
[854]
&C.
F, F', &c., H, H', &c., are functions of t, y, ?/, y", &c., and their differentials [854']
as far as the order i — 1 : it is easy to determine them, when F, V, &c.,
d* y .
are known ; for F is evidently the coefficient of -,— 7? in the differential of
V ; F' is the coefficient of -^, in the same differential, and so on for the [854'^
Others. Likewise H, H', &c., are the coefficients of —-4, -7^* &^c., in [854^
af* at*
the differential of V ; and since the functions V, V, &c., are supposed to be
d^~^y d^~^y'
known, if we take their differentials relative to r— ly* - ._^ , &c., only, [854^^]
we shall have the factors by which we ought to multiply the differential
equations,
" = ^^ + ^= « = 7F+^' ^'^■■^ ^"^
to obtain exact differentials. This being premised.
We shall now resume the differential equations [850],
^ = ^ + ^ + "-^' ^ = 7^ + ^ + "-^5 &C. [856]
If we multiply the first by Fdt^ the second by F'dt^ and so on for the rest ;
we shall have, by adding these products,*
0 = dV+adt.{FQ-^F'q + hQ.\', [857]
the equations [852], and they may be rendered identical by using the appropriate values of
F, F', &;c., requisite to make these sums exact differentials of the form d V= 0,
dV = 0, he. [852].
* (593) Substituting also in the sums, d V, d V^ &£C., instead of the equivalent
expressions, given in the second members of the equations [854].
120
478 INTEGRATION OF [Mec. Cel.
[858]
[859]
in like manner we shall have
O^dV' + udt.lHQ + H'Q + kc.} ;
&c. ;
hence by integration
c — a.fdt.{FQ + F'Q'-{-kc.]=V;
c — a.fdt.{HQ + H'Q'+kc.} = V';
&c. ;
thus we shall have, i n differential equations, which will be of the same form
as in the case where Q, Q', &c. are nothing, [851], with this difference only,
that the arbitrary quantities c, c\ c", &c., ought to be changed into
[860] c—ai.fdt,{FQ+F'Q!-^kc.\ ; d^a.fdt.{HQ-\-H'q-^kc.] ; &c.
Now if in the hypothesis of Q, Q', &c., being equal to nothing, we eliminate
from the in integrals [851] of the order i — 1, the differentials of the
variable quantities ?/, y\ ij\ &c., we shall have the n finite integrals of the
proposed equations ; we shall therefore have these same integrals, when Q,
Q', &c., do not vanish, by changing in the first integrals, c, c', &c., into
[secy] c— «./</^{jPQ+F'Q'+&c.}; c'^a.fdt.{Hq-^H' Q'-\-kc,]', &c.
41. If the differentials
[861] dt.{Fq + F' q-^&Lc], dt.lHQ + H' Q' + Sz.c.\, &c.,
be exact, we shall have, by the preceding method, the finite integrals of the
proposed differential equations [856] ; but this takes place only in some
particular cases, of which the most extensive and interesting is that in which
the equations are linear. Let us therefore suppose P, P', &c., to be linear
functions of y, «/, &c., and of their differentials as far as the order i — 1
[861'] inclusively, without any term independent of those variable quantities ; and
we shall, in the first place, consider the case in which Q, Q', &c., are nothing.
The differential equations being linear, their successive integrals will also be
[861"] linear, so that c = V, c'=V', &c., being the in integrals of the order i — 1,
of the linear differential equations,
[862] o = ~K+P; 0 = ^ + F; &c. ;
V, V\ &c., may be supposed linear functions of y, y', &-c., and of their
differentials as far as the order i — 1 . To prove this, suppose in the
n. V. Ml] LINEAR EQUATIONS. ^ 479
expressions of y, y, &c., the arbitrary constant quantity c to be equal to a
determinate quantity, augmented by an indeterminate constant quantity Sc ;
the arbitrary constant quantity c' to be equal to a determinate quantity,
increased by the indeterminate <5c', &,c. Reducing these expressions into
series, arranged according to the powers and products of ^ c, ^ c', &C., we
shall have, by the formulas of § 21,*
[863]
[863']
&c. ;
Y, Y'f ("T- )? &c., being functions of t without arbitrary constant
quantities. Substituting these values in the proposed differential equations,
it is evident that S c, &c', &c., being indeterminate, the coefficients of the
first powers of each of them ought to be nothing in all these equations ; now
these equations being linear, we shall evidently have the terms affected by
the first powers of (5 c, ^c', &c., by substituting I— )'^c-\-(-ri ) •'^c'-h &c.
for y ; ( -r— ) . ^ c -f f -— - j . 5 c' + &c. for xf ;t &c. These expressions
* (595) The formulas here referred to are [607 — 612]. The general expression of
?».»',«" to [612], corresponds to the coefficient of 5c". ^t/"'. (5c"""&tc. [863], putting
5 c, 5 c', 5 c", &tc. for a, a', a", &c., respectively.
f (596) Since 5 c, (5 c', &tc. are arbitrary, we might put 5 c = 0, 5 </ = 0, in [863],
and the resulting values of y, y, &JC., namely, y=Y, y' = Y', &c., would satisfy the [863o]
proposed equations [862] ; the same equations being likewise satisfied, by substituting the
whole values of y, y, &«. [863]. If we now suppose for a moment, that 5 c, 5 c', Stc. are
infinitely small, we may neglect the powers and products of 5 c, 5 c', &c., in [863], and put
simply.
480 INTEGRATION OF [Mec. Cel.
of y, ^, &c., therefore satisfy separately the proposed differential equations ;
and as they contain the i n arbitrary quantities 5 c, <5 c', &c., they are the
complete integrals of them. We thus see that the arbitrary constant
quantities exist in a linear form in the expressions of i/, i/, &c., consequently
[863"] also in their differentials ; hence it is easy to conclude that the variable
quantities y, y, &c., and their differentials, may be supposed to exist under
a linear form, in the successive integrals of the proposed differentials.
and as the proposed equations [862] are linear, and are satisfied by putting ?/ = Y",
if = Y\ &;c., [863a], they must, from the nature of linear equations, be also satisfied, by
putting for y, y', he. the differences of their two preceding values respectively, that is by
putting
[8636]
[863c]
Again, as the proposed equations [862], are linear, in y, y'. Sec, containing no constant term
[861'], independent of y, y', &lc., the last values of y, y', &ic. [8636], will also satisfy the
equations, if they are all multiplied by a very great constant quantity C, making
Hence it is evident, that for the quantities C 6 c, C 5 c', he, we may put arbitrary finite
quantities e, e', e", he. ; making
in which the arbitrary constant quantities e, e', &z;c., are under a linear form, as in [863'],
where 5 c, 6 c', &ic., are used for e, e', &c.
Now having n quantities y, y', y", Sec, expressed in these functions of t, [863c], and the
in indeterminate constant quantities e, e', e", &c. If we take successively, the differentials
of these expressions, as far as the order i — 1 , we shall obtain i n equations, which will be
linear in e, e, e", &c. ; and they will also be linear in y, dy, d^ y, he, y', dy', he, and
[833rf] by eliminating all the arbitrary constant quantities except e, we shall find e = linear function
of y, y', &ic., and their differentials as far as the order i — 1 . In like manner by eliminating
all these constant quantities except e', we shall find e' equal to a similar function of y, y', &c.,
and their differentials. In this manner we shall have in equations, which will give the values
of e, e', e", &;c., in linear functions of y, y', &z;c., and their differentials, as far as the order
t— 1. These correspond to c=F, c' = F', Stc, [861"].
n. V. § 41.] LINEAR EQUATIONS. 481
Hence it follows, that F, F', &c., being the coefficients of -^4-, -r4- ,
&c., in the differential of F; H, H', &c., being the coefficients of the same
differentials in the differential of V, and so on for others ; these quantities
will be functions of the single variable quantity ^.* Therefore, if we suppose [863'"]
Q, Q', &c., to be functions of t only, the differentials
dt.{FQ + F'Q+kc.], dt.{HQ+H'Q+kc.}, &c., [863-]
will be exact.
From this we obtain a simple method of finding the integrals of any number n
of linear differential equations of the order i^ containing any terms aQ, aQ', &c.,
functions of the single variable quantity t, when we knoio how to integrate the [863 »]
same equations in the case ivhere these terms vanish ; for in this case, if we
take the differential of these n finite integrals i — 1 times in succession, we
shall have i n equations, which will give, by elimination, the values of the
in arbitrary quantities c, c', &c., in functions of t, y, y, &c., and of the
differentials of these variable quantities as far as the order i — 1. We may
thus form the in equations [861"], c^ F, c'= V, he. ; this being supposed,
F, F', &c., will be the coefficients of .J[ , -r— if , &c., in F, [854"] ; [sesvn
H, H\ &c., will be the coefficients of the same differentials in V [854"],
and so on for the others ; we shall therefore obtain the finite integrals of the
linear differential equations
^ = 7^ + ^+"^' ^=77r + ^ + «e'; &c.; [864]
by changing in the finite integrals of these equations, deprived of their last
terms, « Q, « Q\ &c., the arbitrary quantities c, c\ &c., into [860]
c — u.fdt.{FQ + F'Q' + kc.];
c' — a.fdt,{HQ + H'Q' + kc.l; &c.
We shall now, for an example, consider the equation
0 = ^ + «'y + «-«- [865]
* (598) They cannot contain y, y', he, or their differentials, because of the linear form
of the equations [861"].
121
[8641
482 INTEGRATION OF [Mec. Cel.
The finite integral of the equation*
[885T o = ^+a'y,
is
c c'
[866] y = -. sin. at-\- -. COS. a t ;
a a
c and c' being arbitrary constant quantities. Taking the differential of this
equation, we have
[867] ~=c. COS. at — c'.sin. a^.
at
If we combine this differential with the integral itself, we shall obtain the
two following equations of the first order,!
c = ay . sin. at-\--~ . cos. a t ;
I * ^y • ^
c ^ ay. COS. at — ~.sm. at;
[868]
[864a]
* (600) This equation is very much used throughout this work. Its integral [866] is
easily proved to be correct ; for by taking the first differential of [866] it becomes as in
[867], and the differential of this divided by — dt is —— =c a. sm. at -{-c'a.cos. at,
in which the second member is equal to the value of a^y deduced from [866]; hence
— --^= a^V, and 0 = — ^ + a^ v- Its integral y= - . sin. at-\--. cos. a t, may
be put under either of the following forms,
y =z b . sin. (a t-\-(p),
y = b . cos. {at-\- (p),
h and (p being arbitrary constant quantities. For by using sin. (a < -f- <?)» [21] Int., it
changes into y = b . cos. (p . sin. at -{-b . sin. 9 . cos. a t, which becomes identical with
[866], by putting -=b .cos. (p, -=6.sin. 9. The second form of y, [864a], by
developing cos. {at -\- 9), as in [23] Int., becomes
y = b. cos. 9 . cos. at — b . sin. cp . sin. a t,
c . c'
which also changes into [866] by putting - = — b . sin. 9, - = 6 . cos. 9.
f (601) IMultiply [866] by a . sin. a t, and [867] by cos. at; the sum of these
products gives c [868]. Again, multiply [866] by a. cos. a ^ and [867] by — sin. af,
the sum of these products gives c' [868].
n. V. '^41.] LINEAR EQUATIONS, 483
hence we shall have, in this case,*
F = COS. a t ; H= — sin. a t ; [869]
and the complete integral of the proposed equation, will be
C • ^ I C . a. sin. at n^j. . , a. COS. at rr\j^ • ^ ro~«-,
y=-.sin.a^+-.cos.a^ .J (^ at. cos. at -\ ./Qdt.sm, at. [870]
Hence it is easy to conclude, that if Q is composed of terms of the form
K.f^^(mt + s), [87(y]
each of its terms will produce, in the value of y, the corresponding termf
((K sin. , ^ , N
-K i". (mt + s). [871]
m^ — a^ cos.^ ' ^ •■ ■*
* (602) Comparing [865] with [853] we find i = 2, and by the rules [863^], F
d v
and Hare the coefBcients of — in the expressions c= J^, </= V'y [851], which
correspond to [S6S], hence P=cos. at, H= — sin. at, as in [869] ; and the
expressions [864'] become c — a ./Qdt .cos. at, c'-{-a.fQ^dt.sm.atj These
being substituted for c, cf, in [866] give the complete integral of [865],
sin. at ( /. /-v 7 7 1 COS. at ( , , ->-.•. . )
y = jc — a.fi^dt. COS. a < > H . < c -\- a . f Q^dt . sin. at >,
c . . d
as in [870]. IVIoreover, the two first terms - . sin. at^-. cos. a t, may be put under the
form h . sin. (a < + <p), or h . cos. (a < + 9), as was observed in [864a] ; therefore
the complete integral of
_|_|_a2y_|_„Q^0, [865a]
IS
7 sin. /• ■ \ ct . sin. a t „^ . . a. cos. a f «^ ,
y = 6.f|°'(«< + <p) --.fqdt.cos.at + - .fqdt.sm.at. [8656]
cos.
f (603) If ^■= K . sa\. {m t -\- e), the expression '■ — ' — .f^dt. cos. a t
becomes '-— — . fdt .cos. at .sm..{mt-\-s), which by [18] Int.
a
Ka.sm.at
fd t . {sin. [(m -{-a) .t-\-i\-\-sm. [(m — a) • ^ + 01
^ — COS. [(m -}-«)'<+ ^] COS. [(m — «).<-{-£] )
( m'\-a m — a 5 '
4^84 INTEGRATION OF [Mec. Cel.
If m is equal to a, the term
sin.
[8711 ^.^^g (mf + s),
will produce in y* First, ihe term — T~2* *(^^ + 0' which, being
no constant quantity being added to the integral, because it already contains two, c and d.
Now by [19] Int.
2 . sin. a t . cos. [{m -\-a) .t-{- i\ = sin. [(m -\- 2 a) . t -\- s\ — sin. [mt -{- s),
and 2 . sin. a t . cos. [(m — a) .t -\-s] = sin. (w ^ -f- s) — sin. [(w — 2 a) . < + 0)
substituting these in the preceding expression it becomes
K a. . svOi. a, t -, ., ,. Ka. \ • rr \ n \ , \ -\ • / . i k")
. Cdt. cos. a t . sm. (mt-+-B)= - — ; — ^ — - .< sm.r (m + 2a) . ^+£j — sin. {mt-f-s))-
[871a]
Ka
— . \ sin. [mt-\- s) — sin. [(m — 2 a) .t-\-s\ V ,
[8716]
4a.(»i
and by writing ^nr -{-at for a ^, | * being a right angle, we get
— ^-— .fdt.sm.at.sm.{mt-\-B) =~~— -^ . | — sin.[(m4-2a).^+£]— sin.(w^+£) |
+ - — r r . < sin. (mt-\-s) + sin. ffm — 2 a) . ^4-s] f .
4o.(m — a) ( ^ ' / ' u\ / 1 J ^
The sum of the expressions [871a, 5], gives the value of
J i^dt . COS. a ^ -j .J (^dt . sm. a ^,
arising from K. sin. (m ^ -|- s), namely,
— — . sm. [m t-f-s)-\- - — . sm. {mt-\- s),
4:a.{m-\-a} ^ ' 4a.(w — a)
and by reduction it becomes — r . sin. {m t + s), which is like the first form [871].
If in this we write mz'-j- J * for ^ ^» the term of Q, [870'], ^. sin. [mt-\-s) will
become K. cos. (m^-)- s), and the preceding result — ^ i*^^^* (^^~f"0 will become
-. cos. (m ? + s), which is the second form [8711.
m2 — a3 \ 1 / I- J
*(604) When Q = ^. sin. (a< + e), the term
a.sm.at -^ , aiST-sin. af ^ , , . /^ \ s , ■ >
.f(^dt. cos. a^ = 'fdt . {sm. (2 ai + s) -f-sm. sj
aX'.sin.af f — cos. (2 a < -|- s)
nn.at ( — cos. (2 at 4- s) , , . )
. } ' — - 4- t . sm. £ > ,
a I 2a ^ 5'
11. V. §42.] LINEAR EQUATIONS. 4,85
c c'
comprised in the two terms - . sin. at-\-- . cos. a t, may be neglected ;
Second, the term
±^.'=?^-(at + ,); [871-1
2 a Sin. V ' / ?
the sign + taking place, if the term of the expression of Q be a sine, and
the sign — if it be a cosine. Thus we see how the arch t is produced out
of the signs of sine and cosine, in the values of y, t/, &c., by successive [871'"]
integrations, although the differential equations do not contain them under
this form. It is evident that this will occur whenever the functions F Q,
F' Q', &c., H Q, H' Q;, &c., contain constant terms.
42. If the differentials dt.\FQ-\-&ic.\, d t .\H Q-\-kc.\, &c., be not
exact, the preceding analysis will not give the rigorous integrals ; but it
furnishes a simple method of obtaining the integrals by approximation, when
a is very small, if we have the values of y, y', &c., in the case of a being
and
a. COS. at ^^j. . , Ka.co3.at ., -. . ,,, >,
./ k^dt.sm.at^ .fat . Jsm. at . sm. {at-\-s)i
a a
Ka. COS. at ^^ ^ ^^^^ ^ ^^^ ret „ * i ,^? Ka.cos.at ^^ sin. (2 a f -j- s)
2a
Hence we have
rj. c fn . \ \y Ka.cos.at C sin. (2 a < -f s) ")
.fat. |cos. £ — cos.{2at-\-s)] = . j t.cos.s — — —^ ^ .
a. sin. at «y-v , , a .cos. at /.>-.,
.f(^dt . COS. at-] .fQ^dt.sm.at
= — — . ] — sin. a t . COS. {2at-{-e) -j-cos. a t . sin. (2at-\-s) >
+ -^ — • 5 — sin. a t . sm. s -\- cos. a t . cos. s v ;
and by [22] Int. the coefHcient of ^ is = sin. [(2 at -\-z) — o <] = sin. {at-\- s), and
the other coefficient — sin. a t . sin. s -[- cos. a t . cos. s = cos. {at-\- s), hence the
preceding expression becomes — — . sin. {at-\-s)-\- — — . cos. {at-\-e). These are the
terms produced by Q = jK". sin. {mt-\- s), and by writmg i * -|- s for s we obtain
those arising from Q = JST. cos. {in t -f- s), namely,
— ^•cos.(a/ + 6)-.^-.cos.(a^ + e),
as in [871', 871"].
122
^S6 INTEGRATION OF [Mec. Cel.
nothing. Taking the differentials of these values i — 1 times successively,
we shall form the differential equations of the order i — 1,*
[872] c = V; c'=V'; &c.
The coefficients of — -^, --^, in the differentials of V, V, &c., being the
dV dV ' ' > b
values of F, F\ &c., H, H', &c., we must substitute them in the differential
functions [864']
[873] dt.(FQ + F'Q'+kc.); dt.(HQ + H' Q' + kc.) ; &c.
Then, in these functions, we must substitute, for y, y', &c., their first
approximate values ; which will render these differentials functions of t, and
[873'] of the arbitrary quantities c, c', &c. Let Tdt, T'dt, &c., be these
functions. If in the first approximate values of y, y', &c., we change the
[873"] arbitrary quantities c, c', &c., into c — u.fTdt, d — a. f T'dt, &c., we
shall have the second approximate values of those quantities.f
We must then substitute these second values, in the differential functions
[873],
[874] dt.(FQ + &c.) ; dt.(HQ + &c.) ; &c.
Now it is evident, that these functions are then what Tdt, T'dt, &c.,
[874] become, by changing the arbitrary quantities c, c', &c., into c — a./Tdt,
c'—u./T'dt, &c. Therefore let i;, T/, &c., be what T, T', &c., become
by these changes, we shall have the third approximate values of y, y', &c.,
[874"] by changing in the first values, c, c', &:c., into c — a./T^dt, c' — a.fT'^dt,
&c., respectively.
[874"'] In like manner, put T^^, TJ, &c., for the values of T, T", &c., when c, c',
&c., are changed into c — a .fT^dt, c' — a ./T^ dt, &c., we shall have
the fourth approximate value of y, ij , &c., by changing, in the first
approximate values of these quantities c, c', &c., into c — (n.fT^^dt,
[874'»] c' — a .fT/^d t, &c., and so on for farther approximations.
We shall hereafter see that the determination of the motions of the
* (605) These equations are formed in the manner explained in [850", 851].
•}• (606) This method evidently follows from what is said immediately after the equation
[859] or [864].
II.v. §42.] LINEAR EQUATIONS. ^87
heavenly bodies depends almost always on diflferential equations of the
form
0 = ^ + a^2/ + «.e, [875]
Q being a rational and integral function of y, and of sines and cosines of
angles, increasing in proportion to the time represented by t. The following
is the most easy method of finding the integral of this equation.
We must first suppose a nothing, and we shall have by the preceding article [875']
[866] the first value of y.
Substitute this value in Q, which will thus become a rational and integral
function of sines and cosines of angles proportional to t. Then finding the [875"]
integral [870] of the differential equation, we shall have a second value of y,
exact in terms of the order « inclusively.
This last value being substituted in Q, and the integral [870] of the
differential equation being found again, will give the third approximate value [875'"]
of y, and so on for others.
This manner of finding, by approximation, the integrals of the differential
equations of the motions of the heavenly bodies, although the most simple of
any, has however the inconvenience of producing, in the values of y', y', &c., [8751'^]
arcs of a circle without the signs of sine and cosine, even in those cases
where these arcs do not exist in the correct values of those integrals. For it
is easy to perceive, that if these values contain sines or cosines of angles of
the order a t, these sines or cosines must be expressed in the form of series,
in the approximate values, found by the preceding method, since these
quantities are arranged according to the powers of «.* This development of
* (607) For an example of this method, suppose in the equation [865], Q=^(2a-|-a) .y, [876a]
and it will become -- — -|- (a + a)^ • y = 0, which is of the same form as [865'], changing
a into a -{-a, and its complete integral [864a] is y=b. sin. ^ (a -(- a) .t-\- cp], which
being developed by [21] Int. is y = h . {sin. («<-{"?)• cos. a t -\- cos. (a ^ -f- 9) . sin. a /|,
and if for cos. a t, sin. a t, we substitute their values in series, [43, 44] Int. it will
become
488 INTEGRATION OF [Mec. Cel.
the sines and cosines of angles, of the order at, ceases to be exact, when, in
[875 '^j the course of time, the arc at becomes considerable ; and for this reason, the
approximate values of y, ?/, &c., cannot be extended to an unlimited time.
Now it is important to obtain these values in such forms as will include past
and future ages. This is done by reducing the arcs of a circle, comprised in
the approximate values, to the functions which produced them, by their
development in series. This is a delicate and interesting problem of analysis.
The following is a general and very simple method of solving it.
43. We shall consider the differential equation of the order i
[876] o = -^| + P + «e;
d y d^~^y
[876'] a being very small, and P, Q, being algebraical functions of y, -—
and of the sines and cosines of angles increasing in proportion to the time t.*
We shall suppose that we have the complete integral of this differential
and by arranging according to the powers of a,
[8766] y=b . sin. {at-\-<:^)-\-oL.ht. cos. [at-\-(^) — . s\n.{at-\-!^) — — '- — . cos. («<+<?)+ &«;.
and it is under this last form that the integral will appear, when computed by the above
method. For the purpose of illustrating this calculation, we shall compute some terms of
the series, which would be found from putting in [875], Q = (2 a -f- a) . y, following
nearly the method there pointed out. In the first place, putting a = 0, the equation
becomes 0 = — --f-«^y, hence 3/ = 6 . sin. (a / -j~ ?)» [864a]. Substituting this in
Q, [876a], it becomes Q = {2a-\- a) &sin. {fit -\- cp), which being compared with [870']
gives K = (2 a -\- a) . b, m = a, s = <p, and the term of y resulting in [871"], is
- — .bt. COS. (a t -}- 9), or by neglecting a^, abt. cos. {at -\- 99), so that the second
[876c] value of y is y = b . sin. {a t -\- qo) -{- a b t . cos. {at-j- cp), which agree with the two first
terms of [8766]. Substituting this in Q, [876a], it becomes
{2 a -\- a) .b . sin. (a t -{- cp) -}- 2 a . a b t . cos. {at -\- 9),
neglecting terms of the order a*^. This may be substituted for Q, in [865J], and by this
process we may obtain successively as many terms as we please of the series [8766].
«
* (609) By this is meant that the first power only of i is included under the signs of
cosine and sine, the second, third, &ic., powers i^, t^, he, being excluded.
II. V. §43.] LINEAR EQUATIONS. 489
equation, in the case of a == 0, and that the value of y given by this integral,
does not contain the arc t, without the signs of sine and cosine ; vv^e shall [876"]
also suppose, that bj integrating this equation, by the preceding method of
approximation, when a is finite, we shall have
y = X-f «. Y + ^^Z+t^6f+&c. ; [877]
JC, y, Z, &c., being periodical functions of ^, containing i arbitrary quantities
c, c', c", &c. ; the powers of t in this expression of i/, increasing infinitely in
the successive approximations. It is evident that these coefficients decrease [^TT]
with greater rapidity the smaller the quantity a is taken.* In the theory of
the motions of the heavenly bodies, a expresses the order of the disturbing
forces, in comparison with the principal forces acting on them.
If we substitute the preceding value of y, in the function -^ -\- P -\-uQ^
[876] ; it will become of this form, k + kt + kW^ &c. ; A;, k\ k", &c., [877"]
being periodical functions of t ; but by hypothesis, the value of y satisfies
the differential equation [876],
0 = ^f+P + ..Q;
[878]
we ought therefore to have identically,
0=:k + k't + k"t^-\-kc, [879]
If k, k\ k", &c., do not vanish, this equation would give, by inverting the
series, the arc t in functions of sines and cosines of angles proportional to t ;t
supposing therefore a to be infinitely small, we should have t equal to a finite
function of sines and cosines of similar angles, which is impossible ; therefore [879^
the functions /c. A;', V, &c., are identically nothing.
* (610) The computation in [S766] shows that F, Z, S, &tc., are respectively of the
orders a, a^, a^, he, in the example there given.
f (611) This inversion might be made by La Grange's formulas, [629c], which by
changing x into t, and t into a?, to conform to the present notation, become
i=,x-^F{t), and ^{t)=^{x)+F{x).-].'{x)-{-hc., [879a]
and if we put -^ {x) = x, which makes 4^' {x) = 1 , also for brevity, F (x) = X,
this last expression will become
, = ^ + X+--^^+---- + &c., [879i]
123
^90 INTEGRATION OF [Mec. Cel.
Now if the arc t be raised onlj to the first power, under the signs of
sine and cosine, as is the case in the theory of the celestial motions, this arc
will not be produced by the successive differentials of y ;* substituting
therefore the preceding value of y, in the function -~ + P + « Q? the
function k-\-k' t-{- &c., into which it is transformed, will not contain the
[879"] arc t out of the signs of sin. and cos., except as it is already contained in
that form in y ; therefore by changing in the expression of y, the arc t,
without the periodical signs into t — ^, d being any constant quantity, the
[879"'] function k-^-k t-\- &c., will become k-\-k! .(t — ^) + &c. ; and since this
last function becomes identically nothing, in consequence of the identical
equations k=Q, k' = 0, &c., it follows that the expression
[880] y = X+ (t — a) . Y + (t — 6y . Z+ &LC.
will also satisfy the differential equation [876]
[881] 0 = -^^ + P+aQ.
Although this second value of y seems to contain i-\-l arbitrary quantities,
namely & and the i terms c, c', c", &c. ; yet it cannot actually contain more
than i such quantities, which are really independent of each other. It
therefore necessarily follows, that an appropriate changef in the constant
k k"
Now if we divide [879] by K, we shall get t = — - — T7 • ^^ — &c. Comparing this
k k"
with t, [879a], we get a: = — - , and F{t) = — — . i^ — &;c. ; hence
7.// jLW
Substituting this value of X in [8795], we shall get the required value of <, expressed in
terms of k', k", he. When a is infinitely small, tliis value of t would, as in [879'], be a single
finite function of sines and cosines of angles proportional to t, which would be impossible
because there are an infinite number of values of t, corresponding to the same sine or cosine.
* (612) The successive differentials of any term like b . ^'"* {m t + s), taken relative
to t, will not produce t out of the signs sin. and cos., which would not be the case if the
exponent of t should differ from unity, as b . ^^"' (m t^ -}- s), the differential of which,
divided by d t, would contain t without the signs of sin. and cos.
f (613) This consists in supposing c,c', c", &c., to be functions of 6, as is shown hereafter.
U. V. §43.] DIFFERENTIAL EQUATIONS. ^91
quantities c, c', c", &c., will make the arbitrary term 6 disappear from the [881T
second expression of y [880], and in this manner it will be made to coincide
with the first [877]. This consideration furnishes a method of making the
arcs of a circle disappear from the quantities without the periodical signs.
We shall put the second expression of y under the following form :*
y = X-^(t—&).R. [882]
Since we suppose that ^ disappears from y, we shall have f — | j = 0 ;t [882']
consequently
Taking successively the differentials of this equation, we shall have
^ /'dR\ fddX\ , ^^ , fddR\
^\-di)-\^-d^^)-^^'-'^\-din'
^ fddR\ /'d^X\ , ,^ . /d^R\
&c. ;
* (614) The expression [877], by changing as above t into t — 6, becomes as in [880],
and if we put R=Y-{-{t — 6).Z-{- kc.j it will become as in [882].
f (615) This follows from the value of y. [877], which being wholly independent of 6
must evidently give f — j ^ 0. Substituting this in the differential of [882], relative to 6,
-^- (^)=C^)-«+('-^)-Q=o. -- ^=(f)+(-^)-Q'
as in [883]. Taking the differential of this relative to 6, we get
/dR\ /ddX\ /dR\ , , . /d^R\
and by transposing (— ), we obtain the first of the equations [884]. The differential of
this last equation being found relative to 6, and divided by d d, gives
-(f)=(?f)-(fF) + ('-^)-P
and by transposmg — ("^■^j' ^^ S^^ ^^ second of the equations [884], and so on.
[884]
492 INTEGRATION OF [Mec. Cel.
[885]
hence it is easy to conclude, bj eliminating R and its differentials from the
preceding expression of y*
X is a function of t [877'], and of the constant quantities c, c', c", &c. ; and
as these quantities are functions of 6, X will be a function of t and &, which
we may represent by
[885'] X=(p(ty&).
The preceding expression of y is, by the formula (i) ^ 21 [617], the
development of the function cp(t,&-{-t — ^), according to the powers of
[885"] t — ^ ;t therefore y = cp (t, t) ; hence it follows that we shall have y, by
changing 6 into t in the function X [617]. The problem is by this means
[885'"] reduced to the finding of X in a function of t and 6, and it will therefore
require the determination of c, c', c", &c., in functions of ^^.
* (616) Substituting jR [883] in [882], it becomes
Substituting in this the value of ("TTJj deduced from the first of the equations [884],
^ , . . /dX\ . {t—^f /ddX\ . {t-&)3 fddR\ c. u • • •
weget, 2/=X+(^-^).(^-j + -^ . (^--j+^- . {--y Substuutmg m
this the value of (-ty) deduced from the second of the equations [884], we shall find
another value of jt; and, by proceeding in this manner, we shall finally obtain [885].
f (617) Putting, in [617], i = 6, a = t — 6, we shall get
and as t is considered constant, in the differentials of the second member, we may introduce
the term t under the function cp, and write 9 {t, 6) for <p (&), and <p{t,&-{- 1 — 6) for
<p{d-^t — 6), that is, we may write X := (p (^, 6), [885'], for 9 {&). By this means
— J -|- &;c., the
second member of which is the same as in [885], therefore it is equal to its first member y;
hence y = cp (^t, 6 -\- t — 6), and as 6 -\- t — d = t, this becomes simply y = (p(t,t).
Hence it appears that the value of y may be obtained, by changing 6 into t, in X= 9 (t, 6);
[885'].
dd /' \dd J \dd
c, c', &c., in these functions ; hence we get
dX\ rdX\ dc_ fdX\ dd fdX\ ^ , „
dd J \dc J d& \ddj d6 ' \d c" J dd
dY\ /dY\ dc . /"dYX dd . ^dY\ dd'
dd J \ dc
&c.
dc_ fdY\ dd^ f^_\ ^' ! fir
'd6^ \d7j ' d6 "^ VrfcV *^ "^ '
Now it may happen that the arc t is multiplied by some of the arbitrary
quantities c, c', c", &c., in the periodical functions X, Y, Z, &c. ; the
differential of these functions relative t(^^, or in other words, the differentials
relative to these arbitrary quantities, will develop this arc, and make it come
forth, from under the signs of the periodical functions ;t the differentials
* (618) This equation is found by computing [~f:\ from the equation [880], and
putting, as m [882'], (-^^ = 0.
f (619) Suppose, for example, X=c . sin. at-{-h . sin. ct; a,h, being independent
of c, c', Stc, we should have (—) = (—-). sin. at-\-b . f — ) . t . cos. c t, in which the
last term contains t without the sign of cos. c t, being produced in the manner above
mentioned.
124
[887]
n. V. <^43.] DIFFERENTIAL EQUATIONS. 493
For this purpose we shall resume the equation [880],
y = X-^{t — &). Y ^ {t — df . Z + {t — &y . S ^hc, [886]
Since the constant quantity ^ is supposed to disappear [881'] from this
expression, we shall have the following identical equation :*
Applying to this equation, the same reasoning as in the case of
[879, 879'], we shall easily perceive that the coefficients of the successive
powers of (t — ^), ought to vanish. The functions X, Y, Z, &c., contain 5
only as it is included in c, c', &c. [877', 885'"] ; so that to form the partial
differentials (-;— ), \~JTn V^vT)' ^^"> ^^ ^^ ^°^3^ necessary to vary
[887T
[888]
[888']
494 INTEGRATION OF [Mec. Cel.
(^)' O' (4f)' «'<=- will then be of the form,
[889] /rfTN _ y, ^ ^ . y» .
(4f) = ^' + *-^" =
&c. ;
X', X", F, Y", Z', Z", &c., being periodical functions of t [877', 888],
[8890 containing also the arbitrary quantities c, c', c", &c., and their first
differentials divided by d 6, which differentials appear under a linear form in
these functions ;* therefore we shall havef
(iE^=X' + 6.X"+(t->).X";
[890] l^) = T + e.Y" + (t->).Y";
(^^'^ = Z' + i.Z" + (t-»).Z";
&c.
Substituting these values in the equation (a) [887], we shall have
[891] ^(t — (>).lT + 6.Y"+X" — 2Zl
+ (t—6y.{Z'-\-6.Z" + Y" — 3S}+kc, ;
(•
putting the coefficients of the powers of t — 6 severally equal to nothing, t
we shall have
* (620) That the differentials dc, dd, dc", he, appear under a linear form is evident
from the equations [888].
f (621) These values of (~\ (^\ he, [890], are evidently identical with those
in [889], writing 6-\-{t — &) for t, so that when they are substituted in the equation
[887], they may be arranged according to the powers of t — 6, as in [891].
J (622) For the same reason that k, Id, K', &;c., [879], were severally put equal to
nothing, [879'].
n. V. <^ 43.] DIFFERENTIAL EQUATIONS. 493
0 = Z' + ^.Z" — Y;
0 = Y' + 6. Y"-{-X" — 2 Z ; [892]
0 = Z' + d.Z" + Y"-'3S',
&c.
If we take the differential of the first of these equations i — 1 times
successively, relative to t, we shall obtain the same number of equations,
between the quantities c, c', c", &c., and their first differentials divided by [892']
d 6 ; taking the integrals of these equations relative to 6, we shall have these
constant quantities in functions of 6. By merely inspecting the first of the
preceding equations, and comparing separately the coefficients of the sines
and cosines it contains, we may almost always obtain the differential equations
in* c, c', c", &c. For it is evident that the values of c, c', &c., being [892"]
* (623) To show the use of these equations by an example, we shall suppose the
differential equation to be, 0 = —-{-(« + «)^ • y- The value of y deduced from this
by a first approximation in [876c], neglecting a^, is y=b.sin.{at-{-(p)-\-aht.cos.{at-\-(p), the
last term of which contains i without the sign of cosine (a < -{- 9), the arbitrary constant
terras being &, 9, corresponding to c, c', in the above rule. By comparing the expression of
y with [877], we have X=b .sm. {at -\- cp), F= a 6 . cos. (a < + 9), Z, S, he,
being nothing. The differential of this value of X, taken relatively to 6, considering J, 9,
only as variable, gives (— j = (jA . sin. {a t -\- <?)-{- b . (~\ . cos. (a < + 9).
Comparing this with the first of the equations [889], we get
-^' ^ (rfl) • ^^"' (« ^ + ?) + ^ • (^) • COS. (a t + 9),
and X" = 0. Substituting these in the first of the equations [892], namely,
0=:X' + 6X"—Y, it becomes
and as this ought to be identically nothing, we must have ( — ) ^0, b . ( — j — ab=0.
The first gives b constant. The second divided by b becomes ( — ) — a= 0, or
d<p=ad6, whose integral is 9 = a 5 + <?' J ^' being a constant quantity to complete the
integral. Substituting this in X, [892fl], it becomes X=b . sin. {at-{-a6-\- 9'), and
this gives y by changing 6 into t, as in [885"]. Hence y = b. sin. {at-f-at-{- 9'},
which is of the same form as the complete integral [876a, Sic], found by the method [870].
This example is here used merely as a convenient way of illustrating the formulas [892].
[892a]
496 INTEGRATION OF [Mec. Cel.
independent of t, the differential equations which determine them ought also
to be independent of t. The simplicity of this manner of considering the
subject, is one of the principal advantages of the method. In general these
equations can be integrated only by successive approximations, which may
introduce the arc ^, without the periodical signs, in the values of c, c', &c.,
[892'"] even when this arc does not really appear in that form in the complete
integral ; but in this case it may be made to disappear by the method we
have just explained.
It may happen that the first of the preceding equations, and its i — 1
difTerentials in t^ do not give the requisite number i of distinct equations,
[892'^] between the quantities c, c', c", &c., and their differentials. In this case we
must refer to the second equation, and to those following it.
When we shall have determined, in this manner, the values of c, c', c",
[892 V] &c., in functions of ^, we must substitute them in X [885] ; then changing
& into t, we shall obtain the value of y [885"], free from arcs of a circle,
without the periodical signs, when that is possible. If this value yet contain
such arcs, it will be a proof that they exist in the rigorous integral.
44. We shall now consider a number n of differential equations,*
[893] o = -^ + P + «Q; o=^ + P' + «Q'; &c.;
P, Q, P', Q', &c., being functions of y, ?/, &c., and of their differentials as
[893'] far as the order i — 1 inclusively, also of sines and cosines of angles,
increasing in proportion to t, whose differential is considered as constant.
Suppose the approximate integrals of these equations to be
y = X+t.Y+tKZ+t\S+kc. ;
m] y' = X^+t.Y, + t\Z, + t\S^ + kc.;
&c. :
* (624) This method is merely a generalization of that in the preceding article, and the
demonstrations will be easily found by comparing the similar parts of the two articles. Thus
the equations [876] are similar to those in [893] ; [877] is similar to [894] ; [892] is the
same as [895], and this last is of the same form as [896], the letters being accented with an
additional mark, &c.
n.v. §45.] DIFFERENTIAL EQUATIONS. ^^^
Xj Yf Z, &c., X^i y^, Z^, &c., being periodical functions of t, containing the
i n arbitrary constant quantities c, c', c", &c. We shall have, as in the
preceding article [892],
0 = X' + ^.Z"— Y;
0=F + ^.Y" + Z"-2Z; ^895^
0=Z' + ^.Z"+y" — 3^;
Sic.
The value of ?/ will likewise give equations of this form,
0 = Y; + d . Y/' + X/' — 2 Z^ ; [896]
&c.
The values of?/', i/", &c., produce similar expressions. We must determine
from these different equations, by selecting the most simple and approximate
forms, the values of c, c', c", &:c , in functions of L Substituting these values [896^
in X, X, &c., and then changing d into t, we shall have the values of y, i/,
&c., free from arcs of a circle without the periodical signs, when it is possible
to be done.
45. We shall resume the method explained in ^ 40 ; from which it will
be found, that if instead of supposing the parameters c, c\ c", &c., to be
constant, we make them variable, so that we may have*
dc = — adt.{FQ + F'Q'+kc.};
dc' = — udt.{HQ + H'Q + kc.}; [897]
&c. ;
* (625) In all the preceding articles of this chapter it is supposed that the arbitrary terms
c, c', &c., have been found in the form of the equations [851], namely, c=V,
d = V\ Stc, The object of this article is to find c, c', &;c., without being under the
necessity of forming the equations c = F', c' = V\ Sic, as is observed in [906'].
Now whether a be nothing or finite, if we put for dV, d V\ &c., the values assumed
in [854], we shall obtain the equations [857], kc, and if in these we substitute the values
[897], we shall have dc = dV, dcf = d V\ &:c., whose integrals [398] take place
whether a be finite or nothing ; in the former case c, c', &.C., will be variable^ in the latter
constant.
125
[898']
[898"]
[898'"]
[898iv]
4-98 INTEGRATION OF [Mec. Cel.
we shall always have the in integrals of the order i — 1
[898] c=F; c'=V'; c"=V"; &c. ;
as when a is nothing. Hence it follows, that not only the finite integrals,
but also all the equations, in which are found no other differentials except
those of an order inferior to i, preserve the same form, whether a be nothing
or finite ; since these equations can result only from the comparison of the
preceding integrals [898] of the order i — 1. We may therefore in both
cases, take the differentials of the finite equations, i — 1 times in succession,
without varying c, c', c", &c. ; and as we are at liberty to vary them all
at the same time, there will result some equations of condition between
the parameters c, c', &:c., and their differentials.
In the two cases of « nothing and a finite, the values of i/, «/, and of their
differentials as far as the order i — 1 inclusively, are the same functions of
t, and of the parameters c, c', c", &:c. ; therefore let Y be any function
whatever of the variable quantities y, ij, y", &:c., and of their differentials of
an order below i — 1, and put T for the function of t, which Tbecomes,
when we substitute in it the values of these variable quantities, and their
differentials in functions of t. We may take the differential of Y = T,
supposing the parameters c, c', c", &c., to be constant ;* we may even take
n the partial differential of Y, relative to one or more of the variable quantities
y, y, &c. ; provided we vary in T only those quantities which vary with
them. In all these differentials, the parameters c, c', c", &-c., may be
1 considered as constant ; since by substituting for y, i/', «fec., and their
differentials, the corresponding values in t, we shall obtain equations which
are identically nothing, in the two cases of « nothing and of « finite.
When the differential equations are of the order i — 1, we must no longer
suppose the parameters c, c', c", &c., to be constant in taking the differentials.
To find the differentials of such expressions, we shall consider the equation
* (626) By hypothesis Y" does not contain any differentials of the order i — 1; its
differential will not therefore contain any one of a higher order than i — ] , and in finding
differentials of this order, we may, as is observed in [898"], consider c, c', c", &;c., as
constant. The same remark will apply to partial differentials of Y, relative to one, or more,
of the quantities y, y', y", Uc.
[898'
n. V. §45.] DIFFERENTIAL EQUATIONS. 499
(P = 0, (p being a differential function of the order* i — 1, containing the [898'^"]
parameters c, c', c", &c. Let 09 be the differential of this function, supposing
c, c, &c., and the differentials d'~^y, d^~^ ij^ &c., to be constant. Put
S for the coefficient of — fj, in the complete differential of 9 ; S' for the [898^"]
d' v . .
coefficient of y-^j, in the same differential ; and so on. The complete
differential of 9 = 0, will become
[899]
[900]
Substituting, for --,—^37, its value [893], — c?^.{P + aQJ; for -r- 37- , [899^
its value [893], — dt.{P' -[- olQ], &c., we shall have
When a is supposed to be nothing, the parameters c, c', c", &c. [897], will be
constant ; and the preceding equation will become
0 = <5 9 — J^ . {^P + 5" P' + &C.} [901]
If we substitute in this equation, for c, c', c", their values F, P, F", &c.,
[898], we shall have a differential equation of the order i — 1 without [901^]
arbitrary constant quantities, which is impossible, except the equation be
identically nothing. The function
6^ — d.t.{SP-\-S'F + hc.] [902]
therefore becomes identically nothing, by means of the equations c = F,
c = V\ &c. ; and as these equations exist, when the parameters c, c', c", fcc, [902']
are variable [898'], it is evident that in this case the preceding function
* (627) After showing how to find the differential of a quantity of this order, and of this
nature, as in the resuh given in [903'] ; the author applies the method, in [904, 905], to the
investigation of i successive integrals of the finite quantity 4^, and by this means determines
r, d, Sic, as in [906'], without bebg under the necessity of reducing them to the form [898],
500 INTEGRATION OF [Mac. Cel.
will yet remain identically nothing ; the equation (t) [900] will therefore
become
[903]
»=e)-''^+(a-''^'+^^-
Hence we see that to find the differential of the equation 9=0, it is only
requisite to vary in ?, the parameters c, c', &c., and the differentials d^~^ y,
[903'] d^~^y'j &c., and to substitute, after taking the differentials, — a Q, — aQ',
&c., for -~^, --?-, &c.
Let 4^ = 0 be a finite equation between y, y, &c., and the variable
quantity^; if we denote by H, 6^-^, 6^^, &c., the successive differentials
[903"] of ^|.,* supposing c, c', &,c., to be constant, we shall have, by what has been
said, in case c, c', &c., are variable, the following equations :
[904] + = 0, H = 0, ^2^=0....(5'-i4. = 0.
By changing therefore successively, in the equation (x) [903, 903'], the
function (p into 4>, 5-^, 5^4^, &c., we shall havef
* (628) These differentials being divided by dt, dt^, he, respectively, to conform to the
last of the equations [905].
f (629) None of the equations [904], except the last, 5'~^4' = 0, contain the terras
d'~^y, d'~^ij, &,c., on wiiich S, S', he, [898^"'], depend; therefore in all these
equations except the last, we may suppose S, S', Stc, to be nothing, and this would be in
conformity" to the method given in [895^], and we shall then get, by writing successively 4'>
S 4', &£C., for 9, the wliole system of equations [905] except the last, which being derived
from 5'~^ ■]^ = 0, will contain S, S', &;c., and will become
.[905a] 0 = (^^i).rfc + (^-|^).c?c' + &.c.-a^^.^SQ + S'Q' + &c.|
Now a little attention will show that S, [898'"'''], which is the coefficient of d'~^ y in d^~^ 4'>
is also the coefficient of d~^y in d~^-\', or the coefficient of d'~^y in <^'~"'4'j ^^^ so on,
till we get to the coefficient oi dy in d-]^, which gives '^^^(;7~)' ^ '^^^ manner,
S'= [yX ^^' Substituting these values of S, S', &tc., in [905a], we obtain the last of
the equations [905]. .
II.v. §45'.] DIFFERENTIAL EQUATIONS. ^^1
[905]
-«.qQ.(g) + ^.(g)+.c.J
Thus the equations 4> = 0, ^]>' = 0, &c., being supposed to be the n finite
integrals of the diflferential equations
we shall have i n equations,* by means of which we may determine the
parameters c, c', c", &c., without being under the necessity of forming,
for that purpose, the equations c ^ F, d = V\ &c. ; but when the [0Qer\
integrals appear in that form, the determination of c, c', c", &c., will be
more simple.f
45'. This manner of varying the parameters is of great importance in
analysis, and in its applications. To point out a new use of it, we shall
consider the differential equation
0 = ^^ + P; [907]
* (630) Each of the n quantities ■\', •vj^'j ■4'"j ^c., will produce i equations of the form
[905], making in all i n equations.
f (631) When the arbitrary terms c, &, he, are given under the form c= V, c=V't
he, as in the equations [851], which were found by supposing a to be nothing, it is easy to
deduce from them the values of F, F', he, [863'"], and thence the values of F", V, &-c.,
[859], when a is finite. But if the equations do not appear under the form of [851], we
may use the equations [905, &ic.], to determine the values of rfc, d<^,hc., and thence
c, (/, &CC., when a is finite. As an example of this method, we shall apply it to the equation
[865], O^jl + a^y + aQ) already computed. The integral of this, when a is
126
502 INTEGRATION OF [Mec. Cel.
P being a function of U 2/? and its differentials as far as the order i — 1, and
of quantities q, q\ &c., which are functions of t. Suppose we have the
finite integral of this differential equation, when q, g^', &c.,* are constant ;
and let us represent it by <? = 0, (p being supposed to contain the i arbitrary
[907] constant quantities c, c', &c., We shall denote by ^9, (5^ 9, <5^(p, &c., the
successive diflierentials of 9, considering q, q', &c., c, c', c", &c., as constant.
If we suppose all these quantities to vary, the differential of 9 will be
c c
[906a] nothing is by [866] y=- . sin. at-\ — . cos. a t ; and if we consider a as finite, this will
still be the value of y, supposing c, c', to be variable, and to be determined, by the equations
c . c'
[905]. Now from [906a] we get y . sin. at — — . cos. at=0, which is equivalent
to -^=0, [904], hence
c . c' . . d-\. dy . 1 / •
4/ = « . sm. at . COS. at, 04/=^— -=-7- — c. cos. at -\-c . sin. a t,
^ a a dtdt
Now the first and last equations of [905], are
and by substituting the values of -v]^, <5 -^j they become
d c dc'
0= . sm.at . cos. a^ ; 0 = — dc .cos. at-\- dc . sm.at — a.dt . Q.
a a
Multiplying them by a . sin. a t, cos. a t, respectively, and taking the sum of these
products, we get, — dc — adt . Q. cos. a ^ = 0. Again muhiplying them by
— a . cos. a t, sin. a i, respectively, and taking the sum of these products we get
dc' — adt .Q.s\n.at = 0; hence dc=- — ad t . Q^. cos. a t, d c' = adt . Q^. sin. at,
whose integrals are c== C — a f Q. dt . cos. at, c' == C -}- a fQ. dt . sm. at ,
C, C, being constant quantities. These being substituted in y, [906a], it becomes as in
[870], changing c, c', into C, C, respectively.
* (G32) The object of this article is to show how to find the values of y, when q, q', &c.,
vary with extreme slowness, supposing y to be known for the case of c, d, he, constant,
as is observed in [912'].
II. V. §45'.] DIFFERENTIAL EQUATIONS. 503
therefore by putting
f5 (p will yet be the first differential of 9, supposing c, c', &c., q, q\ &c., to be
variable. If we also put
[910]
6^(p, 6^ q> ,... 6' (p, will yet be the second, third, &c. differentials of ?, as far
as the order i, when c, c', &c., 5', 5', &c., are supposed variable.
Now in the case of c, c', &c., 5^, g^, &c., being constant, the differential
equation
0 = ^^|+P, [9,1]
is the result of the elimination of the parameters c, c', 8z;c., by means of the
equations
(p=0; 6(p = 0; (5^9 = 0; ....a»*(p = 0; [9i2]
and as these last equations take place when q, q', &c. are supposed variable, the
equation (p = 0 satisfies also in this case the proposed differential equations,
provided the parameters c, c', &c., are determined by means of the i [912']
differential equations [909, 910] ; and as their integration gives i arbitrary
constant quantities, the function 9 will contain those quantities, and <p = 0 [912"]
will be the complete integral of the proposed equation.
This manner of varying the parameters may be employed with advantage,
when the quantities q, q', &c., change with extreme slowness, because this
generally renders the integration of the equations which determine the ^^^^"^
variable parameters c, c', &c., much easier by approximation.
^04 MOTIONS OF THE HEAVENLY BODIES. Mec. Cel.]
CHAPTER VI.
SECOND APPROXIMATION OP THE CELESTIAL MOTIONS, OR THEORY OP THEIR PERTURBATIONS.
46. We shall now apply the preceding methods, to the perturbations of
the celestial motions, with a view to obtain the most simple expressions of
their periodical and secular equations. We shall resume for this purpose the
differential equations of § 9 [416, 417, 418], from which the relative motion
of m about M can be determined. If we put
1^13] j^^ m! .{xx'-^yy' + zz') m" . {x ocf' ■\- y f -\- z z") ^^ X_ ^
{x'^ + y'^^z'^)i {x"^ + y"^ + zf'^)i »» '
in which by the above mentioned article [412]*
m m' m m"
[914J
\{^'-^r+{y'-yr+{z'-zri^ K^"-^)'+(/-#+(^"-^f}*
m m
^ { {a:"-a^'f-{-{y"-yy + {z"-zyi^
* (634) The above value of R, [913], gives
(dR\ m's/ m"3/' 1 /<^M mx mx 1 /rf^N
m X m X m' xf m" x^', „ ■n.-r -f • ^ ^^ , ■
because 2 .-— = —- + — - + — -- 4- &;c. Now if in ihe term — -—, we substitute
for m its value, [914'], f* — M, we shall have (—-) = ^^-— ^ + S.~ • (^),
\dx/ r3 r3 m \dx/
, Mx . mx 1 /rfM f^x , /dR\ ... ddx , ., , .
hence — --j-2.— . -—) = — --]--— ; addins; -— - to each s:de, and usine
r3 ' r3 m \dxj r3 ' \dxj^ ^ dt^ ' °
[416], we shall get the first equation [915]. The two remaining equations in y, «, are
deduced in like manner from [417, 418], which in fact are the same as [416], merely changing
the axis of x into that of y or «.
[916]
the
Evmbol
n. vi. § 46.] PERTURBATIONS. 505
supposing also [530'% 411],
M + w = M- ; r = ^/^aqr^-qr^s ; r'= ^/^^M^TM^^ *. ^c. ; [9141
we shall have
ddx ^x /dR\
dt^ ^ 7^^\dx J
ddy [hv /dR\ \ „.
0 = ^ + 7l + (^^) }; (P) P15]
_ fZc?^ fjLz / d R\
~d^'^~?^^\dj)
The sum of these three equations, multiplied respectively by 2dx, 2dy,
2dz, gives, by integration,*
the differential dR referring only to the co-ordinates x, y, z, of the body m, [916]
and a being an arbitrary constant quantity, which, when R is nothing, u^^f
becomes, by ^ 18, 19, [596'], the semi-transverse axis of the ellipsisf '^d
described by m about M. [916"]
The equations (P) [915] multiplied respectively by x, y, z, and added to
the integral (Q) [916], will givef
* (635) In finding this integral, we must substitute in the term 2fA . ~^ '
the value of the numerator, xdx-\-ydy-\-zdz = rdr, [57 16], which reduces it
r (It d t 2 M*
to 2(x. — = 2/x. — , whose integral is . Again, as the symbol d, [9 1 6'], only
affects the co-ordinates x, y, r, of the body m, we shall have as in [13&, 14a],
the integral of the double of this second member, which occurs in [916], will therefore be
u,
represented by 2 ./d i?. Lastly, - is the constant quantity to complete the integral.
f (636) When jR = 0, the equation [916] becomes identical with the last of the
equations [572], and in [596'], it is shown that in this. case the transverse diameter is 2 a.
X (637) After making this addition we must put ^ . [x^ -\- -f 4- z^) = ~^.r^ =:^- ^
[914'] ; also
xddx-\-yddy-\-zddz-\-dy?-\-d'f-\-ds?=d.{xdx-{-ydy-\-zdz) = d.{rdr)=^\.^.r^,
[914', 549'].
127
506 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
.., o=i.^-^, + ^^+2/die+..(i^)+,.(4i) + ..(4^). (i.)
Now we may conceive that the disturbing masses m', m", &c., are multiplied
by a coefficient a ; and then the value of r will be a function of the time t
[917'] and «. If we develop this function according to the powers of «, and after
the development put « = 1 , it will be arranged according to the powers
^ry^\>o\° and products of the disturbing forces. We shall denote by the characteristic
(5 placed before any quantity, its differential taken with respect to a and
[917"] divided by d a. After having determined 6 r, in a series arranged according
to the powers of « ; we may obtain the radius r, by multiplying this series
by da, and taking the integral relative to «, then adding to the integral a
function of t independent of «, which function will evidently be equal to the
[917"] value of r when the disturbing forces are nothing, and the body describes a
conic section. The determination of r is therefore reduced to the finding
and integrating the differential equation upon which 6r depends.
For this purpose, we shall resume the differential equation (i?) [917], and
for greater simplicity, we shall put
taking the differential of [917] relative to «, we shall have*
1919] o = ^-^+''-^ + 2fi.dR + S.rR; (S)
Put dv for the infinitely small angle included between the two radii vectores
r and r-^dr ; the element of the curve described by m about M, will be
[919^] \/dT^-{-t^.dv^ [583] ; hence we shall have
[919"J dx^+dif + dz'' = dr^-\-r^.dv'-,
and the equation (Q) [916] will become
* (G38) The differential of ^ r^ relative to a, or, in other words, relative to the symbol S,
IX l^ 0 T Ut V 0 T
is r 5 r, consequently ^.8.d^ .7-^=d^.r8r. Also, — 5.- = —^=———. Substituting
these and [918] in the differential of [917], taken relative to <5, it becomes as in [919].
II. vi. '^ 46.] PERTURBATIONS. 507
Eliminating — from this equation, by means of the equation, (R) [917], we
shall obtain*
taking the differential of this relative to «, we shall findf
2 1-^ .dv .d .Sv r . dd .8r — Sr . d dr S ii,r (W
dt^ dv"
r,SR--R.5r, [922]
* (639) Substituting in [917] the value r R\ [918], and then subtracting from it the
equation [920], it becomes, 0 = | . — -- -|- - -\-rR --—- — '- , and since by-
development ^ . d^ .r^ = rddr-{- dr^, it gives by reduction [921].
f (640) The differential of [921], relative to 5, is
2r^dv.dSv-{-2r8r.dt^ 6r .ddr-\-r.d^5r li'Sr
dfi rf<2 Ja~
2 6r .
and [921] multiplied by is
—2r6r.dv^ —QSr.ddr SfiSr
dfi dfi 7^
adding these two expressions together we get
2r^dv.d6v r .d^6r — 5r .ddr SfA^r
+R .8r-{-r.6R';
2R.6r,
dt^ dl^
-^r,8R-^E .6r,
which, by writing — for — , becomes as in [922]. This, added to three
times the equation [9 1 9], becomes
'ir^dv.dbv 3.d^.r8r-\-r.d^5r-'Sr.ddr , ^ ^, ,„ . ^ ^ -,,, , ^™ ^
—^ = ~-—^ \-6.fS.dR+3.S.rR + r.SR — R'.Sr,
Substituting for 2 .d^ .r Sr, its developed value, 2 . {d^ r . 8 r -\- 2 d r . d S r -{- r d^ 8 r)^
also 2 8.rR'=3.{R .8r-\-r.8 R), it becomes
2r^dv.d8v 2d^r .8r4-Ar .d^8r+6dr.d8r , ^^, ,„ , ^ ^. , _
j^ = —-Tt^-- + Qf8.dR + 4r.8R' + 2R.8r,
of which the terms free from R, R, in the second member are equal to — — — -,
^r^ dv
as is easily proved by development. Substituting this and dividing by -— ^ , we shall
obtain [923]. ^
608
[923]
[923']
[923"]
[924]
IMOTIONS OF THE HEAVENLY BODIES.
II .r 8r
If we substitute in this equation, for
equation (S) [919], we shall have
r3
[Mec. Cel.
, its value, deduced from the
, ^ d.{dr.Sr-\-2r.dSr)-\-dt^.\^fS.dR-\-2r.SR'-^R'.Sr\
r'' .dv
(T)
We may, by means of the equations (S) [919], and (T) [923], obtain as
accurately as may be necessary, the values of ^ r and 6 v ; but we must
observe, that d v being the angle included between the radii r and r -{- dr,
the integral v of these angles is not in the same plane.* To determine the
value of the angle described about Mhy the projection of the radius vector r
upon a fixed plane, we shall denote this last angle by v^, and shall put 5 for
the tangent of the latitude of m above the plane ; r. (1 + ss)—^ [680], will
be the expression of the projection of the radius vector ; and the square of
the element of the curve described by m will bef
r^ .dv^ , , o ■ r^ . ds^
-\~ dr -\-
i-{-ss
{l-\-ssf '
* (641) The mutual attraction of the bodies m and M, would make the orbit of the body
m wholly in the same plane, as is observed in [533"]. The other attracting bodies m, m",
&z;c., not being situated in that plane, their disturbing forces will tend to change the orbit of w,
so that two consecutive infinitely small parts of the orbit, will not be accurately in the same
plane.
f (642) Let M be the place of the body M, supposed to
be at rest ; MB^r, ME==r-\-dr, the radii vectores
of the body m, including the angle B M E = d v ;
CAA'M the fixed plane ; P B A A' B' e a spherical
surface described about the centre M, with the radius
MB, and meeting ME in e. Draw the quadrantal arcs
P B A, P eB' A', perpendicular to the arc A A', and
the arc B B' parallel to A A'. Then putting the latitude
of the point B equal to /, we shall have the angle
AMB = A'MB' = l,
BB' = AA' .cos.ht.=
dv.
BMe = dl, AMA! = dv,,
cos. Z, and since tang. l = s, gives cos. I ■
AA'^rdv,,
1
v/i+.
we
rdv
[924a] shall get BB' = -y==^; eE = dr, and the arc B e = rdl=^
V/l + 55
Now the lines B^', Be, e E
r ds
l-\-ss
[51] Int.
being perpendicular to each other, the sum of their
ILvi. §46.] PERTURBATIONS. ^^^
but the square of this element is r^ . d v^ -\- d r^ [did'] ; we shall therefore [9241
have, by putting these two expressions equal to each other,
^''- 7TT77 •
Hence dv^ may be determined from dv, when 5 is known.
If we take the plane of the orbit of m at a given epoch for the fixed plane,
ds
s and — - will evidently be of the same order as the disturbing forces ; by [9251
neglecting therefore the squares and products of these forces, we shall have
v = v^. In the theory of the planets and comets, we may neglect these [925"l
squares and products except in a few terms of that order, which are rendered
sensible by particular circumstances, and which may be easily determined by
means of the equations (S) and (T) [919, 923]. These last equations
assume a more simple form, when only the first power of the disturbing force
is noticed. For we may then suppose <5r and 6v to be the parts of r and v
arising from these forces ;* dR and S.rR will be what R and rR' become, [925'"]
squares is equal to the square of the distance of the points B, E, and, by substituting the
preceding values of BB', B' e, eE, we shall get for BE% the same expression as in
[924]. Putting this equal to i^dv^-\-dr^j [924'], and rejecting dr^ from both sides
of the equation, we get —j- 1- -— - — - = r^ dv'; multiplying this by — - — ,
transposing the second term and extracting the square root, we get
dv^^^t / dv^ .{\-\-ss) — — |- — , which is easily reduced to the form [925] j [9246]
ds
and by neglecting terms of the order of the square of «, or — , it becomes dv, = dv,
hence v^ = «, as in [925"].
* (643) The radius r being developed according to the powers of a, as in [917'], if we
neglect the second and higher powers of a, it will become of the form r = r' -\- at".
Hence 6r={—\ = r", therefore r = / + a 5 r. Now when the terms depending on
the disturbing forces, or upon «, vanish, we shall have r = /. The difference of these
two values of r, namely, a8r or Sr, will represent the part of r depending on the
disturbing forces, as in [925'"] ; in like manner Sv wiU represent the part of v depending on
128
510 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
when, for the co-ordinates of the body, we substitute their values relative to
the elliptical motion : we may therefore denote them by these last quantities,
subjected to that condition. The equation (S) [919], in this manner, will
become
The fixed plane of x and y being supposed that of the orbit of m, at a given
[926'] epoch, z will be of the same order as the disturbing forces ; and since we
neglect the square of these forces, we may neglect the quantity z. (-r-) • *
[9;^"] Moreover, as the radius r differs from its projection but by quantities of the
order z^ ;t the angle which this radius makes with the axis of a:, will differ
from that made by its projection, by quantities of the same order [9246] ;
this angle may therefore be supposed equal to v, and we shall have, by
neglecting quantities of the same order,
[927] X = r . COS. V ; 2/ = ^ • sin. v ;
the disturbing forces. Tlie quantity R, [913], being of the order m', or a, its development
according to the powers of a, will be of the form R = aR" -\-a^ . R'" -f- &;c., in which
R" evidently represents the value of R, found by using the values of r, v, Stc, corresponding
to the elliptical motion. If we retain only the first power of a, it will become R=uR".
Its differential relative to 5 gives 6R = R", as in [925'"]. In like manner S.rR' is
equal to r R', corresponding to the elliptical motion.
* (644) The values of R, X, [913, 914], give
/dRS _ m'z^ m"z-' , g,^ _ i /^\
z /d\\ _ C m'.jzf — z) J_ 8t ^
«"^ -m • [TzJ — '^-i [(x'-.)24-(3/-3/)2 + (z'-z# + ^^- J '
all the terms of which contain powers and products m' z, m" z, &;c., of the second degree
of the disturbing forces, [926'], therefore the whole expression, in the present hypothesis, will
be of the order of the square of the disturbing forces.
f (645) The projection MD^ fig. page 508, of the radius vector, r==MB, on the
plane of A MA', is by [680], equal to r . (1 —-^s^-j-kc.), which differs from r by
quantities of the order s^ or z^, r being supposed nearly equal to unity.
II. vi. §46.] PERTURBATIONS. 511
hence we deduce*
/dR\ , /dR\ /dR\
fxdy — y dx\
\ dt ;
[928']
consequently
It is easy to prove, by taking the differential, that if we neglect the square of
the disturbing forces, the preceding differential equation will give, by means
of the two first of the equations (P) [915],t
* (645«) The values of a?, y, [927], are the same as those in [371]. These give
(^)""^°^*''^ (S)^'^"'-''^ ^^''^^ r.[j^=r.cos.v=X', r.(^)=r.sin. v = y.
Again, in the last of the equations [371], we have r^ = a;^-j-y2, hence r may be
considered as a function of x, y, consequently \~^)'^^{l~)-\l~)~^\~r~/' ( 7/*
Multiplying this by r and substituting for ^ ' [~rp ^* (^) ^^^^ preceding values x, y,
it becomes as in [928]. The first member of which, neglecting ^ • ("t~)> [926'], is the
same as that of [918], consequently their second members are equal, hence
f (646) We may easily prove that the value of r 5 r, [929], satisfies the equation [926].
For jR being of the order of the disturbing forces, [913], we may, by neglecting the squares
and products of such forces, substitute the elliptical elements in the second member of the
'T (T li 1/ // 'T
formula [929], and then we may put, as in the first equation [572], — ^- — constant
and equal to c. Putting also for brevity, Q = 2/dil + r. {-^\ as in [934], the [929a]
expression [929] becomes
x.fydt.Q — y.fxdt.Q
r8r= -^^ 2^ — ^-^ 3t ^ [92gj^
, - J.-- . , . d.{rhr\ dx .fydt.Q — dy . fxdt. Q . ,
whose first differential gives — ^ — - = =^-^ ^^ -— ^ ; and second
° dt cdt '
512 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
For the co-ordinates in the second member of this equation, we may use their
T* /i 77 .I.I. ■ 7 / // T*
[929^] elliptical values, which makes — ^— ^^ — constant [929a], and equal to
j.^ ., . rf2.(rJr) ddx.fydt.Q — ddy.fxdt.q—QdtAxdy—ydx) ,,,,
differential gives —^^ = ^ -S ^ ^^^^ ^ — ^ \_J__^ — )^ which by
substituting xdy — y d x = cdt, becomes
[-929^1 d^.jrSr) __ ddx.fydt. q — ddy.fxdt. Q
di^ cdt^ "'
Again, the assumed value of r 5 r [9296], gives
ii.rSr (XX fydt.q li'y fxdt.q
r3 r3 c "fi c ^
L 1 ^/^,r^ 1 f*^ ddx /dR\ \hy ddy . /dR\ ,. , , .
but by [915] we have _ = _--_^_j, _-| = _^ + (_j, which bemg
substituted in — ^- — , neglecting the terms containing both quantities Q and d R, because
they are both of the order of the disturbing forces, and their product is of the order of the
- , - , „, a.r6r — ddx. fydt . Q4-ddy . fxdt. Q _, .
square of those forces, we shall have , = ^-^ -^ —- -. This
added to the value of -~-^, [929c], gives -^- + ^— = — Q ;
transposing Q and resubstituting its value, [929a], 2fdR~{-r • (~p)j or by [928'],
/72 T* 0 ?• UO TOT
2fdR + rR', it becomes ^— + ^-^ + 2fdR + rR=0, as in [926]. Hence
the expression [929] satisfies the proposed differential [926], and as the former contains two
signs of integration including two arbitrary constant quantities, it is the complete integral.
Besides this demonstration, it has been thought proper to give the following direct investigation
of the value of r Sr, as it answers the purpose of illustrating the use of the formulas [864'].
Put r8r== Y, and Q as in [929a], then the equation [926] becomes
dt^ ^ 7^ ~ ^'
which may be integrated by the method of § 40, he. Comparing it with the first of the
equations [864], writing F instead of y in [864] to distinguish it from y of this article.
d^Y M-y
For if we suppose R = 0, and Q = 0, the equation will become 0 = — - -j~ — ;
multiplying this by x, and substituting for •— - its value — -— , deduced from the first of
x^z Y Yddx
the equations [915], it becomes — = 0, whose integral, using c [863^'], is
dt^
xdY—Ydx ^^ dx . dY
=-r.-+..
dt dt ' dt
n.vi. §46.] PERTURBATIONS. ^1^
^ ^a.(i^e**), by ^ 19 [596c], ae being the excentricity of the orbit of
m [377"]. If, in this expression of r5r, we substitute, for x and y, their
IT fJ II ■ OJ fj X
values [927], r.cos. v, and r.sin. zj; and for — ^-j-^ — , the quantity [939"]
^iu,a.(l e^) ; observing also that by § 20 [605'], f*=:w^a^, we shall have*
If we multiply by y instead of x, and substitute the value of — , deduced from the second
of the equations [9 1 5] , we shall get 0 = — , whose mtegral gives
ydY-Ydy_ dy dY
dt 'dt'^y' dt '
Comparing these values of c, c', with the equations c= V, <f = V, [863'»], in which
y, y', Sic, are changed into Y, Y', &;c., we shall find F=x, H=:y, these being the
coefficients of -j- in c, c', respectively, and the terms of the equations [864'], 'namely,
c — afd t .F Q, c' — afd t . HQ^, become in the present case, where Q is put for a Q,
c — fx dt . Q^, d — fy dt . Q^. These being substituted instead of c, </, in the two
preceding integrals give
which are the two first complete integrals of [926], and we may even neglect c, and c',
supposing the constant quantities to be included under the terms fx dt.Q, fydt.Q,
by which means we shall have
/. 1 ^ -TT dx , dY fl 1 ^ -wr dy , dY
—fxdt. Q = —Y.—4-x.-—, —fydt. Q = — F. 37 +« .-r- .
•^ ^ dt ' dt J y ^ dt ^ ^ dt
dy — ydx\ t^- • t 1 ^dy — ydx ,
^ if \ T^.„.j — u„ — ~» — J and
dt
... xr ^ , . P X. fydt. Q—y, fxdt. O ..
resubsutuung Y = rSr, we obtam r8r = ^xdy-ydx^ ' ^"^ ^^ [929rf]
\ dt )
resubstituting the value of Q, [929a] it becomes as in [929].
* (647) Neglecting the squares and products of the disturbing forces, as in [929a], we
we may use in the second member of the equation [929], the elliptical values of the elements,
129
Eliminating [jj] from these, we shall obtain Y. This is done by multiplying the first
by y, the second by — a?, and adding the products, from which we get
-y.fxdt.q + ^'fydt.q=Y,(^-^^y Dividing by
51^ MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
V ,fndt,r .sm,v . J2/di2 + rY^') \
a. sin. V ,fndt,r,cos,v. < 2fdR-\-r. (-r— ) >
a . COS.
6r =
f..v/l=:^ ^^^
The equation (T) [923], being integrated, neglecting the squares of the
disturbing forces, gives*
2r .d. Sr -\-dr.Sr 3a
l^ ^-^ f* -^ \drj
[931] a^.ndt . , _. . ,
' « = 7!^ '—^- m
[93r] This expression will give easily the perturbations of the motion of m in
longitude, after those of the radius vector have been ascertained.
X d v ^— ■ w dx - X dv ""■ v d X
[930a] and put — ^-~ — =c=v/fjoa.(l — c2), [596c]. Substituting this value of — ~- —
Or Z Cv E
and those of x, y, [927], in r Sr, [929</], it becomes
^ r .cos.v .fr .sin. v.dt. Q — r .sixi.v.fr.cos.v.dt. Q
dividing this by r, and multiplying the second member by -y= = I, [605'], we shall obtain,
5r, [930].
* (648) By [372] we have r^.dv = xdy — ydx = dt. \/]i.a. (1— e2), [930a],
s
and since ^fx=r=n«^, [605'], it becomes r^ .dv^=--(?ndt .s/i—^. Substituting
this in [923], neglecting the term K , 5 r, which is of the order of the square of the
disturbing forces, because both i?' and 5 r, are of the same order as these forces, it becomes
d .ov = ' 7— V — /, „ , m which the denommator is
constant. Taking the integral of this, we obtain
dr.8r~{-2r.d5r 3 ^ r j n i ^ /• jt n/ j .
Substituting in the two last terms, for — , its value — , [605'] ; introducing the constant
quantity n, under the signs of integration j putting, as in [925'"], 6. R for SdR, R' for
6 R', and then making rR' ==r. (-^\ [928'], it becomes as in [931].
II.vi.§47.] PERTURBATIONS. 516
It now remains to determine the perturbations of the motion in latitude.
For this purpose we shall resume the third of the equations (P) [915].
Integrating it, as we have done the equation (S) [919], and putting z = r6Sf [931"]
we shall have*
a.cos.v.J nat. r. sm.v . ( — — j — a. sm,v. J nat. r. cos.v . ( —— )
''= .^Vw ^-^' (^> p=«i
^ s is the latitude of m above the plane of its primitive orbit. If we wish to
refer the motion of m to a plane which is a little inclined to that orbit, we
may put 5 for the latitude, when it is supposed not to quit the plane of the [932^
primitive orbit, and then s -f (5 5 will be very nearly the latitude of m above
the proposed plane.
47. The formulas (X), (F), (Z), [930, 931, 932], have the advantage
of presenting the perturbations under a finite form. This is very useful in
the theory of comets, in which those perturbations cannot be found, except [932"]
by the quadrature of curves. But the smallness of the excentricities, and
the inclinations of the orbits of the planets to each other, enables us to
develop their perturbations, in converging series, of sines and cosines of
angles, increasing in proportion to the time, and we can then arrange them
in tables which will answer for an indefinite time. Instead of the preceding
expressions [930, 932] of 6 r and (5 s, it will, for this purpose, be more
convenient to use the differential equations, by which these variable [932"']
quantities are determined. If we arrange these equations according to the
powers and products of the excentricities and inclinations of the orbits, we
* (649) When the fixed plane is supposed to be that of the orbit at a given epoch, s will
become S s, and the elevation of the point B, in the figure, page 508, above the fixed plane
A MA' will be nearly r8s, substituting this for « in the last of the equations [915], it
, ^ (P.{rds) , fx..{r5s) , /dR\ ,.\ . ^ ,
becomes 0 = — — 1 — r ( "T~ )» which is of the same form as the equation
[926], changing 5 r into 5 5 and 2fdR + rR, or 2fdR-{-r.(j^) into (^\
and by making these changes in the value of S r, [930],' deduced from [926], we shall
obtain the value of Ss, [932].
^1^ MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
may always reduce the computation of the values of (5r and ^5, to the
integration of equations of this form,
[933] O^^ + n^y+Q;
the integral of which has been given in § 42 [870]. This very simple form
may be given to the preceding diiferential equations in the following direct
manner.
Let us resume the equation (R) [917] of the preceding article, putting
for brevity
[934] Q = 2fdR + r,(^-~y,
by this means it becomes*
In the case of the elliptical motion, where Q = 0, r^ is by ^ 22, a functionf
[OSS'] of e . COS. (n t -\- s — xs), ae being the excentricity of the orbit, and
nt~{'^. — zs the mean anomaly of the planet. Put
e . cos. (nt -\- s — zs') = u ;
r^ = cp (u) ;
* 650) Substituting in [917] the value rR\ [918], and then its value r . (~\ [928'],
neglecting terms of the order of the squares of the disturbing forces, we shall get [935],
using the abridged symbol Q, [934].
[935"]
f (651) If in the value of — , [659], we change ni into nt-\-s — -a, as in [669],
and put for brevity nt-\-s — zi = z', we shall have r in a series of the following form
r = A + B. cos. z'-\-C. COS. 2z' + D. cos. 3z'-\-F. cos. 4 2:' + &c.
Now from [6, 7, &;c.] Int. we find cos. 2 z'=2 cos. z'^ — 1 , cos. 3 z'= 4 cos. z'^ — 3 cos. z',
and so on for cos. 3 2/ . cos. 4 2/, Sic, all of which will be expressed in terms of cos. 2:',
and its powers. The general form of such expressions may be obtained by taliing half the
sum of the formulas [15, 16] Int., from which we get
[935a] cos. nz' = i , {cos. z' -\-\/—i . sin. z'\'^ + J . {cos. z' — \/^ . sin. z'}".
Developing the second member, and putting sin. z'^ = l — cos. z'^ we shall find cos.wz',
in terms of cos. z', and its powers. Therefore the preceding value of r will become a
function of cos. «', or of — , or simply a function of u, as in [935"].
n.vi.^47.] PERTURBATIONS. 517
and we shall have*
-\-n^u. [936]
o = (S)
In case the motion is troubled, we may also put r' = (p (w), but u will not
then be equal to e . cos. (nt-\-z — to) ; but will be given by the preceding [936']
differential equation [936], increased by a term depending on these disturbing
forces. To ascertain this term, we shall observe that if we put
W = v}. (f) ; [936"]
we shall havef
-I7^+^-^ = -7^-^(^)+-^^-^ (^> + ^-^^^)' [937]
4-' (r^) being the differential of 4. (f) divided by d,r^, and ■\l' (r^) the [937^
differential of ■\! (r^) divided by d.r^. The equation {R) [935] gives
' equal to a function of r, increased by a quantity depending on the
disturbing force. J If we multiply this equation by 2r dr, and then
* (652) Having w = e . cos. {nt-\-e — «), its second differential gives
d du „ , , . _
-— = — n-'e . COS. {nt -\-s — -a) = — nru,
transposing — n^w, we get [936]. ^
f (652a) Substituting in the first member of [937] the value of u, [936"], it becomes as
in the second member.
J (653) The equation [935] gives ~=2. 0— ^) — 2 Q=/(r) — 2 Q, puttmg
for brevity, 2.f j equal to a function of r denoted [by /(r). Multiplying this by
2d.r^ or 4rdrj it becomes =2./(r) .d.7^ — QQ.rdr, whose
integral, by putting f 2 f{r) . d .r^ = F{r) is
^--^=F{r)-fsq.rdr or -^^F{r)-^Sfq.r dr.
The parts of these expressions depending on Q are as in [938]. Substituting the complete
values in [937], it becomes
-d^+^'«=^/W-+'(^')+^W-+'X^')+^'4(^)H2Q.^X^)-8.4.V)-/Q-'-^^-
130
^IS MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
integrate it, we shall have equal to a function of r, increased by a
quantity depending on the disturbing force. Substituting these values of
[937"] -^ and —j^ in the preceding expression of -—^-]-n^u; we shall find
that the function of r, independent of the disturbing force, will disappear of
itself, since it is identically nothing, when the force is nothing ; we shall
therefore have the value of ^ + ^^ ^> by substituting in it, instead of
[937'"] ' , and - ' g , the parts of these expressions, which depend on the
disturbing force. But by noticing only these parts, the equation (R) and
its integral will give
d^
[938] 4r^.dr^
df
= ^SfQ.rdr;
therefore
[939] ^ + n" u = — 2Q.^' (r^) — 8V (r^) JQ ' r d r.
[939'] Now from the equation w = v}. (r^) [936"] , we deduce du==2rdr,-l>'(7^);
and r^ = 9 (w) [935"], gives 2rdr = du.(p' (u), consequently,*
[940] ^'(r^) = -l-.
O *• /T *•
Taking the differential of this equation, and substituting cp (u) for — - — , we
shall obtaint
Now when Q = 0, the term f{r).y{r^)-\-F{r).Vir^) + n^'^{r^) of the second
member must, by [936], be equal to nothing, and as this is simply a function of r and
constant quantities, it must be identically nothing, and the preceding expression will become
as in [939].
* (654) Substituting in du=:2rdr.-\^' (r^), the value of 2rdr = du.((! (w),
and dividing by du . cp' (u), we get [940].
t (655) The differential of [940] is 2rdr.y (r^) = ""^^.^"^ » substituting
2i'dr=:du.(p' (m), ' and dividing by du .(p (m), we obtain [941].
n.vi.<^47.] PERTURBATIONS. 519
cp'{uf
^"(^)=-^3; [941]
(p" (w) being equal to ' ^ , in the same manner as 9'(w) is equal to [941']
'_^^^^ . This being premised ; if we put
du
U = e . COS. (nt-\-s — -a) -\-5u, [942]
the differential equation in u will become*
^ d^ .Su . ^ , 4:.(?" {u) .^ J / / N , 2 Q
o = ^^-+«^a«--^yi./Q.<^«.,'(«) + ^; [943]
and if we neglect the square of the disturbing force, u may be supposed equal [943']
to e . COS. (nt-{-s — «) in the terms depending on Q.
T
The value of - found in § 22 [669], gives, by carrying the approximation
to quantities of the order e^ inclusivelyt
* (656) From M [942] we get —— - = — n^e. cos. (n^-j-s — «) -j — t^5 hence
— — -\-n^u = — ~ — [- n^ . 5 u, which bemg substituted in [939], as also the values of
^'{r% 4."(r2) [940, 941], and 2r dr==du.(^' {u), [939'], we shall obtain by
reduction the expression [943].
f (657) In finding 9 [u) or r^, and its differentials to be used in [943], we may, as in
[943'], put u=e . cos. {nt-\-e — «), which gives cos. {nt-\-s — ©) = - , also by [944a]
[6, 7] Int. cos.(2»i + 2s — 2«)==2.cos.2(ri^4-e — «) — 1 = — — 1, and
4 It" 3 u
COS. 3 . {nt-{-s — -cy) = 4 . cos.^ (n i + s — «) — 3. cos. {nt-\-z — ■») = —
c2
e3 e
These values being substituted in r, [659], altered as m [669], namely
t
'2
e2
\-\-\e^ — e.cos. {nt-\-z — «) — -.cos. 2. {nt-\-t — «)!
»• = «•% 3e3
— . [cos. 3 . (n ^ + s — •«) — cos. {nt-\-s — ■«)]
8
itbecomes r = a . J 1 +^ e^ — w — -.T-^ ij — -^.f-^ t) S ' which, by
reduction, is as in [944], terms of the order e* or u\ being always neglected; squaring this
we get r^, which, by [935"], is equal to 9 {u).
620 MOTIONS OF THE HEAVENLY BODIES. [Mec.Cel.
hence we deduce
[945] 7^=aK{l+2e' — 2u.(l—i 6^) — 2^^ — w^} = ?(«).
If we substitute this value of <p(w), in the differential equation in ^w [943],
and resubstitute Q = 2/di?+r. f-^ j [934], and u = e. cos. (nt-\-s — to);
we shall have, by neglecting quantities of the order e^*
[946]
— ^/wf?^rsin.(7i^+s~T3).[l + e.cos.(«^H-s— TO)]J2/di2+r/^Vl.
* (657a) Taking the first and second differentials of cp («), [945], we get
[946a] 9'(«)=a2.{— 2.(1— ie^) — 2w — 3^2^; 9"(M) = a2. {— 2— 6 w|.
This value of cp' (w) gives
neglecting terms of the order e^ or vP. If in this we substitute for u, | u^ their values
[944a],
— M = — e . cos. {nt-^s — to), — hu^= — i e^ — ie^. cos. {2nt-\-2s — 2 «),
it becomes
-i- = — 1 ^l+ie2 — e.cos. (w^ + s — to) — ^e^.cos. (2wi + 2£ — 2 7»)] .
, ,., , . 4(p"(m) _4a2.(_2— 6«) , , .
In like manner we obtam - ^- ^,..[_2.(i_^e^)_2.&c.]p' °'' ^^ "'S^'''^"§
terms of the order e^, w^,
4(p^^(tt) — 4a2.(_2— 6w) 8+24m . — (84-24it)_ 1
'^'{^vif~ a6.^— 2— 2«p ■~"a4.(_2— 2u)3 "~a4.(8+24«)~ a^ *
In the terra f Qdu.q)'{u), we may put du = — e .ndt .sm.{nt-\-s — to), [944a],
(p{u) = a^. ( — 2 — 2n) = a^ . { — 2 — 2 e . cos. {nt-\-s — •zrf)}, and it becomes
fqdu.cp. (u) =fe .ndt. sin. {nt-\-s — zi).a^.{ + 2 + 2e. cos. (w i + s— -sj) }. Q
= 2a^e.fndt. sin. (w ^ + s — -cj) . ^ + e . cos. {nt-}-s — «) |. Q-
Now by substituting, in [943], the preceding values of — -r— ^ ; tttt 5 / Q «^ « • <p' («) ;
and Q [934] we shall get [946], which is exact in terms of the second degree in e, because
-4<P"(m)
<p'{Tt)3 '
of the first degree, so that their product must be exact in the second degree.
the term — rr-r-^j which was computed to the first degree, is multiplied by du.<p' («),
n. vi. §47.] PERTURBATIONS. 521
When we have found ^ u, by means of this differential equation, we shall
have 5 r by taking the differential of r relative to the characteristic 6, which
gives*
6r =— a(5?i.514-feH2e.cos.(n^+s— «) + fe^cos.(2w^+2£— 2t^)|. [947]
This value of 6r will give Sv, by means of the formula (Y) [931] of the
preceding article.
It remains now to determine 5s. If we compare the formulas (X), (Z),
[930, 932] of the preceding article, we shall find that 5r is changed into 5s,
by writing f^\ for 2fdR + r.f^^ in [930]; hence it follows, [947]
that to obtain S s, it will be sufficient to make this change in the differential
equation in 5u [946], and then to substitute the value of Su, given by this
equation, in [947]. This value of 5u we shall denote by Su', and we shall
have, [946, 947], f
^.fndtAsm.(nt-\-s — 73).{l + e.cos.(n^+£ — ^)}'(~JTn ' (^')
6s = ^a5u'.{\ + ie' + 2e. COS. (nt+f.—^)-\-^e'.cos.(2nt+2s—2ti)}.
The system of equations (X'), (Y), (Z'), [946, 931, 948], will give in a very
simple manner the motion of m, taking notice only of the first power of the
disturbing force. The consideration of the terms of this order, is very [948]
nearly sufficient in the theory of the planets ; we shall therefore proceed to
* (658) The differential of the value of r, [944], taken relative to the characteristic S, is
8r= — aSu.{l—^e^-Jr2u-{-^u^], but by [944a], 2m = 2 e. cos. (n / + £ — «) ;
I ^2 __ 9 g2 _j_ 9 g2 ^ j.Qg^ (2 n < + 2 s — 2 «), hence 5 r becomes as in [947].
f (659) That is, we must change in [946], S u into 8 u', and 2/d R-\-r \-T) , into
f — j, this gives the first of the equations [948]. The second is obtained by changing 8 u
into 8u' in the equation [947].
131
[948]
522 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
deduce, from these equations, such formulas as may be convenient, for the
determination of the planetary motions.
48. It is necessary, for this purpose, to develop the function jR in a
series. If we notice only the action of w! upon m, we shall have by
§ 46 [913]
(-9491 R = 5^ ^^ , ^- 1 .
This function is wholly independent of the position of the plane of x and y ;
[9491 for the radical \/ {d — xf + {}/'—yf + («'— zf, denoting the distance of m!
from »?,* is independent of it ; therefore the function
x" + f -\- z" -^ a!"" -\- ip -\- 2!^ ^2xx' — 2yiJ — 2 z z' ,
is also independent of it ; but the squares of the radii vectores y? -{-y^-^-z^,
x'^ + y'^ -\- 2'^, do not depend on the position of that plane, hence the
quantity xx' -\-y'i/ -\-z2f, and therefore the function i?, must be independent
of it. Suppose now in this functionf
[950]
[950a]
X = T . cos. V \ y z= r . sm. v ;
x' = r' . cos. v' \ 2/ = / . sin. v' ;
we shall have
* (660) Developing the terms denoting the square of the distance of m from m', namely,
(^'_a;)2_j_(y_j,)2_j_(^__^)2^ [109a], it becomes
^j^fj^z^-\-x'^-^y"'+z'^ — 2.{xx'-\-yy'^zz),
which ought to be independent of the situation of the plane of a?, y, and as c^-]-if^-\-z^=r^,
x'^-\-yf^-{-:^^ = r'^, [914'], they are also independent as observed above. The
difference 2 . {x x' -\- yi/ ~\- z z'), and xx' -\-yy' -]r zz', must also be independent,
hence every one of these terms of R, and therefore R itself must be independent.
f (661) The values of x, x', y,y', [950], are lil^e those in [927]. They give
a^ + y2 = r2; x'^ + i/^^i"^ ;
xx' -{-y^ = rr' . (cos. v . cos. i/ + sin. v . sin. v') =irr'. cos. {1/ — v),
[24] Int. These being substituted in [949] give [951], by developing {x' — a?)^+(y' — y)^
as in last note.
n. vi. § 48.] PERTURBATIONS. 623
P m' . \r / . COS. {v — v)-\-zz'\ m!
[951']
[953T
(/2_|_^2)| Jr2— 2r/.cos.(»'— ») + r'2 + (z'— 5^)2|^ [951]
The orbits of the planets being nearly circular, and but little inclined to each
other, we may select the plane of x, y, so that z, z', may be very small. In
this case r and r' will differ but very little from the semi-transverse axes a
and a' of the elliptical orbits ; we shall therefore suppose
r = a,{\+u)', r' = a' . (1 + <) ; [952]
u^ and u' being very small quantities. The angles v and v' differ very little
from the mean longitudes nt-{-s, ti! t-\-^ ; we shall suppose [952']
v = nt-\-z-\-v^', v' = n't-^s' J^v' ', [953]
v^ and vl being very small angles. Then by reducing J? to a series
arranged according to the powers and products of w^, v^, z, w/, z?/, and z',
this series will be very converging. We shall put*
-^ . COS. (n't — nt + s'— a)--{a2__ 2 a a' .cos. (n' t — n^ + s' — 0 + «"} ""^
= I A^'^+A^'K COS. (n' t-^nt + s' — s-) -\- A^K COS. 2 (n' t — nt + s' — s) [954]
+ A^^\ COS. 3 (n' t — nt + s' ^s) + ^c.
= 12 . ^». COS.*. (n't — nt-{-s' — s).
* (662) After substituting the values [952] and [953] in R [951], and developing it
according to the powers and products of u^, m/, v^, r/, zf^ . {z' — z^. The part which is
wholly independent of the quantities u^ , m/, &;c., is evidently equal to the first member of -
[954] multiplied by m' ; and the first term of the factor of {z' — zY . f' is equal to the
first member of [956]. The form of the series in the second members of [954, 956] is
evident from the usual rules of development. Now if for brevity we put
T=n't — nt-\-s' — e, and W=ft-\-zi, [954a]
the second member of [954] will become
i .^o)_^^(i), COS. r+^(2) . C0S.2 T4-^3) .COS. 3 T+ &c. ;
and as cos. T = cos. ( — T), cos. 2 T = cos. ( — 2 T), &c., this may be written
M^'^ + i-^'^.{cos. T + cos. (— r)| +i.^(2).|cos.2 T+cos. (—2 T)} +&c.,
and by putting A^^^ = A'^-^\ JP'>— v2(-2>, &c., [954"], it changes into
i ^°> + i JP'^ . cos. r+ 1 S^^ . cos. 2 r + fee. i
+ i S-^^ . COS. (— T) + i .4(-2> . cos. (— 2 T) + &c. V
which is evidently expressed by the general formula ^ 2 . S"^ . cos. i T, taking t from
— 00 to 4" o°> iJicluding i=0.
524 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
[954'] We may give to this series the form, ^."s.A^K cos. i.(n' t — nt-\-z — s), in
uso of the which the characteristic of finite integrals 2, refers to the number i, and includes
2 all whole numbers from i = — co , to z = co ; the value i = 0, being also
[954"1 comprised in this infinite number of terms ; and we must also put A'^~'"^=A^\
This form has not onlj the advantage of expressing, in a very abridged
manner, the preceding series, but it gives also the product of this series by
the sine or cosine of any angle y^+^ ; since it is easy to perceive that this
product is equal to*
* (663) Using T, TV, [954a], we shall have, for the two terms of ^ 2 . ^« . cos. i T,
depending on any integer i, ^ A^'^ . cos. i T-f- ^ •^^"'^ • cos. ( — i T), or as it may be
written A^"^. cos. iT, this being multiplied by sin. TV, and reduced, as in [18] Int.,
becomes M® -sin. (t r+ «^) +i^« . sin. (— i r+ ^), or
A ^« . sin. {i T+ W) +i^(-'>. sin. {—i T+W),
which are evidently the two terms depending on the Same integer i in l'E.A^'\cos. {iT-\-W),
Hence
[9546] sin. ^ . ^ 2 . ^® . cos. i T= | 2 . ./2® . sin. {i T + W),
which is the upper formula of [955]. The lower formula is obtained by writing in the
preceding \ic -\-W ior W, ^if being a right angle, for this changes sin. W into cos. W,
and sin. (i T+ W) into cos. {i T+ W), hence we get
[954c] cos. ?r. i 2 . ^» . cos. i T= i 2 . ^« . COS. (i T+ W).
From these two equations the following may be deduced,
[955a] sin. W . ^ :s . i A^'K sin. i T= — \^.i A^'^ • cos. (iT+W),
[955b] COS. i^. i 2 . i^» . sin. i T= i 2 . i ^» . sin. (J T+ TV),
[955c] sin. TV.^^.i ^® . cos. i T=: J 2 . i A^'^ . sin. [iT+TV),
[955d] COS. TV.ii:. iS^ . COS. i T= | 2 . iA^'K cos. (iT+TV),
For the equations [9546, c], being identical, exist with all values of T; we may therefore
take their differentials relative to T, and divide by — dT and we shall obtain [955a, 6] ;
and if in these we change i T into « T -f- ^ *? we shall get [955c, d\.
In like manner we may take the differentials of the equations [955a— <Z], relative to T,
and dividing by dzdT, we shall obtain the four following equations
sin. W^. I 2 . i2 ^(') . cos. i T = i 2 . i^^W . sin. {i T + TV),
cos. TV.^S. i^A^'^ . cos. i r= I 2 .i2^') .COS. {i T+ TV),
[955e]
[955/]
[955g] sm. fr . i
sin. ?F . i 2 . i2^« . sin. i r= — i 2 . i^^w . cos. {iT-\- TV),
[955h]
cos. TV. in. v'S^ . sin. i 7=^ 2 . t" S^ . sin. (i T+ TV),
and so on for others.
ll.vi.<^48.] DEVELOPMENT OF THE FUNCTION R. ^25
sm.
^^.A^^.l2l{i'(n't — nt + B'^s)Jr.ft + zsl [955]
From this property we may obtain very convenient expressions of the
perturbations of the motion of the planets. We shall also suppose
{a^ — 2aa'. cos. (n' t — nt + s' ^s) -{- a'^} ~^
= J.2.JB».cos.i.(n'^ — n^ + s' — 0 ; ^^^^
B^~'^ being equal to B^'K This being premised, we shall find, from the [956']
theorems of ^ 21,*
* (664) If we develop the terms relative to z, zf, in the value of R, [951], it becomes
„ m'r. COS. («' — v) m! . wIztI
^^ V^ |r2 — 2rr'.cos.K— v) + r'2^i "^ ^3
• COS. {v—v)-\- o C.0 o.^ .r.. u,'.,^A.^.A + &^C.,
[955i]
2/4 ^ >' ' 2.^r2— 2r/.cos.(w'— r)-^ r'sp
each of the terms of which may be further developed by substituting for r, /, v, v', their
values [952, 953J. It has been supposed sufficiently accurate to put in the three last terms
a, a', nt-\- B^ n' t -\- s', for r, /, v, v', and using for brevity T, [954a], they will
, m'zz 3m'az'2 m'.(z'— 2)2
become 27^' '^^'' T + 2.[a^-2aa'.cos.T+a'^)^' If we substitute, for the [956a]
denominator of the last term, its value deduced from [956], the three preceding terms will
become like the three last of the expression [957]. The two remaining terms of R, [955t]
being taken for u, [607, &;c.], we shall have
^^m'r.co3.{v'-v) rn[
/a ^r'2_2r/.cos.(v' — •D) + r2|i' ■"
and if we use the values [952, 953], putting a=au^, 0^ = 0 m/, a" = v,' — v,, we [956c]
may, as in [607 — 612], develop u according to the powers of a, a', a", observing that the
two terms vj — v^, are connected together, because they occur only in this form in w ; by this
means we shall get
[957a]
.=^+.(£-:)+..(^,)+„-.(i5,)+,.„..(i^)+„..(^,)
+i-«--(T.i)+««"-G-^^-)+«'»"-G-^.)+i---(^)+^-
m which Z7= -— . cos. T— TT—Ti — ; ^"T^^i > or hy [954^, [9576]
a'2 |a2 — 2aa'.cos. T-j-a'Sp' ^ *- -" *■ -■
=f'.2.^('\cos. i T, this being the value of m, when a, «', a", are nothing. In the terms
(du\ /du\
Tar Vrfo?/' we must also put a, a', a", equal to nothing, or in Other words r = a, [957c]
r' = a', and v —v=n' t^nt-{-s' — s—T, but as a or a m, is found in «, only as it
132
526 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
[957]
i? = — . 2 . ^ » . c O S . Z . (ll' ^ — n f + s' — s)
+ — . w^ . 2 . a . ( — — J . COS. I. (n t — nt-\-z — s)
+ — .<.2.«'.(^-^^J.cos.z.(?i'i — ni + s' — s)
m'
A ,uf.^.a. — ;-^- ) . COS. I . (n t — nt-{-z — s)
4 ' V "« /
, wi' / . fddA^^\ . ^ ,^ ^ , ,
_|___.^^^^.2.««.|--^j.COS.i.(7i/ — ni+s' — s)
+ — .<^2.a'^(^-^-^j.C0S.*.(n'^— 7lf + £' — a)
5- • (^/ — z?J . w^ . 2 . z a . f — — j . Sin. I. (n t — nt -{-s — e)
— — . (v; — vf . 2 . ^2 . J (^) . COS. i.(n't — nt + s' — B)
+ --73 —,^.cos.(nt — nt + s' — s)
4 ^
+ &C.
If we substitute in this expression of R, for w^, w/, i)^, v', z, zf, their values
relative to the elliptical motion, which values are functions of the sines and
enters into the values of r, [952], we shall have (t")'^!^)*!^")' ^"^
r=:a-\-au, = a-{-a, [952^, gives f-^j:^l, hence (—\=:(—\ Again, if
we compare the functions u, [956J], and U, [9576], we shall find that u is composed of
r, /, v — «, in exactly the same manner- as C7is composed of a, a', T, and it is evident
from a little consideration that (t~ ) will be exactly equal to the value of (t-)j in which
a, a, T, are written respectively for r, r', v — v, as in [957c] ; therefore in the above
value of u, [957a], we must put (—j == (—] = (— j, and in like
manner
U. vi. §48.] DEVELOPMENT OF THE FUNCTION R. 527
cosines of the angles nt-{-s, n' t + s', and of their multiples ;* R will be
expressed by an infinite series of cosines of the form tn!k.cos.(i'n't — int-\-A),
i and i' being whole numbers. [957^]
It is evident that the action of the bodies m", m"\ &;c., on m, will produce
in R terms like those which result from the action of m', and we may obtain
these terms, by changing, in the preceding expression of i?, all the quantities [957"]
which refer to m', into the corresponding quantities relative to w", m'", &c.
We shall now consider any term of R, represented generally by
m' k . COS. (i' n't — int-\- A). [957'"]
If the orbits were circular and in the same plane, we should havef i' = i ;
/du\ /Ju\ /^\ . (du\ / du \ (dU\ /ddu\ /d d u\ /ddU\^
\da:)~\di')~~\da')' \da:j~\dA^^^))~\dT)' \d^)~\d^ )~\d^ P
/ ddu \ fddu\ / ddU\ _ /ddu\ /ddU\ / ddu \ /ddU\ [957d]
Kdada') ~~ \d7d^j ~ \dadaj'^ \daf^)~ \1^'^ ) ' \dada") ~ \dad t) '
/ddu \ _ / ddU\ ^ /^^\ /'^'^ ^\ . ^^'
\d a'd^')~ \da'd t) ' \d d'y ~ \dT^) ' Jsdj o-
which being substituted in m, [957a] we get
Substituting in this the value of ZJ^f ' . 2 ^". cos. i T, [957&], and those of a, a, a", [956c],
it becomes, term for term, like those in the ten first lines of [957]. These, with the terms
in Zj z', above found, [956a], constitute the complete value of R, [957].
* (665) This is evident from the equations [669], or from [659] and [668].
f (666) The orbits being circular, we should have m = 0, w' = 0, [952], also
r = «, r' = cd. ]VIoreover, the motions being uniform, and in the same] plane, we shall
have V/= 0, vj = 0, [953] ; there being no reductions like those in [675, 676'] ; hence
v' — V = n't — nt-{-^ — s= T, Substituting these and z = 0, z'=Oj in [951] we
, „ _ m' a.cos.T m' i. , i f-«„.n .
Shall get R = —^r J^^^^^ZtT^I^^ ^ "^ -'' '' ^^^^
= f ' . 2 ^^'^ . COS. i{n' t — nt-{-^ — s), and in this last expression the coefficient of n' t [957g-]
and n ^ is the same quantity i.
528 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
therefore i' cannot exceed or fall short of i, except by means of the sines
or cosines of the expressions of u^, v^, z, u', v'^ 2', which, being combined
[957i^] with the sines and cosines of the angle n' t — n ^ -j- s' — s, and its multiples,
will produce sines and cosines of angles, in which i' differs from i.
If we consider the excentricities and inclinations of the orbits as very small
[957 V] quantities of the first order, it follows, from the formulas of § 22,* that in the
expressions oiu^^v^, z, oxr s,s being the tangent of the latitude of m, [1027a],
[QST'^i] the coefficient of the sine or cosine of an angle like f .(nt-{-s) is expressed
by a series, whose first term is of the order/; the second term of the order
[957v»] y_[_ 2 ; the third term of the order f-\- 4 ; and so on. It is the same with
the coefficient of the sine or cosine of the angle /' .(n't-\- s') in the
expressions of m/, v', z'. Hence it follows, that if i' and i are supposed
positive, and i' greater than i ; the coefficient k, in the term
[957vi'i] w! k . cos. (i' n' t — int-\- A)
is of the order i' — i, and in the series which expresses it, the first term is
of the order i' — z, the second term of the order i' — ^ + 2, and so on;
[957«] so that this series is very converging. If i be greater than i\ the terms of
the series would be successively of the orders i — ^', i — *' + 2, &c.
Let ^ be the longitude of the perihelion of the orbit of m, ^ the longitude
[957^] of its node. In like manner let w be the longitude of the perihelion of w',
and 6' that of its node ; these longitudes being counted on a plane but little
inclined to that of the orbits. It is evident, by the formulas of ^ 22, that in
the expressions of u^, v^, [669, 952, 953], and z, [679, &c.], the angle
nt-\-s is always accompanied by — w or — 6; and in the expressions of w/,
[957=^'] v', zf, the angle n't-{-s' is always accompanied by — «', or — ^' ; hence it
follows that the term tn! k . cos. (i' n' t — int-{-A), [957'"] is of the
following form :
[958] in! k . cos. (i' 7}! t — int-\- %-! — iz — gzi — ^ vi — ^" ^ — ^" ^'),
[958'] ^, ^, ^", ^'", being whole numbers positive or negative, so that we shall
* (667) The formula [659], altered as in [669] gives u,^ u] ; also [668] altered as in
[669], gives v^, i*/; the reduction to the fixed plane is made as in [675, 676', &€.] Now
by examining all these, it will evidently appear, that the order of the coefficient of the sine
or cosine of any angle, as f{nt-\- e)j is as in [957'*, Stc]
U.vi. §48.] DEVELOPMENT OF THE FUNCTION R. 529
have*
0 = i' — z — g — ^— ^'— ^"; [959]
which also follows from the consideration that the values of R and its different
terms are independent of the position of the right line from which the
longitude is computed. Moreover, in the formulas of ^ 22, the coefficient of
the sine and cosine of the angle -^ [669, 8tc.], has always for a factor the [959']
excentricity e of the orbit of m ; the coefficient of the sine and cosine of the
angle 2 «, has for a factor the square of the excentricity c^, and so on.f
Likewise the coefficient of the sine and cosine of the angle 6, [676', &c.],
has for a factor tang. ^ <?, <p being the inclination of the orbit of m upon the [959"]
fixed plane. The coefficient of the sine and cosine of the angle 26, has for
a factor tang.^^tp, and so on ; hence it follows, that the coefficient k has for
a factor e^ . e'^ . (tang. ^ (py . (tang. ^ cp'y" ; the numbers g, g', g", g"', being
taken positively in the exponents of this factor. If all these numbers are [959"1
* (6G8) It is shown in [957g-], that when m,, w/, v^, v,', s/, z, are nothing, which takes place
when g = 0, g'=0, g" = 0, g'" = 0, the term { — i must be equal to 0. Now
if g' becomes 1, the term i' must be increased by unity, because the change must arise from
multiplying the sine or cosine of an angle like i{nt — nt-\-^ — s), by one depending
on n' t -\-^ — z/, which must increase the coefficient of n' t, as much as it does that of
— z/j and must still leave the expression [959] equal to nothing. By following this method
we shall easily perceive that the equation [959] always takes place for all values of i', i
g, &cc. The same conclusion may also be drawn from the other consideration mentioned
relative to the position of the line from which the longitudes are computed. For the function
jR is not affected by the position of the plane of cc y, [949'], it cannot therefore be affected
by the position of the axis of x, from which the angles s, s', -&, z/, 6, 6', are computed. If
we now suppose the origin to be altered so as to augment these quantities by the angle b,
the angle i' n' i — int -\-i' ^ — is — g-a — g' zs! — g" 6 — g'" 6', of the expression
[958] will be varied by i'b — ib — gb — g' b — g" b — g'" b, which expression ought
to be equal to nothing, in order that the part of R denoted by [958] should remam unaltered.
Putting it therefore equal to nothing, and dividing by b, we obtain
0 = i'—i-g-g'-g"-^g"\ as in [959].
f (669) This appears from the formulas [659, 668], altered as in [669]. The remarks
relative to (tang. J (p'y't appear from the formulas [675, 676'].
133
530 MOTIONS OF THE HEAVENLY BODIES. [M6c. Cel.
really positive, this factor will be of the order i' — i, by means of the
equation [959],
[960] 0==i' — i-^g—g'—g"—g"';
but if one of them as g is negative and equal to — g, this factor will be of
the order i' — i-\-2g.* Retaining therefore, among the terms of R, only
those depending on the angle i' n' t — int, which are of the order i' — i,
[96(r] and rejecting those which depend on the same angle, but which are of the
orders i' — z-(-2, i' — i + 4, &c., the expression of R will be composed of
terms of the form
^^1^ H.e^.e'^ . (tang. I ^Y . (tang. \ ^'Y"
. cos. {i' vii — int-^i' i — is — g.-a — ^ . «' — ^' . 5 — ^"- 0 5
H being a coefficient independent of the excentricities and inclinations of
[961'] the orbits ; the numbers gy g', ^', g"', being all positive, and such that their
sum is equal to i' — i.
If we substitute in R the value of r [952],
[961"] r = a . (1 + M^),
we shall havef
fdR\ fAR\
* (670) Suppose the negative value of ^ to be — G, G being a positive number ; the
factor k will contain the terms e^ . e'^' . (tang. ^ cp)^' . (tang, J c^'Y" , and it will be of the
order G -\- g' -\- g" -\- g"' • In this case the equation [959] will become
0 = i' — i + G — ^' — g" —g'", which gives G +^ +^'+^" = i'_ i + 2 G.
consequently that term will be of the order t — i-\-2 G, exceeding i' — i by the positive
quantity 2 G.
f (671) After substituting r = « . (1 -f u,), [952], in R we must consider u, as not
containing a explicitly, and then the partial differential of r = a . (1 +W/), relative to a,
will give ( y^) = ( 1 + M,)> whence (~j = - . Now, by considering ^ as a function
/dR\ /dR\ /dr\ /dR\ r
of a and then as a function of r, we shall have ['daj'^ \dr) ' \Ja) "^ Vdt) ' a '
whence we easily get [962]. In like manner, since by [953], v=nt-{-s-{-v^, we may
first suppose /J to be a function of v and then of nt-j-s-\-v^, and we shall have
n. vi. § 49.] DEVELOPMENT OF THE FUNCTION R. 531
If in the same function we substitute, for i/^, v^, 2, their values given by the
formulas of § 22, we shall have
provided we suppose s — « and s — 6 constant, in the differential of i2, taken [963']
with respect to s ; for then w^, v^, z, will be constant in that differential ;
and as we have v = nt-\- £-{-v^f it is evident that the preceding equation
takes place. We may therefore easily obtain the values of r . f -y- J , and
which enter in the differential equations of the preceding articles.
m
[963"]
dv
when we shall have the value of R, developed in a series of cosines of angles
increasing in proportion to the time t. The differential dR will likewise be
very easy to determine ; taking care to vary in i2, only the angle n t,
supposing n't to be constant ; since di2 is the differential of R, taken on [963'"]
the supposition that the co-ordinates of m', which are functions of n' t, are
constant [916'].
49. The difficulty of developing jR in a series, is therefore reduced to
that of forming the quantities A^^, J5®, [954, 956], and their differentials,
relative to a or a'. For this purpose we shall consider the function
(a^ — 2aa'. cos. 6 + a'^)~%
and develop it according to the cosines of the angle 6, and its multiples ; if
we put
^ = « ; [963ir]
a
it will become
(a' — 2a a' . cos. & + a'^)-' = d-^' . {1 — 2 a . cos. &-^o?]-\ [963-]
f — j = f— j.f — j; and if we take the differential of r= nf -j- ^-j-^/* [953],
relative to s, without varying v,, we shall have l — \= 1, hence l—-\=z(—y
Now by comparing the second equation [669] with [675, &;c.], it appears that in «, the terra
g always occurs in the form s — ts or s — &, we must therefore suppose e — «, s — d to
be constant, in finding [963] in the method here used.
532 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
Suppose
[964] (\—2a.cos.6 + a^)-' = ^.b^^'> + UlKcosJ-{-bf.cos.2&+¥f.cos.S6-Jrkc.;
b^s\ b^s^ b'-f, kc, being functions of « and 5. If we take the differentials of
the logarithms of both sides of this equation, with respect to the quantity e,
we shall have
[965] — 2 5 . a . sin. 6 _ — 69 . sin. 6 — 2 . 6^^^ sin. 26 — &c.
1— 2a.cos.^ + «' ~~ h • b^^ + b^s . cos. 6 + b^l^ . cos. 2 ^ + &c.
Multiplying by the denominators, to clear from fractions, and comparing the
similar cosines, we shall have in general*
m] /,« _ V^-l).(l+«^).^>r^>-(^ + 5-2).«.^>^^) .
* (672) Putting for brevity JV= ¥p . sin. 6 + 2 6f . sin. 2 ^ + 3 6f . sin. 3 ^ + he.
D = J jf 4- b^^ . COS. 6 + bf . COS. 2 6 + bf . COS. 3 5 + &;c., the equation [965] will
— ^s.a.sm.6 — -^ ht i • i • i- i -rv /^ ^ . , o\
become - — 71 — - = —77- . Multiplying this by V . (1 — 2a. cos. 6 4- a-'),
1 — 2 a . COS. 6 -f- a^ D j. j <d j \ 1 y
and reducing we get JV . ( 1 -}- «^) — 2 JV . a . cos. & — 2 D s . a. sin. ^ = 0.
Resubstituting the values of JV, J), performing the multiplications, and putting as in
[18, 19] Int. 2. COS. ^. sin. m^ = sin. (?;z-|- I) . ^ + sin. (w — 1) . ^,
2 . sin. & . COS. m6= sin. [m-{- 1) .6 — sin. (m — 1 ) . ^,
we shall get the following expression of the different terms of the preceding equation. The
first line is the value of JV. (l-f-a^), the second and third lines are — 2JVa.cos.^,
the fourth and fifth lines are — 2Ds a. sin. 6.
(14-a2).J(i).sin.^+(l + a2).2tf.sin.2^+(l+a2).3if.sin.3^+(l+a2).4i(''>.sin.4d+^
r —u.Up.sm.26— 2 a . if .sin. 3^— 3a . 6f . sin. 4^ — &c. i
(—2a.bf\sm.6— 2a.bf\sm.2&— 4 a . J^"^) . sin. 3^— 5 a. if . sin. 4 a — &c.^
f — 5a.Jf.sin.^ — sa.6f.sin.25 — sa.if.sin.35 — sa.if.sin.45 — &;c.^
^_j_sa.Jf .sin.5+ sa. if .sin.25+ sa.if .sin. 3 5 + s a .if . sin. 45 + Stc. ^
The sum of these three expressions being equal to nothing, the coefficient of each cosine
must be equal to nothing. Now the coefficient of sin. {i — 1) • 5, in the preceding sum,
i being any positive integer greater than unity, is
(1 + a2) . (i— 1) . i(j-i) _ (i _ 2) . a . i^'-2) _ i a . iW — s a . i^'-^) -{-sa.b<^^',
for this is evidently the case if i be 2, 3, 4, or 5, and the law of continuation is manifest.
Putting this coefficient equal to nothing we get [966].
[966a]
II. vi. ^49.] DEVELOPMENT OF THE FUNCTION R. ^^^
we shall therefore have ¥f, 6^?, &c., when we know 6^°\ 6^^.
If we change s into 5 + 1 , in the preceding expression of
(1— 2«.COS.^ + a2)— ,
[964], we shall have
(l_2a.COSJ + a2)-*-^ = 1.6(°^l + 6(^|l.COS.^+6^i.COS.2^ + 6^l.COS.3d+&C. [967]
Multiplying both sides of this equation by 1 — 2a. cos. 6 + a^, and
substituting, for (1 — 2a. cos. & + «^)~'> its value [964], we shall have,
h . fef + b'J^ . COS. 6 + 6f . COS. 2 ^ + &c.
=(1— 2«.cos.^+a2).5i.6(«)^i+6(\i.cos.4+6i%i.cos.2a+6f\.i.cos.3^+&c.};
hence we deduce, by comparing the similar cosines,*
Up = (1 + «^) . 6('V, — « . 6^^/) — « . U;:^p. [969]
The formula (a) [966] gives,t
*'+' (»•-«).« !
the preceding expression of 6® [969], will therefore become
By changing i into ^ "+ 1 in this equation, we shall have
' "" i — s + l
and if we substitute for M:^i\ its preceding value [970], we shall find
(i — 5). (I— 5+1). a
[970]
[972]
[973]
* (673) Putting as in [20] Int., 2 cos. 6 . cos. m6= cos. (m -f 1 ) . « + cos. (to — 1 ) . ^, [969a]
and then making the coefficients of the term cos. i6, equal to each other, in both members
of the equation [968], we shall obtain the expression [969].
f (674) By writing t + 1 for i, and s-j-l for s, in [966], we get [970]. Substituting
this value of h^l in [969], we shall obtain, by reduction [971].
134
^^4 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
These two expressions of 6» [971], bf+'^ [973], give*
[974] s •C^ + ">^- ^- , -"-^^ (6;
0^ + 1— (1— a^/ '
substituting for bi"+^\ its value, deduced from the equation (a) [966], we
shall getf
llZl!) . (1 + «.) , 6(«) + Mi±fzzl) . „. 6('-^)
[975] ^w , = i . i : . (0
an expression which might have been deduced from the preceding [974], by
changing i into — i, and observing that
[975'] 5W = 5(-0.
*(675) Multiplying [971] by ^-^\ (l+a2), and [973] by — ^^^-^ii^— , and
adding the products we shall get
(i-±i).(l+a^).5f-2
(.) „ (•'— £±I)„_j«+«
P7U] =^-|z7-(i+<^)'-(£-7j-P-(i-»)-«''-'-(i + «'')T|-'S-.
+ P « • 7I.T • (1 + " ) — ^ " • 7I.T • (^ +'' ) 5 • **+! '
in which the terms of the coefficient of b^:^\^ mutually destroy each other ; and if we connect
together the terms of tlie coefficient of S^i^j multiplied by (1 + a^)^ the second
member of [974a], will become
or simply (1 — a^)^ . b%i. Dividing this by (1 — o.^f we shall obtain [974].
t (676) Changing i into i+ 1 in [966] gives b^-^'^==''^^'^''''^'f~}^'~^^''''^^~'^'
U 54-1)
Multiplying this by — 2 . . a, it becomes
_2.(i=d:li.„.6(H-.>=_?f.(i+„^).i<o+2.!^i=i>.a.tr";
s S s
substituting this in [974], and reducing, we shall get [975].
Il.vi. <§49.] DEVELOPMENT OF THE FUNCTION R. ^^o
Therefore we shall have, by means of this formula, the values of 6/+\, 6,+ij
6f|i, &c., when those of bi'\ W\ hf\ &c., shall be known.
Putting for brevity,
X = 1 — 2a. COS. d-\-o?, [975"1
and then taking the differential of [964]
X-* = 1 . h^P + up . COS. ^ + 5f . COS. 2 d + &C. [976]
relative to «, we shall get
d ¥°^ d W^ d b^^^
— 2 s. (a — COS. 6) . X-»-i= 1 . — i- + — ^ . COS. 6 + -— ^ . COS. 2 4 + &c. ; [977]
^ da a a a a
but we have*
— « + cos.a=^~"'~^; [978]
therefore we shall have
S.(l-a^) ^ , , 5.X-* dbf^ dUP ^ , _
^^ d.x-*-^ = 1 .— [-— - — . COS. d+&C. ; [979]
a a. da. a a.
whence we deduce generallyf
dUp
= ML-:i^.5(.^,^i.5(o.
da I a.(l — «2) 5 ' 1
If we take the differential of this equation, we shall havet
* (677) Adding — 1 + a^, to both sides of the equation [975"], and dividing by
— 2 a, we shall get [978]. Substituting this in [977] we get [979].
f (678) Substituting in [979] the value X"* [976], also that of X-^^, deduced from the
same formula, namely, ^ J^^ -|- 6^!].i . cos. & + ^' ? then putting the coefficients of
cos. i 6 equal to each other, on both sides of that equation, we shall obtain [980].
f (679) In finding the differential of the coefficient of bf, it will be convenient to put it
J , - C2.(i+s).a , t> , . , . ., , , i-f(i + 2*).a2
under the form ^ — ; — — > , which is evidently equal to ' ■ ' , , and
i 1 — a^ a ) a.(l — a2)
the differential of the first of these expressions, relative to a, being divided by d a, will give
the coefficient of J^'^, [982].
[980]
da
Substituting the value of b^\.i, given by the formula (6) [974], we shall
find
db^ ^^" + (^' + 2.).«^>^^^.^_2^-.-fl)^^,^,^ ^,81^
o36 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
dHl _ C ^+(^• + 2^).«^ ^ ^ . J 2 . (z + ^) . (1 + «^) ____t_ ) .^^
and again taking the differential, we shall get*
S.{i-s+\).a dhf+'^ 4.(t-. + l).(l+3«^) ^,^j^
(l_a2)2 • rfa (1— a2)3 * ^* *
Hence we perceive, that to determine the values of hf, and its successive
[DBS'] differentials, it is sufficient to know those of UP and U^. We shall find
these two quantities in the following manner.
If we put c for the number whose hyperbolic logarithm is unity, we may
put the expression x-* [975"] under this form,t
[984] X-^=(l— a.C "^ ) .\\—a.C ^ ) .
* (680) In finding this differential, it is only necessary to compute the coefficients of bf,
U^'^^\ those of the other terms having been already computed in [982]. Thus if the
coefficient of J^, in [981], were put equal to C, the term Cb^'^ would produce in [982] the
terms C . -^ ~5" vT" ) • ^^^ ' ^^ '^^ ^^^ expression [983], the terms
\da^J^ \daj da~\da^) "
Now C and (— ) having been found in [982], we have only to compute i~d~^)
arising from this term, and the similar one from the coefficient of b^'^^^ in [981].
f (681) Substituting in [975''], 2 cos. ^ = c^'*^^+ c~^'^^^, [12] Int., we shall
get _ _
[984a] >.= l—a.{c +c ) + a^={l—a.c )'{l—a.c ),
whose power — «, gives X-% as in [984]. The two factors of this expression being
developed, by the binomial theorem, become as in [985].
n. vi. <§ 49.] DEVELOPMENT OF THE FUNCTION R. 537
Developing the second member of this equation, according to the powers of
c , and c ; it is evident that the two exponentials c ,
and c , will have the same coefficient, which we shall denote by k.
The sum of these two terms k.c , and k,c , is 2k.cos.i& [984']
[12 Int.] ; this will be the value of b^K cos. id [964] ; therefore we shall
have hf^ = 2k. Now the expression of x~* is equal to the product of the
two series
multiplying these two series together, we shall find, in the case of t = 0,*
fe = 1 + ^ . „^ + rililili J . „4 + &c. ;
and in the case of z := 1 ,
therefore
* (681a) The two factors [985] being multiplied together, we shall find that the terms
free from tf, or in other words, those which correspond to i = 0, are evidently produced by
multiplying each term of the lower series, [985], by that immediately above it, and adding
these products together ; this gives the expression [986]. In like manner, the coefBcient h
of the term kc is found by multiplying each term of the lower factor [985], by the
term immediately following it in the upper factor, and adding these products ; the sum will be
the value of k in [987]. This product contains therefore the terms
k,e'^^^^-{-k.r^'^'^^=2k.cos.6, [12] Int.
Comparmg these with [976], we get, U^^ UP^ as in [988].
135
[985]
[986]
[987]
[988]
•^38 MOTIONS OF THE HEAVENLY BODIES. [Mec. C6L
To make these series converge, it is necessary that « should be less than
unity. This condition can always be satisfied, by putting « equal to the ratio
of the least to the greatest, of the distances «, a' ; having therefore assumed
[988'] a = - [963'''], we shall suppose a to be less than a'.
In the theory of the motion of the bodies m, m', m", &c., we shall have
[988"] occasion to ascertain the values of U^\ UP, when 5 = i, and 5 = f. In these
two cases, these values converge but slowly, unless « be a small fraction.
These series converge more rapidly when s = — i, and we shall then
have [988],
[989]
2 -i ' \2j ' \2aJ ^ \2.4.6y ' \2.4.6.8y T ^^'
wi) S^ l-l 2 A 1-1-3 4 1-3 1.1.3.5 g 1.3.5 1.1.3.5.7 „ . )
-i I 2.4 4 2.4.6 4.6 2.4.6.8 4.6.8 2.4.6.8.10 )
[989'] In the theory of the planets and satellites, it will be sufficient to take the
sum of the eleven or twelve first terms, and to neglect the rest ; or more
accurately, to take the sum of the remaining terms, as a geometrical
progression, whose ratio is 1 — a^.* When we have ascertained, in this
* (682) The expressions [989] may be put under the following form, in which Cj, Cg,
C3, &;c. ; Di, Dg, &c., denote the terms of the series, immediately preceding those, in
which these symbols respectively occur,
i5c.^=i+Q)!a^+g)!«^.c,+Q!«^^^ +(|^)!«^c„+&c.
5-,=-a+Q.«^ + (^^).a^A+(^^).«^A+(^f,).a^A + s.c
Now, when n is very large, the coefficients of 0? C„, o? J)„, are very nearly equal to
unity, and then the terms of the upper series following C^ become nearly
and the similar terms of the lower series are nearly equal to ^ . D^ . Before seeing
this publication I had used this method of finding the last terms of the series [989], and had
computed the values of U^, b^P, he, corresponding to the orbits of the planet IMars and
the Earth, by rapidly converging series, like those in [989].
ll.vi.<^49.] DEVELOPMENT OF THE FUNCTION R. ^39
manner, ¥1\, and 6^\, we shall obtain ¥l\ by making i = 0, and s = — J, [989"]
in the formula (6) [974], and we shall find
If in the formula (c) [975], we suppose i = l, and s = — ^, we shall get
,.^2..n+3.(i+^).n.
With these values of 6^?^, U^^, we may find, by the preceding formulas, the
values of ¥'^, and its partial differentials, whatever be the value of i ; thence [Qgi^
we may determine the values of b§ and its differentials. The values of
6f and 6| may be found very easily by the following formulas,*
Now to find the quantities A^^\ A^^\ &c., and their differentials, we shall
* (683) Putting i = 0, and 5 = J, in [974], we get
f (1 — a2)2 '
and by substituting the values of b^^, 6i , [990, 991], it becomes
f (1— a2)4
7 W
(1_„2)4 • -i— (1_„2)
__(l4.a8).jf + 2a.&f
Putting 1=1 and s = J, in [975], we get 6 3 = — ^; ^^^ ' 2- , and by
2 (1 — crfi
substituting b^^\ bf, [990, 991], it becomes
-(l+a2).{2«.5^!!^+3.(l+«^).g^]+2«.{(l+«^)-g^ + 6«-^-^|
t (l-a2)4
^-3.(l + a2)2 + ]2a2 (i) ^-3.(l-«2)2 oj ^ -3 (i,
(1 _ a2)4 • ^-i (1 _ „2)4^ • ''-i (1 _ a2)2 • ^-i '
as in [992].
54»0 MOTIONS OF THE HEAVENLY BODIES. [M6c. Cel.
observe that by the preceding article [954] the series*
[993] ^ . ^(0) + ^(») . COS. & + ^(2) . COS. 2 ^ + &c.
arises from the development of the function
[994] iL.^_(a2__2aa'.cos.d + 0-*,
[995]
-^
in a series of cosines of the angle d and its multiples. Putting -=«, this
function becomes [964]
vs^hich gives generally
[996] ^« = — i.&«;
a ^
when i is nothing, or greater than 1 independent of its sign. If i = 1 , we
shall have
[997] ^(1)== « _l^.j(i).
a'^ a' '
Hence we getf
[998] (^\=-l.iIi.(^A-
\ da J a' da \daj ^
now we have (-—) = -; therefore
\d a J a!
[999] (_^) = — _.^.;
\ da J a'^ da.
ft tt COS A 1 1
*(684) Putting - = a, in [994], it becomes -^^— ^.(1— Sa.cosJ + a^)-*,
and this, by [964], is equal to
a . COS. 4
■\, • { ^5^+4'^. COS. d + jf. cos.2d + &c. I ,
as in [995]. Now as this is equal to the expression [993], we shall get, by comparing the
terms depending on the same multiple of d, the equations [996, 997].
f (685) Taking the differential of [996], relative to a, always considering J^, as a
a
ftinction of a, and a as a function of a, we get [998], and as « = 3> [963'^], we shall
have (-^ ) = -, which, being substituted in [998], gives [999].
n. vi. §49.] DEVELOPMENT OF THE FUNCTION R. ^^1
and in the case of i= 1, we shall have*
\ da J a'" '
Lastly we have, even when z =: 1,
day a^ { da.)
[1001]
To obtain the diflferentials of A^^ relative to a', we shall observe that A^^ [lOOl']
being a homogeneous function in a and a', of the dimension — 1 , we shall
have, by the nature of such functions,!
"•(■^) + «-(^=-^^^' tl002]
* (686) Taking the differential of [997], relative to a, dividing it by da, and putting
( — ] = -, as in the last note, we shall get [1000]. In like manner, the diflferential of
[999], will give the first equation [1001], which is correct even when i = 1, because the
differential of [1000] agrees with [1001]. The differential of the first equation [1001]
gives the second, and so on, always substituting f — j =- .
f (687) A homogeneous function of fl, a!, &;c., of the degree denoted by m, is a function Homo-
in which the sum of the exponents of a, cK, &c., noticing their signs, is equal to m, in every function.
term of the function ; as for example cP -\- S a^ a' -\- a a a" -] — — , is a homogeneous
function in a, a', a", of the third degree. If we put a = i y, a' = tt/, a" = tt/', &c.,
in a homogeneous function A^'^ of the degree m, it will become of the form A^'^=t'^V, P'bemg [lOOOa]
a function of y, y', y", Stc, independent of t. Taking now the differential of this expression
relative to t, and observing that ^'^ contains t, only as it is found in a, d, &c., we shall get
But a=ty, gives f— j = y; hence t . (-—j z= ty= a; in like manner
Substituting these in [10006], multiplied by t, and in the second member of the equation
putting for m f . F, its value m A^'\ [1000a], we shall get
136
542
MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
hence we deduce*
&C.
[1001a] « • (-^j + « • (-7^ j + ^^- = "^ •^''-
Now it is evident that the first member of [954] is of the order — 1, in a, a', and therefore
its development in the second member, must be of the same order, consequently A^"^ in its
second member, must be a homogeneous function, in a, a', of the order — 1. Putting
therefore m = — 1, in [1001a] we shall get the equation [1002].
f (688) The first of the equations [1003] is deduced from [1002], by transposing
a.f — J — j. The next equation is the differential of this relative to a, divided by da. The
differential of the first, relative to a', being multiplied by — gives
and if in this we substitute the values of a' . \—r~r )» and aa' .{ -= — r—r )> deduced from
\ da / \dada J
the first and second equations, we shall get the third. The differential of this, relative to a,
a'
gives the fourth. The differential of the third, relative to a', multiplied by — gives
\ rfa2 y ' \ da!^ J \ da' J \dada' J \ da^da' J
and by substituting the values a! . (-j-7-)> «' • ( , . X «'* • \—^r\ §'^^" ^^ *^^
three first of the equations [1003], also the differential of the second, relative to rfrt, which is
, fd d S^\ _ c. (dd ^»\ (d^ ^('> \
"^ • V»rf^7 —~'^' \-d^-)~'''Vd^r
we shall get the last of- the equations [1003].
n. vi. § 49.] DEVELOPMENT OF THE FUNCTION R. 543
We shall have B'-'"' and its differentials, by observing that by the preceding
article [956, 963^], the series*
^ . jB(») + B^'^ . cos. 6 + 5(^> . COS. 2 d + &C. [1004]
is the development of the function a'~^.(l — 2a. cos. 6 + ^^j"^) according nocMn
to the cosines of the angle & and its multiples ; now this function being
developed [964], is equal to
«'"' • I i • 4°^+ b^^^ . COS. & + bf . COS. 26 + kc.]; [1005]
therefore we shall have in general
B» = i5.6f; [1006]
hence we find
■^^(0\ 1 db'-^ /ddB(^)\ 1 ddb'^^
Moreover, ^^'^ being a homogeneous function in a and a' [956], of the order [1007']
— 3, we shall havef
Hence we may easily deduce the partial differentials of B'^^ taken relative to
a\ from those of the partial differentials relative to a.
In the theory of the perturbations of m', by the action of m, the values of
A^^ and B^^ are the same as above, t excepting -4^^^ which in this theory
* (689) This follows from [956], putting a =- [963'^], and developing
(1— 2a.cos.d + a2)~^,
as in [964]. Comparing the coefBcient of cos. i^, in the expressions [1004, 1005], we
get S^'\ as in [1006]. Taking the differentials of this, relative to a, we shall get [1007].
f (690) This is deduced from [lOOla], changing A^'^ into ^'\ and putting m = — 3;
it being evident, from [956], that B'^ is of the degree — 3 in a, a'.
J (691) Changing a into a', and a' into a, in the first member of [956], its value would
remain unaltered ; therefore, the second member, or the value of B^'^, would also remain
unchanged. In like manner, the general value of .4?®, found in [954], would remain
544 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
n' 1
[lOOS'i becomes —^ : . W? . Therefore the same calculations of ^^*\ W^^ and
investiga- thclr differentials, will answer in the theory of the motions of both bodies, m
tion of the
{^^nsofthe and m'.
planets,
neglecting
'andT""" 5^- After this digression, upon the development of i? in a series, we shall
theex^n- rcsumc the differential equations (Z'), (Y), (Z'), [946, 931, 948], in order
tionir"" to determine, by means of these equations, the values of ^ r, h v, and (5 s ;
[1008"! and we shall continue the approximation only to quantities of the same order
as the excentricities and inclinations of the orbits.
[1009]
[1010]
If in the elliptical orbits, we suppose
r = a.(l+uy, r'=a'.(\+u;);
v = nt-{-s-\-v^ ; v' = n' t -j- s' -\- vl ',
we shall have, by § 22,*
u^== — e , COS. (nt-\-s — -a) ; ^/ = — ^' • cos. (n' (-{-s' — to) ;
v^ = 2 e . sin. (nt + s — w) ; v^ = 2e' . sin. (n' t + s' — -n') ;
nt-\- Sj n't-{- s', are the mean longitudes of m, m' [543] ; a and «',
[534'], the semi-transverse axes of the orbits ; e and e' the ratios of the
[1010'] excentricities to the semi-transverse axes ; lastly, * and w' are the longitudes
of their perihelia. It is a matter of indifference whether these longitudes be
referred to the planes of the orbits, or to a plane but little inclined to them,
[1010"] since we shall neglect quantities of the order of the square or product of
the excentricities and inclinations. Substituting the preceding values in the
expression of R, § 48 [957], we shall findf
unchanged, except in the case of i=l ; ./2^'^ being affected by the first terra of the first
a!_
a2
member, so that it would become A^^'^ = — ,.b)^ , instead of the value [997], as is
evident from [954, 964].
* (692) The equations [1009] are like [952, 953], from which, by means of [669], we
deduce [1010].
f (693) The first term, or line of the value of R, [957], produces the first term,
independent of e, e', in [101 1]. The terms multiplied by u, and v^, in the second and fourth
lines of [957], produce respectively the two terms of the second line of [1011], multiplied
TI. vi. § 50.] PERTURBATIONS. ^^^
i? = — . 2 . ^('■) . COS. 2 . (?i' i — W ^ + s' — s)
— — .2. <a .[— — j-\-2i.A^""'i.e.cos.{i,(n't — nt-\-s' — s)-\-nt-]-s — 7s\
[1011]
the sign 2 of finite integrals, embracing all integral values of e, positive or
negative, including the value i = 0. Hence we deduce,* [loii']
by e. The terms multiplied by m/ and v/, in the third and fourth lines of [957], produce
respectively, by means of [954c, 95oa], the two terms of the third line of [1011], multiplied
by e'. These two last terms first appear under the form
— f'. 2.5 a'. ^^^) — 2 i.^w^.e'. cos. {i.(w'^—ni + s' — s) 4- n'^ + s' — -5/^,
and by changing i into i — 1, which may be done, because i embraces all numbers, from
— CD to -f~ '^ J including i = 0, it becomes
— 1\ 2 . ^ a' . (^^^)— 2.(i— 1).^^'-" I .e'.cos.{(i— l).(n'i--^<+s'~£)4^'<+g'_ro'|,
which is evidendy equal to
— -'.2.^a'.(^^_J— 2.(i— l).^'-i>^.e'.cos.fi.(»'<— n< + £'— £) + ni + £ — xtf'J,
as above.
* (694) If we exclude the value i = 0 from the terms under the characteristic 2, in
[1011], it will become,
ie=f\2.^«.cos.i.(n'^— n^ + £'--£)4-f'.^W--f'.a.('^Ve.cos.(n<4-s--«)
-^''^'{<''{^)+^i"^'']'e.cos.{i,{n't-.nt+^-s) + nt + s^'a] jioi^aj
~f'.2. Ja\(^pij~2.(i--1).^(-^>| .e'.cos.[t.(ri'<— ni + e'— s) + n<+s--<|,
because A^-^^=A'^^\ [954"]. Taking the differential of jR, relative to nt, we get d/J, [ioi26]
and its integral being doubled gives
137
^^^ MOTIONS OF, THE HEAVENLY BODIES. [M6c. Cel.
m' ( , /ddAm\ , „ /(?^m\ , „ , /rf^o\ , , ,,„) ,
m'
2 -^-^ , 2.(;-l).n ( /Ag-A ^;_ Jj •e.cos.{,.(n<-n(+s'-s)+«(+._o|
t.(»i — n') — 71 ( '\daj ' )-^
. \aa'. — — — - — 2.U — l)rt.( ) i
V i.(n — n) — n i \ da / ^ ^ ) J
2fdR=2m'.g+^. -^ . 2 . ^» . COS. i Jn' t — nt-\- ^ — s)
[lOlSc] —"^.2.^'. f^Pj + 2 ^(1) l.e'. COS. (n i + s — «')
— ^.2.?#^-^4«-('^'W2*.^®l.e.cos.h-.(n'^— n^+s'-£)+ni+£--w^
2 i.(n — n) — ft i \ da / )
_.^'.2 . ^-^. L'. flf;:!!)-2.(z-l).^«-^)L\cos.{qn7-.n<+s'-s)+n<+e-Vi.
2 t.(n — n) — n i \ da / ^ ' ) ^^
Moreover, since r . ( — j =a. ( — j, [962], if we take the difFerential of R [1012a],
relative to a^ and multiply it by a, we shall we shall get
"•(7;)='--w)=¥-^''-W)-™^-'-("^-'"+'-^'
'2 \ da J 2 I \ da J ^ \ da^ J ) ^
2 ( \ da da' / ^ 'Vrfa/)
Adding together the expressions [1012c, c?], and connecting the terms depending on the
same angles, we shall get [1012], equal to the value of Q, [934].
n.vi. §50.] PERTURBATIONS. 547
The sign 2 includes in this and in the following formulas, all the integral
values of i, positive or negative, excepting i = 0 ; the term depending on
i = 0, having been brought from under that sign ; w' g is a constant [lOlQr\
quantity, added to the integral fdR. Now put
' 2.{i.(n — n) — n\ ( \da J n — iv ) l "J
{i—\).n C 3 /J^W\ . ^ . ^,, >
I .{n — n) — n( \ da J 3
« . (n — w ) — n I \ da! J ^ ■' ^
Supposing the sum of the two masses M-\- m to be equal to unity, and
observing that by ^ 20 [605'], ^ = ri^ ; the equation (X') [946] will [loia^
become*
* (696) Putting M-\-m=\ in [605'], we shall get n^a^=\^ which will be [ioi3a]
used hereafter. Also putting, for brevity, (^=2f6.R-\-r.i--\ [1012], the equation
[946], neglecting e^, will become
0= -^ \-n^ .6u — — . \ 1 — e.cos.(n<4-^ — '^) ( -Q j . f ^.ndt .s\n.{nt-\-s — «).
Substituting for — its value n^ a, [1013a], we shall get
0=:—j--^ — \-n^.Su — n^aQ-j-n^aQ^.e.cos.{nt-\-s — -a) — 2 n^ a e./Q^.ndt. sin. (nt-\-s — «). rioi4ai
In the two last terms, multiplied by e, we may for Q, [1012], substitute the part of it
independent of e, e', namely,
Q=2m^+-.a.(^-^j+-.2.5a.(^_j+^^-^,.^0( cos.z.(n'^--n^+a'-e).
^^8 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
dt^ ^ ^ 2 \ da
[1014]
[10146]
— .2 . ^ «^ — — H , . a J« S . COS. i.(r^t—-nt-\-^'-t)
2 ( \ da J n — n ) ^ ^
-{- n^ w! . C . e . cos. (w ^ + s — ■r^) -{- n^ m' . D . e' . cos. (w ^ + s — w')
+ ?i^m'.2. C^'). e.cos.{z. {n't — nt-\-^ — £) + w^ + £ — w}
+ 71^ m' . 2 . J9® . e' . cos. { i . (/l' if — ?Z i + a' -- s) + ?l ^ + £ — ztf' I ;
and by integration*
which gives, by [954c],
n^a Q.e.cos. (n^ + s — •«)= \ 2m'g-j- — . a. (— — j i.n^ae .cos. {nt-{-s — «)
-\-— .n^ae .2.ja.(-T-— )-| ~ .A'-'K. cos. |i. (w'^ — nt-{-s' — s) -\-nt-{-s — ts^.
Also, by [954&],
— Q.ndt .sin. (n^ + ^ — «) = — j2mg-\- — .a.(—- — ji.ndt . sin. {nt-\-e — «)
— ^.2. 5a.('i^)+-^, .^«|. n<?^. sin. fz.(n'< — w^ + £' — £) + n^ + £ — t;j|.
Taking the integral of this last expression, and multiplying it by 2n^a e, we shall find,
— 2n^ae.fQ^.ndt.sm.{nt-{-s — zi)=2n^ae.<2m'g~\- — .a.(—. — j> .cos.{nt-}-s — vs)
[1014c] ^, ,^^(,v , 2n i . 1 / X I . L i
— — .2w^ae.2.— — .\tt'{ —. — IH ■,.A'-'>[.cosAi.(nt — nt-\-s' — s)4-nt-\-s — ts\.
2 i.(n — n) — 71, ( \ da / n — n > * ^ / i i j
no constant terra being added, because it would produce, in [946], a term independent of
Q, which would prevent 5 u from being nothing, when Q == 0, which is contrary to the
principle assumed in § 46. Substituting [10145, c] in [1014a], we shall get, by reduction,
0 = ' -\-n^.5u — n^a .(^-\-m' ,n^ e .\ ^ ag-^-^a^-i — r — ) ( . cos. {nt-\-s — w)
. COS. 1% . {n' t — nt -\-s' — s) -\-nt-\-s — zi^.
If in this we substitute the value of Q, [1012], and connect together the terms depending
on the same angles, using the abridged expressions [1013], it will become as in [1014], as is
very evident, from the mere inspection of the terms.
* (697) The equation [1014] is of the same form as [865], whose solution is given in
[870 — 871"]; putting y=8u, and changing a [865] into w [1014] ; representing also
by a Q, [865], all the terms of the expression [1014], except the two first -j— -\-n^.6u ;
n.vi. '5 50.] PERTURBATIONS. ^9
m' o ( \ da J n — n ) • / /. ^ i / \
.n^.2. -^^ \g . ^ ttt; o -. COS. t.fnt — nt-\-s' — s)
2 z^ . (w — 71 )- W ^ ^
-\-m! .f^. e . COS. (nt-\-B — vi)-{-m' .fl .e' . cos. (nt-\-s — to')
— .C .nt.e. sin. (71^ + ^ — ^) —. D .nt , e' . sin. (?i ^ + s — w') [1015]
J>(i) ^2
+ m'.2. — — -^ — -^. e'. COS. ii. (n't — nt-{-s' — s)-\-nt-\-s — w'}.
\i.{n — n') — n\^ — w^ * ^ ■*
j^ and fl being two arbitrary constant quantities. The expression of Sr by
means of 8uy found in ^ 47 [947], will give* [1015']
observing that if any one of the terms of a Q be denoted by ^.cos. {m t-\-s), it will,
by [871], furnish in 5 m the term ^ ♦ cos. {mt-{-s) ; and by using the appropriate
values of m, namely, m = 0, m=i.(n' — n), m==i.{n' — n)-\-n= — ^i.{n — n') — n|,
we shall get all the terms of Su [1015], except those depending on the angles nt-{-s — •sr,
nt -^ s — -s/. These two angles depend on m = n, and by [871"], they will produce the
terms depending on C, D, [1015], also terms similar to those depending on f^, //, which
might be connected with the constant quantities c, c', [870].
* (698) The equation [947] , neglecting e^. and the higher powers of c, gives
t J.
— = — Su — 2 8 u. e. COS. {nt-]-s — -a), [lOlGa"!
and by substituting, in — 2Su .e . cos. {nt-\~s — is), the terms of the value of S m,
[1015], independent of e, we get
— 2 5u.e.cos.{nt-\-s — zs)= \ — Airtl .ag — rn ,c? .i — — j > .e.cos. (nt-{-e — zs)
(_ \ da / n — n y
i2.(n— n')2_n2
subtracting from this the value of 5«, [1015] we shall obtain
8 T
— 5m — 25m. e. cos. {nt-\-z — ■«) =^ — ,
^ 'a
[1016a], and by putting /;=/', and — / — 4a^— a^. (^°'^ = — /, it wUl
become as in [1016].
138
+ w'n2.2.i ^"-4 '"^;^^ >-.e.cos. li.(n'<— n<+6'— £)+n<+s— «?,
550 MOTIONS OF THE HEAVENLY BODIES, [Mec. Cel.
2 m . a g . a^ .
a 2 \ d a
C o fdA^^\ , 2n .,.,
_f- — . ?l^ 2 . ^ \ da J ' n—n' J> . COS. Z . (/l^ — nt + z' — s)
^ i'^ . (n — n'y — n^
— m! .fe . COS. (nt-\-s — th) — m! .f e' . cos. (nt-\-s — «')
[1016] -{-iTnlC .nt.e. sin. {nt + z — ^) + l7ii! D .nt. e' . sin. {nt-{-z — w')
< ( \ da J n — n
-\-m.w,'s..
(n — n'Y — n^ \i.{n — n) — n\^ — n^
X e . COS. {i .(jt' t — nt-\- z' — z) -\-nt-\- ^ — ^5
D(0
— m TT.'E.-—- ^.e'.cos.h-C^'^ — nt4-s' — s)4-nt-\-e — wJ;
li.[n — n) — «}'* — w^ *■ ■*
[1016] y ^jj^j yv i^gjjjg arbitrary quantities, depending on f^ and ^'.
This value of 6r being substituted in the formula (Y) § 46 [931], will
give Sv, or the perturbations of the motion of the planet in longitude ; but
we ought to observe that as n ^ expresses the mean motion of m, the term
[1016"] proportional to the time t, ought to disappear from the expression of Sv.
This condition will serve to determine the constant quantity g, and we shall
find*
[1017] g=-^^^a.
d a
We might have dispensed with the arbitrary quantities /and f, in the value
of 8r, since they could be supposed to be included in the elements e, w, of
the ellipticalf motion ; but then the expression of ov would have contained
* (699) The calculation of 8 v, from [931], is made in note 702, in which it will appear
that the term, independent of sines and cosines, is Sm! . agnt -\-m' . a^ . I— — j .nt,
[1021/], putting this equal to nothing, as in [1017'], we shall get^, [1017].
f (700) If in the elliptical value of - [669], we neglect e^ with the higher powers of e,
and put for brevity nt -\-s = r, we shall get
- = 1 — e . cos. (nt-4- s — to)= 1 — e. cos. (r — sj)
[1017a] « \ i J
= I — e . cos. -zs . cos. r — e . sin. zi . sin. t, [24] Int.
II. vi. § 50.] PERTURBATIONS. 551
terms, depending on the mean anomaly, which would not have been comprised
in those of the elliptical motion ; now it is more convenient to make the
terms disappear from the expression of the longitude, in order to introduce [1017]
them in the expression of the radius vector ; we shall therefore determine j^
and /' so as to satisfy this condition. This being premised, if we substitute
for a' . (-^^) its value [1003], — A^'-'^ — a . (~^) , we shall
have [1013, &c.],
*
6 T
In like manner, the terms of — , [1016], depending on/ and /', are
— m! fe . (cos. ts . cos. t -f- sin. -sj . sin. t), and — m! f ^ . (cos.'za/ . cos. t -\- sin. t^ . sin. t).
If we add these terms of — to -, flOlTa], and put
a a
e . cos. -si -|- m'/e . cos. ts -|- »i'/' e' . cos. w'= e^ . cos. -us^ ,
e . sin. zi -f vn! f e . sin. « + m' f e' . sin. z/ =6^. sin. zi^ ,
it will become
=1 — e^ . (cos. ts^ . cos. T -f- sin. «^ . sin. t) = 1 — e^ . cos. (t — zi)
= 1 — e^. COS. {nt-\-s — zi), [24] Int.
which is of the same form as the equation of the ellipsis, [1017a], changing e, zi, into e^, zs^.
* (701) The value of C, [1018], is deduced from that in [1013], by substituting g,
[1017), and reducing. The value of J>, [1013], becomes, by substituting the values of
a'.(--—-\ and a', f ^ / )> [1003], the same as in [1018]. Similar substitutions,
in jy-'^ [1013], produce jD^'^ [1018], after making the usual reductions in connecting the
coefficients of the similar terms. These values of C, D, IP, and that of ^, [1017], are to
be substituted in [1016] ; and if we put, in the coefficient of the angle
i.\n' t — nt-{-s'-^s)-^nt-\-z^zi\, [1016],
(. \ da J n — n )
^.{n—n'f—rfi
6 r
we shall get — , [1020]. This value of £^'^ may be reduced in the following manner.
Put for brevity
v=zn — n', T=n't — nt-\-s' — s, W=nt-\-s — zi, W' = nt-\-s — z^,
[1018a]
Gr = a^ . ( — — ) A ■ . a jT-'^;
\ da / n — n
^^2 MOTIONS OF THE HEAVENLY BODIES. [M6c. C61.
[1018] m= (!=i)-.(y-') • %.^..-..+ i'^- ('■-''•)-''J ■ .^ fl41-Ji^
n — i.(ri — n) n — i.{n — n) \ da J
J 3 /ddA(i-^^
* da''
'ddA(''^\
substitute in E^'^ the value of (J'\ [1013], and it will become
E^)-U' /^^^^ , (2i±i) ^, /^^«\ , iinl!^ G
iv — w (. \ da / ) i2v2 — n2
connecting the terms depending on d A^^\ G, we shall get
2i.(t— l).n ^ffl. — ^t^.v^ + ^'^-'^— 1-^^ G^
'T~ i^ — n "• i2v2— n2 • '
observing that the first coefficient of G is reduced to the same denominator as that of the
second, by multiplying the numerator and the denominator by ^i\i-\-^n. Now the above
value of G gives a^ . (— — ) = G — — .a ./2®. This being substituted, we get
\ da J V
[1018c] ^ ^" ^ 2.;»v-n5 i V 5
2i.(i-l).n ^ ^^^.^ -J^t2.v24-tn.v-f.n2 ^ ^^
iv — n ' 1*2 v2 — »2
„ a{j)
The coefficient of -: , in this expression, taking the terms in the order in which they
« -o . I 3n2 .. « . , 3n2 ^ /iy — n\
occur, is — 2i''n — tn-j \-2v^n — 2in= — ^in-\ = — 3n. ( 1,
which being multiplied by -; , produces the corresponding term of jE^*^ depending
TLvi. §50.] PERTURBATIONS. ^^^
Now putting
E^'
= — .: — r> • « ^' ' i JTTZ — zr<2 — Ta
n — n'' i^.{n — n')
( \ da J n — n
2 n2 . £(*■)
n^ — {n — i.{n — n')l
'\n '
[1019]
we shall have*
on A'-'^, namelv .aA^\ The coefficient of -zr—, -, in this last expression of
JEW, is [r-^ V -f- 1 i V — |n|.[t v-f" "•} — I i^ v^-|-in V — | ji^, because the first term can
be reduced to the same denominator as the second, by multiplying the numerator and
denominator by | . 1 1 v -{- n | . Performing the multiplications and reducing, it becomes
Pv^-\-i^n\i — 3 n^ = i^ V . (re -f- * v) — 3 re^, consequently the term of jG^'^, depending on
G, is — ~^~^ ^ • ^» ^^^ the complete value of E^^, becomes as in [1019].
The values of/, /', F^*), G^\ [1018, 1019], are computed by means of S v, in the following
note.
* (702) The value of 5« [1021] maybe obtained from [931], by using the symbols
[101 8rt], and substituting for — , its value [1020]
r
-=1 — e.cos. (n^ + s — ®) = (1 — e.cos. ^), [1018rf]
[1017a]; also f^R and ^ '\J~)^ [1012c,ei); then determining / and /', by
making the coefficients of sin. [nt-\-s — -n), and sin. {nt-\-s — to'), equal to nothing,
[1017'] ; observing that terms of the order e^ are neglected, and ^=.\. This calculation
is rather long, but as the equation is of great importance, it will be proper to enter into a full
explanation of the whole computation. The equation [931], with these conditions, becomes
139
554 MOTIONS OF THE HEAVENLY BODIES. [Mec. Gel.
Perturba = _.a^(" )+ J.j -^ .^ / ^ L \ l.COS.t.fn't^nt + s'—s)
tion.of a 6 \ da J 2 i^.(n — n)^ — n^ ^ ' ^
the radius ^ ^ ■'
— m' .fe . COS. (7i ^ + s — ra) — m' .f e' . cos. (w ^ + s — to)
+ ^ m' . C . w ^ . e . sin. (nt-\-s — zs) -{- ^m' . D .nt . e' . sin. (w ^ + s — t/)
^ ;^;^T(;^::^. ..'.cos. {iK^-r.^V-o+nr+s-.'}
[1020a] Sv=-r^---{--^^-{-Sa.fndt.fdR-^2afndt.r.(~). Each of the four
terms of the second member may be computed in the following manner.
^, ^ Qr.ddr 2a.{l—e. cos. w].d5r 2.d§r 2e.cos.W. d5r
[1021a] The first term -r — - = ; — = ; : and m
a-i.ndt a^.ndt a.ndt a.ndt
the part depending on e, we may substitute the terms of the value of d S r, [1020],
independent of e, e' ; hence we shall get
— 2e. COS. fV.d5r C 2e.cos.fr) mf n^ G . , • - m
= < } . — — . 2 . --— .tv.dt. sin. I T
a.ndt I ndt ) 2 i2v2 — n2
= m'n.:E.^^^.e.sm.{iT+W), [9556].
Hence -r — -- = :,- + m' w . 2 . — — :— . e . sin. (* r+ W). Now if in this, we
a^.ndt a.ndt n^ — i^v^
2.d5r
substitute the value of — — r > deduced from [1020], it will give the following value of
a.ndt L J u
2r.dSr
a'^ .ndt
2r.dSr , iv.G • • m
— --=m'w.2.-— -. sm. li
a^.ndt i^v^ — n^
+ 2 W .fe . sin. W+ 2 m' .f e' . sin. W
+to'. Ce.sin. ^ + w' .De'.sin. W -\-rri .C .nt .e,co%,W-\-m' .D.nt.d ,cos.W'
+ 2m'n.2. <
+ m'n.2.-^^,.6.sin.(iT+^).
„ ^ r- -, . dr.Sr lae. ndt. Bin. W). 6 r &r . „,
The second term of 6 v, [1020a], is ^^—^j = '• -^^-^^ = e . - . sin. }V,
ILvi. ^§50.] PERTURBATIONS. ^^^
dA^^\ . 2n ^,^) ■) Perturba-
2^3. X a\ '^^^ -f -^^, mA^"^
tions in
^ ( i.(n — n7* "^ i.{n—n').\i''.{n—n'f—n^ )
-{-m! . C .nt.e. cos. (w ^ + s — ts) -\-m' .D .nt .e' . cos. (w < + e — -n')
1 ^""7 7x-e.sin.{2.(n'i — n^ + s' — s)+nt-\-s — to} i [-logij
I TT- r,'e.sm.H.(n't—nt+s'-^s)Jrnt+s--^']
n — i.[n — n) ^
Sr
[1018(?], and by substituting, for — , the terras of [1020], independent of e, e', it becomes
by using the formula [9546],
dr.6r m! „ /dA^^\ . ^. , m' rfi G
The third term of 5 V, [1020a], 2 a .fndt .fdR, is easily deduced from that of
2/d/2, [1012c], by multiplying it by -^.ndt, and again taking the integral. It will
not be necessary to add any constant term, the arbitrary term s, in the value of v, [669],
being sufficient ; hence
3a.fndtJdR=3m'.agnt--Sm'.:E.^.aA^'KsmAT—^m\a^.(^l\.e.sm. W
— ^m\\aa\(^J:^\-{-2aSA . e' . sin. W
+|m\2.^p=^^-^' $aa^('^^Il^')— 2.(i— n
The fourth term of 5 v, [1020a], is 2a.fndt.r. (-i-\ found by multiplying ^-(-7- ),
[1012<r|, by 2 a .ndt, and integrating
^5Q MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
[1021'] The sign 2 includes, in these expressions, all the integral positive and
negative values of z, the value i = 0 being excepted [1012'].
Connecting together these four terms of Sv, \_l02lb, c, d, e], we shall have the complete
value of 8 v. The coefficients of these sines admit of various reductions. To obtain these,
we shall compute each separately ; first noting the terms, in the same order as they occur, in
these expressions, [10215, c, d,e], and then making the necessary reductions.
First, The term of S v, independent of cosines and sines, is
3m .ag nt-\-m a^ . ( — . — \ .nt,
fd ./2("A
[1021/] this is reduced to nothing, by putting ^g- = — J a . f —^ — j, [1017].
Second, The coefficient depending on sin. (nt -\- s — w), or sin. W, in S v,
\l02lh,c,d,e] is
2m'.fe-\-m.Ce4- — .a-'e.l —, — ) — # m' . a^ e . ( )
6 \ da J ^ \ da J
This is to be put equal to nothing, by [1017']. Dividing it by — 1 id e, and transposing f,
we get.
Substituting the value of — i C, [1018], namely —^a''.(^:^\—la?.(?^\
and connecting like terms, we get /= § a^ . ( -7 — j + J a^ . f ^ j , as in [ 1 0 1 8] .
TAirJ, The coefficient of sin. (ni-|~^ — '^)j or sin. ^T', in 6v, [10216, c, J, e], is
This being put equal to nothing, in the same manner as with the last coefficient, and then
dividing by — 2 m e, also transposing /', we get
Substituting the value of A [1018], of a' .(i^^, and a' . (^1^\ [1003], it
becomes, without reduction,
/'=_i.^<,^o-<.^(^-^>)-4.3.(^')^+S.^«.[-.'"-<..(^)]+.<.^.>^
n. vi. ^ 50.] PERTURBATIONS. 567
We may here observe, that even when the series represented by
2 . J<') . COS. i,{n't — nt^^ — i)
and by connecting the terms together, it becomes
as in [1018]. In the original, the sign of the last term was positive mstead of negative.
This was corrected afterwards by the author, in vol. iii, [4060].
Fourth^ The coefficient of sin. i T, in Sv, [10216, c, d, e], is
wiw.2.-— - — |m'.2.-— .a^^'^— w .2.-.a2.(!ilf_),
substituting a^ . (—. — j = G .a S"^, [1018a], it becomes
w • 2 . -— . (? — I m . 2 . — . a^^'^— m'.2 . - . ^ G .aS^[,
l9v2_„2 -i tv2 t„ ( V 5 '
and by connecting the terms of ^^'^ and G,
2-^-^l2v2_„2 tv S ' 2 1 iv2 5 "*^^ '
. 1 • "'' 2n3.G , m' rfi ^.. i • i • i ,
or, by reduction -^r -^ ' . — tt,— r>' -r :;: • ^ • ri. - o, •^ > which is the same as the
coefficient of sin. i.(w'< — n t -{- ^ — s), in [1021].
Fifth, The terms of 5j;, [10216], which contain t without the signs of cosine, agree with
the second and third terms of the expression [1021].
Sixth, The terms of Sv, depending on sin. (z T-f- ^), [10216, c, rf, e], are [1021^1
m! ne . sin. (i T-j- W), muhiplied by the following expression H, in which the sign 2 is to
be prefixed to the terms of the second member,
rr_2.(iv — n).£(') iv g, , jnG
„2_(„_tv)2 ""nS— t2v2 ""i^va — »i2
^ {iv — nf i \ da J ' 5 iv — n i \ da^ / '^ ' ^ \ da )\
2 ^ /jj I v)2
This being multiplied by n — % v, and for ^_^ . . FP, substituting its value
2n2
2 E^'> — ; — 1 . i? ^ we shall get
n^ — (n — wy
(n-iv).H=2i:(0_44lI!L + -iV. G_i^
^ '' n2 — (n — ivr n+iv n-\-x'i
Substituting for 2 £('">, in the first term of the second member, its value given in [1018],
which, by the symbols we have used, is
140
^^^ MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
S r
converges slowly, the expressions of — and <5 v, become converging, by
we obtain,
IV — n i \ da J ^ ) ^ \daj n^ — {n — ivf'
and if in this we substitute, for a^-(—, — ), its value G .aA^^\ [101 8a], it will
\ da J V
become
[1021.1 («— )-ff=-7-''-^"^+^— V5=^s — +;H5;-;q5-H7=i' — (2'+i)^-g
+ i4i=')--»J?^.«^('>-2i.«^(')U(2i+l).?=.a^«-4^!!^,.
IV — n (v )'^'''w n2 — (ii — iv)2
The coefficient of a A^"^, in this expression is
V (^ tv — n ^ J *v — n v ( *v — n ^ ^ )
Si.{i—l).n ,. , n ( 3n , , ) 3t.(i— l).w (i— l).n ( 3n , ^ 3iv )
r '—=(i—\).-. \ ^4 [ r —= — . ] h4— ; >
tv — n V (. IV — n ) tv — » v (^ly — » iv — n)
(i — l).7i C 3n + 4.(iv — n) — 3zv ■) (i — l).n (iv — n^ {i — l).n
V ' (_ IV — n ) V ' iiv — n) v
({ \\ n
[1021A;] hence this term of H.{n — iv), is ^^ '—.aA^'\ The coefficient of G, in [10211],
, ^ 2i2v.(n+iv) 2i2v ^ . , 2in j • •
becomes, by putting the first term -^r-- — ^;— = - = 2 z + •; , and rejectmg
2 i — 2L which destroy each other, •: —- z + -: — ; '^-^- — — 1 >
•' IV — n t^v^ — n-* iv-|-n tv — n
connecting the first and fourth terms, also the third and fifth terms, it becomes
^in-\-§n |w 6 w^
t V — n iv-\-n i^v^ — n^
•o Q o -1 hin.(iv^n) — 3n2
by reducing all the terms to the denommator i'^ v^ — n^, it becomes ^ 2_^ '
This part of [102Jt], being connected with that found in [10217c], gives
(t-l).« Ain.(iv+n)-3n2 2 n^ . £«
(W — iv) . H = . a A>'> H -^-r -z . (r — -g — p .-Ts ,
V / V ' i2v2 — n2 n3 — (n — iv)2
which is equal to F« [1019], therefore H=-^. This, by [1021 A], is the
coeflicient of m'ne .sin. {i T -\- W), in S v. It agrees with [1021].
II. vi. § 50.] PERTURBATIONS. 559
the divisors they acquire. This circumstance is the more important, as
otherwise it might have been impossible to express analytically the reciprocal
Seventh, The terms of S v, [I02\b,c,d,e], depending on the angle iT-\-W, are
m' ne' .sin. {iT-\-JV'), multiplied by the following expression H', the sign 2 is to be [10211]
prefixed as above,
g=^ji^■iy.^+^('-')■^5«a^(^4;l')_2.(;_l).„^.-.,|
n2 — (n — iv)2 (tv — n)2 { \ da J )
IV — n i \dada' J ^ ' \ da J y
multiplying this by n — i v, and then putting the coefficient of jD^'^, or ^^
n^ — (n — ivp
under the form 2 — ; r-^ , substituting also the values of ( |,
n2— (n — iv)2 ° \ da' J
(?J^\ [1003], we shall get
\dada J "-
{n—iM).H'=2m -— ni + ^-p ^- ] a •(—7 — + (2 t— l).a^('-» V
^ ' n^ — [n — tv)2 (tv — n) ( \ aa / )
in which, for the first term of the second member 2 D^'^, we must substitute its value [1018],
which is
n — tv n — iv \ da J \ da^ /'
and then, by connecting the similar terms, in the order in which they occur, we get
(„■■■■..) „,_^3.(.--l).(2i-l).n |.ft-l).(2.--l).n> ^y^„
(. n — IV n — iv }
^\2t^v-2n_^.{i-l).n,^.\^^/dA^^'A_ 2n^.D(0
( n — IV n — iv ' )' ' \ da J rfi — (n — iv^ *
which is easily reduced to the form
^ '' 2.(n — iv) ' 2.(n — iv) \ da J n^ — (n — ivf
[1019], hence we get H^= . , and, by [1021?], this is to be multiplied by
m ne' . sin. {iT-\- W), and the sign 2 prefixed, to obtain the corresponding term of 8v,
depending on the angle iT-\- W, which will therefore be
nm'. 2. J^.e'.sin.(ir+ W),
n — t V
as in [1021]. Thus we have proved the correctness of the expressions [1018 — 1022].
^^0 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
perturbations of those planets,* in which the ratio of their distances from the
sun differs but little from unity.
These expressions may be reduced to other forms, which will be useful in
the course of the work, by putting
|.^^^ h = e. sin. ^ ; h' = e' . sin. ^ ;
I = € . COS. w ; i = e' . COS. -a ;
whence we shall getf
a 0 \ da J 2 t'^.(n — ny — n^ ^ ^
[1023] _ m' . (hf+ h'f) . sin. (nt + s)—m'. (lf+ I'f) . cos. (nt + s)
+ ^,{l.C-{-l\D].nt.^m.(nt+B)^~.[h.C+h\D].nt.cos.(nt+s)
Second / +-5^ — 7^75. cos. h'.fW^ nt-\-Z O + ^^ + ^l V
forms of f r? — \n — x.in — n)P * ^ ^ J
the pertur- >. < v / > X
batioiis in - r / t m-\-K n > -n
longitude , ( r. •? S o / " .Af W \ 2 n .,.> / 1
latitude. ^tj=_.2. ^ ^^ ffj(0 I r \ da J n — n' ) > .sin.«.(n'^— ni+s'— s)
[1024J +m'.{h.C+h'.D\.nt.sm,(nt+s)-^m'.{LC+r.D\.nt.cos.(nt+s)
, fin — «.(w — n) cv ^ ^
/ -^ .J /, .cos.li.(n't—nt + s'—s) + nt+s]
\ n — i.[n — n) * ^
* (702a) This will easily be perceived, by examining the terms of — , d v, [1020,
1021] ; it being evident, that as t increases, the divisors of the form i.{n — w'), will
increase, and most commonly also, the divisor i^ . (n — n'Y — w^, &c.
f (703) By [22, 24] Int. we have cos. (If — ts) = cos. H. cos. •ci + sin. H . sin. zs ;
sin. [H — zi) = sin. H. cos. ro — cos. H. sin. zs. Multiplying these by e, and substituting
the values [1022], we shall get
[102:3a] e . cos. {H — zs)=l . cos. H-\-h. sin. H. e. sin. {H — zi) = l . sin. H — h . cos. H.
In like manner e' . cos. {H — z/) = I' . cos. H-j-h' . sin. H, and
e' . sin. (H— z^)=T , sin. /f— /i' . cos. H.
These values being substituted in [1020, 1021], we shall get, [1023, 1024].
11. vi. § 51 .] PERTURBATIONS.
aei
Connecting these expressions of Sr and ^d with the values of r and v [669], [1024']
in the elliptical motion, we shall have the whole values of the radius vector
of m and its motion in longitude.
51. We shall now consider the motion of m in latitude. For this purpose
we shall resume the formula (Z') ^ 47 [948]. If we neglect the product of [1024"]
the inclination by the excentricities of the orbits, it will become
the expression of i? § 48 [957] gives, by taking for the fixed plane the orbit
of m at the commencement of the motion,* [1025']
fdR\_m'.
^ - — .2.5».cos.t.(n'«--w^ + £' — s). [1026]
The value of i comprises all integral positive and negative numbers, including
also z = 0 [954"]. Let y be the tangent of the orbit of m' upon the primitive [1026']
orbit of m, and n the longitude of the ascending node of the first of these
5r
It has been remarked, by M. Plana, that the constant part of — , [1020, 1023],
represented by —.a^.(— — j, does not express the whole variation of the mean
distance a, arising from the disturbing force ; or, in other words, the whole difference between
the values of a, in the primitive orbit, and in the disturbed orbit ; because the use of the
constant quantity^, and the finding nt from observation, [1016", 1021/], produce in the
value of n, some part of the effect of the disturbing force ; and as a is found from the
equation n^a^=l, [1013a], it will also introduce into the assumed value of a, some
part of the effect of this disturbing force. This subject is discussed by IVI. Plana, in vol. ii,
page 326, of the Memoirs of the Astronomical Society of London, and in the same paper, he
has also made several important and interesting remarks on other parts of the Mecanique
Celeste.
* (704) The terms of the second member of [1026], are produced by the terms
-^H -^ — ^.2.5«.cos.».(n'< — n^ + s' — s), of /?, [957] ; rejecting the
terms containing z, after taking the differentials, because m'zisoi the order of the square of
the disturbing forces, [92G']. The other parts of R do not produce any terra in (-r-)-
141
[1028]
^62 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
orbits, upon the second ; we shall have very nearly*
[1027] z' = a'.'r'Sm.(n't-\-s' — u);
which gives
— — .«'. 2 . ^(»-i> . 7 . sm.{i.(n't — nt-{-s'—s)-{-nt+s — n].
The value of z, in this and in the following expressions, includes all
integral positive or negative numbers, excepting i = 0. The differential
[1028] equation in ou' will therefore become, by multiplying the value of (-7—)
by n^aP=l, [1013a],t
* (705) In the figure, page 351, if C be the place of the sun, D that of the planet m,
B its projection on the fixed plane, we shall have BD = z, and by [678', 679'j,
tang. B C D =s, C B = r^. Then, in the rectangular triangle C B D, we shall get
[1026a] J5I>= C^. tang. J5 CD, or z = r^s. If the orbit be but little inclined to the fixed
[1027a] plane, we shall have r^ very nearly equal to r, [680], and z will become z = rs, as in
[10276] [957^']. Substituting the value of s, [679], we shall find z = r . tang. 9 . sin. {v^ — 6).
Accenting the letters we shall get the corresponding expression for the planet m',
z' ==^r' . tang, cp' . sin. (v/ — &') ;
and if the orbit be nearly circular, we shall have / nearly equal to a' ; also tang. 9' = 7,
[669", 1026'], hence z'^a'y. sin. {vj — 6'). But v^ — 6' is nearly equal to
n't -{-^—6', [669'] or n't+s' — U, [1026'], hence 5;' = a' 7 . sin. («' < + s' — n),
as in [1027]. This being substituted in [1026], we shall get, by using [9546],
(^)==!^.y.sm.(n't + ^ — U)—^.a\:^.B^^r.sm.{i.{n't—nti-s'--s)-\-n't-\-^--Ul,
\dz/ a'2 2
and by changing i into i — 1, as in note 693,
\dz J a 2 2
in which B'~^^ includes the term depending on i = 0 ; if we wish to exclude this we must
bring the term, depending on B''-^\ from under the sign 2, and then we shall obtain the
expression [1028], observing that ^<-i> = £(^>, [956'].
t(706) IMultiplying the term ^.(^)» [1025], by w2a3=l, [1013a], it becomes
n^a.(^\ and by using the value of (^-A, [1028], the equation [1025] will take
the form [1029].
n. vi. <^ 5 1 .] PERTURBATIONS. ^^^
at Q,
l_!^Ll!!L.«a'.5(i).y.sin.(n^ + £--n) [1029]
2
m . rr
hence, by taking the integral, and observing that by § 47 [948] hs= — a.%',* [io29']
6s= '- . .7. sin. (n' t-\-s' n) Perturba-
»,2 »,'2 „'2 ' V ' >' tions la
n^ — n'" a
m' . a^ a'
latitude.
. B^^^ .nt.y . COS. (/i « + s — n)
[1030]
to' ot^ „2„/ THi—i)
To obtain the latitude of m above a fixed plane, but little inclined to that of
its primitive orbit, f we shall put (p for the inclination of this orbit to the fixed [1030^
* (707) Put y=(5w', and a = n, in the differential equation, [8G5], and it will
become of the same form as [1029]. Its integral [870 — 871"], will give 6 u', which being
multiplied by — a, gives — aSu' ==8s, [1029'], and the result will be as in [1030] ; no
constant terms c, c', being added, because they are included in p, q, he, mentioned in the
general value of s, [1034].
f (708) Suppose a spherical surface to
be described about the body M, as a centre,
with a radius equal to unity, cutting the
fixed plane, in the arc HAB C G; the
primitive orbit of m, in AD F; and the arv
orbit of m', in BDE. From the point D, let fall the arc D C, perpendicular to HG; and
from any other point E, corresponding to the time t, let fall the perpendicular E F G.
Then taking H for the origin of the longitudes, we shall have HA = 6, HB=6',
HC = n, nearly. HG=nt-{-s, AG=nt + s — 6, and C G = nt-\-e—U,
nearly ; FAG = cp, EB G — cp', tang. EDF=y. Then in the right angled
spherical triangles, A G F, B G E, we shall have
tang. F G= tang. FA G . sin. AG = tang. 9 . sin. {nt-\-s^6),
tang. EG = tang. EBG. sin. BG=z tapg. 9' . sm. (» < + e — &'),
I
[1032]
564 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
plane, 6 for the longitude of its ascending node upon the same plane ; then
this latitude would be obtained with sufficient exactness, by adding ^ s to
[1030"] the following quantity, tang. 9 . sin. (v — ^), or tang. 9 . sin. (nt-i-s — 6),
neglecting the excentricity of the orbit [669]. We shall also put <p' and d',
in the orbit of m', to correspond to 9 and 6 in the orbit of m. If m should be
supposed to move in the primitive orbit of m', the tangent of its latitude
[1030"'] would be tang. <p'. sin. (nt-{-s — 6') ; it would be tang. 9 . sin. (nt-\-s — 6),
if m should continue to move in its primitive orbit. The difference of these
[1030'"] two tangents is nearly equal to the tangent of the latitude of m above the
plane of its primitive orbit, supposing it should move in the plane of the
primitive orbit of m' ; therefore we shall have
[1031] tang. 9'. sin. (nt-\-s — 6') — tang. 9. sin. (nt-{-s — 6) = y, sin. (nt-\-£ — n).
Putting
tang. 9 . sin. 6 = p ; tang. 9' . sin. &' = p' ;
tang. 9. cos. 6 = q ; tang. 9'. cos. 6' =z q' ;
we shall have*
[1033] r.sm.u = p' — p; 'y.cos.u = q'^q;
The first subtracted from the second, gives tang. EG — tang. F G, wliich, by [30] Int.,
is equal to tang. {EG — F G) . { 1 + tang. E G . tang. F G], or simply
tang. {EG^FG) = tang. E F,
neglecting terms of the third order in EG, F G, hence,
tang. EF = tang. 9' . sin. {nt-\-s — 6') — tang. 9 . sin. (nt-{-s — 6),
and this would represent, very nearly, the tangent of the latitude of m, above the plane of its
primitive orbit, supposing it should move in the orbit of ?»'. Now this same tangent
corresponding to the angle nt-{-s, is, by [1027], equal to y . sm. {n t -{- s — II).
Putting these two expressions equal to each other we shall obtain [1031].
* (709) Put for brevity w ^ + £ = t, and the expression [1031] will become
tang. 9' . sin. (r — d') — tang. 9 . sin. (t — ^) = 7 . sin. (t — n).
Developing the sines of r — ^, r — 6, r — n, by [22] Int., we shall get
tang. 9' . [sin. r . cos. ^' — cos. r . sin. (f} — tang. 9 . |sin. r . cos. 6 — cos. t . sin. 6\
=:y . jsin. T .COS. n COS. T.sb. U},
II. vi. § 52.] PERTURBATIONS. 566
therefore if we put s equal to the latitude of m above the fixed plane, we [1033']
shall have nearly*
s = q . sin. (nt-i-s) — p . cos. (nt-{-s)
«' «2«'
Formula
'- . (p' — p) . B^^'' ,nt. sin. (nt-[-s) lautude.
— --^.(9'— g).5(^).?i^.cos.(n^ + e)
62. We shall now collect together the formulas which we have here
computed. Putting (r) and (y) for the parts of the radius vector, and the [1034]
longitude i), upon the orbit, depending on the elliptical motion ; we shall
havef
r = (r)-\-^r \ v ^ {v)-\-^v. [1035]
and by substituting the values [1032], it will become
5' . sin. T — p' . cos. T — q . sin. t + P • cos. t = y . cos. n . sin. t — y . sin. II . cos. <r,
and, as this ought to exist, for all values of r, the coefficients of sin. <r, cos. r, in each
member of the equation, must be equal to each other ; hence we obtain the two equations
[1033].
* (710) If the body m should continue to move in the primitive orbit AD F, in the
figure page 563, its latitude FG would be, as in note 708, nearly equal to
tang. <p . sin. {ni-{-s — 6), [1034a]
which being developed, as in the last note, is q . sin. {nt-\-s) — p . cos. {nt-\- s).
These are the two first terms of s, [1034] ; the other terms are deduced from the value
of Ss, [1030], by similar developments, relative to IT, substituting the values [1033].
Thus
y . sin. (n' t -\-s' — ll) = 5/ . cos. n . sin. {nt-\-^) — 7 . sin. 11 . cos. (n t -f- 0
= (9' — 2) • sin. {n' i + ^) — {p' — p)' cos. (w' t + /),
and so on for the other terms.
f (711) The values (r), (?;), are given in [669], 8r and 5 1> in [1023, 1024].
142
^66 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
The preceding value of s [1034] will be the latitude of m above the fixed
[1035'] plane ; but it will be more accurate to use, instead of its two first terms,
which are independent of m', the value of the latitude, which would take
place if m did not quit the plane of its primitive orbit.* These expressions
contain the whole theory of the planets, when we neglect the squares and
products of the excentricities and inclinations of the orbits, ivhich can generally
[1035"] be done. They have besides the advantage of being under a very simple
form, in which we can easily perceive the law of their different terms.
Sometimes it will be necessary to include terms depending on the squares
and products of the inclinations and excentricities, and even of higher
powers and products. We may determine these terms by the preceding
[1035'"] analysis : the consideration which renders them necessary will always
facilitate their computation. The approximations in which these are noticed,
will introduce other terms, depending on new arguments ; they will also
reproduce the arguments, given by the preceding approximations, but with
smaller coefficients, according to the following law, which is easy to deduce
[I035iv] from the development of R in a series, in § 48 ; an argument which in the
successive approximations, is found for the first time among quantities of an
order r, is reproduced only by quantities of the orders t-\-2, r4-4, &c.t
SI n
Hence it follows, that the coefficients of the terms, of the form t. ' (nt-{-s),
* (711a) These two terms express the tangent of the latitude, which was taken for the
latitude in [1034a], it is therefore more exact to use the latitude itself
f (712) Comparing the values of r, v, [952, 953], with those in [659, 668], altered as
in [669], it will be perceived that the elliptical values of r, v, u,,v^, and therefore of r, i/,
w/, v/, possess the property mentioned in [1035'^], relative to the successive terms of the
series. This law would not be affected by reductions similar to those in [675, 676'], and
a little attention will also show, that z, z', [1027], are affected in like manner. Therefore
all the terms of R, [957], possess tliis property, and the same must evidently take place
with 2fdR-{-r.(—\ ^-(y-)) and (-r-)- Hence it follows that 8r, 5v, Ss,
[930,931,932], are formed in a similar manner, consequently {r)-{-Sr, {v)-\-8v,
(s) -{-^s, or the complete values of r, v, s, must each be expressed by a series, whose
successive terms, depending on the same angle, have the same property as in [1035'''].
n.vi. §52.] PERTURBATIONS. 567
which enter into the expressions of r, v, 5, [1023, 1024, 1030, 1035], are [1035^]
correct as far as quantities of the third order ; that is, the approximation in
which we shall notice the squares and products of the excentricities and
inclinations of the orbits, will add nothing to these values ; they have,
therefore, all the precision that is necessary. This is the more important, [1035»']
because the secular variations of the orbits depend on these coefficients.
The various terms of the perturbations of r, v, s, are comprised in the
form
sin
k. '{i.(n't — nt-\-s' — i^j^mt + is], [i036]
r being a whole number, or nothing ; and k a function of the excentricities
and inclinations of the orbits, of the order r, or of a higher order ;* hence [1036']
we may judge of the order of any term depending on a given angle.
It is evident that the action of the bodies m", m'", &c., produces in r, v, s,
some additional terms, similar to those resulting from the action of m! ; and [1036"]
by neglecting the square of the disturbing force, the sums of all these terms
* (713) From the remarks [957'"], it appears that the elliptical values of r, v, u^, v,,
&c., have the property mentioned in [1036']. The formula [961] shows also that R has
the same property ; for by putting
[lOdoa]
jff' = H. e^ . e'*' . (tang, i 9)«" . (tang* J (pj'",
the formula [961] becomes
H' . COS. {T'—G) = H'. (sin. G . sin. T' + cos. G . cos. T'),
[24] Int. ; and by putting H' . sin. G = k, JEf' . cos G = A/, it changes into
A;.sin. T' + ^.cos. T'. Now if we put T=n' t — nt -^s' — s, and r=«' — i, [10306]
the value of T', [1036a], will be
T'==i'n't—-int-i-i'^—is=i'.{n't---nt-\-^----s)-\-{i'—i).{nt-{-s)=i'T-\-Y.{nt-lrs),
and the expression [10366] will become
k.sm.(i' T-\-T n t -\-Y s) -{- Jc' . COS. {i' T+rnt -{-r s),
which is of the same form as in [1036], and this term of R is, by [961'], of the order r,
or of a higher order. Lastly, as the value of R, and the elliptical values of r, v, he,
satisfy the above condition, it is evident from the equations [930, 931, 932], that Sr, 8v, 5 s,
must also be subject to the same condition.
568 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
will give the complete values of r, t), s. This follows from the nature of the
^ ^ formulas (X'), (Y), and (Z'), [946, 931, 948], which are linear with respect
to quantities depending on the disturbing force.*
Lastly, we shall obtain the perturbations of m', produced by the action of
[I036iv] m, by changing, in the preceding formulas, «, n, A, /, s, ©, p, q, and m\ into
a', n', h', r, s', OT, y, q'j and m, and the contrary.
* (713a) That is, R and its differential are found only in the first power in these
equations.
n.vii.§53.] SECULAR INEQUALITIES. ^^^
CHAPTER VII.
ON THE SECDLAR INEaUALITIES OF THE MOTIONS OF THE HEAVENLY BODIES.
53. The forces which disturb the elliptical motion, introduce into the
d 1)
expressions of r, — , and 5, of the preceding chapter, the time ^, out [1036 v]
of the signs of sine and cosine, or under the form of arcs of a circle ; and as
these arcs increase indefinitely, they will finally render the expressions
defective. It is therefore necessary to make these arcs disappear, by
reducing the series which contain them to the original functions, from which
they were produced by development. We have given, for this purpose, in
Chapter V, a general method, from which it follows, that these arcs arise
from the variations of the elements of the elliptical motion, which then
become functions of the time. As these variations are produced in a very [1036''']
slow manner, they have been called by the name of secular equations. The secular
, /., .. /., . ..., equation*.
theory oi these equations is one oi the most interesting points in the system
of the world ; and we shall here explain the subject with all the fulness its
importance requires.
We have, by the preceding chapter,*
* (714) The expression [1037] is found by adding the values of r, [669], and 5 r,
[1023], putting m S for all the terms of 5r, which do not contain n t without the signs of
sine and cosine ; developing also the term — ae . cos. {n t -\-s — «), of the expression
[669], as in [1023a], by which means it is reduced to the form
— ah . sin. {nt -\-b) — al . cos. {nt-\- s). ri037fll
The other terms, depending on quantities of the order e^, e', &;c., [669], might be developed
in a similar manner, and would produce, in [1037], quantities of the order A^, l^, Stc. ; but
such terms are neglected in the present calculation, as is observed in [1051"]. The value
143
^^0 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
r 1 — h . sin. (nt-[-s) — / . cos. (nt-\-s) — &c.
[1037] r = a.l + f-'.{^.C+/'.i)}.?i^.sin. (w^ + s)
( —'^,{h.C-\-h!.D}.nt.cos.{nt-^^) + m'.S
—- = n-\-2nh. sin. (nt -}-s)-\-2nl . cos. (nt-\-s) -\- &c.
[1038] — m'.ll .C -\-r.D}.n^t, sin. (nt-\-s)
+ m' .\h.C + h' .D\.nU . COS. (n t -{- s) -\- m' . T ;
s = q . sin. (nt-^ s) — p . cos. (nt-\-s)-\- &c.
m
[1039] —.a^a', (p' — p) . 5^') . w i . sin. (nt-\-s)
-.a^'a' .(^ — q). B^'^ .nt. cos. (nt + e) +m'. x ;
[1039] S, T, and Xj being periodical functions of the time t. We shall first consider
the expression of — , and compare it to that of y, ^ 43 [877]. As the
arbitrary constant quantity n is multiplied by t* under the periodical signs, in
— , [1038], is found, by adding v, [669], to (5r, [1024], taking the differential of the sum
relative to d t, dividing it hy dt and putting m' T for all the terms arising from S v, except
those containing n t without the signs of sine and cosine ; developing also, as before, the term
dv
2ne. COS. {ni-\-s — to), of ~, deduced from [669], so as to put it under the form
[1037&] 2 n A . sin. {nt-{-s)-{-2nl. cos. {nt-\- s). The value s is the same as that of [1034],
putting m! x for all the terms of S s, independent of the arcs of a circle. The reason of
dv
using — , instead of v, in [1038], is to render the second member free from t, without
the sign of sine and cosine, except in the terms depending on C, D, arising from the
disturbing forces ; by which means, it becomes of a form that is directly comparable with
the value of y [877].
dv
* (715) The value of y, [887], being compared with — , [1038], gives X and Y, as
/// '}r\
in [1041], Z, he, being nothing. Now by [889] we have (—\ = X'-{-t X", &c.,
the arbitrary quantities n, h, I, s, being considered as functions of 6. Therefore from the
value of X, [1041], we must find ("T7 j? considering «, h, I, s, variable, and we shall
obtainX', X", [1042].
n. vii. §53.] SECULAR INEQUALITIES. 57/
d v
the value of — , we must use the following equations, computed in
^ 43 [892] :
0 = X' + 6,X"—Y;
0=Y' + 6, Y"-\-X" — 2Z ; [i040]
&c.
We must now find what X, X', X", Y, &:c., become in this case ; and if we
dv
It
compare the expression of — [1038], with that of y, in the article just
[1042]
quoted, [877], we shall get
X=n + 2nh. sin. (n t -{- s) -}- 2 n I .cos. (nt+s)-j-m' T ;
rio4n
Y==m'.n\{h.C+h'.D}.cos.(nt+e)—m'.n\{l.C+r.D].sm,(nt+B).
If we neglect the product of the partial differentials of the constant quantities
by the disturbing masses, which may be done, because these differentials are [1041']
of the same order as the masses, we shall have by § 43 [889],
X' = f^^ .{l + 2h. sin. (w^+e) + 2/ . cos. (nt+s)]
+ 2w. f -^\{h. COS. (nt+e) — I. sin. (nt-\-£) I
+ 2n. (j^^ . sin. (nt-\-B)+2n, (^—^ . cos. (nt + s);
X" = 2n.(^\{h.cos.(nt+s)—l.sm.(nt + s)l.
Hence the equation 0 = X' -\-6.X" — Y, will become
0 = fp\\l-\-2h.sm. (nt + s) + 21, cos. (n t + s)]
+ 2n, f—j . sin. (nt + s) + 2n . (j-j . cos. (nt + s)
-\-^^'{''(jP) + {^)\'\h^cos.(nt + s)-Lsm.(nt+s)\
—m'.n\{h.C+h'.D\.cos.(nt+s) + m'.n\\l.C+l'.D\.sm.(nt+s).
[1043]
MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
The coefficients of the different sines and cosines being put separately equal
to nothing,* we shall have
« = © =
[1044'] If ^^ integrate these equations, and in their integrals change 6 into t, we
shall have, by § 43 [885"], the values of the arbitrary quantities, in functions
of t, and we may then efface the arcs of a circle from the values of — , and
r ; but instead of this change, we may, in the first instance, write t for s, in
these differential equations.! The first of these equations shows, that n is
constant ; and as the arbitrary quantity a in the expression of r depends on
* (716) The equation [887] is identical, as appears from [887', 879'], therefore [892]
and [1043], which were deduced from it, must also be identical ; consequently the coefficient
of the different sines and cosines must be nothing, as well as the term independent of those
sines and cosines. Now this last term is (;i— ), which, being put equal to nothing, gives
the first equation, [1044]. This being substituted in [1043], and the whole divided by 2 n,
gives
0 = Q.sin.(r.^ + s) + (^).cos.(n^+s)+(^^).^A.cos.(n^ + s)_Z.sin.(n< + s)]
-^%^. {h C + h'D). COS. {nt + 8) + "^. {I C + l'D).sm.{nt + s).
The coefficient of sin. (?i<-{-e), being put equal to nothing, gives the second equation
[1044], and the third equation is found by putting the coefficient of cos. (nt -\-s) equal to
nothing.
f (717) This may be done because C, D, n, which occur in the second and third of the
equations [1044] are constant, these terms being functions of a, a', as is evident from the
values of C, D, [1018], which are functions of a, a', [954]; and n [1013a] is equal
to a , which is constant. [1044"].
11. vii. § 53.] SECULAR INEQUALITIES. ^^^
it, by means of the equation r^ = -3-, [1013a], a will also be constant. The
other two equations are not sufficient for the determination of h., I, s. We [1044']
may obtain another equation, observing that the expression of — gives by
integration fn d t for the value of the mean longitude of m ;* now we
have supposed this longitude equal to nt-\-z [952'] ; therefore we shall
have nt -\-s=fndti which gives [1044, 1045'], [I044"q
^'dt^dt-^' [1045]
and as -— = 0, we shall also have -— = 0. Thus the two arbitrary rio45'i
at dt *- -'
quantities n and s are constant ; the arbitrary quantities h and I will therefore
be determined by means of the differential equations
^ = -^.i^.^+^'-/>!; (1)
dt 2
dl w! .
'dt^~2
'J- = ^.{h.C + h'.D\. (2)
[1046]
Q Iff
The consideration of the expression of — having enabled us to determine
the values of w, a, h, /, and s ; we see a priori, that the differential equations,
between the same quantities, which would result from the expression of r, rio46'i
must agree with the preceding. This may be easily proved a posteriori, by
applying to this expression the method of § 43.t
* (718) This is evident from the equation [1038]. The differential of [1044'"]
gives [1045].
f (719) Putting a = 71^, in [1037], and comparing the resulting expression of r
with that of y, [877], we shall obtain values of X, Y, of the following forms,
X==ii^ .\ I— h. sin. {nt-\-s) — I, COS. {nt-\-s) — hc.\ -{-m' S;
_o [1046a]
Y^im!.n~^.\n.{lC+TD).sm.{nt-{-s)—n.{hC + h'D).cos.{nt-\-s)\,
which correspond to the equations [1041]. From these we may deduce other expressions
analogous to those in [1042 — 1044]. A very slight attention makes it evident that the
144
^^^ MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
We shall now consider the expression of s [1039]. Comparing it with
that of y, in the article before mentioned, we shall find,*
X^=q . sin. (nt-{- s) — p . cos. (n t -\- s) -\- m' . x '^
[1047] Y = ^ .a" a'. ^('> . (p — /) . sin. (nt + s)
4. ^ ,a^a'. 5(^> . (9 — ^) . cos. (nt + s) ;
n and s being constant, as has been already shown [1046']. Hence we shall
have^ by § 43,
X'=(^).sin.(n/ + s)_@).eos.(.^ + 0;
X" = 0.
The equation 0 = X'-{-6.X" — Y [1040], will by this means become
[1048]
X" = 0.
equations [1046a] will produce, in the terms independent of ' (n t-{-s), an equation
f — j = 0, lilie the first of [1044], which gives n constant, consequently a constant, and s
is then constant as in [1045']. • Supposing now n, a, s, to be constant in the value of X,
and putting (—-j = X' -\-t X", as in [889], we obtain
X' = n"^.^ — (^Vsin.(«^ + £)~(^^).cos.(«^ + s)_&c.|, X" = 0,
and the equation 0 —X' -{- 6 X" — Y, [1040], becomes 0 = X' — Y, or by dividing
by n"^,
• 0 = $ — (^) — |m'n.(ZC + Z'i))|.sin.(«i + £)
+ ^ — (j:-\ + ^m'n.{hC+h'D)l. COS. (nt + s),
from which we get the equations [1046], by putting the coefBcients of sin. [nt-]- s), and
cos. {nt-\- s), separately equal to nothing and changing 6 into t.
* (720) The equations [1047, 1048, 1049], are deduced from 5, [1039], in the same
manner as [1041, 1042, 1043], were deduced from ■^, [1038].
n. vii. § 54.] SECULAR INEQUALITIES. ^"^^
0 := (ii\ . sin. (nt + s) — f^Y COS. (nt+ s)
d6
m' .n
-^.a^a'. 5<^> . (p — /) . sin. (nt + s) ^1049^
[1050]
_ HL^ ,a^a'. 5w. cg __ g'^ . cos. (nt + s).
Hence, by comparing the coefficients of the similar cosines and sines, and
changing ^ into t, to obtain directly p and q in functions of t, we shall get
^ = -^-«^«'-5<'>.(5-g') (3)
'^=-:^.a^a'.B^'K(p-^). (4)
When p and q have been found, from these equations, we must substitute
them in the preceding expression of s [1039] ; then rejecting the terms which
contain the arcs of a circle, we shall have
s = q . sin. (n f -|- s) — p. cos. (n t -\- s) -\- m' . X'
54. The equation — = 0, just found [1045'], is of great importance in
the theory of the system of the world, because it shows that the mean
motions of the heavenly bodies, and the transverse axes of their orbits, are [loSlT
unchangeable ; but this equation is correct only in quantities of the order
m' . h inclusively. If the quantities of the order m' . h^, and of the higher [losi"]
orders, produce in — , a term of the form 2k t ; k being a function of the
elements of the orbits of m and m' ; it would produce, in the expression of v,
the term kf, which, by affecting the longitude of m, in proportion to the
square of the time, would become at length extremely sensible. We should riMe.
d 71
then no longer have — = 0 ; but instead of this equation we should, by [1051"']
[1051]
MeaB mo-
tion and
transverse
axis inva-
dt
IV p.
dt
the preceding article, have —- = 2k;* it is therefore very important, to
* (721) If — , in the formula [1038], should contain the term 2 Jet, the value of Y,
[1041], would be increased by 2^; X, X', X", [1041,1042], bemg unchanged. This
^^^ MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
ascertain whether there exists any term of the form kf in the expression
[I05ii»] of V. We shall now proceed to demonstrate, that if we only notice the
first power of the disturbing masses, however far we may carry on the
approximations, relative to the powers and products of the excentricities,
[1051V ] and the inclinations of the orbits ; the expresssion of v will not contain
similar terms. We shall resume, for this purpose, the formula (JT),
§46 [930],
[1052]
6r =
a . cos. V .fndt.r . sin. v . \ 2/d i2 + r. \-i—] >
— a . sin. V .fn dt .r. cos. v . < 2yd R-\-r A -- — ) >
We shall consider the part of (5 r, which contains the terms multiplied by f^
or, for greater generality, the terms which being multiplied by the sine or
[1052'] cosine of an angle a^+f3, in which a is very small, have at the same time
c? for a divisor. It is evident that by supposing « = 0, there will result a
term multiplied by f ; therefore this second case includes the first.* The
would introduce in the equation 0 = X' -f- ^" • ^ — Y, [1043], the term — 2 k, which
would make the first of the equations [1044] become 0 = — — 21c, and by changing fl
(t 0
into t, [1 044'], it would become 0 = 2 k, or -— = 2k, as above.
at at
* (722) To illustrate this, suppose that S r contains a term, depending on the double
[1052al integral of the expression h .dt^ . sin. (a ^ -j" ^)» ^nd let the integrals be taken so as to
vanish when ^ = 0. The first integral will be .dt . cos. {a.t-\-^)-\- - .dt . cos. |3,
[10526] and the second ^ . sin. {a.t-\- ^)-\-- .t . cos. ^ -| — -. sin. p. If we now develop
sin. (a < + ^), according to the powers of a t, by means of the formula [678o], in which
T is changed into p, and a into a t, we shall get
sm.
hence
(ai + ^) = (l— ^ + &c.).sin.^ + (ai— ^- +&c.).cos.^,
^'Sin. {(xt-\-^)=. ^.sin. p + JJi^.sin. ^ . ^ cos. p -f -g- . «^ . cos. ^-\-hc.
II. vii. § 54.] SECULAR INEQUALITIES. 577
terms which have a^ for a divisor, must evidently arise from a double [I052"j
integration ; they cannot therefore be produced except by the part of 6 r
which contains the double sign* of integration /. We shall first examine
the term [1052]
2 a . COS. V .fn dt .{r. sin. v ./d R)
M-.\/l— «
[1053]
If we fix the origin of the angle v at the perihelion, we shall have, in the [1053]
elliptical orbit, by § 20 [603],
I -j- e. COS. V
consequentlyt
a.(l-.e^)—r a . (I — e^) 1
i,v = — ^ = — ^ ;
cos. V = ^ = ; [10551
er ere
Substituting tliisin the preceding integral [1052i], it will become
^ht^ . sin. ^-\- — .i^. COS. ^ -f- fee,
and by putting a = 0, it changes into i 6 t^ ; which is the same as would be produced
from the double integral of b dt'^ .sin. [at -{-^), putting c = 0, in the firi* instance,
by which means it would become hdi^ . sin. p, whose integral, taken twice, would produce
the term ibi'^ . sin. p, as above.
* (723) By neglecting the square of the disturbing force, the expression of R, [957],
becomes of the form m! i: . JV. cos. {p t -\-p'), JV, p, p, being wholly composed of the
elliptical values of the elements of the orbits of m, m', which, by [659, 668, 675], contain
no terms where t is without the signs of sine and cosine. Substituting this value of R, in 5 r,
[1052], and still neglecting the square of the disturbing force, we may consider r.sin. t?,
r . cos. V, and ndt, as containing only the elliptical values [659, 668], without any term of
the form A t, so that there is nothing but a constant term of the form A' dt in d R, which
can produce in 5 r terms of the form A" t^. Similar results follow from S v, [931 J.
f (724) The formula [1055] is the same as [716']. If we take its differential,
considering r, v, as variable, and multiply it by — r^, we shall get the formula [1056].
Now from t^dv = hdt, [585], A = ^^a.(l— e2), [599], and v//r=na^, [605'],
we get r^dv = dt. v/fji,a.(l— eS) = a^.ndt. y/l — e2, [1057]. Putting this value of
145
^^^ MOTIONS OF THE HEAVENLY BODIES. [Mec. Gel.
and by taking its differential,
[1056] r'.dv. sin. V = ^^-Ili2. .dr:
e
but by § 19,
[1057] i".dv = dt.y/i,a.{i—e^) = a''.ndtyT=T^;
therefore we shall have
rjQ5g-] a .ndt .r . sin. v rdr
Ihe term — 7== — ^, will therefore become
[1059] i^^./;r(?r./di?}, or ^.jr^/di2— /r^.di2|.
As this last function does not contain any double integral, it is evident that
[1059'] there cannot result from it any term which has «^ for a divisor.
We shall now consider the term
[1060]
2 a. Sin. v.fn dt .\r . cos. v ./d R]
«2
M-.V/l— e^
of the expression of Sr [1052]. Substituting for cos.??, the preceding value
in terms of r [1055], it becomes
2 a . sin. V . fn d t Ar — a . (I — e^)\ . fd R
[lOGi] ^ , — y ^-^-^ .
Now we have, by § 22 [659, 669],
[1062] r = «.{l+ie2 + e.x'|,
riOG2'] X being an infinite series of cosines of the angle nt -]- s and its multiples ;
therefore we shall have
[1063] '^—,\r — a.(\—e')}.fdR = a.fndt.{§ei-x']J^R-
r^dv in [1056], and multiplying by /_ , we shall get [1058]. This being substituted
in [1053], it becomes -— .f\2rdr.fdR]. Integrating it by parts, relative to r, we get
-— .|r2./di2— /r2,d/2}, as in [1059].
n. vii. § 54.] SECULAR INEQUALITIES. 579
If we put the integral of fx''^dt=^-x\ we shall have* [1063]
a.fndt.\ie-\--)l\.feiR==%ae.fndt.f^R-^a-)l'.fAR—a,f-)l'^R, [1064]
As these two last terms do not contain the double sign of integration, they
cannot produce any term having a^ for a divisor ; noticing therefore only [1064]
terms of this kind, we shall have [1060 — 1064]t
2a.sva.v.fndt.\r.cos.v.f6iR\ Sa^e .sin. v .fndt.fdR dr Sa ^ , .. _
and the radius r will become
(r) and (— ;7-) being the expressions of r and — — -, relative to the [1066']
elliptical motion. Therefore if we notice only the part of the perturbations
divided by a^, in the expression of the radius vector, it will be only necessary
to increase the mean longitudeX nt-\-z by the quantity — .fndt.fdR, in [1066"]
the expression of that radius relative to the motion in an ellipsis.
* (725) Substituting in [1063] the value x .ndt = d-)(\ [1063'], it becomes
^ae.fndt.fdR-{-a.fd-x'.fdR, and if we integrate by parts the term
a .fd x" . /d R, it becomes a x" ./d i2 — a . // . d 22, as in [1 064]. Neither
x', [1062'], nor x") [1063'], contain t, without the sign of sine and cosine, noticing the terms
as in [1051'].
f (726) The last member of [1065] is deduced from the second member by substituting
for Sa^e.sin. v, its value deduced from [1058], namely ' — y ^ .
J (727) From [1053'] we have -!*:=: 0, and the elliptical value of r, [669], becomes
a function of nt -{- s, which we shall denote by (r) = (p . (nt-{- s), and we shall
suppose that (r) becomes (r) -f- 5 r, by increasing the angle nt by the small
quantity 6 T, so that (r) -{- 8r=(p . (nt-\-s-{-S T). The second member of this
expression being developed, by the formula [617], will be
.■in, + s) + ,T/-^l±^ + ^., or „ + .r.(,9 + 8..
and if we neglect the second and higher powers of S T, it will give 8r = 5 T . (--t~)«
Putting this increment equal to that in [1066], ( --7-) • — •/» dt.fdR, we shall get
^80 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
We shall now examine into the manner of noticing this part of the
perturbations in the expression of the longitude v. The formula (Y) § 46
[I066"'l [931] gives, by substituting — . -^ .fndt .fd R [1066] for 6 r, and
noticing only the terms divided by a^,*
(2r.ddr-{-dr^ )
[1067] I a^.n^dlT"^^] 3a , , _„
''=' ^/T^ '-'-Jndt.fdR;
now by what precedes,!
, ae.ndt . sin. v o i ^ , ,
[1068] a r = /— — -3 — ; irdv = a^,ndt. y/i— -e^ j
hence it is easy to obtain, by substituting for cos. v the preceding value
in terms of r,t
2r.ddr-\-dr^ -
[1069] a^.n^dt^ "^ _ dv
\/l-^ ~n'dt''
ST= — .fndt.fdR. Therefore, if in the elliptical value of (?•), we increase the
mean longitude nt -{-s by — . fn dt .fdR, we shall obtain the value of r, in which,
terms having the divisor a^ are noticed.
* (728) Put for brevity, —.fndt.fdR==W, then Sr=~.W, [1066], this
Lit 7v(LZ
gives d 6r = — — • . W, the term d W being neglected because it does not contain the
double integral. These values o{ d6r, 6r, being substituted, in [93J], we shall get the two
first terms of ^v, [1067], the tliird term of [931] being like the third of [1067].
f (729) The equation [1058] gives the value oi dr, and the value of r^Jv, is as
in [1057].
fit f" Tt dt d v cos V
t (730) The value of dr, [1068] gives ddr — — ' .J ^ , which by
substituting dv = ^i-i^ — li__Ill, deduced from the second of the equations [1068],
jj a^en^ .dt^ .COS. V , 2r.ddr 2ae.cos.v - ,
gives ddr = :s , hence -——--= , and the value ot a r,
II. vii. §54.] SECULAR INEQUALITIES. ^81
noticing therefore only the part of the perturbations, which has the divisor a^,
the longitude v will become
(v) and ( -- J- ) being the parts of v and — ^, relative to the elliptical
motion. Therefore in order to notice this part of the perturbations, in the
expression of the longitude of m, we ought to follow the same rule which we
have given [1066"] for the similar terms of the radius vector; that is, we [iotck]
3 a
must increase the mean longitude nt-\-s by the quantity — .fndt.fdR,
in the elliptical expression of the true longitude. [1070"]
The constant part of the expression of ( — r^ ) » being developed in a series
of cosines of the angle nt-]-s and its multiples, is reduced to unity, as we [i070"]
have seen in ^ 22 f hence there arises, in the expression of the longitude,
the term — .fndt.fdR. U dR should contain a constant term km'.ndt, [ioto^v]
it would produce, in the expression of the longitude v, the term f . — . kn^f.
[1068], gives -^-— -- _- — '— . Substituting these in the first member of [1069], we
Set 6 • COS* V C • Sin.^'B 1
shall get — J^^ + /j_g2a + r[Z^)h ' '^^ ^° ^^^^ ^^^^^' reduced to the [i034al
,, .# e2.gin.2^)-i_i — c2 1 — e2.cos.2t) (1 — e.cos.r).(l+e.cos.'«)
denommator ( 1 — e e) ^ , are = = ' '—\
(l_e2)f (l_e2)f (1— e2)&
... ^ , o.(l — e2) r^^^^T • 1 (1 — e.co3.v).a
substituting 1+e.cos. v = , [1054], it becomes — -==z|— — ,
, . . , , - Soe.cos.v
connecting this with the first term /- — ^ , the sum will be
and this, by means of the second equation [1068], becomes -— , as in [1069].
Substituting this in [1067], it produces the last term of [1070].
* (731) This follows from the value of v, [668 or 669].
146
582 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
The investigation of the existence of such terms in the longitude v, is
therefore reduced to the examination whether d jR contains a constant
term.
When the orbits have but little excentricity, and are inclined to each other
[1070 »] by very small angles, we have seen in § 48 [957], that R may always be
reduced to an infinite series of sines and cosines of angles, increasing in
proportion to the time t. We may represent them, in general, by the
[lO/O'^i] term km! . cos.\i'n't -{-int-\- A], i' and i being integral numbers, positive
or negative, including i = 0. The differential of this term, taken only with
[1070^"] respect to the mean motion of m, is — ik.7n'.ndt.sin.\i'n't-\-int-\-A\ ; which
is the part of dR relative to this term. This cannot be constant, unless we
[1070^"'] have i' n' -{-in = 0 ; which requires that the mean motions of the bodies m
and m\ should be commensurable with each other ; and as this is not the case
[l070ix] in the solar system, it must follow, that the value of dR does not contain
any constant term ; hence, if we take into consideration only the first power
of the disturbing masses, the mean motions of the heavenly bodies will be
ft ji
[1070^] uniform ; or, in symbols, — = 0.* The value of a being connected with
that of 71, by means of the equation n^ = -^ [605'], it follows, that if we
[1070"] neglect the periodical quantities, the great axes of the orbits will be
constant.
Equations If thc mcau motions of the bodies m and m\ without being exactly
of a long ' o J
period, commensurable, are however very nearly so ; there will exist, in the theory
[1070""] of their motions, some equations of a long period, which may become very
sensible, on account of the smallness of the divisor o?. We shall see hereafter
that this is the case with Jupiter and Saturn. The preceding analysis will
* (732) Another demonstration of this proposition is given in §65, [1197'"], and in
the supplement to the third volume, it is proved, that the same is true even when the
approximation is carried on to terms of the order of the square of the disturbing masses,
and Poisson, who first extended the demonstration to terms of the second order of the
[1070a] masses, has also proved that the proposition is true for terms of the third power of the
masses, arising from those of the second order in the disturbed planet, as will be mentioned
hereafter, in the notes upon this supplement.
II. vii. <§ 56.] SECULAR INEQUALITIES. 583
give, in a very simple manner, the part of the perturbations which depends [i(y7o«»]
on this divisor ; since, from what has been said, it will only be necessary to
vary the mean longitude nt-\-^ or fndt, by the quantity — .fndt.fdR,
[1066", 1070"] ; which amounts to the same thing as to increase n, in the
integral fndt, by the quantity .fdR. Now if we consider the [Kno'^i
orbit of m as a variable ellipsis, we shall have n^ = — [605'] ; and the
preceding variation of n will produce in the semi-transverse axis «, the
. . ^ 2a^.fdR
variation* . [icy70«v]
If in the value of — , we carry on the approximation to quantities of the
order of the squares of the disturbing masses, we shall obtain some terms [1070''"]
proportional to the time ; but, by considering with attention the differential Temsof
equations of the motions of the bodies m, m', &c., we shall easily perceive °l^^^^
that these terms are also multiplied by quantities of the order of the squares maswl
and products of the excentricities and inclinations of the orbits. However, [lo^o'^'"]
as every thing which affects the mean motion may at length become very
sensible ; we shall, in the course of the work, notice these terms, and we
shall find that they produce the secular equations observed in the motion o/*[io7o''v»"]
the moon.-f
55. We shall now resume the equations (1) and (2) § 53, [1046], and
shall suppose
(0,1) = -—; [03 = —^—; [1071]
* (734) By [605'] we have n=(x*a~f, the differential of its logarithm is
dn da
n ^ ' a '
Substituting for dn its value [1070""], ./d jR, and multiplying by — % .a, we shall [10706]
get da = .fdR, as above.
t (735) In Book vii, § 23, [5543].
[1073]
584 MOTIONS OF THE HEAVENLY BODIES. [Mec.Cel.
they will become
^ = (0,l).«-E7I].r;
±L = -(o,i).h + \^.h'.
The expressions of (0, 1) and [oTj] may be determined, very easily, in the
following manner. Substituting for C and D their values found in § 50,
[1018], we shall have
(0,1) = ^'\^\-J^) + i^'\-J-a^-)\'^
r—, m'.n ( .,,, 2 /dJ^W\ ^ 3 /ddA^^)\}
Now by ^ 49,*
a^ f^:^\ + 1 «3 fdd:^\ ,. d 6f > _^^s dd 6f .
^5(0) ddU^^
we shall easily obtain, by the same article, — yi-, — r4—^ in functions of
•^ *^ da. dar
bf\ and 6J^^ ; and these quantities are given, in linear functions of ¥^^, and
6ii\ ; we shall thus findf
* (736) Putting i = 0, we shall get in [999],
2 l^l^W — - ^^ = __ 2 ^'
" * W« / a!^' da " • rfa '
and in the first of the equations [1001], ^ a^ . f j = — J a^ . ^ . The sum of
these is as in [1074].
f (737) Putting in [981], 5 = ^, and successively i=0, and « = 1, we get
LlU74aj -^^ l_a2 ^ l_a2-"4 ' ^a a.{l— a2) * 1 — a^ *'
also putting i=2, s = h in [966], we find &f =(^+°'^) '^J"^ " '^^^^ which,
being substituted in rf b^p, gives
^t¥__L+?^ m) ?_ ^(l+«^)>&?-^«-^>g^?__Il}__ 50) -i?
[10746] -rf^ a.(l — a2)' * 1— aa*^ 3„ 5 a.(l— a2)* i^l-cfi'
Putting i = 0, and s = i, in [982], we obtain
ddbf^_a_ dbf, l+«^ z(0)_ 1 ^ l5_ m
do? 1— a2 • da ~^(1 — a2)2' * 1 — a2 * da. [l — a^f ^'
n.vii. §55.] SECULAR INEQUALITIES. ^85
therefore
Put
(a^--2aa'. cos. & -{- a'^y = (a,af) + (a, a')', cos. fl + (a,a')". cos. 2^ + &c., [1077]
we shall have, by § 49*
(a ,«')=!«'. ¥1\ , (a , a')' = «' • &^i » &c. ; [1078]
which, by substituting —1-, [1074a], and —j-^, [10745], becomes
ddbf a ^ abf ¥P >, l + a2 ^,o, 1 (, —¥P , J^^) > 2a¥P
d
a2 I_a2 7i_a2 i —aa^"!" (i_a2)2- i (i_a2)'|a.(l— a2)"^(l— a^)) (1 — a^)
and this, by reduction, becomes
fldb'Sf) 2a2 1 — 3a2
""0^ ^ J(0)_i__i r_ 7,(1)
This value, and that of * , [1074a], being substituted in the second member of [1074],
( a3 a5 •) ,,„, , ( a2 i a2.(l-_3 a^) ) ,.,.
„3 ^(0) 1 /,2 n 4- a2^
reduction _^^ + 3_±^^ i(-) ; and, by substituting 6f, 6^), [990, 991], the
denominator will be (1 — a^)"*, and the numerator,
— a3.{(l+a3).5W^ + 6a.J(!^}+|a2.(l + a2).f2«i(4 + 3.(l+a2).6(l|}.
The coefficient of J^^, in this numerator, is evidently equal to nothing, and that of 6^, is
— 6a^ + f a2.(l+a2)2==3a2.|(i_|.„3)2_4„2|^3„2^(l_„2^2.
3 „2 /J 2\2 WD
therefore this second member of [1074] will be ? — '-^ ^y ~^ , and by rejecting
(1 — a^)^j common to the numerator and denominator it becomes like the second member
of [1075]. Substituting this in (0, 1), [1073], we shall get [1076],
* (738) Putting s = — \, in [964], and multiplying by a', using a=a'a, [963'''], [1074c]
we shall get
^a2 -- 2 a a' . cos. 0 + a'^^-i = a' .{| 5^^ + ¥1\. cos. ^ + &^ . cos. 2 ^ + &c.f
147
586 MOTIONS OF THE HEAVENLY BODIES. [Mec. Gel.
therefore we shall find
.rx 1. Sm^.n c? a' .{a, a')'
[1079] ^ (0,1) = — ^ ^
4.(a'2_a2)2
Now we have, by ^ 49,^
*
Substituting for 6J'^ and its differentials, their values in 6^\, and 6^^^, we
shall find the preceding function to be equal tof
(.081] _3..Ki+^.n+i..6ni,
(1 — a'^y
Comparing this with [1077], we shall obtain the equations [1078], Multiplying the value of
(0, 1), [1076], by — , and substituting a'a = a, it becomes,
^'^"~ 4.(a'2_a2)2 '
and this, by substituting a' . J^|=(a, a')', [1078], becomes as in [1079],
* (739) From [997] we get a S'^ = ■^— ^, . Ul^ =aP — a. h^l^ ; from [1000],
we find — « •(— 1 — 1= ;^ + -;^--T-^= — a^ + a^.—-^, and from [1001]
\ da J a^ a^ da da. ■- -*
we get — ^a^.(__— j =: -[-| a^. -T-^; the sum of these three expressions will
be as in [1080].
f (740) Putting i=l, and s = |, in [982], it becomes
<;a2 a.(l— a2)' rfa "^ ( (1 — a2p a^\' ^ (1— a2)* rfa (I— a2)2" *'
Substituting this in the second member of [1080], which for brevity we shall call S, we shall
get
3a3 dhf__ 3a4
2.(1— a2)' rfa (l_a2)2-^i'
and by reduction
Q^3.(-l + 3a2).a _^ 3^_ ^) 3a3 rf&(2) 3a4
2.(l_a2)2 . ""^ ■t"2.(1— a2)* rfa
,(l_a2)2 • • * ^2.(1— a2)* rfa 2.(1— aS)' da {\ — a^f'^^'
[1082]
II. vii. § 55.] SECULAR INEQUALITIES,
therefore
L^ 2.(1 — a^ys '
Now putting s = i, and i^2, in [981] we shall get
dlf_^±S^ 5_
da a.(l— a2)* ^ 1 — a2* ^*
substituting this and -ji, [1074a], in S, it becomes
3«.(-l + 3a2) 3«s^ ^ Hll^!. t(.)__^ K.) ^
2.(1— a2)2 • i ~2.{l_a2)- ^„.(l_a2j- i i_„2 ' i j
~2.(1— «2) { a.{l_a2) * ' l_a2 * ^ (i_a2)2 ''i »
15 a3 15a2.(l + a2) 15 a3
or, by reduction, S =^;^^^ - bf- ^^^^_^,^ • ^' + i;^^ • ^?' ^ence we get
^il=^M==15„2.5^,)_15„.(,+„2).t(2)_^15„2.J^). ^^^^^^^
Now putting « = 3, and s = ^, in [966], we shall find
^(3)^2.(l+a^).^&f) — |a.&^^^^ ^^^ 15a2.6f=12a.(l4-a2).5f) — 9a2.6^i>,
hence [ 1 080a] becomes 2.(1— a2)2. ^ g „2 ^ ^^) _ 3 „ ^ ^ 1 _^ „2) ^ j(2), Again, the
same formula [966], by putting i = 2, and s = ^, gives ^^^^^ ^ '^ ' °^ / ' ^ g « » t>^ ^
fa
or _3a.(l+a2).6(|) = — (l+a2).{2.(l+a2).t(>)_a.J(0)}; hence
^^^^^=6a2.J<:^) — (l+a2).{2.(l+a2).J^i)_tf.J(0)j
= (_2 + 2 a2 — 2 a4). 6(^) + a . (1 + a^) . bf.
This, substituting the values of bf, ¥l\ [990, 991], and multiplying by (1 — aP)^ gives
2.(l-a2)4.^^ (—2+2 a2— 2a4) . {2 a . t^^ + 3 . (1 + a^) . JO^j
a
+ a.(l + a2).{(l + a2).6(^ + 6a.6a|j,
or by reduction =— 3 . (I — a2)2. a6<^ — 6 . (1 + a^) . (1— a^ja. i^i^. Multiplying
this equation by ^ _ we get -5> = — d * (i— ft2>2 ' -y ^^ m
[1081]. Substituting this in [1073] we shall get [1082]. Multiplying the numerator and
denominator of [1082] by a'^, substituting also the values of ¥% 5^, [1078], and
a'a = a, [1074c], we shall obtain the formula [1083].
588 MOTIONS OF THE HEAVENLY BODIES. [Mec. CeJ.
or
rj^ygg. P — I 3m' . an.\{a^-\-a'^) . {a,a'y-\-aa' .{a,a')l
therefore we shall obtam, in this manner, very simple expressions of (0,1)
and [oTj] ; and it is easy to prove, from the values of 6^\, ¥1\_, given in
[108a'] series, in § 49, that these expressions v^^ill be positive, if n be positive, and
negative if n be negative.*
* (741) The second of the formulas [992] gives Ull = — ^ . (1— a^)'-^ . Ul\ and by
2
putting s = |, in the second formula [988], we obtain i^^ which, being substituted, gives
3 3 3.5
and as every term of the infinite series q "^ o * o"! ' "'^ H~ ^^•' ^^ positive, its sum must be
positive, hence ¥1\ is negative, consequently — ^ '^' ~-, must be positive, therefore
the expression (0, 1), [1076], must have the same sign as n. Again if we substitute the
values ¥^={l—a^f.bf, ¥ll = — ^.{l—a^f.bl\ [992] in [1081], it
[1081a] becomes a . 5 ( 1 + a^) . 6^^ — f a ¥f X ; and putting « = |, in [988] , we shall get
7/n r. (3 , 3 3.5 ... 3.5 3.5.7 . . . )
1 S^\i/^i 52 Vs. 4/ ^3 72 \2A.6/ ^
4 /3\= , 8 /3.5\2 , , 12 /3.5.7\2
Multiplying [10816, c] respectively by — | a^, and a, we shall get
<3 . 5 . 7\2
3 „,(0) 3 „ 3 /3\2 , 3 /3.5\2 3 /3.5.7\2
,(1) 3 „ , 5 /3\2 . , 7 /3.5\3 „ , 9 /3.5.7\3
[1081e]
r=i-«"'+2i2J-»'+3-i2r4;-«°+4-i'''''''
whose sum is
,0) 3 3,(0) , /3\2 2 /3.5\2 3 /3.5.7\2
II. vii. ^ 55.] SECULAR INEQUALITIES. ^^9
Put (0,2) and [oj^, for what (0,1) and [oTT| become by changing a', m', [1083"1
into a", m". Also (0,3) and [oTa], for what the same quantities become by
changing a', in!, into a'", m!" ; and in the same manner for others. Also let [1083"]
h!\ l\ h!\ V", &c., be the values of h and /, relative to the bodies wi", m!",
&c. ; we shall have, by the combined actions of the different bodies, m', ml\ [1083'^]
ml", &:c., on m,*
^=5(0,l) + (0,2) + (0,3) + &c.5.Z-[o7riJ'-EIE.^"-&c. ;
[10841
li = -_{(0,l) + (0,2) + (0,3) + &c.}.^+EID-^' + II3.^'' + &c.
It is evident that -r-, -;-» -; — » -;-» &c., will be determined by similar
expressions to those of — , and — , and it is easy to deduce them from
[1084], by changing successively what relates to m, into the corresponding
terms of m', in!', &c., and the contrary. Suppose thereforef
(1,0), [ITo], (1,2), [TTi], &c., [108.5]
Now multiplying [lOSltT] by a^, we shall get
31(1) 4 /3\2 8 /3.5\2 12 /3.5.7\2 g
Adding this to [1081 e] we shall get the value of the required function [lOSla],
^fi I 2x Ad) 3 j.(o)^ 1.5 /3\2 . , 2.7 /3.5\2 3.9 /3.5.7\2
in which the law of continuation is manifest, and every term is positive, consequently the
whole expression [1081] must be positive. Hence it follows that the value of foT^I,
[1082, 1083], must have the same sign as n.
* (742) The planet w! produces in — the terms (0, 1) . I — [orr] . l\ [1072]. In
like manner m" must produce (0, 2) . I— [oTs] . Z", m'" must produce (0, 3) . I — [o]!] . /"',
&c. The sum of all these gives the complete value of — , as in [1084]. — is deduced
in the same manner from [1072].
t (743) In all these expressions \iTo], \TX\, he., (0,1), (0,2), &c., the first figure
denotes the number of accents on the mass of the disturbed planet, and the other that of the [10836]
disturbing planet.
148
690
MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
to be what
[1086] (0,1), [oTi], (0,2), [03, &c.,
become, when what relates to m is changed into the corresponding terms of
m', and the contrary ; suppose also
[1087] (2,0), [^, (2,1), [2TI], &c.,
to be what
[1088] (0,2), [03, (0,1), [Kl], &c.,
become when what relates to m, is changed into the corresponding terms of
m", and the contrary, and so on for the other bodies. The preceding
differential equations, referred respectively to the bodies m, m', m", &c., will
give, to determine h, I, h', I', hl\ I", &c., the following system of equations,*
System J I
i'=s ^= {(o,i)+(o,2)+(o,3)+&c.i./— EH]./'— [o-2].r-[^3].r-&c.^
of the first dt
degree, to
find the a I
excentrici-
ties.
li==_.{(0,l)+(0,2)+(0,3)+&c.}./i+[^.A' + E:.].A''+[oTi].^'''+&c.
^=[(l,o)+(l,2)+(l,3)+&c.^/'-[^3.z-[^Tg.^-E;g.^-&c.
[1089] ^=— {(l,0)+(l,2)+(l,3)+&c.}./t'+|iI3.A+[iZ2].A''+lir3^ } ' (^)
^'={(2,0) + (2,l) + (2,3) + &c.}.r— [i^./— [2TT].Z'---E3].r---&c.
^=— {(2,0)+(2,l)+(2,3)+&c. 5.A"+ E:o]./i+ [^].h'+ [^].A'"+&c.
&c.
The quantities (0,1) and (IjO), [oTi] and frTo], have some remarkable
relations with each other, which facilitate their computation, and will be
useful hereafter. We have, by what precedes, [1079],
3 m' .na^ .a . {a , a')'
[1089'] (0,1) =
4.(a'2_^2^s
* (744) The two first of these equations are as in [1084], in which the disturbed planet is
m, the accent on which is considered as nothing. Now if in these we change 0 into 1, and
1 into 0, in the expressions (0,1), (0,2), Stc, [oTi], &c., we shall obtain, as was observed
In the last note, the third and fourth equations [1089], corresponding to the disturbed planet
m', and so on for the others. It may be observed that these equations give h, I, &c., exact,
[1089a] except in terms of the order m' e', or m'^.
II. vii. § 55.] SECULAR INEQUALITIES. ^^l
If in this expression of (0, 1), we change rn! into m, n into w', a into a',
and the contrary, we shall have the expression of (1,0), which will
therefore be
^1 n^ ^m.n'a'^.a.{(^,a)\ [1090]
but we have {a, a!)' = («',«)', since both of these quantities result from the [1090']
development of the function (a^ — 2aa' , cos. 6 + a'^y, in a series [1077]
arranged according to the cosines of the angle & and its multiples ; therefore
we shall have*
(0,l)m.n'af=(l,0).m'.na; [i09i]
now by neglecting the masses m, m', &c., in comparison with M, we shall
have
71^ = ^; w" = ^; &c. ; [1092]
therefore
(0,l).m.v/a=(l,0).»i'.v/^; [1093]
from which equation we may easily compute (1,0), when (0,1) shall be
determined. We shall find in the same manner
[^.m.v/^ = [r^.m'.v/V. [1094]
These two equations will also take place when n and n' have contrary signs ;
that is when the two bodies revolve in opposite directions ; but in this case
we must prefix the sign of n to the radical ^, and the sign of n' to the [10941
radical \/a'.-\
* (745) This appears from the equation [1077], the first member of which is not altered
by changing a into a', and a' into a. The values (0,1), (1,0), [1089', 1090], being
substituted in both members of [1091], they become identical, therefore this equation
is correct. Now by [1013'], neglecting m in comparison with M, we get M= v? a^, and
in like manner M. = n ^ a' ^, hence n^ a?=n'^ a' ^, or na. y/a = n' a' . y/a'. [1091a]
Multiplying this by [1091] and dividing the products by nn .a a', we shall get [1093].
In like manner we may find [1094] from [1083, 1085, 1086]. The formulas [1095] are
merely the generalization of [1093, 1094], applying them to other bodies.
f (746) The radicals \/a, \/a', Sec, were introduced into the formulas [1093, &c.] by
means of y/^ deduced from [1091a], which is
y/jtf = n a . v/a'= n' a' . \/o^= n" a" . v/a^' = &c.
^92 MOTIONS OF THE HEAVENLY BODIES. [Mec.Cel.
From the two preceding equations, we can evidently deduce the following :
(0,2).m.v/^=(2,0).m".^'; E3.m.v^ = E3•w^".v/^ ; &c.
[1095]
[1096]
56. Now to find the integrals of the equations (A) [1089] of the preceding
article, we shall put
h = N.sm. (gt-\-^); Z = iV.cos. (gt + ^);
h'=N' . sin. (^ ^ + /3) ; l'=N', cos. (^ ^ + 13) ;
Substituting these values in the equations {A) [1089], we shall have*
iVg=j(0,l) + (0,2) + &c.i.iV— [TTI-iV' — [oT3.iV"— &c.
[1097] iV'^ = j(l,0)+(l,2) + &c.j.iV'— [iToj.iV— [rg.iV" — &c.
iV"^ = |(2,0) + (2,l) + &c.i.iV"— E3--^--Ed]-^'— &c. ^' ^ ^
&c.
Supposing therefore, \/m and the semi axes «, a', a", Sic, to be positive, it will follow from
these equations that the expressions n . y/a, n' . \/a', n' . y/a", &c., must also be
positive, and \/a must have the same sign as n ; y/ai the same sign as «', &i.c.
Therefore if we suppose n to be positive when the motion of m is direct, we must also put
\/'a positive. In like manner we must put fi and \/a! positive, if the motion of ml be direct,
but if its motion be retrograde, n' must evidently be negative, therefore \/a' must also be
negative to preserve the same sign in the quantity n . y/a'. This change of signs of the
quantities \/a, \/af, &;c. when the corresponding values w, n', &tc. become negative, is a
very important consideration, since it will be shown, in § 57, that the permanency of the solar
system depends on these radicals having the same sign.
* (747) The assumed values of A, I, h', I', he. [1096] being substituted in the equations
[1089], produce the equations [1097]. The two first of the equations [1089], produce
[1097a] separately, the first of the equations [1097], The two next produce the second of [1097],
and so on; consequently the number of equations [1097] is just half that of [1089], so
that there will be as many equations [1097] as there are bodies m, w', m", he, which
number is i, the number of the equations [1089], being 2 i. To show, by a simple example
the use of these equations, we shall suppose that there are only two of the bodies m, m',
and the equations [1097] will become
A-g = (0, 1) ; JV— [oTi] . A" ; A*' g- == ( 1 , 0) . JV' — [Ho] . A-.
n. vii. ^ 56.] SECULAR INEQUALITIES. 593
If we suppose the number of bodies m, m' w", &c., to be equal to i, the
number of these equations will be i ; and by eliminating the constant
quantities N, N', &c., we shall have a final equation in g of the degree «,* [1097]
which may be obtained in the following manner.
The first gives JV' = —j==~' • •A'*, which being substituted in the second, put under the
form 0 = 1(1,0) — g I . JV" — [ITo] . JV, becomes divisible by JV*, and gives
{il,0)-g].{[0,l)-g}
which is of the second degree in g, and furnishes two values, which we shall denote by g, g^.
The first of these being substituted in JV', gives its value JV' = '— n— . JV, and
if we assume another arbitrary terra A*i instead of JV, to correspond to gi , the corresponding
(0, 1) o-j
value of JV*', which we shall denote by JV/, will be JVi= V^-^-. . JV*i. Hence in
addition to the values of A, h', Z, Z', in [1096], we may also put A = JV\ . sin. (gi t -\- ^i),
1= JV,. COS. {git + iSi), h' = JV/. sin. {gy_ t + Pj), l' = JV/. cos. {g, t + 3i). and
as the equations [1089] are linear in h, I, h', Z', we may take the sums of these two values
of A, or Z, &c. ; by which means we shall have A=JV. sm. {gt-{-^)-\-J\/'i . sin. (g"i<+Pi),
1 = JV. COS. {gt + ^)+JVi. COS. {g^t + ^i), A'=JV'.sin.(^< + p)+JV/.sin.(^i<+^i),
Zi = JV'. cos. {gt-\-^)-{- JV/ . cos. (gi t -\- ^i), which satisfy the four equations [1089],
and contain four arbitrary constant quantities, JV, J\\ , ^, ^l . They must therefore be the
complete integrals of the proposed equations.
[10976]
* (748) It was proved in the last note that when there are two bodies m, m', the resulting
equation in g will be of the second degree. If there be three bodies, m, mf, m", the first of
the equations [1097], gives JV'= (.^5- + jB) . JV+ C JV", A, B, C, being
coefficients depending on (0,1), (0,2), &,c., and independent of g^ JV, JV', JV". This
value being substituted in the second and third of the equations [1097], they will become
of the forms
0={A^g' + B'g+C').JV+{D'g + E').JV";
[1098a]
0 = {A''g + B").JV"-^{C"g-{-D").JV;
A', R, &c.. A", B', he., being like A, B, &;c., independent of g. The value of JV",
deduced from the first of these equations, being substituted in the second, it becomes of
the form
149
^9* MOTIONS OF THE HEAVENLY BODIES. [Mec. C61.
Let the following function be represented by (p, that is*
(p = iV^w.v/^.;^— (0,1) — (0,2) — &c.j
+ iV'^m'.v/^.i^ — (1,0) — (1,2) — &c.j
+ &c.
[1098] +2N.m.\/2,{[^].N' + [Ti] .iV" + &c.}
-\-2N'.m'.vQ.l\T:^,N"-{- [IT^I . N"' + kc.\
+ 2iV".m".v/^. {[23 '^"'+ KH 'N""+kc.l
+ &C
The equations B [1097], are reduced to the following forms, by means of
the conditions mentioned in the preceding article,!
Dividing this by the common factor JV, and reducing, we obtain an equation in g, of the
third degree of the form
0 = A"" g^ + B"" g2 + C"" g + B",
having three roots, g, gu g^, which is the number required to give the complete integral in
this case of three bodies ; and it is evident, from a little consideration, that if the number of
bodies be i, the number of the equations [1097] will also be i, and they will produce by
elimination an equation in g of the degree i, which will give the number of arbitrary constant
quantities necessary to obtain the complete integrals of those equations ; this agrees with
[1097']. We may observe tliat all the quantities JV, JV', &c., being supposed to be of the
same order, it will follow from either of the equations [1097], that g is of the same order as
[1097c] the quantities (0, 1), (0,2), he, [ojT], &;c., which are of the same order as the disturbing
forces by [1079, 1083, &c.]
* (749) This function 9 is so formed that the coefficients of JV^ .m.\/a, JV'^.m' .^a^
Sic, are the same as the coefficients of JV, JV', he, in the equations [1097], the other,
terms are so formed that the coefficient of any term, as 2 JV^"^, contains only the terms
jYie+i)^ JV<«+2>, &c., whose indices (e+1), (e + 2), &c., exceed that of 2 JV^^^.
f (750) The function 9, [109S] gives
(^^ = 2JVm.v/^.S^ — (0,1) — (0,2)— &c.i + 2m./^.l[oTI].JV'+[oZ2].JV'^^
= 2 m. v/«.{-^^ — [(0> 1) + (0, 2) + &c.] . JV+ [o]T] . JV' + [0T2] • JV + fec.}.
ILvIi. §56.] SECULAR INEQUALITIES. -595
Supposing therefore N, N\ N", &,c., to be variable quantities, 9 will be a [1099']
maximum. Moreover, since <p [1098] is a homogeneous function of these
variable quantities, and of the second degree, we shall have*
therefore, in consequence of the preceding equations, we shall have ? = 0. [iioo']
We can now determine the maximum of the function 9 in the following
manner. Take first the differential of this function relative to N, and
substitute in <p the value of iV deduced from the equation (—— j=0, which [iioo"]
and as the part between the braces is, by the first of the equations [1097], equal to nothing,
— j = 0. The same value of 9, [1098], gives
(^) = 2 JV' m' . v/^ . {5— (1, 0) - (1, 2) - &c.|
+ 2 A'm ./a. [oTT] +2m\v/^. f [TT2] . JV''+ [Ti] . A*''' + &c.},
but from [ 1 094] we have 2 JVm . \/a . [Q.i I := 2 JV* w' . \/V • QTo ] , hence
and as the part between the braces is, by the second of the equations [1097], equal to
nothing, we shall have f — — j =0. Agam, the expression [1098] gives
+ 2JVm.v^.[og4-2JV'm^v/^. [ri]+2»'i''.v/^.{[273].JV'''' + 8ic.|.
But from [1095] we get
2JVm.v/^[oT2] = 2JVm".\/^. [2^], and 2JV'm'. \/Z. [^2] =2JV'm".\/F. [2TT] ;
these being substituted we get
which, by means of the third of the equations [1097], becomes (t^J = 0, and so on
for the rest. These equations [1099] are evidently the same as the well known expressions
for finding the maximum value of 9, as is observed in [1099'].
* (751) This follows from the theorem [1001a], changmg A^'^, a, a', &;c., m, into 9, JV,
JV*', &tc., and 2. Substituting the values [1099] in [1100], we shall get 9 = 0,
^^ MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
value will be a linear function of the quantities N\ N", &:c. ; we shall by
this means have a rational, integral and homogeneous function of the second
degree in N'j N", kc. ; let this function be 9^^\ Taking its differential
relative to N', and substituting in (p^'^ the value of N' deduced from the
[1100'"] equation f j = 0 ; we shall have a homogeneous function of the second
[llOOiv] degree in N", N'", &c. ; let this function be (p^^K Continuing in this
manner, we shall finally obtain a function (p('~^^ of the second degree in
^(i-o^ which will therefore be of the form (N^^~^^y.k ; k being a function
of g and constant quantities. If we put the differential of (p^*~'\ taken
relatively to N^'~^\ equal to nothing, we shall find /c = 0 ; from which we
[1100 »] shall get an equation in g, of the degree i, whose different roots will give
as many different systems of indeterminate quantities N, N', N", &c. The
indeterminate quantity iV^'-'^ will be the arbitrary constant quantity of each
system, and we shall immediately obtain the ratio of the other arbitrary
quantities iV, iV', &c., of the system to this, by means of the preceding
equations, taken in an inverse order, namely
[1101] (l^^\-0' (^^!ill\ = 0' &c
[iior] Let g, gi, g2, &c., be the i roots of the equation iag; also N, iV', N", &c., the
system of indeterminate quantities, relative to the root g; iVi, iV/, iV",", &c.,
the system relative to the root g^, and so on ; then we shall have, by the
Solution of known theory of linear partial differential equations,*
the system
of linear
j|"gg§- h = N. sin. (gt + ^)+Nr . sin. (gj + ^d + N^ - sin. (g^ t + ^,) + Slc. ;
tfe:.^"' h'r=N'. sin. (gt + ^)-{- iV/ . sin. (g, t + ^,)+ Ni . sin. {g^ t-\-^^) + &c. ;
[1102] ^" ^ ]S[\ sin. (^ i + 13) + Nl' . sin. {g, ^ + f3,) + N^' . sin. {g^ ^ + W + &c. ;
&c.;
f (75Ia) Each of the values of g, namely, g, gi, g^, he, will furnish a system of values
of h, h', he, I, I', Sic, similar to [1096], which will satisfy the differential equations [1089],
and as these differential equations are linear, it is evident, as in the example [10976], that
the sums of all the corresponding values, found as in [1102, 1102a], will also satisfy the
same differential equations ; moreover, these sums contain the requisite number of arbitrary
constant quantities, [1102"], they will therefore represent the complete integrals of the
equations [1089].
n. vii. -^ 66.] SECULAR INEQUALITIES. 597
§, Pi, Pa? &.C., being arbitrary constant quantities. Changing the sines into [iioai
cosines in these values of h, h\ h", &c., we shall have the values of /, /', /",
&c.* These values contain twice as many arbitrary quantities as there are
roots g, gii gQ, &.C. ; for each system of indeterminate quantities contains
one arbitrary term, and there are also i* arbitrary terms p, pi, fSg, &,c. ; [1102"1
consequently these values will be the complete integrals of the equations (A)
[1089] of the preceding article.
It now remains to determine the constant quantities N, iVj, &c., iV', A^/,
&c., (3, j3i, &c. These quantities are not given directly by observation, but [1102"]
may be deduced from the excentricities of the orbits c, e', &c., and the
longitudes of the perihelia is, ot', &c., at a known epoch, which give the
corresponding values of h, h', &c., /, /', &c.,t whence the former values may
easily be obtained. For this purpose we shall observe, that if we multiply
the first, third, fifth, &c., of the difierential equations (A) [1089] of the [iioasv]
preceding article, by N.m. \/a, N' .m' . \/a'i &c., we shall have, by means
of the equations (B) [1097], and of the relations, found in the preceding
article, between (0,1) and (1 ,0), (0,2) and (2,0), &c.,t
* (752) These are
l = JV. COS. (g<4- p) + JVi- COS. (g-i < + Pi) + A'a . COS. (5" <2 + Pa) + &^c.,
l'=JV.cos.{gt + ^)+jy,'.cos.{g,t + ^,)-{-JV^.cos.(gt^ + ^,)-{-hc. ["02al
&c.
A little consideration will show that in these values of h, A', Sic., 1,1', Stc, terms of the [11026]
order w'e^, m'^, are neglected; quantities of this kind having been neglected in [1089].
Also gi gi, g<i, Sic, are of the order m', to'', &c., [1097c]. In [1097'] it was observed that
there are i equations [1097], and 2 i equations [1089]. The equations [1097] furnish i
arbitrary constant quantities JV, JV\, JVg, Stc, also i quantities p, ^', p", Sec, making in all 2«
quantities, being the number necessary for the complete integration of the 2i equations
[1089] of the first degree.
f (753) h, h', &;c., I, I', he, are deduced from e, e', he, «, «', &;c., by means of the
equations [1022].
f (754) The first member of the expression arises from the sum of the products thus
found, without any reduction. The products to be added together in the second member are
JVw . v/^. {[(0, 1) + (0,2) 4- &1C.] . Z— [¥77] . r_ [oT¥] . I" — he],
JV'to'.v/Z. {[(1,0) + (1, 2) +&C.] .?'— [TH • ^— [iZ] • ^" — &^-!'
Sic.
150
598 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
ni03i at ^ at ^ at ^
^g.\N.Lm. v/« + iV' . Z' . m' . v/^ + N" . /" . m" . ^7 + kc. \
If we substitute in this equation the above values of A, h\ &c., /, /', &c., we
shall have, by comparing the coefficients of the same cosines,*
which, by substituting the values of (Oj 1 ) + (0, 2) + he, ( 1 , 0) + ( 1 , 2) + &c.,
deduced from [1097], become
m".\/F.r. {JV"g + [EZl ••7V+ [!Z] • -l^'+lhll •JV'+hc.l
— JV"m".v/^. {[2To] .Z+ lirr].r+ I373 ]./'" + Sic. I,
Sec.
adding these products, then connecting all the terms multiplied by g together, and afterwards
those by JV, JV', &;c., we shall get
g.\JV.l.m.^a + JV'.l'.m'.^a'+kc.l
_|-JV'.{— m'.v/a'.[[I7o] -l+U^n 'l"+[bl\ .r+&c.] +m . Z.v/a . [o7I]+m".Z".v/^'.[27T]+&c. \
+ &c.
=g.{JV.l.m.^a+JV'.r.m'.\/a'+J^".r.m".^7' + hc.l
n\(m\v/o'.[I[o]— wi.v/«-[M]) + Z''.(w''.v/^.[2:o]_ff,.v/^. [072]) i
"^ 7 +r.(m'".v/^'.ll^]— wi v«- HHD + s^c.^
+ &C.
each of the factors of I, I', I'', he. in the terms of the second third, he. lines of this
expression becomes nothing, by means of [1094, 1095], and the whole is reduced to the
first line, which is the same as the second member of [1103].
[1105a] * (755) We shall, for brevity, put the values [1102, 1102a], under these forms,
A = 2.A;.sin.(^„^ + pJ, h' = l.JV^^Ksm.{gJ + ^n),
II. vii.§56.] SECULAR INEQUALITIES. 599
[1104]
[1106]
&c.
This being premised, if we multiply the preceding values of A, h\ &.C.,
[1 102], respectively by N .m . ^, N' .m' . ^/o', &c., we shall have, by
means of these last equations,*
N. m.h. v/V + iV' . 7n!,h'. v/^'+ A^" . m!'.h!'. v/^ + &c.
= JiV2w.v/^ + iV'^m'.v/^ + iV"^m".v/^ + &c.^sin. (^^ + ^). [iios]
We shall likewise have
N,m.l.\/^+N' .mlJ' .s/7-\-N" .m!'.r .^7 + k,c.
= {iV^ m. v/^ + iV'^ m'. v/7 + iV"^ m" . v/^ + &c.}.cos. (g^ + p).
and generally U'^='L.M\^ . sin. (^„< +^„). In like manner P^=J. . J\Y> . cos. {gnt + P«)
These being substituted in [1103J, it becomes
JVm./«.2.A;^„.cos.(5-„^ + p„)+JV'm'.v/^.2.A;'^,.cos.(g„< + pJ + &c.
==5-.{JVm.vAr.2.A;.cos.(^„< + pJ+Jrm'.v/^.2.A;'.cos.(5-„< + p„) + &c.^
Transposing all to the first member it becomes
0= JVm.v/a . 2 .{gn—g)'K- cos.{gJ+ p„)+A"m'. ^. 2 .{g—g) . K'- cos. (^„ < + |3„)+&c.
Putting now successively n=l, n = 2, w = 3, &;c., we shall obtain the coefficients
of cos. {git-\- pi), cos. (^2 ^ H~ ^2)1 ^c., which being put equal to nothing, and divided
respecdvely by g^ — ^g-, g^ — g, &tc., give the equations [1104].
* (756) The first member of the sum of these equations gives, without reducuon, the
first member of [1105], the coefficients of sin. {gi t -{■ p^), sin. (^3 ^ "I" fe)* Stc., in the
second member, are respectively equal to the second members of the equations [1104],
therefore they are equal to nothing. The coefficient of the remaining term sin. {gt-\- p),
becomes like that in the second member of [1105]. The equation [1106] is obtained in
like manner from the expressions [1102a] ; or more simply, by changing p, p^, ^35 &ic., into
^ + i'»'j 3i + ^'»'j ^2-\-\'^i &^c., respectively, I -r being a right angle. For the
values of ^, A', &c., [1102], by this means become Z, Z', &c., [1102a]. These changes
being made in [1105], we shall get [1106]. If in these equations we suppose the values
A, A', &cc., Z, Z', &tc. to correspond to the time < = 0, the terms sin. {gt-\- p),
cos. {gt-\- ^) wiU become simply sin. p, cos. p, and then dividing the expression
[1105] by [1106], we shall obtain tang. ^, [1107].
600 MOTIONS OF THE HEAVENLY BODIES. [Mec.Gel.
Fixing the origin of the time t, at the epoch for which the values of h, /,
h\ /', &c., are supposed to be known ; the two preceding equations will
give
[1107] .n„». ^_N.h,7n,x/a^-N'.h'.m'.s/a'-\-N".h;'.w!'.s/^' + hQ,
^* ^ JS .1 .m.>/l-^N' J' .7r^ .^a' + N" .1" .m!' .^-^' + hc.
This expression of tang. f3 contains no indeterminate quantity ; for although
the constant quantities iV, iV', N'\ &c., depend on the indeterminate quantity
[1107] iV"^*-!) [1 100'] ; yet as the ratios of these indeterminate quantities are known
by what precedes, it will disappear from the expression of tang. (3.^ Having
thus found /3, we can find N^^~'^\ by means of one of the two equations
which give the value of tang. |3 ; hence we may obtain the system of
[1107'] indeterminate quantities iV, TV', N", &lc., relative to the root g. Changing
in the preceding expressions this root successively into g^, g^, g^, &c., we
shall get the values of the arbitrary quantities, corresponding to each of
these roots.
[1107'"] If we substitute these values in the expressions of h, I, h', l', &c. [1102,
1102«], we may deduce from them the values of the excentricities e, e', &c.,
of the orbits, and the longitudes zs, w', &c., of their perihelia, by means of
the equationsf
* (757) The first of the equations [1101] is linear in JV^'"", JV('-2>, and gives JV^^\
by a simple equation, of the form JV*^'~2)=a. JV^'-^>. The second of the equations [1 101]
is linear in JV«-J>, JV^^^\ JV^'-^\ and gives JV^'-^) = a' JV«-2> + 6' A*«-", and by
means of the value of JV^'~^\ found by the preceding equation, it becomes JV^'~^^=a"JV*<'~^^
The third of the equations [1101], by similar reductions, gives JV'-''~'^^ = a'" JV^'^^\ &c.,
the terms h', a, a', a", he, being independent of JV, JY', he. Substituting these values of
jy(j-2)^ jy(;-3)^ ^c^^ j^ [1107], the numerator and denominator will become divisible by
JV^'~^\ and we shall get tang. ^, independent of that quantity. Having found |3, we may
substitute it, and also the preceding values of JV^'~^^ JV^'~^\ in [1105] or [1106]; then
putting ^ = 0, we shall obtain the value of JV*^'~^'.
f (758) These are easily deduced from [1022], h = e . sin. -ra, l = e . cos. -w, Sic.
The sum of the squares of h, I, being evidently equal to e^, and the former divided by the
latter gives tang. w.
n. vii. <^ 56.] SECULAR INEQUALITIES. 601
e2 = /i2 + r. e'2 = /i'^ + r; &c. ;
tang. ^ = j I tang. -^ =j,'^ &c. ;
[1108]
[1109]
hence we shall have*
^ = N^ + N{' + N,^ + SLc. + 2NN^.cos.{(g,—g).t + ^, — ^]
This quantity is always less than (7V+ iVi + iVg + &c.)^ when the roots
g, ^1, &c., are all real and unequal, taking the quantities iV, iVj, iVg, &c., [1109']
positive. We shall also havef
* (758a) From A, Z, [1 102, 1 102a], we get
-\- \JV. COS. {gt + ^) + JV,. COS. {g,t + ^,) + kc.\^
= JV^ . {sin.2 {gt + ^)-{- C0S.2 {gt + ^)l
-\-2 JV JVi. {sm. {g t + ^). sm.{git + ^1) + cos.{g t-{-^). COS. {git + ^,)]+hc.
= JV2 + 2^'JV\.cos. {(g-i— ^).< + Pi — p}+&c.;
the coefficient of 2 A' JVi, being reduced by [24] Int. This expression must be symmetrical
in JV, JVi, JV*2, Sec, hence from the terra 2 JV JV*i . cos. f (^g-i — ^) • < + Pi — 3 j 5 it will
evidently follow, that the general expression, corresponding to JV*„ JV^ is
2 a; JV^ . COS. {(^^— gO . t + P,„ — P„},
as in [1109]. If we suppose all the quantities JV, JVi, JV^, he, to be positive, the greatest
possible value, of the second member of the equation [1109], will be when the cosines are
all equal to unity, and then
e2=JV2_^JVi2+A-22 + &c.+ 2JVJVi+2A-A'2+2JViA^2+&c.=(A*+JV*i + A-^
Now when g, gi, g^, he, are all real, unequal, and incommensurable, these cosines cannot all
become unity at the same instant ; therefore the general value of e must be less than
(JV+«^i + '^2 + ^c.), as in [1109']. This is the case with the solar system, as is
observed in ^ 57. It may be remarked that in these values of e, e', &c., terms of the same [IIO80]
order are neglected as in [1102&].
f (759) This value of tang. Ttf is deduced from that in [1108], by substituting A, Z,
[1102, 1102a], and if for brevity we put gt-{-^=T, ^^ ^ -{- ^^ = T^, kc, it
becomes tang. ■cy=— 7~-T7 7:^, — 7;^—, This eives
151
602 MOTIONS OF THE HEAVENLY BODIES [Mec. Cel.
rillOl JV. sin, (g^-f p) -f JVj . sin, {g, t + ^,) -\- JV^ . sin, (g^ i+^,) + he. .
^^^^'''~J^.cos.{gt + ^)+JV,.cos.{g,t + ^,)+J>r,.cos.{g,t + ^,)+hc. '
hence it is easy to deduce
[1111] tang.(« gt ■^;-^^^^_^^3^^^^^_^)^^^^^_^^_^,y^,,o3.^(^,^_^^).,^^^_^|_^&,e.
When the sum iVj + iVg + &c. of the coefficients of the cosines of the
[llir] denominator, taken all positively, is less than iV, tang. (^ — gt — f3) cannot
„ JV. (sin. T— cos T. tang. 7)+ A*, . (sin. T, — cos. T, . tang. T) -f&c.
tang. « - tang. T== JV.cos. r+A',cos. T. + &C. '
also
JV.(co3.r+sin.r.tang.r)4-JV,.(cos.r, + sin.r,.tang. r)-|-&c.
1 , tang, trf . tang. r= A*. cos. T+JV,. cos. T. + Aa .cos. 72 + &c.
tang, •zrf — tang. T , _,> ro/^^
Dividing the former by the latter, we get ^ , ^^ — — -y = ^ang. (^ — i ), [30 J
Int., hence
.V. (sin, r— COS. T. tang. T) + JV, . (sin. T, — cos. T, . tang. T) + &c.
tang, (to— i ) — j^^ (cog, T+sin. T. tang. r)+.Yi.(cos. Ti+sin. T, .tang. r) + &c.
Putting for tang. T its value — —^, and multiplying numerator and denominator by
COS. T, it becomes
JV.(sin.T.cos.r— cos.T.sin.T)+;V,.(sin.T,-cos.T— cos.Ti.sin.T)-l-.\^2-(sin.T2.cos.r— cos.r2-sin.T)+&c.
Al(cos.r,cos.r+sin.r.sin.TH-A*,.(cos.r,xos.r+sin.Tpsin.TH-A2.(cos.r2-cos.r+sin.r2-sin.r)+
JV, . sin. (r,— r) + ^"2 . sin. (Ta— y) + fcc.
— JV-I- JV, . cos. ( r, — T) + JVa . cos. ( Ts — ^y') + &-<=. '
by [22, 24] Int. Resubstituting the values of T, Tj, he, it becomes as in [1111]. The
terms JV, JV', he, are of the order h, N, &;c., [1102], and if the first term JV of the
denominator, [1 1 1 1],' be greater than the sum of all the following coefficients JVi-j-JVa+Scc,
considering them all as positive, the denominator will always be finite and positive ; and as
the numerator cannot exceed this sum of JVj + JVi + JV3 + &:c., the expression
tang. (■5J — T), cannot be infinite; therefore zi — gt — ^, must be less than a right
angle ; now this cannot be the case unless the mean motion of -m be exactly equal to g t,
p being constant. For if the difference, between the mean motion of -a and the angle g t,
were even very small, it would, by increasing, in proportion to the time, finally become
greater than a right angle, consequently the mean motion of the perihelion of m must in this
case he gt. In all these computations, terms of the order mentioned in [11025] are
neglected.
U. vii. <^57.] SECULAR INEQUAUTIES. 603
become infinite ; the angle w — gt — ^ cannot therefore become equal to
a right angle ; consequently the mean motion of the perihelion will be in [lUl"]
this case equal to gU
57. It follows from what has been proved, that the excentricities of the
orbits, and the positions of the transverse axes, are subject to considerable
variations, which in the course of time change the form of these orbits, in
periods depending on the roots ^,, ^2? ^c. ; and as it respects the planets,
these periods include many centuries. We may therefore consider the [nil'"]
excentricities as variable ellipticities, and the motions of the perihelia as not
being uniform. These variations are very sensible in the satellites of [mi''']
Jupiter ; and we shall show hereafter, that they explain the singular
inequalities which have been observed in the motion of the third satellite, [iiiiv]
But the question arises whether these variations of the excentricities are
limited in extent, so that the orbits will always be nearly circular. This [iiii»i]
is a subject which ought to be carefully examined. We have just shown
[1109'] that if the roots of the equation in g be all real and unequal,
the excentricity e of the orbit of m will be always less than the sum
■^+-^1 + ^2+ ^c. of the coefficients of the sines of the expression of A-,
taken positively ; and as these coefficients are supposed to be very small, [ijiirii]
the value of e will always be small. Therefore, if we notice only the
secular variations, the orbits of the bodies m, w', m", &C., may become more
or less excentrical, but they will never vary much from a circular form, [imviii]
though the positions of the transverse axes may suflfer considerable variations.
These axes will be invariably of the same magnitudes, and the mean motions,
which depend on them will always be uniform, as we have seen in ^ 54 [1070^]. [nil«]
The preceding results, founded on the smallness of the excentricities of the
orbits, will always take place, and may be extended to all past or future [iiiixj
ages ; so that we can affirm, that the orbits of the planets and satellites
never were, at any former period of time, and never will be, hereafter,
considerably excentrical, so far as it depends on their mutual attraction. But
this would not be the case, if any of the roots g, g^<, g^, &c., were equal [uiui-i
or imaginary : the sines and cosines of the expressions of h, /, h', /', &c.,
corresponding to these roots, would become arcs of a circle, or exponential
604 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
[iiii«"l quantities ; and as these quantities would increase indefinitely with the time,
the orbits would at length become very excentrical.* The stability of the
planetary system would in this case be destroyed, and the results we have
found would cease to take place. It is therefore very interesting to ascertain
full*'"] o
whether the roots g, g^, g^y &c., are all real and unequal. This we shall
prove to be the fact, in a very simple manner, in the case of nature, where
the bodies m, m', m", &c., of the system, revolve in the same direction.
We shall resume the equations (A) § 55 [1089]. If we multiply the first
by m.\/a .h, the second by m . \/~a . /, the third by m' . \/a' . h', the
fourth by m'.^/a'.l', &c., and then add these products together, the
[iiilw] coefficients of hi, h'V, h"l", &c., in this sum, will be nothing ; the coefficient
of h'l — hi' will be [oTi] m . \/~a — [TTo] . m! . ^/i^, and this will become
nothing by means of the equation [o]!] . m . \/~a = [iTp] . m' . ya'i found in
§ 55 [1094]. The coefficients of h"l—hl%h"l'^h'l% &c., will be nothing,
[uiF"] for a similar reason ; therefore the sum of the equations (^A) [1089], thus
multiplied, will be reduced to the following equation,
/hdh + ldl\ ,_ , /h'dh'-\-l'dl'\ , ._ , p
[1112] \—dT—) • ^ Va + [ Tt ) • ^^ • v^«' + &c- = 0 ;
which is equivalent to the following,!
[1113] 0 = ede.m .^'^-\- e'de' .m' .^a' + &c.
Taking the integral of this equation, and observing that by ^ 54 [1070'''],
the semi-transverse axes a, a', &c., are constant, we shall have
[X114] e^. w.^+e'^.m'. ^7+ &c. = constant. (u)
Now the bodies m, m', m", &c., being supposed to revolve in the same
direction, the radicals ^, y/^, &:c., ought to be positive in the preceding
equation, as we have seen in ^ 55 [1094'] ; all the terms of the first member
riiun ^^ ^^^^ equation are therefore positive, consequently each one of them must
be less than the constant quantity, in the second member ; now if we
* (760) This is shown more fully in [11 14'"— 1 1 18'^].
f (761) The differentials of the values of e^, e'^, &c., [1108] being substituted in [1112]
give [1113].
n. vu. Ǥ 67.] SECULAR INEQUALITIES. 605
suppose, at a given epoch, the excentricities to be very small, this constant
quantity will be very small ; each of the terms of the equation will therefore
be small, and cannot increase indefinitely, so that the orbits will always be [iii4'j
nearly circular.*
* (762) If we substitute in the equation [1114] the values of the masses of the planets,
given in Book vi, § 21 , 22, the terms like e^ m . ^/^, relative to Mercury, Venus, the Earth,
Mars, Jupiter, Saturn, and Uranus, respectively, expressed in fractions of to o o or-ffo-oo o of
unity, will be nearly 50, 1,9,88, 258000, 276000, 95000, whose sum, 629148, is the value
of the constant quantity of the second member of [1114], corresponding to the solar system, riii4a]
neglecting the terms depending on the comets and satellites, and on the very small planets,
Vesta, Juno, Pallas, and Ceres, whose masses are unknown, but which are probably so very
small that they could not sensibly affect the calculation. The constant term of the second
member of [1114] being always equal to 629148, it will follow that the orbits of the three
larger planets Jupiter, Saturn and Uranus, can never be very excentrical ; but it does not
follow, from the same equation, that the orbits of the smaller planets wiU always be nearly
circular ; since they might be very excentrical, or even parabolic, and the equation be
satisfied. For if the orbits of Mercury, Venus, the Earth, and Mars, were parabolic, or
their excentricities equal to unity, the preceding terms 50, 1, 9, 88, would become
respectively, 1 190, 16000, 30340, 10190, and the first sum would be increased by 57572,
but this increment would be wholly balanced by decreasing the excentricity of Jupiter about
72
one eighth part, which would decrease the term 258000 to 258000 . — = 197531, the
difference 60469 being greater than the preceding sum 57572 ; therefore the orbits of
Mercury, Venus^ the Earth, and the four lately discovered planets, might be supposed
parabolic, and yet the equation [1114] would be satisfied. We cannot therefore conclude
from that equation, independent of other considerations of analogy, that the orbits of all the
planets will never vary much from a circular form. It may be observed that some of the nil46]
terms of the order m.^/a .e^, he, neglected in [1114, 1151", &;c.], exceed some of the
terms of the order m . y/a • e^, which are retained ; this is an additional reason why the
equation [1114], should be restricted to the three greatest planets. This subject was mentioned
by me in a paper presented to the American Academy of Arts and Sciences, and published
in Vol. IV of their memoirs. The same defect was observed by La Grange in Vol. 11, p. 148
of the second edition of his Mecanique Analytique. The equation [1114], is also affected
by the small secular equation depending on the attraction of the fixed stars, treated of by the
author, in Book vi, § 47 ; and M. Plana, in the paper mentioned in page 561, has shown that
this attraction prevents the second member of the equation [1112] from becommg nothing,
as will be shown in the notes on that book.
152
606 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
The case we have examined is that of the planets and satellites of the
[1114"] solar system ; since all these bodies revolve in the same direction, and at
the present epoch the excentricities of the orbits are very small. To leave
no doubt on this important result, we shall observe that if the equation, by
[1114*''] which g is determined, contains imaginary roots, some of the sines and cosines
of the expressions of h, I, h', /', &c., will become exponential quantities ;*
therefore the expression of h will contain a finite number of terms of the form
[1114'] P . c^\ c being the number whose hyperbolic logarithm is unity, and P a
real quantity, since h or e . sin. w [1022] is a real quantity. Let Q.c^%
[lll4vi] P'.cf\ Q''C^\ P".c^\ &c., be the corresponding terms of /, h\ I', h", &c. ;
Q, P', Q', P", &c., being also real quantities ; the expression of e^ will
contain the term (P^ + Q^) • <^^' ? the expression of e'^ will contain the
[lll4vii] term (P'^+ Q'^) . c^-^% and so on ; the first member of the equation (u)
[1114] will therefore contain the term
[1115] l(P^+Q^).m.\/a+(F'+Q'').m\^a'+(P''' + Q''').m\y/-^ + kc.}.c'f\
If we suppose c^' to be the greatest of the exponentials contained in A, /,
h', /', &c. ; or, in other words, that one in which /is the greatest ; c^^' will
[11151 ^^® ^^® greatest of the exponential quantities contained in the first member
of the equation [1114] ; the preceding term [1115] cannot therefore be
balanced by any other term of the first member, and to render this first
member constant, it is necessary that the coefficient of c^^* should be
nothing ; hence we get
^11,6^ 0 = (P^ + Q^) . 7^ . V/a + (P" + Q") .m'.^'+ (P'" + Q'") . m" . v^ + &c.
When \/a, \/a', \/a!', &c., have the same sign ; or, which is the same thing,
when the bodies m, ml, 7nl\ &c., revolve in the same direction [1094'] ; this
rillG'l equation will be impossible, except we supposef P = 0, Q = 0, P =: 0,
* (763) As an example of the productions of real exponential quantities, we may observe
that the expressions sin.^^, cos. gt^ [11,12] Int., depend on exponentials of the form
c ~ , and if g become imaginary, and equal to — 7 . \/— 15 these exponentials
,Ty'
will become real, and of the forms c
f (763a) In this case, the quantities y/a, \/a', &tc., being positive, the sum of all the terms
[11 16], cannot become nothing, except each term is separately equal to nothing; hence
[1115a] T2+Q2_0, P'2+Q'2 = 0, &c.
U. vii. § 57.] SECULAR INEQUALITIES. ^^
&c. ; hence it follows that the quantities h, /, h', l, &c., contain no
exponential quantities, consequently the equation in g contains no imaginary [ili6"i
roots.
If the equations [1 1 14] have some equal roots, the expressions of h, /,
h', I, &c., would contain arcs of a circle, as is well known ; and we should
have, in the expression of h, a finite number of terms of the form* P . V. [1116"']
Let Q.r, P'. r, Q.f, &c., be the corresponding terms of /, /i', /', &c.,
Now P, Q, being supposed to be real quantities, [IIH""'], P^, Q^, must be affirmative,
their sum P^ -{" Q^ cannot therefore become nothing, unless we have separately P = 0,
Q = 0. In like manner P = 0, Q' = 0, &;c.
* (764) By [1102],
h = JV. sin. {gt + ^)-\-JVi. sin. (gj < + Pi) +&c.
= JV. (sin. g t . cos. (3 -|- COS. g t . sin. p) 4- •'V'l . (sin. gi t . cos. ^i -{- cos.gi t . em. ^i) -\- &£c.
[21] Int., and if any number of the roots g, gi, fee, be supposed equal, the part of h
depending on these roots will be
(JV. cos. p + -^i • COS. pi + &^c.) . sin. gt-{-{JV. sin. p + -^i • sin. ^l -|- fee.) . cos. g t ;
which, by putting JST . cos. ^ -}~ -^i • cos. Pi -\- &c. = v . cos. b,
JV . sin. p-\~JVi. sin. ^j -f~ &c. = v . sin. h,
will become v . (cos. b . sin. g t -\- ^n. b . cos. g t) = v . sin. {g i -{- b) ; and whatever
be the number of equal roots, the terms of h depending on them, may be thus reduced to
one expression, containing only two arbitrary constant quantities, v, b, instead of JV, JVi, &;c.,
^, ^i, he. The expression of h will not therefore contain the requisite number of arbitrary
quantities to render it the complete integral, but they may be obtained by putting
^1 =5* + "i' i?2=g"+ «2, &c., in [1102, 1102a], developing the quantities according
to the powers of a^, a^, &tc., changing the constant quantities so as to retain the requisite
number, and afterwards putting a^ = 0, og ^ 0, &c. In this manner we shall get, by
using [21] Int., and putting JV*i . cos. Pi = Wi, JV^ . sin. p^=mi,
JV; . sin. (fi-, < -f pi) = JV; . { sin. g^ t . cos. ^^ + cos. g, t . sin. ^, \ =n^ . s\n. gj -\- m, . cos. g^ t
= n^.sm.{gt-\-a^t)-\-m,.cos.{gt-\-a^t),
and by [21,23] Int.
= ni . fsin.^-^.cos. aj^-fcos.g-^.sin. «,<} +^1- {cos.^^ . cos. Oj^ — sin. ^ ^ . sin. a^ <)
= \n^. COS. a^t — ffi, .sin.tti^l . sin. gt -\- [m^.cos. a^t'\- n^. sin. a^t] . cos. gt,
608 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
[1116'*] P, Q, P', Q', &c., being real quantities ; the first member of the equation
(u) [1114] would contain the term
[1117] I (P'+Q') ,m,^+ (P"+Q"). m' . \/d + (P"^+Q"^). m" . v/7 + &c. j . t^\
substituting the values of sin. a^ t, cos. a^ t, in series, like those at the bottom of page
487, we shall get
[1 117a] '^' ' ^^"' ^^' ^ + ^i) = (^1 — . m, a, ^ — I n, a^^ t^ + &;c.) . sin. g t
-\- [m^-^-n^a^t — I m^oL^t^ — &£c.|. cos.^^
In like manner, by putting JV.cos. |3 = n, JV.sin. (3 = m, JVg . cos. p^ = Wg,
JV*2 . sin. ^2 = %j ^c., we shall find
JV*. sin. (g- ^ -f" ^) = w . sin.jg- t-\-m. cos. ^ f,
[11176] -^2 . sin. (^2 < + fe) = {^2—^2 a^t — ln^u^f + Uc] . sin.g f
~f" {^'^S ~f" ^2 "2 ^ 2 '^a "2^ ^^ ^C. } . COS. g t,
&c..
Taking the sura of the expressions [1 117«, &], and putting .^0 = n + w^ -f~ "2 + ^c.,
^j = — wij ttj — m2 02 — &;c., "^a = — 2 "i "1^ — i^ ^2 "2^ — ^c.,
J5 = m -f- W2j -{" ^2 + ^c., 5j = n^a^-\- Wg ag + ^^c-j
J?2 == — ^ Wj ttj^ — 1^ jn^ ^ — ^c.,
we shall get,
JV.sin.(g-^ + p)+JV,.sin.(^i^ + ^0+JV2-sin.(g-2^ + fe) + &^c.
[1117c] = (^0 + ^1 ^ + A^ t"" + Sic.) . sin. ^ ^ + (J?o + 5, < + Sg i^ + &c.) . cos. g t,
in which the arbitrary constant quantities JV, JV*i, &ic., (3, p^, Sec, are replaced by the
same number of arbitrary quantities, Aq, Jl^^ Stc, ^qj -^1' ^^' Thus if there were
three equal roots ^, ^1, gc^, there would be szo? arbitrary quantities JV, JVj, JVg, p, Pi, (82,
depending on them, which would produce six arbitrary quantities n, n^, Wg, m, m^, m.2, whose
places are supplied in the last expression [1117c], by the six arbitrary quantities, Aq, A^, A^,
Bq, jBj, B.2, the terms A^, A^, he, being supposed to vanish by putting a^ = 0,
02= 0, he. This expression [1117c], being substituted in h, [1102], instead of the first
terms of its value, corresponding to the equal roots g, g^, he, will give the complete integral
with the requisite number of arbitrary quantities, connected with angles and arcs of a circle
which are not reducible to a more simple form, and which will be found to satisfy the original
differential equations. Similar remarks may be made relative to the equal roots in the value
of I, [1102a], and it is evident that if the term f occurs without the signs of sine and cosine
in the value of h, it may also occur in those of I, h', I', &.C., combined with the coefficients
P, Q, Sic, as above, which may therefore produce in the value of e% the terms mentioned
in [1117].
II. vii. § 58.] SECULAR INEQUALITIES. 609
If r be the highest power of t, which the values of h, /, h', /', &c., contain ;
t'^"" would be the highest power of t contained in the first member of the
equation (u) [1114] ; to reduce this first member to a constant quantity, it [ni7']
would therefore be necessary to put
0 = (P^ + Q^) . m . v/^ + (P" + Q") . m' . v/a' + &c. ; [iiis]
which would give* P=0, Q = 0, P'=0, Q=0, &c. The expressions [uiS']
of h, /, h\ /', &c., do not therefore contain exponential quantities, or arcs of
a circle ; consequently all the roots of the equation in g are real and plis"]
unequal.
The system of the orbits m, m\ m!', &c., is therefore perfectly stable, as it stawmy
respects the excentricities. The ellipticities of the orbits oscillate about ^y^^""-
their mean values, from which they vary but little, while the transverse axes [ili8"]
remain invariable. These excentricities are always subject to this condition,
that the sum of their squares, multiplied respectively by the masses of the [iiisiv]
bodies, and by the square roots of the transverse axes, is always constant. f
58. When e and zs have been found in the preceding manner, we must
d V
substitute their values in all the terms of the expressions of r and — , given
in the preceding articles, neglecting the terms which contain the time t, [iii8']
without the signs of sine and cosine. The elliptical part of these expressions
will be the same as when the orbit is not troubled, excepting only that the
excentricity, and the position of the perihelion, will be variable ; but the [lliS'S]
periods of these variations being very long, on account of the smallness of
the masses m, m', m", &:c., in comparison with M ; we may suppose these
variations to be proportional to the time, during a very great interval, which,
as it regards the planets, may be extended to several centuries before and
after the time selected for the epoch. It is useful, for astronomical purposes, [1118^"]
to have the secular variations of the excentricities and the perihelia of the
orbits expressed in this manner ; we may easily obtain them from the
* (764a) This follows from [1118], by reasoning as in [1 115a].
. f (765) This is to be understood to take place with the restrictions mentioned in
note 762.
153
610 MOTIONS OF THE HEAVENLY BODIES. [Mec.Cel.
preceding formulas. For the equation e^ = h^-\-l'^ [1108], gives
[1118^"'] e d e = hdh-\-l dl ; and noticing only the action of m\ we shall have,
by § bb [1072],
^^(0,l).Z~EIi]./';
[1119]
|i=_(0,l).^+[oTl]./t';
therefore*
[1120] -^=^I^'\^^—'^^'\ '-»
[1120] but WQ have h' I — hl'=e e' . sin. (^ — «), therefore we shall get
de
[1121] — = [^ . e' . sin. {ts' — to) ;
hence by noticing the mutual action of the bodies ?w, m', m", &c., we shall
find
^e ^
j"" = ["HI • ^' • sin. («' — w) -[- [0,2] . e" . sin. (w" — to) -f &c. ;
f^e'
-7- = [i,o1 . e . sin. (ro — 'zs') + [iTi] . e" . sin. (to" — to') + &c. ;
[1122] «^
jT = [iji] • ^ • sin. (to — to") + IfTi] . c' . sin. (to' — to") + &,c. ;
&c.
The equation tang. « = y [1108] gives, by taking its differential,!
[1123] eKdzs = l,dh — h.dL
* (766) The differential of the first equation [1108] gives ede = hdh-\-ldl.
dJidl
Dividing this by d f, and substitutmg the values of — , — . [1119], it becomes as in
[1 120]. The values of A, /, A', Z', [1022] being substituted in h' I — h /', it becomes
Kl — hi' = ee' . f sin. ■c/ . cos. -ss — sin. -a . cos. to'^ = e e' .sin. («' — to),
[22] Int. Substituting this in [1120] and dividing by e we get [1121], from which the
expressions [1122] are easily deduced by generalization. In these last equations terms of
the order /» e^ are neglected.
f (767) The differential of the equation tang. ■»=-, [1108], is Zi= 72 — *
Multiplying this by e^ . cos. zs^ = P, [1022], we obtain e^ .d'a = ldh — hdl, [1123].
n. vii. §58.] SECULAR INEQUALITIES. 611
If we notice only the action of m', and substitute for dh and dl their values,
we shall have
tll^ = (0 , 1) . (A^ + Z^) — [oTTI . (/i A' + / /') ; [11241
which gives
— = (0,1) — [oTT| . - . cos. (z^i' — «) ; [1125]
therefore we shall have, by means of the mutual action of the bodies
wi, m', &c.,
^=(0,l) + (0,2)+;&c. — [^.-^.cos.(^'— ..) — [^.-^^
^=(l,0) + (l,2) + &c.— [i:o].-^.cos.(t.— «') — [ra.^.cos.(^"— ^')— &c. ; [ii26]
— =(2,0) + (2,l) + &c.— E°]-^-cos.(«— ^')— Ea-7.cos.(x3'— ^')— &c.;
&c.
If we multiply these values of t-» -i-i A-c., -^, -r-, &c., by the
^ -^ dt dt dt dt -^
time t, we shall have the differential expressions of the secular variations of
the excentricities, and of the perihelia ; and these expressions, which are [iiae']
rigorously exact only when t is infinitely small, may however serve for a
long interval as it respects the planets. If we compare these expressions [iiae"]
with accurate observations, made at distant intervals, we shall obtain in the
most correct manner, the masses of the planets which have no satellites.
We shall have, at any time t, the excentricity e equal to
. ^ de . ^ dde , g
e, — , -T-^i &c., being the values corresponding to the origin of the time t.
Dividing this by d t, and substituting — , — , [11 19], we get [1 124]. Now from [1022]
€v Z GiZ
we have hh-\-ll=^ee,
hh' -{-ll' = ee' . \ COS. z/ . cos. is -\- sin. ts . sin. zs] = e e'. cos. (a' — «),
[24] Int. Substituting these in [1124] we get [1125], and by generalization [1126].
612 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
de
or the epoch.* The differentials of the preceding value of — -, will give
d d B d 6
[il26'v] those of -—^i -— ^, &c., observing that «, a', &c., are constant ; we
can therefore continue the preceding series as far as may be necessary, and
by the same process we may obtain the series in ra ; but with respect to
[1126V] the planets, it will be sufficient, in the most ancient observations which
have been transmitted to us, to notice the square of the time, in any of the
expressions of e, e', &c., «, «', &c., in a series.
59. We shall now consider the equations, relative to the positions of
the orbits ; and shall resume, for this purpose, the equations (3) and (4)
§ 53 [1050],
[11-27]
dt 4
Now by ^ 49 [1006], we havef
[1128] a^a'.B^'^ = aKbf;
* (768) This value of the excentricity is easily deduced from the formula [G17], by
supposing the value of e, corresponding to the time t, to be 9 (t), and that corresponding to
de a? d^e
the lime t -\- a. to be in general, e-\-a .—--^■—-. — -\-k.c., as in [1126^*), the time
de d^e
after the epoch being changed from a to ^. In finding the differentials — , — , &c.,
quantities a, a', he, are constant, also (0, I), (0,2), he, [oTT], [0^], he, which depend
on a, a. Sic, [1079, 1083]. In the equations [1 122], terms multiplied by e^, e'^, &c., are
neglected, or, as it may be generally expressed, terms of the order m'e'^; and when we take
the differentials to find ~, &;c., the neglected terms will be of the order m'e'^ . — . But
— , [1 122], is of the order m' e', therefore the neglected terms will be of the order m'^ e'^.
Hence we perceive that in the coefficients of the terms t, t^, Sic, in the general value of the
excentricity, there are terms neglected, which are to those retained, in the same coefficient,
as the squares of tlie excentricities e, e', &;c., to unity.
f (769) Putting i = l, in [1006], multiplying by ^.a^a', and substituting
„2^^^, [963-], we shall get "-^ .a^a'. J5('> = ^ . a^ bp and by using &|>, [992],
II. vii. § 59.] SECULAR INEQUAUTIES. 613
and by the same article [992]
fc(i) ^^-i . [11291
therefore we shall have
-^ • « « • ^'^ = 4.{l-a?f =(0,1). [1130]
The second member of this equation is what we have denoted by (0,1) in
^ 55 [1076] ; therefore we shall have
da t"313
^ = (0.1). (;»-/).
Hence it follows, that the values of q^ p, 9', p', &c., will be determined by
this system of differential equations,
dq
'L = 5(0,1) + (0,2) + &C.} .i?-(0,l) ./- (0,2)./-&c.\ ^3,.
d t \ ential
t
dq'
_ equations
i^=-|(0,l) + (0,2) + &c.| . 9 + (0,1) . </ + (0,2) . g" + &c. I %^
the orbite.
^^=S(l,0) + (l,2) + &c.|./-(l,0).p-(l,2)./-.&c.
^ = -{(l,0) + (l,2) + &c.}.g' + (l,0).9 + (l,2).g"+&c.)- (C)
^={(2,0) + (2,l) + &c.}./'-(2,0).;?-.(2,l).p'-&c.
^ = -{(2,0) + (2,l) + &c.} . g"+ (2,0) . 9+ (2,1) . ^+ &c.
&c.
This system of equations is similar to the system (A) ^55 [1089], and they
would coincide wholly, if, in the equations (A), we should change h, /, h', I',
&c., into q, p, ^, p', &c. ; and should also suppose [o,ij == (0,1),
[r7o]=(l,0), &c. ; so that the analysis, used in § 56, to integrate the
equations [1089], may be applied to the equations [1132]. Therefore we
shall suppose
it will become as in [1130], being the same as the function (0,1), [1076]. Substituting
this in the equations [1127], they will become as in [1 131], and, by generalization, we shall
get the equation [1132].
164
[1132]
[1132']
degree
[1133]
614 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
thtatve"' q=N, COS. (g ^ + /3) + iVi . COS. {g^ ^ + f3i) + iVg . COS. {g^ ^ + ^2) + &c. ;
SSt p = N.sm.{gt-\-^)-\-N,.sm.{g,t + i^,) + N^.sm,{g^t + ^^) + hc.;
gr' = iV'.cos.(^^ + /3)+iV/.cos.(gi^ + ^0 + ^2'. COS. (g2^ + |3,) + &c.;
/ = N', sin. (^ ^ + ^) + N; . sin. (^^ ^ + 130 + iV; . sin. (^^ ^ + W + &c. ;
&c. ;
[1133'] and we shall have, by ^ 56, an equation in g, of the degree i, whose roots
are g, g^ gc^ &c.* It is easy to perceive that one of these roots is nothing ;
for it is evident that we may satisfy the equations [1132], by supposing
[1133"] p^ p'^ p"^ ^c,^ iQ be equal and constant, and also q, q', q", &c. ; this requires
that one of the roots of the equation in g should be nothing ;t which reduces
it to the degree i — 1. The arbitrary quantities N, iVj, N', &c., /3, jS^, &c.,
[1133"'] may be determined by the method explained in § 56. Lastly, we shall find,
by the analysis of § 57, J
[1134]
constant = (p^ + q^) .m .^ + (p'^ + q'^) . m' . v/a' + &c,
* (770) These roots g, gi, g^, he, as well as the values of JV, JV\, JVg, Stc, ^, ^', ^", he.,
are different from those corresponding to the equations [1102,1102a], but the form and
manner of computation are the same.
f (771) If p = j)' =y, he, q = q' = (f, he, the second members of the equations
[1132] will become nothing ; and if all these quantities are constant, we shall have
0 = dp = dp' =r. dp", he, and 0 = dq = dq' ^=d^',he,
which will render the first members of the same equations nothing ; so that the equations
[1132] will be satisfied, by the assumed values of p, p', he, q, q, he, ; and these values
correspond to the supposition that one of the roots, as g, is equal to nothing, for then the
parts of q,p,he, [1133], depending on the angle gt-{-^, will become JV.cos. p,
JV.sin. p, he, which are constant.
J (772) IMultiplying the first of the equations [1132] by 2m.\/a. ?, the second by
2m.\/a.p, the third by 2m .\/F.q', he, adding the products, and making the
reductions, similar to those used in computing [1112], we shall find that the sum becomes
nothing, and its integral is as in [1134]. In the second member, terms of the order
w-v/a-i?'*, he, are neglected, as will also more evidently appear from another demonstration
given in [1151— 1155].
I
II. vii. § 59.] SECULAR INEQUALITIES. 615
hence we may conclude, as in the article just mentioned [1114 — 11 18"], that
the expressions of p, q^ jp', g', &c., contain neither arcs of a circle, nor [1134']
exponential quantities, when the bodies revolve in the same direction ;
therefore the equation in g has all its roots real and unequal.
We may obtain two more integrals of the equations (C) [1132]. For if
we multiply the first of these equations by m.\/li, the third by rr^ .s/a'i the [1134"]
fifth by w!' . \/7', &c. ; we shall have, by means of the relations found
in § 55,*
0 = ^.m.v/^ + ^.m'.v/^'+&c.; [1135]
which gives, by integration,
constant = g^ . w . \/a + g' m! . \/d + &c. ; (1) [1136]
and in like manner
constant =^ . w ^/^ + ;?' . w' . v/^' + &c. (2) [1137]
If we put 9 for the inclination of the orbit of m upon the fixed plane, and [1137']
^ for the longitude of the ascending node of this orbit, upon the same plane ;
the latitude of m will be nearlyf tang. ? . sin. (71^+^ — ^)« Comparing this [1137"]
with the following, q . sin. {nt-\-z) — p . cos. {ntA^ s), we shall get
j9 = tang, (p . sin. ^ ; 5 = tang. 9 . cos. ^ ; [1138]
♦■ (773) The first member of the sum is evidently equal to the second member of
[1135], and the second member of this sum must be equal to nothing. For the terms
depending on (0, 1), and (1, 0), in this sum are (p — y).{m.\/a.(0, 1) — m'-v/a'- (^j^)^,
which by means of the equation [1093], become nothing. In the same manner the terms
depending on (0, 2), (2, 0), Sic, are nothing, hence it follows that the whole sum is
equal to nothing. The integral of the equation [1135] is [1136]. Again, multiplying the
second, fourth, &«., of the equations [1132], by m.\/'a,i m .\/a', &c., respectively,
and taking the sum of all the products, it will be nothing, and its integral will be as in [1137]. [1136a]
In both these equations, terms of the order p^ m . ^/o" are neglected, as will evidently
appear in [1158, &tc.]
f (774) This expression is the same as that of tang. jP G, [1030a], or F G, nearly, and
if we develop it, with respect to 6, by [22] Int., it becomes
(tang, (p . COS. 6) . sin. {nt-\- s) — (tang. q> . sin. 6) . cos. {nt-{-e).
This expresses the part of the latitude depending on the angle nt-]-£, which in the value
616 MOTIONS OF THE HEAVENLY BODIES [Mec. Gel.
hence we deduce
[1139] tang. 9 = sj f + f ; tang. ^ =^ ;
we shall therefore have the inclination of the orbit of m, and the position of
its node, by means of the values of -p and q. Marking the values of tang. ?,
[1139'] tang.^, successively with one accent, two accents, &c., for the bodies m', m",
&c., we shall obtain the inclinations of the orbits of m', w", &c., and the
positions of their nodes, by means of the quantities y, g^, y, g^", &c.
The quantity \/f + f is less than the sum iV+ iVj +iV2+ &c., of the
[1139"] coefficients of the sines of the expression of q ;* and as these coefficients are
very small, since the orbits are supposed to be but little inclined to the fixed
plane, its inclination to this fixed plane will always be very small ; hence it
[1139"'] follows, that the system of the orbits will also be permanent, relative to their
inclinations,! as it is with regard to their excentricities. We may therefore
consider the inclinations of the orbits as variable quantities, comprised between
fixed limits, and the motions of the nodes as not being uniform. These
[ll39'v] variations are very sensible in the satellites of Jupiter ; and we shall see
hereafter, [Book viii, ^ 30], that they account for the singular phenomena,
[1139 V] observed in the inclination of the orbit of the fourth satellite.
From the preceding expressions of p and ^, we obtain the following
theorem :
Let there be a circle whose inclination to the fixed plane is iV, and the
longitude of its ascending node gt^<^ \ upon this first circle, let there be
[1137o], of «, [1039], is put equal to q . sin. (nt-^-s) — p . cos. {n t -}- s). Comparing these two
expressions we get p = tang, (p . sin. *, ^ = tang. <?. cos. ^, as in [1138]. The square
root of the sum of the squares of p, q, is tang. cp=v/p2-[-g(2, and the value of /?, divided
by that of q, gives tang. d = -, as in [1139].
^ (775) The sum of the squares of the values of p, q, [1 133], gives for p^ + 2% an
expression precisely similar to that of e^ or h^-^P in [1109], hence we may prove, as in
[1109'], that \/p^-\-q^, is less than the sum JV+ JVi-\- he, considering all the quantities
JV, JV*i, hc.f as positive.
f (776) This is liable to the same restrictions as in note 762.
II. vii. § 59.]
SECULAR INEQUALITIES.
617
placed a second circle, inclined to the first by the angle Nj, so that the [1139";]
longitude of its intersection with the first circle may be ^i^+Pi ; upon this
second circle suppose a thii^d to be placed, and inclined to the second by the
angle iVg, the longitude of its intersection with the second circle being
ga ^ + 1^25 and so on for others ; the position of the last circle will be the
orbit of m.*
* (777) Let ABC be the great circle
in the heavens, corresponding to the fixed
plane, A. the point from which the angles
gt-\- ^, git-\-^ii &^c., are counted,
A'B'C the first circle, A" B" C" the
second circle, &c., B the intersection oi
the first circle and the fixed plane, B' the ■^"'
intersection of tlie first and second circles, B" that of the second and third, B'" that of the
third and fourth, he. ; and taking tlie arch A C equal to a right angle, or | *, we shall draw
perpendicular to it, the arches A A"", B i/, B"b", B" h'\ &c., CC"". Then by
construction, AB=gt-\-^, Ab'=git-{- ^i, •^ ^" = ^'a ^ + ^2) ^-i and the
angles AB A' =C B C =J\\ A' B' A" =C' B C'^JV^, &c.; these angles being
very smaU, we shall have nearly A' B' = Ab' =gii-{- ^j, A" B' = Ab"=^g2t + %,
&c.;also j5C = i*— (g< + p), 5' C' = 6' 0 = ^^: — (g-i^ + Pi), &c. In
the spherical triangle B A A', we shall have tang. AA' = tang. ABA' . sin. A B, which
on account of the smalbess of the arch A A', and the angle ABA', is nearly
AA' = ABA'X sin. AB = JV.sm.{gt-\- ^).
In like manner, as the spherical triangle B' A' A" is nearly right angled in A', we shall have
A' A" = A' B' A", sm. A' B = JVi. sin. {git -\-fij), and in the triangle A" B" A'" we
shall have A" A'" = A" B" Al" . sin. A" B'=^N^. sin. {g^ t + ftj), &c. The sum
of all these arches gives the value A A"'\ corresponding to the last of the circles, which, in
the present figure is A"" F C"" ] and comparing this sum, with the value of jf>, [1133],
we shall find AA!"'=p. Again the spherical triangle C B C gives nearly
CC'=CBC'.sm.BC=::J^,cos.(gt-{-fi);
155
618 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
Applying the same construction to the expressions of h and /, § 56
[ii39vii] [1102, 1102«], we shall find that the tangent of the inclination of the last
circle to the fixed plane, will be equal to the excentricity of the orbit of m ;
and the longitude of the intersection of this circle with the same plane, will
be equal to the longitude of the perihelion of the orbit of m*
60. It is useful for astronomical purposes, to have the differential
variations of the nodes and inclinations of the orbits. For this purpose, we
shall resume the equations [1139] of the preceding article,
[1140] tang, (p = \/p2 -|- 52 ; tang. & = -.
the triangle C B' C" gives
C C" = C'B' C" . sin. B' C = ^\ . cos, (gi t -f p^), fee. ;
the sum of all these is the value of C C"", which, being compared with 5^ [1133], gives
C C"" = q. This last arch, A"" F C"", being supposed to intersect the fixed plane in F,
we shall put AF = d, FC=i^ — ^, and the angle A FA"" = cp= C F C"'\
The spherical angle A FA"" gives nearly
A A"" = tang. A F A"" . sin. AF= tang. 9 . sin. 6,
and the triangle C F C"" gives
O C "" = tang. C F C"" . sin. F C = tang. 9 . cos. 5 ;
substituting the preceding values of A A"" C C"", we shall get j9=tang. 9 . sin. ^,
q = tang. 9 . cos. &, and as these are the same as the equations [1138], it follows that the
angles <p, 4, or A FA"" and AF, determined by this method, must be equal to those
determined by the equation [1139], that is, they must be equal to the inclination, and the
longitude of the node of the orbit of m.
* (778) In this supposition, the sum of the arches A A', A' A!', &c., will be equal to h,
[1102], and the sum of C C, C C", C" C"\ &c., equal to /, [1102a], instead of p
and q, [1133]. Hence, as in the last note, we shall get A = tang. 9 . sin. 5,
I = tang. 9 . cos. &, the sum of whose squares will give \/W^== tang. 9 ; and the first
divided by the second is tang ^=r> or by [1108], e=:tang.9, and tang. d]= tang. w,
or d = ©, as in [1139^"].
U. vii. ^ 60.] SECULAR INEQUALITIES. ^19
Taking the differentials, we shall have*
d(p = d p . sin. 6-\- dq , cos. d ;
J <i|> . COS. 5 — dq , sin. fl [1141]
tang, (p
If we substitute the values of dp, dq, given by the equations (C) [1132]
of the preceding article, we shall havef
^= (0,1) . tang. (p'. sin. (a — ^) + (0 , 2) . tang. 9". sin. p — ^ +&c. ;
^ = -K0,l) + (0,2) + &c.} + (0,l).^'.cos.(.-0 [1142]
4- (0,2) . ^-^^^ . COS. (« — n + &c.
*(779) The differential of tang, (p = /^q:?^ is _j|_^^^£±l^ ^ and from
pop
tang. ^ = - we get cos. ^=--^==, and sin. 5 = -^^==|. Substituting these, we
d(p
find — = dp . sin. d-{- dq . cos. d, and neglecting terms of the order <p^, we may
put cos.^ <P == 1 J and we shall have d(f) = dp . sin. 6-\-dq . cos. 5, as in [1 141].
Again, the differential of tang. 6 = -, is ^7= — - — ^ — , and the precedine
values of cos. d, sin. d, give q =^ cos. A .\/j]»-[-q% p = sin. & . v/p^+g^.j substituting rjj^j,
these in the preceding equation, and dividing the numerator and denominator of the second
, , / — ; — , „ d() dp. coa.& — dq.B'm.d , . . - -
member by v/p^+g^, we shall get -^^= cos.^^.y/^Ha ? substitutmg for /i^s+gs,
its value [1 140] and multiplying by cos.^ 6, we shall get the value of d&, [1141].
+ (780) Substituting in t7= 37 • sin. ^ + 7? . cos. 6, [1 141], the values -^y -r^ ,
at at dt dt dt
[1132], we shall find
^=[{(0,l) + (0,2) + &c.^p-(0,l).y-(0,2).i>"-&c.].cosd
+ [-K0»l) + (0»2) + &c.}.g + (0,l).5' + &c.].sin.fl,
and the coefficient of (0, 1) is p . cos. fl — p' . cos. fl — q . sin. 6-\-^ . sin. fl, which,
by using the values of^, gr, [1141a], and the similar values y = sin. ^ . \/pi~\.^i, [11416]
^ = cos. d . s/p'^r^i becomes
y/p2_|_^2 . |cos. d . sin. d — cos. 4 . sm. A\ + ^yslfr^ . { — cos. fl . sin. d* -{- cos. fl' . sin. fl},
620 MOTIONS OF THE HEAVENLY BODIES. [M6c. Cel.
We shall likewise have
^=(1,0). tang. 9 . sin. (^' — ^ + (1 , 2) . tang, cp" . sin. (</ — 6") + &c. ;
[1143] ^ = _J(l,0) + (l,2) + &c.} + (l,0).^-Vcos.p'-0
at i\ y \ y \ y tang, (p ^
_|_ (1 , 2) . ^-^^ . COS. (6' — O + &c.
' ^ ' ^ tang.9' ^ ^ '
&c.
Astronomers refer the motions of the heavenly bodies to the variable orbit of
[1143] the earth ; for it is in fact from the plane of this orbit that we make our
or simply \/p'2-|-g'2 . j — cos. 6 . sin. d' + cos. 6' . sin. ^} = V^p'^+q'^ • sin. {6 — 6'),
[22] Int. Now the formulas in p', q', cp', similar to those in [1139] give
V/p'2+9'2 = tang, (f/ ;
therefore the preceding term becomes (0> 1) • ^^"S* ^' • ^^^' (^ — ^')* ^^ ^^^ manner the
term depending on (0, 2), is (0, 2) . tang, cp" . sin. (6 — 6") . he. The sum of all these
terms is equal to the value of — in [1142]. Again, by substituting the values of dp, dq,
[1132], in — , [1141], and multiplying it by tang, (p, we shall get
— .tang.9 = — 1(0, l) + (0,2)4-&;c.}.p.sin.d + (0, l).p'.sin.^ + (0,2)./'.sin.d + &;c.
I1141cl
^ ■' _.^(o, 1) -f- (0,2) + (0, 3) + &c.|. 9 . COS. d + (0, 1) . g^ . cos. 6 + (0, 2) . / .cos. d + &tc.
In which the coefficient of (0, 1), is — p . sin. ^ + p' . sin. 6 — q . cos. 6 -\- ^ . cos. 6,
and by substituting the values oi p,p', q, q', [1141a, 6], it becomes
^p2J^q2 . \ — sin.^ 6 — cos.^ 6] -f-y/j9'2_|_^2 . ^sin. 6' . sin. 6 -\- cos. 6 . cos. 6']
= — s/WA^ + v/p'2+9'2 . COS. (4 — d),
[24] Int. But by [1139] we have ^|,2_^2==tang. 9, y/y2ip^ = tang. 9', therefore
the term depending on (0, 1) will be — (0, 1) • tang. 9 + (0, 1 ) . tang. 9' . cos. (d — d').
In like manner the term depending on (0, 2) is
— (0, 2) . tang. 9 + (0, 2) . tang. 9" . cos. (^ — d"), &ic.
The sum of all these terms, representing the value of — . tang. 9, [1141c], is the same
as the second member of the last of the equations [1142], multiplied by tang.9. The
equations [1143] are easily derived from [1142], by changing the accents, &c.
ILvii. §60.] SECULAR INEQUALITIES. 621
observations. It is therefore necessary to ascertain the variations of the
nodes and of the inclinations of the orbits with respect to the ecliptic. Suppose
now that it was required to determine the differential variations of the nodes
and of the inclinations of the orbits, referred to the orbit of one of the bodies
m, m', w", &c., as, for example, the orbit of m. It is evident that
q . sin. {'n!t-\-s') —p . cos. (n't+s') [1143"]
would be the latitude of m' above the fixed plane,* if it moved in the orbit
of VI. Its real latitude above the same plane, is
^ . sin. (n' t+s') — p'. cos. (n' t-^s') ; [il43'"i
and the difference of these two latitudes is nearly the latitude of m' above the
orbit of m ; putting therefore ?/ for the inclination, and 6' for the longitude
of the node of the orbit of m', referred to that of m, we shall have, by what [1143^^]
precedes,
tang. 9/ = ^{p'-pf + {^-gf ; tang. ^ = ^^ • tn44]
If we take for the fixed plane that of the orbit of m, at a given epoch, we
* ("731) The general expression of the latitude of m above the fixed plane, is represented
in [1 137a] by q . sin. {nt -{-s) — p . cos. {nt -\- s), and by accenting these quantities, we
shall obtain the latitude of m, above the same plane, q' . sin. (n t-\-^) — p' . cos. {n't-\-^).
Now from the 6rst equation it follows, that if a body move, on the plane of m, its latitude
corresponding to the longitude n' t -{- s', will be q . sin. {n' t-{-s') — p • cos. {nf t-\-^),
nearly. Subtracting this from the preceding expression, the remainder will represent, very
nearly, the latitude of m', above the orbit of m,
(q — q) . sin. {n't-\- s') — (p' —^p) a>sin. (n' t-^s') ;
and this must be equal to tang, (p/ . sin. (n' t -^ ^ — ^/), which is similar to the expression
used in [1137"], changing the accents. Now if we compare these expressions
iq' — 5') . sin. («' t-\-^) — {p' — p) . sin. {n' t -\- ^) and tang, (p/ . sin. (n' i -f s' — &,')
with those in [1137"],
q . sin. {nt -j- s) — p . cos. {nt-{- s), and tang. 9 . sin. {nt-\- s — 6),
we shall find that the two former may be derived from the two latter, by changing p, q, n, s,
op, B, into p' — p, ^ — q, n', e', <?/, 4/, respectively, and if we make the same changes
in [1139], we shall get the equations [1144]. The same changes being made in [1141],
we shall obtain the expressions [1 145], observmgthat tang. 9/ becomes tang, (p', if p=0, q=0.
156
622 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
[1144] shall have, at this epoch,* ^ = 0, ^ = 0; but the diflferentials dp and dq
will not vanish, and we shall have,
^jj^g^ d 9/= {dp' — dp) . sin. &'Jr{dq' — dq). cos. ^ ;
, , [dp' — dp) .COS. ^' — {dq — d q) .sin. ^
' tang. <p'
Substituting for dp,dq, dp', dq',^c., their values given by the equations
(C) [1132] of the preceding article, we shall have,t
^ = 1(1, 2)- (0,2)! . tang. 9". sin. (^-f)
+ j (1 ,3) — (0,3) ! . tang. cp'". sin. (^—6'") + &c. ;
[1146] ^=_[(l,0) + (l,2) + (l,S)+&c.i — (0,1)
+ i(l,2)-(0,2)j.|^'.cos.0'-O
+ 1(1 ,3)-(0,3) i . ;^". COS. (i-n + kc.
* (781a) The fixed plane being taken for the primitive orbit of m, we shall have, at the
origin of the time, (p = 0, [1137'], and in this case we shall have, by [1138], at that
time, p = 0, q = 0.
f (782) Putting ^ = 0, q = 0, in the four first equations [1132], and deducing
therefrom the values -^—z — - , , we get
dt ^ dt ^ ^
^^ = -^0,l)+(l,0) + (l,2)+(l,3) + &^c.^^' + Kl,2)-(0,2)i.9"*
+ Kl,3)->(0,3)1.2'" + f(l,4)-(0,4)}.r + &^.
[1146a] ^^_^(o,l)+(l,0)+(l,2) + (l,3) + &c.|.y-Kl,2)-(0,2)|./'
-K^3)-(0,3)^p'"-|(l,4)-(0,4)^y"'-&c.
Multipljring the first by sin. ^', the second by cos. ^', and taking the sum of the products
we shall get the value of — ^, [1145]. The terms depending on p', q, are
{(0,l) + (l,0) + (l,2)+&c.|.(jp'.cos.a' — ^'.sin.d');
[11466] hut in [1 138, Sec], we have, p' = tang, (p' . sin. &', q' = tang. <p' . cos. ^ ; hence
p' . cos. 6' — 9' . sin. d becomes nothing. The terms depending on p", 5", are
.1(1,2) — (0,2)}. (/.sin.d'—/.cos.^),
U. vii. <§6I.] SECULAR INEQUALITIES. 623
It is easy to deduce, from these expressions, the variations of the nodes and
the inclinations of the orbits of the other bodies w", m'", &c., to the variable
orbit of m,
61. The integrals of the preceding differential equations, by vt^hich the
elements of the orbit are determined, are merely approximate values, and the
relations which they give, between all these elements, take place only upon [ihgt
the supposition, that the excentricities of the orbits and their inclinations are
very small. But the integrals [430, 431 , 432, 442] which we have obtained
in § 9, will give the same relations, whatever be the excentricities and
__. CC (L It I II 1J fi fP
inclinations. To prove this, we shall observe that — ~t-^ — is double
^ at
the area,* described by the planet m, during the time dt, by the projection [1146*1
which, by substituting p"= tang, cp" . sin. 6f', q" = tang. 9" . cos. 6", [1 138, &tc.], become [1146c]
I ( 1 , 2) — (0, 2) } . tang. <?" . (sin. d' . cos. 6" — sin. 6" . cos. ^)
= {(1, 2)— (0,2)1 . tang. 9" .sin. (^—6").
In like manner the terms depending on p'", q"' become
{(1,3) — (0, 3)} . tang. <p"' • sin. (d' — r) ;
and in like manner for the others, so that the whole expression becomes as in [J 146].
r ^P' — ^P ^ ?' ^Q r /•Ti / J •/
Again, multiplymg the values of • — — — , — — — , [114Da],by cos.tf, and — sm.6,
respectively, and adding the products, the sum will be, by the second equation [1145], equal
to d &! . tang. (?', which is therefore equal to
-{(0,l) + (I,0) + (l,2) + 8ic.^(^.cos.fl'+y.sin.O + {(l,2)-(0,2)j
X {q" . COS. 6' +p" . sin. &') + \ (1, 3) — (0, 3) \ . {q'" . cos. 6' +/' . sin. ^ + &c.
If we use the values of p', 5', p", he, [11466, c], we shall find,
q . COS. 6' -{-p . sin. ^ = tang. 9' . | cos.^ 6f -j- sin.^ ^ \ = tang. 9' ;
/ . cos. fl' + p" . sin. 6' = tang. 9" . j cos. fl" . cos. 6f-\- sin. ^'. sin. fl' j =tang. 9". cos. {6^—6") + &c.
Substituting these in the preceding value of d 9/ . tang. 9', and dividing the whole by
tang. 9', we shall get -r^, as in [1146].
* (783) This is proved in [167'J. From the first of the equations [572] we have
c= — . Substituting the value of c = \/(xa.(l — c2), [596c], and putting /x^^l,
which may be done, by neglecting the mass of the planet, in comparison with that of the sun,
taken as unity [1013'], we shall obtain the formula [1147].
624 MOTIONS OF THE HEAVENLY BODIES. [Mec.Cel.
of its radius vector upon the plane of x, y. In the elliptical motion, if we
neglect the mass of the planet in comparison with that of the sun, taken as
unity ; we shall have, by § 19, 20, relative to the plane of the orbit of w,
To refer to the fixed plane the area described in the orbit, we must multiply
it by the cosine of the inclination 9 of the orbit to this plane ;* therefore we
shall have, as it respects this plane,
we shall likewise have
[1149] — ^-yf- — = I / ^^ \ ;
&C.
These values of xdy — ydx, x! df/ — 't/ dx\ &;c., may be used, when we
[1149'] neglect the perturbations of the motions of the planets, provided we suppose
the elements e, e', &c., <?, (?', &c., to be variable, in consequence of their
secular inequalities ; the equation [430] will then becomef
[1150]
V l + tang.2(f3 Y l+tang2<|/
* (784) This is evident from the principles of the orthographic projection, by which any
area, in a given plane, being projected upon another plane, is reduced in the proportion of the
cosine of the inclination (p of tlie two planes to the radius. This gives the two first
expressions [1148], the third is found, by substituting for cos. 9 its value —======.
The formula [1 149] is deduced from [1148] by merely accenting the letters.
t(785) PutUng M=l, and using the values of ^JMJZI^ ^ iMirl^ ^ &c.
dt d t
[1148, 1149], we shall get from [430] the expression [1150], which is exact. The
expression [1151] is exact in all terms of the order ?»', but not in those of the order
m'^. In the other expressions [1153, 1154, 1155], terms of the order m . \/a . e^ are
neglected.
n.vii.^6].] SECULAR INEQUAUTIES. 625
Neglecting this last term, which is of the order mm!, we shall have [1150']
c==m,\/^I^^^^-{-m!.\/?^-^-\-hc. [1151]
V l + tang2(p V l+tang.aq/
Therefore whatever changes, in the course of time, may be made in the [usrj
values of e, e', &c., ?, (^, &c., by means of the secular variations, these
values ought always to satisfy the preceding equation.
If we neglect the very small quantities of the order e^, or e^?^, this equation t^^^^'l
will become
c = m .\/li-\-m' . \/a' + &c.
— im.v/«-{e' + tang.M — i»i'.v/a'.{e'2 + tang.%'} — &c.; ^"^^^
therefore, if we neglect the squares of c, e', 9, &c., we shall have
constant = m . \/a + 7w' . y/a' + &c. [1152']
We have already proved, [1070"], that if we notice only the first power of
the disturbing force, each of the quantities «, a', &c., will be constant ;
therefore the preceding equation will give, by neglecting the very small [1152"]
quantities of the order e^, or c^^^
constant = m . \/7 '{^ + tang.^ ^\-\-'m! . \/^ • {e'^ + tang.^ 9'} -f- &c. [ii53]
If we suppose the orbits to be nearly circular, and but very little inclined to
each other, the secular variations of the excentricities of the orbits, will be
determined, in § bb [1089], by means of differential equations which are [1153^
independent of the inclinations, and therefore of the same form, as if the
orbits were all in one plane ; now, in this hypothesis, we shall have 9=0, [ll53*^
9' = 0, &c. ; and the preceding equation will become
constant = e^ .m. \/a + e'^ . m' . \/a' + e"^ . m" . \/7^' + &c. ; [i 154]
which we have already obtained in § 57 [1114].
Likewise the secular variations of the inclinations of the orbits, are, in
^ 59 [1132], determined by means of differential equations independent of [ilM^
the excentricities, and which are therefore of the same form, as if the orbits
were circular ; now, in this hypothesis, we shall have e = 0, e' = 0, &c. ; [1154'']
therefore we shall get
167
626 MOTIONS OF THE HEAVENLY BODIES. [Mec. Gel.
[1155] constant = m .\/a. tang.*^ © + m' . \/7 . tang.^ 9 -|- m" . ^/o^ . tang.^ cp" + &c. ;
which equation is the same as was found in ^ 59 [1134].*
If we suppose, as in the last article [1138],
[1156] p = tang, (p . sin. ^ ; q = tang. 9 . cos. 6 ;
it is easy to prove, that if the inclination of the orbit of m to the plane of
Xy y, be 9, and the longitude of its ascending node, counted from the axis of
[1156'] ^^ bg ^ . ^iig cosine of the inclination of this orbit to the plane of a:, z,
will bef
g
[1157] V/i+tang.2(p'
* (789) Substituting in [1134] the value ;/^2^52 = tang. (?, [1 139], and the similar
values of vZ/^ + g'^) ^c., it becomes as in [1155].
t (790) In the adjoined figure, let D H, DE, D G, be the
axes of X, y, z, respectively ; GH C E, a spherical surface y^^T\
described about the origin D, with the radius 1 , cutting the plane X \
of the orbit of m in the great circle FOB, which plane intersects / .. -7 ■„ \f
the plane oi xy, in the point C ; that o( xz, in the point B, and ■^^/cT / ^^-^^^^E
that of z y, in the point F, and the inclinations of that orbit to ^ "t"
those planes will be represented by the spherical angles F C E, CBH, CFE, respectively.
To find these two last angles we have F CE = (p, H C = 6, C E = ^i:' — 6, ^ ir
being a right angle. Then in the right-angled spherical triangle C HB we have
COS. CB H= cos. C H . sin. H C B= cos. S . sin. 9,
and by the second equation [1 138] or [1 156], we have cos. ^ = , hence
q.sin.'p
cos. C B H= = a . cos. <p,
tang. 9
which is evidently equal to -y=~===, as in [1157]. Again, in the right-angled
spherical triangle C EF we have
COS. CFE = cos. CE .sm.F CE=sm.d.sm.(p,
p
and by substituting the value sin. 6 = —^ — , [1 156], we get
COS. v± E = = p . COS. o = - >--r-
tang.2(p
n. vii. §61.] SECULAR INEQUALITIES. 627
cc u V li d 00
Multiplying this quantity by — =~-^^ — , or by its value \/a.{l—e^) [1147],
X d z I z d X
we shall obtain the value of — ; the equation [431] will therefore [1157]
give, by neglecting quantities of the order m^
^ y l + tang.29 ^ V^ l+tang2(p' ' [1158]
We shall likewise find, from the equation [432],
^ V l + tang29 ^ |X l+tang2(p'^ L^isyj
If in these two equations, we neglect quantities of the order e^, or e^(p, they [1159']
will become
constant = mq. \/~a + wi' q . ^^ + &c. ;
constant = mf . v/a + ??i'y • \/a' + &c. ; [H^l
which equations we have before found in ^ 59 [1136, 1137].
Lastly, the equation [442]* will give, by neglecting quantities of the ^ ^
X du — y dx
The whole value of — , corresponding to the supposition that the axes x, y, are
situated in the plane of the orbit B C F is, by [1147], equal to \/a.(l— c2). JVIultiplying
this severally by the cosines of the angles F C E, C B H, C F E, that is, by cos. <p,
v/l-hLg.29^ /l^^' "^^ '^^'' ^^^^^ principles of the orthographic projection,
, . , , <. , . . xdy — ydx xdz — zdx ydz — zdy . ,
obtain the values 01 the projections — , — , — , respectively,
the two last being substituted in [431, 432], putting M= 1, neglecting terras of the order
m to', we get c', c", as in [1158, 1159]. Developing these in series, and neglecting terms of
the order e^, e^ (p, m m', we evidently obtain the equations [1160].
* (791) The equation [442] contains X, which, by [397] is of the order mm' ; now by
neglecting such terms, and putting M==l, this equation becomes
(rfx2 + rf»/2 4-rf^2) ^
n = 2 . m . ; ,* 2r . — ,
at r
which, by the last of the equations [572], may be reduced to the form A =: — 2: . ro . - ,
and this, by means of [530'^], will become A = — 2 . to . - , neglecting terms of the order
to'. This agrees with [1161].
628 MOTIONS OF THE HEAVENLY BODIES [Mec. Cel.
order m7n', and observing, that by ^ 18 [572],
a r df
[1161]
m , m, m , 0
constant = — — 7-4 — jr-j- &c.
a a a
All these equations take place in relation to the inequalities of a very long
[1161] period, which might affect the elements of the orbit of m, m', he. We have
observed, in ^ 54 [1070''"], that if the ratio of the mean motions of these
bodies be nearly commensurable, it may introduce into the expressions of
the transverse axes of the orbits, considered as variable, some equations
whose arguments are proportional to the time, and which will increase very
[1161"] slowly ; these equations having for divisors the coefficients of the time t
in these arguments, may therefore become sensible. Now it is evident,
that by noticing only terms which have such divisors, and considering the
orbits as ellipses, whose elements vary in consequence of these terms, the
[lier] integrals [430, 431, 432, 442], will always give the relations we have just
found between these elements ; because the terms of the order mm', which
we have neglected in these integrals, in finding these relations, have not for
[UG1'»1 div^isors the very small coefficients we have mentioned ; or, at least, they do
not contain them except they are multiplied by a power of the disturbing
forces, superior to that we have taken into consideration.
62. We have observed, in § 21, 22, [167'% 180, &c.] of the first book,
[1161V] that in the motion of a system of bodies, there exists an invariable plane,
preserving always a parallel situation, which might at all times be found by
this principle, that the sum of the products, formed by multiplying each mass
of the system, by the projection of the area described by its radius vector, in
[1161 vi] a given time, is a maximum. It is chiefly in the theory of the solar system,
that the investigation of this plane is important, on account of the proper
motions of the stars, and of the ecliptic, which makes it very difficult for
astronomers to determine with precision the motions of the heavenly bodies.
[1161 "»] If we put 7* for the inclination of this invariable plane to the plane of x, y ;
* (792) The equations [178, 179] give
c" — c"
sin.d.sin.4.=.7-=-==^= , sin.a.cos.4- = ./ „ , .orr^ > cos.d=
Invariable
plane.
yc2 + c'2+c"2 ' ^'"•"•"""•^-'V/C2 + C'2-R^' ^ c^J^e'^J^c"^
11. vii. § 62.] SECULAR INEQUALITIES. ^29
and n for the longitude of its ascending node ; it will follow, from what we
have demonstrated in ^ 21, 22, of the first book, that we shall have,
c" c
tang. 7 . sin. n = -- ; tang, y . cos. n = - ; [1162]
c " c
consequently
tang
ctjucuiijr
m.\/a. (I — e^) .sin.(p.sin.^ + OT'.\/a'.(l — e'2).sin.(?'. sin.6'+&c.
'■'* * m.\/a.{). — e^j.cos. 9 + Wl'.^a'.(l — e'^) . COS. (p' + &c.
, . , , , . , , , o [11621
m.i/a.ri — e2').sm.(p.C0S.^ + W .\/ d .{\ — e'2).sm.(p .cos.^ 4-&C.
tanff 7 cos n == '' . *. '■ '' i
^'^* * m.\/a.(l— e^j.COS. 9 + w'.\/a'.(l — e'2).cos.9'+&c.
We may easily compute, by means of these values, the two angles y and n ;
and it is evident that to determine the invariable plane, we must know the [ii62"]
masses of the comets, and the elements of their orbits. Fortunately these
masses seem to be very small, and it appears that we may, without sensible
error, neglect their action on the planets ; but time alone can give us the
requisite information on this subject. We may also observe, that as it
respects this invariable plane, the values of ^, ^, j9', ^, &c., do not contain
c . . c
Dividing the first and second by the third, we get tang.^ .sin.4/=— , tang.^.cos.l= ,
c c
in which 5 is the inclination of the fixed plane of a?^^, , y^^^ , to the plane of x, y, and by
note 81, the longitude of the ascending node P, of the fixed plane, in the figure, page 112,
is <B' — 4., -r being two right angles. To conform to the notation in [1161"''], we must put
6=7, 11 = * — -v^, or 4' = * — n; substituting these in the two last equations,
we shall get [1162]. The equation [1151], by putting cos. <?, cos. 9', he, for
c = m. v/a.{l — e2) . cos. <p-^m' . v/a'.(l — c'2) . cos. 9' + he. [1162a]
Substituting the values [1156], and the similar values of p', q, he, in [1158, 1159],
observing also that ^j!^-^ = sin. 9, ^;^^= = sin. 9', &:c., they will
become c' =^m. \/a.{l — t^) . sin. 9 . cos. d-\-m! . v/o'.(l — e'2) . sin. 9' . cos. 5' + he, [11626]
c" = m . v/a.(l— c2) . sin. 9 . sin. 4 + &c. Substituting these in [1 1 62], we shall get the
equations [1162'], which are exact in terms of the order m, neglecting m^.
158
^^^ MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
[i\m"'] any constant terms ;* for it is evident, by the equation (C) § 59 [1132], that
these terms are the same for p, p', p", &c. ; and that they are also the same
for q, q', q", &c. ; and as it respects the invariable plane, the constant terms
[il62iv] of the first members of the equations [1136, 1137] are nothing; therefore
the constant terms disappear from the expressions of p, p', &c., q, q', Slc,
by means of these equations.
We shall novs^ consider the motion of tv\^o orbits, supposing them inclined
orbit"s?'''" ^o each other by any angle ; we shall have, by ^ 61,t
c' = sin. 9 . cos. ^ .m. \/a . ( i — e^) + sin. 9' . cos. ^' . m! . s^a' .{\—e^) ;
[11631 // • • . / ■ — , • / . / / ,
c =sin. 9. sin. Km . /a.(I — e^) + sm. 9' . sm. ^ . m . ^a'.(i — e'^).
* (793) That isp, p', he, q, q', he, do not contain terms like j5 =/, r/=f'^ &,c.,
q=l, q' = l', &tc., in which /, /', Sic, I, V, &;c., are constant quantities, independent
of the time. For the substitution of these, in [1132], would make the first members of
those equations vanish, so that they would become
0=K0.0+(0.2) + kc.}./-(0,l)./'-(0,2)./" + &c.
0=-f (0, 1) + (0,2) + hc.].l-\-{0,\).V-{- (0, 2) . I" + &c,
and we should have as many linear equations in /, /', &z;c., as there are different quantities
f, f, ho,., and from these we should obtain, by the usual rules of elimination of algebraic
equations of the first degree, the values of the quantities /, /', &,c. It is easy to perceive
that these values may be obtained, by putting all the constant quantities /, /', /'', &;c., equal
to each other. In like manner, by putting all the constant quantities Z, V, &:c., equal to each
other, we may satisfy the linear equations in I, V, &;c., so that if we notice only the constant
terms of the values of p, p', Sic, q, q', Sic, we shall have p =/, p'=f, p"=f, he,
[116;^] q = l^ q =h 9" = ^? he, and the slightest inspection will show, that these values
will satisfy the equations [1 132]. Now, as it respects the invariable plane, we have c'=0,
c"=0, [180'], therefore the first members of the equations [1158, 1159, 1160] must
vanish, and if we substitute in [1160] the values [1163a], they will become
[11636] 0 = Z . j OT . v/a + w' . \/a' + &;c. j , 0 =/. j m . \/a-\- m' . \/a' + &c. j ,
but the terms m . \/a, m' .\/a', &c., [1114'], have all the same sign, therefore
m.\/a-\- m! . \/a', must be a finite quantity, in which case the equations [1 1636] will give
1=0, f= 0, consequently, the constant terms must disappear from the values of
p,p', &;c., q, q, &ic.
f (794) These values were computed from [1 158, 1159], reduced, as in [11626].
II. vii. ^ 62.] SECULAR INEQUALITIES. 631
We shall suppose the fixed plane, to which we refer the motion of the orbits,
to be the invariable plane just mentioned, with respect to which the constant
quantities of the first members of these equations are nothing, as we have [1163]
seen in § 21, 22 [180'] of the first book. The angles 9 and 9' being positive,
the preceding equations will give* [1163"]
m . \/a.(l — e^) . sin. <p = w' . i/a' . ( i — e'^) • sin. <p' ;
smJ = — sm.5 ; cos.^^ — cos.ff; ^ ^
hence we deduce 6' = ^ -f- the semi-circumference ; therefore the nodes of [ii64']
the orbits are upon the same line ; but the ascending node of the one
coincides with the descending node of the other ; so that the mutual [1164"]
inclination of the two orbits is 9 + <p'«
We have, by § 61 [1162«],
c = m. \/a . (1— e^) • cos. 9 + ?«' • \/a'.(l — e'2) . COS. 9' ; [iir,5]
by combining this equation with the preceding between sin. 9 and sin. 9', we
shall findf
2.mc. COS. 9 . \/a.{\ — e^) = c^ + w^ . a . (1 — e~) — m'^ .a' .(\ — e'^). [uee]
* (795) Put c' = 0, c" = 0, in [1163], and we shall obtain
sin. 9 . COS. d .m. \/a.{l—e^) = — sin. 9' . cos. &' .m! . v/a'.(l— e'2),
sin. 9 . sin. 6 .m . y^aT{l^^e^) = — sin. 9' . sin. d .m' . \/a'.{\ — e'2).
Dividing the second equation by the first, we shall get tang. ^=tang. 5', which corresponds
to &' =&, or d' = * -}- &. The first value cannot be used, for by substituting it in the
first of the preceding equations, it would become divisible by cos. ^, and would give
sin. 9 . Wi . \/a.{l — e2) = — sin. cp .m' . \/a'.{l — c'2),
now by [1 114'], the radicals \/a, \/af, or \/a.(l — e2), ^a'.[l—e'^), must have the same
sign, and as 9, 9', are both positive and acute, [1163"], their signs must be positive, the first
member of the preceding equation will therefore be positive, the second negative, they cannot
therefore be equal to each other, so that we cannot use the first value of 6', and must take
the second 6' = d -\-if, which gives, as in [1 164], sin. 6 = — sin. 6', cos.^= — cos.^j
substituting these in the two equations [1164rt], and dividing them respectively by cos. ^,
sin. &, we shall get m . v/a.(l — e2) . sin. 9 = to' . v/a'.(l— e'2) . sin. 9', as in [1 164].
f (796) From [1 165], we get c — m. \/a.{l—e^) • cos. 9 = m' . v/a'.(l— e'S) . cos. 9' ;
squaring both sides, and substituting cos.^ 9=1 — sin.^ 9, cos.^ 9' =: 1 — sin.^ 9',
it becomes
c« — 2m.c.\/o.(l-c2).cos.9 + m2a.(l— e2).(l— sin.29)=m'2a'.(l_e'2j.(i_sin.Y^.
[1164a]
^32 MOTIONS OF THE HEAVENLY BODIES. [Mec. Gel.
If we suppose the orbits to be circular, or so little excentrical that we may
neglect the squares of their excentricities, the preceding equation will make
[lies'] cp constant ; and for the same reason, cp will be constant ; the inclinations of
the planes of the orbits, to the fixed plane, and to each other, will therefore
be constant ; and these three planes will always have a common intersection.
[1166"] Hence it follows, that the mean momentary variation of this intersection is
always the same, since it cannot be expressed but by a function of these
inclinations. When they are very small, we shall easily find, from ^ 60,
and by means of the preceding relation between sin.tp and sin. 9, that for the
[1166'"] i[ixiQ t, the motion of this intersection is* — 1(0,1) -\- (1 ,0)] .t.
The position of the invariable plane to which we have just referred the
motion of the orbits, is easily determined for any instant ; it being only
necessary to divide the angle of the mutual inclination of the two orbits, into
[il66'v] two angles, 9 and 9', so that they may satisfy the preceding equation between
adding to this the square of the first of the equations [11 G4],
m^a . (I — e^) . sin.^ cp^m'^ a' . {i — e'^) . sin.^ 9',
we shall get c^ — 2 m . c . \/a.(l — e^) . cos. cp-{-m^ a .{\ — e^) == m' ^ a' . {I — e'^),
which, by transposition, gives [1 166] , and if e, e, are so small, that we may neglect their
squares, this equation will give cos. 9 = -_- — , in which each term ol the
second member is constant, consequently 9 is constant, as in [1 166'].
* (797) The second of the equations [1 142], in this case, where there are only two
bodies m, m', becomes ^ = — (0, 1) + (0, 1) . *^^- . cos. (^ — ^'). Now by [1164'],
' ' dt \ ' / I \ ' / tang. 9 ^ ^
cos. {6 — &') = cos. ( — ir) = — 1 , and the first of the equations [ 1 1 64], neglecting terms
of the order e^ 9, becomes m . v/a . sin. (p = m' . \/^. sin. 9' ; or, by neglecting terms
_ _ , tang. 0/' m . \/a
of the order 9^, m.y'a. tang. 9 = m' . /a' . tang. 9 hence {^^ = ^j^T^, '
consequently — = — (0, 1 ) — (0, 1 ) . ^^^j— ; but from [ 1 093] we have
(0,1). "^=(1,0), hence ^ = _ {(0, 1) + (1, 0)} ;
Multiplying this by dt^ and integrating, we get 6 = — {(0, 1) -f (0, 1)} . ^
as in [1166']".
II. vii. § 62.] SECULAR INEQUALITIES. 633
sin.<?3 and sin.?'. Denoting, therefore, this mutual inclination by t^, we shall
have,*
m^v/a^ (1 — 6^2) .sin.^a
* (798) Put cp-\-(p' = 'a or (p' = 'n — 9, hence
sin. (ff = sm. -a . cos. 9 — cos. « . sin. (p, [22] Int.
Substituting this in the first equation [1164], we get ,
m . ^a.(l— e2) . sin. cp = m' . \/a'.{l — e'2) . | sin. « . cos. 9 — cos. ^ . sin. 9 1 ,
Avhich, being divided by cos. <p, becomes
m . v/a.(l — e2) . tang. 9 = w' . v/a'.{l— c'2) . | sin. w — cos. a . tang. 9^ ;
transposing the last terra, and dividing by the coefficient of tang. 9, we get [1167].
169
^^* MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
CHAPTER VIII.
SECOND METHOD OF APPROXIMATION OF THE MOTIONS OF THE HEAVENLY BODIES.
63. We have seen, in Chapter II, that the co-ordinates of the heavenly
bodies, referred to the foci of the principal forces which act on them, are
determined by differential equations of the second order. These equations
have been integrated in Chapter III, noticing only the principal forces, and
it has been shown, in this case, that the orbits are conic sections, whose
elements are the arbitrary constant quantities introduced by the integrations ;
and as the disturbing forces produce but small variations in the motions, it is
very natural to endeavor to reduce the disturbed motions of the heavenly
bodies to the laws of the elliptical motion. If we apply to the differential
equations of the elliptical motion, increased by small terms arising from the
disturbing forces, the method of approximation explained in ^ 45, we may
suppose the motions which are performed in oval or returning curves, to be
elliptical ; but then the elements of these motions will be variable, and we
may find the variations by that method. These differential equations being
of the second order, their finite integrals, and also their integrals of the first
[1167"] order, will he the same as if the ellipses were invariable ;* so that we may
Important take the differential of the finite equations of the elliptical motions, supposing
method, the elements of these motions to he constant. It follows also from the same
method, that in the equations of this motion, which are differentials of the
first order, we may again take the differentials, considering as variahle only
■ the elements of the orhits, and the first differentials of the co-ordinates ; provided
* (798) This is conformable to the remarks [898'], the term i being here equal to 2,
[UBTal consequently, by what is there said, the functions of the first order will be the same in both
ellipses.
n. viii. -5 63.] SECOND METHOD OF APPROXIMATION. 635
that instead of the second differentials of these co-ordinates, ive substitute the [1167'"]
part of their values, depending upon the disturbing forces.* These results
may also be deduced from the consideration of the elliptical motion.
For this purpose, suppose an ellipsis to pass through a planet, and through
the infinitely small arch which it describes; the centre of the sun being in its
focus. This ellipsis is that which the planet would invariably describe, if
the disturbing forces should cease to act upon it. Its elements are constant
during the time dt, but they will vary from one instant to another. Therefore ^^^^"^
let V=0 be a finite equation of the invariable ellipsis, F being a function of
the rectangular co-ordinates x, y, z, and the parameters c, c\ &c., which are
functions of the elements of the elliptical motion. This equation will also
take place in the variable ellipsis ; but the parameters c, c', &c., will no [1167^]
longer be constant. However, since this ellipsis appertains to the infinitely
small part of the curve described by the planet during the instant d t ; the
equation V=0 will take place for the first and last point of this infinitely [li67»i]
small arch, supposing c, c', &c., to be constant quantities. We may therefore
take the first differential of this equation, supposing only x, y, z, to be
variable, and we shall get.
Hence it is evident, that if we have a finite equation of the invariable
ellipsis, we may take its first differential, supposing the parameters to be
constant, and it will nevertheless correspond to the variable ellipsis. In like [lies']
manner, every differential equation of the first order, in the invariable ellipsis,
will take place also in the variable ellipsis. For let V = 0 be an equation
d oc d ij d z
of this order; V being a function oi x, y, z, -j-, -^, -j-, and of the [ii68"i
at at at
parameters c, c', &c. It is evident, that all these quantities are the same, in
the variable ellipsis, as in the invariable ellipsis, which coincide with each [1168"]
other, during the instant dt.
Now, if we consider the planet, at the end of the instant d t, or at the
commencement of the following instant, the function V will not vary, from
* (799) This method of difFerentiation is proved in ^ 45, in the equation [903J and in the
remarks immediately following it.
636 MOTIONS OF THE HEAVENLY BODIES. [MecCel.
[1168'^] the ellipsis relative to the instant dt^ to the consecutive ellipsis, except by the
variation of the parameters, since the co-ordinates iP, ?/, z, corresponding to
the end of the first instant, are the same for both ellipses ; the function V
being nothing, we shall have
This equation may also be deduced from the equation F=0, supposing all
the quantities a:, y^ z, c, c', &:c., to be variable ; for if w^e subtract the
equation (^) [1168], from this equation, we shall obtain the equation
(z') [1169].
Taking the differential of the equation (i) [1168], we shall obtain another
[1169^ equation in dc, dc\ &c., which, with the equation (z') [1169], will serve
to determine the parameters c, c', &c. It was by this method, that the
mathematicians who first attempted the computation of the theory of the
perturbations of the heavenly bodies, determined the variations of the nodes
and of the inclinations of the orbits : but we may simplify this method in
the following manner.
We shall consider generally the differential equation of the first order
[1169"] F' = 0 ; this, as we have just seen, corresponds both to the variable and the
invariable ellipsis, which during the instant dt coincide with each other. In
the following instant the same equation corresponds to both ellipses, but with
this difference, that c, c', &/C., remain the same in the invariable ellipsis, but
[1169"'] vary in the variable ellipsis. Let V" be what V becomes, when the ellipsis
is invariable ; F/ what the same function becomes, when the ellipsis is
[1169^''] variable. It is evident that to obtain V'\ we must change in P, the
co-ordinates x, y, z, corresponding to the beginning of the first instant d t,
into those, corresponding to the beginning of the second instant ; we must
[1169^1 ^^^^ increase the first differentials dx, dy, dz, respectively by the quantities
ddx, ddy, ddz, corresponding to the invariable ellipsis, the element of the
time dt being supposed constant.
Moreover, to obtain F/, we must change in V, the co-ordinates ar, y, z,
into those corresponding to the beginning of the second instant, which are also
fli69^n ^^® same in the two ellipses ; we must then increase dx^ dy, d z, by the
quantities ddx, d dy, ddz, respectively ; lastly we must change the
parameters c, c', &c., into c-i-dc, c'-\-dc\ &c.
n. viii. §63.] SECOND METHOD OF APPROXIMATION. 637
The values of ddx, ddy, ddz, are not the same in both ellipses ; they
are increased in the case of the variable ellipsis, by quantities arising from [ii69vjij
the disturbing forces. We see therefore that the two functions V" and F'
differ only in this respect, that in the second expression, the parameters c, c',
&c., increase by dc, dc\ &c. ; and the values of ddx, ddy, ddz,
corresponding to the invariable ellipsis, increase by quantities arising from t^'^-'J''"']
the disturbing forces. We may therefore compute F/ — V", by taking the
differential of V, supposing x, y, z, to be constant, and dx, dy, dz, c, c', fee,
to be variable, provided we substitute in this differential, for d d x, ddy, [iiGd"]
ddz, &c., the parts of their values arising only from the disturbing forces.
Now if in the function V" — V, we substitute for dd x, ddy, ddz,
their values corresponding to the elliptical motion, we shall have, for the
- • TTn TTi • • r dx dy dz
function y — y, an expression m terms ot x, y, z, — , -^, — , c, [1169»]
</, &c., which, in the case of the invariable ellipsis, will be nothing ; this
function will therefore be nothing, in case the ellipsis is variable.* We
evidently have, in this last case, F/ — V = 0 ; since this equation is the [1169«]
differential of the equation F' = 0 ; subtracting from it the equation
V" — F'=0, we shall get F/ — F"=0. Therefore we may, in this case, take
the differential of the equation V = 0, supposing only dx, dy, dz, c, c', &c.,
to be variable, and substituting for ddx, ddy, ddz, the parts of their values l"*'^'"!
corresponding to the disturbing forces. These results are exactly the same
as those we have obtained in § 45, by a pure analytical method ; but on
account of the importance of the subject, we have thought it proper to
* (800) This function V" — V, after the substitution of the elliptical values of ddx,
ddy, ddz, becomes a differential function of the first order, which must therefore, by
using the method explained in [1167"], be the same for the variable as for the invariable
ellipsis. On the contrary, F/ contains d c, dc', &;c., and the values of ddx, ddy,
ddz, corresponding to the variable ellipsis. These last values of ddx, ddy, ddz, may
be considered as consisting of two parts, namely, the elliptical values, and the parts arising
from the perturbations ; and as V" contains the elliptical values, F/ — V" = 0, must
contain only the parts oi d dx, ddy, ddz, arising from the pertui'bations. Hence we see
the reasojo of the method of differentiation [1169''"].
160
^^^ MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
deduce them also from the consideration of the elliptical motion. This being
supposed,
64. We shall resume the equations (P) ^ 46 [915],
^ ddx iJ^x /'d R
dt^ r^ \ dx
11170]
« = ^+tI+(47) = (^)
_ ddz , fi-z I Z' d R
dt^ ' r" ' \dz
If we suppose R = 0, we shall obtain the equations of the elliptical motion,
^ J which we have integrated in Chapter III [545]. We have obtained, in
§ 18 [572], the seven following integrals,
X dy — y dx , x dz — zdx „ y d z — z dy
'dy^-\-ds?\ ) ydy.dx zdz.dx
c = — —-- — ; c — ^ ; C — -
dt ' dt ^ di
dt^ ' dt^
ii7r
_ ^,, , C fA /dx^ -\- dy^\ l , xdx.dz , ydy.dz
fji, 2 f/. /dx^-{-dy^-\-dz^\
As these integrals express the arbitrary quantities in functions of the
co-ordinates and their first differentials, they are under a very convenient
form, for computing the variations of the arbitrary quantities. The three first
integrals give, by differentiation, supposing only the parameters c, c', c", and
the first differentials of the co-ordinates to be variable, as in the preceding
article [1167'"],*
X ddv — V ddx ,, x ddz — z d dx ,,, y ddz — z ddy
[„„1 rf, = y__^ ; dd= ^^ ; dd'='—^^ ?.
Substituting, for ddx, ddy, ddz, the parts of their values, arising from
* (801) These difFerenliations are made, and the equations [1173] deduced, upon the
principles mentioned in [1167'"] or [1169''"].
n. viii. <§ 64.] SECOND METHOD OF APPROXIMATION. 639
the disturbing forces, which are easily deduced from the equations [1170],
namely, -dt\ (^) , -de. (^^) , -df. (^^J) ; we shall [n«l
find,
"'-"■{'■m-mh
[1173]
We have seen, in § 18, 19, [591, 599, 575'], that from the parameters c, cf,
c", we may determine three elements of the elliptical orbit ; namely, the Lilys']
inclination ip of the orbit to the plane of x, y, and the longitude ^ of its node,
by means of the equations* [591]
V/c'2 + c"2 C"
tang. 9 = ^ ; tang. ^ = - ; [1174]
c c
also the semi-parameter of the ellipsis o.(l — e^) [3785], by means of the ni'^^n
equation
f.«.(l — e^) = c2 + c'^ + c"^ [1176]
These equations take place also in the variable ellipsis, provided we
determine c, c', c", by means of the preceding differential equations. We shall
thus have the parameter of the variable ellipsis, its inclination to the fixed [ii75']
plane of x and y, and the position of its node.
From the three first of the equations (P) [572], we have deduced, in
^19 [579], the finite integral 0 = c"x — c'y-\-cz; this equation takes [1175^
place in case the ellipsis is disturbed [1167"], and its first differential,
0 = c" . d X — c' d y -\- c . d z, found upon the supposition that c, c', c", are
constant, also takes place.
If we take the differentials of the fourth, fifth and sixth of the integrals
(p) [1171], supposing only the parameters /, /', /", and the diflferentials
* (802) The equations [1 174] are the same as [591]. The equation [1175] is deduced
from ^2= c2 -}- (/^-f c"2, [575'], substituting A^ = ,* a . (1 — e^), [599].
640 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
dXy dy, dz, to be variable ; and then substitute for d d x, ddy, ddz,
[117^"] the values [1172'] —df.(^\ ^dtK(~^\ —diKf-^X we
shall find*
+ (y dx — X dy) . (-^^ + (z d X — X dz) . (^-^y
... ./=...5..Q-..(|)j+...j..(^)-,.(^-f);
J^(^xdy — ydx),(^^ + {zdy — ydz).(^^',
+ (xdz-zdx).{^^ + {ydz-zdy).(^^y
\\v7m L^s^^J' ^^® differential of the seventh of the integrals {p) [1171], taken in
the same manner, will give the variation of the semi-transverse axis a, by
means of the equationf
[1177] J.^ = 2.d72;
* (803) The differential of the fourth of the equations [1171], taken as above directed,
and observing that r = \/x^-\-y^-\-z% is
{dy.ddy-\-dz.ddz) {ddy .dx-\-dy .ddx) {ddz.dx-\-dz.ddx)
0 = dj — 2x. — \-y. ~ [-Z. ~ ,
or, as it may be written,
_- - / — yddx-\-xddy\ , , / — zddx-A-xddzX , . , > ddy . ddz
df= dy . ( J -)+dz . ( -^f- y{ydx-xdy) . J-{zd:v-a:dz).- ;
substituting, for ddx, ddy, ddz, their values [1175'"], we shall obtain df, [1176]. The
fifth of tbe equations [1171], may be deduced from the fourth, by changing /into/', x into
y, and y into x. The sixth may be deduced from the fifth, by changing /' into/", y into z,
and z into y. The same changes being successively made in df, [1 176], we shall get df.
df", [1176].
* (804) Taking the differential of the last of the equations [1171], in the abovementjoned
. „ , (* I ^ dx.ddx-\-dy .ddyA-dz.ddz i . • .»
manner, we shall get 0 = <? . - +2 . d¥^ ' substituung ddx.
II. viii. § 64.] SECOND METHOD OF APPROXIMATION.
641
the differential dR refers only to the co-ordinates a;, y, z, of the
body m.
The longitude of the projection of the perihelion of the orbit, upon the
fixed plane, and the ratio of the excentricity to the semi-transverse axis, are
determined by means of the values of /, /', /". For 7 being the longitude [ii77'j
of this projection, we shall have, by § 19 [594']
f
tang. 7=-^-; [ii78j
and e being the ratio of the excentricity to the semi-transverse axis, we shall [1178']
have, by the same article,*
fjL e = ^y2_|_y'2_j_y"2. [1179]
This ratio may also be determined, if we divide the semi-parameter
a . (1 — e^), by the semi-transverse axis a : the quotient subtracted from [1179']
unity, will give e^.
The integrals (p) [1171] have given by elimination, in § 19 [582], the
finite integral 0 = m- r — h^ +/ar +/' y +/" z ; this equation takes place [1179"|
also in the disturbed ellipsis [1167"], and it determines, at each instant, the
nature of the variable ellipsis. We may take its differential, supposing /,
/', /", to be constant, and we shall get
0 = ij.dr -\-fd X -{-/' dy~{-f"dz. [1180]
The semi-transverse axis a gives the mean motion of m, or more accurately,
that which, in the disturbed orbit, corresponds to the mean motion in the
undisturbed orbit ; for we have, by § 20 [605'] n = a ^ . v/fl ; moreover, [iisc]
if we put ^ equal to the mean motion of m, we shall have, in the invariable
ellipsis d^ = ndt [1044', &c.] ; this equation generally takes place in the [ii80"j
ddy, ddz, [1172'], we shall find
the second member of which Is evidently equal to 2 d /?, the characteristic d being supposed •
to affect only the co-ordinates of the body m.
* (805) We have p -\-f'^+f"^ = P, [574"], and I = (>.€, [597'], hence
fji- e = vZ/M^TH^T^j 3s in [1179]. The value of e may also be determined, as in
[1179'], from that of fxa . (1 — e^), found in [1175], and the semi-transverse axis a,
deduced from [1177].
161
642 MOTIONS OF THE HEAVENLY BODIES [Mec. Cel.
variable ellipsis, since it is a differential of the first order. Taking its
[1180"'] differential, we shall have dd^ = dn.dt; now^ vv^e have*
[1181] dn = --—-.d.~
2 |x a fi.
S an. dt .dR
f*
therefore
[1182] dd^
and by integration,
3
[1183] ^ = -.ffandt,dR.
f*
Lastly, we have seen in § 18, that the integrals (i?) [1171] are equivalent
only to five distinct integrals, and that they give, between the seven
[lies'] parameters, c, c', c", /, /', /", a, the two following equations of condition,!
0=fc"~f'c'-hf"c;
these equations take place also in the variable ellipsis, provided the parameters
are determined in the preceding manner. Which may also easily be proved
a posteriori.
We have thus determined five elements of the disturbed orbit, namely, the
inclination ; the position of the nodes ; the semi-transverse axis, which gives
[1184'] the mean motion ; the excentricity, and the position of the perihelion. It
now remains to find the sixth element of the elliptical motion, being that
* (806) The differential of the logarithm of n=a^.^'il, [1180'], is
dn da 3a , 1 3a , ju,
n a 2 * ' o 2jx * * a '
3an
5et rf n =
value of rfn, [1181].
multiplying by n, we get dn = -^~.d.-; substituting [1177], we find the second
f (807) The first of these equations is given in the same form in [574'], the second is
deduced from [578], substituting for P, h% their values [574", 575'].
i
[1187T
n. viii. § 64.] SECOND METHOD OF APPROXIMATION. 64S
which, in the undisturbed ellipsis, corresponds to the position of tw, at a given
epoch. For this purpose, we shall resume the expression of dt § 16,*
dt.\/]: _ dv.{\—e^f [1185]
^f ~ \\-\-e. COS. {o—"si)\^'
This equation being developed in a series, as in that article, becomes
ndt = dv.{\+ E^'K cos. (z? — «) + E^^^Kcos. 2 . («; — ^) + &c.} [ii86]
Integrating this equation, supposing e and xs to be constant, we shall get
fndt-^e = v-{- E^'Ksin. (?; — x^) + ^. sin. 2. (t) — «) + &c. ; [ii87]
s being an arbitrary constant quantity. This integral corresponds to the
invariable ellipsis : to extend it to the disturbed ellipsis, we must make its
differential agree with the preceding, when all the terms, including even the
arbitrary quantities s, e, «, are supposed to be variable ; hence we getf
ds = deA ^-^Vsin. (i; — «) + ^.r-^Vsin.2.(?; — -51) + &c. I ^nss]
— 6? «.{£('). COS. (v^zs) + E^^ . cos. 2 .(«; — «) + &c.}
V — a is the true anomaly of m counted upon the orbit, and ^ is the longitude [1188']
of the perihelion, counted also upon the orbit. We have already found the
longitude / [1178] of the projection of the perihelion upon the fixed plane ;
now we shall have, by ^ 22 [676'], changing v into to, and v^ into /, in the
expression v — 13 of that article,t
zi — |3 = / — 6 + tang.^ ^ <p . sin. 2.(1 — 6) -\- &c. [1189]
* (808) This is the equation [535], multiplied by ^^, and it is developed [542], in
the form [1186], whose integral is [1187].
f (809) Take the differential of [1187], supposing all the quantities n, s,v, c, is, to be
variable and [E^^\ E^^\ he, to be functions of e; subtract from it the equation [1186], the
difference will be [1188].
X (810) The longitude / is given by the formula [1178]. With this value of /we may
find that of -a, by changing in [676'], v into «, v^ into /, fi-ora which we get [1189]. Putting
t; = 0, v^ — 0, in [676'], it becomes — ^ = — d -f tang.^ J 9 . sin. {—2 6)-\- &c.,
or ^ = ^+tang,2J9.sin.2 5 + &iC., asin"[1190].
644
MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
Supposing V, v^, to be nothing, in the same expression [676'], we shall find
[1190] ^ = ^ + tang.^ ^ (p . sin. 2 6 -f &c. ;
therefore*
t3 = /+ tang.' I (p . {sin. 2 d + sin. 2 . (1—6)}-^ &c. ;
hence we get
d^ = dl. {1+2 tang.' a 9 . cos. 2 . (1—6) + &c.}
+ 2 c? ^.tang.'^9.{cos. 2^ — cos. 2 . (7 — ^ + &c.|
[1191]
[1192]
i^^_^|J_^.5si„.2^ + sin.2.(/-^) + &c.}
COS.'' J 9
The values of dl, d^, and 6^9, having been determined, by what precedes,
we shall, from [1192], get the value of <Zw, and then, from [1188], the value
of ds.
Hence it follows, that the expressions in series, of the radius vector, and
its projection upon the fixed plane, the longitude of the body in its orbit, or
referred to the fixed plane, and the latitude, which we have given in § 22,
[liga'] for the case of the invariable ellipsis, take place also in the variable ellipsis ;
provided we change nt into fndt, and determine the elements of the
variable ellipsis, by the preceding formulas. For the finite equations
between r, v, s, x, y, z, and fn d t, are the same in both cases ; and the
[1192"] series of § 22, result from these equations by analytical operations, wholly
independent of the constancy or variableness of the elements ; therefore it
is evident, that these expressions also take place when the elements are
variable.
When the ellipses are very excentrical, like the orbits of comets, we must
alter a little the preceding analysis. The inclination of the orbit to the
fixed plane 9, the longitude of its ascending node ^, the semi-transverse axis
[1192"'] «, the semi-parameter a. (I — e'), the excentricity e, and the longitude /of
the perihelion, upon the fixed plane, may be found as before. But the
* (SIX) The sum of the two expressions [1189, 1190] gives tu, [1191], and its differential
is [1192]. The values of d I, d6, dcp, are found from the differentials of the equations
[1178,1174], substituting the values of the differentials of [1173, 1176J. This value of
£? TO substituted in [1188], and also the value of de, deduced from [1179,1176], will give
the value of t? e.
n.viii. §65.] SECOND METHOD OF APPROXIMATION. 645
values of xj and <Zw, being given in series arranged according to the powers [ii92'»]
of tang. ^ (p, we must, in order to render them converging, make choice of
the fixed plane so that tang. ^ 9 may be very small ; and the most simple
method of doing this, is to take for the fixed plane, the orbit of m at a given
epoch.
The preceding value of ds [1188], is expressed by a series, which
converges only when the excentricity of the orbit is small ; it cannot [1192^]
therefore be used in the present case. To find a substitute for it, we shall
resume the equation [1185],
^ n + e . COS. (« — «) i^ ' [1193]
If we put 1 — c = a, we shall find, by the analysis of ^ 23, in the invariable [ii93^
ellipsis,*
3 3
t+r= ^^°_;['-^.tang4.(».-^). jl+l^. tang.H- (»'-«)-&c.j CU94]
T being an arbitrary constant quantity. To apply this equation to the
variable ellipsis, we must take its differential, supposing T, the semi-parameter
a.(l — e^), a, and w, to be the only variable quantities. We shall thus have
a differential equation, which will determine T\ and then the finite equations, [1194']
which take place in the invariable ellipsis, will take place also in the variable
ellipsis.
^b. We shall now consider particularly the variations of the elements of
the orbit of m, when the excentricities of the orbits, and their inclinations to
* (8 12) In the equation [690] the angles ^, r, are supposed to commence together, but
if we suppose t to be equal to — T, when r = -sj, the first member of the equation will
become t-\-T^ and the angle «, in the second member, will become v — ta. Substituting
D = aaj [681"], in the factor / , [690], it becomes
2 J. J 2g^.a^.(2 — «)^ 2a^.{«-(^ — «)l^ 2j.{{l—e).{l-\-e)]^ __ 2 a^J^t-ff
(2-.a)i.v/]L~~ (2--a)a.v/i^ — {2—af.\/^ ~ {2—afyiL — (2— a) Vf^ '
These substitutions being made in [690], it becomes as in [1194]. We may observe that Hl^^^l
no terms are neglected in ^ 64, so that the equations of that article are accurate.
162
646
MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
each other are small. We have given in § 48 [957] the method of developing
the value of R, in this case, by a series of sines and cosines of the form
[1194"] m' Jc . cos. (i' n' t — int-\- A), k and A being functions of the excentricities,
inclinations of the orbits, positions of the nodes and perihelia, longitudes of
the bodies at a given epoch, and transverse axes. When the ellipses are
variable, all these quantities must be supposed to vary, in the manner already
explained ; we must also change, in the preceding quantity, the angle
[1194'"] i' 1^' t — int, into i'fn'dt — ifndt^^ or w^hich is the same thing, into
Now we have, by the preceding article [1177, 1183],
[1195]
^=fndt = - .//« ndt .dR.
The differential d R being taken, supposing only the co-ordinates x, y, z, of
[1195'] the body m to be variable, we must, in the term m'k . cos. (i' ^' — i ^ -{- A)
of the expression of R, developed in a series, consider as variable, only those
quantities which depend on the motion of this body ; moreover, R being a
finite function! of x, y, z, x', y', zf, we may, by § 6S, [1167"], suppose the
elements of the orbit to be constant, in the diflferential dR; therefore it will
be sufficient to vary ^, in the preceding term, and as the differential of ^ is
[1195"] ndt [1180"], we shall have im'.k.n d t . sm. (i' ^' — i^-\-A), for the part of
d R, corresponding to the preceding term of R ; and if we notice only this
term, we shall have [1195]
l = ^l^,fk.ndt.sm.(i'^' — i^ + A);
^=^^^^.ffak.nHf.sm.{i'^—i?,+A).
* (813) This change of the angle int — int, appears evident, by comparing the
value of R, [951], with that of [957] ; it being easy to perceive that the last value would
'■ more accurately conform to the first, and to the principles above explained, by making this
substitution.
f (814) This follows from [913, 914]. The differential of R, being of the first order,
we may, in finding it, suppose the arbitrary quantities a, e, &c., to be constant, conformably
to [1167"].
n. viii. §65.] SECOND METHOD OF APPROXIMATION. ^^.7
If we neglect the squares and products of the disturbing masses, we may, in
the integration of these terms, suppose the elements of the elliptical motion [1196']
to be constant, which will change 2, into ntf and ^' into n't ; and we shall
obtain i
- = jTr-, r-r • COS. (int lUt^ A) '^
a \i..{in—in) " ' \{wr\
^ = TT~, — r-T^ • sm. (I'n't — int-{- A).
{h.iin inY Great ill
equalities
Hence we see, that if in' — in does not vanish, the quantities a and <^ will ^I^p^I^^^
contain only periodical inequalities, provided we notice only the first power mouon.
of the disturbing force ;t now i' and i being integral numbers, the equation [ng^j
i' vl — in = 0, cannot take place if the mean motions of m and m' be
incommensurable, which is the case with the planets, and may be admitted
generally, since n and n' are arbitrary constant quantities, susceptible of all rjig?"]
possible values, and the supposition that this ratio can be exactly defined in
whole numbers, is in the highest degree improbable.
We are therefore led to this remarkable result, that the transverse axes of
the orbits of the planets, and their mean motions, are subjected only to
periodical inequalities, depending on their mutual configuration, and by [1197"']
neglecting such quantities, these axes will be constant, and the mean motions
will be uniform ; this result is conformable to that we have found, by another
method, in ^ 54 [1070"'].
If the mean motions nt, n't, without being exactly commensurable,
approach very nearly to the ratio of i' to i, the divisor i'n' — in, will be [1197*']
very small, and there may result in ^ and ^ some inequalities, which vary so
slowly, that observers may be induced to suppose the mean motions of the
two bodies m and m' not to be uniform. We shall see in the theory of
Jupiter and Saturn, that this has happened relative to these two planets :
their mean motions are such, that twice that of Jupiter is nearly equal to
five times that of Saturn ; so that 5n' — 2n is but the seventy-fourth part [1197^]
of n [381 8f?]. The smallness of this divisor renders the term of ^, depending
upon the angle 5 n't — 2nt, very sensible, although it is of the order i' — i,
f (814a) This is true, even if we include the second power, and some terms of the
third power, of the disturbing masses, as has been observed in [1070a].
648 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
[1197"] or of the third order relative to the excentricities and inclinations of the
orbits, as we have seen in ^ 48 [957""']. The preceding analysis gives
the most important part of these inequalities. For the variation of the
mean longitude depends on two integrations, whilst the variations of the
other elements of the elliptical motion depend but on one integration ;
[n97vii] therefore the terms of the mean longitude only can have the square
(i'n' — iny for a divisor ; so that if we notice only those terms which, on
account of the smallness of this divisor, ought to be the most considerable,
[iigr'"'] it will suffice, in the expressions of the radius vector, the longitude and the
latitude, to increase the mean longitude by these terms.*
When we have found the inequalities of this kind, which the action of m'
[1197«] produces in the mean motion of w, it will be easy to deduce the corresponding
inequalities produced by the action of m in the mean motion of m'. For, if
we notice only the mutual action of the three bodies i)f, m, and m\ the
formula [442] will give,t
* (815) This also follows from the equations [1066, 1070], as well as from the method
here used.
f (816) Dividing the equation [442] by JH + S . to, or M + m-^-m', we get
. 2 . m .
M-^-^.m Jkf+S.m dt^
, (,{dx'-dxfMdy'-dyf+(dz!-dz)^}
z.mm.< -jZiz ' c m
-J- I ^£ J — 2Jf.2.- — 2X,
' JW+2.m r
M 2.ni . (m-\-m') . .
and as -rr; = 1 — tt, = 1 — ,, ,' — , it beconies
Jk/-f-2.m M+Z.m Jlf-fS.m'
h dx^-^dy^+dz^
— : — — == 2 . TO . TTil
Jtf-f2.7tt ar
' (M-\-m-\-m').dt^
— 2Jlf.2.-— 2X.
r
The term of the second member, having the denominator {M -}- to + to') . </ 1%
may be put under a more simple form, since the terms of the numerator, depending on x, a/,
are mm .{dcd — dxf — (to + to') . (to . rf a;^ + to' . d a/ 2), which by reduction become
U. viii. § 65.] SECOND METHOD OF APPROXIMATION. 649
^ ^ (do^ + df+dz^) , m'.{djf^-\-d'!/^-\-dzf'')
constant = m . ^^ ,\ H ^^ —r4^ — '
dr dr
Umdx-^m'dx'f4-(mdy4-m'dy'f4-(mdz-{-m'd2fY\ , .
{M -\- m -\- m') . d t^ v / l
2 Mm 2 Mm 2 mm'
"^ V/^+FT^ "" V/^^TF^T^ ~ ^'(a^' — xf + (y — yf + (z' — z)2 *
The last of the integrals (p) [1171] of the preceding article, substituting
for - the integral 2/di2 [1195], gives*
d_^+dl±d^ 2_.Gg+^ _
If we put R for what R becomes, when we consider the action of m upon [1199']
m', we shall have
n,_, m.{xa/-{-yy'-{-zz') m
^ ' ^ ' '' [1200]
the differential characteristic d' affects only the co-ordinates x\ 3/, 2:', of the [1200']
body m'. Substituting these values of ^J^Jt±^ , d^^+J^p^^ ,^,
in the equation (a) [1198], we shall havef
— {m.dx-{-n^ .da/y, and as the numerator is symmetrical in x, x', y, y', z, z', the whole
numerator will be — {m.dx-\-m' .dx')^ — (m .dy-\-m' .dy'Y — (m .dz-\-m'. dz')^;
substituting this, and putting ^ = ^/(7:z^ ■ J^ p . (^^^^ > [397], it becomes as
in [1198].
* (817) Substituting - = 2/d/2, and {J. = M+m, [530^^], in the last of the
equations [1171], we shall get [1199]. The value of R, [949], changing the terms relative
to m', into those relative to m, and the contrary, becomes the same as R', [1200]. Similar
changes being made in [1 199], we shall get the second equation [1200].
^x2-4-rfw24-rfz2 rfa/24-rfi/2J-rfz'2
f (818) After substituting the values of — 1^^ — ' ~f^' '
transposing the terms 2m.fdR, 2 m' .fd' R, dividing by 2 and reducing, the
expression becomes as in [1201] ; the lastterm, depending on m, m', was accidentally omitted
in the original work.
163
650 MOTIONS OF THE HEAVENLY BODIES. [M^c. Cel.
[1201]
m.fdR+m'.rd'R= constant— ^ -^ TrTlr i ■ >V S
m'^ m m'
~^ \/l^+f + z^'^ s/x"" + y'2"4_ /2 ^(a;/_a:)2 -|- (y— yfJ^ {z'—zf
It is evident that the second member of this equation contains no terms of
the order of the squares and products of the masses m and m', which
[1201'] has i' n' — in for a divisor ;* and by noticing only these terms, we shall
have
[1202] m ./d R + 7n'.fd'R'= 0 ;
therefore, by considering only the terms, which have (i' n' — iny for a
divisor, we shall getf
Sffa'ndt.d'R m.(JII + w). a'n' Sffandt.dR ^
[1203]
M-^m' m'.{M ■\-m').an M-\-m
* (819) The terms having in' — in for a divisor [1066, 1070, Stc], must be those
arising from the disturbing force, and they will therefore be of the order w or to' ; that is,
the parts of a?, y, z, dx, dy, dz, x', &;c., depending on sucli angles, must be of the order
m or mf ; these parts being substituted in the second member of the equation [1201], will
produce terms of the third order, as it respects the powers and products of the masses m, mf ;
and by noticing only terms of the second power, we may put that second member equal to
nothing, as in [1202].
f (820) R, [949], is of the order m', hence m .ffdR .dt, is of the order m m',
and the like is to be said of m . ffd' R .dt, and if we neglect terms of a higher order,
we may, from [1196], write — , lor m.JJdt.dJi., and
m'.ffa'n'.dt.^J^^ ^^^ m' . //^ ^ . d' 22'. JMultiplying the expression [1202] by ^dt,
integrating, and making the preceding substitutions we shall get,
3m.ffan.dt.dR , 3m' .ffa'n' .dt.A' R
an ' an
The constant quantity of the second member is put equal to nothing, because no terms,
except those depending on the angle, i' n t — int are here noticed. IVIultiplying the
numerator and denominator of the first of these terms by M-^m, and those of the second
by M-\-m', we shall obtain, by reduction, the formula [1203]. The equations [1204] are
deduced from the second equation [1195], putting successively, it/=M~\-mj ii.=M-{-m'.
Substituting [1204] in [1203], we get ^' = — "^/ [f"! ' •<?, from which we easily
n. viii. <§ 65.] SECOND METHOD OF APPROXIMATION. 651
now we have
Sffandt.dR ^, Sffa'n'dt.d'R'
therefore we shall obtain
m' . (If + m') .an.^' + m. (M+ m) .a'n'.^ = 0. [1205]
We then have [605']
n =
1
n'
m!
neglecting
therefore
m and m\
in comparison
with H, we
shall find
m,\/\
j . ^ + w' . v^"' •
r
= 0;
or
^ m.\/a
'I'
[1206]
[1207]
[1208]
Therefore the inequalities of ^, which have {i n! — i ny for a divisor, will
give those of ^' which have the same divisor. These inequalities will [I^OS'J
evidently be affected with contrary signs, if n and n' have the same sign ; ^^^^^1^^'
or, which is the same thing, if both bodies m, m', revolve in the same Zlamel
• • , 01 a long
direction ;* they are, moreover, in a constant ratio to each other : hence it fikj°^j„g
follows, that if these inequalities appear to accelerate the mean motion of m, ° and "*'
r T 1 1 Saturn.
they will appear to retard that of m', according to the same law ; and the
apparent acceleration of m, will be to the apparent retardation of m', as [i208"j
obtain [1205] ; now an = — p^ , a' n = — ^^> [1206], these being substituted
in [1205], it becomes m'. (JWf+m') .^^^. ^' + m . (J)f + w) . ^^^^' .^ = 0, or by
reduction m . \/M-j-m' • \/af. 8,' -{-m. \/M-^m . \/o • |= 0, and if we neglect m and m'
in comparison with M, we may divide it by \/M-{-m, or \/M-\-m', and we shall get
[1207], from which [1208] is easily deduced. This beautiful theorem is frequently used
by the author, particularly in the third volume, where it is applied, without restriction, to all
terms of the order of the square of the disturbing forces ; which has been objected to by
M. Plana, in a paper published in Vol. II, of the Memoirs of the Astronomical Society of
London, as will be more particularly stated in the notes upon that part of the third volume of
this work.
* (821) Conformably to note 746.
652 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
m' . \/Z' to m . \/^. The acceleration of the mean motion of Jupiter, and
the retardation of that of Saturn, discovered by Halley, in comparing ancient
and modern observations, are very nearly in this ratio ; and. I inferred, from
[1208"] the preceding theorem, that they were produced by the mutual action of
those planets upon each other ; and since it is evident, that this action cannot
produce any alteration in the mean motions, independent of the configuration
of the planets, I did not hesitate in the belief, that there must exist, in the
theory of Jupiter and Saturn, an important equation of a very long period.
[1208'''] Taking also into consideration, that five times the mean motion of Saturn,
minus twice that of Jupiter, is nearly equal to nothing [3818c?] ; it appeared
to me, to be highly probable, that the phenomenon observed by Halley, was
produced by an equation, depending on this argument. The calculation of
this equation proved the conjecture to be true.
[1208 "] The period of the argument i' n t — int being supposed very long, the
elements of the orbits of m and m' vary so much during this interval, that it
is necessary to notice this circumstance in the double integral
ffa k.n^df. sin. (i' n't — int-\- A).
For this purpose, we shall put the function k . sin. (i' n' t — int-\- A) under
[1208"] the form* Q.sm.(i'n't — int~{- i' e' — i s) + Q' . cos. (^' n't — int-\-i's' — i s) ;
* (822) In [1]94"] the general term of R is put equal to m'k. cos. [i'n'i — int-\-A).
Comparing this with the expressions [958, 961], we shall get
A = i's'~is—gTS —g' zs' —g" 6 —g"' (f,
and k equal to a function of a, a', e, e', (p, (p'. If for brevity we put
K=i'n't — int-\-H — is, JV' = ^« + ^'is'+^'^+^"^',
the corresponding term of k . sin. {i' n t — int-\- A), will be
k . sin. {JV— JV')—k. {sin. A*, cos. JV' — cos. A*, sin. A"|,
and if we substitute Q, = k . cos. JV", Q' = — k. sin. JV", it will become
Q.sin.JV-l- q.cos.JV,
as in [1208'''], Q, Q', being functions of a, a', e, e', <p, (p', &;c. During the period of the
argument i'n' t — int, the quantities e, e', cp, cp', he, will vary by reason of the secular
[1209ol inequalities ; therefore, Q, Q', must vary for the same cause ; a, a', n, n', not being liable
to such variations [1044', &c.].
II. viii. § 65.] SECOND METHOD OF APPROXIMATION. 653
Q and Q being functions of the elements of the orbits ; we shall then
have
ff a k . n^ df . sin. (i'n't — i n t + A) =
ri'a.sm,{i'n't-4nt-]-i'^—is) ^ 2dq 3ddq 4d^q ^ )
n^a.cos.{i'n't—int-\-i'.^^s)i 2dq 3ddQ; 4d^q
{i'v! — irif
S 2dq 3ddq Ad^q >
i ^ "^ {i!n'-^n).dt {ifn'—inf.dt^ {ifn'—infJfi ^ ' S
If we substitute the value of k . sin. {i' n' t — int-\- A), in the first member of [1209]
we shall get
ffakn^ .dt^. sin. {i'r^ t — int-\- A) =ffa ri^.dt^.iq. sin. JV+ q . cos. JV}.
And it easy to prove, by integrating by parts, that if A, B, are any functions of t, we
shall have
ffAB.dt^ = ApB.dt^ — 2.^.pB.dfi+2.^.pB.dt'—4.^^.pB.dt'-{-hc. [12096]
For if we take the differential of this equation, and connect the similar terms of the second
member we shall get
fAB.dt^ = AfB.dt'' — ~.rB.dfi-\-^.pB.dt*—kxi., [I209c]
d t being constant. Again, taking the differential, all the terms of the second member will
be destroyed, except the first term, A B . dt^, which is the same as in the first member.
Putting in this formula A= q, B = an^ . sin. JV, it becomes
ffan^.dt^. q. sm. JS^= q.pan^ .dt^ .sin. jy
^'^^ P anK d fi .sin. JV+^^ .P an"" . d tK s\n, JV—kc,
dt -^ ' dfi
Taking the integrals of the second member, it becomes equal to
Qarfi.8in.JSr 2dQ arfi.cos.JST , ScPQ an2.sin.JV , 4</3Q on^. cos.JV , „
*(i'n'— in)2 dt '(i'n'— inp ' d<2 {i'n'—in)'^ ' dt? '{i'n' — in)^~ '
In like manner, putting A= q, B = ar? . cos. JV, we shall obtain
ffar^.df.q.cos.K===q.pan'.dt'^.cos.J\'—^^^.parv'.di^.cos.JV-\-Uc.
_,, arfi.cos.JSr , 2d& arfi.ain.JV , 3d^& arfi.coB.JSJ"
^ (I'n'— in)2 ~ dt '{i'n' — in^~ dp (^n' — tn)4 *
adding these expressions, and connecting the terms depending on sin. JV, also those depending
on cos. JV, we shall have the value of ffakn^.di^.sm.{i'n't — int-\-A), as in
the second member of f 1209].
164
654 MOTIONS OF THE HEAVENLY BODIES [Mec. Cel.
As the terms of these two series decrease very rapidly, on account of the
slowness of the secular variations of the elliptical elements, we need only
retain the two first terms of each series. Then substituting the values of
the elements, arranged according to the powers of the time, and retaining
[1209'] only the first power ; the preceding double integral may be transformed into
one single term of the form*
[1210] (F+E.t). sin. (i'n't — int + A+H. t).
[iSiC] With respect to Jupiter and Saturn, this expression will serve for several
centuries before and after the time selected for the epoch.
The great inequalities we have just mentioned, produce similar ones
[1210"] among the terms depending upon the second power of the disturbing masses.
For, if in the formula [1196],
[1211] ^ = — .ffa k.ri'df. sin. (^' ^,'—i^,-^ A),
* (823) The terms Q, — , &;c., vary very slowly, and their values may be arranged
in a series, proceeding according to the powers of the time, D -{- U t -\- U' t^ -\- &c.
If we retain only the first power of t, the coefficients of the sine and cosine of
{i' n' t — int -{-i' ^ — i s),
in the second member of [1209], may be put under the forms E' -{-E" t, F' -f- F" t,
E" and F" being very small in comparison with E ', F', so that
ffakn^.sm.{^n't — int + A) = {E' + E"t). sm. JV+{F'-i-F"t). COS. jy.
If we now put F' = F.sm.A', E' = F.cos.A', E" = E .cos. A' — FH. sia. A',
F'' = E . sin. A' -\- FH. cos. A', the preceding expression will become
{ F. cos. A'+E ^ cos. A'—FHl sin. A' ] . sin. JV+ { F. sin. ^'+E^.sin.^'+ZH^cos.^' } .cos.JV,
or, as it may be written,
{F + Et).{cos.A'.sm.JV+sm.A'.cos.JV]-}-FHt.\—sm.A'.sm.JV-\-cos.A'.cos.J\'\
= (F -f E 0 . sin. (.Y + A') -\-FHt. cos. (JV+ A'),
[21, 23] Int. In this last term we may write F-\-Et, for F, neglecting terms of the
order t^, and then it will become
{F + E t). {s\n.{JV-{- A') +Ht. COS. {JV-{-A')]=^{F + Et). sin. {JY+ A' -\-Ht),
[60] Int. If we neglect E, H, this ought to agree with 2,, [1197], which would give
J\r+A'==i'n't^int-i-A,
and the preceding expression would become as in [1210].
II. viii. § 65.] SECOND METHOD OF APPROXIMATION. 655
we substitute for ^ and ^' their values,*
Si.m'an^.k . ,., ,. • . , a.
nt TT—, — 7-Tx . sin. (i nt — int-\-A);
^ ^ [1212]
nt-\ T^-, — ^^^.^.sm.nnt — int-\-A);
fi . (t n — I ny \/a'
there will arise, among the terms of the order m^, the following if
— o 2 r' > — ^-^T • > /-, ^^^ • Sin. 2.(i'n't — tnt + A). 1213]
* (824) It follows from [1197], that the value of «^, which is n ^, when the elliptical
elements are constant, becomes as in the first formula, [1212], when they vary by terras
depending on the angle {i' n' t — int-\-A). The corresponding variation of i^', orn'f,
[1212], is found by means of [1208], multiplying the value of the decrement of ^, [1197]
, m.s/a
f (825) Putting for brevity i' n' t — int-\-A:=^JX, V-^^"" \ ~ *' ^^
expressions [1212] will become ^ = nt — J.sin..^'; ^' = n'^+^'~T~P'Sin..4'.
These values being substituted in i' E,' — i ^ -{- .4, it becomes
^ ^ ' ' ' m!.\/a!
= .^' +i:^liV:^±li^Li^ . & . sin. ^',
m' .\/a!
and as the part depending on I is very small, we shall get, by [60] Int,
sin.(^-g--^^ + ♦^)=sin.^-+|^•^^VJ + ^;^V^6.sin..3^|.cos.^-
= sin. ^' +^^^^^^^^±4^^-^. & . sin. 2 ^',
2 w! . \/a'
substituting this in [1211], we shall find
„ 3iwi' /•/. 7 2 J ^2 1 '^ ai \ i • tn' . \/a' -\- i' , m . \/a i • n a}
P= fakinr, dr.<sm.A-\- ^ — t^—!= — .6.sm.2,/2>.
^ fx •'*' i ' 2m'.v/a' S
In the part depending on &, we must resubstitute the values of A, b, and it will become
[i?.{in'—inf 2ml. \/Q *'*' ^ ' ^'
and the integrals being taken we shall get [1213]. The correspondmg term of ^' is found,
as in [1208], from multiplying the preceding expression by ^—^ , which gives [1214],
771 •y of
656 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
The value of ^' contains a corresponding term, which is to the preceding, in
[1213'] the ratio of m . \/7 to — ml , \/a',
QQ. It may happen, that the most remarkable equations of the mean
motions will occur only in terms of the order of the square of the disturbing
[1214'] forces. If we take into consideration three bodies, m, m\ m", revolving
about M ; the expression of d i?, as it respects terms of that order, will
contain equations of the form k . sin. (int — i' n' t-{- i" n" t-\-A); now if
[1214"] we suppose the mean motions nt, n't, n"t, to be such, that in — i'n'-\-i"n"
is an extremely small fraction of w, there will result a very sensible equation
[1214'"] in the value of <^. This equation may even make the quantity iri — i'n'-\~i"n"
rigorously vanish, which will establish an equation of condition between the
mean motions, and the mean longitudes, of the three bodies m, m', m". As
[1214'»] this very singular case occurs, in the system of the satellites of Jupiter, we
shall here investigate the analytical expression of this equation.
[1214^] If we put M for the unity of mass, and neglect m, m', m", in comparison
with M, we shall have [1206]
[1215] n"^-', n" = ^', n"^ = ^,.
We shall also have [1195"]
[1216] d^=^ndt; d^' = n'dt; d^" = n"dt;
therefore*
dd?, , 1 da ddl' 3 A da\ dd^," 4 da"
dt " " •'" * a2 ' dt a'2' dt a
"9 *
* (826) The differential of d^ = ndty [121 6], is dd^ = dn.dt; and the
differential of n=a~^, [1215], is
-4 , -i da i da
hence dd^ = -^^n^.^, as in [1217]; the values of ddi,', dd^", are found
in the same manner, by merely accenting the letters.
n. viii. <^ 66.] SECOND METHOD OF APPROXIMATION. 667
We have seen in § 61 [1161, 1161'], that if we notice only equations of a [1217']
long period, we shall have
. ^ mm' m"
constant = _ + - _|- _ ; fi2i8j
which gives, by taking the differential,
„ da , , da' , „ daf
We have seen in the same article, that if we neglect the squares of the
excentricities, and of the inclinations of the orbits, we shall have [1152']
constant = m . \/2 + «*' . \/'d + td' . \/~d' ; [1220]
hence, taking the differential,
-, da , , dd , „ da"
From these equations it is easy to deduce*
-77--— 2-w •■^'
4
*
dd^' 3 m.n'^ (n — n") da^
4
ddi" _ 3 m.n"3 (71 — n') da
(828) The first of the equations [1222] is the same as the first of [1217]. Multiply
the formula [1221] by , and add it to [1219], we shall get
da f, a,^\ . , da' (^ a^\
3 1 a 1 5^ 1
now a^= - , a'^ =- , a"^=— , [1215] ; substituting these we get
n
da f n"\ , , da' f n"\
0 = m.— .(1— -)+m'.-— .(1 7),
dt£ m n' n — n" da ,. . , , /• ,
hence —n: = ; • — • -, 7, • ~t ; and 11 we substitute this m the second of the
£r2 ni n n — n o^
equations [1217], we shall get the second of [1222]. The equations [1217, 1219, 1221]
being symmetrical, as it respects the elements of the orbits of ?»', w", we may change, in the
equation just found, w', »', into m", n", and the contrary,,tp get the last equation [1222],
165
^^^ MOTIONS OF THE HEAVENLY BODIES. [Mec.Cel.
Lastly, the equation ^ = 2fdR, § 64 [1177], gives, by taking its
differential,*
It now remains to determine dR.
[lags'] We have in § 46, neglecting the squares and products of the inclinations
of the orbits,t
171 T
R ^= -7^ . cos. (v' — v) — m' .\r^ — 2 y / . cos. (v' — v) + r'^}-*
[1224]
-f — .COS. (^"— -y)— ^".{r^ — 2rr".cos. (7^'—v)-\-7"^-K
* (829) The equations [1217 — 1222], are defective in terms of the order of the square
of the disturbing forces, but [1216, 1223] are correct.
f (830) Neglecting the squares and products of z, 2/, z", in [913, 914], we shall get
__ m'.{xx'-\<yyf) m".{xx"-\-yf) m^
[1224a] {^'^+y'4 {^'^-Vf4 ^^*'-^)' + (2/'-#F
\{^"-^f+{]/'-yf\'~rn.{{x"-x'f+{y"-^f\^ '
Substituting the values of a:, y, x', y', [950], and the similar values a" = r" . cos. t;",
y" = r" . sin. v", which give
xx -{-yy' = rr' . cos. {v' — v) ; x x" -\~ y y" = r r" . cos. {v" — v) ;
(a/ — ^Y -\-{y' — y)^ = rr — 2 r / . cos. (v' — v) +^^ J [950a],
[1224&] {x"—xf + {y"—yf = r r — 2 r /' . cos. {v"—v) + r"2 ;
This value of R will become like that in [1224], with the addition of the terra ai-ising from
— ■ — rrTi 1\9 I / // 7\9Ti , which term may however be neglected, because this value
m.\[x' — xy-\-{y — y)s
of R is only used for finding d R, and the characteristic d does not affect x', x", y', y, so
that the result of this term in [1226] will be nothing. Now if we use a notation, similar to
that in [1077], putting
T
■~T . COS. (v' — v) — 1 7^ — 2rr'. cos. (v' — v) -\-r'^~^
[1225a] /2 V ^ i V y 1 J
= i {r, r'Y'^ + {r, /)<'> . cos. {v' —v)-}- (r, r" f'> . cos. 2 . (v' — «;) + he. j
using also a precisely similar expression in r", v", found by changing, in [1225a], r'into 7*",
and v into v", we shall obtain the value [1225], whose differential relative to d is in [1226].
II. viii. § 66.]
SECOND METHOD OF APPROXIMATION.
659
If we develop this function in a series arranged according to the cosines of
the angles (y' — v), (v" — v), and their multiples, we shall have an expression
of this form,
i? = - . (r, rj'^ + m' . (r, rj'^ . cos. (v' —v) + m'. (r, r^^ . cos. 2 . (v — v)
+ m' . (r, r'y^^ . cos. 3.(v'^v) + &c.
_!_!!! . (ryy'^ + m'\(ryy'Kcos.(v''---v) + m''.(r,r''y^\cos.2.(v''--v)
+ m".(r,r73).cos.3.(i;"— iJ) + &c. ;
hence we deduce
\dr.'
_|- ^n' . (ii.'^ip-\ , cos. 2.(v'—v)+ &c.
dR-.
+ m". ( ^f^^ ) . COS. 2 . K — ^) + &c.
-{-dv.
m'. (r, r')w. sin.(?;'— ??) +2m'. (r, r')^^). sin. 2 .(«;'— 'y)+&c.
+ m".(r,r")^'^sin.(t?"— tj)+2m".(r,r")^^-sin.2.(v"— i;)4-&c.
[1225]
[1226]
Suppose now, in conformity to what appears to be the case in the system
of the three first satellites of Jupiter, that n — 2n' and n' — 2n" are very
small fractions of n^ and that their difference (n — 2 n') — (n' — 2 ft"), or [1226]
71 — 3n' -{-2 n", is incomparably smaller than either of them.* It will
S T
follow from the expressions of — and ^v k^bO [1020, 1021], that the action
*(831) It is shown in Book VHI, §20, [6782, Sic], that w = n"' . 9,433419,
n' = n'" . 4,699569, w" = n'" . 2,332643, hence w — 2 n'= n'" . 0.034281, [i226a]
n' — 2 w"=w'" .0.034283, so that either of these quantities is much smaller than
n, n', n'. Also (w — - 2 n') — {n — 2 n") = — n!" . 0,000002, which is also [12266]
incomparably smaller than either of the preceding quantities.
660 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
[1226"] of m' will produce in the radius vector and in the longitude of m, a very
sensible equation depending on the argument* 2. (n't — nt-\-s' — s). The
terms, corresponding to this equation, have for a divisor 4 . (?i' — ny — n^
* (832) First, This term of — arises from the part of the expression [1020],
depending on cos. i. [n t — nt-{-s' — s), putting i ^ ± 2 ; the same angle, in
[1021], gives the corresponding term of Sv. Second, If in these expressions of — , 5 v, we
change what relates to m into m', and the contrary, we shall have the parts of — , 8 v',
arising from the action of m upon m! ; and by putting i = ± 1, in the terms depending
on . ' i .(nt — n' t-\- £ — s'), they will have the divisor {n' — w)^ — n' ^, as above.
Third, Changing in [1020, 1021], the terms relative to m, m', into those of m', in",
respectively, we shall have the values of -7 , S v', arising from the action of m" upon mf ;
and the terms depending on ^P^' i . {n" t — n' t -\- ^' — s'), will, by putting i = ± 2,
furnish terms having the divisor 4 . (w" — n'Y — n'^, or {n — 2 n') . (3 n' — 2 n").
Fourth, Changing in [1020, 1021], the terms relative to m into those relative to w", the
terms depending on ^?^' i . (n" t — n t-{-s" — s'), for tlie case of t = db 1, will
have the divisor {n" — n'Y — n"^, or n' .{nf — 2 n"). All the preceding terms have
the small divisors of the order n — 2 n', or n' — 2n", and give for Sr, S v, he.,
expressions of the same forms as in [1227]. The object of the present calculation is merely
to find the greatest terms o^ Sr, S v, he, of forms similar to [1227], so that their substitution
might produce terms depending on the angle [1227'], therefore it is not necessary to
introduce any terms multiplied by the excentricities, which are very small, and would
generally produce angles different from those which are here noticed, as is observed in
[1228c]. It is easy to prove that there are no other terms similar to those above retained,
which have those small divisors n — 2n', or n' — 2n". For /?, w', n", are nearly to
each other, as 4, 2, 1, [1226a], respectively, and if we put w= 4, n' = 2, n"=l,
the proposed divisors n — 2n', n' — 2n", will become nothing. Those m^eg-raZ values
of i only, ought therefore to be retained, which make the divisors become nothing, by putting
1 1226c] n=4, n'=^2, n"=l. Now in the action of ml upon m, the angle i.{n't — nt-\-^ — s),
produces the divisor i^.{n' — rif — n^, or Ai^ — 16, which, being put equal to
nothing, gives i=±2; in the action of m" upon m, we have the angle i.{n"t — nt-\-^' — s),
and the divisor i^ . {n" — w)^ — n^ =: 9 i^ — 16, which, being put nothing, will not give
an integer for i, and it must therefore be neglected ; this corresponds with 6r,bv, [1227].
II. viii. § 66.] SECOND METHOD OF APPROXIMATION. 661
or (w — 2 /i') . (3 n — 2 n') ; and this divisor is very small, on account of [1226'"]
the smallness of the factor n — 2 %'. We perceive also, in examining the
same expressions, that the action of m produces, in the radius vector, and in
the longitude of m\ an inequality depending on the argument {r^t — nt-\-z' — s), [1226>»]
w^hich, having {n! — nf — ri!^ or n.(n — 2w') for a divisor, is very sensible.
We also find that the action of m!' upon m! produces, in the same quantities,
a considerable inequality, depending on the argument 2. (n"t — n!t~\-i" — s').
Lastly, we find that the action of m! produces, in the radius vector, and in [1226 »]
the longitude of m", a considerable inequality, depending on the argument
n!'t — n't-\-s" — e'. These inequalities w^ere first discovered by observations;
we shall fully develop them, in the theory of the satellites of Jupiter : their [l226vi]
magnitudes, in comparison with the other inequalities, permit us to neglect
these in the present case. We shall therefore suppose
6r = m'. E'.cos.2.(n't — nt-{-s'—s) ;
5v = m' .F' .sin.2. (n't — nti-s'—s) ;
6r' = m". E". C0S.2. (n"t — n't -\-s"—s')-{-m.G.cos.(n't — nt + B' — s) ;
6v' = m". F" . sm.2. (n"t — n't + s" — s') -{-m . H. sin. (n t — nt + s' — b) ;
5r"= m'.G'. COS. (n"t — n't + e" — s') ;
6v"= m'.H'. sin. (n"t — n't + s"— /).
We must now substitute, in the preceding expression of d R, for r, v, /, v',
r", v", the values of « + <5 r, ti ^ + s + <5 1;, a' + Sr', n't-{-s'-{-5v', a"-\-5 r".
Again, the action of m upon mf, depending on the angle i . (n' t — nt-\-s — s), produces
the divisor r^ . [n' — n)^ — n'^= 4 i^ — 4, which becomes nothing, by putting i = ± 1 ;
the action of m" upon m', depending on the angle i . {n" t — n' t-\- s" — s'), has the
divisor v^ . [n" — w')^ — n'^ = i^ — 4, v^rhich becomes nothing, by putting i = dr 2 ;
these furnish the two terms of 5r', 6v', [1227]. Lastly, the action oim upon m", depending
upon the angle i . {n" t — nt-\- ^' — s), has the divisor v^ . (n" — n)^ — 71"^ = 9 i^ — 1 ,
which gives a fractional value of i, and must therefore be neglected ; the action of m' upon
m", depending upon the angle i . {n" t — n' t -\-s" — /), has the divisor
which gives i=zbl, and furnishes the terms 5r", 5v", [1227]. Therefore the terms
above noticed, are the only ones necessary to be retained,
166
[1227]
[lQ26d]
662 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
n"t -^ ^' -{- 5 v"* and retain only the terms depending on the argument
[12271 nt — Sn' t + 2 n" t + s -^ 3 s' Jf- ^ ^" ; and it is easy to perceive that the
substitution of the values of ^r, 5v, Sr", Sv'\ can produce no similar terms ;
but this is not the case with the substitution of the values of 5r', 5v' ; since
the term m' . (r,r'y^K dv . sin. (v' — v) of the expression of di2, produces
the following quantity,
^1228] _^^i^iiZi^ Ae". ('lLi^^\ -^F".(a,aT I .sm.(nt^3n't+2n"t+B^3e'+2s") ;
* (833) In the value of dR, [1226], we may neglect terms of the order m^. Now
r=a-{-§r, gives dr = d8r, which, by the first of the equations [1227], is of the
order m', also dv = ndt-\-d8v, and the last term dSv, by reason of the second of the
equations [1227], is also of the order m'. If we now put, for brevity,
(r, /)(" . sin. {v' — v)-\-2. (r, rj^^ . sin. 2 . (v' — zj) + &tc. = | 2 . i . (r, r'f\ sin. i . (t/— v) ;
and use also similar expressions for the terms in r", &;c., we shall have from [1226],
[1227a] -^ndt.{m'.:E.i.{r,T'Y'^.sm.i,{v'--v)+m''.:s.i.{r,r"yKsia.i.{v"—v)l
-\-dSv.{m' .:E.i.{r, r'Y> . sin. i.{v' — r) + m" . 2 . i . (r, r'J'^ . sin. i . (tj"— t) » .
Substituting the values of r, r', r", v, «', «", in the preceding equation, and rejecting terms of
the order ?n'^, we may, in the terms multiplied by d6r, d 8v, put a, a', a", nt-\-s — «,
&c., for r, /, r", v, Stc, respectively ; because these terms are multiplied by m' or m", and
the terms dSr, d5v, are of the order m' or m', neglecting the excentricities and inclinations,
as we shall hereafter find, may be done [1228c] ; but in the term multiplied by ndt, we
must develop (r, Z)®, sin. i . {v' — v), he, to terms of the order m. Thus the development
of (r,/)^'^, made by means of the formula [610], becomes
and
(r,0»= («,«")» + (WE) . Sr+{±^^) . S /',
Again, as in the formula [678], we have
sin. i.{v' — v) = sin. i . (w' ^ — nt-\-^ — s)-{-i.{Si/ — 8v) . cos. i.{n't — n < -f* ^ — 0 >
sin. i . («" — t;)=sin. i . {n"t — nt + s"— e) + i. (3u"— 5 v) . cos. i .(n" /— n < + 6*— s)*
II. viii. § 66.] SECOND METHOD OF APPROXIMATION. 663
it is the only quantity of this kind which the expression of dR contains.
The expressions of — , <5d, [1020, 1021 ], applied to the action of m" upon m', [1228]
hence
2dR=d5r
[12276]
7»'.2.fll%^^Vcos.i.(n'< — n^ + s' — £)"
\ da J ^ I
V — {a,a'Y^.Bv.cos.i.{n'i — nt-\-^ — i)-\-{a,a'y^.ihv'.cos.i.{n't—nt-\-^ — s)
+m". n rf^2.i.; J^l^A^yr", sm.i. {n"t-.n t+s"-e)
(__(a,a")(0.i5t,.cos.i.(n"^— w<+s"— £)+(a,a")®.i5t;".cos.t.(n"<— ni+s"— s)
•^'.d8v.^A.{a,c^Y^.sinA.{n't--'nt-\-^--s)-\-m\d8v.^A.{a,a'ysm.i.{n"t — nt-|-s"— ?).
Now if we substitute the values of Sr, 6v, [1227], and their differentials, in the preceding
expression, and reduce it, by the formulas [9545, 955&, c], the angles resulting will all be of the
form {i-{-2).{nt—nt-{-s—.s), or i.{n"t — nt-{-^'—s)-{-2.(n't — n^ + s'— s); in the
first of these forms the coefficients of n, n', are the same, therefore the angle cannot be of the
proposed form nt — 2n' t-\-2n"t-{-s —S^-^2^', [1227'] ; and in the second form the
coefficient of n' t is 2, which cannot be of the proposed form, so that we may neglect 5r, «5«,
and their differentials. In like manner, if we substitute the values of 5 r", 6 v'\ [1227], in
2di2, and reduce the angles, by the formulas [954&, 955&, c], they will become of the form
t.(w'7 — nt-\-^' — s)-\-n"t — Vkt-\-^' — s', in which the coefficient of n't is — 1, which cannot
agree with the proposed form, [1227'], so that we may neglect 5r", 5t?", and as the terms
of 2diJ [12276], independent of 5r, 5r', 5r", 6v\ &£c., cannot produce the proposed angle,
there will only remain the terms depending on 5/^, 8v', namely
, , ( f^li^y5/.sin.i.(n'^ — ni + 8' — £) >
( -\-{a,aJ^ .i . Sv' . COS. i . {n' t ^nt -\- ^ — s) )
The parts of 5/, 5 r', [1227], depending on G, H, produce the angle
(i-}-l).(n'^— ni + 6' — e),
^^^ MOTIONS OF THE HEAVENLY BODIES [Mec. Cel.
give, by retaining only the terms that have n' — 2 n" for a divisor, and
observing that n" is very nearly equal to \ n'*
___^'2 l \ da J n' — n" ^ ^ ) .
[1229] a! ' (n'— 2n").(3n' — 2n") '
which does not contain n", consequently it cannot be of the proposed form, [1227']. The
part depending on E", F'', will produce the angle
i . {n' t — n t + s' — s)-\-2 . (n" t — n' t + ^' —s'),
which, by putting i= — 1, becomes of the proposed form. Substituting therefore
[12286] 8 r' = m" .E" .C0S.2 . {n"t — n' t + s" — ^)', 8v'=m" .F" .sm.2 .{n" t — 7i' t-j-^'^s');
we shall have
ri)
( . ^ (d-'{a,a)\^„-^„^ ^,Qg_2 .(n"i—n'i+^'—^'). sin Un't—n t^^—B) )
2dR=m'.ndt.2.2 \ da' J ^ i ; v ('
I -\-{a,afv'.m".F".sm.2.{n"t—n't-{-s"—^).cos.i.{n't—nt+s'—s) )
and if we reduce the angles by the formulas [9555, e], it becomes
2dR = m\ndt.:LAi.(^'^^^fy\m\E\+iK{a,aJ^.m\F''^
X sin. li.{n't — nt + s' — s)-{-2. {n" t — n' t -{- s" — s') ] .
Now putting i = — 1, and (a, a'Y'^'' = {a, a'Y^\ which is similar to [954"], we shall
get 2dR, corresponding to <^7?, [1228].
As it respects the terms multiplied by the excentricities and inclinations, mentioned in the
last note, it may be observed, that the terms of 5r, 8r', he, depending on the pure elliptical
motion, will in general be of the form
5 r = i:. E^'^. COS. {knt + A'), 8v = :s .F^'K sin. (knt + A'), he,
k being an integral number, as appears from [669, 675, 676', fee] These values being
substituted in 2 d -R, [1227a], would produce terms depending on the angles
[1228c] \i.{n'—n)^kn\.t-]-i.{^—$)-{-Jj:', \i .{n"—n) -\-kn\ .t-^i .{b"—s)+A', &c.,
and it is evident that no integral value of i will reduce this to the form [1227'], particularly
as this term, would generally introduce the longitudes of the perigee and node into the
argument, so that on every account these terms are to be neglected.
*(834) Comparing the expression [954] with [1224, 1225], we easily perceive that
^(2) = (a, a'Y^, and if we put another accent upon a, a', we shall get the value of A'^\
corresponding to the action of the planet m" upon m', namely, A'^^^ = (a', a"Y\ Now the
II. viii. § 66.] SECOND METHOD OF APPROXIMATION. ^^
therefore we shall have
mf .m" .ndt
dR =
2
^ ^, C 2.{a,aT _ ^d.{a,aT\ 7
X sin. (nt — 3n't + 2n"t + s—3 s' + 2s") = — i . ^.
a
[1230J
values of m".E'\ ■m".F", which occur in 5r', Sv', [1227], are easily deduced from
the terms of — , Sv, [1020,1021], depending on the angle i . {n' t — nt-\-^ — s),
putting another accent on n, n', «, a', w', &tc., by which means we obtain the parts of
— , 5 1/, arising from the action of m" upon m', and these become
5«' = '^.2. .i \i^i !^ZZ!^ i.sin.i.(w"i — n' < + £"—£'),
using w3^'2) = .4^~2^ = («', rt")^-^, and retaining only the values i = 2, « = — 2,
corresponding to tlie angle 2 . {n" t — n' t-\- %" — s'), we shall get
^ = m".n'2.i ^ ^ ,/ 4~1 ^ -i.cos.2.(n"^-n'^-f3"-0.
a 4.(n — n"Y — n2 ^ ' -"
5t;'=m" L }, ^" ,, /, "~" ,,> >.sin.2.(n"<-7i'< + £" — /).
Substituting in these, for 4 . (n' — n")^ — n % its value (n' — 2 n") . (3 n' — 2 n") ;
then comparing the value of — , with that in [12285], — 7— .cos.2.(n'7 — n'^-f-s" — s'),
E"
we shall obtain the expression of — , [1229]. Again the values [1226a], give nearly
n' — n" = i n', and if we substitute this in S v', [1228e], the coefficient of
sin. 2. (n"i — n' < + £"—£'),
. . " (5 r' 2 E"
win become equal tb'twicle the preceding coefficient of -7, or m" . — —, as in [12291.
Now substituting the value of jP", [1229] in [1228], we shall get dR, [1230], which
is equal to — i.-r, [1223], . , ,
167
^^6 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
Substituting this value of -^ in the values of -— ^, -— ^, — 2_^ and
putting for brevity*
[1231] ^=.l,E\ I 2.(«,a')^'^-«'-(-j^) I • \ lr^'.rn!'+\.m.m"+^,.m.rn! \ ;
we shall have, because n is very nearly equal to 2w', and n' very nearly
equal to 2 ?fc",
[1232] ^^-_3.^' + 2.^'=:^.n^sin.(w^ — 37l'^ + 2w"^+a_3s'+20;
or more accurately,!
* (835) Substituting the value of — --, deduced from [1230] in [1222], and putting
for brevity,
[1230a] G = E\[^-A^^-(^^S^^^
we shall get ^ = f . m'm" . w^. G; ^' = _ s . ^ ;«".„'^.!!Z::!^; . G;
-— |- = f .mm'.n"^ .^— — 7^. G; but the values n, n', n", [1226a], give nearly
n — n" ^ n — n'
7,= 3, 7, =^2, therefore
n — n' n — n
ddP' „ // /f Av ddP" cy , A n
a t'^ "^ at''
hence we get
[12305] ^ — S.^-lr2.^==§G.[m'm".n^ + 9mm".n'^-}-4mm'.n"^]',
but from [1215] we obtain n'-^ = n^.n ^=n^a, also n''^ = n'^a, w"^ = n"^a",
and from [1226a], we have nearly n'^^n, n" = in, therefore we shall have
i. A
n'^ =^n^a' ', n"'^ = Jg- n^ a" ; substituting these in [12306], we shall get
^_3.^'+2.^ = #G.n2J7n'ff2".a + f .mm".a' + imm'.a"|;
resubstituting the value of G, [1230a], and using ^, [1231], we shall obtain [1232].
f (S35a) It is evident that the value of — -r, [1230], depending on the con6guration
of the planets, will be more accurately expressed, if we change, as in [1194'"], nt into fndt,
[1232o] or ^ ; also n' t into ^', and n" t into ^" ; since this part of R will be rendered more accurate
by these changes, as has been observed in [1195a].
n. viii. ^66.] SECOND METHOD OF APPROXIMATION. 667
^-3.^ + 2.^ = ^.«\sm.(?-3r + 2|" + .-3s' + 20; [1233]
and if we put
F=^ — 3^' + 2^" + £— 3£' + 2£"; [1234]
we shall find by substitution in [1233],
-— — = Q .n^. sin. V. [1235]
The mean distances a, a', «", vary but little, and the same may be observed
relative to the quantity n ; we may therefore suppose ^.n^ to be constant, [1235']
in this equation, and then, by integration, we shall find*
a t = / ~ ^ rr ' - [1236]
c being an arbitrary constant quantity. The different values, which might
be given to this constant quantity, furnish the three following cases.
If c be positive and greater than ± 2 (3 . n^, the angle V will always [1236]
increase,^ and this must happen, if at the commencement of the motion
(n — 3 n' + 2 n"y exceed rb 2 (3 . ?i^ . (1 ={= cos. F), the upper signs taking [1236"]
* (836) JMultiplying [1235] by 2dV, we get ^^~— = 2^.n^dV. sin. F,
whose integral, supposing ^, w, to be constant, is
— = c — 2p.n2.cos. F, fl236„]
hence c? < = 77 ^^^ « 2 f ' ^^ ^^ [1236]. The supposition that p, n, are constant
is allowable in this integration, because it has been proved in <^ 54, that the values of a, a', a",
are constant, if we neglect terras of the order m^, and in the appendix to the third volume
it will be shown, that the same is true, if we neglect terms of the order w^, and higher
powers.
f (S36rt) Because tlie denominator of the value of d t, [1236], will always be real,
oscillating between »/c_2s.n2, and v/c-r2s.n2, so that df^-y, ^ , and its
Kit-' ^ y/c-|-2(3.n3
rfcF
integral will give t > -r^ ^ , the angles f , F, being supposed to commence
together.
668 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
place if |3 be positive, the loioer signs if |3 be negative.^ It is easy to prove,
and we shall particularly show, in the theory of the satellites of Jupiter, that
the value of |3, corresponding to the three first satellites of Jupiter, is a
[1236'"] positive quantity ;t supposing therefore ^tz ^='!s — F, n: being the
semi-circumference, we shall get [1236]
y c-\- 2^ .11^ .COS. vi
In the interval from ^s = 0 to t3 = -, the radical ^/^ + 2 ^ . n^ . cos. -ss
At
^ ^ would exceed y/2^.nS provided c should be equal or greater than \/i ^ . n^ ;
in which case we should have, during that interval,! «>/i?.v/2^; and the
* (837) Taking the differential of the equation [1234], and substituting the values of
dZ^, A I', d^", [1216], it becomes dV= dt .{n — 3 /i' -f- 2 n"), hence the equation
[1236a] will give
[12366] (n — 3w'4-2n")2 = c — 2^.n2.cos. F;
subtracting c — 2 p . n^, we get
[n — ^n'-\-2n!'f—{c — 2^.n^) = 2 (3 . n^ . (1 — cos. V),
in which the terms (n — 3 «' -}- 2 n"Y, n^, 1 — cos. F, are evidently positive ; also by
hypothesis, c is positive, and greater than 2 j3 . w^, therefore c — 2 p . w'^, is positive ;
and, if ^ be positive, the preceding equation will give
(n — 3 n' + 2 «")2> 2 ^ . w^ . (1 — cos. V).
Again, if we subtract c + 2 (3 . n^ from the equation [12366], we shall obtain
(^ _ 3 ^' -|- 2 n")2 — (c + 2 (3 . n2) = — 2 ^ . w2 . (1 + cos. V),
and if ^ be negative, tlie three terms
(n _ 3 n' + 2 w")2, (c-f2|3.w2), — 2^.7i2.(l + cos. F),
must be positive, hence (n — 3 w' + 2 n"f > — 2 ^ . w^ . ( 1 + cos. V). Both tliese
cases are included in the form [1236"], (/i — 3 »' + 2 n"f > ± 2 ^ . n^ . (1 q= cos. V).
f (838) Comparing [1235] with BookVIII, §15, [6611], we find ^=h and by
[12:36c] Book VIII, § 29, [7272], we have h = 0,000000607302 = |3, which is positive.
J (839) When w=0, \/ci^J7n^^s:^, becomes \/c-\^2^.rfi, and when
« = ^ir, it becomes y/c", hence if c be equal to, or greater than, \/2 /3 . n^, the quantity
v/c + 2(3.n2.cos.'Ky, will exceed v/2^.n2, or w . \/2^, whilst -cj varies from 0 to | ir,
and within these limits the equation [1237] will give d t < ^ ^^a' ^®"^®
II. viii. § 66.] SECOND METHOD OF APPROXIMATION. 669
time t required for the angle « to increase from 0 to a right angle, would be
It
less than ^ -j=. The value of p depends on the masses m, m', in!'. [1237"]
The inequalities observed in the motions of the three first satellites of
Jupiter, of which we have spoken above, give the ratios between their
masses and the mass of Jupiter ; from which it follows, that ^ 7== is [1237"1
^ -^ /I . \/2 ^ ■• ^
less than two years, as we shall see in the theory of the satellites [7274] ;
therefore the angle -a would, on this supposition, require less than two years
to increase from nothing to a right angle ; now from all the observations of
the satellites of Jupiter, since the time of their discovery, * has been found [1237^*]
to be nothing, or insensible : the case we are now examining does not
therefore correspond with that of the three first satellites of Jupiter.
If the constant quantity c be less than ±2/3. n^ the angle V [1236]
will merely oscillate ; it will never attain to two right angles, if /3 be
negative, because then the radical \/c — 2 p . n^ . cos. V would become [1237^]
imaginary. On the contrary, V will never vanish, if |3 be positive. In the
first case, its value will be alternately greater and less than nothing ; in the [1237"*]
second case, it will be alternately greater and less than two right angles. [I237^ii]
All the observations of the three first satellites of Jupiter prove that this
second case is that which corresponds to these bodies ; therefore the value of [1237""]
|3 must be positive, as it respects them ; and as the theory of gravity gives |3
positive^ we may consider this phenomenon as a new confirmation of that [1237«]
theory.
We shall now resume the equation [1237],
d t = / I - o • [1238]
\/c-\-2^.n^. COS. « ^
whose integral is zs'^nt . \/2^, supposing m to be nothing at the commencement of the
time t. If we suppose T to be the value of the term corresponding to ts = J *, the
preceding expression will become ^ -jr ]> n T. \/2^i or T <[ h 7f=» as in [1237"].
The value of ^ — '-y^ , is computed in Book VIQ, <§ 29, [7274], and found to be about [1237a]
401 days, which is considerably less than two years, as is mentioned above.
168
670 MOTIONS OF THE HEAVENLY BODIES. [Mec.Cel.
[1238'] The angle -a being always very small, as appears by observations, we may
suppose cos, Trf = 1 — ^ TO^, and the preceding equation* will give by
integration,
[1239] ^ = X . sin. (nt.^+r),
[1239'] X and 7 being the two arbitrary constant quantities, which can be determined
only by observation. No inequality of this kind has yet been discovered ;
which proves that it must be very small.
From the preceding analysis, we obtain the following results. Since the
[1239"] angle nt — 3n't + 2n"t + s — Ss'-\-2s", oscillates about two right angles,
its mean value will be equal to two right angles ; therefore if we notice only
[1239'"] the mean motions, we shall have,* n — Sn'-j-2n" = 0 ; that is, the mean
^notion of the first satellite, minus three times that of the second, plus twice
* (840) Tlie general value of cos. ts, [44] Int., neglecting ©*, and higher powers of «,
is cos. w = 1 — i -zs^ ; hence { 1 237] becomes dt= , , ^ — „ ==^ . If we
put c-\-2 ^ .n^ = ^ .n^ .>?, and multiply the preceding value of df ^ by n . \/^, we shall
get ndt . v/p = -y^ — =1 whose integral is nt .\/^-\-'y==2xc.{sm. — \ or
sin. (n t . v/^ -}- 7) = "~ > hence we easily obtain -sf, [1239].
f (841 ) As the mean value of the angle (n — 3 n' + 2 n") . < -f- s — 3 / + 2 s",
[1239"], is always two right angles, or -jt, the coefficient of the time t must be nothing,
otherwise the angle would constantly increase or decrease, therefore n — 3 n' -f- 2 w" = 0,
[1239a] as in [1239'"] ; substituting this in (n — 3 n' + 2 n") . ? + s — 3 / + 2 s" = ir, we
shall get e — 3 s' -f- 2 s" = *, [1239^]. The mean motions of the satellites m, m', m",
in the time t, are nt, n't, n" t, respectively, hence the mean motion of the first, minus three
times that of the second, plus twice that of the third, is
[•X2395] n < — 3 n' < + 2 n" <= (n— 3 w' + 2 n") . ^ = 0,
as above. The mean longitudes of the same satellites are nt-{-s, n't-{- s, n"t-\- s",
hence the mean longitude of the first satellite, minus three times that of the second, plus
twice that of the third, is equal to (n — 3 n' + 2 n") . < + s — 3 s' + 2 s", which is equal
to two right angles, [1239a]. These beautiful laws, discovered by La Place, have been
confirmed by Delambre's elaborate computations of several thousands of observations of these
satellites.
II. viii. § 66.] SECOND METHOD OF APPROXIMATION. 671
that of the third, is accurately and invariably equal to nothing. It is not Fir«t
necessary that this equality should take place at the commencement of the LaPiaw,
•^ 1 J 1 relative
relative
to the
motion, which would be highly improbable ; it will be sufficient for it to be motVnsof
. • Jupiter's
nearly correct, so that n — 3n'-\-2n" may, independent of its sign, be less satellites.
than X . ?i . ^^ ;* and then the mutual attractions of the three satellites, will [12^'»]
render the equation rigorously exact.
We have also s — 3s-\-2s" = two right angles ; hence the mean longitude [1239^]
of the first satellite, minus three times that of the second, plus twice that of [1239^"]
the third, is accurately and invariably equal to two right angles. By means ^^""f
of this theorem, the preceding values of 6 r' and 5 v', are reduced tof r»iative '
motions of
Sr' = (mG — m"E") . cos. (n't—nt~\- s'— s) ; iZ^
6v'=(m H— m" F") . sin. (n!t-^nt-\- /— s). [1240]
The two inequalities of the motion of m', arising from the actions of m and
m", are, by this means, reduced to one, and will be always united. It follows
also, from the same theorem, that the three first satellites cannot be eclipsed
at the same instant ; neither can they be seen at the same time, all in [l24(ri
conjunction or in opposition with the sun, when viewed from Jupiter. For
* (842) Substituting the value of «, [1239], in the equation [1236'"], rp «= * — F,
we get F':= * zt: X . sin. (n < . \/^ -f~ 7), whose differential is
dV - _
— = ± X n . /^ . cos. {nt .^^-{- y).
Taking the differential of V, [1234], and substituting the values of d^, d8,', d^", [1216],
d V
we obtain -7- = n — 3 n' -|- 2 n" ; hence
dt
n — 3n'-\-2n" = ±:Xn.^^.cos.{nt.^-\- V);
and as the cosine of the second member never exceeds unity, the second member,
independent of its sign, must be less than dr X n . \/^, therefore the first member
n — 3 n' -|- 2 n". must be less than X n . ^/^, to render the equation possible.
t(843) Having nt — Sn't-]-2n"t-\-s — Ss'-\-2s" = '^, [1239a], we get by
transposition 2 . {n" t — n't-\-s" — £') = * + (n'< — nt-}-^ — s), therefore
COS. 2. {n" t — n't -{-^' — a') = — cos. (n < — n < -f- s' — s), and
sin. 2 . [n" t — n't-\-^' — ^) = — sin. {n' t — nt-\- ^ — s),
hence 5/, 6v', [1227], become as in [1240].
[1240a]
^^^ MOTIONS OF THE HEAVENLY BODIES [Mec. Gel.
the preceding theorems take place relative to the mean synodical motions,
[1240"] and the mean synodical longitudes of the three satellites, as it is easy to
prove.* These tvro theorems take place, notwithstanding the alterations in
the mean motions which may arise, either from a cause similar to that which
[1240'"] alters the mean motion of the moon, or from the resistance of a very rare
medium [5715]. It is evident that these causes would add to the value of --^
a r
[1235], a quantity of the form -t-^> which could only become sensible
[1240*^] by the integrations ;t supposing therefore F = * — w, and w to be very
small, the differential equation in V [1235] will become
* (844) Let the mean longitude of the sun, as seen from Jupiter, be JVt-{-E, the
mean synodical motions of the satellites m, mf, m", n, t, n' t, n'/ t, respectively, put also
£ = £ + £,, £' = jE: + s;, s" = £ + £/'. Then n=JV+n,, n'=JV-\-n;,
n" = JV-{- n'l'. Substituting these values of n, 7i', n", in the equation [12396],
{n — on' -\-2 n") . ?; = 0, it becomes (» — SnJ-\-2 n") .t = 0, and the values of
s, s', s", being used in s—3s' -{-2 s" — ir, [1239a], we get s^ — 3 s/ + 2 s/' = r.
[1240a] Hence {n^ — 3 n/ -j- 2 n/') .t-{-s^ — 3 s/ -[- 2 s/' = «r. From which equations it follows,
that the laws of La Place take place, when the synodical motions and longitudes are used.
The elongation of the satellites from the sun, as seen from Jupiter, being represented by
n^t-{-s^', n' t-\- s/ ; n'' t -\- s'/ ; these quantities cannot be nothing, at the same time,
as is evident by substituting nothing for each of them, in the preceding equation [1240a],
[12406] put under the form {n^ t-\-s) — ^. {nj ^ + s/) + 2 . (n" t + s/') = -r, therefore the
three satellites cannot be in conjunction at the same time. The same elongations being put
equal to if will not satisfy the equation [12406], therefore the satellites cannot all be in
opposition at the same time.
f (845) For if F change into V-\--\' by means of these secular equations, ddV [1235]
will increase hy dd^^, whilst the change in |3 w^ . sin. F" may be neglected on account of the
smallness of pw^ and ^^j [1241']. ;
Otherwise, in Book VII, § 23, [5543], it is shown that the secular equation of the moon is
of the form a'i^-\-a"i^, he, a, a", being very small ; or rather, as in [1052'] this equation is
of the form 2 . A; . sin. (a t -f- p), in which a is very small, and ^is a quantity which has been
much increased by the integrations. This last expression being developed according to the
powers of t, will produce the first. The resistance of a very rare medium would prevent
U. viii. '^ 66.] SECOND METHOD OF APPROXIMATION. 673
The period of the angle nt.\/'^ being a very small number of years [1237a],
J J J
whilst the quantities contained in ^ , are either constant, or include [1241]
several centuries ; we shall have, very nearly, by integrating the preceding
equation,
Hence the value of ^ will always be very small, and the secular equations of [1242']
the mean motions of the three first satellites, will always be modified by the
mutual action of these bodies, so that the secular equation of the first, plus [1242"]
twice that of the third, will be equal to three times that of the second.
The preceding theorems give, between the six constant quantities w, 7i',
n", s, s', s", two equations of condition, which reduce these arbitrary quantities [1242"]
to four ; but the two arbitrary constant quantities x and 7, of the value of «
[1239], supply their places. This value is apportioned between the three
satellites, so that by putting p,p\p", for the coefficients of sin. (nt.^^-{-y),
in the expressions of v, v', v" ; these coefficients will be in the ratio of the
the motion from being uniform, and the change produced might also be expressed by a series
proceeding according to the powers of t, connected with very small coefficients. Causes like
these operating upon the satellites would produce similar terms in n t, n't, n"t, or in n, n', n",
or by [1215], in a, a', a", and their differentials, therefore terms of the form aV-j-a'^-f-fcc,
or S.Jc. sin. (a t + js), would be produced in the values ^^, ^^, '^^^" ,
[1217, 1222], and also in the values of i^ — 3.^ + 2.^2, [1232,1233], or
in , [1234, 1235] ; so that the equation [1235] would become
= ^.n^ . sin. F'-f- 2 . ^ . sin. (a < + p).
and by putting V= * — -a, it would become
0 = — — + |3 . w^ . sin. zi--\-^.k. sin. (a < + p).
As Ttf is very small, we may write th for sin. -ss, and we may also put
2.^.sin.(«^ + ^) = ^,
and then it will become 0 = -^ + ^ . n^ . ^ -f -^ , as in [ 1 241] ; whence we may
169
674 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
, -. ddP ddP' ddP"
preceding values of y^, — ^ , -jj- '•> moreover, we have
[l242'v] j?- — 3y + 2p"=x.* Hence there arises, in the mean motions of the three
Libration first satcllitcs of Jupiter, an inequality which differs for each of them, merely
safemtM ^y *^^ coefficient ; and which forms, in these motions a species of libration,
[1242V] whose extent is arbitrary. Observations have shown that this libration is
insensible.
67. We shall now consider the variations of the excentricities and of the
perihelia of the orbits. For this purpose, we shall resume the expressions
[1242vi] of df, df\ df'\ found in § 64 [1176] ; putting r equal to the radius vector
of m, projected upon the plane of x and y ; v equal to the angle which this
[1242"'] projection makes with the axis of x ; and 5 for the tangent of the latitude of
wi, above the same plane ; we shall then havef
obtain «, by the formula [S65, 871] ; any term of —j^, represented by Tc . sin. {a.t-{-^),
k
will produce in •!* the term — . sin. (a t -f- p), [871] ; and as a is excessively small
this will become nearly r . sin. (a t -f- p), which is equal to the term of -j-^ ,
(3 . w-^ dt'^
1 ... rfrf-f
multiplied bv — 1 so that the whole correction to be applied to zi will be — -— ,
^ "^ /3.n2 ^^ ^.n^.dV'
as in [1242], this quantity being dependent on constant quantities, or angles of a very long
period, [1241'].
* (846) Substituting
p . sin. {n t . \/^ -{- y), / . sin. (n ^ . \/^ + 7), p" . sin. {n t . \/J -\- y),
for the secular librations of v, v', v", or ^, ^', ^", the corresponding libration of
F= <^— 3 ^' + 2 ^" + s— 3 s' -I- 2 s",
[1234], will be {p — 3p' -{-2 p") . sin. (n ^ . y/^ + 7), the coefBcient of which being
put equal to that of the same angle in V= tf — ts, [1240'^], or by [1242],
^=^_X.sm.(«<.v/^ + 7) + -;;r^2'
we shall get p — 3 p' + 2 p" = — X, which, independent of its sign, is as stated
above.
I (847) These values of x, y, are found as in [371], corresponding to S X, P X, of
the figure page 240, P being the projection of the place of tw, upon the plane SXP,
and the tangent of the angle mSP being*, we shall have mP = jPS.tang.m SP,
or z = rs.
n. vm.<§67.] SECOND METHOD OF APPROXIMATION. 675
x = r . COS. V \ y "= If • sin. t? ; z = rs ; [1243]
hence it is easy to deduce*
/dR\ /dR\ /dR\
-(4! )-(")=^^+^>— (4?)— ©+-^-(S ■' ^'^^
/<?R\ fdR\ ., , „, . f'dR\ . /dR\ /dR\
* (848) From the equations [1243] we obtain r = ^x2-f-^, tang. 1;=:-,
Z
J = -y ^ , ^ . This value of r eives
(dr\ X r . COS. v /dr\ y r. sin. » . /dr\
V QV X d V ^~~ V d X
The differential of tang. »=- is — — = — ^-~ — , substituting a?, y, [1243], and
tXf COS* Jj X
^ u- ^ • u 9 ux • J dy . cos. » — dx . sin. « ,
multiplying by cos.'* v, we obtain dv = — ^ ; hence
/'dv\ sin.w /dv\ cos.u /dv\
[7-.)=-—' {ry)=-^' y^"- f'=^i
,. 1 rir.,ion ^ J '^^ rfa:.5.cos.« dv .s.sm.v ,
and the values [1243], we get ds = — — ^ , hence
/ds\ s . cos. V /<^ *\ s . sin. v /d s\ 1
Uj = -— 7— ' W=""-T-' lrzj = r-- [1243.]
Now considering iR as a function of a?, y, 2:, and then as a function of r, v, 5, we shall get
/rf.R\ /'dR\ /dr\ /dR\ /dv\ /dR\ fd «N
\dx)~\dr) ' \dx) '^\dv) ' \dx) "^ \ds) ' \dx)'
and by using the preceding values it becomes
\dx/ ' '\dr/ r '\dv/ r '\ds/'
in like manner,
/dR\ . /dR\ . cos. tJ /dR\ s.sin.v /dRS
fdR\_l /dR\
\dz)~~r'\da)'
^^e • MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
We have also, by § 64 [1171]
[1245] xdy — ydx = cdt ; xdz^zdx==c'dt ; y dz — z dy = c"dt ;
therefore the differential equations in /, /', /", [1176], will become*
df=.-dy.(^^^-dz. j (l+s^).cos.«.(^)-r..eos.«.(^)+..sin.^.(^) j
jf \ ' fdR\ cos.ti fdR\ 5.sin.v fdR\\ c'dt fdR\^
df^dx.(^^-^dz. I (H-s^).sin...(^)-r5.sin...(^)-5.cos...(^) |
, ( /dR\ sin. t> /tZ/?\ 5.C0S.I; /^\ > _f^^ /"^^ .
( ' '\drj r '\dvj r '\dsj) r '\dsj''
[1246]
Hence
a; .( — )=r.cos.«.f — - j=r.cos. v.sin.v.f — l-[-cos.''«.( — j — s .cos.v.sm.v. I -j" )
/dR\ . /dR\ . /(ii2\ , . _ /dR\ , . /dR\
The sura of these two expressions, putting cos.® v -j-s'm.^v = 1, is
/dR\ /dR\ /dR\
^•u;-2'-Uj=u>
as in [1244]. Again,
/dR\ /dR\ /dR\
/dR\ /dR\ /dR\ , . /dR\ , - /dR\
whose sum is
as in the second of the equations [1244]. The third is found in a similar manner, or more
simply, by changing in the preceding equation x into y, which, as in [1243], is the same as
changing » into v — J*, that is, cos. « into sinv, and sin. r into — cos.t>.
— ) >
/ — j, (-j-)i found in the last note, we shall obtain, without reduction, the equations
[1246].
SECOND METHOD OF APPROXIMATION.
n. vm, § 67.]
df" = dx. \ (\-\-^).cos.v .(——\ — rs .COS.V, (-^\-\-s.s\ik.v, \-i—) \
j (l + s').sin.«.(^)-rs.sin.j,. (4v)-*-fo=-«-(l7) S
S/dR\ sin. t) /dR\ s.cos.v /dR\ )
COS.. . (^— j -. (^— j . (^—j ^
( . /<Z/?\ cos.t? /<?i2\ s.sin.u f^^W
\ ' ' \dr J r ' \dv J r ' \ds J )
677
+ dy
+c'dt
-\-d'dt
[1246']
The quantities c', c", depend, as we have seen in § 64 [1174], upon the
inclination of the orbit of m to the fixed plane, so that these quantities would
be reduced to nothing if the inclination were nothing ;* moreover, it is easy
to see, by the nature of R,\ that (-7—) is of the order of the inclinations [1246"]
of the orbits ; neglecting therefore the squares and products of these
inclinations, the preceding expressions of df and df will become
df=—dy.
dR
dv
cdt
dR\
cos.^
r
sin. t>
dR
dv
")!•
?^
[1247]
now we havej
dx= d.(r .cos.v) '^ dy = d .(r.sm.v) \ cdt = xdy — ydx^=r^dv ; [1248]
* (850) Putting 9 = 0, in the first equation [11 74], it will become 0 = V'^^-\-'^ ? ^ [1245a]
and as c is finite, we must have 0 = c'^ + c"^, which cannot be satisfied with any real
values of c', c", except c/ = 0, c" = 0.
f (851) Substituting z^rs, [1243], in R, [913,914], and then finding (-7-) j
it will evidently be of the order m! z', ml' z!\ &c., that is, it will be of the order of the
disturbing forces, multiplied by the inclinations of the orbits. From the first of the equations [1246a]
[1174], d and c" will be of the order of the inclination of the orbit of m, and if we neglect
terms of the order of the square of the inclinations, in the equations [1246], they will
become as in [1247].
t (852) The difierentialsof a?, y, [1243], givet^a:, <?y, [1248], or
dx = dr . COS. v — rdv. sin. u ; dy = dr . sin. v -{-rdv . cos. v.
Substituting these values of x, y, and their differentials in cdt^= xdy — y dx, [1245],
170
[1249]
678 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
therefore we shall get
df= — [dr . sin. v -\-2r dv .cos.?;} . (-v— ) — r^dv . sin. v . (-7— ) ;
df'^= [dr . coB.v — 2rdv. sin. v] . (— — j -}-r^ dv . cos. v . (-7— )•
These equations will be more exact, if we take for the fixed plane of x, y,
[1249'] the orbit of m at a given epoch ; for then c', c", 5, will be of the order of the
disturbing forces ; therefore the neglected quantities will be of the order of
the square of the disturbing forces, multiplied by the square of the mutual
inclination of the orbits of m and m'.
The values of r, dr, dv, (-7— )> (~7~)» evidently remain the same,
[1249"] whatever be the position of the point from which the longitudes are counted ;*
now if we decrease v by a right angle, sin.?? will become — cos. 17, and
cos.i; will become sin. v, and the expression of df [1249] will, in this
manner, change into that of df ; hence it follows, that when we shall have
developed df in a series of sines and cosines of angles, increasing in
[1249'"] proportion to the time, we may obtain the value of df, by decreasing, in
this series, the angles s, /, ^, 'o', ^, and ^', by a right angle.f
we shall get r^ .dv, as in [1248], being the same as in [372a]. Substituting these in
[1247] we shall get [1249]. Now if we talie, for the plane of a?, y, the orbit of m at a
given epoch, the values oi z, dz, s, \—\ c', c", will be, as in the last note, of the order
rri z\ consequently the neglected terms in [1249] will be as the square of that quantity, that
is, the neglected terms will be of the order of the square of the disturbing forces, multiplied
by the square of the inclinations of the orbit of the disturbing planet, as is remarked
above [1249'].
* (852a) This follows from the function R, being wholly independent of the plane of
X, y, as is remarked in [949'], and in note 668.
f (853) The value of R is composed of terms of the form [958],
m' k . COS. {i'n't — int + iW — is—g-i^—g''a' — g" 6 —g"' 6').
Substituting this in [930, 931, 932], and for r, cos.v, sin.v, their elliptical values, [669, &;c.],
and neglecting terms of the order m'^, we shall get expressions of Sr, Sv, 5 s, depending on
II. viii. § 67.] SECOND METHOD OF APPROXIMATION. ^79
The position of the perihelion and the excentricity of the orbit are [1249'']
determined by the quantities / and /' ; for we have seen, in § 64 [1178],
that
tang. /=-,; [1250]
/ being the longitude of the perihelion, referred to the fixed plane [1177'].
When this plane is the primitive orbit of m, we shall have, if we neglect terms [1250']
of the order of the square of the disturbing forces, multiplied by the square of
the respective inclinations of the orbits, 7=^, -a being the longitude of the [1250"]
perihelion upon the orbit.* We shall then have
/'
tang. ^=j; [1251]
which gives
similar angles, therefore the complete values of r, v, s, will depend on lilce angles ; so that
we may assume, for v, an expression of the form
v = -S.K.cos.{ifn't — int-\-i'^ — is—gzi — g'z/—g^'6—g"'d').
This equation must exist, whatever be the origin of the angles [1249"]. Suppose now the
origin to be moved forward, by a quantity equal to a right angle, the angles v, s, s', -zs, -5/, 6, 6',
will all be decreased by a right angle ; therefore, if in this value of v, we decrease the
longitudes s, s', •zrf, w', 6, (f, at the epoch, by a right angle, the value of v, resulting in the
preceding equation, will also be decreased by a right angle, as is observed in [1249"].
Making these changes in df, we shall obtain df, corresponding to [1249], in the manner [1249o]
mentioned in [1249'"].
* (854) The inclination 9 of the orbit of m to the fixed plane, at the epoch, being
nothing, we shall get from [1032] p = 0, §' = 0; hence by [1034] 5 is of the order
m! p or m' q ; therefore the greatest latitude of m, or the greatest inclination to the primitive [1250a]
orbit, will be of the same order mf p', or m' q'. Substituting this quantity for (p, in [1191],
and neglecting quantities of the order {Tn'p)% {m' q')% we shall find zrf=:J, and then
from [1250] we shall obtain [1251] ; whence we easily deduce [1252], from the well
known formulas sin. zi = /, , ' = , cos. « = -7=====. .
V/1 + tang.2 th y/1 -|-taDg.2 -a
680 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
We then have, by § 64 [1179, 1184],
[1253] ^e = \/pj^f'^J^f"^', fn_f'^-fc .
now since d, c", are, by the preceding supposition [1246a], of the order
of the disturbing forces,* /" will be of the same order ; and if we
neglect terms of the order of the square of these forces, we shall have
[1253'] ,Ae = ^/2-j-/'2. If vve substitute for ^pj^f^, its value M-e, in the
expressions of sin. w, and cos. vs^ [1252], we shall get
[1254] fjL e . sin. TO =f' ; i^e . cos. w ^f.
From these two equations we may determine the excentricity and the
position of the perihelion, and may easily deducef
[1255] i^'.ede =fdf-{-f' df ; ^Ke^d^s =fdf' -f df.
[1254] Taking the orbit of m for the plane of a;, ?/, we shall get, from § 19, 20,
when the ellipses are invariable, [1054, 1056, 1057],
a.ii — e^) , r^dv .e .sva.iv — to)
If i L • H f i — i. •
[1256] 1+e.cos. (v — to:) ' a.{l—e^) '
7^dv = a^.ndt. \/i—e^ ;
and by § 63, these equations take place when the ellipses are variable ;t
these expressions of df, df, will therefore become^
* (855) From [1246a], c', c", are of the same order as tlie inclination of the orbit of m,
and from [1250a] this is of the same order as the disturbing force ; hence, by the last
equation [1253], /" is of the same order j and if we neglect terms of the order of the
square of this force, we may neglect /"^, in the value of luoe [1253], and we shall get, as
[1253a] in [1253'], jjue =\//2-|-/2. Substituting this in [1252] we shall obtain [1254].
f (856) The differential of /x^ e^ =/2-f-/'2, [1253a], gives the first of the equations
[1255]. The differential of [1251] is __^^/^/W^/ multiplying it by
(fAC . cos. to)^ ==/^ [J 254], we shall get the second of the equations [1255].
I (857) These equations being either finite, or of the first order, must take place also
when the ellipsis is variable, as was observed in [U 67"].
/ § (858) The equations [1257] were deduced from [1249] by substituting the values
[1256]. For the coeflicient of (y-), in the value of c?/, [1249], is
— dr . sin. v — 2rdv. cos. v ;
U. viii.§67.] SECOND METHOD OF APPROXIMATION. 681^
^/= — 7!=^ • 5 ^ • ^°^- ^ + i ^ • COS. « + ^ e . COS. (2 1) — 7.) } . ^— J
— a^ .ndt . \/i — e2 . sin. «? • ( -p ) ;
^/' = ~7!=p • 5^ • ^^^- ^ + 1^ • sin. t« + ^ e . sin. (2 v — «)} . (^
[1257]
+ a^ . n (Z ^ . ^1— e^ . COS. ^ • ( y ) 5
therefore*
but
~^" /*^ tt V 6 Sin ( V I -jj I , sin* t?
and from d r [1256], we get — dr . sin. v= ■ ' — ' ,, ' \ — — '— ,
a.(l — e")
sin. (» — w) . sin. •»= ^ cos. « — J cos. (2 1> — «),
[17] Int., and r^dv = a^.ndi. \/\ — t% [1256], hence
--rfr.sin.v = — -7==. {| cos. to — Jcos. (2 «— -zs)!. [1257a]
Again, from r^ rf v, [1256], we get r d v= — I -K ^ . Substituting r, deduced
from the first of the equations [1256], it becomes rdv= >' ^ . { 1 + e . cos. {v — «) | ;
multiplying this by — 2 cos. v, we get
— 2rdv. COS. v = — / ^2 . cos. v-{-2e.cos. « . cos. {v — to)^
= — ' /izri' ^^ • COS. V + e . COS. ■!* + e . COS. {2v — «)| ;
adding this to the value of — dr . sin. », [1257a], we obtain
— dr . sin. v — 2rdv .cos. v = /yhT'^ '^^ ' cos. v + 1 e • cos. ■!* + ^ « • cos. (2 t? — ts)]^
which is equal to the coefBcient of ("T") in df, [1257], The coeflScient of ( — ) in df,
[1249], is — t^dv . sin. v, and if we substitute the value of T^dv, [1256], it becomes
as m [1257]. Lastly, if in the coefficients of (-^jj (■T~)> we decrease the values of
«, «, by a right angle, it will give the corresponding coefficients in df, [1257], as is remarked
in [1249a].
* (859) Substituting the values of/, /', [1254] in [1255] we obtain
iiP .ed e = ii'e . {df. cos. zi -\- df . sin. ot|, and
[)?e^.d-si=-^e . {df .cos.zi — df. sin. zi}.
Dividing by ii? e, we get
de=- .{df. COS. -a -\-df .sin. vs], edzi=- .{df .cos. zi — df.sm.zsl. [I257c]
171
682 MOTIONS OF THE HEAVENLY BODIES. [Mec. Gel.
[1258]
, a^ .ndt , , , /dR\
+ -j-.^l-e-.cos.(v-«).(^-y,
, a.ndt .^ . N,, o, N, /dRX
^^ = -";:77l^2-{2.cos.(i; — ^) + e + e.cos.^(i; — t.)J.(^— J
a^ .ndt , . , . /dR\
Before substituting the values of df, df, we shall observe that they will become more
symmetrical, if in the coefficient of (y Jj [1257], we write V — | -ff, for v, by which
means the term — sin.w will become cos. V, cos.t; will become sin.F, and the value of df
will consist of terms depending on cos. v, cos. ^, cos. (2 « — zi), cos. F^; then df
will contain the sines of the same angles multiplied by the same coefficients, so that if we
denote any one of these angles by W, and its coefficient by k, we shall have
[1257rf] df= 2 . ^ . cos. W, df' — :E.k. sin. W ;
substituting these in [1257c], we get
c?e = - .Z.kAcos. W.cos. zrt + sin. W . sin. '5i^= -.2.^. cos. (TV — -ss),
and
ed'ui=- . I sin. W. cos. trf — cos. W .sm.zi] =- .2 .k. sm.(W — zs) ;
hence we may obtain de, ed-m, from df, df, [1257], by substituting v^= V — Jcr, in
the coefficient of (-r-), then decreasing each of the angles w, ■5J, 2v — ■:*, V, by w,
putting COS. (trf — -2^) = 1, sin. [yi — «) = 0, and dividing by f^. Hence we shall get
, a^.ndt . ,y^ . (dn\
+ -^-.V/l-e^.COS.(r-^).(-j,
, a^.ndt ^ . .^^ . /dR\
substituting
i COS. (2 1> — 2 7s) = cos.2 {v — zi)^i, I sin. (2 « — 2 *) = sin. {v — th). cos. {v — -us),
[6, 31] Int., and V= i * + v, they will become as in [1258].
U. viii. «^ 67.] SECOND METHOD OF APPROXIMATION. 683
This expression of c? e may, in some circumstances, be put under a more
convenient form. For this purpose, we shall observe that*
substituting for r and dr their preceding values, we shall find
r^J^.e.sin. (2^— w).('^') = «.(l — e^).di2 — a.(l — e^).<Z2J.('^^ ;
now we have
'rdv = a.nat. \/i — e^ ; dv^ ^- — ^ ^~^
[1259]
(1— e^>
thereforef
a.(l— e^)
a^ .ndt . \/i — e^ . sin. (v — ^) • ( "T" )
ii:L£l^.|,+,.eos.(.-,,)f.r^V
[1260]
[1261]
In the appendix to the third volume [5788, 5789], the author reduced these expressions
of e rf •cs and de \o the following simple forms, which are demonstrated in the appendix,
, /dR\
edzi== — a .ndt .\/i — eS • I "T~ )j
de= — ^ (1 — t/i_e2).d/?+ — .ndt .[—-],
*(860) The general value of di? is dR:=(j^ydr+(j£) .dv + (^.dz,
and as the plane of x, y, is the orbit of m, at a given time, [1249'], z will be of the order of
the disturbing force, and i — j, will be of the same order [913, 914], hence dz. {-7-) ,
will be of the order of the square of the disturbing force, and if we neglect this term we shall
as in [1258']. Substituting dr, [1256], and multiplying by a.(l — e^), we shall get
[1259]. The value r^dv, [1260], is the same as in [1256], and if we divide it by the
square of r, [1256], we shall obtain dv [1260].
. f (861) Dividing [1259] by e, and substituting r^dv, dv, [1260], we shall get [1261].
Multiplying the value of de, [1258] by e, and substituting the values [1261], we shall get
[1262]. For the terms depending on cos. (v — w), and cos.^ (« — irf), will mutually
destroy each other, and tlie other terms, by reduction will become as in [1262].
684 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
hence the preceding expression of de will give
[1262] ^^^_a.yi^^v/T=r? /^i^y ^-tJ-^^) dj^.
We may also find this formula in the following very simple manner. In ^ 64
we have*
dc /dR\ /^dR\ /"dR
dt ^ \dx J \dyj \dv
[1263] but by the same article,! c = ^^a.(l — e^), which gives
[1264] dc= ^«Vm-«-('-^^) _ ^de. y/^ ^
2 a V/i— ^^ '
therefore
[1265] c f? e = v^ -j- a . ( 1 __ e2^ .
we then have, by § 64,
[1266] (l^==__(li2 ;
hence we shall get, for e d e, the same expression as above [1262].
* (862) Tlie first of the equations [1173], compared with tlie first of [1244] gives
[1263].
f (863) cf and c" are of the order m', [1246a]; and if we neglect quantities of the order
m'^, the equation [1175] will become fji-a.(l — e^)=c^, hence c = v/(xa.(l— eS),
as in [1263'] ; its differential gives [1264], and if in this we substitute the value
dc = — ^^•("3~)» [1263], it will become
7, /dR\__da.\/^a.{i—e^) ede. y/jZ^
'\dvj~ 2a \/T=:^ '
multiplymg this by -^7= , we get ede = -y=^ . (^_ j + ( i _ e^) . -.
3.
Substituting -^ = Vl^ [605'], in die coefficient of (-t-Jj and putting o-g-^ for
— , in the other term, it becomes ede=^ ^^ '^ ~~^ '\1~) ~\~^ '{^ — ^) •n~Q*
2a fA \dv/ 2a8
as in [1265], but from [1177] we easily obtain ^ = — d /?. Substituting this we
finally obtain the expression [1262].
11. vui. <^68.] SECOND METHOD OF APPROXIMATION. 685
68. We have seen, in § 65 [1197], that if we neglect the square of the
disturbing forces, the variations of the transverse axes, and of the mean [1266']
motion, will contain only periodical equations, depending on the configurations
of the bodies m, m', m", &c. This is not the case with the variations of the
excentricities and of the inclinations ; their differential expressions contain
terms, independent of this configuration, which, if they were rigorously
constant, would produce, by integration, terms proportional to the time.
This would at length render the orbits very excentrical, and make them [1266"]
very much inclined to each other ; therefore the preceding approximations,
founded upon the smallness of the excentricities, and of the inclinations of
the orbits to each other, would be defective, and might be wholly inaccurate.
But the terms which enter the differential expressions of the excentricities
and inclinations, though they appear to be constant, are really functions of
the elements of the orbits, varying with extreme slowness, by reason of [1266'"]
the changes thus introduced. Hence we see that there may result, in these
elements, considerable inequalities, independent of the mutual configuration
of the bodies of the system, and the periods of these inequalities will
depend upon the ratios of the masses m, m', &c., to the mass M. These
inequalities are those we have before named secular equations, which we [1266'^]
have considered in Chapter VII. To determine them by this method, we g^cuiar
shall resume the value of df of the preceding article [1257], equations.
^/= — 7!^" • ^^ • ^^^' ^ + 1 ^ • COS. ^ + he. cos. (2v — '^)]. (-—^
— a^ .ndt. \/i—e^ . sin. v . i -— j .
[1267]
We shall neglect, in the development of this equation, the squares and
products of the excentricities and inclinations of the orbits ; and among the
terms depending on the excentricities and the inclinations, we shall retain [1267]
only those which are constant ; we shall then suppose, as in ^ 48,
[952, 953],
r = a.{\+u); / = «'.(!+<);
[1268]
: 172
686 MOTIONS OF THE HEAVENLY BODIES [Mec. Cel.
This being premised, we shall substitute for R, its value, found in § 48,
[957], observing that by the same article [962],
we shall also substitute, for m^, w/, v^, v', their values, given in § 22 [1010],
u,= — e . COS. (nt -j-s — «) ; u'= — e'.cos. (n't-\-B — w') :
[12691 V i / ' / VI y 1
v^= 2 e . sin. (nt-{-s — u) ; t)/ = 2 e' . sin. (n't + s' — «') ;
retaining only the constant terms depending on the first power of the
excentricities, and neglecting the squares of the excentricities, and of the
inclinations ; hence we shall find*
* (864) The terms u^, w/, v^, v', are of the order of the excentricities ; if we neglect
the squares of these quantities, we shall have, as in [60, 61], Int.,
sin. V = sin. {nt -{- s)-\- v, . cos. {nt-\-i)', cos. v = cos. {nt -\- s) — v, . sin. (w t-\-s)',
e . COS. (2 V — vs) = e .COS. {2nt-\-2s — zj) ; also, f — j^|l — "/^•(t~)-
Substituting these in [1267] we shall get
df= — a.ndt.\'2, COS. {nt-\-i) — 2v .s\n.{nt-\-s)-\-le . cos. 'ui-\- ^ e . cos. {2nt -\-2 s — «)}• (— )
[1266a] ' .^^. ^^^^
— a^ .ndt .^ sin. {nt -{- s) -{-v^ . cos. {nt -\-s) — u^. sin. {n t -\- s)]. l—L
using the values of u^, v^, and reducing by [17 — 20] Int., we shall have
— 2v^. sm.(nt-}-s)= — 4 e . sin. {nt-\-s). sin. {n t-\-s — «)= — 2 e. cos.zi~\-2e . cos.(2w^-|-2s — -zs) ;
v^ . cos. (n ^ -f- ^) =2 e.cos. {nt-\- s) .sin. {nt -f-£ — zi) = e.sm. {2nt -\-2s — w) — e.sin.-Kf;
— u^. sin. {nt-\- s)=e . sin.(n t-\-s) . cos. (n t-\- s — ■«) =:J e . sin. {2nt -\-2 s — -sj) -|-| e . sin. ss) j
hence [1226a] becomes
df= — andt .f2.cos. {nt-{-s) — | e .cos. to + I* . cos. {2 nt -\-2s — zs)] . (-^j
[12r.7a] .^^. "
— a^ndt. {sin. (w ^ + s) — J e . sin. ■cJ + |* . sin. (2 w < + 2 s — to) } . ( "T~ ]•
If we now substitute the above values of u,, m/, v,, v/, in R, [957], it will become by means
of [954c, 955a],
i2=™'. 2 .^». COS. i.{n't—nt-\-^—B) — f '. e .2.a . (^^)- cos. {i . (nV— n ^+£'— s)-]-n t-\-s—zs)
— f' . e' . 2 . a' . ^^^^ . COS. \i . {n' t — nt -]- s' — b) + v! t + s —i^\
— ^' . 2 e . 2 . i . .^^^ . COS. \i.{n' t — nt-\-s' — s)-\-nt -\- s — -sj|
+ ™' . 2 e' . 2 . i . .^« . COS.- {i.{n t^nt-{-B' — s)-\-n't-\-^ —T^l,
II. viii. § 68.] SECOND METHOD OF APPROXIMATION. ^87
— am'.ndt.^. < i.A^'^-j-^a.l— — j > .sin.{i.(n't — nt-\-s' — s)-\-nt-\-s} ;
and if we connect the terms depending on the same angles
— m'e.S.^i^W + ia.^^^^.cos. {i.(n'< — wr+s' — f) + n< + £ — Ttfj
+ mV.2.$i^« — ^a'.('^V.cos. [i ,{n't—.nt-\-^ — s)-\-n't + s' — z/}.
Hence we may find ( — jj and also (-j-) ', observing, as in [063], that (~7~)==(t~)'
supposing s — zi and s — 6 to be constant, in the differential relative to s,
{—) = 5' . 2 . i ^('> . sin. i.{n't — n < + s' — s)
— w'e. 2 .<iS''^-\--^a. (— — ji .1 .sin. ^.{n' t — nt-{-^ — s)-\-nt-{-s — ml
[12675]
+ m'c'.2.^i-^«— ia'.(^^)?.i.sin. \i .{n' t — nt+ s' — s) +n't +^^i^\',
-m'e.J.^(i+J).(^') + ia.(^)^.cos.|i.(»'(-«(+s'-e)+«(+a-oj
Substituting these values in df, [1267«], and reducing by [18, 19] Int., there will arise terms
depending on sin xrf, cos. -a, of the order e, which are to be retained, also other terms
depending on angles containing the time t, which have coefficients containing terms mdependent
of e, e', connected with terms depending on e, e', but these last being much smaller than the
others may be neglected. Moreover, we shall, in the terms depending on sin. ttf, sin. -a',
retain only the terms depending on the first power of e, e'. Therefore, in the value of df,
we may, in the terms multiplied by e, take only the first term of (~7~) • (t~) instead of
their whole values, bv this means we shall have
688 MOTIONS OF THE HEAVENLY BODIES. [Mec.Cel.
[I27(r| the sign 2 in this expression, as in that of R, § 48 [954'], includes all
integral positive and negative values of i, also the value i = 0.
We shall obtain, as in the preceding article [1249'"], the value of df,
df= — andt .2 . cos. {nt-j-s) . ( — )
— andt.{ — ie .cos.'a-\-^e.cos.{2nt-\-2s — ■a)j.{^'.:z .iA^^.sm.i.{n't — nt-^-s — s)\
— a^nat. sm. \nt-\-s) . ( — j
-^a^ndt . i —ie. sm.zi-{-§e . sin. (2 n t-{-2 s— s) J . ^ f '. 2jz^\cos.i.{nt—ni-\~s—s) I .
In the terms of this expression containing e, we must, as in [1267'], retain only such as are
independent of the time, corresponding to i = 0, which makes i A^'^ = 0, and
, , fdA^<\ . , ,, , , , X 1 , {dS'\
and we must also neglect the term multiplied by | e . sin. (2 n i( + 2 s — -zs), because it
will contain the time ^, hence we shall get, from [1267c],
, , /dR\
df= — an at. 2 . cos. {nt~\-s) . \-^]
-~a2«cZ^.sm.(n^ + s).(^— j + — ^-.e.sm.^.ia.(^--j.
If we substitute in this the values of (—\ (—\ [12676], the terms of the order e, e',
will depend on the sine or cosine of the angles
i^(^n't — nt-\-^ — s)-\-nt + s — T^, i . {n' t —^n t -]- s — s) -^ n' t -\- s' — -a',
and as these are multiplied, in [1270a], by cos. (w^+s), or sin. (n< + s), they
cannot produce terms independent of t, except i is taken, so as to make the former angles
depend on nt-{-s, now this is done, by putting in the first angle,
i.{nt — nt-\-s' — s)-{-nt + s — '!!i, i = 0,
and in the second i.{n't — nt — s' — s)-{-n't-{-^ — to', i = — 1,
and as ^(-" = ^('>, we may use, instead of (^), (^), [1267&], the following
values,
(|?)=J„'...(^').cos.i.K.-». + .'-s)-»'e4i.('?^)+i«.(^')^
U. viii. §68.] SECOND METHOD OF APPROXIMATION.^^ 689
from decreasing the angles s, e', «, w', by a right angle, in the value of <//; [1270"]
hence we shall get
- -«.......cos... 5 A^^^^,a.(^-£-) +K. (^) -fi-'. (S) S ^-^
+ ««i'.7ic;^.2. j i.A^-\-^a.f—z—j i . cos.{z.(w'^ — nt-]-s' — s) -\- n t -\- s] .
We shall, for brevity, put X equal to the part of the expression of df [1270], [1271]
contained under the sign 2 ; and Y for the part of the expression of df
[1271], contained under the same sign. We shall also put, as in ^ 55^
[1073],
(0,1) = — y-.^«^(^ + ^a3.^-—
[1272]
We shall then observe that the coefficient of e'd t.sin. to', in the expression
of df [1270], would become [oTH, if we should substitute, for the partial [1272']
Multiplying these expressions by —andt .2 cos. {nt-{-e), — a^ n d t . sin. {n t -\- s),
we shall get, by means of [9556, c, &c.], and retaining only the angles and terms as
aboveraentioned,
— andt . 2 COS. {nt-]-s) .(—-j = — am' .nd t . S .iA^'^. sm.\i . {n' i — nt-\-^ — £)+»^4~^l
-{-am'ndt.e'. sin.«'. i ^(»> + J a' . (^)\y
— a^ndt. sin. (n t-\- e) . (-^ =—am' ndt.S.^a. f^plXsm. \ i.{n't—nt-]-^—s)-\-nt-\-e j
substituting these values in [1270a], we shall obtain df, [1270].
173
690 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
differentials of A^'\ in terms of a\ their values in partial differentials relative
to a ;* lastly, we shall suppose, as in ^ 50 [1022],
e . sin. Ts = h ; e' . sin. rs' = h' ;
e . COS. zs = I ; e' . COS. to = Z' ;
[1273]
which gives, by the preceding article, f=iil, f = ^j.h ; or simply, f=L
[1373'] f ==z h ; taking M for the unity of mass, and neglecting m with respect to
M, we shall find
i^ = (0 , 1 ) . / — [^ . /' + « m' . w . Y ;
[1274] ^^
^^^(0,l).h + \^i].h'—am'.n.X,
at ^ —
[1274'] Hence it is easy to conclude, that if we put (Y) for the sum of the terms
analogous to am' .n.Y^ arising from the action of each of the bodies m', m",
[1274"] &c., upon m ; (X) for the sum of the terms analogous to — am'.n.X,
* (866) From the two first of the equations [1003], we get
substituting tliese in the term depending on sin. '^', [1270], it becomes
which by [1272] is equal to e' . sin. -s/ . dt . [o^], and if we substitute, in the term
depending on sin -m, [1270], the value of (0, 1), [1272], and put
—am'ndt.:E.^iA^^+ia.(^^\ ] . sin.{i . (n't — n t + s'—s) + n t -]-s\ = -^am'ndt . X,
the value of df, [1270], will become
[1272a] df= — (0, 1 ) . c^ < . e . sin. -rrf + pM] .dt .e' . sin. ■c/ — am'ndt . X,
and in like manner df, [1271] will become
[12726] elf = {0,l) .dt .e. cos. th — [oTi] .dt.e'. cos. -^ -\-ard ndt . Y.
Substituting in [1254], the values of e . sin. zi. e . cos. «, [1273], we shall obtain (i l=^f,
fih =/', and if we put, as in [1273'], M=l, and neglect m, m', he, in comparison
with M, we shall get ii = M-}-m=l, [914']; hence Z =/, A = /', consequently
dl=dfj dh= df. Substituting these in df, df, [1272a, b], also the values of
e . sin. TO, &c., [1273], we shall obtain [1274], which, by generalization, produces [1275].
II. viii. '^ 68.] SECOND METHOD OF APPROXIMATION. 691
arising from the same forces ; and mark successively, with one accent, two
accents, &c., what the quantities (X), (Y), A, /, become, relative to the [1274'"]
bodies m', m", k,c. ; we shall obtain the following system of differential
equations,
^=-{(0,1) + (0,2) + &c.| . h+\oZ\.h'+\o:^]. h" + kc.+(X) ;
[1275]
^ = j(i,o) + (i,2) + &c.}./'~[rTo]./-^[rr.].r~&c.4-(F);
~=-{(l ,0) + (1 ,2) + &c.|. A'+ [773 .h + \r^.h"+kc. + (X') ;
&c.
To integrate these equations, we shall observe, that each of the quantities
h, /, h', r, &c., is composed of two parts ; the one depending on the mutual [1275']
configuration of the bodies m, m', Slc. ; the other independent of that
configuration, and which contains the secular variations of these quantities.
We shall obtain the first part from the consideration, that if we notice that
part only, h, /, h', Z', &c., will be of the order of the disturbing masses ; [i275"]
consequently, (0,1). h, (0,1). /, &c., will be of the order of the squares
of the masses ; and if we neglect quantities of this order, we shall have,*
dh' dV tl276]
| = (F); |=(Z0;
* (867) Denoting ihe periodical parts of A, I, N, I', &£C., by h^, l^, h*, he, the secular
parts by A , , l^^ , hj, &lc., we shall have h = h,-\-h^,, / = Z^ + Z,^ ; V 7=1^ -\- Z^^', &c.
These being substituted in [1275J, produce equations of the form
^-=Ko.i)+(o,2)+&c.}.(z,+zj-lIlIj.(/;+z;)-[o3.(z/'+z/)-^^^^
&.C.
Now these equations are linear in h, hf, I, Z', he., and the secular and periodical parts will be
satisfied separately. The periodical parts become
^'=1(0,1) + (0,2) + &c.5.z,-[o7r].z;-[oTi].z;-&c.-f(r), &c.,
^'=!(i,o)+(i,2)+&c.}.z;-[r7o].z,-[T:E].z;-&c. + (F'), &c., ;;
692 MOTIONS OF THE HEAVENLY BODIES. [Mec. Gel.
therefore
[1277J h=f(Y),dt; l=f(X).dt; h'=J(Y'),di; &c.
If we take these integrals, without noticing the variableness of the elements
[1277'] of the orbits, and put Q for what f(Y).dt then becomes ; also «5Q for
the variation of Q, arising from the variation of the elements, we shall
have,*
[1278] f(Y).dt = Q-^f6Q;
now Q being of the order of the disturbing masses, and the variations of the
elements of the orbits being of the same order, 6 Q will be of the order of
the square of these masses ; if we neglect quantities of this order, we shall
have
[1279] f(Y).dt=q.
Therefore we may take the integrals f(Y).dt, f(X).dt, f(Y').dt, &C.,
[1279'] supposing the elements of the orbits to be constant, and then consider these
elements as variable in the integrals ; we shall thus obtain, in a very simple
manner, the periodical parts of the expressions hj /, h', /', &c.
To obtain the parts of these expressions which contain the secular
inequalities, we shall observe, that they are given by the integration of the
and as (Y), {Y'), he, are of the order of the disturbing forces, it will follow from these
equations that h^, h^, he, l^, Z/, he, are of the same order ; and since (0, 1), (0,2), &c.,
[Ml, [02], hxi., [1272], are also of the same order, the terms (0, 1) . l„ (0,2) . Z,, &c.,
[oTT] . Z^', &c., must be of the order of the squares of the disturbing forces, and, if we
neglect terms of this order, we shall find -—L=z{Y), —^={Y'), &c., as in
[1276], whose integrals give [1277].
* (868) Let cZ' Q be the differential of Q, supposing the elements of the orbit to be
constant, 5 Q the differential of Q, supposing these elements only to be variable, and d Q the
complete differential, we shall have cZQ = cZ'Q-|-5Q, or <Z'Q = «ZQ— (5Q. Then
[1277a] as the integral of {Y).dt, taken upon the supposition that the elements are constant, is Q,
[1277'], we shall find, if we again take its difierential relative to d', (Y) . <Z< = tZ' Q,
hence {Y) .dt = d Q^ — (5Q, whose integral, considering all the quantities as variable, is
f{Y).dt=Ql—f8q, as in [1278].
I. viii. §69.] SECOND METHOD OF APPROXIMATION. 693
preceding differential equations, deprived of their last terms,* (Y), (X),
&c. ; for it is evident that the substitution of the periodical parts of h, /, [1279"]
h', /', &c., w^ill make these terms disappear. But if w^e deprive these
equations of their last terms, they vv^ill become like the differential equations
(A), ^ 55 [1089], vrhich we have considered before in a very full manner.
69. We have observed, in ^ 65 [1197'] that if the mean motions nt, n't,
of two bodies m and m', are nearly in the ratio of i' to i, so that i'n' — in [1279"']
is a very small quantity, there may result, in the mean motions of these
bodies, very sensible inequalities. This ratio of the mean motions may
also produce sensible variations in the excentricities of the orbits, and in
the positions of their perihelia. To determine these variations, we shall
resume the equation found in § 67 [1262],
II \avj (X
It follows from what has been said in § 48 [961, 926'], that if we take the
orbit of m, at a given epoch, for the fixed plane, we may neglect in R, the
inclination 9 of the orbit m to this plane ; then all the terms of the expression
of i2 depending on the angle i'n't — int, will be comprised in the following
form [958], t
m'k . COS. (i'n't — int-\-i's' — is — gT^ — g'z/ — g" 6') ;
* (869) This is evident from [1276a, Stc], where we find
^'= { (0, 1) + (0, 2) + &c. S . Z, - [oTD . IJ- &c. ;
^' = -j(0,l) + &c.i.A,+ [o7r].A;+[oTl.A;' + &c.;
&c. ;
which are of the same form as [1275], neglecting the last terms ( Y"), (X), &c.
f (870) The term 6, of the expression [958], is to be neglected, because the orbit of m
at the origin, is taken for the fixed plane, [1280'], hence g'', [958] is to be put equal to
nothing, and if we change g"' into g", that expression will be as in [1281] ; moreover, the
equation [959], if we make the same changes, will be 0 = i' — i — g — g' — g", as
above. If the same changes be made in the coefficient of [961], putting also Q for H, it
will become e^ . e'®*. (tang. ^(p')«". Q, the term tang. J 9 being neglected for the same
reason that 6 was omitted.
174
[1280]
[1280']
[1281]
694 MOTIONS OF THE HEAVENLY BODIES. [Mec. €61.
i', i, g, g', ^', being integral numbers, such that 0 = z' — / — g — V — ^',
[12811 [960]. The coefficient k has for its factor e^.e'^. (tang, i 9% g, ^, g",
being positive in the exponents [961'] ; again, if we suppose i and i' to be
positive, and i' greater than z, the terms of R which depend on the angle
i' n't — int, will be of the order i' — 2, or of an order superior by two, by
four, &c., as has been shown in § 48 [957'"'] ; therefore, if we notice only
[1281"] terms of the order i' — ^, k will be of the form e^. e'<(tang. ^?)^".Q, Q
being a function independent of the excentricities, and of the respective
inclinations of the orbits. The numbers g, g\ ^\ contained under the sign
COS., are then positive ; for if one of them, for example, g, were negative,
[1281"J and equal to — /, k would be of the order f-{-g'-i-g"l but the equation
0==i'—i—g—g'—g", [1281'], would then become 0=i'—i-\-f-—g'^g'\
whence f -\- g' -\- g" = i' — * + 2jr; therefore k would be of a higher order
than i' — i, which is contrary to the supposition. This being premised, we
[l28l'v] shall have, by ^ 48, [963], (-7—) = (~3~)' provided we suppose s — m
to be constant in the last differential ; the term of ( -- — J , corresponding
to the preceding term of R [1281], will therefore become*
[1282] m' .(i+g) .k. sin. (i' n' t — i n t -]- i' s' — i s — g zs —g^z/ —g"^).
The corresponding term of di2 is
[1283] m' .ink.dt .sin. (i'n't — int-\-i's' — is — gts — g zs' — g"if) ;
[1283'] noticing therefore only these terms, and neglecting e^ in comparison with
unity, the preceding expression of e^e [1280] will give
[1284] de = .^— . Sin. (t nt — tnt-i-t s' — is — gzs — g'-sj' — g"6') ;
now we havef
[1285] ^==g.e^-K e'^ . (tang, i 9')^' • Q = (f^) ;
* (871) Putting i-{-g — g for i, in [1281], it becomes
m'k. COS. \in't — {i + g).nt-\-i'^ — {i-\-g).s-\-g. [n t -\- s — zi) — g' zi'—g" 6' ^,
and if we talce the differential relative to £, supposing g '{nt-{-s — zi), to be constant,
we shall get the expression [1282].
f (872) In [1281"] we have k = €« . e'«' . (tang. J 9^*" . Q, hence
^=g.e«-^e''^'.(tang.|<?')^'.Q,
n. viii. §69.] SECOND METHOD OF APPROXIMATION. 695
therefore we shall have, by integration,
e== —- — ^^. ( -r-].cos.(i'nt — mt + t's' — ts — £-o — V-n' — sf'd'). [i286]
fj- . (i n — 111) \dej ^ ^ ° Of
Now the sum of all the terms of 72, which depend on the angle i' if^ t — int^
being represented by the following quantity,*
m'.P.sin. (e'?i7 — ^wi + ^'s' — z£) + m'.P'.cos. (i'n't — int + i's-^is), [1287]
the corresponding part of e will be
— m'.an C /dP\ . .., . . ^ , .,, .. , /dP'\ ,., ,^ . ^ , .,. .. )
\i..\%n — m) l\dej ^ \de J ^ }
This inequality may become very sensible, if the coefficient i'n' — in be
very small, as is the case in the theory of Jupiter and Saturn. It is true [1288']
that the divisor is only of the first power of i'n' — in, whereas the mean
motion has the second power of that quantity for its divisor, as we have
seen in § 65 [1197] ; but (-p) and (-7— )> being of a lower order
and from the same value of A; we get ^— j =^ . e^^ . e'«' . (tang. \ 9)^' . Q, therefore
— = ( — ), as in [1285]. This value being substituted in [12841 we get
m'.andt /dk\ •/•,,, • . \ -i j • > t n ai\
de = • ( 7" ) • ^^°- (*^^ — int-\-t's — is — g zs — g vi — g ff),
whose integral is [1286].
[1284al
* (873) Supposing for brevity
i'n't — int + i'^ — is=T; g vs -{• g' z/ -{- g" 6' = W, [i286a]
the value of R, depending on the angle Twill be represented by R=m'k. cos. {T — W),
[1281], or R = m'k. cos. W. cos. T-\-mk. sin. W. sin. T, [24] Int., and if we put
k . sin. W= P, k . cos. W=^ P', we shaU get R = m' P . sin. T-\- m! F . cos. T, [12865]
as in [1287]. Applying the same notation to the expression [1286], it will become
e= !^?V:.f?V {cos. T.cos. ^+sin. T.sin. W},
moreover, if we use the values P, P', [12866], which give (t") *^^°* '^'^('T") ' [1286c]
(— ) . cos. W= i-j-\ it will become as in [1288].
696 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
[1288"] than P, P', the inequality of the excentricity may be considerable, and even
exceed that of the mean motion, if the excentricities e and e' are very small ;
we shall see some examples of this in the theory of the satellites of
Jupiter.
We shall now determine the corresponding inequality of the motion of the
perihelion. For this purpose, we shall resume the two equations,
[,289] ede^fM+ril, ^d.=fML-Lif,
which we have found in § 67 [1255]. These equations give*
^ ^ df = ^ . de . cos. TO — [t'C . dm . sin. « ;
hence if we notice only the angle i'lnlt — int-\-i's' — is — gzs — g'-a' — ^'^',
we shall find
[1291] j^_^/^ a.ndt. [-1-)' cos.-ss . sin.(iV^ — int-\-i'i' — is — g-si — g'z/ — g"6') — ii.e.dm.sm. w.
We shall put
[1292] ^m',andtA ^^^+A;' | .cos.(^'?^7 — zri^ + zV — ^s— ^to— ^to'— ^Y),
for the part of it^e . drs depending on the same angle,t and we shall get
df=m'.andt. \ (^^ + h^] .sin.{^V^— zw^+iV— f£--(g— l).^— ^^'— V'a'}
[1293] ( \«e/ >
— ^ '^^ — M' . mi.\i' r^ t — int-\-i'^ — is — (^ + 1).« — ^^. — g"^.
* (874) Multiplying the values of ede, e^d-m^ [1289] by / and — /', respectively,
and adding the products we shall obtain fede — /' e^d-si = — -~- • df, but by [1253',
1254], f^-\-f'^ =■ (m- c)^ /= (Ji- e . cos. zs, f'=fi.e. sin. -a, hence
[ue^ .de . cos. -a — [t^^ .dvi . sin. zs = — — - . df',
dividing by e^, and reducing, we shall obtain [1290]. Substituting in it the value of de,
[1284a] we shall get [1291].
f (875) It will be seen that this substitution tends to simplify the computation, by finally
rejecting the term A/, on account of its smallness, [1293']. If we put
T' — i'n'i—'int-\-iW^is—gzi-^g'z/—g"^,
II. viii. ^ 69.] SECOND METHOD OF APPROXIMATION. 697
It is evident, from the last of the expressions of df, given in § 67 [1257],
that the coefficient of this last sine has for a factor* e^+^. e'^. (tang. ^9)^' ; [1293']
k' is therefore of a higher order by two than ( j- ) » and if we neglect it,
in comparison with ( t- ) > we shall have
. ( 7- ) . cos.(i n t — int-{-t s — IS — g-ss — gvf — go).
[1294]
and substitute the value of f* e <?xs [1292] in [1291], we shall get
df=m' .andt. (~) • cos. ■n. sin. T' -\-jr^ .andt .\ (t")+^ ( . sin. -a . cos. T'j
putting, as in [18, 19] Int.,
cos. « . sb. r'= J sin. (T' + «) + J sin. (T' — -a),
sin. -s. cos T' = ^sin. (T' + w) — ^sin. (T' — «),
and reducing, we get
rf/= m'. antZ^. ^ (^)4-n'|. sin. (r' + «)—^—.^. sin. (T'-^),
as in [1293].
M876) Having (f) = (l +„,)-. Q, [962,952], f^) = (f), [963], [,««„,
and v = nt-\-z-\-v^, [953], we get by substitution in «?/, [1257],
'^/=— 7^ -^^ cos. (n <+£+«,)+! e. cos. 7tf+Je. sin. (2n^+2s—^ + 2«J?.^—)
/dR\ [^293al
in which the terms containing zi explicidy, are multiplied by e. If we now develop the terms
containing v,, according to the powers of r^, as in [678a], and then substitute the values of
u^, v^j deduced from [669], writing 1 + m, for - , as in [952], and ni-{-s -\~v,y for t?,
supposing the series [669] to be continued, as in [659, 668], to higher powers of c, and to
multiples of the angle •bj, it will appear that wherever the angle -a occurs, in the coefficient of
f— j, or (-7") J [1293a], it will be multiplied by the quantity e ; the term depending on [12935]
the angle 2 zi will be multiplied by a coefficient of the order e^, and in general the angle g -ss,
will be connected with a coefficient of the order e^. Again, from [961], the value R, as
175
698 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
for the term of ed'a, which corresponds to the term
[1295] m'k . COS. (i'n't — int-\-i'^ — is — g zi — ^ ^ — g!' ^') '■>
of the expression of i?. Hence it follows, that the part of w, which
corresponds to the part of i2, expressed by [1287],
[1296] ?w' . P . sin. (%' n't — int-\- i' s' — i s) -\- m' . P' . cos. (i' n't — int-^-i' s — i s),
is equal to
[1297] —TTn — ^T- '{{-r ]'Cos,(i'nt — int~\-u' — is) — ( -— - ].sm.(t'nt — tnt4-i's' — is) \ .
I*. (in — in).e (\dej \de J ^ -^ )
We shall therefore have, in a very simple manner, the variations of the
excentricity and of the perihelion, depending on the angle i'n't — int-\-i'i' — is.
They are connected with the corresponding variation of the mean motion (^,
in such a manner that the variation of the excentricity is*
1 fdd2^
[1^8] JTn'KjJdt
well as those of ( — ], ("t~)) must also have the same property, therefore df, [1293a],
must also have the same property, and the term of [1293],
which contains the angle (^-f- 1) . zs, must have for its coefficient a term of the order
e^+^ and as e' and tang. | (p' do not occur in the value of df, [1293tt], except as they are
found in R, these quantities must be of the same form in df as in jR, [961], namely,
e'^ . (tang, i (j/)^', according to the form assumed in [1293']. Hence the value of K must
contain the factor e^+^ . e'^ . (tang. ^ (f)^'. The other term of df, [1293], has for a factor
(— \ and this, by [1285], is of the order e^-^ . e'^ . (tang. | (p^*", in which the exponent
of e is less by two than in the other term A;' ; therefore we may neglect ^, and the value of
ihed-a [1292], will give e </ w, as in [1294], and if we use T, W, [1286a], we shall get
edvi= . ( -T- ) • cos. ( T — W)= •(-;-• (cos.T. cos. W-\- sm. T. sm. W)
fA \de / ' (A \de/ ^
mandt ( fdP\ . ^ /dF\ ^
[1286c]. Dividing by e and integrating, we get zs, [1297].
* (877) Putting in ^, [1 197], A = i'^ — is — gzi—g^z/ — g" 6', to agree with
[1281], and using T, TV, P, F, [1286a, 6, c], we shall find
n. viii. § 69.] SECOND METHOD OF APPROXBIATION. 699
and the variation of the longitude of the perihelion is
i'n' — in fd ^
o in . e \" e
The corresponding variation of the excentricity of the orbit of m', arising from
the action of m, will be*
1 fddr\
and the variation of the longitude of the perihelion of wi', will be
(i'n' — in) fdl''
2i'n' .e' ' \d7) '
[1301]
and as we have by ^ 65 [1208], ^' == — — -^.^, these variations
will be
Si
■'n'.m'.^'\de'dtj " 3i'n' . e'. m' ,vA' '\de'J' ^ ^
^ = -:r-r-r-^.^ -8111.(7— ?F)= — ^ — ^^.k.\^m.T.Qos.W—cos.T.sm.Wx ri298al
3i.m'o7i2 , ,
From this last we obtain
/d^\ 3i.m'an2 C /dP\ ^ /dP'\ . ^ )
\dedtj fJ'.(t'n'— tn) i\dej ' Vrfe/ >'
hence we easily deduce
{i'n'— in) /d2\ m'an C /dP\ _, /rfP'X . ^ )
-r-^ -• (3^ =-7^7-^ — ^-^-5 ("i-J-cos.T— ( — ).sin. r h
and the second members of these expressions are equal respectively to those of [1288,
1297], representing the corresponding variations of e, ts ; the first members must therefore
represent those variations, which is conformable to [1298, 1299].
* (878) The formulas [1300, 1301] are deduced from [1298, 1299], by changing
n, e, (^, e, &c., into n', e', ^, s', he., and the contrary, considering m' as the disturbed planet, [I300a]
and m as the disturbing planet ; and in order that the angle T may remain unchanged, it will
also be necessary to write — i' for i, and — i for i\
700 MOTIONS OF THE HEAVENLY BODIES [Mec. Cel.
When the quantity i' n' — in is very small, the inequality depending on the
[1302'] angle i' n! t — int, produces another sensible equation in the expression of
the mean motion, among the terms depending on the square of the disturbing
masses ; we have given the analysis of it in § Qb [1213, 1214]. The same
inequality produces, in the expressions of de and dns, some terms of the order
of the square of these masses, which are functions of the elements of the
orbits only, and have a sensible influence on the secular variations of these
[1302"] elements. For if we take into consideration the expression of de, depending
upon the angle I'tiI t — i n t, we shall find, by what has been said*
[1303] de= — ] . f — J .COS. (I'nt — int-\-i's—ii)—{ -j- \ . sm. (i'n7— mt + I's — ^s) \ .
From § Qb, the mean motion oi nt ought to be increased byf
[1304] ——, — '—-rz — . \ p. cos. (i 'n't — int + i V — i s) — P'. sin. (i 'n't — int + i 's — is)],
[I'n—my.li' * ^
and the mean motion n't ought to be increased by
[1305] — v^, — r^^ . , ^,_ . \ P.cosJi'n't — ^n^+^ £ — iB)—P,sm.(i'n't — i7it-\-i'^ — is) ] .
In consequence of these increments, the value oi de will be increased by
the functiont
'3>'ni .a^in^.At ,.,,_, ., ^, ( j. ( AF\ ^, fdP\\
* (879) The differenlial of the part of e [1288], relative to d t, gives d e, [1303].
f (880) In [1212] the increment of n < is ji--, — ^-^ . k . sin. {i'n't— int-^ A),
v(?hich is the same as ^, [1298a], and this was in [12986], reduced to the form
\P . cos. T — F' . sin. I \,
lt'.{i'n' — inp
as in [1 304] ; IMultiplying it by — ^7^> [1208], we shall get ^', [1305].
J (881) JMultiplying [1305] by i', [1304] by —i, and adding the products we shall get
the increment A of the angle T=i'n't — int-{-i'^ — i s, [1286a], arising from
these terms, which will be
II. viii. § 69.] SECOND METHOD OF APPROXIMATION. 701
and the value of d^ will be increased by the function*
^ a /- r. 1 r-^^. e.w'.v/^+2'.m.v/^ . \ i"- "j- ) + i^ • ( -7-" ) \ ' ti307]
2 iiT . ^o' .(in — my .e *■ ^ i. \" ^/ \ " ^ / )
[1306a]
A= — -r— — r-Tz — .— — r .\r.cos.T — I'. sin. 1 \ — -—, — t-^ — .\P.cos.T — r.s\n.r\
[in — inf.\i> m.\/a' ^ [i n — inf.fx * '
_— ^an^.i.{i'm.\/7i + i m' . \/Z'\ jp ^^^ T—P' sin. T\.
(i'n' — inf.it'.y/a! 'I ' ' ' ' S'
Now if we increase the angle Tby .^ in the expression [1303], it becomes.
de= ^ .^— j.cos.(r+^)-(^— j.sin.(r+^)j; [1306i]
from [60, 61] Int., neglecting w2^, we get
cos.(T+w3)=cos. T— ^.sin. T; sin. (T + ./2) = sin. T + w^ . cos. T; L1306c]
substituting this in [J 3066], and retaining only the terms depending on A, we shall find
..= l^-.^.^(lf).si„.T+(^.eos.T].
Substituting now the value of A, [1306a], and reducing the angles, retaining only the terms
independent of T, so that sin. T .\P . cos. T — P' . sin. T\, produces — ^ P', and
cos. T .{P . cos. T — P' . sin. T\, becomes \ P, we shall get
m' an.dt C — 3 an^
de = -
Jn'-inf.it..v^7 i'l ^ \dej'^^ \dejy
and if we arrange the terms differently it will become as in [1306].
* (882) Taking the differential of xrf, [1297], we get
and if we change, as in the last note, T into T-\-Ay and develop the expressions as in
[1306c], retaining only the terms depending on A, it becomes
^'.^.{-(f).oos.T+Q.si„.r|,
substituting the value of j1, [1306a], reducing the angles and retaining only the terms
independent of T, in the same manner as in the last note it becomes
(*« 'I (i'n'-tnp.M-.v/^' Si ^ \de) ^ \de)l'
which is evidently equal to the expression [1307].
176
^^^ MOTIONS OF THE HEAVENLY BODIES. [Mec. Gel.
we shall likewise find that the value of de' will be increased by the function*
and that the value of d ra will be increased by the function
Zixr .a .{i n — my.e*- * i \" ^ / \ " ^ / 5
[1309]
[1309'] These different terms are sensible in the theory of Jupiter and Saturn,
and in that of the satellites of Jupiter. The variations of e, e', w, and w',
relative to the angle i' n' t — int, may also introduce some constant terms,
[1309"] of the order of the square of the disturbing masses, in the differentials de,
de', dzi, and d^^, depending on the variations of e, e', w, w, relative to the
same angle ; it will be easy to take notice of them by the preceding analysis.
Lastly, it will be easy, by this analysis, to find the terms of the expressions
[1309'"] of e, o, e', and v/, which depend on the angle i'n't — int-\-i's' — is, but
have not i'n' — in for a divisor, and those which depend on the same angle,
and on the double of that angle, which are of the order of the square of the
[I309i»] disturbing forces. These terms, in the theory of Jupiter and Saturn, are of
* (883) These increments of d e', dzi', corresponding to the part of R, [129G], may be
deduced from those of de, dvs, by making the changes mentioned in [1300a], hence we get
[1307a]
, , 3m.a'2.i'n'3.rff .. , _ , ., /-> ^ z, fdP\ . ^ /dP'\}
^ - = 2]i^7^:F^?=i^^^
in which terms the factor — /~'~i ^^7 he put under another form; for n^a?=ii=M-]-m,
[605'], and if we neglect m in comparison with M, and put M=i 1, we shall get
n^ a^=zl = n'^ a'^, hence
a'^i'n'3 i'n'.{a'3n'^) i'n'.ja^n^) i' n' . a^ . \/a . n^
and as i n — in is very small, we may put i' n' = i n, and it becomes
a'^ . i' n'^ i n.a^ . y/o . n^ a^ . \/a . t w^
V/a a' a'
Substituting this in [1307a],' we get for dc', dz^, the values [1308, 1309].
II. viii. § 70.] SECOND METHOD OF APPROXIMATION. 703
sufficient importance to be noticed ; we shall develop them as far as it shall
be found necessary, when we shall treat of that theory.
70. We shall now determine the variations of the nodes, and of the
inclinations of the orbits ; for this purpose, we shall resume the equations of
§64 [1173],
dc' ^dt.\ z .( --—] — X.
^dx -^'KdzJS' [1310]
If we take notice only of the action of m', the value of i?, § 46 [913],
will give,*
'•(!r^^'''(!T^^^^^ [1311]
(884) The expression of jR, [913, 914], depending on m' is
D mf . (x of -{- y r/ -}- z z') m'
hence
^a/^j^y'^-^z'y {{a/^a^f + iy'-yf + iz'-zn^
/dR\ ^ m' x' ni .{x' — x) . [l^l^^l
and if we put
r= \ ,
it becomes < f — j = m' a/ . W^-}- ;n' a; . F", in like manner if we change a;, «', into y, y',
[iHOftJ
704 MOTIONS OF THE HEAVENLY BODIES. [MecC^l.
If we now put
[13.2] 'l^p. 7=9;
the two variable quantities p and q will give, as in § 64, the tangent of the
inclination 9 of the orbit of m, and the longitude ^ of its node, by means of
the equations*
[1313] tang. 9 = \/f 4- 52 ; tang. ^ = ^ .
We shall put p', c[, p", 9", &c., for what^, q, become, relative to the bodies
m', m", &c., and we shall find, from § 64,t
[1314] z = qy — p X ; zf = q'i/ — p' a/ ; &c.
and the contrary, (~j = m'y'. W-\- m! y . V', changing also x, x', into z, z', and the
contrary, we find f — j = mV . W-\- m' z . V. Substituting these in the first members
of the equations [1311], and then using [1310], we obtain
dc dd dc"
[131I0] ^ = »»'-(«'y — ^y)- ^; — = 7n'.(j;'2r — a?2/).?F; — =m' . (y'z — 2/*') . ^T;
the terms depending on V destroying each other. The remaining terms agree with the
second members of the equations [1311, 1311a].
* (885) The assumed values of j9, 5^, [1312], give c"=^pc, d=^qc, substituting
these in [1174] we get tang. (p = ^^2-fg2, tang.^ = -, as in [1313], which might
also be reduced to the form of those in [1032], tang, (p . sin. ^ =p, tang. 9 . cos. 6 = q.
[I312o] For the sum of the squares of these last gives tang. 9 = \/p^-\-q^, and if we divide the
first by the second we get tang. 6=-, therefore the values of p, g', [1312], are equivalent
to those in [1032].
f (886) The equation [579], 0 = c" x — cy-\-cz, being divided bye, gives
d d' d' d .
z=-.y .x; and if we substitute the values of -, -, [1312], it becomes
zr=^qy — p X, as in [1314], and from this we get z' = ^ y' — p'x', &c., by accenting
the letters in the usual manner.
U. viii. §70.] SECOND METHOD OF APPROXIMATION. 705
[1315]
The differential of the preceding value of p [1312] gives*
dp 1 C dd' — p dc )
~dt~ c' \ ~dt 5 '
substituting for dc, d c", their values, we shall getf
we shall also havej
dq
di
m' { 1 1 )
=-. I {p'-p).^x'-j-{q-^).a^ i 'I (^2+y 2+^2)f-" ^^^_^).+(y_y)._|_(^._^).^^|- [1317]
* (887) The differential of the first equation - =p, divided by dt is,
dp 1 Cdc^' ^ dc\
dt c 'idt c ' dty '
and if we substitute the value of - = p, it becomes as in [1315].
dc dd'
f (888) The values of — , — , [1311a], being substituted in [1315] it becomes
T7 = — -jy^ — y z! — P'{x'y — xy')^.W, and by usmg the values «, 2:', [1314],
^ = ~ '{y' '{qy — P^) — y • (?' y — P' x') — p • (*^ y — * y') } • ^> or by reduction
^=7-1(9'— 9')-yy'+(i>'—p)-«'y|-^, [i3i6a]
as in [1316].
X (889) The differential of the value of q, [1312], divided by d t, gives
dq 1 (dd d dc^ 1 ( dd_ dc^
dt c'idt c'dtS c'idt ^'Jty
dd dc
[1312]. Substituting the values of —, — > [1311a], we get
■^ = ^'\^z--xz'—q.{3[/y — xi/)\.W'f
and the values of z, «', [1314], make it
■^^='^.ja/.{qy—px) — x.{^y' — p'a/) — q.(x'y — xi/)}. W -,
which by reduction becomes
^=7-{Cp'— i^)-*«^ + (? — 9')-^2''l -^5 [13166]
as m [1317].
177
[1318]
706 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
If we substitute, for x, y, a/, i/, their values [1243], r. cos. v, r. sin.tJ,
r' . COS. v', r' . sin. v', we shall find*
(q-- q') -yy' + (p'-'P) ' 3f y = (^^ J .rr' ,\cos. (v' + v) — COS. (v' — v)]
+ (- — ^ j .r/.fsin. (v' -^v) — sin. («/ — v)\ ;
(P' — P)' x^+Cq — ^) '^y = [^-^}''''^'{^^^' (^' + ^) + cos. (v' — v)]
+ f^^Yrr' .{sin. (v' + v) + sin. (v' — v)}.
Neglecting the excentricities and inclinations of the orbits, we getf
[1319] r = a ; v^nt-\-s ; r' = a! \ v' = 7}! t-{-^ ;
* (890) The values x = r .cos.v, y = r.sin. v, [1243], and the similar values
a/ = / . COS. v' ', yl z=r' . sin. v' j give
y y' = r / . sin. V . sin. v' = \rr' . \ cos. {v' — v) — cos. («' + «>) | ,
and in like manner
x' y = rr' . cos. v' . sin. v =r.^rr'. | sin. {v' -{-v) — sin. {v' — v)]',
[1317a] xa/ =:rr' . cos. « . cos. v' =^ r /. | cos. {v' + f ) + cos. {v' — v)];
xy =-rr' . cos. v . sin. v' =^rr' .\sm. {v' -f- v) -4" sin. {v' — v)].
Substituting these in the first members of [1318], vee shall obtain the second members of the
same equations.
f (891) The expressions [1319] may be deduced from [1268], neglecting w,. m/, v^, v',
vehich are of the same order as the excentricities. Neglecting z, z\ which are of the order
of the inclinations, and substituting the values of x^ y, x', y', of the last note, v^^e shall get, as
m[1224J], a;'2 4-2/2_|_^'2^^/2_|_y2^^'2^ and
(a/— xf + {y'— yf + (;2' — zf = (a?'— xf + (y'— y)2= r^ — 2 r / . cos. («'— v) + /^
and if we use the values [1319], we may put x''^ -\-l/^ ~\- ^'^ = <*'^
(a;'— xf + {if—yf ■\-{z!—zf==a'—2aa'. cos. («' ^ — n < + e' — s) + a\
substituting these in the first member of [1320], or in W^ [1310&], it becomes like the
second member of that expression j hence if we use the values [1319, 1321], this expression
[1320a] of W will become W = -^ — i^ 2 . -B^'' . cos. i.{n' t — w < + ^ — 0*
ILvUi. §70.] SECOND IVIETHOD OF APPROXIMATION. '^^'^
hence we obtain
1 111 |-1320]
we have also, by ^ 48 [956],
3 = i2. 5«.cos. i.fn't — ni+s' — 0- [^32i]
ja2 — 2aa'.cos.(n'i — ni + s' — £)+a'2j5 ^
The symbol 2 includes all integral values of z, positive or negative, also the M32in
value i = 0 ; therefore, if we neglect the terms of the order of the squares
and products of the excentricities and inclinations of the orbits, we shall
find,*
* (892) The terms p^ q, p', q', being of the same order, as the inclination of the orbits
[1313], we may, if we neglect the square of these quantities, substitute in [1318], the
values r, v, r, v', [1319], and if we put T= n' t — nt -{-^ — s, we shall find
r/-\-''y = n't-\-nt-\-^ + s = T-\-2ni-{-2s, i/ —v — n' t — nt -{- ^ — s=T.
and the expressions [1318] will become
{q — q).yj/ + {p'—p).oo'y = h'{^—q)'aa''{cos.{T+2nt+2e)—cos.T]
+ -|-(p' — ?)•««'• fsin.(T+2n^ + 2e) — sin. T];
[p'— p).x a/ -\-{q — ^).xi/ = i.(^—p). a of .{cos.{T-\-2nt-{-2s) -{-COS. T}
+ i.{q'— q'). a a' .{sm. (T4-2 nf + 2 s) + sin. T];
substituting these in [1316a, J], and using W, [1320a], we shall get,
[13206]
[1320c]
^ = (^=py m\a a!. {cos. {T+2nt + 2s) — cos.T] A -^--^^.B^.cosAtI
+ (^^Ym\a(^.\sin.{T+2nt-{-2s) — sm.T].i-^--i:s,BKcos.iTl;
^^ = (^^ym'.aa'.{cos.{T+2nt-\-2s)-{-cos.T].\-^---ii^.B^'^.cos.iTl
-\-(^-^A.m\aaf.{^n.{T+2nt + 2s)J^sm.T}.i-^--i^.R'>.cos.iT\ .
The factor — of the expression of ?F, produces in these values of — , — , the same
terms as are found in [1322], independent of ^'^ The terms depending on B^'^ can be
amplified, observing that by the formulas [954c, 6], we get
708 MOTIONS OF THE HEAVENLY BODIES. [Mec.Cel.
, (?'— 9) . . , mi) S cos.[(t + l).(n'^-n^ + s'-£)]
4c ^—cos. [(i+l).(n'i — w^+s'— £) + 2w^+2s]
, iP'-p) ^ ^^ 2 ^(0 \ sin.[(i+l).(n'^-n^ + a'_a)] .
4c * * * * •^— sin. [(i+l).(n'^— n^ + s'— £) + 2n^ + 2£]^ '
[1322]
, (;>—/) / / r»«. ^ cos.[(i+l).(n'^— ni + s' — s)] ^
, («/— <7) , / -^>r^ ^ sin. [(i + 1) . (n'^ — w^ + s'— s)] ^
The value 2 = — 1, gives, in the expression of -j-. the constant quantity*
[1322'] ii — g/ ,fpi .ad , B^~^^ ; all the other terms of the expression of -^, are
4c * at
COS. (r+2n <+'2s) . ^2 . ^') . COS. ir=|2 .S('\ COS. j (i+1) . r+2 n < +2ej;
COS. T . I 2 . 5» . COS. i T= 1 2 . ^« . COS. (^ + 1 ) . T;
sin. (r + 2 n ^ +2 s) .1 2 . ^('> . cos. i r= | 2 . J?^'^ sin. j (i +1) . T-{- 2 n < + 2 s j ;
sin. T. ^ 2 . 5(^> . COS. i T=^ 2 . £(«> . sin. (i + 1) . T;
hence, by substitution, we obtain the terms of [1322], depending on jB^'^
* (893) Because the term cos. {i-{-l) . [n' t — nt -\-^ — s), then becomes 1, and
the part of ^ 2 . B^*^ . cos. (i + 1 ) . T, becomes | JB^~^^, or J ^^\ which produces
in ^, ^, [1322], the parts depending on B^^^ in [1323, 1324].
II. viii. § 70.] SECOND METHOD OF APPROXIMATION. 709
periodical : if we denote their sum by P, and observe, that J5^~'^ = jB^^\ [1322"]
§ 48 [956'], we shall get
^ = l^Illi.-^'.«a'.jB(i)4_P. [1323]
dt 4c
In the same manner if we denote by Q, the sum of all the periodical terms of [1323']
the expression of — ^, we shall find
d^^{p_-zJl . m\aa'.B^'^^q. [1324]
at Ac
If we neglect the squares of the excentricities and of the inclinations of the
orbits, we shall get, from § 64,* c = ^jra *, then supposing fji'= 1, we shall [1324^
1 . m' .aa' . B^^^
have n^a^=l ; hence c== — , and the quantity — '-— will
Ct 71 4 C
become — '- j — , which, in § 59 [1130], is equal to (0,1) ;
therefore we shall get,
^ = (0,l).(5'-g)+P;
[1325]
^ = (0,1). (;,-/)+«.
Hence it follows, that if we put (P) and (Q) to denote the sums of all the [1325^
functions P and Q, relative to the action of the different bodies m', m", &c.,
on m ; if we likewise put (P'), (Q'), (P"), (Q"), &c., for what (P) and (Q) [1325"]
become, by changing successively the quantities relative to m, into those
relative to m', m", &c., and the contrary ; we shall have, to determine the
* (894) From the first equation [1313], p, q, are of the order of the inclinations,
therefore c', c", [1312], are of the same order, and if we neglect quantities of the order of
the square of the excentricities and of the inclinations, the equation [1175] will become
ixa = c^, or c=\/'ir^, as in [1324^, and if we put |x = l, we shall get c=v/a"; but m324o-]
from [605'], \/o = — , hence c= — , as above : substituting this in — . ,
'■ ^ an an 4c
it will become *"' ' ^^ "| ' ^ -^'^ = (0, 1), [1130], and the equations [1323, 1324], will
become as in [1325].
178
710 MOTIONS OF THE HEAVENLY BODIES. [M^c. Cel.
variable quantities, p, 9, p', 9', p", q", &c., the following system of differential
equations,
^=-{(0,1) + (0,2) + &C.} .9 + (0,1) .9' + (0,2) . /+ &c. + (P)
d q
d
[1326]
dt
^ = {(0,l) + (0,2) + &c.}.;7-(0,l).y-(0,2)./-&c. + (Q)
dt
^=-~;(l,0) + (l,2) + &c.}Y + 0,0).g + (l,2).^' + &c. + (P')
pL = J(l,0) + (l,2) + &c.}./-(l,0)./?-(l,2)./'-&c. + (Q')
&c.
The analysis of § 68, gives, for the periodical parts of /?, q, p\ q[, Silc.*
P=f(P)-dt; q=f(Q).dt;
p'^f{P).dt; q'=/(Q').di;
we may obtain the secular parts of the same quantities, from the integration
of the preceding differential equations, after effacing the last terms (P), (Q),
[1327'] (P'), &c. ; by which means they will become like the equations (C), § 59,
[1132], which we have already discussed, with much care, so that it will
not be necessary to say more on the subject.
71. We shall resume the equations ^ 64 [1174],
[1328] tang, (p = '^ ~ ; tang. 0 = -7 ;
c c
hence we shall obtainf
(f c"
[1329] - = tang. <p . cos. 6 ; - = tang. ? . sm. 6 ;
* (895) If in the equations [1275], we change A, A', h", &c., into q, q', q", &;c.,
I, r, r, he, into p,p',p", &c., {X), (F), (X'), he, into (P), (Q), (P), he, they wiU
become lilce [1326] ; and if we malte the same changes in the equations [1277], we shall
obtain [1327]. The metiiod of finding the secular equations [1279"] is the same as
in [1327'].
t (896) Already found in [590, 591].
n.vm.^71.] SECOND METHOD OF APPROXIMATION. 711
taking the differentials, we get*
d . tang, (f = - ,^d c' . cos. 6-\-dcf' . sin. 6 — d c. tang. cp\ ;
I [1330]
J^.tang. 9 = -.^dc". COS. 6 — dcf . sin. 6],
c
dc dc dc"
If we substitute, in these equations, for — , -j-, -3-, their values [1310],
at at at ^
/dR\ /dR\ /dR\ /dR\ /dR\ /dR\
y\-d^)-'\-di)^ '\-d^)-''{j7)^ '\-di)-y\-d-^)'^ [13301
and for these last quantities, their values given in ^ 67 [1244] ; observing
also that s = tang. <p . sin. (v — 6) [679], we shall getf [1330"]
* (897) The differentials of [1328] are
COS.24 C' C'2
Multiplying the first member of this last equation by cos.^4 . tang. 9, and the second member
by - . COS. 6, which is equal to it, by the first of the equations [1329], we get
d 6 . tang. 9 = - . < rf c" . cos. 6 7 . cos. 6 .dcf >;
c"
but the second of the equations [1328], gives — .cos. fl= tang. d. cos. d^=sin.d; substituting
this we get the second of the equations [1330]. Again, if we substitute in the equations
[1329], the value of tang.9 = \^2!±^, [1328], they become - = V^5Z±£?. cos.5,
_^v2E±Zl.sm.6; hence ^;^=|=cos.^, and ^^^^^^ = sm.^, these
values, and that of ^ — ^^— -, given in the first of the equations [1328], being substituted
in d . tang. 9, [1329a], it becomes like the first equation of [1330].
* (898) If we substitute in the equations [1310], divided by dt^ their values computed
in [1244], we shall obtain
dc /rfiTv
dt \dv)'^
^=-(l + .2).cos.t,.(^ + r,.cos.t;.(^-5.sin.t;.(^; [1330a]
^ = -(l + ^).sin.t;.(^) + r*.sin.t;.(^4-,.cos.t,.(^;
dd'
d\
712 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
J , d t. tang. (p. COS. (v— 6) C /dR\ . . . , /^dR\ . . >
A . tang. 9 = , '[r- [j-^).sm.(v-^) + (^^J. cos. (v—s) ^
^^.tang.<p= , \ r.{j-^yin.(v-6) + {^-ycos. (t;~0 ^
[1331]
. . . . . dd' dd .
substituting these m — . cos. ^ — — . sm. 6, and connecting together the terras depending
°" (^)' (^)' (^)' ^^^hallget
dc" , dd . ^ ,t I 9N /<^-R\ < • A • .>
— .cos. ^ — — . sin. ^= — (1 ~r s ) ' [ — ) • Jsin. V. cos. d — cos. v . sin. 61
Oft dt \(i s /
-|-?*5 . f — j . {sin. u.cos. 6 — COS. v . sin. 6\ "i"* •("T") -{cos. v . cos. & -|~sin. r . sin. ^| ;
multiplying by dt, and reducing, by means of [22, 24] Int., we get
dc" . cos. d — dc'. sin. 6 = ^ (I 4- s^) . dt . sin. (v — 6). (-A
[1331a] \ds/
-\-s dt . sin. (u — 6) . r . ( — j -{- s d t . (—) . cos. (z? — 6),
and as dc', dc", [1330a], do not contain 5 explicitly, we may change Hnto 6 — J -r, ^ir
being a right angle ; hence we shall obtain
d c" . sin. 6-^dc'. cos. 6 = — {l-{-s^).dt. cos. {v — 6). (^~\
[13316] /dR\ /dR\
-i-sdt.cos.{v'-6).r.i^— j — sdt.i^— y sin. {v — 6).
s is the tangent of the latitude [1242""], which is equal to tang, (p . sin. {v — 6), [679].
Substituting this in the two last terms of [1331a], and the result in the second of the
equations [1330], we shall get the second of [1331]. The same value of s being
substituted in the two last terms of [13316], and the resulting value, together with that of
--^ = — (-7-), [1330a], being substituted in the first of the equations [1330], we shall get
d . tang. 9 = -
— (i -\- s^) . d t . COS. {v — 6). ("T~) + ^^ .tang. 9. sin. {v — 6)
X cos. (« ■— 5) . r . (^— ^ + T-— y rf < . tang. <p . [— sin.2 (v — ^) -}- 1 ]
and if we put — sin.^ (v — d) -|- 1 = cos.^ {v — 6), it will become like the first equation
[1331].
n. viii. '^ 71.] SECOND METHOD OF APPROXIMATION. 713
From these two differential equations we may determine directly the
inclination of the orbit, and the motion of the nodes ; they give*
sin. (v — 6) .d , tang. 9 — d6. cos. (v — 6). tang. 9 = 0 ; [1332]
an equation which may also be obtained from 5^ tang. 9. sin. (ij — 6) [1330"].
For this equation being finite, we may, as in § 63 [1167"], take its differential,
supposing <p and 6 to be constant, or we may consider both these quantities to [1332^
be variable ; therefore the differential of this value of 5, supposing 9 and 6
only to be variable, must be nothing ; hence results the preceding differential
equation.
Suppose now the inclination of the fixed plane to the orbit of m to be
extremely small, so that we may neglect the squares of s and of tang. 9, we
shall findf
J . dt . ^ /dR\
a . tang. 9 = cos. (v — V • ( "T" ) '
a 6 . tang. 9 = . sm. (v — V • ( "3" ) 5
and if we again put, as in [1312«],
p = tang. 9 . sin. 6 ; q = tang. 9 . cos. 6 ; [1334]
* (899) Multiplying the first of the equations [1331] by sin. {v — 6), the second by
— COS. (« — 6), and adding the products, the terms of the second member destroy each
other, producing the equation [1332].
f (901) Substituting z = rs, [1243], in [951], we get
[1333]
R
m
' . \r r' . COS. {v — v)-^rsz'] m'
C/2 I ^'SnI {r^—2ri'.cos.{i/—v)-i-t^-\-{z'—rsfji '
[1333o]
so that (~r~]j ("7~)» are of the order w' ; these quantities are multiplied, in [1331], by
tang. 9, which may also be considered as of the order m', the fixed plane being the primitive
orbit of m, therefore these products are of the order m'^; and if we neglect them, and also
the quantity s^, which is of the same order, the two equations [1331] will change
mto [1333].
179
714 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
we shall obtain the following equations, instead of the two preceding [1333],*
, dt /dR\
da = .COS. V . — — :
c \ds J
[1335] ^ ^
, dt . /dR\
[1335'] Now we havef s = q . sin. v — p . cos. v, which gives
'dR\ 1 /'dR\ /'dR\ 1 /dR
[1336]
dsj sin. v'\dq J ^ \ds/ cos.?; \dp
* (902) The differentials of [1334] are dp = sin. 6 .d . tang. (p-\-dd . cos. d . tang, cp,
dq= — dd. sin. 6 . tang, (p -|- cos. d . c? . tang, (p, which, by means of [1333], and using
[21,23] Int., become
dp = — — .{sin.d. cos. {v — ^)4-cos. ^ . sin. {v — 6)]. (-7-)
=__.sm.{^ + (.-^)|. (-)=--. sin.. .(-j,
dt r . . , V , / .X) /dR\
dq = .| — sm. 6 . sm. [v — 6) -\- cos. 6 . cos. [v — ^) ] . [-r )
dt .^ . . ^,, /dR\ dt /dR\
as in [1335].
f (903) The value assumed above for s, [1330"], is
s = tang. <p . sin. {v — ^) = tang, (p . | sin. v . cos. 6 — cos. v . sin. 6 1 ,
which by means of [1334] becomes s = q. sin. v — p . cos. v, as in [1335']. Now j?, q,
do not occur in R, [1331a], except through s, therefore
/^\ _ /dR\ /dj\ /dR\ _ /dRX /d_s\ ^
\dq)~\ds)'\dq)' \dp)~\ds)'\dp)'
and the preceding value of s gives ( — j = sin. v; ( 7~ ) "^ — ^°^* ^ ' hence
,„.?N /dR\ . YdR\ /dR\
/dR>
\dq,
as in [1336] ; substituting these in [1335] we get [1337]. If we take the primitive orbit of
m for the fixed plane, we may, as in [1824a], neglect c% c"^, being of the order of the
square of the disturbing mass, and the expression [1175] will give c=\/fr^ . ^1—e^, but
il335a] from [605'] , n^a^ — fx, hfince /^ a = w o^ = -^ = — , and if we put ft = 1 ,
n. viii. § 71.] SECOND METHOD OF APPROXIMATION. 715
therefore
dt /dR\
, dt /dR\
[1337]
We have seen, in § 48 [949'], that the function R is independent of the
position of the fixed plane of x and y ; supposing therefore all the angles of
that function to be referred to the orbit of m, it is evident that R will be a
function of these angles, and of the inclination of the two orbits to each
other, which inclination we shall denote by 9/. Let ^/ be the longitude of [1337]
the node of the orbit of m', upon the orbit of m ; suppose also that
m' k . (tang. 9/)= . cos. (i' n' t — int-\-A — ^5/), [1337"]
is a term of jR, depending upon the angle i'n't — int ;* we shall have, by
it will become \/(x a = — , hence c = — . s/l—e^- This value being substituted in [1337al
(t ft CL 7h
r.^o»T 1 11 ^V andt /dR\ dq andt /dR\
[1337], we shaU get - = _j7=.(-j; _ = -^^.^_^; [133,j,
which are the same as the formulas [5790, 5791], in the appendix to the third volume, being
accurate in terms of the first order of the disturbing forces.
* (904) This term of R is deduced from [961], observing that g" = 0, because the
fixed plane is the primitive orbit of m, so that the inclination and longitude of the node of mj
upon the fixed plane, must disappear, if we neglect terms of the order of the square of the
disturbing masses ; and the term of R becomes
He' . e'<^. (tang. | <p')«"' . cos. {i'n't — int -\-i' ^ — is— gi^i — g' zs' — g'" (f].
To conform to the preceding notation we must change 9' into 9/, and ^ into ^/ and if wr out
Hc« . e'*' = 2«"' . m' A:, i' ^ — is — gzi — g'z/ = A, this term will become, [1337cl
m'k.{2. tang. J (plY" . cos. {i' n' t — int-\-A—g"' 6/],
and by neglecting, as above, the cube of 9, we may put 2 . tang. ^ 9/ = tang. 9/ ; lastly
to simplify the notation, we may put g instead of g"', and it will become
m! k . (tang. 9/)^^ . cos. {i'n' t — int -{-A — g fl/),
as in [1337"].
^16 MOTIONS OF THE HEAVENLY BODIES. [Mec. C61.
§60 [1144],*
[1338] tang. (?/ . sin. ^/ = p'—p; tang. ?/ . cos. ^/ = ^ — ^ ;
whence we deducef
(tang. ,:r . sin. g V = i'>'-9 + (p'-p)V=ii--W-V-{p'-p).,/^\^ .
[1339] ^
(tang. ,;y. COS. ^.; = J9--? + (/-rtV^ij^+h'-v-(/-,)V=Ti^^
Noticing therefore only the preceding value of R, we shall havef
f ^^ ==--g. (tar\^, cp;y-\ m' k. sin. {i'n't — int+ A — (g--l).yi;
[1340] ^^^^
( — j = — g . (tang. %y-^ .m'k. cos. { ^'n'/ — int-\-A — {g — 1) . ^; | .
* (905) The equations [1 144], tang. 9/ = y'{p'—pfJ^{(^—qf ; tang. ^/ ='j-^ ,
are of the same form as those in [1313], and may be derived from them, by changing
jp, q, 9, 6, into p' — p, q' — q, (?/, ^/, respectively. The same changes being made in
the equations tang, cp . sin. 6 =p, tang. 9 . cos. 6 = q, deduced in [1312a], from the
equations [1313], we get the expressions [1338].
f (906) IVfultiplying the first of the equations [1338] by ± v/^j and adding the
product to the second, we shall get
(tang. 9/) . {cos. 6,' ± sin. 6,' . \/^^} = q'— q ± {p'—p) • s/'—i J
raising this to the power g, and using [15, 16] Int., we shall find
(tang. 9/)^ . \ COS. g dj ± /=! • sin.^ ^; j = j ^'_ ^ ± (p'—p) . v/=T j ^ ;
taking the sum and difference of these two equations, depending on the different signs db,
we shall find
(tang. 9/)^ . 2 . cos. g6;={q'- q + (p'^p) .^/=i|^-}-{^_gr — (p'_p) . v/=T^;
{t2^^.(p!y.2.^/:ri.sm.g6;={q'-'q + {p'—p).v'=^Y—W—9--{p'—p)V-i]'y
dividing these by 2, and 2 . v/^^j respectively, we shall get the expressions [1339].
[1340a] t (907) If we put T = i' n't — int-\-Jl, we shall get, from [24] Int.,
COS. {in't — int-{-A —g&',) = cos. {T—g &',) = cos. T. cos.g6j + sin. T. sm.g 6^,
hence the term of R, [1337"], will become
R = rnk. (tang. 9/)* . \ cos. T . cos. g 6,' + sin. T. sin. 5- ^/} .
II. viii. <^ 71.] SECOND METHOD OF APPROXIMATION. ^^'^
If we substitute these values in the preceding expressions of dp and dq^
observing that we have, very nearly,* c = — , we shall obtain [1340^
Substituting the values [1339], we shall get,
hence
4 W kg .{[q'—q + (p'—p) .v/=i]^^+ y—q— {p'—p) • \/-l]^^| • sin. T,
and
/dR
m' kg
^) = -.jm'A:^.{[^-5+(y-p).v/=ri]-i+[^--5_(y_p).v/=ri]«-i|.cos.r
Changing g into g — 1, in the formulas [1339], we shall obtain the values of
W—q + {p'—p)-i^^V-'±W-'q—{p'—p)V-il'~'.
hence by substitution we shall get
\^)'=^'^S'{^^^- O'^'-sin. {g—iy6;.cos.T— m'hg. (tang. c^D'^K cos. (^— 1) • V- sin.T,
\Jq)==—'''^^S\^^'^^'^lT"-<^Q^-{g—^)'^!'C0S.T--m'kg.{i2.u^.^^^^^
If in the first we substitute for sin. {g — 1 ) . «/ . cos. T — cos. {g — 1 ) . d/ . sm. T, its value
— sin. \ T— {g — 1) .&l\, [22] Int., and in like manner, in the second, for
— cos. {g — 1 ) . V • COS. T — sin. {g — 1) . 4/ . sin. T,
its value — cos. { T — {g—i ) . ^ f , [24] Int. they will become
(^) =-"*' ^^ • (^^"S- <P;)^-^. sin. { T- (g-~l) . ^;|,
(^) = — *^' '^^ • (^^°S- <p;r^ COS. \ T- {g—i) . bll
resubstituting T= i' n' t — int-\-A, [1340a], they will become as in [1340].
* (908) Neglecting the square of the excentricity, we shall have, as in [1335a], c= — ;
an
substituting this, and \-T~\ (rf~)' [1340] in [1337] we get
dq^-S-rr^k.an.dt ^ ^^^^^^ ^,^^,^ ^.^^ \i! n' t-int -\- A- {g-l) . d/} ;
^^ g-m'k.an.dt f „, , ,., , . , ^ , , ,, [1341o]
dp = 6 (tang. (p;)'^-». cos. {t'n'< — ini+^ — (^— 1).0/} ;
whose integrals give g-,;?, [1341]. i..
180
■718 MOTIONS OF THE HEAVENLY BODIES. [Mcc. Gel.
[1341]
[1342]
s^ . tn k . an , /\„i . r 'i i , • ,, a , -s
^ = ;r(7^^=:i^ • ^*^"^- ^'^ • ''''^' S* ^' ^-^ ^ ^ + ^ - (^ ~ 1) . ^;i.
Substituting these values in the equation s = q. sin. v — p . cos. v [1335'],
we shall get*
This expression of 5 is the variation of the latitude, corresponding to the
preceding term of R [1337"], and it is evident that it is the same, whatever
be the fixed plane to which we refer the motions of m and ?w', provided the
[1342'] inclinations of the planes of the orbits to the fixed plane be small ; we shall
thus obtain the part of the expression of the latitude, which becomes sensible
by means of the smallness of the divisor i' "d — in. It is true, that this
inequality of the latitude contains only the first power of that divisor, and it
[1342"] must, on this account, be less sensible than the corresponding inequality of
the mean longitude, which contains the square of that divisor ; but on the
other hand, tang. <?/ is not raised to so high a power by unity ; which
is analogous to the remark we have made in ^ 69 [1288'], upon the
[1342"'] corresponding inequality of the excentricities of the orbits. Hence we see
that all these inequalities are connected with each other, and with the
corresponding part of i?, by very simple relations.
If we take the differentials of the preceding expressions of j? and q [1341],
and then, in the values of -/-, -p, thus obtained, augment the angles nt^
at at
and n't^ by the inequalities of the mean motions, depending on the angle
[I342iv] i' ri! t — int\ there will be produced, in these differentials, some quantities,
* (909) The values p, q, [1341], being substituted in s, [1335'], it becomes.
-g.m'k.an , „ ( sm.{i' n' t — in t-]- A — {g-\) .&',]. cos. v^
s = ,., , • , ' (tang, cpy-' .< };
''•(^"-^") (—cos.{i'n't--int + A — {g—]).6;}.s\n.v)
the terms between the braces may be reduced to sin. {i' n't — int-{-A — v — {g — 1)«^/|>
£22] Int., by which means the value of s becomes as in [1342].
[1343]
U. viii. §72.] SECOND METHOD OF APPROXIMATION. 719
which will be functions of the elements of the orbits only,* and which may
have a sensible influence upon the secular variations of the inclinations and
of the nodes, although they are of the order of the square of the disturbing
masses ; which is analogous to what we have said in § 69 [1302', &c.], upon
the secular variations of the excentricities and of the aphelia.
72. It now remains to consider the variation of the longitude s of the
epoch. We have, by ^ 64 [1188],
rf s = rf e . / ( — — J . sm. (y — '^)-\-h-[ —i — J • sm. 2 . (v — to) + &c. v
— d^.{E^'^ . cos. (v—vs) + E^^^ . COS. 2 . (i? — «) + &c.} ;
substituting for E^^\ E^^\ &c., their values in series arranged according to
the powers of e, which series may easily be deduced from the general
expression of £^'^ § 16 [541 ],t we shall find
* (910) Substituting the value of A, [1337c], in the angle i'n't—in t-\-A—{g—l)Jl,
it becomes {i' n t — int-\-i'^ — is) — \[g — 1) • ^/ +5"^"l"i§r''^}j and if we put
i' n't — int-{-i's' — is = T, {g — 1) . 6^ -{- g zi -{- g' z/ =^ W, it will change into
T—W, and since sin. [T—W) = cos. ^. sin. T— sin. W. cos. T,
COS. {T—W) = cos. T. COS. ^+sin. T. sin. W,
the values of dq^ dp, [1341a], will become, by putting,
(77) = -^ ^ • ^^^"S- <?/?"' • COS. W', (~-^ =gk. (tang, (p/)^^ . sin. W ;
, m'andt ^ /dP"\ _, /dP'\ . ^>
^i=^--\ icir) ■ ■=-• T- (-.7) ■ - ^ ■'
which are of a similar form to that of d e, [1303], and if we increase the angles n U n't,
by the expressions [1304, 1305], respectively, the increment of the angle T will produce, in
the preceding values of d q, dp, terms similar to those in [1306], which depend upon
the elements of the orbits only.
f (91 1) If we neglect e^ we may put v/^=p =1— 1 e^, in [541], hence
720 MOTIONS OF THE HEAVENLY BODIES. [Mec. Gel.
ds=i — 2de . sin. (v — w) -|- 2 e . t? ^ . cos. (v — w)
[1344] +cc?e.{| + ^e^+&c.i.sin.2.(iJ— ^)— e^£/^.{|+ie'4-&c.;.cos.2.(tJ~^)
+ &C.
If we substitute, for de and e d'ss, their values, given in § 67 [1258], we
shall find, by retaining only quantities of the order e^ inclusively,*
a^ .ndt
a 3 ^ IjJUl^ . ^HZia. {2 — I e . cos. (i, — «) + e^ cos. 2.{v—^)], (^\
and the denominator ^z:^i = 2'.(l-ie2)i = 2^ * ^ ^ + ^ * ^^^ ' therefore
£(')=
± e' . (i + 1) ± e'+2. ^ i . (i_- 1) ^
putting successively i=l, 2, 3, 4, &c., and using the signs, as in [541, Sic], we obtain
E(" = — 2e— fee; £(2)^|g2^|g4__sjc,. JE^^^ = — e^ — &c. ; hence
substituting these in [1343) we get [1344].
* (912) Neglecting terms of the order e*, in [1344], we get
ds== de . { — 2 . sin. {v — zi) -\-^e . sin. 2 . (u — zs) — e^ . sin. 3 . {v — w) |
-\- e d-a .\2 . cos. {v — Ts) — | e . cos. 2 . (« — •») -|- e^ . cos. 3 . (« — -sj) | .
Substituting the values of <? e, e <?•!*, [1258], we find in cZs, terras multiplied by
andt /dR\ , a^ndt ^ /dR\
=z=r.{-7-), and • v/l^^ • ( -7- )•
The factor of . (-7~)> as it first appears, without reduction, and putting for
brevity x = v — zi, is
(2 . cos. a; -j- e -f- e . cos.^ x) . ( — 2 . sin. a? -f- f e . sin. 2 a; — e^ . sin. 3 a?)
~\- (2 . sin. a? + e . sip. x . cos. a?) . (2 . cos. a? — f e . cos. 2x-\-e^ . cos. 3 a?).
II. viii. § 72.] SECOND METHOD OF APPROXIMATION. 721
The general expression of d s contains some terms of the form
m' k.ndt . cos. (^' n' t — int-\- A) ;
therefore the expression of ? contains terms of the form
m'kn . .., ,, • . , ^\
. sm. (t nt — int-\-A) ;
[1345']
in — tn
multiplying these factors together, and arranging the terms according to the powers of e,
observing that the terms independent of e mutually destroy each other, the product will become
e .[3 . (sin. 2 a; . cos. x — cos. 2 x. sin. a?) — 2 sin. x . (1-j- cos.^a; — cos.^a?)|
-\-e^. \ — 2. (sin. 3 x . cos. x — cos. 3a; . sin. a;)-}-|. cos. x .(sin. 2a? . cos. x — cos.2a;.sin.a;)-|-|.sin.2a? ]
=6.53. sin. (2a; — x) — 2.sin.a?j-(-^^-{ — ^2. sin. (3 a; — a?)-|-f .cos.a;.sin.(2a; — a;)-j-f. sin. 2a; |
=e . { 3 . sin. x — 2 . sin. a; | -f~ ^^- { — 2 . sin. 2 a; -{- f • cos. x . sin. x-\-§ . sin. 2 a; | ,
and since sin. 2 a; = 2 . sin. a: . cos. x, this will finally become
e . sin. X -{-^e^ . sin. x . cos. a?,
resubstituting for x its value v — ts, we shall get the coefficient of . ( — ) ,
fx.v/i-e2 \dvj
as in [1345].
Again, the factor of . y/l— e2 • (~7-)> in the expressions of ds abovementioned is
— sin. X .( — 2 . sin. a;-f | e . sin. 2a; — e^. sin. 3a;)-|-cos. x .(2. cos. x — |e . cos. 2a;-f-e^. cos. 3a;),
which, being arranged according to the powers of e, is
2.(sin.^a;-j-cos.^a;) — | e .(cos. x . cos. 2 x -f sin. x . sin.2a;)-|-e^.(cos. 3 x . cos.a;-{-sin. 3a?. sin.a;)
= 2 — |e.cos. (2 a; — x)-{-e^. cos. (3a; — a?) = 2 — § e . cos. a? -f e^. cos. 2 a?
= 2 — I e . cos. {v — to) -|- e^ • cos. 2 . (« — «),
as in [1345].
In the appendix to Vol. III. [5787], it is shown thatrfs is expressed by the following
formula, which includes all terms of the first order of the disturbing force, /ji. being equal to
unity,
7^ andt.\/Tir^ ,. . , /dR\ , ^ „ /dR\
ds=.^ ~ (^""V/l-e3)-(77J + 2a2.(^— j.TiJ^ [1.344a]
181
722 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
but it is evident, that the coefficient Ar, in these terms, is of the order* i — z,
therefore these terms are of the same order as those of the mean longitude,
depending on the same angle, and as these have for a divisor the square of
[1345"] i' n' — in, it is evident that we may neglect the corresponding terms of s,
in comparison with them, if i' ni! — i n be a very small quantity.
If in the terms of the expression of ds, which are functions of the elements
of the orbits only,t we substitute, for these elements, the secular parts of
* (913) Substituting in [1345], for (—^ and ('—V their values [1292a],
(1 -f-M,)~^« (-T-)j and ("T")} also v = nt-\-s-\-Vi, it becomes
ds=z^^.^lZ^^.\2—^. e . cos.{nt^z—'ui-^v)-\-e^. cos.(2n^+2£— 2^+2t?J |.( 1 +w,)-^ (~\
f* \da/
[1345a]
andt . , I 1 \ r I 1 /I I SI fdR\
•~f^./i=^ • ^ • ^^"- {nt-\-^—^-\rv)-\l-\-^e. cos. [nt + s — zi^v)]. ^^— J.
This expression oi ds is similar to that of df, [)293a], as it regards the order of the
coefficients, and it will appear, from what is said [12936], that this value of c?£ has the same
property, relative to the form and order of the terms, as the value R ; now in [957^'"], it is
shown, that if the value of R is composed of terms of the form m' k . cos. (i' n' t — int-\-jl),
the coefficient k will be of the order i' — i, therefore the part of d s, depending on this
angle will be of the same order, and its integral will give s, of the order i' — i, with the
divisor i' n' — in. But the part of ^, [1197], depending on the same angle, is of the order
i' — i, and has the divisor {i' nf — iny, therefore this part of ^ must be much larger than
the corresponding part of s, if i' n' — i n be very small.
f (914) To give an example of the manner in which such terms are formed, we may
take in the value of d s [1345a], the term
[13456] -j-.yT:=rr^.Ue.cos.{nt-Jrs^zi + v,)].u^.i^~j;
in which cos. (w^-j-f — 'us-\-v^), being developed [61] Int., has for its chief term
COS. {nt-\-s — ttf) ; multiplying this by w^ = — e . cos. (n < + ^ — ''^)5 [1010], we
shall get — ^ e — ^ e . cos. 2 . {nt-\-s — -zs), and if we retain only — | e, and put
1 for v/l— e2, in [13455], we shall obtain the term . | e^ . (-t-)- Now the first
term of ("7~)> deduced in [957], from i = 0, is f-f-T — )) therefore tZs contains
U. viii. §72.] SECOND METHOD OF APPROXIMATION. 723
their values, it is evident that there will be produced some constant terms,
and other terms depending on the sines and cosines of the angles, on which [1345'"]
the secular variations of the excentricities and inclinations of the orbits
depend. The constant terms will produce, in the expression of s, some
terms proportional to the time, which are included in the mean motion of [1345'v]
m. As it respects the terms affected with the sines and cosines, they will
acquire by integration, in the expression of s, very small divisors, of the same
order as the disturbing forces ; and as these terms are both multiplied and
divided by these forces, they may become sensible, although they are of the [1345 v]
order of the squares and of the products of the excentricities and inclinations.
We shall find, in the theory of the planets, that these terms are insensible ;
but they are very sensible in the theory of the moon [3543], and of the
satellites of Jupiter, [Book VIII], and it is upon these terms that their [1345^1]
secular equations depend.
We have seen in ^ Q>b [1195], that the mean motion of m has for
3
expression -//« .ndt.dR, and if we notice only terms of the order of [1345^"]
the first power of the disturbing masses, d R will contain only periodical
quantities [1197']. But if we consider the squares and products of these
, a^ndt
the terra . %
m . f — — j, which is a function of the elements of the orbit [1345c]
only ; A being a function of a, a', [954]. If we substitute in e^ = h^-\- P, [1108], the
values of h, I, fee, [1 102, 1 102a], we shall obtain e^ = £2 _|_ j . O . cos. (^ ^ + p) ;
JE? being the constant term of e^ and g a term of the same order as g, g^^ kc., [1 102], or [l345d]
of the order of the disturbing forces. Substituting this in [1345c], we shall obtain in ds,
the terms — ^^-^ . f m' . (-^— ) .{IP + ^.C. cos. {g t + ^)l and by integration [i345e]
we shall get the corresponding part of
in which the first terra is proportional to <, and is comprised in the mean motion ; the last
terras are divided by ^, which is of the order m', [1345d], so that they are both raultiplied [1345/]
and divided by terms of the order m', which might render them sensible. It is shown
however, in Book VI, that in those planets where this ought to be most sensible, they are not
of any importance.
^'^ MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.
[laio^Hi] masses, this differential may contain terms which are functions of the elements
of the orbits only.* If we substitute in them the secular parts of the values of
these elements, there will result some terms affected with the sines and cosines
of the angles, upon which the secular variations of the orbits depend. These
terms will acquire, by the double integration, in the expression of the mean
motion, very small divisors, which will be of the order of the squares and of
[I345i''] the products of the disturbing masses ; and being both multiplied and divided
by the squares and products of these masses, they may become sensible,
although they are of the order of the squares and of the products of the
[]345x] excentricities and of the inclinations of the orbits. We shall also find that
these terms are insensible in the theory of the planets.
73. The elements of the orbit of m, being determined in the preceding
manner, we must substitute them in the expressions of the radius vector,
longitude, and latitude, which we have given in § 22 [659, 668, &c.] ; we
[1345"] ^^^^^ *^^^^ obtain the values of these three variable quantities, by which
astronomers usually determine the positions of the heavenly bodies. If we
develop these expressions in terms of sines and cosines, we shall get a series
of inequalities, which we may arrange in tables, and by this means we may
compute the position of m, at any given time.
This method, founded upon the variation of the parameters, is very useful
in the investigation of those inequalities which in certain ratios of the mean
[i345"'l motions of the bodies of the system, acquire great divisors, and on that
account become very sensible. Inequalities of this kind chiefly affect the
elliptical elements of the orbits ; therefore if we determine the variations of
the elements arising from these inequalities, and substitute them in the
rij45«in expression of the elliptical motion, we shall obtain, in the most simple
manner, all the inequalities which these divisors render sensible.
f (916) That is, terms similar to tlie quantity computed in [1345c], which may be
reduced, in like manner as in [loA5d, e], and the double integral being taken, the coefficient
%£ the term depending on the angles of the form {gt-{- j3), will be of the order of the
square of the masses, divided, on account of the double integration, byVg^, which divisor is
also of the order of the square of the disturbing masses, [1345/]. These terms, notwith-
standing the smallness of their divisors, are insensible, as is shown in Book VI.
n.viii. §73.] SECOND METHOD OF APPROXIMATION. 725
The preceding method is also useful in the theory of comets. These
bodies are visible only in a small portion of their path, and observations
furnish merely that part of the ellipsis which coincides w^ith the arc of the [1345*'']
orbit they describe during their appearance ; therefore if we determine the
nature of the orbit, considered as a variable ellipsis, we shall obtain the
changes in this ellipsis during the interval of two successive appearances of
the same comet ;* and we shall thus be enabled to predict its return, and [I345»v]
upon its reappearance, we may compare its theory with the observations.
Having thus given the methods and formulas, to determine, by successive
approximations, the motions of the centres of gravity of the heavenly bodies,
it now remains to apply these formulas to the different bodies of the solar
system ; but the ellipticity of the heavenly bodies having a sensible influence [i345xvi]
upon the motions of several of them, it is proper, before we make the
numerical calculations to examine into the forms of these bodies ; a subject
which is quite as interesting as the theory of their motions.
* (917) This method is explained in Book IX.
182
APPENDIX, BY THE TRAJfSLATOR.
DEMONSTRATIONS OF THE FORMULAS OF THE INTRODUCTION.
The formulas [49 — 59] are found in almost every book treating of the differential calculus,
and they may be easily demonstrated. Thus if we put x for the cosine, y the sine, and t the
tangent, of an arch of a circle z, whose radius is 1, we shall have cc^ -\-y^=\, whose
y dy
differential gives dx = . Substituting this in \/rfx2-[-rfi/2, which expresses the
differential of an arch of any plane curve, whose rectangular ordinates are x, y, we shall get, [1345J]
X d X
by reduction, the expression [49]. The substitution of dy = , gives in like
manner [50] ; the negative sign being prefixed, because x decreases when z increases.
The expressions [49, 50] are equivalent to [52, 53]. Substituting these values of d .sin.z.
d . COS. z, in the differential of tang. z= — — , we get
° COS.Z °
. COS. 2 . rf . sin. z — sin. z . rf . COS. z rfz.(cos.2z+sin.2z) dz . . , « ,
d.tang.2r= == — ^ -L -'= — - =rfz.(l+tang.2z),
C0S.2z C0S.2z C0S.2z ^ O /'
as in [54] ; hence dz = ^ ■ ^"^'^ = -^j^ > as in [51], If we develop the formulas
[49, 50, 51], according to the powers of y, x, t, respectively, and take the integrals, so as to
commence with z = 0, we shall obtain [46 — 48].
U x = log. y, we shall have x-{-dx=^ log. {y + d y). Subtracting the first from
the second, and observing that the difference of the logarithms of two numbers is equal to
the logarithm of their ratio, we shall have
dx = log. (3/ + ^ y) - log. y = log. (^) = log. (l + ^).
Developing this, by Taylor's theorem [617], according to the powers of — , and retaining [1345l!|
only the first power, it will be of the form dx = (log. 1) + a • — , a being a constant
quantity, representing tlie modulus of the logarithms, and as log. 1=0, it will be
d,x== a . — . If a=l, It will become dx=-^, [^9], corresponding to
y y
hyperbolic logarithms.
728 APPENDIX, BY THE TRANSLATOR.
If we put y = 1 + a?, in [59], we shall get
d . hyp. log. [l-{-x) = — ■— = dx — xdx-\- x^ dx — &;c.,
[13453] whose integral is [58]. If we suppose hyp. log.y=2r, and multiply it by l=hyp. log.c,
we shall get hyp. log. y = z hyp. log. c = hyp. log. c% hence y = c% and dy=d.c';
but from [59] we have dy = y .d . hyp. log. y==ydz = c^dz, or d .c^ = c' dz,
as in [57].
Changing x into ±:z m [607c], we shall obtain [55,56], From [p01d,e] we get
[43,44]. Dividing [43] by [44], the resulting equation is [45]. Putting ±z\/'^, for
X in [607c], we get
= i+^v-i-i:^-r:^.v/=T+f:^i+&^c.;
[13454]
Hence,
^ ~^ =«— r^T-Ts + T-lT-^-r-r — ^c. = sin.;r, [607cZ] ;
2.^IIl ~" 1.2.3 ' 1.2.3.4.5
[13455] ^ .
2 1
^ + TTiTiTi "" ^^' "" *'°^" '^' ^^^'^'^*
These expressions of sin. ar, cos. ^r, are as in [11,12]. If we multiply [11] by
±\/=T, and add the product to [12], we shall get the formulas [13, 14]. Raising these
to the power w, we shall obtain [15, 16].
If n be an integral number, and we raise the formula [12] to the power n, and connect
together the positive and negative powers of c, which have the same exponent, we shall
obtain an expression of the following form,
- . C n2./=l , -~nz.s/—i\ , -n r { (n— 2).z./— 1 , -(n-2).z./=I J
[13456] +C.^{c("-^)-'-^^+c-(^-^^-'-^'^} + &c.
= A . cos. nz-\-B . COS. (n — 2) .z-{- C . cos. (n — A).z-\- Sec,
and if we put successively n = 2, 3, 4, 5, 6, we shall get the formulas [6 — 10]. If in
these we change z into z — ^ *, -n- being the semi-circumference of a circle whose radius
is unity, we shall obtain the formulas [1 — 5].
From [1 1] we get sin. a = ^ t/ZTf » ^**°* ^"^ 2~v7— 1 *
SPHERICAL TRIGONOMETRY.
729
The product of these two expressions is
sin. a . sin. h = \ .
2
-i
2 ''
= i . cos. (a — b)-\-l . COS. (a -|- &),
as appears from formula [12]. This is the same as [17]. Changing & into i'!f-{-b, we
get [18, 19]. If we change a into iir' + ^j '^^ C^^]' ^^ ^^^^^ S^t [20].
The sum of [17,20] is [24], and if we change the sign of b, it becomes as in [23].
Putting a — ^'Tf for a in [23, 24], we shall get [21, 22]. Changing a into ^ .{a-{- b),
b into ^ . (a — 6), in [18], and multiplying the result by 2, we get [25]. Putting in this
— b for b, it becomes as in [26]. Increasing the angles a, b, [25, 2G], by a right angle
and observing that sin. 4 . (a — b) = — sin. | . (6 — a), we obtain [27, 28].
sin. a . COS. & -f-cos. a . sin. 6
Dividing [21] by [23], we get tang. {a-\-b) =
COS. a . COS. 6 — sin. a . sin. 6 '
dividing
also the numerator and denominator by cos. a . cos. b.
. . sm. a
and putting — -=tang.«,
-1- = tang, b, we obtain [29]. Changing the sign of b it becomes as in [30]. Putting
b = a, in [29], we get [30^, &c. Making b = a, in [21,23] we get [31,32].
Substituting, in [32], cos.^ a = 1 — sin.^ a, sin.^ a := 1 — cos.^ a, we obtain [33, 34],
Dividing [26] by [25], [26] by [27], [25] by [27], [25] by [28], [28] by [27],
we shall get the formulas [35 — 39] respectively. Putting a = 0, in [39] we get [40],
also 6 = 0, in [36] produces [41, 41 n, and changes [38] into [42, 42']. Formula [60]
is derived from [678a] ; changing in this z into i * -{-2^, we get [61].
SPHERICAL TRIGONOMETRY.
All the formulas of spherical trigonometry, used in this work, may
be derived from the theorem [172i], and it has been thought expedient
to investigate, in this appendix, some of the most important of these
formulas, and to give the following additional demonstration of this
theorem.
Let ABC be a spherical triangle, described on the surface of a
sphere, whose centre is D, and radius DA^l, Draw the lines
DBB', DC C, to meet the lines A B, A C, which are drawn
through the point A, tangents to the arcs AB, AC, respectively,
and forming the plane triangles D B' C, A B' C. Then the angles
of the spherical triangle, being denoted by A, B, C, and their opposite
183
B.-
\ xC
^^^ APPENDIX, BY THE TRANSLATOR.
sides by «, h, c, respectively, we shall have DA = l, D C' = sec. h, A C' = tan*, h •
[13457J D B' = sec. c, AB' = tang, c ; angle B' A C = spherical angle BAC = A,
angle 5' D C = a ; and from [471] or [62] Int. we shall have, in the plane triangle
BAC,
J5' C" 2=^5' 2—2 ^S'. ^ C. cos.5'^ C '+.^ C'2=tang.2 c— 2 tang, c . tang. J.cos.v24-tang.86,
and in the plane triangle B' D C, we shall have, in like manner,
B C'^=^DB'^^2DB' .DC'.cos.B'DC'-\-DC'^
= sec.2 c — 2 sec. c . sec. h . cos. a -\- sec.^ b,
subtracting the first expression of B'C'^ from the second, and reducing, by putting
sec.2 c — tang.2 c = I , sec.^ b — tang.^ 6=1, we shall get
0 = 2 — 2 sec. c . sec. b . cos. a -j- 2 tang, c . tang. 5 . cos. j1.
Multiplying this by | cos. c . cos. 6, and putting cos. c .sec. c=l, cos. c . tang. c = sin. c,
&;c., we shall obtain the formula [172i], 0 = cos. c . cos. b — cos. a -f- sin. c . sin. b . cos. A,
or COS. a = cos. A . sin. 5 . sin. c -{- . cos. 6 cos. c. If in this we change each letter of the
triplets, a, b, c'j A, B, C; into the next in succession, recommencing the series, when the
letter c or C is to be changed, we shall get the following system of equations,
A.
[134531 COS. a = cos. A . sin. b . sin. c -\- cos. b . cos. c,
^ /A
[13459] cos. b = COS. B . sin. c . sin. a -j- cos. c . cos. a,
c
[I34510] COS. c = cos. C . sin. a . sin. b -f- cos. a . cos. b.
rrii c . f .r .• • /I cos. a — cos. J . cos. C r
Ihe first 01 these equations gives cos. A= — — . — . — . , hence
^ ° sin. 0 . sin. c
1 zi o • Q 1 /I (cos. & . COS. c + sin. 5 . sin. c) — cos. a cos. (5 — c) — cos. a
1 — COS. A = 2 sin.2 * ./2 = i ! — , — : ^- = ^-^ — ^.
sin. 0 . sin. c sin. 0 . sin. c
2 sin. ^ (g — b-{-c). sin. | (a -f ^ — c)
sin. b . sin. c '
and if we put s = \{a~{'b-\-c), we shall get,
rno.nn e.-^ 2 1 4 _ siu. j (tt — & + c) . sin. j (a + 6 — c) _ siu. (^ — &) . siu. (^ — c)
[134511] sm. 2^ sin.6.sin.^ sin. 6. sin. c
The same value of cos. A gives
, , /» o iJi /J cos. « — (cos.J.cos.c — sin. J. sin. c) cos. a — cos.{b-\-c)
1 +COS. .^ = 2cos.^iwi = ^^ ■ , ■ ^-=^ ■„ . ■ \
' sm. 6.sm. c sin. 6. sin. c
2 sin, i {b -\- c -{- n) . sin. \{b-\-c — a)
sin. h . sin. c
SPHERICAL TRIGONOMETRY.
731
hence
^„ 2 1 /? _ sin- i ' (b + c + a).sin.^.(6 + c — g) _ sln.g.sin. (^ — «) [134512]
•^ sm. 6 . sin. c sin. 6 . sin. c
Dividing the preceding expression of sin.^ ^A by that of cos.^ J A, we get
sin.i.(a — b-{-c).s\a.h.{a'{-b — c) sin. (^ — 6) . sin, (j? — c)
tang. ^•^=gin.^,(5_|_c4-a).6i^iT{6 + c^~~ d^. « . sin. (« — a j" * [134513]
These three formulas to find | A, are very convenient for the use of logarithms.
If we multiply together the formulas [1345"'^^], putting
sin.2 1 A . cos.2 iA = (sin. I A . cos. i Af = {\ sin. Af,
, ,, ^ 1 • Q /I sin. 5. sin. (5 — a) . sin. (.? — J) . sin. (* — c) ^x. .,. ,. ,
we shall get, t sm.^ A = ^^ / , , .% '- ^^ '- . Dividmg this by
° ' sin.2 6 . sin.2 c ^ •'
^ sin."* a, and taking the square root, we shall find,
sin..4 2 . y/sin.s .sin. [s — a).sin.(* — 5).sin.(« — c)
sin.a sin. a . sin. h . sin. c * [1345 ]
The second member of this expression is symmetrical in a, J, c, and without altering its
value, we may change A, a, into jB, i, or C, c, and the contrary ; by which means we
shall obtain,
sin. A sin. B sin. C
sin.a sin. 6 sin.c ' [134o'5]
which Is the well known formula, that the sine of the side is proportional to the sine of the
opposite angle.
Substituting cos. c, [1345'°] in [1345®], and then putting cos.^J = l — sin.^6, we
shall get,
cos. a = cos. A . sin. 5 . sin. c + cos. h . {cos. C . sin. a . sin. h -\- cos. a . cos. h\
= cos. A . sin. h . sin. c + sin. a . sin. h . cos. h . cos. C + cos. a. (1 — sin.^ b).
Rejecting cos. a, which occurs in both members, with the same sign, and dividing by sin. b,
we shall obtain the first of the two following equations ; the two others being deduced from
this, by changing successively, each letter of the triplets a, b, c, A, B, C, into the next in
succession, as before.
cos. a . sin. b a=i cos. A . sin. c -|- sin. a . cos. b . cos. 0, [134516]
cos. b . sin. c =: cos. B . sin. a -}- sin. b . cos. c . cos. A^ [T345i~]
cos. c . sin. a = cos. C . sin. b -j- sin. c . cos. a . cos. B. [134518]
Substituting In the first of these equations, the value sin. c = — ' , [13451^ :
sin. .4 1- ■* ^
^^^ APPENDIX BY THE TRANSLATOR.
COS. a COS. .^
J* 'J* 1 • • uus. t* COS .//
dividing by sm.a, putting -- = cot.«, and -J=:cot.^, we get the first of
the following equations, from which the other two may be derived, by the change of letters,
as above.
[134519] cot. a . sin. h = cot. A . sin. C + cos. b . cos. C,
l^^^~^] cot. b . sin. c = cot. B . sin. .4 + cos. c . cos. .^,
[134521] cot. c . sin. a = cot. C . sin. B + cos. a . cos. 5.
If we change A, a, into O, c, and the contrary, in [I345i'], we shall get
cos. b . sin. a = cos. B . sin. c + sin. 6 . cos. a . cos. C
Substituting this value of sin. a . cos. b, in [1345'^], it will become
cos. a . sin. b = cos. A . sin. c + cos. C . (cos. B . sin. c -f- sin. b . cos. a . cos. C), or
COS. a . sin. 6.(1 — cos.^ C) = cos. .^ . sin. c + cos. B . cos. C . sin. c.
Substituting in the first member 1 — cos.^ C = sin.^ C, and using
sin. b . sin. 0= sin. c . sin. 5, [1345^^],
the whole will become divisible, by sin. c, and we shall get
COS. a . sin. C . sin. B = cos. A + cos. B . cos. C,
which is the same as the first of the three following equations, the second and third being
derived from it, by changing the order of the letters as above,
1 134522] cos. A = COS. a . sin. B . sin. C — cos. B . cos. C,
[134523] COS. B= cos. b . sin. C . sin. A — cos. C . cos. A,
[I34521] COS. C = COS. c . sin. A . sin. B — cos. A . cos. B.
The whole of spherical trigonometry is comprised in the formulas [1345^^^], but in some
cases it will be convenient to use an auxiliary angle, in the manner hereafter to be explained.
If we compare the two formulas [1345^'^], we shall find that they become identical, by
[134525] changing o, 6, c. A, B, C, into •;!' — A, ir — B, if — C, it — a, it — b, ir — c,
respectively, * being two right angles. We may therefore substitute the one of these
men?at triauglcs for the other. This second triangle is called the supplemental triangle. The sides
and angles of the first being changed respectively into the supplements of the angles, and the
supplements of the sides of the second. This is a well known property of spherical
triangles.
Supple-
mental
triangle.
SPHERICAL TRIGONOMETRY. "^^^
If JBbe a right angle, the equations [1345^'^^' ^~. 23, 21, 22 j^ ^ju give the six following equations, Rec^tan-
comprising the whole of rectangular spherical trigonometiy. f^gonom-
etry.
cos. b = cos. a . cos. c, [134527]
. - sin. a sin.^ [134528]
SiU. U . — -^ J
tang, c = tang, b . cos. j1, >
tang, a = tang, b . cos. C, j
cos. b = cot. A . cot. O, ^--^ " 1 [13453«]
cot. C = cot. c . sin. rt, >
cot. A = cot. a . sin. c, 3
cos. .4? = sin. C . cos. a, ^
COS. C = sin. A . cos. c. S
[134529]
[134531]
[134532]
In several cases of oblique trigonometry, it will be necessary to introduce a subsidiary angle
to facilitate the computation by logarithms.
1 . If 6, c, C, be given to find a, we may assume the auxiliary angle a, so that
tang, a' = cos. C . tang, b ; [1345^3]
which by putting tang, b = — '— , will give cos. b . tang. a'= cos. C . sin. b. Substituting
this in [1345^"], we gel
, , , . , . COS. 5 t • J • f t \ cos.&.cos.fa — a!\
COS. c= COS. 6 . (tang. a . sm. a + cos. o) = , . ( sm. a. sm.a + cos. a . cos.a )= ^ -;
hence,
, ,. COS. c. COS. a'
COS. (a — fi ) = ; :
^ ' C0S.6
from which a — a', and then a may be determined.
The same process answers for the case where J, «, C, are given to find c. For having
, , / 1 1 COS. 6. COS. (a — a')
computed a, we get a — a, and then cos. c = ; . [13453'^]
It is evident from [1 345'-^' 33] that this subsidiary angle cl is equal to the segment CP of
the base, formed by letting fall the perpendicular A P upon the base B C. [134536]
If we change A, a, into C, c, and the contrary, in [1345^], we shall get the following
formula, to determine B by a, b, C, cot. b . sin. a = cot. B . sin. C + cos. a . cos. C,
, -. cot. 6. sin. a ,>, ^ , . . ■, ^
nence cot. B = r -— cos. a . cot. C. SubsUtuting cot. 0 = cos. C . cot. a ,
184
^^- APPENDIX. BY THE TRANSLATOR.
[1345^3], it becomes
cot. B = cot. C . (cot. a' . sin. a — cos. a) = -r-^ . (cos. a' . sin. a — sin. a' . cos. «)
' sm.a ^ ^
cote .
= -: — ; . Sin. [a — a),
sin. a ^ '
therefore,
[134537J ; cot. B = <^°t.C.sin.(a-«') ^
sin. a'
2. If J, A, C, are given to find c, we shall get, by changing, in [13452°], c, C, into 6, 5,
and the contrary, cot. c . sin. b = cot. C . sin. A + cos. b , cos. ./2 ; or
cot. C.sin.w2 , ,
cot. C= : — [- cot. O . cos. .^. -
sm. 0 -4
Now assuming the subsidiary angle w2', such that
[13453S] cot. .4' = COS. b . tang. C ;
we shall get cot. C = cos. J . tang. w2', hence,
COS. 6 . tano". j3' . sin. ^ , , . , , ^. . ^ ,
cot. c= — -rf- cot. 0 . cos. Ji=: cot. 0 . (tang. A^ . sm. .^ -f cos. A)
<^°*'* /•/!/• /I I /I ai\ cot. fc.cos.(.^— .^')
= T . (sm. A'. sm. v2 + cos. A . cos. A) = ^-, >
cos. w3' ^ ' ^ cos.^'
that is,
cot?>.cos. (./2— w?')
[1.345^9] cot. c = ^ .
■" cos. A
If &, c, C, are given to find A, we may use the same subsidiary angle w2', and then the
preceding formula will give,
[ 1345^0] cos. {A — A') = cot. c . tang, b . cos. A' ;
from which we may compute A — A' ; with which, and A', we shall obtain the value
of ^.
It may be oliserved, that the subsidiary angle A' is the same as the angle CAP, formed
by the side C A, and the perpendicular A P, let fall upon B C.
If c, A, B, are given to find C, we may take an auxiliary angle, such that
cot. A" = cos. c . tang. B, whence cos. c . sin. B = cos. B . cot. A''^
substituting this in [13452'*], we get
cos. C = cos. B . (sin. A . cot. A" — cos. A)
COS. J? , . ^ ^,, ^ . ^„. COS. B. sin. (A— ^'}
ri3454il = -: — :;;. • (sm. A . cos. A' — cos. A . sin. A ) = : — — ,
■■ ■• sin. A ^ ' sin..^'
and it is evident that the subsidiary angle A", is the same as the angle BAP.
SPHERICAL TRIGONOMETRT. 7^5
If it be required to find B from J, ^, C, we may use the subsidiary angle A', [1345^^],
from which we shall get also A — A'. Now if we substitute, in [1345^^], the expression
[1345^], cos. h = cot. A' . cot. C, we shall get
COS. B = cot. A' . cos. C . sin. A — cos. C . cos. A = cos. C . (cot. A' . sin. A — cos. A)
COS. C ai • a • ai a\ cos. C. sin. (.4— ./3')
= - — — . (cos. A . sm. A — sm. A . cos. A) = ' ,
hence,
„ COS. C . sin. (^ — •^)
COS. JB := r-^ . ti;M5421
If b, B, C, were given to find A, we must find A', as before, and then
■fa at\ COS. B. sin. .^
sm. (^ —^ ) == — ^-^ , [134543]
from which we may compute A — w2', and thence A.
From [1345^31 ^^ get ,ang. J ./2 = — ^ r— — ^— 7- ) , and by changmg A, a,
*• -^ ° ° \ sin. « . sin. (.? — a) / ^ o o » ?
into 5, 6, and the contrary, which does not affect 5 = | . (a -f- ^ + c), we find
, ^ /sin. (« — a), sin. (* — c)\i
tang. ^ 5 = ^ ■ , \, ) . [134544]
° \ sin.*, sin. (« — 0) / *■ ■■
sin is '•'" c\
The product of these two expressions is tang. \ A . tang. J J? = — '- ; and if we
change B, 6, into C, c, and the contrary, it will become tang. | A . tang. | C = - '. -■- ; [1345451
again changing A, o, into B, b, and the contrary, we shall obtain from this last expression
tang. J jB . tang, i C = — . . Taking the sum and difference of these values, we
sin. 5 '
shall get
/ t » I 1 T»\ 1 r^ sin. {5 — &)4-sin. (j — a)
(tang. hA + tang, i B) . tang, i C = — ^ ~-r^ ■ »
, 1 ,» 1 -nx 1 /-I sin. (5 — h) — sin. (a — a)
(tang. I ^ — tang. I B) . tang, i C = — ^ '- ^^ ^ ,
1 + tang. \ A '. tang. \B =
1 — tang. J A . tang, i 5 =
sin.s [134546]
sin.s-f"sin.{s — c)
sin. s '
sin.5 — sin,(s — c)
tang. Mrb tang. ^jB
Substituting these values in the expressions tang. h 'iA-±.B) = — ^^ — .
^ ^ 8 2 V =e:^; 1 rp tang. i^. tang. as'
736 APPENDIX BY THE TRANSLATOR.
[29, 30] Int., we shall get
^ /■ /» I T>\ 1 /-. slnAs — 6)4-sin.(* — a)
^^^^^ .a„s.J.(^ + B)..a„g.JC=— -J±„;^-',
.ang.4.(^-B) ..a„s.i C = ?i5Ji--^^tf^-JL),
° ^ / o Sin. *-|-9in. (« — c)
but from [25, 26] Int. we have
sin. {s — 6)-f-sin.(s — a)=2sin.|.(25 — a — J).cos.|.(cf — h) =2sin.^c. cos.J. Ca — b),
sin. 5 — sin. (s — c) = 2 sin. ^ c . cos. ^ .{2 s — c) = 2 sin. ^ c , cos. J . (a + 6),
sin. (5 — 5) — sin. («— a) = 2 sin. J. (a — i). cos. J. (2 5 — a — &)=2sin. J.(a — &).cos.^c,
sin. s -\- sin. (s — c) = 2 sin. | . (2 5 — c) . cos. | c = 2 sin. | . (a -{" ^) • ^^s. i^ c.
Substituting these in [1345'*'''], and rejecting the factors common to the numerators and
Napier»g denominators, we shall obtain the following formulas of Napier,
formulas. •
1 / /» I -nx 1 ^ COS. i.(a — b)
(1345«] tang. i.{A + B). tang, i C = -;^'^^; ,
['345«] tang. J . (.4 — B) . tang. J C =
sin. i .(a — h)
sin.^.{a'-['b)
If in these formulas, and in [1345^^'""], we change the values as in [1345^], so as to
correspond to the supplemental triangle, we shall easily obtain, by slight reductions, and
putting S=l . {A-{-B -\- C), the following expressions, of which tlie two first were
discovered by Napier,
[134550J tang. ^a + b). cot. i c = Z'^aIb) '
formulas. . ^
[134551] tang. 1 (« - 5) .cot. h c = '^^-:^^^ ,
», cos.K-^— ■S+C).cos.i(.^+J5— C) cos.{S — B).coUS—C) '''^'
[134552] COS. i«= sin. 5. sin. C = sin. i?. sin. C '
rvo.«3i sin n « __-cos.K^ + ^ + C).cos.Hi^ + C-^)_- cos. S .cos.{S-A)
[134553] sin. ia- sin. 5. sin. C " ^inr^T^hTC '
2^ _~cos.^f^ + .B4-C).cos.^(^+C— ^)_ —cos. 5. cos. (5—^)
[134554] tang, ^a— cos.i(A — B + C).cos.h{'^-i-B—C)-~'cos.{S—B).cos.{S—Cy
If in the preceding formulas we suppose a, b, c, to be infinitely small in comparison with
the radius of the sphere, or unity, we may put sin. a = fl', sin.Z>=Z', sin. c=c,
COS. a = 1, cos. b= If lang. a = a, &tc., and we may, by this means, obtain several
SPHERICAL TRIGONOMETRY. 737
me
[134511-^5], the following,
theorems of plane trigonometry. Thus if s = J (a -}- 6 -|- c), we shall get from piane trig-
onometry.
sin.2 1 A = js-h) {s-c) ^ ^^^^^
be
cos.2i.^=£jLii=L£),
be
[134556]
tang.2 ^A = i'i:^}AL-:^ , [134557]
s.{s — a)
sin. A = ^'\/^-i'-<i)'is-b).{s-c) ^ [mS^T]
be
s'm.A sin.B sin. C
If we retain the second powers of a, b, c, we may put, as in [43, 44] Int., sin. 5 = J,
sin. c = c, COS. a = l — J a^, cos. 6=1 — J 6^, cos. c = 1 — ^ c^.
Substituting these in [1345^], it will become
1 — Ja2=:6c.cos.^ + (l— |62),(i__ic3)^ ^134558-,
whence by reduction cos. A = — -^ , which is the same as [62] Int. In like
manner we may obtain other formulas.
Many other combinations of the angles ^ A, | JB, | C, | a, ^h, J c, may be
found in several works on Trigonometry. Gauss published several formulas of a nature
somewhat similar to lliose in [1345^^~5ij^ which he has often used, though the common
formulas would answer the same purpose, and sometimes be shorter. Delambre in his
Astronomic, Vol. I, p. 164, observes that he had given several of these theorems in
the Connoisance des Terns, 1808, before the publication of the work of Gauss, and that he
had suppressed the demonstrations, supposing the theorems would not be very useful.
Considering tlie remarks of Delambre as essentially correct, and wishing to abridge this part
of the work, I have not inserted any of these formulas, which may however be easily
demonstrated, if it should be found necessary, by the methods here given.
For the more easy recollection of the formulas of spherical trigonometry, Lord Napier Napier»»
supposed a rectangular spherical triangle to consist of Jive circular parts, namely, the two parts.
sides, the complement of the hypotenuse, and the complement of the two oblique angles, which ["134559]
he named adjacent, or opposite, according to their position with respect to each other. In
this method the right angle is not considered as one of the circular parts, neither is it supposed
to separate the sides. In all cases two of these parts are given to find the third. If the
three parts join, that which is in the middle is called the middle part ; if they do not all
join, two of them must, and that which is separate is called the middle part, and the other
185
738
APPENDIX BY THE TRANSLATOR.
[I3456OJ two opposite parts, as in figures 4, 5. Then putting the
radius equal to unity, the equations given by Napier will
become
[134561]
Sine of middle part = Rectangle of the tangents of the adjacent parts
=^ Rectangle of the cosines of the opposite parts.
It may be of assistance in remembering these rules, that
the first letters of the words adjacent and opposite, are the
same as those of the second letters of the words tangents
and cosines, with which they are respectively combined.
The demonstration of the equations [1345^^] may easily
be obtained, by applying them to the preceding formulas
Thus in [ 1 345^^] the complement of the hypotenuse \'rc — h,
is the middle part, and a, c, opposite parts, as in fig. 4 ;
J* being a right angle. In [1345^^], second formula,
if we put, as in fig. 5, I* — C, for the middle part,
a and Jir — h will be the adjacent parts. In [1345^"],
fig* 4, i -^ — h is the middle part, and I* — Jl,
[134562] ^<;r — C adjacent parts. In [1345^^], first formula, a is
the middle part and ^ * — C, c the adjacent parts.
Lastly, in [1345^^], first formula, ^■tt — A, is the
middle part, and ^ * — C, a, the opposite parts. In
this way the expressions [134561], will be found to include
all the preceding formulas in rectangular spherical trigono-
metry, except [1345^], which depends on the well known
formula, that the sine of a side is proportional to the sine of
its opposite angle.
This method may be applied to the solutions of cases in
oblique spherical trigonometry, by dividing the triangle
A C B, fig. 6 — 9, into two rectangular triangles, C P Jl,
B P A, by means of a perpendicular A P, let fall from
the angular point A, upon the opposite side or base B C ;
the perpendicular being so chosen as to 7nake two of the
given quantities fall in one of the rectangular triangles,
or in other words, the perpendicular ought to be let fall
from the end of a given side as C A, and opposite to a
given angle C ; so that all the parts of this triangle are
either given or may be computed, by the formulas
[134527-^], or the equivalent ones [1345*5i]. Each of
these rectangular triangles Q P A, B PA, contains,
Applica-
tion of
Napior'a
circular
parts to
oblique
trigonom-
etry.
[134563]
ud
SPHERICAL TRIGONOMETRY. 739
as above, five circular parts, the perpendicular A P being counted in each, and hearing in
both the same name ; therefore the parts of each triangle, similarly situated with respect to the
perpendicular, must have the same names, as is evident from the inspection of the figures.
If in the triangle APB, we put M for the middle part, A for the adjacent part, and F
for the opposite part; also w, a, p, for the corresponding parts of the triangle CPA;
supposing the perpendicular AP to be an adjacent part, the rules of Napier, [1345^^],
^ „ sin. M ., ., n n a j . a Tt ^*"- "*
will give tang.^P=, -, m the triangle CPA-, and tang.^P = — — -,
o ° tang..^ tang, a
sin. JW sin.m , «
in the triangle BPA-, hence ■^-:j^^^^^ therefore
sin. M : tang. A : : sin. m : tang . a. [134564]
-, . , „, 1 r^^^,.cn /J -n sin.Jlf sin.w
But if .^ P be an opposite part, we shall have by [1345^^], cos. ^P= -_^ = — — ,
hence sin. M: cos. B : : sin. m : cos. b ; and we shall have, for solving these cases of
oblique spherical trigonometry, this rule, the sine of the middle parts are proportional to the
tangents of the adjacent parts, or to the cosines of the opposite parts. Therefore we shall
have, for solving all the cases of rectangular spherical trigonometry, and all except two cases
of oblique angled spherical trigonometry, the following formulas,
C __ J) Tangents of the adjacent parts.
Sine middle part \ c r>, - /• 7 • [1345651
( OC 5 Cosines of the opposite parts.
These expressions, when applied to rectangular spherical triangles, denote, as above, that the
sine of the middle part is equal to the product of the tangents of the adjacent parts, or to the
product of the cosines of the opposite parts of the same triangle. When applied to an
oblique angled spherical triangle, they denote that the sines of the middle part are
proportional to the tangents of the adjacent parts ; or that the sines of the middle parts are
proportional to the cosines of the opposite parts of the same triangle ; observing that the [1345^6]
perpendicular being cornmon to both triangles APB, AP C, and bearing the same
name in each of them, must not be used in these proportions, nor counted as a middle part ;
it not being necessary to compute the value of the perpendicular in making these calculations.
The two cases not included in the formulas [1345^^}, are. First, where the question is
between two sides and the opposite angles, which can be solved by the noted theorem [1345^7]
[1 345'^]. Second, where three sides are given to find an angle, or three angles to find a
side, this last being included in the former by using the supplemental triangle. These
calculations may be made by means of the formulas [1345^^"^^] or [1345^^"^].
The manner of using Napier's method in rectangular trigonometry is well known. The
rules for oblique trigonometry are the same as were given in a paper I published in the third
volume of the Memoirs of the American Academy of Arts and Sciences, and may be
illustrated by the following examples.
740
APPENDIX BY THE TRANSLATOR.
First, Let AB, AC, and C be given, to find B C, and the angles A, B. In the
[134568] first rectangular triangle QPA, we must compute the segment C P, hy means of A C,
C, [1 34529]. jviark the three parts AC, C P, A P, of the first triangle, as in fig. 6,
and the second triangle APB, in a similar manner. Then the rule [1345^],
sin. mid. oc cos. opp. gives
sin. (co. A C) : cos. C P :: sin. (co. A B) : cos. B P, or
[134569]
cos.^C:cos. CP'.icos.AB'.cos.BP,
being the same as [134534]. Having C P, B P, we get BC=CP+BP,
noticing the signs, and then the angles A, B, may be found by [1345^^].
If we mark the triangle as in fig. 7, and use the rule, sin. mid. oc tang, adj., we shall
get sin. CP : tang. (co. C) : : sin. B P : tang. (co. B), or
[134570] °
sin. C P : cot. C : : sin. B P : cot. B,
as in [134537].
If the side J5 C be not required, but merely the angle A, we may compute the angle
CAP, [13453"], instead of the segment C P. Then marking the triangles as in fig. 8,
we shall have, from the rule tang. adj. oc sin. mid.,
tang. (co. A C) : sin. (co. CAP):: tang. (co. A B) : sin. (co. B A P), or
[134571] cot. A C : cos. CAP:: cot. A B : cos. BAP,
as in [1345^9]. Having the segments CAP, BAP, we easily obtain the angle
C AB= C AP -\-B AP, noticing the signs ; and we may then mark the triangles as
in fig. 9, and the rule cos. opp. oc sin. mid., will give
cos. (co. CAP): sin. (co. ACP):: cos. (co. BAP): sin. (co. A B P), or
[134572] sin. CAP:cos.A C P ::sm. B AP :cos. A BP,
as in [134541].
Second, Let AC, B C, and the included angle C, be given, to find A B and the
angles A, B. Having found the segment C P, as above, [1345*5^], we shall get
BP = BC — C P, noticing the signs. Marking the triangles as in fig. 7, and using the
rule, sin. mid. OC tang, adj., we shall find, as before, [134570],
[134573] sin. C P : tang. (co. C) : : sin. B P : tang. (co. B),
as in [1345^7], and as in [1345^9-]^ ^g shall get, by using fig. 6,
cos. C P : sin. (co. AC):: cos. B P : sin. (co. A B) ;
then A may be found as in [1345^^].
SPHERICAL TRIGONOMETRY.
741
ad
Third. Given jB, C, and the side AC', io find the rest.
The segment CP being found as before, we get, in fig. 7,
by using the rule tang. adj. oc sin. mid.,
tang. (go. C) : sin. CP:: tang. (co. B) : sin. B P, [1345-'']
[1345^^]. If we mark the triangles as in fig. 6, and use
the rule cos. opp. oc sin. mid., we get
COS. C P : sin. (co. C A) : : cos. BP : sin. (co. A B),
[1345^^]. Otherwise we may compute as before, [1345^^],
the angle CAP, and then marking the triangles as
in fig. 9, and using sin. raid, oc cos. opp., we shall
get
sin.(co. C) :cos.(co. C AP) : :sin. (co.B) :cos.{co.C AP),
or cos. C : sin. CAP:: cos. B : sin. BAP, [1345^6]
[1345^^], hence we get the angle
CAB=CAP + BAP,
noticing the signs. If we mark the triangles as in fig. 8,
we shall get, by using the rule sin. mid. oc tang, adj.,
sin.(co. C^P):tang.(co../2C)::sin.(co.^.4P) :tang.(co../2jB)
or COS. C^P: cot. ^ C:: COS. jB.^P: cot. ./2^,
[134539].
Fourth, Given the angles A, C, and the included side
A C, to find the rest. Having computed as before,
[1345^0], the angle CAP, we shall have also, the other
segment BAP=CAB — CAP, noticing the
signs ; then marking the triangles as in fig. 9, and using
the rule cos. opp. oc sin. mid., we get
cos.(co. C^P) : sin. (co. C)::cos. (co.5^P):sin.(co.J5),
or sin. CAP: cos. C : : sin. BAP: cos. B, [1345«]. i'^^^'"']
If we mark the triangles as in fig. 8, we shall get, by
using the rule sin. mid. oc tang, adj.,
sin.(co. C./2P) :tang.(co../fC) ::sva..{co.BAP):tm^.{co.AB), [1345771
or cos. C^Prcot.^ C:: cos. BAP :cot.AB,
[134539].
186
^^"^ APPENDIX BY THE TRANSLATOR.
The computation of a spherical triangle, in which the sides a, h^ c, are very small in
comparison with the radius of the sphere, and the angles opposite to those sides are
respectively A, B, C, may he reduced to the computation of a plane triangle, having the
same sides a, h, c, and the opposite angles A', B', C, respectively. For if we neglect
terms of the fifth order in a, b, c, we shall have, by formulas [43, 44] Int.
[134578] I ^i ' 2 -rjj ,
cos. c = 1 — J c^ + 2?j c^ sm. b = b — ^P, sin.c = c — ^c^.
„,.... n COS. a — COS. 6. COS. c ,- ^ o-. , „
SuDstitutmg these m cos. A = -. : , [1345^1, we shall cet
sin. 6. sin. c u j' ^
^^^•'^— hc.\i—\.{b^-\-(?)\ '
or by reduction
ri34579i ^^,^^_^'^+<^--«^ {2aH^Jr^a^c'-{-W^-a^-.¥-^c^) ^
^ -' 26c 246c
If we suppose the radius of the sphere 1, to be infinitely great in comparison with the
sides a, b, c, the terms of the second order will vanish in the preceding expression, and Jl
will become A', corresponding to a plane triangle. The expression of cos. A' thus obtained
will be the same as in [1345^'] or [62] Int., namely,
62+c2 — a2
[134580] COS. A' = —~ ,
abc
from which we get,
[134581] sm.2 ^' = 1 — C0S.2 A' = HI -^^^
Substituting these in [1345'''^], we get,
[134582] ^°^' *^ "^^ ^^^' '^ — i^^' sin.^ A\
Now in this plane triangle, the perpendicular let fall from the angular point C, upon the
opposite side c, is evidently =b.sm.A' ; multiplying this by half the side c, we shall
[134583] obtain the area of the triangle 5 = | & c . sin. A'. Substituting this in [1345^2], it becomes
cos../2 = cos.w2' — ^s.sin.^', hence by [61] Int. cos. .^ = cos. (^' + J s), or
A = A'-^is, and as the area s does not change by putting B ov C for A, and B', C,
for A', respectively, we shall have the following system of equations,
A = A' + is,
B=:B'-\-hs,
[134584]
triaoglet.
SPHERICAL TRIGONOMETRY. 743
The sura of these, putting for Jl'-{-B'-j- C its value 180'' or *, is .5-}-jB+C=ir-f »,
hence
« = ^ + B+C-*. [134585J
Therefore if we. denote hy s, the excess of the sum of the three angles of a spherical triangle [134586]
above two right angles, and subtract one third of this excess from each of the spherical angles, , q^
A, B, C, toe shall obtain the corresponding angles A',B', C, of a rectilinear plane triangle, thfonm
the sides of which are equal in length to those of the spherical triangle. This beautiful spherical
theorem, discovered by Le Gendre, is much used in geodetical operations, for reducing the
calculations of small spherical triangles to the common operations of plane trigonometry.
The area of a spherical triangle ABC, may be found
in the following manner. Suppose the radius of the sphere
to be 1, corresponding to the circumference of the great
circle 2 tt = 6,2831 .... Then by a well known theorem,
the whole spherical surface will be 4 *. This also follows
from the expression of m, [2756], which gives AitdR,
for the mass of the spherical shell of the thickness d R,
included between the radii R and R-{- d R, when
i?=l. Now if we suppose any semicircle of this
surface, as B C D B to revolve about its diameter B B'
till it make a complete revolution, or 4 right angles = 2 ir,
it will pass over, during this revolution, the whole spherical surface 4 if ; the ratio of these
quantities 2 if and 4 if being as 1 to 2 ; and it is evident that the same ratio will obtain,
between any otlier angle as AB C = B, and the corresponding space passed over
B D B' A B, which will therefore be represented by 2 B; so that, in the present notation,
the spherical surface, included between any two semicircles of the sphere, will be represented
hy twice the angle of inclination of these semicircles. *■ '
Continuing the side of the triangle BC so as to complete the great cu-cle BCDBCE,
also the sides BA, € A^ till they cross this great circle in B', C, and meet again in the
opposite hemisphere at A' ; we shall evidently have the arc
BAB' = ABA' — 2i semicu-cle.
[134588]
also the arc, CA C =^ A C A' = 3i semicircle ; subtracting from these AB, A C,
respectively, we shall get AB=A'B', A C=A'C', and as the angle BAC=B'A'C',
we shall have the triangles BAG, BA'C, equal to each other, therefore BC=BC',
and the surface ABC equal to the surface A' B' C.
Putting s = the surface of the triangle AB C, or A' B' C, a = surface AC DBA, [134589]
'7^ APPENDIX BY THE TRANSLATOR.
[134590] 5 ^ surface B'AC, c = surface ABEC, A = angle BAC, B = angle ABC,
C= angle A C B, we shall have, by using the theorem [1345^^],
surface jB C D B AB = s -^ a=2 B,
[134591] surface CBEC'AC = s + c = 2C,
surfaces B A C -{- B' A C = surfaces B'A' C' + B'AC
= sur(^ce A C A' B' A = s-{- b = 2 A.
Adding these three quantities together, we shall obtain 2s-{- a-{-h-\- c=2{A~{-B-\-C).
Now the hemispherical surface 2 'T, is evidently equal to
s -\- a -{- b -\- c, or s-\-a-{-b-\-c=2ir.
Subtracting this from the preceding equation, we get 2s=2{A-\-B-\-C — ir), and
finally,
[134592] s = ^ + 5+C — *.
Therefore the surface of a spherical triangle, expressed in squares of the radius, taken as unity,
[134593] *^ equal to the spherical excess of the sum of the angles of the triangle above two right angles,
expressed in the above notation, in which 180'' is represented by 3,1415....
As the quantity s is always positive, we shall have A-\- B -{- C^'x. That is, the sum
[134594] of the three angles of a spherical triangle exceeds two right angles ; and since each of these
angles is less than two right angles, the sum of the three angles must be less than six right
angles.
If we substitute, in the expression of the area of a plane triangle ^b c . sin. A, [1345^*3],
the expression of sin. A, [1345^^'] we shall get, for the area of a plane triangle whose sides
are a, b, c, the expression
[134595] area = /«.(« — a).(s — 6).{»— c) = I . \/(a-f 6-fc).(— a+&+c).(a— 64-c).fa + & — c).
ON THE SYMBOL /=!•
The imaginary symbol \/—i occurs frequently in this work, particularly in the use of
circular arcs, and as the principles, upon which the use of it is founded, are not commonly
explained in the elementary works in this country, it may be proper to make a few remarks
on the application of it to the calculus of sines and cosines of circular arcs, from which the
propriety of employing it will very evidently appear.
If we, for brevity, denote this symbol by e, so that e = \/—i, we must always put
e^= — 1 , e^ = — v/— i> e* = 1 1 e^ = v/— 1> ^nd generally, e^ •" = 1,
ON THE SYMBOL v/=i. '^^^
g4«+i __ i/lTi, g4m+2 __ — i^ g4m+3 ^^ — \/'—i-i vfi being any integral number
whatever. Tiiis is conformable to the usual rules of multiplication in algebra, and must be
considered as a definition of this symbol, and of the manner of using it, and not as a
demonstration of its properties. It is also to be understood, as a part of this definition,
that in all cases the symbol v/^^» or e, is to be operated upon by addition, subtraction,
multiplication, division, &c., according to the usual rules of algebra. Thus the sum of a and
b \/.iri, is a-{-b \/—i j the product of a by b s/—i is ab v/^ ; and the quotient
of b \/—i, divided by a, is — ^— , or - . \/^. In like manner the product of the
binomials a-\-b^—i by c-\-d^—i, or a-}-&e by c-{-de, is
ac-\-{ad-{-bc) .e-\-bde^ = ac — bd-\-{ad-\- bc).^—i'j
these operations being evidently conformable to the principles and definitions here used.
Again, from [607c], we have c* = 1 -f a; -f- — + ^-^ + &tc., and if ±:ze had
been used instead of a;, in the development of this exponential, by the common processes of
algebra, considering z e as a real quantity, we should have
&;c..
z2e2 z3 e3
^•-l + - + 1.2 + 1.2.3 + «--'
,-.. 1 ., , ^'^' ^'^' ,
^ —^ --' ' 1.2 1.2.3 '
gie __ g— re
I '''' 1 '"'" 1 Zlc
° 2e
' 1.2.3 ' 1.2.3.4.5 ' ^''
-+--H
, r2e2 24e4
^1.2 + 1.2.3.4 + ^-'
[134597]
and if we now substitute the above values e^ = — 1, e^ = 1, he, in the second members
of these expressions, they will represent the values of sin. z, cos. z, [607 d, e], which had
been found independently of the use of the symbol \/—i. Therefore we shall have,
2. /ITT —z.v/ITI z.yCri —z.^'—i
c — c c +c
sm. z = -= , cos.z= ! ,
2.v/=i 2
as in [1345^] ; and these must be considered as nothing more than abridged values of sin. z,
COS. z, reduced to simple analytical forms, extremely convenient in many trigonometrical
calculations. Hence we perceive the real import of these expressions to be nothing more
than that if the quantities c ' " , c *" , or (^% c~'% be developed according
187
"7-^6 ON THE SYMBOL \/^.
to the powers of z e, by the usual rules of development of real quantities, the analytical
expressions — g- , — ^ , putting e^ __ — j^ after the development, will
accurately represent the values of sin. z, cos. z, respectively, in real finite quantities,
independent of v/^ > and there is in fact, no more mystery in the use of this imaginary
symbol e, in this manner, and for this purpose, than there is in substituting the abridged
expression (1 — a?)*, instead of its equivalent values in an infinite series
1 — ^x — |a?2 — j^^x^ — he.
Having obtained these abridged analytical values of sin. z, cos. z, we shall, by changing
z into X, get similar analytical expressions of sin. x, cos. x, and if we wish to obtain the
products, or powers, or any functions whatever, of such sines or cosines, we may use these
analytical formulas, as has already been done in [1345^, Sic] What has been said will serve
to illustrate briefly the logical principles upon which the use of this symbol is founded, and
any one who wishes to pursue the investigation, may consult a valuable paper of Mr.
Woodhouse, published in the Philosophical Transactions of London for 1801, in which this
subject is fully discussed.
Ira Berry and Lucius A. Thomas, compositors.
Considerable pains have been taken to print this volume as correctly as possible ; but
several errors of the press have been discovered, by a young friend who has read the work
before the publication. Most of these mistakes have been corrected with a pen. The reader
is requested to make the following additions and alterations,
Page 43, line 2, for by read through.
Page 47, line 9, for proportional to read equal to.
Page 55, [82o], in IT . (», c, (p), the factor a.{c,<^) ought to be in the denominator.
Page 98, [143'], read, d(^ = J.m .{P . dx -^ (^. dy -\- R . d z).
Page 133, line 14, read^ homogeneous cylinder, of an elliptical base.
There may be other mistakes, which have been passed over without notice, as it is extremely
difficult to print a work of this kind free from error.
5 5"^
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