HI BOOK 52 1 1.L3 1 c. 1 LAPLACE » TREATISE OF COLESTIAL MECHANICS 3 T1S3 DDl^TbbD 1 A TREATISE OF CELESTIAL MECHANICS, BY P. S. LAPLACE, MEMBER OF THE NATIONAL INSTITUTE, AND OF THE COMMITTEE OF LONGITUDE, OF FRANCE ; THE ROYAL SOCIETIES OF LONDON AND GOTTINGEN; of THE ACADEMIES OF SCIENCES OF RUSSIA, DENMARK, AND PRUSSIA, &C. PART THE FIRST— BOOK THE FIRST. TRANSLATED FROM THE FRENCH, AND ELUCIDATED WITH EXPLANATORY NOTES. BY THE REV. HENRY H. HARTE, F.T.C.D. M.R.LA. DUBLIN: PRINTED FOR RICHARD MILLIKEN, BOOKSELLER TO THE UNIVERSITY ; AND FOR LONGMAN, HURST, REES, ORME AND BROWNE, LONDON. 1822. O. CBAISBBBRY, FRniTEB TO IHE VNrrERSITT. TO THE REV. CHARLES WILLIAM WALL, THIS TREATISE IS DEDICATED, BY HIS FRIEND, AND FORMER PUPIL, HENRY H. HARTE, A 3 PREFACE. It has been made a matter of surprise, that considering the great capabilities of many individuals in these countries, so few are conversant with the contents of a work of such acknowledged eminence, as the Celestial Mechanics. Without adverting to other causes, it may be safely asserted, that the chief obstacle to a more general knowledge of the work, arises from the sum- mary manner in which the Author passes over the intermediate steps in several of his most interesting investigations. To re- move this obstacle, is the design of the present treatise, in which the translator endeavours to elucidate every diffi- culty in the text, and to expand the different operations which are taken for granted. He has not attempted to follow the principles into all their details; but he has occasionally adverted to some useful applications of them, which occur in different Authors. He is aware that those conversant with such subjects will find much observation that may be dispensed with ; but when it is considered that his object was to render this work accessible to the general class of readers, he trusts that he will not be deemed unnecessarily diffiise, if he has insisted longer on some points than the experienced reader may think neces- sary. As many of the propositions which Newton announced se- parately are so many different results, which are all comprised VI PREFACE. under the same general law analytically investigated, he has also taken occasion to notice, in the notes, those propo- sitions of Newton, which are embraced in the general analysis of the text, which he was induced to do, in order to show the great superiority of the analytic mode of investigating problems. The Work will be divided into five parts, which will be published in separate volumes. The first volume contains the first book, which treats of the general prin- ciples of the equilibrium and motion of bodies. The number of notes which was necessary for the elucidation of these prin- ciples is much greater than will be required in any of the subsequent volumes. The second volume will contain the second and third books of the original ; the third volume, the fourth and fifth books ; the fourth volume will contain the sixth, seventh, and eighth books ; and the last volume will contain the ninth and tenth books, together with the supplement to the tenth book. Trin. Coll. April, 18S2, TABLE OF CONTENTS. BOOK I. Of the generallaws of Equilibriwn, and of Motion. .... page 1 CHAP. I, Of the Equilibrium and of the composition of Forces which act on a material point. ............ ibid. Of Motion, of Force, of tlie Composition and Resolution of Forces. - Nos. 1 and 2 Equation of the Equilibrium of a point soUicited by any number of Forces acting in any direction. Method of determining, when the point is not free, the pressure which it exerts on the surface, or against the curve, on which it is constrained to exist. Theory of Moments. .......... No. 3 CHAP. II. Of the Motion of a Material Point. 21 Of the law of inertia, of uniform motion and of velocity. ... No. 4 Investigation of the relation which exists between the force and the velocity ; in the case of nature they are proportional to each other. Consequences of this law. Nos. 5 and 6 Equations of the motion of a point sollicited by any forces whatever. - No. 7 General expression of the square of its velocity. It describes a curve in which the integral of the product of its velocity, by the element of this curve, is a minimum. No. 8 Method of determining the pressure, which a point moving on a surface, or on a curve, exerts against it. Of the centrifugal force. ...... No. 9 Application of the preceding principles to the motion of a free point actuated by the force of gravity, in a resisting medium. Investigation of the law of resistance necessary to make the point describe a given curve. Particular examination of the case in wliich the resistance vanishes. ......... No. 10 Application of the same principles, to the motion of a heavy body on a spherical surface. Determination of the duration of the oscillations of the moving body. The very small oscillations are isochronous. ..-..-- No. 11 VUl CONTENTS. Investigation of the curve, on which the isochronisra obtains rigorously, in a resisting medium, and particularly, when the resistance is proportional to the two first powers of the velocity. -...-.,.,.. No. 12 CHAP. III. Of the Equilibrium of a system of bodies. .... "75 Conditions of the equilibrium of a system of points wliich impinge on each other, with directly contraiy velocities. What is understood by the quantity of motion of a body, and by similar material points. --..... No. 13 Of tlie reciprocal action of material points. Reaction is always equal and contrary to action. Equation of the equilibrium of a S3-stem of bodies, fi-om which results the principle of virtual velocities. Method of determining the pressures, which bodies exert on the curve cr on the surface on which they are subjected to move. . - , No. 14 Application of these principles, to the case in which all the points of the system are in- variably connected together ; conditions of the equilibrium, of such a system. Of the centre of gravity. Method of determining its position; 1st, with respect to three fixed and rectangular planes ; 2diy, with respect to three points given in space. - No. 15 Conditions of the Equilibrium of a solid body of any figure whatever. - No. 16 CHAP. IV. Of the Equilibrium 0/ fluids. ...... 99 General equations of tliis equilibrium. Application to the equilibrium of a homogenous fluid mass, of which the exterior surface is free, and which is spread over a fixed solid nucleus, of any f gure whatever. ....... No. 17 CHAP. V. The general principles of the motion of a system of hodiei, - 108 General equation of this motion. ....... No. 18 Developement of the principles which it contains. Of the principle of living forces. It only obtains, when the motions of the bodies change by insensible gradations. Method of estimating the change which the living force sustains, in the sudden variations of the motion of the system. ......... No. 19 Of the principle of the conservation of the motion of the centre of gravity ; it subsists even in the case in which the bodies of the system exercise on each other a finite action in an instant. No. 20 Of the principle of the conservation of areas. It subsists in like manner as the preceding principle, in the case of a sudden change in the motion of the system. Determination of the system of coordinates, in which the sum of the areas described by the projections of the radii vectores vanishes for tw6 of the rectangular planes formed by the axes of the coordinates. This sum is a maximum for the third rectangular plane ; it vanishes for every other plane perpendicular to this last. ... No. 21 Tlie principles of the conservation of living forces and of areas, obtain also, if the origin of the coordinates, be supposed to have an uniform and rectilinear motion in space. In this case, the plane passing constantly through this point, and on which the sum of the CONTENTS, IX areas described by the projections of the radii is a matimum, remains always parallel to itself. The principles of living forces and of areas may be reduced to certain relations between the coordinates of the mutual distances of the bodies of the system. The planes passing through each body of the system, parallel to the invariable plane drawn through the centre of gravity, possess analogous properties. .... No. 22 Principle of the least action. Combined with the principle of living forces, it gives the general equation of motion. ........ No. 23 CHAP. VI. Of the laws of motion of a system bodies, in all relations mathematically possible between the force and the velocity. ...... 152 New Principles which, in this general case, correspond to those of the conservation of living forces, of areas, of the motion of the centre of gravity, and of the least action. In a system which is not acted on by extraneous forces, 1 °, the sum of the finite forces of the system, resolved parallel to any axis, is constant ; 2° the sum of the finite forces to make the system revolve about an axis is constant ; 3° the sum of the integrals of the finite forces of the system, multiplied respectively by the elements of their directions, is a minimum s these three sums vanish in the case of equilibrium. - - No. 24 CHAP. VII. Of the motion of a solid body of any figure xjohateverm - • 159 Equations which determine the motion of translation, and of rotation of a body. Nos. 25, 26 Of the principal axes. In general a body has only one system of principal axes. Of the moments of inertia. The greatest and the least of these moments appertain to the prin- cipal axes, and the least of all the moments of inertia belongs to one of the three prin- cipal axes which passes through the centre of gravity. Case in which the solid has an infinite number of principal axes. -.....- No. 27 Investigation of the instantaneous axis of rotation of a body : the quantities which de- termine its position relatively to the principal axes, determine at the same time the velocity of rotation. ........... No. 28 Equations which determine in a function of the time, this position and that of the principal axes. Application to the case in wliich the motion of rotation arises fi-om an impulsion which does not pass through the centre of gravity. Formula for determining the distance of this centre from the direction of the initial impulsion. Example deduced from the planets, and particulariy from the earth. ...... No. 29 Of the oscillations of a body which turns very nearly about one of its principal axes. The motion is stable, about the principal axes of which the moments of inertia are the greatest and the least ; it is not so about the third principal axis. - • No. 30 Of the motion of a solid body about a fixed axis. Determination of the simple pendulum which oscillates in the same time as the body. ..... No. 31 CHAP. VIII. Of the motion offuids. 222 Equations of the motion of fluids ; condition relative to their continuity. - No. 32 b X CONTENTS. Transformation of these equations ; they are integrable, when the density being any function o!" the pressure, the sum of the velocities parallel to the rectangular coordinates, re- spectively multiplied by the element of its direction, is an exact variation. It is demon- strated, that if this condition obtains at any one instant, it will always obtain. No. 35 Application of the preceding principles to the motion of an homogeneous fluid mass, which revolves uniformly about one of the axes of the coordinates. ... No. 34 Determination of the very small oscillations of an homogeneous fluid mass, which covers a spheroid having a motion of rotation. ...... No. 35 Applicatior. to the motion of the sea, on the supposition that it is deranged from the state of equilibrium by the action of very small forces. . . - . No. 36 Of the terrestrial atmosphere considered at first in a state of equilibrium. Of the oscillation* which it experiences in the state of motion, and considering only the regular causes whicli iigitate it ; of the variations which those motions produce in the heights of the barometer. ........,-- No. 37 A TREATISE OB CELESTIAL MECHANICS, &c. &c, JN EWTON published, towards the close of the seventeenth century, the discovery of universal gravitation. Since that period. Philosophers have reduced all the known phenomena of the system of the world to this great law of nature, and have thus succeeded in giving to the theories and astronomical tables a precision which could never have been anticipated. I propose in this present treatise to exhibit in one point of view, these theories which are scattered through a great number of works, of which the whole comprising the results of universal gravi- tation, on the equilibrium and motion of the bodies both solid and fluid, composing the solar and similar systems, constitutes The Celestial Mechanics. Astronomy, considered in the most general manner, is a great problem of Mechanics, of which the arbitrary quantities are the elements of the motions of the heavenly bodies j its solution depends, at the same time, on the precision of the observations, and on the perfection of analysis ; and it is of the last importance to banish all empiricism, and to reduce it, so that it may borrow nothing from observation, but the indispensable data. The object of this work, is, as far as it is in my power, to accomplish this interesting end. I trust that, in consideration of the difficulties and importance of the ^ xii subject, Philosophers and Mathematicians will receive it with indulgence, and that they will find the results sufficiently simple to be employed in their investigations. It will be divided into two parts. In the first, I will give the methods, and formulas, for determining the motions of the centres of gravity of the heavenly bodies, their figures, the oscillations of the fluids which are spread over them, and their motions about their proper centres of gravity. In the second part, I will apply the formulse which have been found in the first, to the planets, the satellites and the comets -, and I will conclude with a discussion of several questions relative to the system of the world, and by a historical notice of the labours of Mathematicians on this subject. I will adopt the decimal division of the quadrant, and of the day, and I will refer the linear measures, to the length of the metre, determined by the arc of the ter- restrial meridian comprised between Dunkirk and Barcelona. ERRATA. Page Line 3, 23, Jbr the new forces, read these forces. 6, 12, Jbr reluctant, read resultant. 12, 15, for {c) read {b). 17, i>Jor equation, read equations. 23, U,forq>{f),read 'CfJ- 2i, 14i, for a and b, read c and b ; and line 17, for angle, read jangled. 32, lOj^or di/*dz^, read dy- -^dz-. 33, ' 2, fiom bottom, after A^, add — ; and last line, after the differential of the, add square of the, and for s''ds, read . 35, 2, from bottom, for first the order, read the first order. 40, 18, after centrifugal, add force. 47, last line, for dt constant, read dx constant. 49, ll,yb>- 2/i. COS. 6. read Ih. cos. -i. ; and in lines 21, 22, dele the 2 which occur in the Den". 50, 14, dele the 2 by which the values oidt, dz, dx, are multiplied. 51, 1\, for git, read gf^. 65, \,fordsi,readds'%;\m&lQ,for\o^n.{s-\-q) — ), readXog. 7i.{s-\-q)) — ; h'ne 17, for{s'-\-q^') reads'-\^. 82, 4, for P, read —P. 83, 2, for they, read it. 84, 23, /or i, k, k, read R, R, R. 86, 3 from boitom, for -^ read -r— . ox OX 94, 2, a/ier centre, rearf of gravity. 99, 12, for figure, rearf figures. 105, for Sg, read Sp ; 20, after each, rwrf other. ERRATA. Page Line 138, 16, Jbr sin. C. sin. -J'.+cos. ■^. sin. = 1 and x -zz z. cos. K^. If we suppose x to vanish, then z ■=. y, and 9 — — w ; cos. K^ being then equal to nothing, K *must be equal to 2«4-l, n being an integral number; and in this case .r will vanish as often as 9 will be equal to ^^ ; but x being no- thing we have evidently 9 zz A-ct ; therefore 2«+l zz 1, or n zz o, consequently X = z. COS. 9. From which it follows that the diagonal of a rectangle described on the right lines which represent the forces x and y, represents not only, the quantity but also the direction of their reluctant. Thus we can substitute for any force whatever two other forces which form the sides of a rectangle, of which that force is the diagonal ; and it is easy to infer from thence that it is possible to resolve a force into three others, which form the sides of a rectangular parallelipiped of which it is the diagonal. t Let therefore a b and c represent the three rectangular coordi- nates of the extremity of a right line, which represents any force what- ever, and of which the origin is that of the coordinates ; this force will be represented by the function s/a*-\-h''' -\-c*, and by resolving it * In this case K6 is some odd multiple of -■— and therefore K must be of the form 2n-|- 1 . f Tlie given force being resolved into two, of which one is perpendicular to a plane given in position, the other being parallel to tliis plane, if this second partial force be decom- posed into two others, parallel to two axes situated in this plane, and perpendicular to each other ; it is evident that the three partial forces will be at right angles to each other, and that the sum of the squares of the lines representing these forces, will be equal to the square of the line representing the given force, therefore this last force is the diago- nal of a rectangular parallulUpiped, of which the partial forces constitute the sides. PART I.— BOOK I. 7 parallel to the axes of a o£ b and of c, the partial forces will be ex pressed respectively by these coordinates. Let a', b', &, be the coordinates of a second force ; a-\-a', b-\-b', c+c', will be the coordinates of the resultant of the two forces, and will represent the partial forces into which it can be resolved parallel to the three axes, from whence it is easy to conclude that this resultant is the diagonal of a parallelogram, of which the two forces are the sides.* In general a, b, c ; a', b>, C ; a", b«, &' ; &c. being the coordinates of any number of forces ; a -{■ a' ■{■ a" -^ , &c. b+b'+b''+, kc.c-\-c'-\-c"-\- &c. will be the coordinates of the resultant ; the square of which will be equal to the sum of the squares of these last coordinates ; thus we shall have both the quantity and the position of the resultant. t * The coordinates of the extremity of this diagonal are evidently equal to n+a', h-\-b', c+c, therefore tliis diagonal must be equal to the resultant of the two forces. We are enabled to derive an expression for the cosine of the angle, contained between the given forces, in terms of the cosines of the angles which these forces make with the coordi- nates, for calling the forces S and S', and the angles which S makes with the three axes, A, A', A", and B, B\ B", the angles which S' makes with the same axes we have o=Scos. A, b=S COS. A, c=S cos. A",a'=-S cos.B,c'=S cos. 5', c' = S' cos. B' ; the square of the line connecting the extremities of S and S = S* — Q,SS. cos. il+.S ' ; ^ being the angle contained between S and S, the square of this line is also equal to (S cos. A—S cos. BY + {S cos. A'—S' cos. £')=+ (S cos. A'—S cos. B')* ; =» S' + S"^— 2 SS' (cos. A. cos. B + cos. A. cos. 5'4-cos. A', cos. B",) consequently we have cos. A = cos. A. cos. B + cos. A . cos. B + cos. A", cos. B', therefore when the two forces are perpendicular to each other, the second member of this equation is equal to nothing. t Let S S' S", &c. represent the forces of wliich the coordinates are respectively a, 6, c ; a, V , c' ; a", b", c", &c. then by what precedes a-\-a', b-\-b', c+c', are the co- ordinates of the resultant of S and S', a+a'+a", b-\-b'-\-b", c+c'+c', .are the coor- dinates of the resultant of this last force, and the force S" &c. : therefore the resultant f of any number of forces is the diagonal of a rectangular parallelipiped of which 8 CELESTIAL MECHANICS, 2. From any point whatever of the direction of a force S, which point we will take for the origin of this force, let us draw a right line, which we will call 5, to the material point M ; let x, y, z, be the three rectangular coordinates which determine the position of the point M, and a, b, c, the coordinates of the origin of the force ; we shall have If we resolve the force S parallel to the axes of .«■, of i/, and of z, the corresponding partial forces will be by the preceding number S r S ' S ' \izJ ySl/J WS^ the coordinates are equal respectively to the sum of the coordinates of the composing forces, V F»= (a+a'+a" +&c.y- + (6+*'+*" &c.)* + {c+c +c' + &c.)'. Let m,n, p = the angles which V makes with the rectangular axes a4-fl'+o"+ &'C h + b' + b"+&c. c+c' + c" + &c. COS. m = ^ y — cos. m = ^ ^^ JE cos. p = _L_X,_L — •. • we have both the quantity and direction of the resultant. From the preceding composition of forces it follows, that if a polygon is constructed, of which the sides, (which may be in different planes) are respectively proportional to these forces, and parallel to their directions, the last side of this polygon represents the resultant of all the forces in quantity and in direction. * S being considered as a function of x, y, and k, S« ^ ( T~ J ^''''^ { s" J ^■^''' ( T~} ^^ and when s = V{x-ay+(y-by+(z-c-' [jj = -j- j^ = '-^ ' jr =-^ ' &c. are evidently the expressions for the cosines of the angles which s s s makes with the coordinates x, y, and s, since PART I— BOOK I. 9 ■ >;> -i, expressing according to the received notation Ws ^ _ { is tlie coefficients of the variations of Sx, Sy, Sz, in tlie variation of the preceding expression of s. If, in like manner, we name s' the distance of M from any point iu the direction of another force iS', that point being taken for the origin of this force ; S'. \——l will be this force resolved parallel to the axes I SxS of .r, and just so the rest ; therefore the sum of the forces S, S', S", kc. V '■■{ |) = M-^) + s-(t; ) + *'(t^) + '"■ by iiuiltiplying these equations by Sx, 3y, h, respectively, and adding them together, we get >'■'«= K(^).''+(|>'.+(t:)'-0 +.S'Y (l!l)ix+/!^^ 3y ■)-( ^)y Sz+&c.=Sls-t-.S'?.v' + .S'3,v"+ &c.=-Z.S.h. Now since these equation have phice whatever be the variations 3x, Si/; h, one of then; may exist while the other two vanish, therefore the equation (a) is equivalent to the tliree Kiuations which precede it. We shall see liereafter that the introductio)i of the coeffi- cient ( Y^ ) is of the greatest consequence, for from the equation (4) which tbllows immediately from the equation (a), we deduce the equation (/) of No. li, which involves the principle of vertual velocities, and this principle combined with that of D'Alembert, lias given to Mechanics all the perfection of which it was susceptible, for by means of it tTie investigation of the motions of any system of bodies is reduced to the integration of differential equations. .See No. 18. 10 CELESTIAL MECHANICS, resolved parallel to this axis will be 2. S.( — ), the characteristic 2: of (is \ ( Ss' ) Sx ( Sxi Let F be the resultant of all the forces S, S', &c. and u the distance of the point M from any point in the direction of this resultant, which is taken for its origin ; V. < > will be the expression of this re- ' aX J sultant resolved parallel to the axis of x; therefore by the precedhig number we shall have V.< — '-> = H. S.< ( Sx ) l we shall have in like manner Ss_ \ Sy ^ ^ SyS ' Ws ^ ^ cT^ ♦ from which we may obtain, by multiplying these equations respectively by to, Sy, Sz, and adding them together VJu = 1. S. is; As this last equation has place whatever be the variations Sx, Sy, Sz it is equivalent to the three preceding. If its second member is an exact variation of a fuction , and consequently Stp which indicates that the sum of all the forces resolved parallel to the axis of ■>■- is equal to the partial difference ) — ^ i . * This case ob- * If we mult;ply h the variation of any quantity by any function of that quantity, such ^ as -a — ' ^.s™, &c. the product is evidently an exact variation, however this is not true of every species of function, for there are some transcendental and exponential functions. such as which are not exact variations. log. .?. PART L— BOOK I. U tains generally, when the forces are respectively functions oF the dis- tance of their origin from the point M. In order to have the resultant of all these forces resolved parallel to any right line whatever, we shall take the integral S. /." S. is, and naming

will be the resultant I Sx ^ of the forces S S' S", &c. resolved parallel to the right line x. 3. When the point AI is in equilibrio, in consequence of the action of the forces which solicit it ; their resultant vanishes, and the equa- tion (a) becomes O = ■£. S. Ss {b) which indicates, that in the case of the equilibriiun of a point acted on by any number of forces, the sum of the products of each force by the element of its direction is nothing.* c2 * Since the forces parallel to the coordinates .c, y, z, are independant of each other, It follows from the notes to the preceding number, that M'hen the point M is in equilibrio -• -S -J -— J- 2. S. -! —1 !• 2. i>. I __ I. are = respectively to nothing. t. c. ,S. cos. A^S. COS. B-^S" COS. C+ etc. = S. COS. A'-\-S' cbs. 5'+S" cos. C'+&c. = 0. a. COS. A''-\-&'. COS. £"+ S'' COS. C" = 0. {A, A', A" ; B, B', B", &c. are the angles which the direction* of ,S', S>, &c.niake with J-, y, z,) ; these are the equations of equilibrium of a system of forces applied to a mate- rial point which is entirely free. The independence which exists between these equations is ejttremely advantageous, it only obtains \\hen the forces are resoh'ed paralle! to three rectangular coordinates. 2. S. ) -z— > =0 indicates that M is at an invuriable dis- t Sx 3 tance from the plane of y, z ; in this case the forces are reducible to two rectangular ones, in the plane y, z. When the point M is in equilibrio any one of the forces acting on it is equal ami contrary to the resultant of all the remaining forces, for naming V the resultant of the forces .S', S"-)-itc. and n, h, c, the angles which it makes with the coordmates x, y, z, by 12 CELESTIAL MECHANICS, If the point M is forced to be on a curved surface, it will experi- ence a reaction, which we will designate by R. This reaction is equal and directly contrary to the pressure with which the point presses on the surface ; for by conceiving it acted on by two forces R and — R, it is possible to suppose that the force R is destroyed by the reaction of the surface, and that thus the point presses the surface witli the force R ; but the force of pressure of a point on a surface is perpen- dicular to it, otherwise it might be resolved into two, one perpendicular to the surface, which would be destroyed by it, the other parallel to the surface, in consequence of which the point would have no action on this surface, which is contrary to the hypothesis ; consequently if r be tlie perpendicular drawn from the point ^/ to the surface, and termi- nated in any point whate\''er of its direction, the force R will be di- rected along this perpendicular ; therefore it will be necessary to add R.Sr to the second member of the equation (c) which thus becomes O = Z, S. Ss-{-RAr {c) — R being then the resultant of all the forces .V, S', &c. it is perpen- dicular to the surface. If we suppose that the arbitrary variations Sx, Sij, Sz belong to the cui'ved surface on which the point is subjected to remain, we shall have h- — O, since r is perpendicular to the surface, therefore RJr vanishes from the preceding equation, in consequence of which the equation (b) obtains in this case, provided that one of the three variations Sx, St/, Sx, be eliminated by means of the equation to the surface ; but then, tht- what precedes we shall have V'. cos a = S' cos. B-\-S" cos. C^&c. /'' cos. c = S' COS. B'-fS". COS. C'-\-&c. and since S. cos. A+S. cos. B+S". cos. C-f&c. zz 0. We have v. cos. a= — 6'. cos. A ; in like manner it may be shewn that f" cos. b= — S. cos. B, and v. COS. c = — S. cos. C ; if we add together the squares of these equations we shall obtain F'"=S*, because cos. *a -f cos. *i -|- 'c = 1 = cos. -4+ cos. 'B 4- cos. *C.-. we have cos. a = — cos. A &c. •.• a = 200' — A, iu like manner it follows, that b = 20O — B, c = 200— C, v the forces S and V are equal, and act in opposite directions. PART I.— BOOK I. 13 equation (b) which in the general case is equivalent to three, is only equivalent to two distinct equations, which may be obtained by putting the coefficients of the two remaining differentials separately equal to nothing. Let m = be the equation of the surface, the two equations Sr—0, and SuzzO will have place at the same time ; this requires that h- should be equal to \Su, N being a function of x, i/, and z. Naming a, b, c, the coordinates of the origin of r we shall have to determine it from which wc may obtaui -^ — > + < — , + { j — I, and consequently therefore by making I X :r R ^ (- Sx ^ ^ Sy ^ ^ Sz ^ the term R.Sr of the equation (c) will be changed into xiu, and this equation will become r: I. iS'. h-irxiii in which equation we ought to put the coefficients of the variations Sx, ii/, iz, separately equal to nothing, which gives three equations ; but on ac- count of the indeterminate quantity a, which they contain, they are equi- valent to only two between x, y, and a. Therefore instead of extracting from the equation {hi) one of the variations Sx, Sy, Sz, by means of the differential equation of the surface, we may add to it this equation multi- plied by the indeterminate quantity a, and then consider the variations Sx, %, and Sz, as independant. This method, which also results from 14 CELESTIAL MECHANICS, the tlieory of elimination combines the advantage of simplifying the calculation with that of indicating the force — R with which the point il/ presses the surface.* * When the point M is on a curved surface, then all that is required for its equilibrium is, that the direction of the resultant of all the forces which act on it should be perpen- dicular to this surface, but the intensity of this resultant is altogether undetermined, since the reaction is equal and contrary to the pressure of the point on the surface, by adding to 2. S. is the quantity R. Sr we may consider the material point as entirely tree. 3r vanishes because the perpendicular is the shortest line which can be drawn from a given point to the surface. Since the same values of x, y, and z, satisfy the equations 2r = S« = 0, it follows from the theory of equations that 'N = is a function of .r, y, and z, "ill this function it follows from the expression that is given for Jr,' that the cosines of the angles wliit-h the noi-mal makes with the coordinates are equal respectively to iV. < " ' A'. > Jf i .V. j 1: ^ . iix' ^ »y ^ ^" See notes to No. 9. \%z S \^z S \ %z ) then 2. S. S.s -f y.%u=Ci will be equal to X. ix-\- Y. Jy+ 2. 5-- + and on account of the independance of the variables x, y, z, we shall have eliminating a we liave the following equations : y. ??i-x.i!i=o, Z. ^JL-X.h =0. 3x ly Ji ^~ PART I.— BOOK I. 15 Let us conceive this point to be contained in a canal of simple or double curvature ; the reaction of the canal which we will denote by k, will be equal and directly contrary to the pressure with which the point acts against the canal, the direction of which is perpendicular to its side ; but the curve formed by this canal, is the intersection of two sur- faces of which the equations express its nature, therefore we may con- sider the force k as the resultant of two forces R, R', which express the reactions of the two surfaces on the point M ; since the directions of the three forces li, R', /., being respectively perpendicular to the side of the curve they are in the same plane, therefore by naming h; Sr' the elements of the directions of the forces R, R', which directions are respectively perpendicular to each surface ; we must add to the equation (A) the two terms RSr, R'Sr, which, will change it into the following : Q-^.SSs + R.Sr-^R'.h'. (dj These are the equations of equilibrium of a material point solicited by any number of forces S, S, S', and constrained to move on a curved surface : if the position of M on the sur- face is not given, then the two equations, resulting from the elimination of a, combined with the equation of the surface, m:=0, are sufficient to determine the three coordinates of the point. Wlien the forces and position of the point are given we obtain a by means of one of the three preceding equations, from which we can collect immediately the value of R, and consequently the pressure ; the investigation of R would be considerably abridged f ^" 1 ( ^" I by making the axis of j- to coincide with the normal, for then a. < y- >• , x.-\ |r~ ( ' are equal respectively to nothing, and a -! -i— r = i? A'. 4 — > = /f , for in this case ""•Vi] = li] = ^= ^'"^'^ ^il'r ^{4}' '"^ = '" ""''^^' '^^ '•*'" have y = 0, Z = 0, which indicate that the forces resolved respectively parallel to two lines in the plane which touches the surface in the given point, are equal to nothing ; this also follows from considering that the resultant of the forces is necessarily perpendicular to the surface. If the variations 5.r, ly, Iz, are supposed to belong to the surface then we shall have XS-r-J- Y'^y\-Z'iz = 0, and substituting for Sz its value in terms of ?x and 5^, which we get by means of the equation \ -r- f • ^-^ + ^ "j" !" • ^^ + "! "sT ( ' ^^ ~ ^• we can obtain immediately the equations of condition ix dy dx iz 16 CELESTIAL MECHANICS, If we deterniiue the variations ix, Si/, Sz, so that they may appertain at the same time to the two surfaces, and consequently to the curve formed by the canal ; Sr and Sr' will vanish, and the preceding equation will be reduced to the equation (^b) which therefore obtains in the case where the point is constrained to move in a canal ; provided that we make two of the variations ix, Sy, Sz, to disappear by means of the two equations which express the nature of this canal. Let us suppose that u = 0, ?/— are the equations of the tAvo surfaces whose intersection forms the canal. If we make R V(|) Su \- / hi \ * , , Su ^ Sx Sy oz '■ the equation (d) will become 0=2. S. Ss. + A. Su + x'.Su', in which the coefficients of each of the variations Sd; Sy, Sz, will be se- parately equal to nothing ; thus three equations will be obtained, by means of which the values of a and x' may be determined, which will give R and R' the reaction of the two surfaces, and by composing them we shall have the reaction k of the canal on the point AI, and conse- quently the pressure of this point against the canal. The reaction re- solved parallel to the axis of a: is equal to ^ Sx' ' ^ Sx ^ Sx ^ Sx •' * When the point is forced to be on a canal of simple or double curvature there is only one equation of condition, which is obtained by eliminating A and >' ; this equation combined with the equations m = 0, «' = are sufficient to detmiine the coordinates of the PART L— BOOK I. 17 therefore the equation of condition u = 0, u'=0, to which the motion of the point M is subjected, express by means of the partial differentials of functions, which are equal to nothing in consequence of tliese equa- tions, the resistances with which the point is affected in consequence of the conditions of its motion. It appears from what precedes that the equation (/>) of equilibrium obtains universally, provided, that the variations Sj:, Sy^ Sz, are subjected to the conditions of equilibrium. This equation may be made the foun- dation of the following principle. " If an indefinitely small variation be made in the position of the " point M, so that it still remains on the curve or surface along which " it ought to move, if it is not entirely free ; the sum of tiie forces " which solicit it, each multiplied by the space through which the " point moves in its direction, is equal to nothing, in the case of an " equilibrium."* The variations Sx, Sy, iz, being supposed arbitrary and independant, it is possible to substitute for the coordinates .r, y, z, in the equation (a), three other quantities which are functions of them, and to equal the coefficients of the variations of these quantities to nothing. Thus naming p the radius drawn from the origin of the coordinates, to the D point of the canal where the given forces constitute an equilibrium, in this case it is only required for the equilibrium of the point that the resultant of the forces should exist in a plane perpendicular to the element of the curve on which the point is situated, from whence it appears that the position of the resultant is more undetermined than when the point exists on a curved surface. See Notes to No. 9. We might simplify the investigation of the pressures and obtain immediately the equation of equilibrium between ths forces by taking two of the axes in the plane of the normals of the surfaces whose intersection constitutes the curve, for then we shall have at once Z—O, the third axis is in the direction of the tangent to the curve formed by the intersection of the two given surfaces. * The equation (b) obtains universally, but under different circumstances, according at tlie point is free, or constrained to move on a surface ; in the former case V the resultan: of all the forces vanishes, and vS.S.Ji. r= V.hi must vanish; in the latter case Fhasfe 16 CELESTIAL MECHANICS, projection of the point M, on the plane of x and y, and it the angle formed by p and the axis of x, we shall have T=p. COS. TT ; yzzfi. sin. v. If, therefore in the equation [a), we consider 2^, s, sf as functions of - f. It, and 2 ; and then compare the coefficients of Si?, we shall have _1 S^ ^ is the expression for the force V resolved in the direction o( the element p. S-ar. Let V be this force resolved parallel to the plane of X and y, and P the perpendicular demitted from the axis of z on PV' direction of V', parallel to the same plane ; will be a second ex- P pression for the force V resolved in the direction of the element f Jw ; therefore we shall have PF.= V. Su \- If we conceive the force V to be applied to the extremity of the per- pendicular P, it will tend to make it turn about the axis of Z ; the product of this force, by the perpendicular, is denominated the moment of the force V with respect to the axis of z ; therefore this moment is equal to V.\ —1 ; and it appears from the equation (e), that the moment of the resultant of any number of forces is equal to the sum of the moments of these forces.* finite value, but its direction being perpendicular to the surface or the variation of this per- pendicular must be equal to nothing, and consequently in this case also ^.Sis^ Viu must vanish. * The force V resolved parallel to the axis of a = -^-^ =, by substituting for PART L— BOOK I. I9 X its vftlue V.^ '■ ) this last force resolved in the direction of the elemeni u f. dv, i. e. perpentBcuter to ^= V.— l^_i'-^ tr (by substituting for 1/ its value) u ^ V. iSl '. • sin. a- in like manner if we resolve the force V parallel to the axis of «, and then this last force in the direction of 5 3»-, it frill be equal to V. U'^'";""— ). ^j^g^ ^ ti These forces in the direction of j. Stt act in opposite directions, therefore their difference ^^ ((?■ sin. a- — I)). COS. T — (j. cos. ?r — a), sin. ?r)_) is the expression for that part of die force V in the direction of the element ^.Jjr, which is really efficient, this expression """■{■£■}' ^""^ "'" ^^' ^"^' '^— ")"+(?•*'"• ^—l>'+(z—cy (by substituting for J and 1/ their values) ; therefore taking the derivitive function, a- being considered as the variable, we shall have, u. i -r— f = — g. sin. a-. (5. cos. a- — 0)+^. cos. v. (5. sin. 3- 6y. ••• — \ ^\ = — U<^"^- '^- («• ^'°- 'r— 6J— sin. T. (?. COS. ^—a)),= for conceiv- g t OTT J U p ing the force V to be resolved into two, of which one is perpendicular to ^ , tlie otlier being in the direction of 5, the triangle constituted by tliese forces will be similar to a triangle, two of whose sides are ^ and P, and the third side = F' produced to meet P, ■■■ that part of the force V wliich is perpendicular to 5 is to V as P to 5 •.• it is equal to PV From the definition that has been given in this No. of the moment of a force with respect to an axis, it appears that it can be geometrically exliibited by means of a triangle, whose vertex is in this axis, and whose base represents the intensity of the , force, it vanishes when the resultant V vanishes, and also when P vanishes, i. e. when the resultant piisses through the origin of the coordinates. See Notes to No. 6. Let X and Vindicate, as in the preceding notes, the force V, resolved respectively pa- rallel to the axes of x and 3^, X=F.ifi:±l, Y^ V.^MlI^, the expression for these u u forces resolved perpendicular to e=F. i^^llii-- -, V.liUJ. f, their difference « J " f = = ^ ; we are enabled by means of tliis expression to deduce the equa- tion of the right line, along which the resultant is directed, for the equations of its pro- d2 20 CELESTIAL MECHANICS, jection V on the plane of x y is y— ^ = — — . ( x — a), Xy — Xh = Yx — Ylc. Let L be A. equal to Yx — Xy, and the preceding equation will become b = — .a " we might \ A derive similar expressions for the projection of V on the planes of r and 2, and y and z, from whence it is easy to collect the equation of the right line along which V is directed, — Y" indicates the distance of the origin of the coordinates from the intersection of V with the axis of y, and -^ indicates the distance of the origin of the coordiaates from the intersection of the resultant V with the axis of x. Yx — Xy= Ya — Xh shews that it is indifferent what point of the direction of V is considered. Yx — Xy = when V = 0, and also when its direction passes through the axis o z. PART I— BOOK I. 21 CHAPTE21 II. Of the motion of a material point. 4. A point in repose cannot excite any motion in itself, because there is nothing in its nature to determine it to move in one direction in pre- ference to another. When solicited by any force, and tlien left to itself, it will move constantly, and uniformly in the direction of that force, if it meets with no resistance. This tendency of matter to persevere in its state of motion or rest, is what is termed its inertia ; it is the first law of the motion of bodies. The direction of the motion in a right line follows necessarily from this, that there is no reason why the point should deviate to the right, rather than to the left of its primitive 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 should continue without intermission or not. Indeed, as a body is in- capable of exciting any motion in itself, it seems equally incapable of producing any change in that which it has received, so that the law of inertia is at least the most natural and the most simple which can be imagined ; it is also confirmed by experience. In fact, we observe on the earth that the motions are perpetuated for a longer time, in pro- portion as the obstacles which oppose them are diminished ; which induces us to think that if these obstacles were entirely removed, the motions would never cease. But the inertia of matter is most remark- able in the motions of the heavenly bodies, which for a great number of ages have not experienced any perceptible alteration. For these rea- sons we shall consider the inertia of bodies as a law of nature ; and when we observe any change in the motion of a body we shall conclude that it arises from the action of some foreign cause. €2 CELESTIAL MECHANICS, In uniform motions the spaces described 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. From these differences has arisen the idea of velocity, which, in uniform motions is the ratio of the space to the time employed in describing it. Thus s representing the space, / the time, and v the velocity, we have v— — . Time and space being heterogeneal and consequently not comparable quantities, a determinate interval of time, such as a second, is taken for a unit of time, and in like manner a portion of space, such as a metre for an unit of space, and then time and space become abstract numbeis, which express how often they contain units of their species, and thus they may be compared one with another. By this means the velocity becomes the ratio of two abstract numbers, aiad its unity is the velocity of a body vi^hlch describes a metre in one second. 5. Force being only known to us by the space which it causes to be described 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 cause to be described in a second of time, a space equal to the sum of the spaces which each would have caused to be described separately in the same time, or in other words, that the force is proportional to the velocity ; but of this we cannot be assured a p7~iori, in consequence of our ignorance of the nature of the moving force. Therefore it is ne- cessary on this subject also to have recourse to experience, for whatever is not a necessary consequence of the few data which we have on tke nature of things, must be to us the result of observation. Let us name v the velocity of the earth, which is common to all the bodies on its surface, let f be the force with which one of these bodies. M is actuated in consequence of this velocity, and let us suppose that V ~ f'9{,fy is the relation which exists between the velocity and the force, ^f) being a function oi f which must be determined by expe- rience. Let a, b, c. be the three partial forces into which the force / may be resolved parallel to three axes which are perpendicular to each other. Let us then suppose the moving body M to be solicited by s PART L— BOOK I. 23 new force, f, which may be resolved into three others a', h', c, pa- rallel to the same axis. The forces by which this body will be soli- cited parallel to these axis will be a-\-a', b-\-b\ c-{-c', naming F the sole resulting force, by what precedes we shall have F = y^^'l « + (6+i,')« + (c-t-cO* If the velocity corresponding to i<'be named U ; * —■ — will be this velocity resolved parallel to the axes of a, thus the relative velo- city of the body on the earth parallel to this axis will be -^ — — — — '— or(a + «')' 'P'i.P) — <^' ff- The most considerable forces which can be impressed on bodies at the surface of the earth being much smaller than those by which they are actuated in consequence of the motion oi' the earth, we may consider «', 7/, c', as indefinitely small quantities relative to f; therefore we shall have F ^ f •\ „ t and ? (F) = .(/') will be constant, and the velocity will be pro- portional to the force ; it will be also proportional to it if the function ^ ,t for during an inde- ^ dt dt dt *' * By thus referring the position of a point in space to rectangular coordinates, all curvilinear motion may be reduced to two or three rectilinear motions, according as the curve described is of simple or double curvature. For the position of the moving point is completely determined when we are able to assign the position of its projections on three rectangular axes, each coordinate represents the rectilinear space described by the point parallel to the axes to which it is referred, it will consequently be a given function of the time ; and if we could determine these functions with respect to the three coordinates, the species of the curve described might be assigned by eliminating the time by means of the three equations between the coordinates and the time. t The space being a function of the time, dx = v.dt is the limit of the value of the incre- 30 CELESTIAL MECHANICS, finitely small portion of time, they may be considered as uniform, and therefore eijual to the elementary spaces divided by the element of the time. Consequently the velocity with which the moving body is solicited at the connneii; enient of a new instant, is —+P.dt ; ^ +Q.di; Jl+Rdt; dt ' dt dt or ^+d.^-d.^+P.dt; dt dt dt ±+d.-^-d.^ + Q.dh dt dt dt ^ , ^A.d.J^^d. — +R.dt; dt^dt dt ^ but in this new instant, the velocities with which the moving body is actuated parallel to the coordinates x, y, z, are evidently dx , dx dii J dy dz , ^ dz 1- d. ; — ^ + d. -i^; +«• —z ; dt dt ' dt dt ' dt ^ dt' ment of the space, when dt becomes indefiiiitely small ; we can assign tJie actual value by means of Taylor's theorem ; for if i receive the increment dt, then {x=f{t) becomes x'=f{t^di) ^ , s dx , d-x df^ , d'x dl^ , ^ , ,. ... x'-x = / {i+dt)-f (t) = — . dt+--r- . _+--•—-+ &c. by malong dt' ^ df 1.2 ' dt^ 1.2.3 I sinrp ^ dt dx dt indefinitely small all the terms but the two first may be rejected ; and since -— is the d^x CoefiScient of dt it represents the velocity, and since is the coefficient of dt^, it is proportional to the force; consequently if the action of the forces solicit- d^t ing the point should cease suddenly —j^ would vanish, and the point would move d 'jc d^x with an uniform velocity, if instead of vanishing -— • became constant, then — — , and all subsequent coefficients would vanish, and the motion of the point would be composed of an uniform motion, and of one uniformly accelerated, both commencing at the same instant. PART I.— BOOK I. ' SI therefore the forces -d.— + V.dt, — d -^+ Ci-dt, -d. ^+B.dt, dt - dt dt must be destroyed, so that, if the point was actuated by these sole forces it would be in equilibriuiu. Thus if we denote by So:, St/, Sz, any varia- tions whatever of the three coordinates jt, t/, z, which variations are not necessarily the same with the differentials d^, dy, dz, that express the spaces described by the moving body parallel to the three coordi- nates during the instant dt, the equation {b) of No. 3, will become 0=,^;^. \d. ——P.dl.l -\-Sy, \d. !k—Q.dt.l +Sz.\d. ^—R.dtX. (/)* i dt i i dt 3 L dt ) We may put the coefficients of ^.r, Sy, Sz, separately equal to nothing ; if the point M be free, and the element dt of the time being supposed constant, the differential equations will become dt ■ ' dt" dt» * From the equation (J~) it appears that the laws of the motion of a material point may be reduced to those of their equilibrium, we shall see in No. 18, that the laws of the mo- tion of any system of bodies are reducible to the laws of their equiUbrium. f If P, Q, R, are given in functions of the coordinates, then by integrating twice we shall obtain the values of x, y, and z, in functions of the time ; two constant quantities are introduced by these integrations, the first depends on the velocity of the point at a given instant, the second depends on the position of the point at the same instant. If the values of the coordinates x, y, z, which are determined by these integrations, give equations of this form, x=a.f{t), yz:^b.f (t), z=c. fit), the point will move in a right line, the cosines of the angles which the direction of this line makes with x, y, and z, are respectively equal to — — . . — . the constant quantities «, b, c, depend' on the nature of the function y(0. if/(0 ='; a, b, c, re- present the uniform velocities parallel to x, y, and z, the uniform velocity of the point = \/a*+F+cs" if/(0 = -i^> -^ , respectively, (see second note to at at (it the preceding number) consequently the motion of the point will become uniform, and its direction rectilinear, •.• if v express this velocity we will have, by first note to No. 6. t PART I.— BOOK I. S3 c being a constant quantity. — ^ T-_!: is the square of the ve- locityofi/, which velocity we will denote byu; therefore if Pdx,+ Q.dij, + i2(/2, is an exact differential of a function ?>, we shall have This case obtains when the forces which solicit the point M are func- tions of the distances of their origins from this point. In fact, if ^, 5', &c.* represent these forces, s, s', being the distances of the point M F (See Lacroix Traite Elementaire, No 139.) The rectilinear direction is that of the tan- gent, for if A, B, C, denote the angles which this direction makes respectively with x,y, z, we shall have v. cos. A = -j-, v. cos. B = —^ , v. cos. C = —^ , by substituting dx for V. its value, which has been given above, and tlien dividing we obtain cos. A = — j ds COS.B =-j— , COS. C = — ^; but these are the cosines of the angles which the tangent makes with the coordinates ■.• the tangent coincides with the hne along which the point moves when the forces cease. * If P.dx-\-Q.dy + Rdz =f[x, y, z, ) then u^ =c+2,/ ( j, y, z,) let A be the velocity corresponding to the coordinates a,h,c; then A = c+2. y (a, b, c,) •.• v- — A':m'2. f {x, y, z) — 2./(n, h, c,) •.• the difference of the squares of the velocities depends only on the coordinates of the extreme points of the line described ; consequently when the point describes a curve, the pressure of the moving point on the curve does not affect the velocity. The constant quantity c depends on the values of v, and of x, y, z, at any given instant. When the moving point describes a curve returning into itself, the velocity is always the same at the same point. If the velocities of two points, of which one describes a curve, while the other de- scribes a right line, are equal at equal distances from the centre of force in any one case, they will be equal at all other equal distances. If the force varies as the ^i* power of the distance from the centre, then s and / be- ing any two distances, (p or f(x, y, z,) := /' + ^' •.• v*—A\ s" '^^ — s'"'^^ . In tills case also the differential of the velocity r= s.^ds, therefore by erecting ordinates 34 CELESTIAL MECHANICS, from their origins ; the resultant of all these forces multiplied by the variation of its direction will, by No. 2, be equal to X.SJs ; it is also equal to PJ-v + QJ/j + RJz; therefore we have Pj:i:+QJij + E.Sz=^l.S.Ss. and as the second member of this equation is an exact variation, the first will be so likewise. From the equation (g)* it follows, 1st, that if the point M is not proportional to s", we can exliibit the figure which represents the square o^ the velocity, \rhen n is positive the figure is of the parabolic species, when negative it is hyperbolic. It' the distaiices increase in arithmetical progi'ession, while the lorce decreases in geo- metric progression, the figure representing the square of tlie velocity will be the logarith- mic curve. See Principia Matthematica, lib. 1, prop. 40, 39. If P(/j;-+-Q.(/y-l-iJcfe be an exact differential, then -; — ZZ — — ; — ;— — — ;— ■+■ &c. "^ dj/ dx dz dx P,Q,R, must be functions of ^, y, and z, independant of the time •.• if the centres to which the forces were directed had a motion in space, the time would be involved, and conse- quently P.f/j4- Q.r/y-f/i.f/;:, would not he an exact differential, for then the equations dP dR ^ — 1- &c. would not obtam. dz dx When the forces P,Q,R, arise from friction or the resistance of a fluid, the equation P.dx-\- Q,.dy\-R.dz, does not satisfy the'preceding conditions of integrability, for since P.Q,R, de- pend on the velocities > -j^, — in tliis case ; it is evident that P.dx-\- Q.dt/-\-Rdz cannot be an exact differential of a function of x, y, and ;::, considered as independant varia- bles ••• to integrate P.dx+Q.dy-[-R.dz, we should substitute the values of these va- riables and their difTerentials in a function of the time, which supposes that we have solved the problem ; consequently when the centre to which the force is directed is in mo- tion, and when the force arises from friction or resistance, the velocity is not independant of the curve described. * The velocity is constant when/ (x,i/,z) is constant ; and also when f{x,y,z,) vanishes; when the point is put in motion by an initial impulse, the motion is unifonn, and its direction rectilinear, a.ndv'^ — A'^, = c, — •- = c. -^ =.c", for then -I — — > = P, (it dt dt \ I'f ) {d'lJ 1 I d'z ~> —j-~- ( ^^ Q-'\ ■ I 2 ( = ^^ are equal respectively to nothing. The velocity lost by a body, in its passage from one plane to another, is proportional to PART I.— BOOK I. - 35 solicited by any forces, its velocity is constant, because then iprzO. It is easy to be assured of this otherwise, by observing, that a body moving on a surface or on a curved line, looses, at each rencounter with the indefinitely small [)lane of the surface, or indefinitely small side of the curve, but an indefinitely small part of its velocity of the second order. 2dly. That the point M, in passing from a given point with a given velocity, will have, when it attains another point, the same velo- city, whatever may be the curve which it shall have described. But if the point is not constrained to move on a determined curve, then the curve described possesses a singular property, to which we have been led by metaphysical considerations, and which is, in fact, but a remarkable consequence of the preceding differential equations. It con- sists in this, that the integral. /t'.r/5 comprised between the two extreme points of the curve described, is less than on any other curve if the point is free, or than on any other curve subjected to the same surface if the point is not entirely free. To make this appear we shall observe, that P.dx-^Q.dy + Rdzhemg supposed an exact differential, the equation (^'•) gives f.J'u = P.Sx-irQJy+RSz. in like manner the equation {f) of the preceding number becomes, dx dii d" = §x.d.-^ + Si/.d.-^ + Sz.d. — —v.dt. Sv. dt ^ dt dt naming ds the element of the curve described by the moving point, we shall have v.d(=ds ; ds = .ykt'^+dj/^-tdz*, f2 the tliffercnce between radius and cosine of the indination of the planes, i. e. to the versed sine, or to tlie square of the sine ; and when th.e curvature is continuous the sine is an indefinitely small quantity of first the order, •.• the velocity lost, is an indefinitely small quantity of the second order. 36 CELESTIAL MECHANICS, consequently =li:d.~- + Sy.d. JL + Sz.d.-^ ds.Sv, (li) dt dt dt ' V. / by differentiating with respect to <5', the expression for ds, we have ds . r dx . J , dy . ^ dz ^ r dt dt ' dt ^ dt. The characteristics d and S being independant, it is indifferent which precedes the other ; therefore the preceding equation may be made to assume the following form : , , , (dxSx + dy.Sy + dz.Sz) , ^ f/^r ^ r dy , r dz v.S.ds=d. ^ -^ -^ — ^x.d. — Si/.d. -^ —Sz.d. -— , , dt dt ^ dt dt by substracting from the first member of this equation the second member of the equation (//) we shall have . . , , d. (dxJj: + dy.hi + dz.SzY S irds) zr ^ This last equation integrated with respect to the characteristic d, gives I. fv.ds = const.+ ^^•^^■+^.^%+^^-~-^^ ^ *Ford.i±if±±M±if)^ jx , <: dx.lx-X-dy.h/-i-dz-^=\ prescribed m the text, v.e obtain v.d.ds-j-ds.iv=i.{v,dsi= d. i ^ < • This equation being integrated with respect to the characteristic d gives/. d.{v.ds.) const.-)- '^^•'■'+('?Ay+dz-^z _ ^^j^^^^ ^j^^ ^^^.^ extreme points of the curve are fixed, the variations 3x, Jy, 2z, ot the coordinates must be equal to nothing at these points ; con- sequently the variation o{/.(v.ds) is equal to nothing, and •.• r(v.ds) is either a maxi- muni or niininium ; but it is evident from the nature of function /. (v.ds.) that it does not admit a maximum. PART I.— BOOK I. 37 If we extend this integral to the entire curve described by the moving point, and if we suppose the extreme points of this curve invariable, we will have S.J'v.ds = 0, that is to say, of all the curves, which a point solicited by the forces P, Q, R, can describe in its passage from one given point to another, it describes that in which the variation of the integral yt'.cf*, is equal to nothing, and in which, consequently, this integral is a minimum. If the point moves on a given surface without being solicited by any force, its velocity is constant, and the integral fv.ds becomes v.fds. Therefore in this case the curve described by the moving point is the shortest which it is possible to trace on the surface from the point of departure to that of arrival.*. 9. Let us determine the pressure of a point moving on a curved surface. Instead of eliminating from the equation [J") of No. 7> one of the variations Sx, Sy, Sz, by means of the equation to the surface, we can by No. 3 add to this equation, the differential equation of the sur- * Wlien the velocity is constant the integral fv.ds, becomes v. f. ds=v.s ; and since s is a minimum, the time of describing s, which is proportional to s in consequence of the iinifomiify of the motion, will be a minimum in like manner. Since the equation l.J.{v.ds.) =0, has place when Pdx-\-Qfly-\-R.dz is an exact differential, it belongs to all curves that are described by the actions of forces directed to Jixed centres, the forces being functions of tlie distance fiom those centres ; and if the fomi of these functions was given we could determine the species of the curye described, by substituting for v its value in terms of the force, (which we have by a preceding note), and then investigating by the calculus of variations, the relation existing between the coordinates of the curve which answers to the minimum of the expressiony(Ti.rf«). If S the force varied as — ,- by making use of Polar coordinates we would arrive at the polar equation of a conic section, in which the origin of the coordinates would be at the focus of the section ; if S was proportional to s the resulting equation would be also that of a conic section, the origin of the coordinates being at the centre of the section. From the preceding property the known laws of refi-action and reflection have been deduced. Mr. Laplace has also suc- cessfully applied it to the investigation of the law of double refraction of Iceland chrystal, which was first announced by Huyghens, and afterwards confTrmed by the celebrated ex- periments of Malus on the polarization of light. See a paper of Laplace's in the volume of the Institute for the year 1809. 38 CELESTIAL MECHANICS, face multiplied by the indeterminate — xdt, and then consider the three variations Sx, Sjj, Sz, as independant quantities. Therefore let ii = be the equation of the surface, by adding to the equation (J') the term —aSu, (It. the pressure will, by No. 3, be equal to / \du } I dx ) « C f/M / « ) dii )^ I At first let us suppose that the point is not solicited by any force ; its velocity » will be constant, and since v.dt=ds; the element of the time being supposed constant, the element ds of the curve will be so like- wise, and by adding to the equation (./) the term — xJu.dt, we will obtain the three followino- : = V. d'^x { du } ^ , d~y S ^" ds» from which we may collect , d"y \ du I du I ds"- ( dx ) ds^ <^ dy n » d'Z = w. ax but ds beino- constant, the radius of curvature of the curve described by the moving point is equal to ds' t • By substituting for iW its value -r-^^ we eliminate the time i, if the resultin-r equations be squared, we obtain, by adding their corresponding members, _ _. ''■ ^^ j ^ \ dy S "^ \ dz f ■ f This expression for the radius of the osculating curve may be thus investigated : let a, b, c, express the coordinates of the centre of this circle, its radius being equal to r, PART I.— BOOK I. 39 ••• by naming this radius r we shall have * c r then r*={x—aY-\-{ij—bY-^[z—cY ; dx. {x—a)-{-dy.{y—b)-\-dz. (2— c), the differential of tills equation is equal to nothing, as any one of these coordiiuites may be considered as a function of the two remaining, we can obtain the following equations of partial dif- ferences dx. {x—a)Jr^z. (z— (•)=0, cii/.{j/—b)+dz. (z—c) =0, (the values of dz in these equations are evidently different,) consequently we have cf'x. (j — a)-{-d^z. (z — c)+dx* +dz^-=0;d-y {y—b)J^d--z.{z-c)-\-dy-^ + dz'==0, V (x-«)=— £ {z—c), (y-b) = dz . . -7—, (z — e), and since ds is supposed to be constant, we have d^x,dx-\-d''y.dy+dfz.dz =0, (d'z in this equation refers to the entire variation of rfz,) consequently z being consi- dered as a function of x and y, we obtain d^x.dx+d^z.dz==0;d^y.dy+d^z.dz = 0; ■.■^=-p^; ^ — ^^ '^ these values being substituted in place of -7^ -; in the preceding equations we shall dx dy have d"^ X d^y •.• by adding together the two preceding differential equations of the second order, sub- stituting for (x — a) (y — b) their values, and observing that the whole variation of z is equal to the sum of the partial ones in these equations, we obtain, ~ . (z — c)+dx- -i-dy- -\-dz'=0, consequently - (dx-+du^+dz'')^ d^ x^ d*y~ by substituting for (.r — aY {y—b^ their values -—-;-. )s — c)- ;■ J^^ .(z — c = ), whicli have been given, we obtain {x-a)-+{y-by-M—cy=.^^ ^fli'^^T , . {d^x^'-^d^y^^H') d^x^-\-d'yi-d- 1 ds'' 4,0 CELESTIAL MECHANICS, consequently the pressure which the point exercises against the surface is equal to the square of the velocity divided by the radius of curvature of the curve described. If the point moves on a spheric sui'face,* it will describe the circum- ference of a great circle of the sphere,, which passes through the pri- mitive 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 circle ; therefore its pressure against the surface, or what amounts to the same, against the circumference which it describes, is equal to the square of the velocity divided by the radius of this circle. If we conceive the point attached to the extremity of a thread desti- tute of mass, having the other extremity fiistened to the centre of the surface, it is evident that the force with which the point presses the circumference is equal to the force with which the String would be tended if the point was retained by it alone. The effort' which this point would make to tend the string, and to increase its distance from the centre of the circle, is denominated the centrifugal force ; there- fore the centrifugal is equal to the square of the velocity divided by the radius. The centrifugal force! of a point moving on any curve whatever is * If the point move on a spherical surface, the motion will be necessarily performed on a great circle, for the deflection can only take place in the direction of radius, and in the plane in wliich the~body moves. -f- If the body moves on any curve whatever, the centrifugal force =: — , this force acts in the direction of a normal to the curve, and if all the acceleratiag forces which act on the point be resolved into two, of which one is in the direction of the normal, and the other in the direction of the tangent, the resultant of the centrifugal force, and of the former of these decomposed forces, is the entire pressure with which the point acts against the curve, and the resistance of the cui^ve is an accelerating force equal and con- trary to this resultant. If we denote this normal force by L, and if A, B, C, be the angles which it makes with the coordinates x, y, z, respectively, then by the equation (y) and No. 3, we have '^=P+L. cos. A; -^ = Q+L. cos. B; ~ = R+L. cos. C; - PART I.— BOOK I. 41 equal to the square of the velocity divided by the radius of curvature of the curve ; because the indefinitely small arc of this curve is confounded with the circumference of the osculating circle. Therefore we shall and since -J—, —--' —j- , express tlie cosines of the angles which the tangent makes" with X, y, and s, ^-. cos .<4+-,- . cos.i?+— '^. cos. C.=0; because the tangent is per- ds - lis as ' pendicular to the normal. (See last note to No. 1). We liave also cos ^A-\- cos. -B-\- cos. ^C=l, and the four undetermined quantities L, A, B, C, being eliminated between the five preceding equations, the resulting equation will be one of the second order be- tween x, y, I, and / ; this equation combined with the two equations of the trajector}' which are given in each particular case, are sufficient to determine the coordinates in a function of the time. See notes to No. 3, and No. 7. The elimination of L, A, B, C, might be effected by one operation ; for multiplying the three preceding equations by dx, dy, dz, respectively, and adding them together, we obtain the following equation : ^' dt^ ~' '^ ' ~ ^•'^^+ ^•^'/•+ ^'^--i- ^- ('^os- ■^•(1^+ COS. B.dy+ cos.C.ffe.) (the latter part of this second member is equal to nothing, as has been already remarked ;) and since ds-=dx~~ -\r d ij\ ^ dz'- , d''s.ds=d''x.dx-{-d^y.dy+d^z.dz, ;• we shall have d's _„ dx dy d^ df^-^-ir^^-'dT^^-ds ' from this last equation it appears that tlie accelerating force resolved in the direction of the tangent, is equal to the second differential coefficient of the arc considered as a func- tion of the time, •.• this expression for the force has place whatever be the nature of the line along which the point moves. See Notes to No. 7. In like manner it appears that the expression for the force in the dii'ection of the tangent is altogether independant of L. d's It is also evident, that when there is no accelerating force -j-j- = 0, this also follows from the circumstance of the velocity being uniform when P, Q, R, are equal to notliing. Let V denote the resultant of all the accelerating forces which act on the point, and 6 the angle which this resultant makes with the normal, then V. cos. 6 will be the ex- pression of the resultant resolved in the direction of the normal ; and when all the points of the curve exist in the same plane, the entire pressure will be equal to the sum or dif- ference of -- — , and V. cos. 6, according as these two forces act in the same or in con- 42 CELESTIAL MECHANICS, have the pressure of the point on the curve which it describes by add- ing to the square of the velocity, divided by the radius of curvature, the pressure produced by the forces which solicit this point. *t traiy directions, •■• +/,=: =fc: \. V. cos. «. We can express this pressure otherwise by means of the rectangular coordinates ; for since P, Q, are the expressions for the force V resolved parallel to x and _?/, these forces resolved in the direction of the normal are equal respectively to P. -j-; Q. —z—, (the signs of — -, and -j-, are evidently dif- ferent) consequently we have r. cos. 6=JfP.-f- + Q.— , and v L = — -h P. -f- -fQ. -j-, as (Is r as as therefore if we know the equation of the trajectory, and if we have also the values of P and Q in terms of tlie cooi-dinates, we can determine the velocity, and consequently L, and d'x d^v d^z substituting this value of L in the expressions for -j— ' , ' — ~ , which liave been given in the foregoing part of this note, we might b)' integrating determine the velocity in the direction of each of the coordinates, and also the position of the point at a given moment. If the point be attached to one extremity of a thread supposed without mass, of wliich the other extremity is fixed in the evoluf e of the curve described, then the point receiving such an impulse, that the string remaining always tended, may unroll itself in the plane of the evo- lute, it will describe the given curve ; the direction of the string is always perpendicular to the curve, and its tension is equal to the normal pressure on the trajectorj^, and conse- 1 ■"* . P^du + Qdv „ ... . - ^ ^ ,. quently equal to 1 ^ . By equating this expression ot L to notmng, we can derive the equation of those trajectories in wliich the motion is fi-ee, or in which the trajectory may be described freely, i. e. it is not necessary to retain the point on the curve by means of a ' thread, or a canal, or any perpendicular force. * If the motion is performed in a resisting medium, this resistance may be considered as a force acting in a direction contrary to that of the motion of the body, consequently it must tend to some point in the tangent. If we denote tliis resistance by / its moment is equal — 7.Jj (j = >/{.T- 0' + (.y — "')'+C~ — ")*' ^' "'j "> ^6 the .coordinates of the cen- tre of the force 7. therefore 3J = ^-^^. Sx-f-^i!=^^. ^y+ ^^^^. h; if we suppose .r — I dx the centre of force in the tangent, then i— \/dx'+di/^+ds'' =ds •: — ~ — "^ ' •*,. PART I— BOOK I. 43 y— m^_ J) _ z—i_ __ _ ^^j j^. _ j^_^ — 1^^ J _^ — ^ ^ j^^ .^^j^g resisting nie- i ds t as as as as dium was in motion, its motion must be compounded with the motion of the body, in order to have the direction of the resisting force. If da, db, dc, be the spaces described by the medium, wliile the body describes ds, these quantities must be added or subducted from dx, dy, dz, in order to have the relative motions, and as ds = y/dx'' +dij*-{-dz^, if Me dx — da . make d(r ;^ -/{dx — da)'--i-{cli/ — db)'-]^(dx — dc)*, we shall have Si = — '^ °'^+ ^ . jwJ — 1 —, ^z. Tlie resistance / in general= i|/ (v), a function of the ve- dr ■^ da- locity, in this case it is a function of the relative velcoity. By the preceding investigation we ai-e enabled to apply our general formula to motions made in resisting mediums without entering into a particular consideration of this species of motion. However the analysis becomes very complicated when the forces which com- pose P, Q. R, exist in different planes, and as in this case, the causes on which the va- riation of the velocity depends, arise in some measiu-e from the velocities themselves, we are not permitted to regard P.dx+ Q.d^ i-R.dz, as an exact differential of three inde- pendant variables, which facilitates our investigations when the motion is performed in a vacuo. See Notes to Nos. 8. We might also reduce to our general formula, the differential equations of motion, when the retardation arises from the friction against the sides of the canal. f If the body moved on a surface we might, as before, abstract from the consideration of the surface, and consider the material point entirely free by adding to the given forces anotlier accelerating force, of lohich the intensity is unknown, and of which the direction is normal to the surface, •.• if this force be denoted by L we shall have, by the equation (y ) of No. 7, and by No. 3, the following equations : (m^O is the equation of the suface. See Notes to No. 3). If we eliminate L between these three equations, N will also disappear ; and if the two differential equations of the second order, which result from this elimination, be combined v/ith the equation tc=:0 of the surface^ we can detennine the tliree coordinates of the point in a function of the time. If we multiply the preceding equations by dx, dy, dz. respectively, and then add together the corresponding members, we will obtain d^x.dx+d'^M.du+d'z.dz „ , ^ , „ , ,t ^ f ^« 7 , . f ^" 1 > ■ -~^^ = P.dx+Q.dy+Rdz+N.L. \ f^ \ '^'■'+ \-^ \ '^V^ f 3u 1 \ Y7 W« ; but the last part of the second member is = to nothmg, ' g2 44 CELESTIAL MECHANICS, When the point moves on a surface,* the pressure due to the centri- fugal force, is • qual to the square of the velocity, divided by the radius of the oscuiati .g circle, and multiplied by the sine of the inclination of the plane of this circle, to the plane which touches the surface ; therefore, if ■ ve add to this pressure, that which arises from the action of the forces which solicit the point, we shall hiive the entire force with which the point presses the surface. |sincerfi< = 0, and -! -^ | = i -1 V v if V.dx-\-Q_.dy+R.dz is an exact difFeren- tial, we shall have —r;= P-—, — h Q.~- -\-R.^, as before, and— — = r- = C4- dt^ as dt ds dt^ y{P.dx-\-Q.d^-f-R.dz), audif P,Q,R, and consequently v were given in tenns of the coordinates, we might obtain immediately the differential equations of the trajectory by d^x f ^71 i ' multipljing the equation ---^;-= P+L.A^. ■} -r — j- , by di/ and dz successively, and cit~ (. dx ) then subducting it from the two remaining equations multiplied by dx ; by concinnating ds the resulting equations, substituting for dt its value — , and for i; its value m a function of the coordinates, we obtain two differential equations of the second order, fi-om which eliminating the quantities LN there results a differential equation of the second order be- tween the three coordinates z,y,z, solely; this equation, and the equation «^0 of the surface will be the two equations of the trajectory. * If a point moves on any curve the centrifugal force is always directed along the radius of the osculating circle ; and since the pressure on the surface is always estimated in the direction of a normal to the surface, (see No. 3) if the plane of the trajectory is not at right angles to the surface, the radius of the osculating circle will not coincide with the normal to the surface, and consequently the part of the centrifugal which pro- duces a pressure on the surface is equal to , multiplied into the cosine of the an- gle which the radius makes with the nonnal, but this angle is evidently the comple- ment of the angle which the plane of tlie osculating circle makes with the plane which touches the surface. If the forces soliciting the point are resolved into two, of wliich one is perpendicular to the trajectory, then the resultant of this last force, and of the ccntrifijgal force, will express the whole force of pressure on the curve; if this curve was fixed, it would be sufficient for the pressure to be counteracted, that its direction was in a plane perpendicular to this curve , but if the curve be one traced on a given surface, then, in order that the pressure should be counteracted, it is necessary that the resultant of the forces should be in the direction of a nornml to the surface. Sec note to page 16. PART I.— BOOK I. 45 We have seen that when the point is not solicited by any forces, its pressure against the surface, is equal to the square of the velocity, di- vided by the radius of the osculating circle ; therefore the plane of this circle, that is to say, the plane which passes through two consecutive sides of the curve described by the point is then perpendicular to the surface. This Curve on the surface of the earth is called the perpen- dicular to the meridian ; and it has been proved (in No. 8) that it is the shortest which can be drawn from one point to another on the surface.* * If we make the axis of one of the coordinates to coincide with the normal to the surface, we can immediately determine the inclination of the plane of the osculating circle to the plane touching the surface ; for if we denote by A, B, the angles which the radius of the osculating circle makes with the normal and witli the coordinate which is in the plane of the tangent, and by »i, n, I, the angles which the resultant V of all the forces makes with the three coordinates, the force _ll resolved parallel to these coordinates is equal to - — . cos. A, —. cos. B, A '-. COS. 100°, (because the angle between the radius and r r r tangent to the curve is equal to 100') in like manner the force V. resolved parallel to these coordinates equals V. cos. m, V. cos. n, V. cos. I, since A and m denote the inclination* of the radius of curvature, and of the resultant to the normal, . cos. A-\-V. cos. m, r express the pressure of the point on the siuface, V. cos. n-\-- — cos. 100°, or V. cos. n is the force by which the body is moved ; and since this motion is performed in the di- rection of the tangent, V. cos. l-\ . cos. B, which expresses the motion perpendicular to the tangent must vanish; consequently we have V. cos. /-|- cos. B~0, '.' if F. I, V, and r were given we might determine B, which is = to the inclination of the plane of the osculating circle to the plane touching the surface, it also foUoTvs, that when the point is not soUcited by any accelerating force, cos. B=0, •.• B= 100", or the plane of the osculating circle is perpendicular to the surface, which we have previously established from other considerations. Ifthe plane whose intersection with the surface produces the given curve is not ■perpen- dicular to the surface, then the radius of curvature is equal to the sine of the inclination of the cutting plane to the plane touching the surface, multiplied into the radius of cur- vature of the section made by a plane passing through the normal to the surface, and tlirough the intersection of the plane touching the surface and the cutting plane. See LacroLx, No. SSi. ••• the pressure is the same whether the point move in a greater or 46 CELESTIAL MECHANICS, 10. Of all the forces that we observe on the earth, the most re- markable is gravity ; it penetrates the most inward parts of bodies, and would make them all fall with equal velocities, if the resistance of the air was removed. Gravity is very neai'ly the same at the greatest heights to which we are able to ascend, and at the lowest depths to which we can descend ; its direction is perpendicular to the horizon, but on ac- count of the small extent of the curves which projectiles describe rela- tively to the circumference of the earth, we may, without sensible error, suppose that it is constant, and that it acts in parallel lines. These bodies being moved in a resisting fluid, we shall call b the resist- ance which they experience ; it is directed along the side ds of the curve which they describe ; moreover we will denote the gravity by g. This being premised, let us resume the equation (fj of No. 7, and suppose that tke plane of x and y is horizontal, and that the origin of ^ is at the most elevated point ; the force b will produce in the^^direction of the coordinates .r, y, z, the three forces — b. — , — b.-~ , — b.-^ •.' by ds ds ds No. 7 we shall have F=z—b. ~ ; Q--^b.^ ; R-^b.-^ +g-. * dx ^ , du -o 1 dz ds ds as and the equation CJ]) becomes 0=^;..^^. 't+b.^di.l+Sy.\d.± +b.^dt.l I dt^ ds S ^ ^ dt ^ ds ) ^■Sz.\d.^x.b.~ dt.—g.dt. I * I dt^ ds ^ $ less circle, for the sine of inclination occurs both in the numerator and denominator of the expresiion ; this also follows from considering the proportion of the sagiita of curva- ture in a peipendicular and oblique plane. The investigation of the shortest line which can be drawn between two given points on a curved surface, whose equation is u—0, by the method of variations, leads us to the same conclusions. ?ee Lacroix. The consideration of the shortest line which can be traced cm a spheroidical surface is of great importance in the theory of the figure of the earth. (See Book 3, No. 38.) * Since the force b acts in tlie direction of the tangent or of the element ds of the curve (see note to No. 9,)^ this force rs'solved parallel to the three coordinates jc, y, z, dx dy dz ^ dx di) dz , . ^ , = "•~-ri"-~T'i O'—j— , \ot —r- , -y- , — — are =: to the cosmes of the angles as ds ds ds ds ds " PART I— BOOK I. 4? If the body be entirely free we shall have the three equations , d.v , dx , ^ 1 dy , , dij ,. Q—d,—— +b.---.dt; = d.-~- +b. -^. dt^ dt ds dt ds = d.-^ +b.~.dt—g.dt, dt ds The two first give ±. d.±-^. d. <^ = 0. dt dt dt dt from which we obtain by integrating, dx^=.fdy, f being a constant arbi- trary quantity. This equation belongs* to an horizontal right line, therefore the body moves in a vertical jjlane. By taking this plane for that of x, z, we shall have ?/=:0, the two equations, , dx ^ , dx , ^ , rfs . , dz ,^ J, = d.- hb. -r-.dt; O^d.—-. +b.—-~. dt — g. dt, dt ds dt ds will give, by making dx constant, , ds.d»t ^ d*z dz.d*t , , dz , '=-dF~' ""—dT rfF- +'• ^•^^-^••^^^- From* which we obtain g.dt* = d*z, and by taking the differential which the tangent makes with the three coordinates ; they are affected with negative sign* because they tend to diminish the coordinates. ^ _. ... dt/ , dx dx , dii , tfi/* . , * Dividing -f^. d.— . a.— T- =: 0, by -^ it becomes ^ dt dt dt dt ^ df^ dx ^'\~, ^^^^' ■•■ '^y integrating -r— =y" and dx==f.dy, since tlie equation of the ~dt projection of the line wliich the projectile describes on a horizontal plane, is that of a right line, the body must have moved in a vertical plane, otherwise its projection on an horizontal plane would not be a right line ; this circumstance we might have anticipated from the manner in which the forces act on the body. ^ -wc 1 ', , , dx dx dx,d^t * It we make dt constant in the equation d. ^- + i. -=- . di= 0, we get -; dt ds dt- 48 CELESTIAL MECHANICS, . ^g.dt.dH=d'z, if we substitute for rf»nts value Af[fl, and for dt* hi ds d*z - value . — —, we shall have b ds.d^z i~\% g 2(rf This equation gives the law of the resistance b, which is necessary to make the projectile describe a given curve. If the resistance be * proportional to the square of the velocity, b is ds* equal to h, , h being constant, when the density of the medium is dt* ' , unifoiTO. We shall have then b h.ds* h.ds* g g'dl* d*s. d^z • ■ d*" therefore h.ds= , which gives by integrating — ~ = Sa.c^.t 2.d*z da.* + J.— . dt]= 0, '.• b — — j — , by substituting tliis quantity in place of b, and differen- tiating, we get the expression d^z dzJH ds.d't dz , , d^z dz.d^f , dzJH , -Jt dir +-11^ '■d^'^-s-^' = —t dF- + -d? -"'^'•= d'z -g.dt= 0, •■•by differentiating we obtain d^z ='2g.dt.d^f, and substituting for dt d't its value 4. — ~, and for dt' its value — , we arrive at the following equation, ds g 2g.b. C d'z -\ ^ b dH ds I g i g ^d^r * The value of the consrant coefficient /; is obtained by experiment ; it is different in different fluids, and when bodies of different figures move in the same fluid. ds^ t Since the square of the velocity is equal to -^-j-, the resistance is expressed by ds' , d'-z hds^ ds.d^z d-z h.-j^, vby substituting for rf<= its value — ^ 'WT^ 2{d>z)* '' "'" "" "rf»7 ' PART I.— BOOK I. 49 a being a constant arbitrary quantity, and c being the number whose hyperbolic logarithm is unity. If we suppose the resistance of the me- H ••• 2A.4 = log. d^z^log. F :■ g^"^ ^F.d'z, v -— — = -— . (Let 2a= -^r— ) and we r.ax^ ax- F.d.r:^ shall have 2ac-*' = ^ ; dx being constant it is permitted to introduce dx- as a divisor. The constant quantity a depends on the velocity of piojection, and on the angle which its direction makes with the horizon ; for by substituting — g.dt^ in place of d-z we dx^ n- dx^ shall have ■ — ■= -2 — .c— 2'« is the velocity of the body in the direction of the axis of x at the end of the time t ; let ii be the velocity of projection, and i the angle which its direction makes with the horizon, we shall have at the same time t = 0, .T=:0, 2=0, and —r- = u. cos. 6, :■ ii^. cos. *tf = — -~- . Let h be the height due dt 2a ° to the velocity u, u^ =%/'> *•■ by substituting for u- its value, we deduce a= 4:k cos.^# By making dz=pdx, ds becomes equal to dx.^i-i-p'- , •■• — c-''K ds=2h. cos.* t. dp. c d^z 7 c2*s. VI +i^%i""' ''P = -J^ 5 *•• by integrating — — \-C=2h. cos.^ S.fdp.^l+p\ I =2A. COS.' 6.f-^^J=^ + 2h. cos. e.f J^^ } ='''• COS.' 6. log. (p + ^/H:^), -|-.A. COS.- 6. p. \/l-[-p-, the constant quantity C is easily found; for since p is the derivitive function of z considered as a function of x, at the commencement of the motion, when 4=0, p~ the tangent of the angle of projection wliich is given, '.• C is equal to h. cos.=^ 6. } (log. (tan. <-f-sec. 6) +tan. 6. sec. 6.) i + -y-. By substituting for ; — its value, which we obtain from the equation r— = 2.. 2A 2n /5^ ■•■ ^^ ™'^S'''^S "^^ S^^ ^'="- p' - =" "^ '''^- ■?'' '^^ + 2P" • log.;?, the first equation indicates that.j; has a limit, the vertical ordinate increases inde- finitely, but in a less ratio than ;;, therefore'the descending branch has a vertical asymp- tote. By eliminating log. ;j in the expression for t we get an expression for z fiom which ■we may collect, that according as the direction of the motion approaches towards the vertical, the motion of the body tends to become uniform. When the angle of projection is very small, we can find by approximation the relation which exists between .t, and 2, for that portion of the trajectory which is situated above the horizontal axis ; in this case the tangent is very neai-ly horizontal, •.• p is very small, and Vl+p =l,y.;;.-.-f/« = r/aVr+7- =dx, q.p. and s=,t, for they commence together, and substituting x in place of ^, we have ~ = — gT^^ ^"^ " ^^'"^ ^^ hypothesis very small, cos.^ (t =-- 1, •.• dp =—^-. dx, by integrating this equation, when we know the Ik value of the constant arbitrary quantity which is introduced by the integitition we obtain the value of p and •.• of 2 =j p.dx. See a memoir of Legendre's in the Transactions of the Academy of Beriin for the yeai- 1782. • In this case-rr =5ia, v -^=2ax-t-i, v s=ax'-|-i.«+c. d.X' dx PART L— BOOK I. 51 which we may obtain t-zix. V ■ — +f'- If *> •s, and t, commence to- gather, we shall have c= 0, f'= O, and consequently g which gives 2 '2a These three equations contain the whole theory of projectiles in a va- cuum ; it follows, from what precedes, that the velocity is uniform in an horizontal direction,* and that in the vertical direction the velocity is the same as if the body fell down the vertical. If the body moves from a state of repose b will vanish, and we shall have dz _ _ 1 therefore the velocity acquired increases as the time, and the space in- creases as the square of the time. It is easy bymeans of these formula to compare the centrifugal force with that of gravity. For v being the velocity of a body moving in the circum- ference of a circle, of which the radius is r, it appears from No. 9, that its ^» centrifugal force is equal to . Let h be the height from which the body must fall to acquire the velocity v ; by what precedes we shall have v' = 2g.h ; from which we obtain — «•. zJ—. The centrifugal dx I fJ^ * ^^^ Hr ~ ^^ velocity in an horizontal direction =\/ - --, and -^ = the velocity V ^ci (It in a vertical direction = gt ,b.\/ _£.. 52 CELESTIAL MECHANICS, force will be equal to the gravity g, if h^ Therefore* a heavy body attached to the extremity of a thread, which is fixed at its other extremity, on an horizontal plane, will tend the string with the same force as if it was suspended vertically ; provided that it moves on this plane, with a velocity equal to that which the body would acquire in falling down a height equal to half the length of the thread. 1 1. Let us considej the motion of a heavy body on a spherical surface, denoting its radius by r, and fixing the origin of the coordinates at its> centre, we shall have r' — r' — j/' — ^'=0; this equation being com- pared with that of z^— 0, gives u = r' — a'" — y'' — z" ; therefore if we add to the equation (X) of No. 7, the function Su multiplied by the indeterrai- nnte quantity — x.dt. we shall have 0=Ss. S'd. — + 'ixx.dt. I + ^i/. I d.-^ + 2x.ij.dt. l + Sz. I d. —^ Q.xz.dt—g.d/. I * In this equation we can put the coefficients of each of the variations Sx, Sy, Sz, equal to nothing, which gives the three following equations : Q^d. — + 2\.xdt, dt = d. ^ + 2x.y.dt. dt + ^' d^ O = d. — ^o- 2k.z.dt — g.dt, dt * The plane of the motion behig horizontal, the force with which the string is tended arises entirely from the centrifugal force. t Po.{^|=-...{^ ;=-.,. m=_.. PART I.~BOOK I. 53 The indeterminate a makes known the force with which the point presses on the surface. This pressure by No. 9 is equal to consequently it is equal to 2xr ; but by No. 8 we have _ dx^+dy'^+dz' c+2gz ^ * dt* c being a constant arbitrary quantity ; by adding this equation to the equations {A) divided by dl, and multiplied respectively by x, y, z, and then observing that x.dx+y.dy-\-z.dz =0, x.d''x+y.d^i/-{-z.d'z + dx^ + di/''+dz9=0, are the first and second diflPerential equations of the surface, we shall obtain* * For performing these operations we get c-{-2gz = d? + "rfF ^d^"^ -5^+2^-(^=+y'+s')-g2, therefore we have SAr'' = c-\-?igz, and '2xr = ^^ , •.• the pressure is equal to "*" ° , when the ini- tial velocity c vanishes, the tension of the pendulum vibrating in a quadrantal arc is, at the lowest point, = to three times the force of gravity ; — = the cosine of the angle wliich the r radius r makes with the vertical, therefore it follows that when a body falls from a state of rest, the pressure on any point is proportional to the cosine, of the distance from the lowest point, it is easy to collect, in like manner, that the accelerating force va- ries as the rigiit sine of the angular distance from the lowest point ; we might from the preceding expression for tlie pressure deternune the point where this pressure is in a given ratio to the force of gravity. 54 CELESTIAL MECHANICS, If we multiply the first of the equations (A) by — t/, and add it to the second, multiplied by x, and then integrate their sum, we shall have dr.rfj/ — V'dx _. * dt "^^ d being a new arbitrary quantity. Thus the motion of the point is reduced to three differential equa- tions of the first order, x.dx-\-y.dy = — z.dz, x.dy — y.dx = c'.rf?, dx* >- di/' + dz* ,a -J—J- = C + 22-Z. By squaring each member of the two first equations,t and then adding them together, we shall have i^^+y^) (dx^ + dy*) = c''dt' + z'dz\ * x.dy — y.dx = c .dt shews that the area described by a body moving on a spherical surface, and projected on the plane x, y, is proportional to the time ; the same area pro- jected on the plane x, 2, or y., z, is not constant in a given time ; for if we add to the first of the equations {A) multiplied by — z, the third multiplied by x, and then integrate x.dz — z,"x y , their sum, it becomes equal to = c' ■\- f.{gx.d(), this might have been anti- cipated, as the force ^ does not pass pei'pelually through the origin of the coordinates, ••• x.dz — z.dx, y.dz — z.dy are not proportional to the time, but as there is no force acting parallel to the horizontal plane, x.dy — y.dx must be proportional to the time, f For we have in this case x'^ .dx'' ■\-y'^ Ay'' .\-2x.y.dx.dy = z'^.dz''. x^dy^+y .dx^ — ^x.y.dx.dy = c'*rfr. \'(x''+y'^){dx'' + dy'') =^~c'''dF+z^Mz^. dx^-{-dy'^ :• by substituting for x'^+y*, and 'J~^ ' ^^"^ values we obtain (r* — «») . PART I— BOOK I. S5 If we substitute in place of.r'+3/% and '^' ' -^ ^ their respective dz* values r* — z\ and c + 2^'-2r ~rr''> "^^ ^'^^ '^^^^ on the supposition that the body departs from the vertical dt — ^(r'—z'). (c + 2gz)—c'\ The function* under the radical may be made to assume the form (a — z). (b — s).(2o's+y) ; a, b,f, being determined by the equations {cdt'+2gz.dt^—dz^) = c'^dt-+z\dz^, therefore (r«— =*) . {c+2gz) — c ^). dt^ = r'.rfz^ +Z-&* — z'^.dz'^, consequently — r.dz dt = ^/{r^—z').(c+2gz)—c'^, dz is affected with a negative sign, because tlie motion commencing when the body Is at the lowest point, ^ decreases according as t increases. * If we multiply the factors of the expression, and range them according to the dimensions of z, .we get — 2gz^ — c2^+2r^.g2+/-'c — c'S if the same operation be performed on the expression (a— z).(z — b) . (,2gz+f) we will obtain — 2gz^+ {2g (a + b) — /). 2- + (y. (« + 6) — 2g.ab) z—^fab, these two expressions being always equal, their corresponding terms must be identical, consequently, by comparing the coefficients of z, we have y = 2g. - — -- — - by comparing the coefficients of s', and substituting for f its value we get 2.g -( (a+A) --r— J- = — <^ '•■ ^y concmnaung a-'-\-2ab-lfb--—r''4-ab „ f r^—a^ah—b^ 1 2a. 1 — ! — — C = 2g. < — — ; f the comparison of the absolute quantities, gives, by substituting for / and c their values, which ha*e been already found, —r'-.b^-+>-^.ab4-a''b- ) „ (r'—a').{r- 22. < ' ' J- = Jg. = ' <:; 56 CELESTIAL MECHANICS, a+b We can thus substitute for the arbitrary quantities c and d, a and b, which are also arbitrary, of which the first is the greatest value of z, and the second the least. Then, by making • « /^=^ sin. 9= V r ' « — Z> the preceding differential equation will become dt= r.^lTiTV) d^ these values of^ c, c' being possible, we are permitted to substitute the expression {a~z) .{z—b) .{<2.gz-\.f) in place of (r^— 2;-) . (c+2o;r)— e'S therefore ^ = — v/(^a — 2^ . [z—b) . (^ ^^; ^rJ)> ^ being a function of/, this differential coefficient vanishes when a=s, and also when z=b; — = — ^ (^a — z).{z — u) K^g^-rJ) — 0, has at least txao real roots ; for as the point is constrained to move on the surface of the sphere, the trajectorj' has necessarily a maximum and a minimum ; and as impossible roots enter equations by pairs, it follows that all the roots are real, moreover it is manifest from the variations of the signs, tliat one root is negative: _lf expresses the velocity of the point in the direction of the vertical. * The transformation sin . « = a/ is made in order to facilitate the integration. ^ a — b sin.* i = °~^ -, and cos.* 6= ^^^^ v z=:a. cos. ' «+6.(l— cos.* ^) = « cos.^ d-\-b sin.^ t, a — 6 a — b PART I.— BOOK I. 57 y* being equal to (a-\-b)i+r* — b', ~ The angle 6 gives the coordinate z by means of the equation ; z=a. COS.* 9+ i. sin.* 6, dt. COS. «r: ■ == •% —dz=: 2dl. ^/ (a~z) . (z—6) —r.dz and ' - ^^^ ^ (substituting for /its value) -/ (a-zXz-b).{2gz+f) ^ J . ] 2rJe;^(a—z).(z—f>) 2r.d« (substituting for e its value a. cos.M+5. sin.= «) we obtain 2r.de \/a+b \/2g.(a".cos. ^S-j-ab.cos.'6^ab.sin.^e+b~.sin.*i+r*'i-ab 2r.de.^/a + b 2r.d6.\/7+b v/2g.(a-t-6)'-)-l?--— 6^)+(A^— a'). sin.»«)' for 4*— a - in the preceding expression we shall have dt = ""*''*" ^'=(7HiH^»^=)' ^'-'''=-((«+^)'-+('-«-6')).yS .-. substituting 2r.di.^ya+b v/%U«+<' J-i-C-'— 6--)— ((a-t-6)^4-(r-~6^)).y».sin.M. _ r.v/2.(»+i) di 58 CELESTIAL MECHANICS, and the coordinate z divided by r, expresses the cosine of the angle which the radius r makes with the vertical. Let TB- be the angle which the vertical plane passing through the radius r, makes with the vertical plane which passes through the axis of J j w« shall have XZZs/l^ 2*. COS. -ar; * y — oj T^—H^SWi. w; which give xdy — ydx=.{r'^'—!?'). d-cr, .'. the equation xdy — ydx—c'dt will give d.dt dTszz. r'—z^ we will obtain the angle Ts-in a function of S, by substituting for z and dt their preceding values in terms of 6 ; thus we may know at any time whatever, the two angles 9 and -cr, which is sufficient to determine the position of the moving point. T Let us name, — , the time employed t in passing from the greatest * X = the product of the projection of r, on the plane x, y, into the cosine of the angle which x makes with the projected line, .•. as, \f r' — z'- =rr so projected, and » = the angle which xmakes with ^Z r'- — 2^, x=icos. w. \/r» — i^, rfx= —\'r'^ — s'. sin. w. zf/z. cos. OT / — . , /-:; 1 , *dz. sin. v» d^, — . , y =■ \f r'^ — z'.sin. w, .•. dy=\/r^ — s^.ttm. cos. o — 2 . , , xrfz. sin. «7. cos. 33- , , , ., , xdy—ydi=\j^—z^) d'a cos. »ot. h (r«— 2-)rf«. wa z.dz. sin. ra cos. «r. , , , ^ j f For evolving the expression for dt into a series, it becomes, i.V«.sin.'«.rf«+|^V*.sin.*.«.... . 2^g '••^^'"+^'' t/ Supposing the point suspended at the extremity of a thread without mass, of which the other extremity is firmly fixed ; if the length of the thread is r, the motion of the point will be the same as in the interior of a spherical surface ; it will constitute with the thread a pendulum, of which the cosine of the greatest deviation from the vertical will be — . If we suppose that in this state, the velocity of the point is no- r thing J* it will vibrate in a vertical plane, and in this case we shall * — expressing the cosine of the angle which the radius makes with the verti- tical, when the deviation from the vertical is the greatest, z is then least, and consequently it is equal to b, .: >- is the cosine of the greatest deviation, and as generally '— =: 1 — COS. A . , . . . r — b . , . • /. 1 • ■ » , in this case it is r: to -jr— , y" — this quantity, lor making a = r in the expression for v', it becomes r-— Z^" (r—b){r+b) r—b The pendulum described in the text is merely ideal, as every body has weight. How- ever, philosophers have given a rule, by means of which we are able to determine th« length of the imaginary pendulum, such as has been described, from the compound pen- dulum which is isochronous with it. (See No. 31 of this book.) From the equation cfar. (r* — z*)=c'.(/< it follows that the angular velocity is inversly as the square of the distance; this is universally true, whenever the areas are propor- c dt tional to the times, for we have then ^•.d'a.= ddt .: d-n = -^. See note to No. 6. dx-A-dv'^ 4-dz'^ ds'' '^^ From the equation c-f-Sgz = — ■ , ■ = -r—, we derive dt == / , "■■■ — '^ dt dt^ \/c+2ga, ds when the velocity — vanishes before the tangent becomes a second time horizontal. PART I.— BOOK I. 61 have, a = r; v* = The fraction is the square of the sine of half the greatest angle which the thread makes with the vertical j the entire duration Tof an oscillation of the pendulum will therefore be T=: ;■ J) If the oscillation is very small, is a very small fraction, which may be neglected, and then we shall have therefore the very small oscillations are isochronous, or of the sarrie' da- ration, whatever may be their extent ; and by means of this duration, and of the corresponding length of the pendulum, we can easily deter- mine the variations of the intensity of gravity, in different parts of the earth's surface. Let z be the height through which a body would fall by the action of gravity in the time T; by No. 10 we shall have 2z:=g T^, and conse- quently ^ = ^tt.^ r j thus we can obtain with the greatest precision, by means of the length of a pendulum which vibrates seconds, the space through which bodies descend by the action of gravity in the first se^ cond of their fall. It appears from experiments, very accurately made. ds the point describes only a part of a circle of the sphere, but if y be finite, when the tangent becomes a second time horizontal, then the point describes the entire circumference. These circumstances may be determined by means of the equation dx"- 4- di/'^+dz' 62 CELESTIAL MECHANICS, that the length of the pendulum which vibrates seconds is the same, whatever may be the substances which are made to oscillate. From which it follows that gravity acts equally on all bodies, and that it tends, in the same place, to impress on them the same velocity, in the same time. • When the oscillations are very small T = t,*/ — , and if a body vibrated in a cy- cloid whose length was equal to 2r, the time of an entire vibration would be equal to "•• f./ — , ivhatever be the amplitude of the arc, for the equation of this curve is s'^zaz. o ,_ dz , (See Lacroix Traite Elementaire, No. 102) .". dszz\/ a — =' ^"'l V 2g (A z) = V 5r — -r-j (^i equal to the value of r when ^rrO) .'. dt= — . ~—^ = —,\//^^^ \ '^^ V^i vh-7 '^ V/ (y; ^ k/ h >' ' *'''~*' V V~~V arc COS. f -^^-— j + C,ifwetake this integral between the limits r=j^, z= 0,— = - . ^y/ — , ,*, if 2a=r, i, e, if the radius of the osculating circle be equal to 2a, the small oscillations in this circle are equal to the oscillations in the cycloid, and sipce /( does not occur in this integral, the time of describing all arcs of the cycloid are equal, provided one extremity of these arcs be at the lowest point. It appears from the foregoing investigation, that the time of vibration in a cycloidal arc, is the hmit to vhich the time in a circular arc approaches, when the latter becomes inde- finitely small. W hen great accuracy is required, all the terms after the two first in the series expressing the time in a circular arch are rejected, and then the expression for T'= V. w — S 1-j- f-^y (-n~)'' [ '^•"" which it appears that the aberration from isochronism varies, as the square of the sine of half the amplitude. We might determine the time of describing any given arc of a circle, if we knew the coordinates a and b, and also z the coordinate of the extremity of the arc required, for then the angle 6 w ould be determined. We might also, derive a general expression for the time of describing any given arc of a ci/cloid. For if in the initial velocity be such, as w ould be acquired in falling down a height equal to A, we shall have at any point in the ds — __— cycloid «»= 2g. {IHh—z) consequently — = \/'^g(,H-\-h — 2) .♦. dt = ===== (by substituting for ds its value /T"(-- \ PART I.— BOOK I. G3 12. The isochronism of the oscillations of the penduhnn, being only an approximation ; it is interesting to know the curve on which a heavy body ought to move, in order to arrive at the point where the motion ceases, in the same time, whatever may be the arc which it shall have described from the lowest point. But to solve this problem in the most general manner, we will suppose, conformably to what has place in nature, that the point moves in a resisting medium. Let s repre- sent the arc described from the lowest point of the curve ; z the vertical abscissa reckoned from this point ; dt the element of the time, and being any function of this velocity. By No. 7 the differential of this velocity will be equal to —g. -^ denoting (a^j dt dt* by i}/'(s') the differential oi ^ {s') divided by rf/; and by f (s') the dif- ferential of ^'(s'") divided by ds\ we shall have ds ds' ,,, ,. di=dt- ■'-('' the equation (?) will become • Substituting for ^. (c?i) its value in the equation (i), it becomes substituting these values for -r- , and — — , we shall have * dt- dt^' A^r' /?/2 //c'* rft* ff'.rfl PART I.— BOOK I. 65 We make the term multiplied by — — , to disappear by means of the equation which gives by integrating ^{s') = \og. S(A(5'+y)^} zz-s; h and q being arbitrary quantities. By making / commence with s we shall have hq '^ = \, and if, for greater simplicity we make, hzzl, we shall have ^—c"' — 1.* K V dividing all the terms by ^'.[s) and concinnating we obtain * From the value of d^s which has been already given, we get ds'^ ds* ds and by integrating we obtain, log. ds — log. ds'-{-ns = e or log. -^y = e — m ; V -Tr= —^ J and ds = , mtegratuig again we shall nave s' + y = , V \o^. (ji.{s' +q))—ns—e or dividing both sides by n; . °^ '"^^ — ^ )=((log-(«-(*+9)»^)) n ' = « , and if — , be made equal to — ^ we obtain log. ((A. (/-f-y")) T = *. If we , 1 suppose s to commence with s, they are = to at the same instant, •.• log. A.y n = 0, at this instant, and consequently /i.y » = 1, y must be equal to uniiy since n is a constant indetermined coefficient, •.• log. (i'+l)n = s = ^j.(«'), and / = c"" — 1. 66 CELESTIAL MECHANICS, c being the number whose hyperboHc logarithm is tmity ; the diffe- rential equation (/) becomes then dh' , ds' , „ dz ,^ ,., ^ = ^+"-^+"^-^-(^+^^*- By supposing s very small, we may develope the last term of this equation into a series ascending according to the powers of 5' which will be of this form, ks'+ls'' + , &c. ; i being greater than unity ; the last equation then becomes at^ (It mt This equation multiplied by c~^. (cos. y? + y/_i. sin. yt), and then in- tegrated, becomes (7 being supposed equal to w^y^- "'* _) Jl f -) r (Is' f c«-jcos. yt-\-^-\. sin. yt[. j-T^- + ("f- — rV— ^ • ^ C = — l.J^'dt. c^ ^cos. yt-\-\/—\.. sin. 7^. ( — &c.t * For since sf = C" — 1, -77 = A'i.^^ = — ;jr "^ . ,,. ■•' ■\'\^V - ds ^ ' tLc"" n.(l+s) ^^ ' K'.(l-j-/)=' , , , d's' '^^ . dz •.•the equation (I) becomes ——— + w. -7- +""^--77-' (!+*)> when ^ is veiy (IZ CtC tto small the variable part of the last temi of this equation may be expanded into a series proceeding according to the ascending powers of s', for substituting in place of s' it be- dz ... comes = — - . c""*, when *• is very small s' is also veiy small, as is evident from the equa- tion s' = c'" — 1 ".• — p = the sine of the inclination of the tangent to the horizon is ds very small, and as all the terms which occur in the expression -r— . (l-f-^')' are very small it can be developed in a series of the form given in the text. f Cos.yi+\^—\- sin. yi = c'' '^-' _ ggg Lacroix Traite Elementaire, No. 164.,) ••• by substituting c'''v~' in place of the circular function, we obtain PART I.— BOOK I. 67 By comparing separately the real and imaginary parts, we will have els' two equations by means of which we can eliminate — — ; but it will be k2 &c. If we multiply both sides of this equation by dt, and then partially integrate, we shall have (the integral of rf/c(l +yV-i)'- ^ ^,^(|+yvri) *. substituting this value of /rff'.c(-1+''v/-0f- j^ jj^^ ggj,^jjj ^^^^jj ^^f ^1,^ preceding inte- gral, and for k its value y«-4 , we obtain 4 = (-^_;»yV=i.+ v'.y.'. cit ^ A y z= — l.Js" dt.c~^ . sin. yt — &c. the integrals of the second member being supposed to commence with t. ds Naming T the value of t at the end of the motion, when —r- vanishes, at that instant we shall have c . s'.< sin. yT — y. cos. yT. ' cos. yt.c » . s'i —, sin. yt.c 2 . /«, by substituting ' 4y* ^ 4y» 7)1 for cos. yTits value — . sin. yT, the first expression becomes 2y h, l + I^il 4y- sin. y T^.c". «', which is equal to tlie second with a contrary sign, consequently 7nt it follows that whatever be the magnitude /; — ^.yi',-. dtxT". sin. yt = 0, when the integral is taken from f— to t=zT. The same reasoning applies to the other terms of the series, which contain powers of s' superior to i. I being independant of /, if it is equal to nothing when s' is very small it will be al- mt ways equal to nothing ; and since neither sin. yt, nor c T change their signs from f=0, to mt t=. T, it is evident that the evanescence o{Js''.c~2. sin. yt can only arise from I being equal to nothing, in this case also the coefficients of the powers of s' greater than /'. i. e. the subseqnent terms of the series vanish. 70 CELESTIAL MECHANICS, this value of tang. yT has place for any arc whatever, therefore what- ever be the value of *', we have mt 0= /.//', dt.c~. sin. 7'/ + &c. the integral being taken from t—0 to t=T. If we suppose s' very small the second member of this equation will be reduced to its first mt term, and it can only be satisfied by making / =0 ; for the factor c~ . sin. yt, being constantly positive from /— to t= T, the preceding integral is necessarily positive in this interval. Therefore the tauto- chronism is only possible on the supposition of ds' which gives for the equation of the tautochronous curve g.dz-= (1— c J 71 In a vacuum, and when the resistance is proportional to the velocity, n * Substituting for 1+i' its value C", and ds' its value n.ds.C", we obtain n'^S-dz. „.. , , _. _ , k.ds n.ds.C^ 1.2ns = ^(cnj _ J ) ._. gj^ _ J±± jl — c""" i , •.• when the body moves in a vacuo, or in a medium of wliich the resistance is proportional to the velocity, n=zo :• gdz — ' (1 — c-"*) = ks. — , but if we express c-"* in a se- nes Jt becomes = l ~-\ — — -, &c. •.• the general expression for * n \ ^1 1.2 ^ 1.2.3 ; , when 7! = 0, lc.ds.s. From this equation it follows that k = , , this is also g.p true, ds.s when n has a finite value, if s be taken very small, as is evident from the preceding series. PART I.— BOOK I. 71 is nothing ; and this equation becomes g.dzz=.hs.ds ; which is the equa- tion of the cycloid. It is remarkable * that the coefficient n of the part of the resistance, which is proportional to the square of the velocity, does not enter into the expression of the time T; and it is evident from the preceding analysis that this expression will be the same, even though we should add to the expression for the law of the resistance, which has been given above, the terms, ds' , ds* c. p. + q + &c. ' dt' ^ dt If in general, R represents the retarding force along the curve, we, shall have s being a function of t, and of the entire arc described, which conse- quently, is a function of t and of s. By differentiating this last function, we obtain a differential equation of this form, dt V being a function of / and of s, which, by the conditions of the pro- blem must vanish, when t has a determinate value, which is indepen- dant of the whole arc described. Suppose, for example, V ■=. S.T, S * Since the value of T is the same when the terms P. —H 4. n. — ^ + &c. are added rff 3 ^ " tit* ' to VI. ——■ -}- n.—— , it follows that the generality of the conclusion is not affected by ds ds^ ( ds ^ substituting m. -7-+ n. -— - in place of ip ] -7- j • 72 CELESTIAL MECHANICS, being a function of s only, and T being a function of t only ; we shall have d-s „ dS ds „ dT dS ds^ ^ ^ dT ^ = 1 . — ; — • — r— + O. = . — - + *J« , • j dt"^ ds dt dt S.ds if dt but the equation — — = ST, gives t, and consequently — — equal to a dt ^ at function of -, which function we will denote by -—-, 4' » tttt i therefore we shall have d's ds' { dS ,1 ds dt^ S.dt^ {f + *(^)} = -^- Such is the expression for the resistance which corresponds to the ds differential equation — = ST; and it is easy to perceive that it involves the case of the resistance proportional to the two first powers of the velocity, multiplied respectively by constant coefficients. Other differential equations would give different laws of resistance.! * S being a function of x, which is a function of t, tlie differential coeflacient of S, with resDect to ^ = — '— '—r > and substituting for T its value ^ , ■ we obtain •^ ds dt ii-as d^s dS ds"^ „ dT dt^ ~ S.ds dt' ' dt ' f In the precedhng investigation the body is supposed to ascend from the lowest point, and the curve which then satisfies the condition of tautoclironism is U7iiqtte in a given medium ; but if the body descended from the highest point, then it would oscillate at the other side of the point where the tangent was horizontal, and the problem becomes somewhat more indeterminate, in this case k may be announced more generally thus ; to find the lines, the time of describing which will be given, whatever be the amplitude of the arcli described ; the discussion of this problem is too long to be inserted here, the reader will find a complete investigation of it by Euler in the Transactions of the Academy of Petersburgh for the years 1764 and ITS*, he demonstrates tiiat the arcs at each side •f the lowest poijit are not necessarily equal and similar, however, the sum of these arcs PART I.— BOOK I. 7S is proportional to the square root of the vertical coordinate, •.• the curve whose length is equal to the sum of these arcs will be the common cycloid, in like manner, if we have the differential equation of one of these arcs, we can determine the differential equation of the other ; if the first arc be a cycloid, the second will also be the arc of a cycloid : in this case the time of describing each of the cycloidal arcs will be constant, howevef the generating circle of the second cycloid is not necessarily equal to that of the first. If we combine the condition of tautochronism, with the condition of the two branches at each side of the lowest point, being equal and similar, the curve will be then the vulgar cycloid, therefore this is the only plane curve in which the sum of the times of the ascent and descent is always the same in a vacuo ; but this property belongs to an indefinite number of curves of double curvature which are formed by applying the cycloid to a vertical cylinder of any base, the altitude of the curve above the horizon remaining the same as before, for «* ds^ + ds =: -— = c — 2gz, '.' dt= — , consequently the value of t depends on the initial '^** \/c—2gz velocity, and on the relation between the vertical ordinates and arc of the curve •.' what- ever changes are made in the curve compatible with the continuity, the value of dt will not be changed, provided the preceding relation remains ; and it follows conversel}', that the projection of any tautochronous curve of double curvature, on a vertical plane, will be a cycloid with a horizontal base. In the cycloid, if a body falls freely, the accelerating force along the tangent varies as the distance from the lowest point, for 4*=4a2, '•' g--f- (= accelerating force =: -^ , I Us ^(t J the pressure arising from gravity = g. — — , and the pressure which is produced by the centntugai torce — - for radius of curvature = I.kj a{a — 2), and the square of 2.\/(i(a — 2) V \ / 1 the velocity = 'i-g-{a — r), see No. 9, (the coordinates of z are reckoned from the lowest point ;) it follows from the preceding expression that the ivhole pressure at the lowest point, and consequently the tension at this point of a body vibrating in a cycloid is r= to twice the gravity. When a body describes a cycloid, the accelerating force varies as the distance from the lowest point, as has been stated above ; and if a body was solicited by a force varying according to this law, the time of falling to the centre will be given, for we have ^ = — As V '^^2= — ^-5 ■^. ••• "' = — -^«* + C, v=0, s = S, V Cz=AS', ••• dJt "i at v=A. ^ s«_ii &^ •*= - 7-. ^ , V ^*/ = arc. cos.^, and when s—O, t=T V^ O S* i 74 CELESTIAL MECHANICS, 32,—^ , consequently the time of descent to the centre, is the same from whatever point, the body begins to fall. From the preceding expression, it follows, that the time of de- scribing any space s, varies as the arc, and the velocity acquired varies as the right sine. Se« Princip. Mat. Prop. 38, Book 1st. PART I.— BOOK I. IS CHAPTER III. Of the equilibrium of a system of bodies. 13. The simplest case of the equilibrium of several bodies, is that of two material points meeting each other with equal and directly con- trary velocities ; their mutual impenetrability evidently annihilates their motion, and reduces them to a state of rest. Let us now suppose a number m of contiguous material points, arranged in a right line, and moving in its direction with the velocity u, and also another number ?«' of contiguous points, disposed in the same line, and moving with the velocity u', directly contrary to u, so that the two systems may strike each other ; there must exist a certain relation between u and u', when both the systems remain at rest after the shock. In order to determine this condition, it may be observed that the system m, moving with the velocity u, will constitute an equilibrium with a single material point, moving in a contrary direction with the velocity mu ; for every point of the system would destroy in this last point, a velocity equal to u, and consequently the m points would destroy the whole velocity mu ; we may therefore substitute for this system a single point, moving with the velocity mu. In like manner we may substitute for the system m', a single point moving with the velocity m'u' ; now* the two systems being supposed to constitute an equilibrium, the two points which are substituted in their place, ought to be also in equilibrio, therefore their velocities must be equal j consequently we L 2 * These two systems of contiguous material points, may be supposed to represent two bodies M, M', of different masses, equal respectively to the sum of all the ms, and in',s. 76 CELESTIAL MECHANICS, have for the condition of the equilibrium of the two systems, mu-^m'u'. The mass of a body is the number of its material points, and the product of the mass by the velocity, is what is termed its quantity of motion ; this is also what we understand by the force of a body in motion. In order that the two bodies, or two systems of points whicli strike each in contrary directions, may be in equilibrio, the quantities of motion or the opposite forces must be equal, and consequently the ve- locities must be inversely as the masses. The density of bodies depends on the number of material points which tliey contain in a given volume. In order to determine their ab- solute density, we should compare their masses with that of a body t which has no pores ; but as we know no such body, we can only deter- mine the relative density of bodies, that is to say, tlie ratio of their density, to that of a given substance. It is evident that the mass is in the ratio of the volume and density ; therefore, if we denote the mass of the body by M^ its volume by C/", and its density by D, we shall have generally M= U. D ; in this equation the quantities AI,D,U, relate to the units of their respective species. In what precedes, we suppose that bodies are composed of similar material points, and that they only differ in the relative situation of these points. But the intimate nature of matter being unknown, this supposition is at least very precarious, and it is possible that there may be essential differences^ in their integrant molecules. Fortunately, the truth of this hypothesis is of no consequence to the sci- ence of mechanics, and we may adopt it without any apprehension of 7 Distilled water, at its greatest density, is the substance which has been selected for the term of comparison, as being one of tlie most homogeneous substances, and tliat which may be readily reduced to a pure state. X By the integrant molecules of bodies, as contradistinguished from their constituent parts, we understand those which arise from the subdivision of the body, into minuter por- tions ; by the constituent parts are understood the elementary substances of which a body it composed. PART I— BOOK I. 77 error, provided that by similar material points, we understand points which, when they meet with equal and opposite velocities, mutually con- stitute equilibrium, whatever their nature may be.* 14. Two material points, of which the masses are m and »/, can only act on each other in the direction of the line joining them. Indeed, if the two points are connected by a thread passing over a fixed pully, their reciprocal action cannot be directed along this line ; but the fixed j)ully may be considered as having at its centre a mass of infinite den- sity, which reacts on the two bodies, so that their mutual action may be considered as indirect. Let p denote the action which is exerted by in on rri by means of the right line which joins them, which line we suppose to be inflexible and without mass. Conceive this line to be actuated by two equal and op- posite forces p and — p ; the force — p will destroy in the body m a force equal top, and the force/? of the right line will be communicated entirely to the body rri. This loss of force in m, occasioned by its action on m', is termed the reaction of m' ; therefore in the communication of motions, the reaction is ahvays equal and contrary to the action. It appears from observation that this principle obtains for all the forces of nature.! * It' there be actually essential differences in the integrant molecules, then it is noc inconsistent to suppose, with some philosophers, that the planetary regions are filled with a very subtle fluid destitute of pores, and of such a nature as not to oppose any resist- ance to the motions of the planets. We can thus reconcile the permanency of these motions, which is evinced by observation, with the opinion of those philosophers who regard a vacuum as an impossibility ; however the plenum, for which De-Cartes contended, is not confirmed by the preceding hypothesis, as he held that all matter was homogeneous, and that the ether, which, according to him filled the planetary regions, differed from other substances only in the form of the matter. See Princip, Math. Book 2, Prop. 4-0 ; Exper. l*, and Book 3, Prop. 6, Cor. 2 and 3 ; Newton's Optics, Queiy 18; and Systeme de Monde, page 166. However, as extension and motion are the only properties which are taken into accoimt in Mechanics, it is indifferent whether matter be considered as ho- mogeneous or not. f This equahty does not suppose any particular force inherent in matter, it follows ne- cessarily fi-om this, that a body cannot be moved by another body, without depriving this body of the quantity of motion which is acquired by the first body, in the same manner as when two vessels communicate with each other, one cannot be filled but at the expense of the other. 78 CELESTIAL MECHANICS, Let us suppose two heavy bodies m and m' attached to the extremities of an horizontal right line, supposed to be inflexible and without mass, which can turn freely about a point assumed in this right line. In order to conceive the action of those bodies on each other, when they are in equilibrio, we must suppose the right line to be bent by an indefinitely small quantity at the assumed point, so as to be formed of two right lines, constituting at this point an angle, which differs from two right angles by an indefinitely small quantity w. Let J' andj' represent the distances of m and m' from the fixed point ; if we resolve the weight of m into two forces, one acting on the fixed point, and the other directed towards nz', this last force will be represented by ^ , g being the force of gravity. In like manner the action of ni on m will be re- presented by — '^ . , the two bodies constituting an equilibrium, these two expressions will be equal, consequently we will have 7n/=m'J" ; this gives the known law of the equilibrium of the lever, and at the same time, enables us to conceive the reciprocal action of pa- rallel forces. Let us now consider the equilibrium of a system of pointsactuated by any forces whatever, and i-eacting on each other. Let^representthe distance of m from ni \f' the distance of m from m\f" the distance of wj' from /«", &c. * Gravity must be distinguished from weight ; the weight of a bedy is the product of the gravity of a single particle^ by the number of particles. If we conceive a line drawn from the fixed point, parallel to the direction of gravity, meet- ing a line connecting ni and ?»', this last line will be q.'p., horizontal, and therefore perpen- dicular to the vertical line, which will *.• be equal toy multiplied into the sine of the angle whichy makes with the horizontal line, but as the sides are as the sines of the opposite angles, we liave the sine of the angle whichy makes with the horizontal line, to the sine of u, or its supplement, as, f':f-{-J' •.■ it is equal to :i-^ — j^:=q.p.-4——-,,nowi£tiie weight be represented by the vertical line, then mg divided by sine of the angle whichy makes with the horizontal line, i. e, — '^ ., ■ will be the force in the direction of/. "J PART I.~BOOK I. 79 also let p be the reciprocal action of mon m' ; p' that ofm on m'^ ; p'' that of m' on m", &c. and lastly, let mS, m'S', rri'S", be the forces which act on m, rri, m'' ; &c. 5, /, s", lines drawn from any fixed points in the di- rection of these forces, to the bodies m, m', rd\ Sec. ; this being premised, we may consider the point m as perfectly free, and in equilibrio in con- sequence of the action of the force mS, and of the forces, which the bodies m, ni, m\ communicate to it ; but if it was subjected to move on a curve or on a surface, it would be necessary to add to these forces, the reaction of the curve or of the surface. Therefore, let Ss be the varia- tion of s, and let S^ f, denote the variation of y, taken on the supposition that rri is fixed. In like manner let S^f, be the variation of ^', on the supposition that iti' is fixed, &c. Let i?, B!, represent the reactions of the two surfaces, which form by their intersection the curve on which the point is constrained to move, and let J'r, Sr' be the variations of the di- rections of these last forces. The equation {d) of No. 3, will give : Qz=.mS.Ss +p.S,f-\- p'JJ'+kc. + mr + R'Sr' + &c. In the same manner m' may be considered as a point perfectly free, re- tained in equilibrio by means of the force niS', of the actions of the bodies m, rri, iri', and of the reactions of the surfaces on which ni is constrained to move, which reactions we will denote by R", and R". Let, therefore, the variation of s' be called Ss', and the variations of^ and/^', taken on the supposition that m and m" are fixed, be respectively S,,f, J,y""; in like manner, let Sr", Sr'", be the respective variations of the directions of R", R'", and we shall have for the equilibrium of ni O = m'S, Ss' +pJ„f+ f.SJ" + &c. + R'.Sr^' + R'aJr"'. If we form similar equations relative to the equilibrium of m'', and ml", &c. by adding them together, and observing that ifv=i,f\i„f;iS' = lf^S,f',* &c. Sf, and Sf\ being the total •3/"=3^/'+3„/; 3/"' - 3,/'+S„/'/ + &c. ; for/ and y are respectively functions of the coordinates of their extreme points, and when these are moved by an indefinitely fmall quantity, all the powers of the increments of the coordinates, after the first may be rejected, and then the entire increment of _/ is equal to the sum of the partial incrementa 80 CELESTIAL MECHANICS, variations ofyandy'+&c. we shall have 0=-z.m.SJs + xp.if-^j:RJr; (k) in this equation, the variations of the coordinates of the different point? of the system are entirely arbitrary. It should be observed here, that in consequence of the equation («) of No. 2, we may substitute in place of mS.Ss, the sum of the products of all the partial forces by which m is actuated, multiplied by the respective variations of their directions. The same may be observed of the products m'Sis' ; If the distances of the bodies from each other be invariable, i. e. if J^y',jr'''', + &c. are constant, this condition may be expressed by making {/=0, Sf' = 0, &c. The variations of the coordinates in the equation (k) being arbitrary, they may be subjected to satisfy these last equations, and then the forces p, p',p'', &c. which depend on the reciprocal action of the bodies composing tl:e system, will disappear from this equation ; we can also make the terms li.Sr, R'Ji'. •+ &c. t to disap- pear, by subjecting the variations of the coordinates to satisfy the equa- tions of the surfaces, on which the body is constrained to move. The equation (/.) will then become 0=T.mS.Ss; (I) from which it follows that in case of equilibrium, the sum of the varia- which are due to the separate variation of each coordinate, ••■ the entire variation of y is equal to the sura of the partial variations, which correspond to the characteristics 3, and i^. * From this it appears,' that the conditions of the equilibrium of a system of bodies, may be always determined by the law of the composition of forces ; for we can conceive the force by which each point is actuated to be applied to the point in its direction, where all the forces concurring, constitute an equilibrium when the point is entirely free, or which constitute a resultant, which is destroyed by the fixed points of the system, when the point is not altogether free, f See Notes to No. 3. Tlie equation (/) obtains, whether the points are all free, or are subjected to move o* PART L— BOOK I. 81 tions of the products of the forces, into the elementary variations of their directions will be equal to nothing, whatever changes be made in the position of tlie system compatible with the conditions of the con- nection of the parts of the system . We have arrived at this theorem, on the particular supposition of the parts of the system being at invariable distances from each other ; how- ever it is true whatever may be the conditions of the connection of the parts of the system. In order to prove this, it will be sufficient to shew that when the variations, of the coordinates, are subjected to those con- ditions, we have in the equation (Z,) = I..p.Sf-^I..R.h- ; but it is evident that Sr, Si^, &c. are equal to nothing, when these con- ditions are satisfied ; therefore it is only necessary to prove that in the same circumstances we liave = i:.p.Sf. Let us therefore suppose the system actuated by the sole forces j9, pf, p, &c. and that the bodies are subjected to move on the curves, which they can describe in consequence of the same conditions ; these forces may be resolved into others, some of which q, q', q", &c. acting in the direction of J,' J', f", &c. will mutually destroy each other, without producihg any action on the curves described ; others will be perpendicular to those curves ; and others again will act in the direction of tangents to those curves, by the action of which the bodies may be moved ; but it is easy to perceive that the sum of these last forces ought to be equal to nothing ; since the system being by hypothe- sis at liberty to move in their directions, they are not able to produce either pressure on the curves described, or reaction between the bodies ; M curved smfaces ; in the former case, the forces S, S', S", constitute an equilibrium ; in the latter case, these forces have a resultant, of which the direction is perpendicular to the surface. (See Note to page 17.) 0*- ro 8J CELESTIAL MECHANICS, consequently they cannot constitute an equilibrium with the forces — P' — P'f — F^'t ^^' y> ?'» I"y ^^- T, T', T" ; therefore they must vanish, and the system must be in equilibrio in consequence of the sole forces py—p',—p", &c. ; q, q', q", &c. ; T, T', &c. Now, if Si, Si', &c. repre- sent the variations of the directions of the forces T, T', Sec. we shall have in consequence of the equation (A-) = l.(q—p)Jf + i:.TJi ; but the system being supposed to be at rest, in consequence of the sole action of the forces q, q', &cc, without any action being produced on the curves described, the equation (/c) gives us also — l.qJJ';* conse- quently we have = ■s:.pJf—I,.TJi ; but as the variations of the coordinates are subjected to satisfy the con- ditions of the curves described, we have Si, = 0, Si', = 0, &c. ; therefore the preceding equation becomes O =. l.p.Sf;f as the curves described are themselves arbitrary, and are only subjected to the conditions of the connection of the system, the preceding equa- tion obtains, provided that we satisfy these conditions, and then the equation (k) will be changed into the equation (/). The following principle, known by the name of the principle of virtual velocities, when analytically expressed, is represented by this equation. It is thus an- * 0:= 2 y.S/, for q, q, q", are directed along the lines yjy'.y""; and are supposed to destroy each other without producing any action on the curves described. f The object of the second part of this demonstration is to shew, that if the system is at rest, and acted on by the sole forces /), //, ■p"-, these forces may be so decomposed as to afford forces equivalent to the reciprocal actions of the respective bodies, and that the remaining portions of the forces, as well as these reciprocal actions, will balance each other, in case of etjuilibrium, according to the terms of the proposition. Since the equation {k\ is reduced to the equation (/), when we subject the variations of tlie coordinates to satisfy the equations of the surfaces, on which the bodies are con- strained to move, it follows that it is not necessary to compute the forces f, p, &c. in order to derive the equations of equilibrium in each particular case. PART I.— BOOK I. 83 nounced: " If we make an indefinitely small variation in the position* of a system of bodies, which are subjected to the conditions they ought to fulfil, the sum of the forces which solicit it, multiplied respectively by the space that the body to which it is applied, moves along its direction, should be equal to nothing in the case of the equilibrium of the system." This principle not only obtains in the case of equilibrium, but it also insures its existence. Let us suppose, in fact, that whilst the equa- tion (0 obtains, the points m, m', &c. acquire the velocities v, v', in consequence of the action of the forces mS, m'S', which are applied to them. The system will be in equilibrio in consequence of the action of these forces, and of — 7nv, — m'x/, &c. ; denoting by Sv, ix/, &c. the variations of the directions of these new forces, we shall have in con- sequence o£ the principle of virtual velocities = l.mS.SS'—^.mvJv, but by hypothesis l.mS.Ss.zzO, therefore we have 0=l.mv.5v. We may suppose the variations Sv, Sx/, &c. equal to v.dt, i/dt, &c. since they are necessarily subjected to the conditions of the system, and then we have = I,.mv\ and consequently v=0, v' = 0, &c. that is to say, the system is in equilibrio in consequence of the sole forces mS, m',S', &c. The conditions of the connection of the parts of the system may be always reduced to equations between the coordinates of the several bo- dies. Let M = 0, m' = 0, &c. be these different equations, by No. 3, we can add to the equation (Z), the function xSu, x'Suf, &c. or l\hi ; \, x', being indeterminate functions of the coordinates of the bodies, the m2 * When an indefinitely small change is made in the position of the system, so that the conditions of the connections of the points of the system may be preserved, each point advances in the direction of the force which solicits it by a quantity equal to a part of this direction, contained between the first position of this point, and a perpendicular deraitted from the second position on this direction ; these indefinitely small hnes are termed the virtual velocities ; they have been denominated vertual, because the system being in equilibrio, these changes may obtain without the equilibrium being disturbed. 84 CELESTIAL MECHANICS, equation will then become = I..mS.Ss-\-I,.xSu ;* in this case the variations of all the coordinates are arbitrary, and we may equal their coefficients to nothing ; which will give as many equations, by means of which we can determine the functions x,x'. If we com- pare this equation with the equation {k) we shall have l.xJu = l.pJf+l.RJr; by means of which we can easily determine the reciprocal actions of the bodies m, m', &c. on each other, and also the forces — R, — R', with which they press against the surfaces on which they are constrained to move. 15. If all the bodies of the system are firmly united to each other, its position will be determined by that of three of its points which are not in the same right line ; the position of ' each of these points de- pends on three coordinates ; this produces nine indeterminate quan- tities ; but we can reduce them to six others, because the mutual dis- tances of the three points are given and invariable ; these being sub- stituted in the equation (/)» will introduce six arbitrary variations ; by supposing their coefficients to vanish, we shall obtain six equations, which will contain all the conditions of the equilibrium of the system : let us proceed to develope these equations.! * By means of the formulae which are given in the notes to No. S, page H and 15, we can determine A, >! , &c. when S, S^, S*, are given for each individual point ; and there- fore ;;, p, p", k, k', ¥, by means of the equation 2.A. Jie = l.p^f-^- 'S.R.^r ; in the equa- tion Z ?n.SJ«-f- 2. a.5m, m, m', m", &c. may be considered as entirely free ; and if we put the coefficients of the variation of each variable equal to nothing, and then eliminate the indeterminate quantities, A, /', A-^, &c. between these equations, the expressions which re- sult, will give the relations which must exist bstween S, Sf, S", &c. and the coordinates, when the system is in equilibrio. f It follows immediately, from the demonstration of the principle of virtual velocities, that it has place for all the indefinitely small motions which can be given to a solid body, which is either free or constrained to certain conditions, for in all these motions the re- spective distances of the points of the body remain the same. PART I.— BOOK I. 85 For this purpose, let x, y, z, be the coordinates of m ; x', i/, ^, those of m' ; x", y", z", those of m''^ ; &c. ; we shall have then f'= x/{^'^—x)*+(i/"—i/)*+iz'-zy and if we suppose ix = jy = sjo" = &c. $y = Sy = Sy" — &c. ; Sz — Sz' = Sz'' = &c. ; we shall have $f= 0, Sf'= 0, Sf^^=^ O, &c. j* the required condi- tions will therefore be satisfied, and from the equation (/) we may infer we have thus obtained three of the six equations, which contain the conditions of the equilbrium of the system. The second members of these equations are the sum of the forces of the system, resolved pa- rallel to the three axes of x, y, and z, therefore each of these sums must vanish in the case of equilibrium. And as the number of the equations of equilibrium, which are derived from the principle of virtual velocities, is always equal to the number of possible motions, this number being equal to six, in the case of a solid body, or of a body whose parts are invariably connected, the number of equations of equilibrium will be six in like manner. consequently when 3x'=5x, 3j/'=;3y, Sz'=h, &c. Sy=0, therefore 2m.S. -j — | = 0, 2^.S, S ^ I = 0. &c. ; for when ix=i3f - Jx" ; iy—^Z^y ; 3^ == ?-' = ^^'= ^'^- • 80 CELESTIAL MECHANICS, The equations Sf^ =0, if' = 0, $/"== 0, &c. will be also satisfied, if we suppose, z, z', z", invariable, and then make Sx = ySw ; Sy = — x.Jw ; Saf — yi.iu, &c. iy == — x'.Su, &c. SiAt being any variation whatever. By substituting these values in the equation (/)» we shall have .0 = ..„*.|^.(|)_,.(|)|. It is evident that we may, in this equation, change either the coor- dinates x, x', x", &c. or the coordinates y, y', y", &c, into z, z\ z", which will give two other equations, and these reunited with the pre- ceding equation, will constitute the following system of equations : «=x»,5.^^.(A)_,.(|.)^i 2i».S.3*=:0, is equivalent to SmS. S -i \ .S^ — 0, 2mS. ^ -j^ | . J^ = 0, 2.OT.S. -! y- f -'^ :;^0. See Note to No. 2, page 9. * In like manner, if we suppose, 3x=y.3«r, 'ixzzy'^n, Sy = — «?«, 3y:= — x'i», 'if, If, &c. = 0, for substituting in the preceding expression for 'if, which has been given, for >x, ix', iy, 'iy, and it becomes ^ (xW).0,'-y) + (y-3,).(x-xO ^^^ ^^^ j^^p^ By substituting in the equation, lm,Sis:z.O, for 2x, iy, &c. their values it becomes -M» -11 -'{-¥}'-=»■ When all the forces are applied to the same point, the three first equations suffice for the equilibrium ; but when these forces act in different points of space, or when they are PART I.— BOOK I. 87 by No. 3, the function smSi/.] — i is the sum of the moments of all the forces, parallel to the axes of x, which would cause the system to c is ^ revolve about the axis of z. In Hke manner, the function ^m.S.r.) ■—[ is the sum of the moments of all the forces parallel to the axes of i/, which would cause the system to revolve round the axis of z, but in a direction contrary to that of the former forces ; therefore the first of the equations CnJ indicates that the sum of the moments of the forces is nothing with respect to the axis of z. The second and third equations indicate, in a similar manner, that the sum of the moments of the forces is nothing with respect to the axes of ^ and x, respectively. If we com- bine these three conditions with those, in which the sum of the forces pa- rallel to those axes, was nothing with respect to each of them ; we shall have the six conditions of the equilibrium of a system of bodies inva- riably connected together.* If the origin of the coordinates is fixed, and firmly 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 moments of the forces which would make it turn about the three axes, be equal to nothing, with respect to each of them. t applied to different parts of the same solid body, it is also requisite that the moments of the forces with respect to axis of x, y, and z, should be respectively equal to nothing. * If all the points exist in the plane of x, y, then 3z, J2', Si", are equal respectively to nothing, consequently the equations of equilibrium are reduced to the three following : ..,.s.{^| =0, ...-...{^jro. -..s.{,.(|)} -^.(l-j} f When the origin of the coordinates is fixed and invariably attached to the system, the number of possible motions is reduced to three, therefore the number of equations of equilibrium will be three ; this also appears from considering that the number of inde- terminate quantities may be reduced to three, because the distances of any three assumed points in the system, not existing in the same right line, from the fixed origin of tht coordinates, are given. $8 CELESTIAL MECHANICS, f In this case, the resultant of all the forces which act on the body passes through the fixed pomt, which resultant is therefore destroyed by the resistance of the fixed point, and it expresses the force with which this point is pressed. (See last note to No. 3.) WTien there are two points of the system fixed and invariable, then the only possible motion, which can be impressed on the body, is that of rotation, about the line joining the given points, consequently if this line be taken for the axis of z, there will be but one equation of equilibrium, i. e. ^.mS. ^ i/. (-5—) — x. ( -r—) \ = 0, this is also manifest from the circumstances of the indeterminate quantities, wliich were six in number when there was no fixed point, being reducible to one, when the origin of the coordinates, and also another point of the system, were fixed and invariable. The forces parallel to the axes of z cannot produce any motion in the s)'stem, ".' it is only necessary to consider those which exist in the plane of x, ^ ; and as to those, it is evident, fi'om the equation 2m. S. -' „. (-r-) — X. f J-) j- = 0; that their resultant passes through the origin of the coordinates, its direction will be perpendicular to the axis of z, and its intensity will express the force with which it presses on this axis. When the number of fixed points is three, there is evidently no equation of equilibrium. If the forces S, S', S'', &c. do not constitute an equilibrium, in order to reduce them to the least possible number, we should resolve them into three systems of forces, parallel respectively to the axes of x, oiy, and of ;:, then reducing the forces parallel to the axes of X, and ^, to forces — to them respectively, but acting in the same plane, which is always possible, if this last system of forces, and also the forces parallel to the axis of z, have separately unique resultants ; and if these resultants exist in the same plane, we can compose them into one sole force, which will be the resultant of the given forces ; but if the forces directed in the plane x, y, can only be reduced to two parallel forces, not re- ducible into one, then if" we combine them with the force parallel to the axis of z, the en- tire system of forces, will be reduced to two parallel ones acting in different planes, conse- quently irreducible into a unique force. Denoting S.m.S. |-^l ; S.m.S. |.^| ; 2.)n.S. | -i- i, by P, Q, R; respectively, and -X. fii}]. 2...S. |,.]^} -=.{^}}, by L, M,N.,,x„ ,, be the coordinates of that point in which the resultant of all the forces meets the plane of the axes of x, y, we shall have by the last note to No. 3, P.y,, — Qx^, n L ; Ra^., M N = M; — Q.y^, — N; therefore x., = — tt-; v., =: — „ > substituting thtse ex- R R pressions for x^ and y^, In the equation P-y,, — Q-x/, =^ L, we will obtain the equa- PART I.— BOOK I. 89 •tion L.R+M.Q-{-N.P^O, which may be considered as an equation of condition which must be satisfied, when the forces which act on the different points of the system, have an unique resultant. We must however except the case where P, Q, R, are res- pectively equal to nothing; for then the forces are reducible to two parallel forces zz., but not directly opposed to each other. If only P, and Q vanish, then in order that the preceding equation may be satisfied, it is necessary that L should vanish, consequently since P, Q, and L vanish, the forces which are directed in the plane, .r, y, constitute an equilibrium, •.• the unique resultant of the forces S, S', S", &c. must be the same with the resultant R, of the forces parallel to the axes of z, •.' we conclude that if L does not vanish when P and Q vanish, the forces have not an unique resultant, since the forces in the plane of x, y, are in this case evidently irreducible to one sole force ; if however only one of the three sums P, Q, R, vanish, then the forces in the plane .r, i/, and those parallel to the axes of z, would have respectively unique resultants, consequently the pre- ceding equation of condition would apply to this case. When the forces have an unique resultant, it is very easy to determine its position with respect to the coordinates, for if we denote this resultant by V, we shall have V^ ^ P'-\-Q^-\- R-, and—jT-, —, — = the cosines of the angles which V makes with the axes of x, y, and z, respectively, and — — , — , are the distances of the in- H R tersection of V with the plane of x, y, trom the axes of j: and y, respectively. Supposing the system to revolve round the axis of z, the elementary varia- tions of X and y, Sic. are r: respectively to y'^a, — rSa ; if y be made the axis of rotation, and J

x2p z=.y'^^. And consequently Sa- := 0, Sy =^ 0, Jz ^ ; '." this point and all others which have the same property are immoveable, during the instant the point describes the angles 3^, ^, S«, by turning round the axes of x, y, and z ; all points possessing this property exist in a right line passing through the origin of the coordinates, see No. 28, as the cosines of the angles m, n, I, which this hne make with the axes of x, y, and 2, are ■= in this case / r in mis waBc — ^-===;;^^=:= V S« t- -i:t-+ d± -i . .y _ C Sf } N 90 CELESTIAL MECAHNJCS, Let us suppose that the bodies m, m', m", are subject to the sole force S ii I :• the right line which makes with the axes, angles whose cosines are equal to those ex- pressions, is the locus of all the points, which are quiescent during the instantaneous ro- tation of the system. Making S«=-y/3 = S«. cos. m\ ^p ~ J*. cos. n ; S« = S«. cos. ^; consequently Sx = {y. cos. l—z. cos. w). Sfl; ij/ = (z. cos. m — X. cos. /.) ^6 ; Ss. = (x. COS. 71 — 7/. cos. m.) ^S, substituting for Sx, Si/, h, these values in the expression Sx* + Sy*-)"^2^> which is equal to the indefinitely small space described by the point whose coordinates are x, y, z, and observing that cos.* / + cos.' j«-)-cos.^ n= 1, it becomes equal to {x*-\-i/^+z^ — (x. cos. m-\-y cos. }i-\-z. cos. l.y). Ss^ x. cos. l-\-y. cos. m+z. cos. n. is proportional to the cosine of the angle which the line whose coordi- nates are x, i/, z, makes with the right line which makes the angles /, in, n, with the axes of X, y, z, '.' when the line drawn from the origin of the coordinates to the point whose coordinates are x, y, z, is perpendicular, to the instantaneous axis of rotation, the elementary space described by a point so circumstanced — ^x»-)-^* -f-z-. Ss, tliis agrees with what is demonstrated in No. 28. If we suppose d^, 2p, Sv, proportional to Z,, A/, A^, and make H = ^ L^ + M'+N', then L S^)/ M S(p N iu , ~7y = -rr = COS. m ; ——-= -r— = cos. n. -— = -r— = COS I. H Ss H Si H Se ■•• 1' = H.cosm; M = H. cos. n; N = H. cos. I; ••• if // = i, m = 0, n = 100°, / = 100° ; •.• L, the moment of the force is a maximum when = H, and the moments whose axes are perpendicular to the axis of H, will be equal to nothing. Tliis will be more •fially explained in Nos. 21, and 28, it is mentioned here in order to shew how the conditions of the equilibrium of a solid body may be expressed by means of the greatest momt?nt, and unique resultant; if this resultant, and this moment respectively vanish,then ij»=0, H=0, i. e. P,Q,R ; L,M,N, which are equivalent to the equations (j») (re), are equal re4)ectively to nothing ; consequently the evanescence of H and R contains the six equations of the equilibrium of a system, whose parts are invariably connected ; and as by No. 3, the sum of the moments of the composing forces with respect to an axis, is equal to the moment of the projection of the resultant of these forces ; this resultant must necessarily exist in that plane, in which the moment is the greatest possible, •.• the perpendicular to tliis plane L M N must be at right angles to the resultant, consequently, as — — , —ry, -77 , are equal to H H H the cosines of the angles which the axis of the greatest moment make with the axis of P Q It X, y, and z, and as -rrr-, — , — , are equal to the cosines of the angles which V, the unique resultant makes with the same axes ; by note to No. 2, page 7, we have LR+MQ + NP=^0, wliich is the equation indicating that the forces have an unique resultant PART I.— BOOK I. 91 of gravity, as its acts equally on all bodies ; and as we may con- ceive, tiiat its direction is the same, for all the bodies of the system, we shall have S, =S', = S^=8cc.; whatever may be supposed the direction of s, or of the gravity, we shall satisfy the thi'ee equations C^J' by means of the three following :* O = S.m.-y ; O = l,.m.i/ ; = ^.m.z ; Co J N 2 • The force of gravity being uniform, and the direction of its action being always the sa™e,5=S'=S'-&c.;|-^}={|;[=&c.{^}={|,}, (for these quan. ties I -r- I &c. indicate the cosines of tlie angles which the directions of gravity makes with the three coordinates,) the three equations (n) may be made to assume the following form : they are satisfied by means of the three following: 0=2.mx; 0=^S.my ; 0=^2.mz. The equations (m) will be reduced to the following 92 CELESTIAL MECHANICS, The origin of the coordinates, being supposed fixed, it will destroy parallel to each of the three axes, the forces by composing these three forces, we shall obtain an unique force, equal to S.T.m. i. e. to the weight of the system. This origin of the coordinates about which we suppose the system in equilibrio, is a very remarkable point in it, on this account, that being supported, the system actuated by the sole force of gravity remains in equilibrio, whatever position it may be made to assume about this point, which is from thence denominated the centre of gravity of the system. Its position may be determined by this property, that if we make any plane whatever pass through this point, the sum of the products of each body,* by its distance from this plane, is equal to nothing ; for this S 5 S these forces admit a resultant, see note to pace 89, and as -r— , .^ , -^ , are equal to the ex dy dz cosines of the angles which its direction makes with the axes of .r, of y, and of z, com- bining those three expressions, the resultant is evidently = to Sim ; consequently the force with which the fixed origin is pressed, in this case equals the weight of the bodies com- posing the systems. S.lm. answers to the expression tng. in the first note to page 78. It follows from note to page 88, that the resultant of all the forces must pass through the ori^n for 2,rax ; l.my ; 2,n?2 ; are equal respectively to nothing. If another point in the system \)e$ides the centre of gravity was fixed, then = S.\ — V i.my r— . l.mz. > is the sole equation of equilibrium ; in this case the fixed axis of rotation must be vertical. * If Ax' -{■ By'-\-Cz'= 0, be the equation of a plane passing through the centre of gra- vity, the cosines of the angles which this plane makes with the plane of the axes x y, of X z, and of y z, respectively, i. e. the cosines of the angles which a perpendicular to this plane makes with th,e axis of .r, and of y, of 2 = ABC see LacroLx, tom. 1. No. 269, in like manner the cosines of the angles, which lines drawn, from the point, whose coor- dinates are x, y, z, make with the axes of x, of y, and of z, • PART I.— BOOK I. 93 distance is a linear function of the coordinates x, y, z,, of the body ; consequently by multiplying it by the mass of the body, the sum of these products will be equal to nothing in consequence of the equa- tions. CoJ In order to determine the position of the centre of gravity, let X, Y, Z, represent its three coordinates with respect to a given origin j let x, y, z, be the coordinates of m with respect to the same point ; 'Z', y', z', those of m', &c. the equations (oj will then give O = x.m.(r — X.) but we have s.w.X=Xz.w, ■z.m being being the entire mass of the system, therefore we have y 7:.m.x we shall have in like manner s.wi.j/ ^.m.z j ^ — s.w ' -z.m ' :• by note to No. 2, page 7, the cosine of the angle which the perpendicular to the given plane, makes with the line whose coordinates are x, y, z, xA+yB +zC xA+yB+zC let this angle = a and ^/x=-f^^+s» x cos. a'= ./ ^a i P2 , /-z = ^^^ distance ot the point from the given plane, consequently, the sum of all the distances multiplied res- pectively into their masses A. ^.mx-\-B. J.my+ C. S.wz in consequence of the equation (o). 94 CELESTIAL MECHANICS, thus, as the coordinates X, Y, Z, determine only one point, it follows that the centre of a system of bodies is an unique point. The three preceding equations give this equation may oe made to assume the following form :* the finite integral s»zwj'[(y — or)* + (?/' — y)*+(:' — s)'] expresses the sum of all the products similar to that, which is contained under tlie charac- teristic s, and which is formed by considering all the combinations of * The square of the sum of any number of quantities, being equal to the sum of the squares of tho.-e quantities, and twice the sum of tUt; products of all the binary combi- nations of the different quantities, we iiave (2(m.r)) ' = 2()n^.T^ ) -|-2 ^[mm'.xx) ; Smm/. (x — x')') denotes the products which are obtained, by taking on one part all the binary combinations of the bodies mm , &c. in which the quantities mm are affected with different accents, and then multiplying these by the square of (x — x), in which the terms have respectively the same accents as tlie bodies which they are multiplied by, thus 2.(x — x')' = r^4-x'^-)-x*^ + &c. — 2xx' — 2xx" — 2x'x" — &cand ^{mm' x — .t'y) = mm'x'+mm'j/^ -\-mm".x' 4- nim"^"^ + m'm"x'^-j-ni'ni'..i"'-^-S:c. — Imm'xx — Imn'xx" — Im'm" xfx" ; &c. zz'S^mm'.x') — 22( )H///'.(xx')) and as 2(mx*) =, mx* +jn'x'^ +m"x"^ +&c. 2(ot^- ). 2m. =(jnx^+w't'*-; m''2x"» I &c.)-(»n+w'+''-f m"'-t-&c.) ■=:m-x^ + m"'x"--\-m"^x"' +&c, + mmx- -f-mHj'x'+jwm'x'' +m"7n'.c'^-^7nm".x''' ■f-m'm'.x"^ -\-&c. =2(m*x')4. 2(wm'x2) •.• (_;„.i) = = 2(wn = )+ 22(mm'xa') = 2(mx*). 2w — 2(»nm'x') — 2.ram'.(x— j;*)' -j-2;wm')x^) == 2(inx-).2m — 2OTm'(x — x )-, (by substituting lor 2(«i»r^) its value 2()nx")2m — 2'wm'.(x^;, and for 25(?wm'.xx'). its value 2(mni'.(x2),) — 2(^mm'.{x — x')\) •-• the value of Jt* (Smx)' (2»nx') 'Smm'^x — x)' (2jnj* "" im (2?h)« ' we might derive corresponding expressions for Y^, and Z*. This method gives the position of the centre of gravity of any body of a given fonn> •without being obliged, to refer the position of its molecules to coordinate planes. PART I.— BOOK I. 95 the different bodies of the system. We shall thus obtain the distance of the centre of gravity from any fixed point, by means of the dis- tances of the bodies of the system, from the same fixed axis, and of their mutual distances. By determining in this manner the distance of the centre of gravity from any three fixed points, we shall have its position in space ; which suggests a new way of determining this point. The denomination of centre of gravity has been extended to that point, of any system of bodies, either with or without weight, which is determined by the three coordinates X, Y, Z. 16. Is is easy to apply the preceding results to the equilibrium of a solid body of any figure, by conceiving it made up of an indefinite number of points, firmly united together. Therefore let dm be one of these points, or an indefinitely small molecule of the body, and let X, y, z, be the rectangular coordinates of this molecule ; also let P, Q, R, represent the forces by which it is actuated parallel to the axis of r, of J/, and of z, the equations (w?) and (ri) of the preceding number will be changed into the following : =fP.dm ', O =fQ.dm ; ^fR.dm-* ^/C-Pi*— Q-^)- dm ; =f{Pz—Rx). dm ; O =f{Ry—Qz). dm ; The sign of integration f is relative to the molecule dm, and ought to be extended to tlie entire mass of the solid. * 5 "V~ f being the cosine of the angle which the direction of the force S makes with the axis of x, S. ) — > = the force resolved parallel to the axis of x, ••• it is equal to P; and as^.m=/dm, 2m. -S.^-r^f =/ P.dm, and since 2.S. ^-^i.i/m=fPi/.dm; OX t 0X ^.m.S. { y- { ^ } - ^- { -^ } =/ {P})- Q^) dm, &c. From the values which have been given in the text for the coordinates of the centre of gravity, it is manifest that the position of this centre remains unaltered, whatever change may take place in the absolute force of gravity, •.• when bodies are transferred from one latitude to another on the surface of the earth, though the absolute weight varies, still the position of the centre of gravity is fixed. 96 CELESTIAL MECHANICS, If the body could only turn about the origin of the coordinates, the three last equations will be sufficient for its equilibrium.* * When any system of homogeneous bodies is in equilibrio, the centre of gravity is then the highest or lowest possible ; this is immediately evident from the principle of virtual velocities, for let the weights of any number of bodies m, ?h', m", be denoted by S, S', S"» &c. and let y, s, s", &c. represent lines demitted from the centres of the several bodies m, m', m", &c. on the horizontal plane; now if the position of the system be disturbed in an indefinitely small degree, we shall have, when the bodies of the system are in equili- brio, the equation of virtual velocities Sh+ S'.h'+S.h"+&c. = 0, consequently the quantity of which this expression is the variation, i. e. Si + S'«'-f-S'V'-t. &c. (= the entire weight of all the bodies composing the system, multiplied by the distance of the centre of gravity of the system from the horizontal plane, = s^.S.'^m.) is a maximum or mioimum, and as the weight of all the bodies of the system is always given, the distance of the centre of gravity of the system from the horizontal plane must be either a maximum or a minimum when the system is in equilibrio ; this being established, it is interesting to know the equation of the curve, in which the centre of gravity is lower than in any other curve whose points of suspension and length are given; the'investigation of this curve, which is termed the catenary, is very easy, it occurs in all the elementary treatises, the differential equationis of the following form [i^-\-g).dx~g. cos. c.vdx' -\-y'. It might be proved conversely, that when the distance of the centre of gravity from an horizontal plane is the greatest or least possible, the system is in equilibrio, for we shall have SJ.<-(-i)'.Js' + &/''3.^''/-^-&c. = G.S.« , =0, however there is an essential difference be- tween their states of equilibrium ; in the first case, the equilibrio is denominated instable, in the second, it is termed stable, in order to determine these two different states, we should attend to the species of the motion when the centre deviates by an indefinitely small quan- tity from the vertical, see Xo. 30. * In Physical and Astronomical problems, the method that is generally employed, to determine the mean value between several observed ones, of wliich some are greater, and some less than the true one, is to divide the sum of all the observed values by their number. Tliis comes, in fact, to determine the distance of the centre of gravity from a given plane. For if z, z, z", &c. represent the observed quantities, then , &c. n is the expression for the mean value, but if ^, 2', 7!', denote the distances of the centres of zm-\-z'mi-\-z"m."-\- gravity of n masses, equa each to m from the plane, then — , arc. = PART I.— BOOK I. 97 the distance of the centre of gravity of the system of in masses from this plane — '" , &c. = the required mean value. If several forces concurring in a point constitute an equilibrium, then supposing that, at the extremities of lines, in the directions of these forces, and respectively proportional to them, we place the centres of gravity of bodies equal to each other, the common centre of gravity of these masses will' be the point where all the forces concur. For since the forces are by hypothesis represented by lines taken their direction, and con- curring in one point, it is evident that by making this point tlie origin of the coordi- nates, we shall have the sum of the forces parallel to the three rectangular axes propor- tional to 2(j), 2(!/), 2(3), these sums are ••• by the conditions of the problem = to nothing, see note to page 11 ; and since the masses are all equal we shall have 2(x).m = 2(»a) = 0, this also obtains for the other axes, consequently we shall have 2(ot.j-) = 0, ~{m^) = 0, 'Z{mz) = 0, ••• the origin of the coordinates coincides with the centre of gravity of the system of masses respectively equal to vi. The centre of gravity of a body, oc system of bodies, is that point in space from which if lines be drawn to the molecules of the body, the sum of their squares is the least pos- sible. For if X, Y, Z, represent the coordinates of such a point, then the sura of the squares of the distances of all the molecules of the system from this point is equal to 2((j; — X) *-f-(y — ^*(z — Z)'^)> 'f ^e take the differential of tliis expression with respect to each of the coordinates, and multiply each of the terms of the sums which are respec- tively equal to nothing, by the element of the mass, we shall have 2.m.(.i- — .Y) = 0, 2.»7i.(^— Y)=0, 2.«4z— Z,) = 0, ... A'=r ^Oa^. Y_ ^^^ ■ Z— ""'" - 2w 2»J ' 2ffi and from what has been demonstrated in the preceding note it follows, that if we apply to all the points of the system, forces directed towards the centre of gravity, and propor- tional to the distances between those points and the centre of gravity, these forces will constitute an equilibrium ; consequently when several forces constitute an equilibrium, the sum of the squares of the distances of the point of concoiu-se of these forces, from the extremities of lines representing these forces, i. e. the sum of the squares of these Unes, is a minimum. From the preceding property it appears, that if several observations give different values for the position of a point in space, the mean position, i. e, the position which deviates the least from the observed positions, is that in which the sum of the squares of its dis- tances from the observed positions is the least possible. The problem is altogether similar when we wish to combine several observations of a7iij kind whatever; for the distances of the points correspond to the differences between the particular results and their mean value ; and since it is impossible entirely to exterminate these differences, we are obliged to select a mean result, such that the sums of the squares of these differences may be a mi- O 98 CELESTIAL MECHANICS, niniuni ; this is the principal of the method of the least squares, which was devised by Le Gendre to combine the equations of conditions between the errors deduced from a. comparison of the astronomical tables with observation ; it comes in fact to find the centre of gravity of the observations which we compare together. The general form of the equations of condition is as follows : ~ a-\-bx-\-ci/-\-dz-\-&c. when we pass into one member all the terms which com- pose them, a, b, c, are given numerical coefficients, if all these equations could be satis- fied exactly, by the values of x, y, z, their first members would be necessarily reduced to nothing by substituting for x, y, z, their values, but as this substitution does not render them accurately equal to nothing, let E, E', E", represent the errors which remain, then we shall liave E— a-\-bx-\-cy-\-dz-^&:z. ; E~ a'-hi'x-f c'y + i/'s-f&c. ; E":=.a" \V'x-\-c'iy^ &c. the quantities x, y, z\ &c. are to be determined by tlie condition that the values E, E', are either nothing, or very small ; the sum of the squares of the errors := i;=-fii=+£'-4-&c.= (a=-fa'--fa'''=4-&c.)-i-{i'+i'*+i''''*).x2+(c--f (■' = -ff»'*-f&c.)/- -f.(rf*-ha!"- -\-dff ' -f &c.)« ' -1- ; 2{abJro:b'^a"b"->r&c.)xJ(- 2 {acJra'c'+a"c«)y{- 2 {ad-\-dd'+a''d''-\-8cc.)z; ■\-%bc+b'c' -\-b/'c"J^&c.) xy-{- ^(bd + b'd'-^b"d'/)-{-^z + &iC. the minimum of this expression, with respect to x, will be = 2.n6+x 'S-.h'^+y l..bc^z. 2.M-f &c. the minimum with respect to j/ = 2.ac-|-x2.6c-|-y2c*-l- x.2.rfc= 0, we derive a cor- responding value for the minimum of z, hence in order to form the equation of the minimum with respect to one of the unknown quantities, we must multiply all the terms of each proposed equation by the coefficient of the unknown term in that equa- tion, and then put the sum of the products equal to nothing. Though this method requires more numerical calculations, in order to form the particular equation relative to each unknown quantity, than the method suggested by Mayer ; it is more direct in its application, and requires no tentation on the resulting equations. Laplace has shenii in his Theory of Probabilities, that when we would take the mean between a great number of observations of the same quantity, obtained by different means, this is the only method which the theory permits us to employ, see Le Gendres Memoir on the determination of the orbits of the comets, ard Biot's Astronomic Physique, tome 2. page 200. PART I.— BOOK I. 99 CHAPTER IV. Of the equilibrium of fluids. 17» In order to determine the laws of the equilibrium, and of the motion of each of the molecules of a fluid, it would be necessary to ascertain their figure, which is impossible ; but we have no occasion to determine these laws, except for fluids* considered in a mass, and for this purpose the knowledge of the figures of their molecules is useless. Whatever may be the nature of these figures, and the properties which depend on them in the integrant molecules, all fluids, considered in the aggregate, ought to exhibit the same phenomena in their equilibrium, and also in their motions, so that from the observation of these pheno- mena, we are not able to discover any thing respecting the configura- tion of the fluid molecules. These general phenomena depend on o2 * Although the figure of the molecules of fluids are unknown to us, still there can be no question but that they are material, and consequently that the general laws of the equili- brium and motion of solid bodies are applicable to them. If we were able analytically to express their characteristic property, to wit, extreme smallness, and perfect mobility, no particular theory would be required in order to determine tiie laws of their equilibrium and motion ; they would be then only a particular case of the general laws of Statics and Dy- namics. But as we are not able to effect this, it is proposed to derive the theory of their equilibrium and motion from the property which is peculiar to them, of transmitting equally, and in every direction, the pressure to which their surface is subjected; this property is a necessary consequence of the perfect mobility of the molecules of the fluids. In the definition wliich has been given in the text there is no account made of the tena- city or adhesion of the molecules, wloich is an obstacle to this free separation ; this adhe- sion exists however between the molecules of most of the fluids with which we are ac- quainted. 100 CELESTIAL MECHANICS. the perpect mobility of these molecules, which are thus able to yield to the slightest force This mobility is the characteristic property of fluids ; it distinauishes them from solid bodies, and serves to de- fine them. It follows from this, that when a fluid mass is' in equilibrio each molecule must be in equilibrio in consequence of the forces which * solicit it, and of the pressures to which it is subjected by the action of the surrounding particles. Let us proceed to develope the equations whicii may be deduced from this property. For this purpose, let us consider a system of fluid molecules, consti- tuting an indefinitely small rectangular parallelepiped. Let x, y, z, denote the three rectangular coordinates of that angle of the parallele- piped, which is nearest to the origin of the coordinates. Let dx, dy, dz, represent the three dimensions of this parallelepiped ; let p repre- * When a fluid is contained in a vessel, the pressure to which it is subjected at its sur- t'ace is transmitted in every direction, as has been just stated, but since the molecules ar^ material, 'they must have weight, therefore it also presses the sides of the vessel with a force arising from the weight of the molecules, and different in every point of the sides ; and if the fluid is contained in a vessel closed in every side, when the molecules are solicited by any given accelerating forces, then the pressure is different for every particular point, its direction is always perpendicular to the surface, since by No. 3, when the resistance of a surface destroys the pressure on it, the direction of this pressure must be normal to the sur- face. The intensity of this pressure depends on the given forces, and on the position of the point. Therefore it appears, that in the equilibrium of a fluid contained in a vessel, the entire pressure in each point of the sides is the sura of two pressures altogether distinct ; one of which arises from the pressure, exerted on the surface, and is the same on all the pomts ; the other is owing to the motive forces of the particles of the fluids, and varies from one point to another. Fluids are generally distinguished into two classes, incompressible, and elastic; with respect to the last class, they may press against the sides of the vessel in which they are enclosed, although no motive forces act on the particles, or without any pressure urging the surface of the fluid. For from their elasticity they tend perpetually to dilate them- selves, which gives rise to a pressure on the sides of the vessel : however this is a constant pressure in tlie same fluid ; it depends on the matter of ll>e fluid, its density and tem- perature. PART I— BOOK I. 101 seut the mean of all the pressures, to which the different points of the side dy. dz of the parallelepiped, which is nearest to the origin of the coordinates, is subjected ; and let f' be the corresponding quantity on the opposite side. The parallelepiped, in consequence of the pressure to which it is subjected, will be urged in the direction of x\ by a force equal to (j) — ;p')- dy.dz ; p — p is the difference of f, taken on the hy- pothesis that .r alone is variable ; for although the pressure f acts in a direction contrary to ^j, nevertheless the pressure to which a point is subject being the same in every direction, p'*-p may be considered as the difference of two forces infinitely near, and acting in the same di- rection ; consequently we have* p'^^zz < -J-^\.dx, and [p — p'). dy. dz = — \~J^ \' ^'^' ^y' ^~' Let P, Q, /»*, be the three accelerating forces which solicit the mo- lecules of the fluid, independently of their connexion, parallel to the axe^ of X, of y, and oi z ; if the density of the parallelepiped be denoted by />, its mass will be equal to p. dx. dy. dz. and the product of the force P by this mass, will represent the whole motive force, which is derived from • Since p, 5, P,Q,Il, generally vary from one point to another of the fluid mass, tliey must be considered as functions of x, 7/, z. We distribute the fluid into parallelepipeds, in order more easily to express in analytical language the fact of the equality of pressure, which, as has been stated, is the fundamental principle from which we deduce the whole theory of thtir equilibrium, and by supposing these parallelepipeds indefinitely small, we ite permitted to consider all the points of the same side as equally pressed, and also ^ > Pt Q, R, as constant for each side respectively, by means of which we are able to detennine the pressure p. x, y, z, being the coordinates of the angular point next the origin, and p being a function of these coordinates, we shall have the coefficient ) ~- [ = < -^ \ &c. they are taken negatively because they tend to diijamish the coordinates. Sz JOS CELESTIAL MECHANICS, it ; consequently this mass will be solicited parallel to the axes of x\ by the force jpP— f— f-1 C, dx.dy.dz. For similar reasons it will be so- (. \ dx SS licited parallel to the axes of j/, and of a, by the forces ip.Q — \-~- 5 f . dx.dy.dz. and S^.R — 1-^|^. dx. dy. dz. &c. therefore, by the equation [b) of No. 3, we shall have or ip = p(P.~C dy i . * Wheng (P.Jx+Q.Sy+/?.Ss.) is an exact difFerential, | — ^| — 1= < -~ \ &c. (see Lacroix Traite Elementaire, Calcul. Differential and Integral, No. 261.) '''~~clf"^ Ihj dx "^ dx ' dz "^ dz ~ dx "^ dx ' SL—L . _^?:iL__ ± 1 i_Lj if we multiply the first equation by R, the second by dz ~ dz dy dy — Q, and the third by P, we shall obtain, ^.R.dP: JhP.d^ R-i-dQ R.Q.d^ ^.Q.dP Q.P.di _ ^.Q.dR dy ^ dy ' dx dx ' dz dz dx R.Q.d^ ^.P.dQ P.QJj _ ^.P.dR I R.P.di dx ' dz dz '^ dy dy ' PART I.— BOOK I. 103 from which we may obtain This equation expresses the relation which must exist between the forces P, Q, and R, in order that the equilibrium may be possible. If the fluid be free at its surface, or in certain parts of this surface, the value of p will be equal to nothing in those parts 5 therefore we shall have Sp z: 0, provided that the variations Sx, Sy, iz, appertain to this surface ; consequently when these conditions are satisfied, we shall have O = PJx + QJy + RJz. If Su — O, be the differential equation of the surface, we shall have PJx + QJt/ -jr R.iz = \Ju, X being a function of r, 1/, z', from which it follows, by No. 3, that by reducing all the terms in which Jj is involved to one side, and then adding them toge- ther, we get ( RMP R.dQ Q.dP , Q.dR P.dQ P.dR y. dy dx dz dx dz dy _RP^ RQ.d^ QP.d^ RQ.d^ PQ.3g RP.d^ _ dy '^ dx ■*■ dz ~ dx ~~ dz + dy "^ ^ * by coacinnating This equation shews whether the equilibrium is possible, though we are unable to as- certain the density ;. 104 CELESTIAL MECHANICS, the resultant of the forces P,Q,R,* must be a perpendicular to those parts of the surface, in which the fluid is free. Let us suppose that the variation P(}'x 4- Q.(r3/ + i?.(?s; is exact, this is the case when P,Q,R, are the result of attractive forces. Denoting this variation by J'. Therefore the pressure is the same, for all molecules whose density is the same ; thus Jf> must vanish with respect to those strata of the fluid, in which the density is constant, and with regard to these surfaces, we have, 0=PJx+QJy+RJz.* consequently, the resultant of the forces, which solicit each molecule P be at an infinite distance the surface will degenerate into a plane, ••• if the planets were 1 originally fluid, and if their molecules attracted each other with forces, varying as -— they would assume a spherical form. See No. 12, Book S**. • If P.3x-f-(J.Jy+/?.32is an exact variation, Sip, Jprr^Jip, •.• j must be some function of , consequently, if the P " K.(?2-j-l ' ("+ l)"" powers of the distance be taken in arithmetic progression, the pressures and the densities proportional to them, will be in geometric progression, •■• if n is negative, and if in the radius, ordiaates be erected proportional to the pressures or densities ,the locus of their extremities will be a curve of the hyperbolic species, and the radius produced, will be an asymptote to the curve, if n is positive, the locus of the extremities of the coordinates, will be a curve of the parabolic species, if n:;0, i. e. if the force is constant, the locus will be the loganthmic curve. See Princip. Matth. Liber 2. Prop. 22, et Scholium. p2 108 CELESTIAL MECHANICS, CHAPTER V. The general principles of the motions of a system of bodies. 18. We have, in No. 7,* reduced the laws of the motion of a point, to those of its equilibrium, by resolving the instantaneous motion into two others, of which one remains, while the other is destroyed by the action of the forces which solicit the point ; we have derived the diffe- rential equations of its motion, from the equilibrium which subsists be- tween these forces, and the motion lost by the body. We now proceed to employ the same method, in order to determine the motion of a system of bodies m, m', m \ &c. Thus, let mP, mQ, Em, be the forces which solicit ?« parallel to the axes of the rectangular coordinates *, y, z ; let m'P', m'Q', m'R', be the forces which solicit m, parallel to the same axes, and so on of the rest ; and let us denote the time by /. The partial forces m.—-—,m.-^,wz.—^ of the body m at any instant at at at whatever will become in the following :t • The principle established in this number, has been termed in P^^'® "^ ^^'^ increase 110 CELESTIAL MECHANICS, and as the forces rfj , dx dy , , dy dz ^ , dz ni. 1- m. d. —— ; m. -~+ in. d. -^- ; m. —— + m. d. — — ; dt dt dt dl dt dl of the velocity, tliat the changes in the motions of tlie body arc made by insensible degrees. The inspection of the equation (/') sliews tliat it consists of two parts entirely dis- tinct, of which one is the quantity which wc ought to put equal to nothing, when the forces P, (I, It, J'', itc. which are applied to the diflbrent points of the system, constitute an equilibrium, the other part arises from the motion which is produced by the forces /', .(P.di + Q.^i/+i2.(/^) ; (Q) c being a constant arbitrary quantity introduced by the integration. If the forces P, Q, R, are the results of attractive forces, directed towards fixed centres, and of a mutual attraction between the bodies ; the function "L.fm.^P.dx + Q.dy+Rdz)* is an exact integral. For the Q * In fact, the accelerating force of m, produced by the action of m in the direction of the line_^ zsm F, (Fis always a given function ofjl) '.' the components of tliis force pa- rallel to the axes of x,i/, z, are m'F. - — - — , m'F. , m'F. ~ , •.' the parts of P.dx+ Q,.dy-\-R.dz, which answers to this force alone are ni'F.{{x' — x).dx-^{y' — y).dy-\-{z' — z).dz), and as the accelerating force of m', arising from the action of m, resolved parallel to the coordinates x, y, z, respectively = m.F. — 4- m.F. ^^~y + VI.F. ~~ , the corresponding part of FJx-t Q'.dy'+R'.dz, is, F.w. |if=ii.rf/+l^^:^.%'+if=il. dz\, therefore in or- der to have the motive force ai'ising from the mutual action of the bodies m and m' we must multiply the first expression by m, and the second by »n', and adding them toge- ther, they will become mm'.F. (x'—x). dx + (y'—y).dy+(z-zJ.dz+{x—x').dx^-\-{y—y'). dy+iz—z), d^)= mm'.F.fd/, for asf^ = {x-^y+{y-yY +{z-z'y,fdf:^ {x-x').(dx-dx') + {y-yWy-dy) + {z-z).{dz-dz), consequently as F is given to be a function of y; ^f-dj. is an exact differential. If the .centres to which the forces are directed Jiave a motion in space, then P.rfx+ Q.di/^Rdz, is not an exact differential, though the law according to which the forces vary should be a function of the distance, see Note to page 34. The sum of the living forces at any instant will be given by the equation ( Q), when we know the value of this sum at a determined instant, and the coordinates of the bodies composing the system in the two positions of the system. And when the system returns to the same position, the living forces will be the same as before. 114. CELESTIAL MECHANICS, part whicli depends on the attractions directed towards fixed points, are exact integrals by No. 8. This is equally the case, with respect to those parts, which depend on the mutual attractions of the bodies com- posing the system ; for if we name^ the distance of m from m', m'F, the attraction of ot' on »z ; the part of m(P.ds + Q.di/ + R.dz) which arises from the attraction of m' on m, will be, by the above cited No. equal to — mra'Fdf, the differential df being taken on the supposi- tion, that the coordinates x, y, z, only vary. But reaction being equal and contrary to action, the part o?m'{P'.dx'-\-Q'.dy'-]rR'dz'^ which is due to the attraction of m on m', is equal to — mm'.Fdf, the coordi- nates x', y\ z', being the only quantities which are supposed to vary, consequently df being the differential of y on the supposition that both the coordinates x, y, z, and x', y', z', vary simultaneously, the part of the function 1.77i(^P.dx-{-Q.dy -h Ji-dz) which depends on the reciprocal action of m on vi' is equal to —nim'.F.d/i Therefore this quantity is an exact differential when F is a, function of f, or when the attraction varies as some function of the distance, which we shall always suppose ; consequently the function 1.7n.(P.dx+Q.dy-{-R.dz} k an exact dif- ferential, as often as the forces which act on the different bodies of the system, are the result of their mutual attraction, or of attractive forces directed towards fixed points. Let then d(p represent this differential, and naming v the velocity of 7n, t/ the velocity of ni', &c. we shall have I..mv' = c + 2 —r-, trom dt dt dz *' dt dt dt one instant to another ; differences which become finite, when the motions of the bodies undergo finite alterations in an instant. In this The equation (/-") may be made to assume the following form : in which tlie changes that are produced in the motions of the bodies composing the system, are made by insensible degrees, as is evident from the circumstance, that the differential of the velocities is expressed by — — , see note to page 30 ; now, if instead of this gradual di- 1 • • dx dy ininution, bodies experience abrupt changes m their motions A. -^. A.— -, &c. express- ing those changes, the preceding expression will be changed into the following : -2.n;.( P.Sj+ Q.?^ -J- jR- Jy ; dx Sx ~di" It dx and as in this case wA. — ^ is the variation of the force of the body, on the supposition dt that it is entirely free, and m.P.dt is the variation which actually takes place in conse- quence of the action of the bodies of the system, the reasoning in No. 18 is applicable to this case, consequently the preceding expression may be put equal to nothing ; and since the values of dx, dy, dz, are changed in the following instant into rfx+A.tfor, dy-\-^-dy. dz+^dz, we shall satisfy the conditions of the connection of the parts of the system, by making tlie variations 3x, iy, h, equal to these expressions respectively ; and then the preceding equation will assume this form f dx _ dx t dx , ( dy dy \ dy { dz_ Jz_\ dz^ ' dt"^ dt ] ^' dt 2.»n.(P. ( (ix+A.rfx ) + Q. ( rfy-f- A.rf^) + i?.((/z+ A.(/£ ), =0, PART I— BOOK I. 117 equation we may suppose Sj:=:dx+A.dx ; $y zz dy + A..dij; Sz = r/^+A. dz; because the values of dx, dy, dz, being changed in the following in- stant into d.T + £^.dx, dy + £:i,dy, dz+A>..dz, these values of Sx, Sy, Sz, satisfy the conditions the connection of the parts of the system ; there- fore we shall have \^ dt^ dt ^ dt^^dt^ dt ^ dt C dz dz > dz } i-dF^ ^•~dfy^~di\ ^ X.m.(P.(.dr+A.dx) + Q.(dy+A.dy+B.(dzi- A.dz)) This equation should be integrated as an equation of finite differences relative to the time t, of which the variations are infinitely small, as well as the variations of ^, y, z, a/, &c. Let 2, denote the finite inte- grals resulting from this integration, in order to distinguish them from the preceding finite integrals, which refer to the aggregate of all the bodies of the system. The integral of mP/dx + A.dx) is evidently equal to JinP.dx f therefore we shall have const.— dx*+dy' + dz» ^^ [ ( ^ dx* ^, / dy* \ , dz\-i* ^■''- IP +^'-'"r^-5F)+^^-ir) + (--^)( —21.fm.(P'dx + Q.dp + R.dz) ; dx * In this equation, though the value of A. — — may be finite, still dx-\-i^.(Ix, and the variation of the time may be indefinitely small, and V integrating with respect to this C dx dx "J dx^ quantity, 2,2.»n. I ——.A.— V = 2.>n. —j-j-, or it may be otherwise expressed thus, A.(x') :;^(see Lacroix No. 344) 2xh-{-h'^, and if /« be made equal to Ax, it becomes 2:r.Ax-|-(A.x)*, •.• 2. 2.Xj;A..r+(A.x)*) = 2,.(2a:.A.l-l-(Aj;)= )-(- 2,.(Ax)« — x^ + 2,.(Ax4), consequently, if we multiply the preceding equation by two, and substitute dx in place of r, and then integrate, we obtain the expression which has been given in the text. *ia CELESTIAL MECHANICS, therefore v, v, v" denoting the velocities of m, m', in!', &c. we shall have s.^.'=const.-E,2.«.UA.^r+ { A.^r4- u.^xx ' (^ dt f \ dt S '- dt S i + 2I.j:m.{P.dx+Q.dj/ +R.dz'). The quantity contained under the sign Z^, being necessarily positive, we may perceive that tlie living force of the system is diminished by the mutual action of the bodies, as often as during the motion, any of the variations A.— — ,A.— ^, &c. are finite. Moreover, the preceding Civ at equation affords a simple means of determining the quantity of this diminution. At each abrupt variation of the motion of the system,* the velocity * At every abrupt change in the motion of the system, the velocity is not always di- minished for every body, but the expression which is here given may be considered as general, by supposing that when the velocity is increased, a negative portion of it has beeti destroyed, and the square of the velocity after the shock is equal to ^■'"' dt'' and as ^ 2.dxi\.dx + '2{A.dxY + 2.dy^.dy + 2{A.dyy +2dz:^.dz+2{ A.dz)*, =0, by subtracting this equation from the preceding, we obtain the square of the velocity after the shock, equal to .„ (dx'+dy'+dz- ) ^___ (S.dx)^MM>/y-^(^-^=:)' dt'' — "• • dt' and as the square of the velocity before the shock is equal to l.mv'' = 2.W. — ^ — ■^ ^, the square of the velocity lost by the shock =2.ot. F* _v« Jj^±y±{±_clyy+(A.dz)_' . dt' ' consequently the loss which the living forces experierjce, is equal to the sum of tlie living forces, which would belong to the system, if each body was actuated by that velocity which it loses by the shock. PART I.— BOOK I. 119 of m, may be conceived to be resolved into two others, of which one v subsists in the following instant, the other V being destroyed by the action of the other bodies, but the velocity of vi before the decomposi- . , . \/dx* + dy*-{-dz*, ■, , . n tion being -^ , and changing afterwards into dt it is easy to perceive that ( dt S t dt S c dt S ' consequently the preceding equation may be made to assume the fol- lowing form, 2.m'« = const.— 2^.2.7??. V'—2.1.fm.{P.dx + Q.di/ + .dz),* * The variation of the vis viva of the system, is equal to 22m,{P.dx-{-Q,.di/+ R.dz) consequently when this expression vanishes, i. e. when f/.2.(mt'*) vanishes, the vis viva of the system, equal to 2.()??u=), is a maximum, or a minimum; but it appears from the principle of virtual velocities, that 2m.(P.Sx-[- Q.?!/-\-R.h) is equal to nothing, when the forces P, Q, R, P", constitute an equilibrium ; and since the differentials dx, dy, dz, may be substituted for the variations 'hx, tij, S^, when they are subjected to satisfy the condi- tions of the connection of the parts of the system, l.m.[P.dx-\- Q,.dy-\-R.dz) is equal to nothing, in the same circumstances ; •.• when the forces P, Q, R, Pi, constitute an equi- librium, the vis viva of the system is a maximum or a minimum. And as it appears from note to page 96, that the positions of equiUbrium of a system of heavj' bodies, correspond to the instants, when the centre of gravity is the highest or lowest possible, the sum of the living forces is always a maximum or a minimum when the centre ceases to ascend, antl commences to descend, and when it ceases to descend and commences to ascend. The value of the vis viva is a minimum in the first case, and a maximum in the second, for 'Zm.{P.dx-\-Q.dy-{-R.dz) corresponds to the expression S.Ji-f- S'.Ss'-f S'''3i''''-|-&c. in page 96, and •.• by substitution we have Imv'^ = c-\-s,.S'^m. con- sequently 5.?)!ii» is a maximum or minimum, when s, is a maximum or minimum. When S.mii* . is a maximum, the equilibrium is stable ; when a minimum, the equilibrium is in- stable. For from the definition of stability, (see No. 28) it appears that if the system is only agitated Sy one sole species of simple oscillation, the bodies composing it will perpe- 120 CELESTIAL MECAHNICS, 20. If in the equation (P) of No. 18, we suppose, Sxf =z Sx-\-ix/ ; Sy' = Sy-\-Sy; ; Sz' zz Sz-\-Sz; ; tually tend to revert to the position of equilibrium, consequently their ve'ocities will di- minish according as their distance from the position of equilibrium is increased, and •.- tlie sign of the second differential of ^ will be negative, consequently 2 wd^. will be a maximum in this case ; and it may be shewn by a like process of reasoning, that the vis viva of the system is a minimum, when the equilibrium is instable. From a comparison of this observation with the note to page 96, it appears that in a system of heavy bodies, when the vis viva is a maximum, the centre of gravity is the lowest possible, and highest when the vis viva is a minunum. This may be more strictly demonstrated thus : if the system be disturbed by an indefinitely small quantity from the position of equilibrium, by substituting for P, Q,R, P', &c. their values in terms of the coordinates, and then expanding the resulting expressioninto a series ascend- ing according to the variations of these coordinates, the first term of the series will be the value of tp, when the system is in equUibrio ; and since it is given, it may be made to coalesce with the constant quantity c, which was introduced by the integration ; the second term va- nishes by the conditions of the problem ; and when 2.»nv" . is a maximum, the theory of max- ima and minima shews that the third term of the expansion may be made to assume the form of a sum of squares, affected with a negative sign, see Locrobc, No.lSl; the number of terms in this sum, being equal to the number of variations, or independant variables ; the terms whose squares we have assumed, ai'e linear functions of the variations of the coordinates, and vanish at the same time with them ; they are therefore greater than the sum of all the remaining terms of the expansion. The constant quantity being equal to the sum of c, and of the value of 'S.mv'^. when the forces P,\Q, R, P', &c. constitute an equilibrium, it is necessarily positive, and may be rendered as small as small as we please, by dimi- nishing the velocities ; but it is always greater than the greatest of the quantities whose squares have been substituted in place of the variations of the coordinates ; for if it were less, this quantity being negative, would exceed the constant quantity, and therefore render the value of S.mi)'. negative, consequently these squares, and the variations of the coordi- nates, of which they are linear functions, must always remain very small, v the system will always oscillate about the position of equilibrium, and this equilibrium will be stable. But in the case of a minimum it is not requisite that the variations should be always constrained to be very small, in order to satisfy the equation of living forces when

'■ {■'^-^\^--'':-{'-^- p} the terms in this expression which are multiplied by Sx, Sy, Sz, respectively, are by adding them together and being independent of the conditions of the connection of the system, they must be put ieverally equal to nothing. t SuiceX= , Y= —, &c. -r5- = 2.7W.-— = , 2.WJ 2.m di' dt- 2.?k 2to because ^.m.-—^ — 2m.P=0. PART I.— BOOK I. 123 as if all the bodies m, m', &c. were concentrated in this point, the forces which solicit the system being applied to it. If the system is only subjected to the mutual action of the bodies which compose it, and to their reciprocal attractions, we shall have = -z.mP ; = -z.mQ ; O = -z.mR j for p designating the reciprocal action of m on m', whatever its nature may be, and y' denoting the mutual distance of these two bodies ; we shall have, in consequence of this sole action, „ (x — x) „ (y — y') p [z — z') mP=.p.^^—^ — -; mQ=p. ^- . ' •,mR=p. - — ■, — - ; mF = p. ~ . ; mQ=p. '--^ •^ - ; mR=p. ^ ; from which we collect Oz=mP+m'P' ; 0=mQ + m'Q ; 0-mR + m'R'; * and it is evident that these equations obtain, even in the case in It 2 V • / — X, y — ■)/, z' — z, being the coordinates of nt relative to the new origin of the forces, and the action of p being directed along the line the part of mP, which corresponds to the force p resolved parallel to the axis of x=p — , the analogous parts of otQ, and mR, axe p. \. >p- respectively, in like manner the forces soliciting m' parallel to the coordinates, arising from the action of p, -p. J ,P- J- 'P- f > .-. when the sole force soliciting tn and m' arises from p, which expresses the reciprocal action •f m on m', we have mP-^-m'P',—p\JI' ' — =0. Action being equal to reaction, and its direction being contrary thereto, when two bor 124 CELESTIAL MECHANICS, which the bodies exercise on each other, a finite action in an instant. Their reciprocal action disappears from the integrals ^.mP, ^.mQ, 'Z.mR, and consequently, these expressions vanish, when the system is not so- licited by any extraneous forces. In this case we have and by integrating X^a + bt: r~a'-{-b't; Z=a"+b"t;* a, b, a', b', a", b", being constant arbitrary quantities. By eliminating the time t, we shall have an equation of the first order, between either X and Y, or X and Z ; consequently the motion of the centre of gravity is rectilinear. Moreover, its velocity being equal to v/{?F^ If F^ {f }■ or to v' b b'--^b'\ it is constant, and the motion is uniform. It is manifest, from the preceding analysis, that this invariability of the motion of the centre of gravity of a system of bodies, whatever their mutual action may bc,t subsists even in the case in which any one dies concurring, exercise on each other a finite action in au instant, their reciprocal action will disappear in the expressions S.jwP, 5.mQ, &c. in fact, as we can always suppose the action of the bodies to be effected by means of a spring, interposed between them, which endtavours to restore itself after the shock, the effect of tlie shoclc will be produced by force* of tlie same nature with ;;, which, as we have seen, disappear in the expressions 'S.mP, S.otQ, S.mii. ♦ By integrating once we get — = b, .: dX:z bdt, and X~ ht+a; the constant quantities Clt a, a', a", are equal to the coordinates of the centre of gravity when / ■=. 0, and b, V , b", are equal to the velocity of the centre of gravity resolved parallel to the coordinates. See notes to pageSl. ■\ In fact, from what has been observed, in the note to page 116, it is evident that the principle of D'Alembert is true, whether the velocities acquired by the bodies be finite, after a given time, or indefinitely small, or whether the velocities be partly finite, and partly infinitely small, such as arise from the action of accelerating forces, and both PART I.— BOOK I. 125 of the bodies loses in an instant, by this action, a finite quantity of motion.* 21. If we make Sjif-=z ^ + Sx ; Sx =^ h^< ; &c. y y Sy= + Sy/,Sij'= + Sy ; Sy'= + Sy;' ; &c. t y y y , tiie variation J'.r will again disappear from the expressions iJ/, ^', ^f", &c. ; therefore, by supposing the system free, the conditions relative before and afler the impact, we have 0= Trj-> 0= -rr^t *c. and also — . S.m = (/<» ' df^ ' ' dt dx . . . ■ 2.m. — , &c, = the quantity of motion, and sinc^' ' by hypothesis the quantity of motion dx lost, equal to the difference between Sm.— before and after impact, should be = to nothing, such as would cause an equilibrium in the system, it follows that -t^.S.jb. before and after impact must be the same, but 2.?n being given, — equal to the velocity of tlie centre of gravity, will be the same before and after impact. * As the centre of gravity of a system, moves in the same manner as a body equal to the sum of the bodies would move, if placed in the centre of gravity, provided that the same momenta were communicated to it, which are impressed on the respective bodies of the system, the motion and direction cf the centre of gravity, may be always determined by tlic law of composition of forces. If the several bodies of a system were only subjected to their mutual action, then they would meet in the centre of gravity, for the bodies must meet, and the centre of gravity remains at rest. t The fractional part of these expressions for Sy, Sx'/, Sy, ?/, Sj/'^, &c. arises from the ro- tatory motion of the system about an axis parallel to z, for it appears from Nos. 22 and 25, that when the direction of the impulse does not pass througli tiie centre of gravity, the body acquires both a rotatory and rectilinear motion, now if the only motion impressed on the ' system was that of rotation, then the element of the angle described by the body m, is equal to the variation of the sine divided by the cosine =- — ^. Sj, the elementary angle de scribed by 126 CELESTIAL MECHANICS, to tlie connection of the parts of the system will only influence the va- riations Sf, Sf" &c. ; the variation Sx is independent of them, and entirely arbitrary ; thus by substituting in the equation (P) of No. 1 8, in the place of Sx, $x", Sx"\ &c. Sy, Sy", Sf, &c. their preceding values. = \/£2+^ ^'^J''+^/' . _"^^-'^"'+.y'* . ix- .'. the variation of j^ will be equal to t:.^.. v^'g' ^-)-y'~ v' V ^ ' — , Sj,=- . dx the same may be proved of the other variations ix', 3x'' y ^/x'^^-y^ y ^ ^ ' ^^ ' +y' = the distance of m from the axis of r, .*. , , - is equal to the sine of tlie angle which Vx^+i/' makes with y. If the expression — i— — -^ be consi- y a: dered with respect to the cosine^, the variation o_y = — ?x. ~ "^-^ . y Vx^'+^: ix.X „ , . ) tor the variation of the cosme is equal to the variation of the arc affected with a negative sign, and divided by the sine, and as the variation of the angle described hr , \/x--\-y"' ... . . Vx^^'+Z'^ "' -^ . ox, this expression being referred to the cosine is equal to -^ — .• . 3x.= . 3x. If in the expression we substitute for ^x', ^.i", Sj/, Sy', Sy , &-c. their values, it becomes f 3(i/''ix xi/.ox , » . » , » , » , y'x.'.dx y.x''^x . y'xHx y y y y y ~-—+yhl—y-h!~y h.^y^yrrf^ j^ therefore the variation Jx disappears from the expressions "if, "if, &'C. Making the same substitutions in the equation (P) it becomes PART I.—BOOK I. 127 we should put the coefficient of Sx separately equal to nothing, which gives 0='SM. ^^ ^~~^ —^-+-z..m. {Pt/ — Qx) ; from which we deduce by integrating with respect to the time t, c=z.m. C-^^i^— -V^-^) -|-s>?.(Py— Qx). dt; c being a constant arbitrary quantity. In this integral, we may change the coordinates y, i/, &c. into z, s', provided that we substitute in place of the forces Q, Q, &c. parallel to the axis of ^, the forces B, R', 2)arallel to the axis of z, which gives, d = z.m. (^dz-zclj/) ^ s/.m.(P3— i?^). di ; will be the angular distance of the second of these axes from the same equinox. We will name these three new axes, principal axes. Let.r,,_y^, 2, represent the coordinates of m referred, first to the line drawn from the origin of the coordinates, to the- equinox of spring ; x being reckoned positive on this side of the equinox ; 2dly, to the pro- jection of the third principal axis on the plane of x and of y ; Sdly to the axis of z, we shall have PART I.— BOOK I. 131 xz=.xi. COS. ^-\-y,. sin. 4* » y=j/,. COS. v|/— T^ sin. 4/ ;• » ^ */• Let j,^ y,t, Zii, be the coordinates referred, 1st to the line of the equi- nox of spring ; 2dly, to the perpendicular to this line in the plane of the equator ; Sdly, to the third principal axis ; we shall have ^1 = x„ \ y, = y,!. cos. %-\-Z/,. sin. 6 ; z^ = Zi,. cos. 6—3/,,. sin. 6. Finally, let x,„, y ^^„ ^„,, be the cooordinates of m, referred to the first, • s 2 • As the axes of the coordinates .r,, y., exist in the plane of x, y, and as the angle wliich the axis of x makes with the axis of x_, is equal to -i^, we have by the knowii formula; for the transformation of one system of rectangular coordinates, into another system existing in the same plane, x=x,. cos. ■\'-\-yi- sin. •4' ;yr:_y,. cos. -i^ — x. sin. ij/ ; and because the axis of 2 coincides with the axis of s , we have z=z. Comparing the coordinates, * ,^,,z,, with the coordinates x^,y^^,z^, it appears that the axis of .Ty coincides with the axis of x^,, and consequently x,=t// ; and as the axis of »/, is in the plane of the ecliptic, perpendicular to the line of equinox of spring, and aa the axis of ^,, exists in the plane of the equator perpendicular to the same line, it is manifest that the angle formed by these axes is equal to the angle i, the inclination of the two planes, and that these two lines and the axes of z^ and z^^, which are respectively perpendicular to those planes, exist in the same plane, consequently we have, as before, yr^Vii' '^o*' ^-\-'^ir s'"- *> -/=-//• cos. i. — _!/„ sin. i. Lastly, it appears that the axis of z,i coincides with the axis of s,,^, and consequently that z,,'=z,i,, ; and as the axis of x„ and 2/,„ and of x,,^ and^„, are in the plane of equator; and as by hypothesis, ij) is equal to the angle which the axis of*,,, makes with the line of equinox of spring, which line is supposed to coincide with the axis of x^,, we have x^,=x^„. cos.

H".y/- sin- ^ = (-r//- cos. ■^■^y,,- cos. 6. sin. ilz+z,,. sin. «. sin. 4-) = (r„,. cos. *. COS. •vj/ — ly,^^. cos. -vj/. sin. cos, t. sin.^/. sin. (f -[-z,,,. sin 6. sin. 4"), •." by concinnating we obtain *=x„^(cos. t. sui. 4'. sin. ip+cos. (p. cos. ■^) -\-y„, (cos. i. sin. 4'. cos. ip. — cos. 4'- sin. ?)-J-z^,^. sin. i. sin. i^, which is the expression given in the text ; by a similar process we could derive values for y and z. 132 CELESTIAL MECHANICS, second, and third principal axes ; we shall have x^ = x^^. COS. (p — 1/ , . sin. ip ; From which it is easy to deduce X = «'/^/.(cos. 9. sin. vf/. sin. ip + cos. ^. cos. (p) + y^^,.{cos. 6. sin. ■^. cos. ip — cos. vj/. sin. ip) 5;^,/. (sin. 0. sin. 4^) ; «/ =07/^^. (cos. 6. cos. ■]>. sin. (p — sin. \|/. cos. (p) + ^//^.(cos. G. cos. 4/. cos. 9 + sin. i}/. sin. (f) +z^^^ (sin. 6. cos. ^'); z = z,^. COS. — j/j,^. sin. 6. cos.

}', is equal to the cosine of the angle which the axis of 2„,, makes with the axis of PART L—BOOK I. 133 ■Xw, in the preceding expressions ; we shall have, by adding them toge- ther, x^^^ = a:. (COS. 9. sin. 4/. sin. p+cos. v|/. cos. — sin. 6. sin. ■i|'. cos. 4-. cos. ^) — r. sin. I. sin. t'le coefficients of y^,, will be equal to cos. ^6. sin. (p. cos. $r-sin. ip. cos. ^-j-sin. ^$. sin. ij. cos. ip:=0, in like manner the coefficient of z^^ =sin. 6. cos. 6. sin. (p — sin. 6. cos. d. sin. tp^O ; the terms at the other side are those which have been given in the text. In like manner to obtain the value of y^,i, a corresponding multiplication gives x.(cos. (. sin. ■^. cos. ^ — cos. i^. sin. f) = x^,^.(cos.*^. sin. *4/,sin. ^.cos. ?i-j-cos. *. sin. 4/. cos. 4. cos. 'f — cos. 6. sin, 4. cos. -v^. sin. *f— cos, 'if- sin. (p. COS. (p) •]-yf,J[cos, "e. sin. ^T^. COS. '(p-J-cos. ^•4, sin. -?i— 2. cos. 6. sin. 4. cos. ^. sin. f. cos. ^\ + z,„(sin. ). cos. e. sin. ^il'- cos.

-{-sin. ^6. sin. (p. cos. 9)=0, +j/„^(cos. 2^. cos. *?i+sin. ^^-f sin. ^i. cos. *(p)=r^„,, + r„,. (sin. 9. cos. *. cos.

)=0, +3„,.(sin. s^.+cos. -«)=3„^. * When we substitute for the expression Xndym — yuM^u,' the respective values of jr„,, 'i^iii'y,,!' ^Viii' "^ functions of *, dx,y, dy, and of the angles 6,-^, and (p, it is not ne- cessary to take into account any expression, in which the variable part is the product ot a coordinate into its own differential, because this expression occurs again, affected witl% a sign, the opposite to that, with which it was affected before. By performing the pre- scribed multiplication of the value of a:,,, into the value of c/y,,, of y,,^ into dx„, we obtain x„_-dy ii,-=xdy.[cos. -6. sin. -ij/. cos »|/. sin. . cos. (p — sin. 6. cos. ^f/. sin. "ip), — 3.rfy.(sin. ^. COS. 6. COS. ij'. sin. ifi. cos. ?> + sin. «. sin. ■4'. sin. '. cos. ?)-j-sin. 6. cos. •J', cos. -ip), — y.dz.{siu. 6. cos. S. cos. ■^. sin. ^. cos. (p — sin. 6. sin. t^. cos. "if), y^^,.rfr,„.rr«.rf_y.(cos. 2«. sin. il/- cos. il/. sin. (p. cos. ip — cos. 6. sin. s-J.. cos. *ip — cos, t. cos. yvf^. sin. -.p+sin. i^, cos. i^. sin. (p. cos. ip), -}-y.rfx.(cos, 2«. sin. ij/, cos. -J/, sin. (p. cos. ~; ,„ — , which gives c COS. 9 = / „ ., .,, we shall have * 3c,.dii . — y„,'dx , , . 2WJ. dt ' ■ = \/c*^c'*-^c"* + •/ subducting x,iidyi,, from y„,.t/x,„, and making the terms whose variable parts are the same coalese, we obtain ^m-dyn, — .y/„.rfa;,„ = (x.rfy — y.dx). cos. «-{- (xdz — zdx), sin. #. COS. i//-|-(^.t?2 — ^z.dly), sin. 6. sin. •vf/; and substituting for x.dy — y.dx, s.dz — z.dx, &c. their values d ,c" ,, we obtain c. cos. i d. sin. «. cos. -^/^-^fd'. sin. ^. sin. 4'; =x„,.f^y,„ — ViiA^iiii ^y ^ similar analysis we arrive at the expressions for Xm-dz^,! — z^^-dx^^,, ym-dz^i, — ~iii-^y,i,> which are given in the text. c"« J. c" ' * For sin. ' 6. sin.* i^ -f sin. * 6. cos. * i|/= sin. - « = vTv a j.'X'a' **' ''°^' ' ^'~^ — *'"' * * c^+d^+d" f For substituting in place of cos. 6^, sin. 6. cos. xj',. sin. S. sin. •vJ/„ these values, we shall have rf< := V c*4-c'--|-c'S and if we substitute for c, c', c", their values, ■\/c^-J-c'^-f c*", cos. «, — 'V^c*+c'*-j-c"*, sin. «. cos. 4". +v^c2+'c*-f-"c», sin. «. sin. ■!•, the expression "•"'• ill ~" will become v6'-\-c^-{-c', (sin. 6. cos.S. cos.i?, — sm.^.sm.if. ISS CELESTIAL MECHANICS, ^^^^ x„,dz„r-z,Ar,„ ^ ; ^.ILid^H^j:^, = ; dt dt ,'. the values of c' and c" vanish with respect to the plane of x^,, and ^„,, determined in this manner. There exists only one plane, which pos- sesses this property, for supposing that it is the plane of x and i/, we shall have Z;«. "'-^-^--^-•^"- = c. Sin. 9. COS. , ; Im. y'-d^''^-'-^^'" = dt _ dt — c. sin. 9. sin. ip ; If these two functions are put equal to nothing, we shall have sin. 9=0, which shews that the plane of a-,,, and y,,,, then coincides with V dii 1! • dx the plane of x and y. Since the value of "Zm. ' '"' " fji "' — ~ is equal to \/c*+c* + c"*, whatever may be the plane of x and y, it follows that the quantity c* + c'*+c"« is the same, whatever this plane may be, and that the plane of x„ and y,„, determined by the preceding , , , p . X dy — y .dx analysis is that, with respect to which the function s/h. '" '^"' " — - is a maximum ; therefore, this plane * possesses these remarkable pro- cos. ■\. sin. (J) — sin. i. cos. ^. cos. --i^. cos. --^ >. COS. 6, dt j Since c, c', c", are constant quantities, and proportional to the cosines of the angles which the plane on which the projection of the area is a maximum, makes with the coordinatfe planes, it follows, that the position of this plane is always fixed and invariahle ; and as the quantities c, c', c", depend on the coordinates of the bodies at any instant, and on the velocities dx -J—, &c. parallel to the coordinates, when these quantities are given, we can determine the position of this invariable plane ; we have termed this plane invariable, because it depends on the quantities c, c', c'', which are constant during the motion of the system, provided that the bodies composing it are only subjected to tliis mutual action, and to the action of forces directed towards a fixed point. (See page, 128.) Since the plane ^, ^ is indetermined in the text, we conclude, that the sum of the squares of the projections of any area, existing in the invariable plane, on any three coordinate planes passing through the same point in space is constant ; consequently,, if we take on the axes to any coordinate planes y, z; x,z; x, y, lines proportional to c, c', cu, then the diagonal of a parallepiped, whose sides are proportional to those lines, will represent the quantity and direction of the greatest moment, and this direction is the same whatever three coordinate planes be assumed, but the position in absolute space is undetermined, for the projections on all parallel planes are evidently the same. The conclusions to wliich we have arrived, respecting the projections of areas on coordinate planes, are in like manner applicable to the projections of moments, since as has been observed in Note, page 28, these moments are geometrically exhibited by triangles of which the bases represent the projected force, the altitudes being equal to perpendiculars let fall from the point to which the moments are referred, on the direction of the bases. When the forces applied to the different points of the system have an unique resultant, V; then smce the sum of the moments of any forces pr 2.m + 2.jn. — 1— i--^;~I— i_ &c=c+2?. If all the bodies were concentrated in their common centre of gravity, X/, i/j ; dx^, dyj ; would vanish, therefore the second part of the first members of the preceding equation would .^ J ,,, X.dY—Y.dX dX'- + dY' +dZ^- vanish, and we would have 2.»». = c, -j-^ '2m~c-^-2r:y±\ _ A.^m.dy+B.Zmdx ^^^^ ^ dt \ dt § dt ' v J t-™ J jxr^ jxr „ AY.mdy-\-Bzm.dx , A.dY4-B.dX „ 2m. dy, Zm. dx=d Y 2m, dX. 2m ; — -^ becomes — -^ 2.)?j. 144 CELESTIAL MECHANICS, posed, 1st of quantities which would obtain, if all the bodies of the system were concentrated in the centre of gravity ; 2dly, of quantities relative to the centre of gravity supposed immoveable ; and since the first described quantities are constant, we may perceive the reason why the principles in question have place with respect to the centre of gravity. Therefore if we place the origin of the coordinates at this point, the equation Z, of the preceding number will always subsist ; I'rom which it follows that the plane which constantly passes through x.dy — ij,dx this centre, and with respcet to which the function S.ot. < - dt .'. the projection of the area on the plane x, y, with respect to the new origin becomes ,^ . B. dX—A.dY, , . . equal to c-j . Sw;, and similar expressions may be derived for the pro- jections on the planes x,z,y,z, From tliis it appears, that for aJl points in which B.dX—A.dY ."Lin = the value of c will remain constantly the same, but it is evident that this equation will be satisfied, if the locus of the origin of the coordinates be either the right line described by the centre of gravity, or any line parallel to this line, consequently for all such lines the position of the invariable plane will remain constantly parallel to itself; however, though for all points of the same parallel the position of the invarialile plane is constant, yet in the transit from one parallel toanother the direction of this plane changes. If the forces which act on the several points of the system are reducible to an unique resultant, by making the origin of the coordinates any point in this resultant, the quantities c,c',c", and therefore the plane with respect to which the projection of the areas is a maximum, will vanish, if the locus of the origin of the coordinates bo a line parallel to this resultant, the value of the projection of the area with respect to this line on the plane ar,_y, will be constant and equal to — '■ . -^m for c in this case vanishes, if the locus of the origin of the coordinates be a right line diverging from this resultant, the expression BdX—AdY -J. • £»2 IS susceptible of perpetual increase. From these observations it appears that when the forces admit an unique resultant, that point with respect to which the value of x/ c* + 1' - c" - is least of all is a point so circumstanced, that the axis or perpendicular to the plane of greatest projection passing through this point, is parallel to the direction of the unique resultant ; PART I.— BOOK I. H5 is a maximum, remains always parallel to itself, during the motion of the system, and that the same function relative to every other plane which is perpendicular to it, is equal to nothing. The principles of the conservation of areas, and of living forces, may be reduced to certain relations between the coordinates of the mutual distances of the bodies composing the system. In fact, the origin of the coordinates r, j/, z, being supposed always to be at the centre of gravity; the equations (Z) of the preceding number, may be made to assume the following form (■ (It \ C dt S' c".^.m = ^.mm'. S ^y-y)'d^''-dz)j^i^'-zUdy'-^dy) }^^^ It may be remarked, that the second members of these equations u * This expression is proved to be true with respect to three bodies in the following man- ner and as the same reasoning is applicable to any number of bodies whatever, it may be considered as a general proof ^ / , Cx'.dij — y'.dx'l ""• j-^V— } + "'"• {t^ 5-"""- I d/ } \ dt S ^ I dt 5/^ L tit i sad as in the case of three bodies „ ( x". dy"—i/'- da!' 1 ^ / . , . //^ ^ f =^dy—y.dx \ _|.m". J dt \ ■'■ '^' ^"'—'' (»»+»»+>») ~"^ { — ^^ \ + ^V ^'::f^t^}+m".^ ^^jMlr-^^J^-^ + m,n' {^^ ^ ^, ix".dy"-y".dx" \^ , „ { J>'.df-f.di!' \ + «"^' \—ht — ) + """ { — Jt — r By the nature of the centre ofgravity we have w«4- >"''»''+"'" *"=0 2nd also mdy A- m' dtf ^td'.dy" — /. their product vanishes z, e, m''x'Jy+m'^x'di/-\-m."-x".dy"-\-mm'x.dy +mm". x". (fy-j- mm'^di/ ■\-m'm".a!'dy'-i-m.m",xdi/'+m"m'.x'.dy"=0 .'. we have m » x3y +m''t^ulj/-\-m''.'-a!'.dy''s=—mm'.x'dy—mm''.x''.dy—mm'.xdy'—m'm"'.x''dy—mmf'j:dif''. — m'.m'^di/', and by multiplying my+m'i/+m"i/', into mdx-\-m'dj/-\-ni"dx"; — ot' ydx PART I.— BOOK I. 147 By applying to the preceding equations, the analysis of No. 21, it will appear, that the plane passing constantly through any of the bo- dies of the system, and with respect to which the function ( dt 5 is a maximum, remains always parallel to itself, during the motion of the system, and that this plane is parallel to the plane passing through _fn" i/.dx'—m"^7/'. dx" = + mm'.y'. dx + mm" i/'. dx + mm'y.d^ + m'm"y"dx! + mml'.ydx!' ■\-m".m'.y'Jx", .•. adding these quantities together we obtain L dt i l dt S ' X dt i .-. if in the expression for c{m+m'+ m") we substitute in place of the sum of the functions whicli are multiplied by the squares of the masses, the quantities wliich are equivalent to theni we shall obtain c {m + m' -f m")= - nun' \^:i!jii±^ I _ „„„" i f:^^/:^! _ „„,, r ..^^^^ 7 (. dt } I dt y \ dt s which is equal to to the expression which has been given above for the value of Im.rh': i JX'—x) d,/ —d, /)—,/— y (dx'—dx \ ■ I It ) 148 CELESTIAL MECHANICS, the centre of gravity, and relatively to which, the function s.ffj. ^ ' ■ — '- is a maximum. It vpill also appear that the se- cond members of the preceding equations vanish with respect to all planes passing through the same body, and perpendicular to the plane in question. t The equation (Q) of No. 19, can be made to assume the form* I dt •z.fmrd. Fdf; this equation respects solely the coordinates of the mu- * When there are but three bodies S.w.rfx ^=:»iix* + »2' is a maximum, makes with the three coordinate planes, it is evident that the values of the angles which the invariable plane makes with three coordinate planes, is the same in both cases, from these considerations it appears that the invariable plane may be determined at each instant by means of the relative velocities of the system, without a knowledge of their fliio/if/e velocities in space. (See Notes to page 139.) PART L— BOOK I. 149 tual distances of the bodies, in which the first member expresses tlie sum of the squares of the relative velocities of the system about each other, considering them two by two, and supposing at the same time that one of them is immoveable, each square being multiplied by th<> product of the two masses which are considered. 23. If we resume the equation {R) of No. 19, and differentiate it with respect to the characteristic etaphysical truth, and was applied by Maupertius to the discovery of the laws of reflection and refraction, however it ought not to be deemed ajinal cause, for we can infer analogous results from all relations mathematically possible between the force and the velocity, provided that we substitute in this principle, in place of the velocity, that function of the velocity by which the force is expressed, (see next chapter, page 154,) and so far from having been the origin of the laws of motion, it has not even contributed to their discovery, without which we should be still debating what was to be understood by the least action of nature. 152 CELESTIAL MECHANICS, CHAPTER VI. Of the laws of motion of a system of bodies, in all the relations mathematically possible between the force and the velocity. 24. It has been already remarked in No. 5, that there are an infinite number of ways of expressing the relation between force and velocity, which do not imply a contradiction. The simplest of all these relations is that of the force proportional to the velocity, which as we have ob- served, is the law of nature. It is from this law that we have derived, in the preceding chapter, the differential equations of the motion of a system of bodies ; but it is easy to apply the same analysis, to all relations mathematically possible, which may exist, between the force and the velocity, and thus to exhibit under a new point of view the general prin. ciples of motion. For this purpose, let F represent the force and v the velocity, we have F zz (y^ ; (p (y) being any function whatever of v ; let — > — f are multiplied by dt dt dt the function A_i, which in the case of the force proportional to the velocity, may be assumed equal to unity. However this difference renders the solution of the problems of mechanics very difficult. Notwithstand- ing, we can obtain from the equation (S), principles analogous to those of the conservation of living forces, of areas, and of the centre of gravity. By changing Iv into dx, Sy into dy, Sz into dz, &c., we shall have 2.J«. V. dv. dt. ) = const. + x j (T) which equation is analogous to the equation (i?) of No. 19, into which it is changed in the case of the law of nature, or of 9' (tr)zzl . Therefore, the principle of the conservation of living forces obtains in all laws mathe- matically possible between force and velocity, provided that we under- stand by the living force of a body, the product of its mass by double the integral of its velocity, multiplied by the differential of the function of the velocity which expresses the force. If in the equation (5), we make Sx'—Sx-^Sxl, Sy' = Sy-\-Syl, i^= Sz + iz', Sx" = Sx+^x",j &c. we shall have by putting the coefficients of Sx, iy, Sz, respectively equal to nothing becomes 2.«. {^. rf. { ^.

dtdv. f ' (v). PART I.— BOOK I. 155 These three equations are analogous to those of No. SO, from which we have inferred, the conservation of the motion of the centre of gravity, in the case of nature, when the system is not subjected to any forces but those of the mutual action and attraction of the bodies of the system. In this case l.m. P, ^.m. Q, "Z.m. R, vanish, and we have dx a(v) . „ dy a(v) const. = l.m. -—. -^-i ; const. = 2w«. -^. ^-^^-^ ; dt V dt V _ dz {Qz^Ry).dti c, c', c", being constant arbitrary quantities. If the system is only subjected to the mutual action of its component parts, we have, by No. 21, Im. [Py — Qx) = ; s»z. [Pz^-Rx) = O sw. CQz — Ry)=0; also m] x -^ — y. —i.'^^ is the moment of C dt dti V the finite force by which the body is actuated, resolved parallel to the plane of x and y^ which tends to make the system turn about the axis of z J therefore the finite integral s.w. J-^fc^^ LfM is equal to the sum of the moments of all the finite forces of the bodies of the system * Tlie integral of this expression is equal to 2.ot i x. -j- .^^ — /dx. (-^. --^^' \ dx s not constant. 15S CELESTIAL MECHANICS, being supposed fixed, the term which is not affected by the sign/ must disappear in this equation; therefore we shall have in consequence of the equation (S), i. z.fm.(f{v).ds = x.fm.Sv.(p'{v).ds—^.fmdt{FSx + Q.Sy + R.iz) but the equation (T) differentiated with respect to S gives ^.Jm.Sv.— P3»).cfm+S.(Q.«'— P/).cf»", .-. by substituting for the expressions S. P. «?»», S. Q.rfw, to which they are respectively eqtjal ds appears from the equations {A), and freeing the quantities d »«/, «?^«, «, y, from the sign S, the preceding equation will be changed into the following ». &.QL. dm—y. SP. dm + ^^. Sx'Jm + «. ■S-^-<^'«— ^- -V-'^"' -y.S.^.dm +S i^L^l-fj^X . rfm:^. S. Q.d*-3^. S.P.rf»« + S {Q.x*—Py*). dm, and omitting quantities which destroy each other, and also those which by the nature of the centre of gravity, vanish, we will obtain the equation this equation involves the principle of the conservation of areas, for if the forces which s<*cit the Tndlecuks arise from their mutual action, and from the action of forces directed towards fixed points, S[Qx' — Py.) dm=0. PART L—BOOK I. 163 this equation integrated with respect to the time, gives S. S ""'^y'—if-^^' \.dm= S.fiQx'—Py'). dt. dm ; the sign of integration /being relative to the time t. From what precedes it is easy to infer that if we make SJ{Q.x'—Pij'). dt. dm=N-y S.J\R!^—Pz'). ^t. dm= N'i S.f{R^—Q.z'). dt. am=N"i we shall obtain the three following equations I dt S S.l ^.dm=N;y. ^s) these three equations contain the principle of the conservation of areas j they are sufficient to determine * the motion of rotation of a body about its centre of gravity ; combined with the equations (A), they completely determine the motions of translation and rotation of a body. Y 2 * In our investigations relative to the invariable plane in the 5th chapter, we have seen that when a body or system of bodies are not solicited by any extraneous forces, the motion may be distinguished into two others, of which one is progressive and the same for all points, the other is rotatory about a point in the body or system, the first determined by the equation {A), and the second by the equation(£) ; by thus distinguisliing the motion into two others, we can represent with more clearness the motion of a solid body in space,for these two motions are entirely independent of each other, as is evident from the inspection of the equations which indicate them, so that the equations (A) may vanish, while the equations (B) have a finite 164 CELESTIAL MECHANICS, If the body Is constrained to turn about a fixed point ; it follows from No. 15, that the equations {B) are sufficient for this purpose ; but then it is necessary to fix the origin of the coordinates x', tf, z', at this point.* value or vice versa. The centre of the rotatory motion may be any point whatever, how- ever when we would wish to determine these two kind of motions it is advantageous to assume for this point, the centre of gravity of the body, because in most cases its motion may be determined directly, and independently of that of the other points of the body. Dividing the equations (^) by m, we may perceive by a comparison of the resulting expressions, with the equations of the motion of a material point, which have been given in No. 7, page 31, that the motion of the centre of gravity is the same, as if the entire mass of the body was concentrated in it, and the forces of all the points and in their respective directions were applied to it ; this rectilineal motion is common to all the points of the body, and the same as the motion of translation. * If a solid body is acted on by forces which act instantaneously, in general it acquires the two kinds of motions, of translation and of rotation ; which are re- spectively determined by the equations (^A) and {B) ; when the equations (/^ ) vanish, the forces are reducible to two parallel forces, equal, and acting in opposite directions, when the rotatory motion vanishes the instantaneous forces have an unique resultant passing through the centre of gravity, see notes to page 143, when the molecules of the body are solicited by accelerating forces, their action in general will alter the two motions which have been produced by initial impulse, however if the resultant of the accelerating forces passes through the centre of gravity of the body, the rotatory motion will not be affected by the action of these forces, this is the case of a sphere acted on by forces which vary as the distance, or in the inverse square of the distance from the molecules, see Ne^vtou prin.Vol. 1 . Section 1 2, or Book 2, No. 12, of this work, consequently if the planets were spherical bodies, the motive force arising from the mutual action of the sun and planets would pass through the centre of gravity, and the rotatory motion would not be affected, but the direction of this force does not always pass accurately through this centre, in consequence of the oblateness of theplanets, therefore the axis of rotation does not remain accurately parallel to itself, however the velocity of rotation is not sensibly affected, see Systeme du Monde, Chapter 14, Book 4, and Books, No. 7 and 8. It is in this slight oscillation of the axis of the earth arising prin- cipally from the attractions of the sun and moon, that the phenomena of the precession of the equinoxes and of the nutation of the earths axis consist. (See Nos. 28, 29. If the body be moved in consequence of initial impulses, the directions of the forces, their intensities and points of application been given, we might by the formula of No. 21, de- termine the principal moment of the forces with respect to the centre of gravity, and the direction of the plane to whicli this moment is referred, which would completely determine the moment of rotation obout the centre of gi'avity, and it is evident that the same data would be sufficient to determine the rectilinear motion of the centre of gravity, and consequently the motion of translation of the system, see No. 29. PART I— BOOK I. 165 26, Let us attentively consider these equations, the origin of the coordinates being supposed fixed at any point, the same or different from the centre of gravity. Let us refer the position of each molecule to three axes perpendicular to each other, fixed in the body, but moveable in space. Let fl be the inclination of the plane formed by the two first axes to the plane of x, y, ; let (p be the angle formed by the line of inter- section of these two planes and by the first axis ; finally, let \|/ be the complement of the angle which the projection of the third axis on the plane of x, y, makes with the axis of x. We will term these three axes principal axes, and we will denote the three coordinates of the molecule dm, referred to those axes by x', y", z", ; then by No. 21, the following equations will obtain x'=x". (cos. S. sin. ^. sin. ip+cos. ^. cos. 9)4- y". (cos. 6. sin. vj/. cos. q> — cos. ^. sin. (p) + z". sin. 6. sin. ;J/ ; y = x". (cos. 9. cos. »}/. sin. (p — sin. vj/. cos. f) + y". (cos. 6. cos. 4'- cos. ?> + sin. \J/. sin. (p)-\-z". sin. 6. cos. ■^ ; a'= 2". cos. fi — y". sin. 9. cos. (p—x". sin. 0. sin. (p. By means of these equations, we are enabled to develop the the first members of the equations {B) in functions of 9, 4-,

+ Br. sin. 6. cos. (p'—Cp. cos. fl = — N ; Cos. i)/. [Aq. cos. 0. sin. — Aq. cos. } — sin. ^.{Aq. COS. fi. sin.ip+jBr. cos.6. cos. (p+Cp. sin.S] = — iV^" , that the coordinates «", y , z", are constantly the same for the same molecule, and vary only in passing from one molecule ^o another, but the coordinates s! i/ ^ vary witli the time .•. they are fiinctions of the time, as are also the angles 6, ■]/, cos. 4'. cos. ip. cos. « — rf. cos. S —d^. sin. (p. cos. •4'. cos. 6-{-d-^. cos. 4'- sin. tp 4- rf(p. cos. 9. sin. 4) -^'.{dt.cos. 6. COS. ■4'— d-^' sin. 4- sin.«) «?/=: — !i'.dl.6\n.6—fy". di. COS. 6.jeos. :q>+y". dif>. sin. p. sin. 4 — y. cf«, cos. 6. sin. ip — a". dip..cas.Jl>. sin. « /. ^4/= (of', cos. S, sin. 4'. sin. ^ 4" ^^ oos..4'- cos. 9+;y '• cos, *. sin. 4'. cos. ip — .y. cos. 4*. sin. ip+z". sin. 6. sin. 4) X <— *"'rf<. sin. e, COS.4. sin, ^— «". rfif" s™- 'J'- sin. . cos. * "j-jt". *d^. sin, (p. cos. Ip. sin. \J/. COS. ■4'. COS. '6,— «".-rf. sin, ij/, cos. 4-. cos. '(. — y'.'^d'p. sin. -0 sin. 'tl'-cos. « 4-y .*rf<|'. sin. if" cos. 4'' sin. +Aq, sin. ip) — d. {Cp. cos. fi) = — cfiV^ ; )/. (£r. sin. 9 — Aq. cos. ?>) — (fO. sin. 6. {Br. cos. ip H- ^y. sin. (p)+cos 9. d. (Br cos.

) — d^. cos. 9. (£r cos. (p+ Aq. sin. 9) — Qj.rf" 4/. sin. = -- dN" making Cp=p'', Aq= j'j 5r=/; z .". observing the terms which coalesce and those whicli destroy each other in the expression for y^iy — y^*') tWs function_becomes equal to — x".* dS. sin. I. sin. 9. cos.^— «".'rf4'' *'""• '?• cos. ^i — x''.^d. sin. 't. sin. ^(p) — di. sin. i. sin. (p. cos. (p ") , (y + ^'Y I -^r— —5 + {x"^ + s"^). (d^f'. sin. »S. cos. "ip + tfO. sin, <. sin. ?>.cos. ^) — (^"* +«/''*) (<^ip. cos. 5 — dif'. cos. *«), _ equal by making all the quantities by which y,*z",' i",* are respectively multiplied coalesce so that they may be respectively factors of these coordinates 170 CELESTIAL MECHANICS, " these three diflPerential equations give the following ones * dp'+ \—-j—\q'r'.df=dN, COS. ^-—dN'.&m. 0; dr C—B CB A—C7 „, , dq'+ S i .r'p'.dtzz — (dN. sin. 9 + dN'. cos. 0). sin. ^ -\-dN". COS. (p ; h(i>) '+ ^ — j-rii 'P'q'-dt= — (dN. sin. 6 + dN'.cosJ). cos.^. — rfiV^". sin. (p. y.*rf^ (sin. ^«. sin. «(fi-|-tos. *^) — .y. -dd. sin. «. sin. ?. cos. (p — 7/". 'dip. cos. ♦ = i/''^.d4'. cos. '. COS. -«-}-^",»c?t|'. sin.'ip — ^"*. rf«. sin. *. sin. (p. cos. (p~-i/'*.dp. cos, # r''^.^^". sin.*^. sin.*(p — 3"*.c?«. sin. «. sin. ip. cos. p. sin. 6, (Br. sin. (p — ^y.cos.^) + cos.»«. d. ( Cp) — d6. sin. 6. cos. 6. {Cp)+sin. *6. d(Cp)+d6 sin. 6. cos. (.(Cp); = PART I.— BOOK I. 171 these three equations are very convenient for determining the motion of rotation of a body, when it turns very nearly about one of the principal axes, which is the case of the celestial bodies. 27. The three principal axes to which we have referred the angles z 2 by substituting for r and q their values — — B.{di.d-i^.'im.^. cosJ (p+clS.* sin. deserve particular consideration ; we now proceed to determine their position in any solid whatever. From the values oi of 1/ z', which have been given in the preceding number we may obtain the following expressions by No. 21. x"-:z.af (cos. 6, sin. »J/. sin.

}/. cos. ^)* + (cos.^9— sin.'9;. (g. sin. ;}/+/«. cos. \j/). * r''. COS. $i=x'. (cos. *. sin. il'- sin. this expression being extended to all the molecules of the body, will give by substituting for S-r/^dm Sj/.^dm,&c. their respective values a-,b^,/,g,h, &c,the expressionm the text, in like manner sin. (p. x" z" 174 CELESTIAL MECHANICS, by equalling the second members of these two equations to nothing, we shall obtain . /?. sin. x|/ — ";. COS. J/ (a^ — b'^y sin. \j/. cos. ^■\-J. (cos.'^ij/ — sin."* ■i/) J ^. sin. il/+/;. COS. »|/ 2" tan. 2? „ ., • o , , o u , ,^ ,. : — i x~ ' "^ r — a-. Sin. "4/ — 6 . cos.-.\}/ — 'ilj. sm. »J/. cos. 4' but we have always tan. 6 i tan. 26 = 1— tan.-°fl ' by equalling these two values of tan. 29, and substituting in the last ex- pression, in place of tan. S. its value, which has been given in a function of }'.cos.4'— s'«'-sm. = «. sm.>J' -f-a't/'. sin. 6. cos. 6. sin. 4'- cos. ^+1/'." sin. 6. cos. 6. cos. '4'~^!/ *'"■ "*• cos. i^ + ;V. cos. ^6. sin. -^-^z'y'. cos. *«. cos. ij/— z'.' sin. «. cos. 6 ziz sin. «. cos. 6. (a'.'* sin. °^+y.» cos. '-^ — z'^)+'2x'y'. sin. 6. cos. «. sin. ■4'. cos. 4') + (cos. ««— sin. ««.) («'*'. sin. T^.-f^Y- cos.^}..) by extending this expression to all the molecules and substituting a^,i*, c', /',y5, &cact. tor Sx'V»K and S/ ^ f/w &c. we shall obtain the expression which has been given in the text. * The second members are put equal to nothing because by the conditions of the problem, the first members respectively vanish, consequently we have = ( (flS— 6*). sin. ■4. cos. 4+/. (cos^^— sin. ^4)). sin. 6-\-(g. cos. 4— /(. sin. 4). cos. < ; 0*- sin. 6. cos. i. (a- sin. ^-\-b? cos. ^ — c2+2/ sin, ■^, cos. 4) + (cos. !'«— sin. H). (g. sin. 4+/'- cos. -f) ; PART I.— BOOK I. 175 As this equation has at least one real root we may perceive that it is always possible to make these two subsequent expressions, and con- sequently the sum of their squares, to vanish at the same time sin. 6 h sin. 4' — g- cos. ^ •■• ^^e" *^"" * ~ (««— 6-). sin. 4..cos.4'+/(cos.2^^— sin.«^J.)' sin. e sin. <■ cos. 9 _ COS.0 ^!^=: i tan. 2(» = COS. *(»— sin. «* — sin. ^6 l—UmM COS. -i g. sin. ■^.-\-h. COS. -^ (? — a.-sin.24/ — 6.2cos.-^J' — 2/ sin. ^. cos. i^ lhe«e fractions being divided cos. ij', become by substituting u in place of sin. •vj/ hu — g gu-{-h COS. 4'' ((a2_6»). u-{-f. (1— u-)). cos. 4- ' ( (c- (l+;r)— «- tfi—t^—'lfu).cos.-^ if we call the factors of cos.ij/ in the denominators of these respective fractions m and n we shall have tan. «.= "~^ . .-. i. tan. 2« = ?w. cos. y 1 C ^"-g 1 ' = m. cos.>V -hu-y ~ «• c°«- ^ •■• C_hu-g_l (_m. cos.^^J by reducing we obtain {hu—g). »m. cos. 'i^ = (gtt+h). ( .(m cos\|'f — (Au— f )') and consequently = cos. ^. (m. (hu—g). n— (gM+A).>»)+(A«— ^).* (^«+^,) now {hu—g)n.=>. by substituting for n, {c?Q.-\-u^) — a^u^—h^ — 2fu) and then multiplying h(~ u+h(? 1^—ha^ uS—hH u—2fhi^—g(?—gc' ui+gahi^+glr'+^/gu, in like manner 176 CELESTIAL MECHANICS, COS. (p. S. x"^'.dm — sin. ^. S. ^'.z".dm ; sin. f. (S.yy.fi^wj + cos. (p. S.y"z'.dm'^ and this requires that we should have S, x"z».d'm ; Sy"z".dm separately equal to nothing. The value of u gives that of the angle 4'j and consequently the value of tang. 6, and of the angle 6. It is only now required to determine the angle

+i/'. sin. ?>. cos. ip = x'. cos. J. sin. ^. sin. ?-j-y. cos. «. cos. ^. sin. f — z' sin. «. sin. ^ = Q. sin. ^ .-, jr". = x'.(cos. t. sin. ■\J'. sin.

^j Aj become when a;' coincides with x", and as tan. 2[S + 100)— tan. (2(1+200) = tan. 2«, it follows that the other two axes must be taken in the plane y, 2', one making the angle i and the other the angle fl -f 100 with the axis of y, now if we made the axes of y", and s", to coincide with the axes of y and z' respectively, 6, and .•. /;' would vanish, and consequently S{y'z'.)dm would be equal to nothing. But if h' remaining equal to nothing, b' and c' would be equal to each other then // tan.2« = — — p-j would be equal to — .•. « would be indeterminate and every line in the plane y' z', and passing through the origin of the coordinates would be a principal axis, see notes to page 184. 180 CELESTIAL MECHANICS, have been named principal axes of rotation, on account of a property which is peculiar to them and which will be noticed in the sequel. The sum of the products of each molecule of the body, into the square of its distance, from an axis, is called the momeiit of inertia of a body with respect to this axis. Thus the quantities A, B, C, are the moments of inertia of the solid, which we have considered, with respect to axis of -f", of 1/", and of ^'. Naming C the moment of inertia of the same solid with respect to the axis of z', by means of the values of y, y, and z*, which are given in the preceding number, we shall find C = A. sin.* 9. sin.« (p+B. sin.* 9. cos.* ip+C. cos.* 6. * The quantities sin.* 0. sin.* )dm +S.Z//.* sin.*tf. sin.2

+(B— C). sin.^«. cos.2(p, in this case also the two last terms of the second member are negative, . • . C is less than C ; the moment of inertia C is greater than the least of the three principal moments, for if A be the least of the three moments which refer to the principal axes, we have as before C'—A + {B—A). sin.s«. cos ■> + (€— A). cos.2«, and as the differences are on the present hypothesis affirmative, C is greater than A, let C be the least of the three moments, and we have C'=C+(/l-^C). sin.^O.sin.'?! + (B—A).sin.\ cos.V, the terms which compose the second members are always affirmative, . ■ . we conclude that C is greater than the least of the three moments, A, B, C, From what has been established in the preceding note, it appears that when the three principal moments of inertia are unequal there is only one system of principal axes, for let there be another system and make A', D', C, the moments of inertia relative to these axes, then we shall have at the same time A "^ A' and A' "^ A which is impossible, see note to page 178. t For S{J—XY.dm^Sx'.^dm—2X.Sx'.dm-\.X^m—S3;.^dm—21!^-\-X^mM Sx'.dm = X.m. and as the quantity— w. (X^+ Y^) is essentially negative, the moment of inertia witli respect to the centre of gravity must be less than the corresponding moment for any axis not passing through the centre of gravity. If the moments are referred to an axis passing through a point different from the centre of gravity and of which the coordinates are a, b, c. 182 CELESTIAL MECHANICS, Let X, Y, Z, be the coordinates of the centre of gravity of the solid, relatively to the origin of the coordinates which we fix at the point about which the body is subjected to revolve, if it is not free ; x'-^X, y' — Y, z'—'Z, will be coordinates of the molecule of the body, with respect to the centre of gravity ; therefore the moment of inertia, relative to an axis passing through the centre of gravity, and parallel to the axis of zf will be s.^(x'-x)*+0'—Yy } dm: but from the nature of the centre of gravity, we have S. x'.dm=mX, S.y'.dm=mYi .'. the preceding expression will be reduced to Consequently we shall have the moments of inertia of the solid, with respect to an axis passing through any point whatever ; when these moments are known for axes passing through the centre of gravity. At the same time it appears that the minimum minimorum of the moments of inertia appertains to one of the three principal axes, passing through this centre. Let us suppose the nature of the body to be such, that the two moments of inertia A and B are equal, then we shall have C'=^. sin. *e+C.cos.'6: * the value of the moment of inertia with respect to this point is equal to It is evident from an inspection of their values, that the greatest moment of inertia with respect to any point, is less than the sum of the other two moments. * When A=B the moment of inertia with respect to any other axis = A . sin.^f + C. cos.^l, and as neither 4- or (p occur in this ejcpression, the moment of inertia for all axes making the same angle, with the axis of z are equal, and if « be a right angle C-:z.A, therefore in this case there is an indefinite number of principal axes, but they have all a common axis z\ when «=100* we have a^=-h^ and/= i, e, Sx'MmzzSi/.-dm and Sxy.dm=0 this also PART I— BOOK I. 183 and by making S equal to a right angle, wluch will render the axis of ^ per- pendicular to the axis of z", we shall have C=A ; therefore the moments of inertia relative to all axes situated in the plane perpendicular to the axis of z'' are then equal to each other. But it is easy to be assured that we have in this case for the system of the axis of z^', and of any two axes perpendicular to each other, and to this axis, S. afy'.dm= ; S. afz«,dm= ; S.i/'2^'.dm= ; for if we denote by x" and y the coordinates of a molecule /'. sin. t. ; consequently we have S. x'y'.dm = S. x"y".dm (cos.»£ — sin.' i) 4. S. (j/"«— .or"^). dm. sin. i. cos. 1 =0 we shall find in like manner .S". .rV.rf?w = 0; S.T/sf.dm-^O'f therefore all axes perpendicular to the axis of z", are in this case principal axes ; and in this case the solid has an infinite number of similar axes. follows from the equations x':=x". cos. e + ^° sin. e, y =r y". cos. i — x ' sin. s for S. x'Mm ■=. S.{x" ? cosS-\- Sy". - sm..^i). =S.x"?dwz:Sy'. -dm, since Sx"y".dm-=zO, in. the case of an ^ipsoid generated by the revolution of an ellipse above its minor axis, we have always two of the principal moments of inertia equal, the moment which is the greatest is referred to the minor axis. 184. CELESTIAL MECHANICS, If we have at the same time A = B=C; we shall have generally Cz=.A ; * that is to say, all the moments of inertia of the solid are equal, but then we have generally, S.x'y'. dm=0; S.x^.dm=0 ; S.i/'z.dni= ; whatever may be the position of the plane of x' and of i/' ; so that all the axes are principal axes. This is the case of the sphere, and we shall see in the sequel that this property belongs to an infinite number of other solids of which the equation will be given. t * Since by hypothesis ^=i?=C, Sxff.-din=Sy"Mm=SzV.2dm, .-. if in the expression for 2'2 in terms of :r//,°y',- zV and of the angle 6,4'>~rf«. sin. ^$. cos. ip)+^". £/«. sin. l. cos. I. rfi:'. cos. 6= — x".(d6. cos. ^6. sin. ?> -(- rfip. sin. $. cos. «. cos. ip) — i/'.{di. COS. ^^. cos. ip — rfip. sin. 6. cos. «. sin. ip) — i". c?«. sin. «. cos. 9 /. adding and concinnating we have — x".{d6. sin. (p + d^: sin. 1 cos. (p)—y".{dL cos. <>— ff-v}/. sin. «. sin. ?)= — x'V-f-y'. y. * The equations p/'—y2"—0—&c. are the equations of the projections of the line, relatively to which dx' dy' are equal to nothing at any instant, on the planes x" z" , y" je", &c. .*. the cosines of the angles which this line makes with the axes areVespectively For these cosines are equal to 9^' P I 1 I T ~ P P' and the same is true of the other cosines. From the preceding analysis it follows, that the locus of all the points whose velocity is nothing at any given moment is a right line, whose position with respect to the principal axes is determined by p, q, r, :. the preceding equations both evince the existence of such a line and indicate its position, and a body revolving about a fixed point may be considered as revolving about an axis determined in this manner, but as in general;;, q, r, vary from one instant to another, being functions of the time, the position of this axis will also vary, and hence it is that this axis has been termed by some authors the axis of instantaneous rotation ; whenp, q, r, are constant, the axis of rotation will remain immoveable during the motion of the system. PART I.— BOOK I. 189 Therefore this right line quiesces, and constitutes the real axis of rotation of the body. * * The values which have been given for px" — jz", pi/" — rz", qy" — rj", enables us to determine the linear velocity of each point resolved parallel to the axes of x' y and d for if we multiply the first of the preceding equations by cos. i. cos. (f. the second by cos. i. sin. f. and the third by sm. i. we shall obtain by adding them together — dJ . cos. i. sin, ip. cos. ip ■\-dy'. cos. '^i. cos. ' — rfz'. sin. i. cos. ^. sin. 2(p + ay. sin. ** 4-rfi'. sin. i. cos. «=:(// — (px" — ^z'). cos. «. cos.

. cos. ip + rfs'. sin. 6. sin.(p. cos.ip..^«'' cos. *^ + rfy. cos. «. sin. (Ji. COS. (p-—dz. sin. d. sin. ip. cos. ip = dx' =—{px"— gz"). sin. ip+( pi/' — rz" ) . cos. ? ; multiplying px" — jaf ' by — sin. «. cos. 9, pi/' — rz" by — sin.fl.sin.ip. and qy" — rx" by cos.^, we shall obtain dx'. sin. 6. sin. '. When A=.C the expression for dt becomes AB.Cdp' ^{A- k-—H- +A.{B—A)iP). {H''—BAIr) this expression may be reduced to the form C,- — 1 — ^^ ( in which C, is equal to - \ ' ^1F+^^ ' ^ A^B-A).{m-BA.k ) Ak^—H2 and a-=- A.{B—A) the integi-al = C,. log (/+ ^/ a2+;.« If2?=Cthenc/^. ^ ^'^P' _c.—±^PL '/(ACJ.^—H'^j(W—B-k'—Bi(A—B)).p^ ' s^ a^—p'* and the integral will be arc sine =p' rad = a ^, ^ . , AB' C, Dem£; equal to " / • ' ^ ^ <^AC/c^—hK(H^—B''k^) and a^ = iJCIf- H% jlP- m^ —B.(A—B) if /lC.4* = fl2theo PART I.— BOOK I. 195 this equation is only integrable in one of the three following cases, B=A, B=C, A=C. The determination of the three quantities p', q, r', involves three arbitraiy quantities, H.\ Ji^ and that which the integration of the pre- ceding differential equation introduces. But these quantities only give the position of the instantaneous axis of rotation of the body, on the surface, /", e, with respect to the three principal axes, and its angular velocity of rotation. In order to have the i-eal motion of the body, about the fixed point, we must also know the position of the principal axes in space ; * this should introduce three new arbitrary quantities c c 2 ,o A.{B—C).p'2 ,^ ABC dp' ^ t{A—B) ^ A(,B—C)if\H'i—BClr—B.[A-C)iy- ^adv = c,:t - — in which 2C - AEC V (/i{B—C) H'— BC. /c^^ ~ —B,(A—C) its integral will be equal to C,, log. ^ v " —p a + sja^ — i)" See Lacroix, page 256, No. 174.. and if m=BC.lc^ then dt^ -'^^•^/'' \/ ACIfi—H^+A{B~C)pfi)[—B{A—C)p'^) ^^ and t= C. W. '^'^W^a =C,. and t~ C,. log. the constant quantities vanish for these integrals, because as has been already mentioned p'=0 when t vanishes. The value of dt cannot be exhibited in a finite foim except in the cases already specified, and when all the moments of inertia are equal, in every other case, the value of the integral of dt must be obtained by the method of quadratunes. * From the quantities/, g', r', we can collect the values of p, q, r, which are in a given ratio to them, and from these last quantities we obtain the cosines of the angles whicli the axis of instantaneous rotation makes with the principal axes, but as these axes though fixed in the body are moveable in space, we must know the position of these axes at the com- 196 CELESTIAL MECHANICS, which depend on the initial position of these axes, and which i-eqviire three new integrals, which being joined to the preceding quantities will niencement of the motion, in order to have the real motion of the body, which gives three constant quantities. Substituting in the values o{—N^ — N', — N", p' for Cp, q for Aq, r' for Br we shall have q', sin. 6. sin, (p + r. sin. «. cos.

— ^'. sin. ^. sin. 4- -}-r . sin. (p. cos. ■v}'^?'- cos. . cos. '4/+2/>V'. sin. S. cos. ^. cos. -^-2p'r' sin. S. cos. «. sin. 'if'' cos. (p -^2p'q' sin. <. COS. S. sin. ^■^. sin. if) •j-/.* sin. "ip. COS. '4'+o'. ' COS. ^$. cos. '4' — 2yV. sin. (p. cos. ip cos. *4=-^"" /. adding the first members of these equations together we obtain q'.- sin.'*, sin. •tp-^-q','^ cos. ^6. sin. ^ip+y'.^ cos. '?>=(y'^ )-{-/.» sin, '^. cos. *? 4- r'.^ COS. '^. cos. 2(p + /,» sin. '(?= (}•'*) +j3'.' cos. '6-\-p'.^ sin. *«. cos. '4^ -{■p'.' sin. «^, sin, ^4'=/'';* the parts of these squares which are the products of two different quantities vanish when added together and in the expressions for A^',* A'*",* we omit the product {q'. cos. 0. sin. p cos.4' + »•' COS. (. cos, ip, cos. 4' •{•p'- sin. i. cos. 4')'('''' sin. (p. sin. 4'— ?• cos, ^. sin. 4^) for this pro, d,uct occurs in A^" and Nff* affected with contrary signs, .•, it must vanish from A^'2+ A"'* .*. we shall have p'*+q'*+r'^ Zi + N'+N'^+N"^. * PART L— BOOK I. 197 completely solve the problem. The equations ( C) of No. 26, involve three arbitrary quantities N, N', N", ; but they are not entirely distinct from the arbitrary quantities H and k. In fact, if we add together Ihe squares of the first members of the equations (C), we shall have ;/' + f 4- r'"- =N"-+N" + N" ' ; and consequently The constant quantities N, N', N", correspond to the constant quan- tities c, c, c", of No. 21, and the function t. t. \/p'^+q'~-\-r- expresses the sum of the areas described in the time t, by the projection of each molecule of the body on the plane relatively to which this sum is a maximum. iV', N", vanish with respect to this plane, .•. if we put their values, which have been found in No. 26, equal to nothing we shall have = Br. sin. (p — Aq. cos. f ; 0—Aq. cos. 9. sin. (p + Br. cos, 9, cos. (p+Cp. sin. 9 ;* ♦ From the equation Bi: sin. ip—Aq. cos. ^ rr we obtain by substitution tan. (p = ~ .-. COS. //H!7'^+r' sin, 6. sin. 9= ■ .-„^ . •:= > Vp'+^'+r"'' sin. V. COS. = By means of these equations, we can determine the values of S and

in a function of p' ; thus the three angles e, ip, and ip will be determined in functions of the variables^', q', r', which will be themselves determined in functions of the time ^.t Consequently we can have at any instant the values of these angles with respect to the plane of x', and y, which we have considered, and it will be easy by means of the formulae of spherical trigonometry, to » If we multiply the values of qdt, rdt, given in page 166, respectively by sin. (. sin. f, sin. i. COS. , — sin. 6. cos. p, are the cosines of the angles which a perpendi- cular to the fixed plane or the axis of z' makes the principal axes, see page 180, and P' 9' -^ are the cosines of the angles which the principal axes, a''-', i/", x", make with the axis of the plane, on v/hich the projection of the area is a maximum, consequently the cosine of the angle wliicli the axis of the plane on which the projection of the area is a maximum makes 200 CELESTIAL MECHANICS, . * determine the values of the same angles with respect to any other plane this will introduce two new arbitrary quantities, which combined with the four preceding quantities will constitute the six arbitrary quantities, which ought to give the complete solution of the problem which we have discussed. But it is evident that the consideration of the above men- tioned plane simplifies considerably this problem. The position of the three principal axes on the surface, being supposed to be known ; if at any instant, the position of the real axis of rotation on this surface, is given and also the angular velocity of rotation, we with the axis of the fixed plane, (see note to page 7). ■p' . cos, 6 — /. sin. 6. sin. — ?■'. sin. 6. cos.ip N, we might by a similar process shew that the cosine of the angle which the axis of the plane of greatest projection makes with i/', and x', are respectively proportional to N' and A'''', consequently the position of this plane with respect to the fixed axes oi x' y', and z is given, therefore this plane remains fixed during the motion, and the values of N, N', N'', are the three quantities which determine the position of the fixed axes, with respect to the plane of greatest projection. , * The determination of f^,q'y, which give the position of the instantaneous axis of rotation requires three arbitrary quantities and (he determination of 6, ■^, (p, which give the position of the principal axes with respect to the fixed axes requires three more arbitrary quantities, these are, H, k, and the constant quantities which are introduced by the integration of dt and d-^, the two remaining quantities are determined by the values of cos, 6, sin S. sin. ^-\-f^r'" ; consequently we have If at the commencement of the motion we know the position of the principal axes with respect to the invariable plane, i, e, * the angles 6 and (p ; we shall have at this commencement the values of p' q and r and consequently those of /?, q, r, therefore at ani/ instant we shall have the values of the same quantities, t CQUStitvite an equilibrium with the force mv consequently the principal plane i, e, the plane with respect to which the moment is a maximum is the plane passing through tlie centre of gravity, and the direction of the primitive impulsion .•. the sum of the areas described in the timef =\.t.mfv. * The constant quantity k ^ m.fv ; in order to determine H, it may be remarked that the position of the principal axes at the commencement of the motion, with respect to the pkme passing through the fixed point and the direction of the impulse being given, we have the the values of f q, r, being proportional to the cosines of the angles which the principal axes make with the axis to the invariable plane. Consequently we have the constant quantity the third constant quantity will be determined by integrating the value of dt, which will be equal to a function of p'-\- a constant arbitrary quantity ; // which is proportional to the cosine of the angle which the axis of 3" makes with the axis to the plane of greatest moment lias a detemiined value when <=;0 . • . by means of this value we are enabled to find the value of the third constant quantity ; with respect to the fourth constant quantity which arises from the integration of the value of d-^, this gives il' = to a function of p plus a constant quantity, p' being proportional to cos. (, we shall obtain the fonrth constant quantity which is necessary to complete the solution of the problem, if we know what value of 4' cor- responds to a given value of i. f \Mien a solid" body is not eolicited by any accelerating forces and can revolve freely about a point we shall have dx.zs.yi'S'^^i^ydyzi.id-^—Kdin, dxp=,sd!pr-yd^, &c. PART I.— BOOK I. 203 By means of ttis theory, we are enabled to explain the double motion of rotation and of revolution, of the planets, by one initial impulse. In fact, let us suppose that a planet is an homogenous sphere whose radius is D D 2 See page 89, if we multiply the equations (^Z) of No. 21, by rfw d^ d-^ ^' df' W respectively we shall obtain d-a , d

denote the density of one of those stratas of which the radius is equal to R, we shall have ^_ 2ot fp.RMR , 3 Jp.R.-dR ' * The moment of inertia for a sphere is calculated in Book V. No. II. in a general manner, but as it involves some steps which are demonstrated in the second and third books, it will be necessary to give here a special demonstration, let there be two concentrical circles, whose radii are q, q-\-dq, the circumference of the interior is equal to '2-jr.q, and the area of the annulus contained between the peripheries of those circles is equal 2v,q.dq .'. 2%.q?dq is equal to the moment of inertia of this annulus and l^.q,* is the moment of inertia of a concentrical annulus of a finite breadth, .*. when the preceding integral is taken between the limits y=0, q = R the expression becomes ^ttR*, which is the moment of inertia for the entire circle, now in order to obtain the moment of inertia for the entire sphere, let us conceive a plane parallel to the axis of rotation cutting the sphere at a distance from the axis equal to x, its intersection with the surface of the sphere will a lesser circle of the sphere, let ^=: the radius of this circle, the moment of inertia of this circle with respect to its centre is equal by what precedes to ^ vy* ,•. the moment of inertia of an indefinitely small slice is equal to ^■!r.y*.dxz:i Itt (2Rx-~x*)- .dx, for i/^zz2Rx—a:^ R being the radius of the sphere, .•. btegrating we have -^^>-Wro} — the moment of inertia of a spherical segment and this integral being taken between the limits_a;=0, and x=R gives — the moment of inertia of the entire sphere with respect to a diameter, and it is very easy by means of the expression which has been given in page 180, to obtain the moment of inertia for any axis parallel to the diameter, if R is supposed to be variable in the last expression, and if § the density varies from the centre to the circumference, the moment of inertia of any spherical stratum whose radius== R is ^.t^RUR 10 206 CELESTIAL MECHANICS. (p being a function of R). If, as is very probable, the denser strata are nearer to the centre j the function /' p'g ,„ ■ will be less than , consequently the value off jf' Jx, uK 5 will be less than in the case of homogeneity. 30. Let us now determine the oscillations of a body when it turns very nearly, about the third principal axis. We might deduce them from the integrals which we obtained in the preceding number ; but it is .-. the moment of inertia of a sphwe composed of concentrical strata is equal to g- Ty g./Z* dR, m like manner m=the mass of the sphere =i4tir./^R*dh m am d f~ ^'P 8w pf^R*dR _ Jp.f^.R*.dR_ Vs-R*'^^"" 7n.TU ^3Atn.Tlir^.E'dR'-' 3rUfi,R^.dR we obtain the ratio of U to p from knowing the period of the earth and the time of its rotation, for the angular velocities are mversely as the angles described in the same time, 5 being by hypothesis a function of R where the density increases towards the centre e^ — =- .•. the fraction in the text becomes R\dR [R) Rs R3.d')• "^—^ — we obtain by adding and subtracting I,, \ .("*+y)V^ — («<4-y).\/Zr cos. {ni-\-y) = c 4- c ^ '^ ' i sin. {nt -1-y) = c 2 («<+y). v/ZIl — («<+7)V3r if n is imaginary the preceding exponential expressions will become — nl^y s/ 1 «* — yVIZx — "'-fy\/ZI7 nt — yV~\ c Arc c _c '2 2v'zr in these exponential expressions, the part which is not affected with the radical sign, is PART I.— BOOK I. 213 axes of which the moments of inertia are the greatest, and the least j but not so about the other principal axis.* proportional to the time, and therefore the values of q and r, will increase indefinitely vvitli the lime, .•. though they may have been indefinitely small at the commencement of the motion, still as there is no limit to the increase of the exponential expressions, they will at length exceed any assigned magnitude. * It might be shewn di;-ectly by means of the equations Crp^+A^'-^-B-r-^k"; /iBC-.p^-\-A^BCq^'\-AB^Cr-=fP, that there is a limit to the increase of q and r when C is the greatest or least of the three quantities A,B,C, for if we multiply the first equation by AB, and then subduct it from the second we obtain A-.B(C — A)q'-i-AB~.{C — B).r^= Hi — AB.k^, if at any instant the quantities q, r, are very small If^ — A/fi which is constant will be very small, consequently in all the changes wliich r and q undergo they are sub- jected to the same condition, and this condition requires that r and q siiould be always very small when C — A and C — B are of the same sign, because then both the terms of the first member of the preceding differential equation will be either positive or negative, and the expressions m—AB.k^ m—ABlfi A-.B.(C—A) ' AB\C—B) ' are the limits to which the respective values of q and r can never attain. If C — B and. C — A are of different signs, then the terms of the first member of the equation will be of different signs, audit is only the difference of the quantities AiB(^C — A).q^-\-ABi, (C—By^, that is indefinitely small /. since tliis difference depends on the relative values of these quantities, q and r may be very great, though the preceding residual is' a quantity indefinitely small. Pliilosophers have distinguished the equilibrium of stability into two species absolute and relative, in the first case the stability obtains whatever may be the oscillations of the system, ;n the second case it is necessary that the oscillations should be of a certain description, in order to insure the stability of the equilibrium. If a body revolving about afixed axis passes through several positions of equihbrium, these will be alternately stable and instable. For if a system deviates from a position of stable equilibrium, from the nature of this equUibriura it tends to revert, but according as the system deviates more and more from its first position, this tendency will diminish, and at length it will tend to deviate from the original position, but previous to tliis change of tendency there must have been a position in which the systenc neither tended to revert, or to deviate from its original position, consequently this is a position of equilibrium, but this equilibrium is evidently one of instability, for previous to the arrival of the system at this position it tended to revert to its primary position, and when it passed this position, it tends to deviate from the primary and consequently from this second position of 214 CELESTIAL MECHANICS, Now, in order to determine the position of the principal axes in space, we shall suppose the third principal axis to coincide very nearly with the plane of x' and of y', so that 9 will be a very small quantity of which we may neglect the square. By No. 26, we shall have d(p—'d^ =pdt * and by integrating we obtain ^ = (p — -pt — i E being a constant arbitrary quantity. If we afterwards make sin. 0. sin. ?' = s j sin. 6. cos.

-:r.q.dt ; t?!?. sin. «. cos. ip— p. sin. fl cos.ip. dt ■\-d6. sin. ip = r.rff, ; substituting 6 in place of sin. 6, to which it is very nearly equal since the higher powers of * may be neglected, we obtain — d ii=sm. 6. cos. ip = j^ — , i, e, Cp C.p q . r' Sin. 6. sin. ® = r, sm. 6. cos. ?> = j, V P and those are values of the cosines of the angles which the principal axes of id' and if" make \vith the axis of the invariable plane, see notes to page 198. In this case s -^M , , . AM , ^ , , — =tan.

,/• 2 L . ^ ^-^J" , -O-M* / A.(C—A) \ . ,^ „«+s^-s.n. -«. (sm. ^?+cos. ^):^~^^+ -^j- {'b:^c-B) ) = ''"• '' by substituting for (sin. -?-|-cos. - By substituting these values in the equation (2^), we may put the coeffi- cients of Sa, Sb, Sc, separately equal to nothing ; which will give three equations of partial differences between the three coordinates of the molecule x,y, z, its primitive coordinates a,b,c, and the time t. It remains to satisfy the condition of the continuity of the fluid.* For this purpose, let us consider at the commencement of the motion, a rectan- gular fluid parallelepiped, of which the three dimensions are da,db,dc. If we denote its primitive density by (p), its mass will be equal to {p).da.db.dc. Let this parallelepiped be represented by {A), it is easy to see, that after the time t,i it will be changed into an oblique angled parallelepiped ; for all the molecules which in the primitive situation existed on any face of the * In order to determine the condition of a fluid mass at each instant, we must know the direction of the motion of a molecule, its velocity, the pressure p, and the density g, but if we know the three partial velocities parallel to the coordinates, we shall have the entire ve- locity, and also the direction, for the partial velocities divided by the entire velocity, are pro- portional to the cosines of the angles which the coordinates make with the direction, see Note page 26, and page 227. Three of the equations which are required for the determination of those sought quan- tities, are furnished by the equation (F) ; another equation from the continuity of the fluid, for though each indefinitely small portion of the fluid changes its form, and if it is com- pressible, its volume during the motion, still the mass must be constant, consequently the pro- duct of the volume into the density must be the same as at the commencement,. • . by equating those two values of the mass, we obtain the equation relative to the continuity of the fluid. f After the time t, the coordinates of the summit of the parallelogram, whicji were a, b, c, at the commencement of the motion, will be j:, y, z, ory(a 6c<), ^{a i c <), ip (a 6c/), the coordinates of that point of which the initial coordinates were a, b, c-^-dc, will be y PART I.— BOOK I. 225 parallelepiped {A) will still be in the same plane, at least if we neglect quantities indefinitely small of the second order ; all the molecules si- tuated on the parallel edges of (A) will be found on small right lines, equal and parallel to each other. Denoting this new parallelepiped by (B), and conceiving that through the extremities of the slice constituted, of those molecules which in the parallelepiped (A) compose the side dc, we draw two planes parallel to the plane of x and i/. Then producing the edges of the second parallelepiped to meet these two planes, we shall have a new parallelepiped (C) contained between G G f(a, b, c+dc, t), F(a, b, c+dc, t), (p (a,b, c-(-dc, t)z= respectively to . the difference between these coordinates and x, y, s, are and the square root of the sum of the squares of these three quantitities, is the value of the side of the parallelepiped which answers to the side dc of the primitive parallelepiped; extracting the square root, and neglecting the third, and higher powers of dc, this side becomes equal to dz drz , Jc-'^^+a— •^'^' in like manner it may be shewn that the quantities which in the original parallelepiped are equal to da, di, become the opposite sides of the figure are equal to these ; for the value of x, y, z, which corresponds to the primitive coordinates a-^-da, b, c, are/(a+da, J + c t)F{a-\-Aa + b,ct,) ?i(a+da, b c f)= x+J.da+^.da^^+|^.da+g-,.da^ .-j- ^.da+^.da^ da ^%da^ ^ ' da ^2.da^ ' ^ da 2,dai 226 CELESTIAL MECHANICS, those planes, and equal to (B) ; for it is manifest that what one of these planes takes from the parallelepiped (B), is added by the other plane. The two bases of the parallelepiped (C) will be parallel to the plane j', t/, : its altitude contained between its bases will be equal to the difference of ~, taken on the hypothesis that c * only varies ; consequently this altitude will be equal to | — V dc. the values of x, y, z, which answers to the primitive coordinates a-\-da, b, c-j-dc, will be y"(«-j-da, b, c+dc, t) F(a.+da, b, c+dc, <) ifi(a-j-da, b, c+dc i)r: da 2. da- ' dc ^ <2..d(? ' ''da ^ ^da- 'dc idc^ dz , d-z ^ g , dz , d*z . , '~'^do^'+2d^-^^'^d-a-^^+2i;^- ^^' .'. the difference of the coordinates of these points = -7-.dc+ — -. dc^ -f. dc+ -^_. Ac-, -. d<;+— =-:-. dc?, dc IMc- .dc ^ Idc^ dc ^2c?c* and as these differences are equal to the corresponding differences of the opposite side of the figure, it follows that these sides must be equal, being equal to the square root of the sum of the squares of these differences, in like manner it may be proved, that the other sides are respectively equal to those to whichjthey are opposed ; and the parallelism of theee sides is a necessary consequence of their equality, fiom which we infer that the figui-e wliich the molecules assume is a parallelepiped. The equation of the line connecting the points whose respective coordinates are f{a,b,c,t), F{abcl), (p(a bct),f(a-\-da,b ct), F(_a+da, b, c t,), (p)a-^-da,b ct), will be that of a right line, if we neglect the indefinitely small quantities of the second order, and the same is true for all lines parallel to this line, of the sum of which the face may be conceived to made up, .•. this face may be considered as a plane. * The difference between the values of z corresponding to the expressions .=,ia,b,c,t),-J=^,.^abc+dct)^'^. dc+ ^£}.g=5|}.dc PART I— BOOK I. 227 We shall obtain its base, by remarking that it is equal to a section of (B) made by a plane parallel to the plane of a:, y, ; let us designate this section by ({)• The value of z will be the same for all the molecules of which this base is constituted, therefore we shall have °= &3-^- tiM'^rX- ''■ Let Sp, Sq, be two contiguous sides of the section (e), of which the first is made up of molecules which existed on the face Ab. dc. of the paral- lelepiped {A), and of which the second is composed of molecules which existed on the face da. dc. If we conceive two lines to be drawn through the extremities of the side Sp, parallel to the axis of .r, by pro- ducing them to meet that side of the parallelogram (f), which is parallel to Sp, they will intercept a new parallelogram (x) equal to (t), of which the base will be parallel to the axis of x. The side Sp being composed of molecules which existed on the face d6. dc, and relatively to which the value of ~ is constant ; it is easy to perceive that the altitude of the parallelogram (x) is the difference of y, on the supposition that a, z, and t are constant, consequently we have ((Iz ) db+ 5 7— f . dc; (dc i G G 2 by neglecting quantities indefinitely small of the second order. For all the molecules situated on the edge, which corresponds to dc in the original parallelepiped, projected on the axis of z, the values a and 6 remain the same, nor do any molecules which occur in the face daM enter in the constitution of this perpendicular, therefore it is equal to dz on the hypothesis that c only varies. * If we conceive the molecules of the face db.dc relatively to which dz is constant, to be projected on the axis of y, it is evident that the projected Lne is equal to the difference 228 CELESTIAL MECHANICS, from which may be obtained a this is the expression for the altitude of the parallelogram (x). Its base is equal to a section of this parallelogram by a plane parallel to the axis of X ; this section is composed of those molecules of the parallelepiped [A), with respect to which z andj/ are constant ; its length will be equal to the differential of x taken on the hypothesis that z, y, and t are con- stant, which gives the three following equations "'-XTi^'^YiV^^tW- Ida C Idb^ idcy -{i}-ii}-^^+{^l-- of y, on the hypothesis that a is constant, for this projection is the same for every series of molecules, which exist on the face which corresponds to the primitive face di.dc, and rela- tively to which z is the same. We obtain the expression which is given in the text for d^ by eleniinating dc between the two preceding equations. * Since the parallelogram (a) exists in the plane parallel to the axes of «, y, the value of z will be constant for this parallelogram, and since the base of (a) is a line parallel to the axis of a the value of y ■ndll be the same for all molecules situated in this base, but since in this base molecules occur which belong to the faces da.dh, da.dc, db.dc, a,b,c, will vary for these molecules. PART L— BOOK I. 229 In order to abridge, let us make ~ Ida ^- Idb^'^dc \ lda\'ldc\'ldb^ idxl (di/l (dz^l , ^ Idb^'ldc^'lda^ • Multiplying the second equation by j j- f > and the third by -! ;r f . and then subtracting we shall eliminate dc ■■■ |(i-)(i)-(|)L^)}-«+ |(l)(S)-(|)(|)}-a*.=o " ' , , ^ , , .da in like manner we can obtain {{7:}-{l}-{l}-{^i}'-+{{l}-{|}-{|}-{f}}-=o .•.dc= \ di>s'\d^s~ \Ey\da s ^^ dx \ da' ^dz\ Sdj\_UyX idz^y-Xdb\ ^ XdcS'XdbS \dcS'\dbs 230 CELESTIAL MECHANICS, ldb\'lda\'ldc\+ ldc\-lda\'ldb\ Ucylm'idcS we shall have Q.da dxzz IdbS Idc) Idcf'l dz_ db this is the value of the base of the parallelogram (a) j therefore the isur- face of this parallelogram will be equal to ^Aa.Ab \dc) This quantity also expresses the surface of the parallelogram {i), if we idz' multiply it by4(—/dc we shall have ^AaAbAc for the volume of the \db J • ^f/a5 \dly If/a 5 ( dx~\ Sdz-x Sd_y \_ Sdy\ frfc-i ' \ del \dc\'\db j \dci '\dh] , da= {l^}{S}{i}-{^:}-{l}.{i}+{|}.{f}.^,t] • XdcS'XdbS Xuci'Xdh) Q. da \dcj\db\ idcfidbi = the base of the parallelogram (a), this expression being multiplied into the value of di/ gives the^area of (a), and this area being multiplied by the altitude gives the volume of (Q PART I.— BOOK I. 2Si parallelepipeds (C), and (5). Let p represent the density of the paralle- piped (A), after the time /; we shall have its mass equal to p Q.da.db.dc j and by equating this to its primitive mass {f).da.db.dc we shall have pe = (p); (G) for the equation relative to the continuity of the fluid. 33. The equations (i^) and (G) may be made to assume another form, which is in certain circumstances of more convenient application. Let u, V, and V be the velocities of a molecule of the fluid, parallel to the axes of X, of y, and of z j we shall have {ll = -{'f|=-l.T} = ^- By differentiating these equations, u, v, , V being considered as functions of the coordinates x, ?/, z, of the molecule, and of the time t, we shall have c?'.r>_ (du\ V, are respectively unknown functions of x,y, z, and /, they depend on the coordinates X, y, z, because for a given value of t, the velocity is different in different molecules, they depend on t, because for the same values of x,y, z, the velocity varies every instant, •••-=m-'M<^}'^+{|}''-+ {£}•*■ and since dx=udt, dy = v.dt dz='\dt, substituting and dividing by dt, we obtain , . dx da d^x but u = — .: — = — . dt dt dt^ n .1, 1 J-*'" dv dW . . , trom the values of^,^ ,— , given m the text, it appears how the increment of each of the three velocities depends on the two other velocities. F we were able to determine the 232 CELESTIAL MECHANICS, consequently the equation (F) of the preceding number will become. In order to have the equation relative to the continuity of the fluid ; let us conceive that in the value of S, of the preceding number, a, b, c, were equal to t, i/, z, and that j;, y, z, were equal to x + udt, y+vdt, z+V.dt, which is equivalent to assuming the primitive coordinates a, b, c, indefinitely near to cc, i/, z, j we shall have value of !4 in a function of x, y, z,t, we could by means of the equations ~jj'— "' ^ = '"' dz_ It' position of this molecule, and also what function of x i/zt, uv\ are, for substituting in the dx dy dz _ ^V determine the position of a molecule at any instant, provided we know the initial dt iition I equations — = ti, — = v, -j =V the values of t« v, V, in functions oixyxt, and integrating, ^ dt dt dt we would obtain the values of «, y, x, respectively in a function of i, the constant arbitrary quantities which are introduced are the values of «, y, z, at the commencement of the motion which by hypothesis are given, consequently the values o( x y z will be completely deter- mined for any instant. Eliminating t between values of x, y, z, to which we have arrived, we would obtain the two equations of the curve described by the molecule, but since the initial position of each molecule is different, the form of this curve will also be difterent, as will be in like manner, the position. PART I.— BOOK I. 233 :dV-) HH • The fii-st coordinates being assumed indefinitely near to x,y, ~, we shall have da — ds, and the quantity which corresponds to rfa:=to (/«+*«.*, in like manner we shall have dx,+du.dt dx+diuft di/-\-dtrdt di/+dv.dt dz+dV.dt dz+dV.dt "~d^ ' dy ' dz ' dz ' dy ' dx respectively indefinitely small, because when t—O these quantities vanish, /. the product of any two of these quantities may be neglected, making these substitutions the expression for C becomes equal to (dx + duJl\ f di/ + dv.dt \ (dz+dYdt} \ dx ]•{' dy / ■ t "~d2 ) _ ( dx+du.dt \ fd ^-hdv.dt \ (dz + dy^t\ \ dx M" rf-' ri dT i , cdxj-dtudt-i (dy +dv.dt y cdz+dV^l "^ t dy r\ dz ]l dx i f dx-\-du.dt ■) f dy+dv.dt ■> r dz+dV.dt 1 1 dy rl dx i'l dz y ( dx-\-dn.dl \ ( di/+dv.d t \ cdzJ-dVJl) "^1 dz r\ dx i'\ dy i i dx-\-du.dl -i ^ dy+dv.dt \ f dz-^dY.dt \ I dz \ \ dy ]\ dx \ the first term of this expression = by neglecting quantities indefinitely small \dx dy dzl the other terms of this expression vanish. It appears from what precedes that €j is a con- stant quantity independent of the time, when the fluid is incompressible S=l. 234 CELESTIAL MECHANICS, the equation (G) becomes, If we consider p as a function of x, y, z, and t, we shall have therefore the preceding equation will become * The density {, the pressure^, may be shewn to be functions ol xy z, t, by reasoning, analogous to that, by wliich u, v, V, were proved to be functions of these quantities ; is the increment of g on the supposition that t is constant, "'• {1} -«•*■ {1} -* {|} -^- {S} is the variation of § on the hypothesis that x, y, z, t, vary .*. their difference is the differential of the equation (fi) taken with respect to the time ; PART I.— BOOK I. 3SS this is the equation relative to the continuity of the fluid, and it is easy to perceive that it is the differential of the equation (G) of the pre- ceding number, taken with respect to the time t. The equation (H) is susceptible of integration in a very extensive case that is, when uSa: + vJj/ + YJz is an exact variation of *, t/, z, p being any function whatever of the pressure p. Therefore if we re- H H 2 when the fluid is incompressible, we have for in this case both the magnitude, and density are constant, .•.*+ £-^-*+ E-^-'-- 1-^'=£- -"'^ t-""'^Z -'•■•" «« dz -^ ' dz dz dz c/z ^ ' dx- now substituting udl, vdt, Ydt, in place of dx, dy, dz, and remarking that, j^— j-, v=z —, &c. and also that 3.-? = _1^ we shall have dt dt := the sum of the last members of the preceding equations, but these by concinnating, and dividing by dt are evidently equal to the second member of the equation ( H). Since the integration is only made relative to the characteristic S, it is evident that the time is not in- volved in this expression. When the fluid is homogenous —&c.=0.*. the equation of con- tinuity is reduced to the second term, by means of this equation, and the equations uzz. -j-, PART I.— BOOK I. 237 It is necessary to add to this integral, a constant quantity, which is a function of t ; but we may suppose that this function is contained in the function (p. This last function gives the velocity of the molecules of the fluid parallel to the axes of .r, of y, and of z ; for we have The equation (AT) relative to the continuity of the fluid, becomes consequently, we shall have in the case of homogenous fluids, It may observed, that if the function u^s + vS^ + YJz is an exact va- riationof a:, t/, z, at any one instant, it will always remain so. In fact, let us suppose that at any instant whatever, it is equal to Sep, in the sub- sequent instant it will be equal to '^^^'■p\-^^i\'>'^m>^] . * D=— ?, V= — , and the value for y"-^, r: in this case — , we can determine (p and p and dy dz 5 5 consequently u, w, V, in functions oi xy z. • From the value of V — f. — it appears that the pressure of a molecule, of which the e density is constant, diminishes when the velocity which is equal to 238 CELESTIAL MECHANICS, therefore it will be an exact variation at this new instant, if r}''-i:f}-^+{.?}- :clu ^m'^Yi\'+\t] is increased. substituting this value of S. J ^ < b the expression for JV— £ we obtain and since each of the terms, of the second member of this equation, are exact variations of m, y, z, the first member will also be an exact variation, we suppose g to be a function of p. is the differential of S0, on the supposition that the time only varies. Consequently, we are not obliged to determine ip in j:,y, z, in order to know whether it is an exact differential or not. .•. It appears tliat if ii^x.\-v1y-{-V .^z be an exact variation, at the subsequent instant tts increment will bean exact variation, .-. S?i + this increment will be an exact variation. As in general we know the condition of the fluid at the commencement of the motion, if at this moment t(Sx+ uJj/^-V.Si is an exact variation, it will be an exact variation when *^ ± df, t~ ± 2dt, &c. and in general whatever may the value of t. ?s of the small oscillations of the waves of the sea, is yet a desideratum in Physics. Philo- sophers have been much more successful in investigating the oscillations of the pulses of the air, and in the determination of the velocity of the propagation of sound. Tlie integration of which is the equation relative to the continuity of the fluid, when wJx+v.Sy+VS* is an exact variation, and when the fluid is homogenous, which is consequently the simplest possible form, is extremely difficult, however it has been completed effected by Antonie Parseval, PART I.— BOOK I. 241 these two equations contain the entire theory, of the very small un- dulations of homogeneous fluids.* 1 1 • If the fluid which makes small oscillations be water, by making the axis of z vertical, fl>z=g.3r,g representing the force of gravity, Pdx, Q3y are= respectively to nothing, in like manner we may cortteive it to be homogeneous and incompressible, consequently we shall have /i=£....,._fe=,.{*)=,.,^'A.,^ f = O- at the surface p vanishes, •'•-="• 1 j7 I ' consequently when the form of ? is deter- mined, we can derive the equation of the part of the fluid in which p=0, i, e, the equation of the surface of the fluid. We determine ^Mm =»■ ^"•^— ■'^■'- ^--{^} con this equation is of great celebrity in the history of the integral calculus, it was first in- tegrated by D'Alembert, in an analysis of tlie pfoblem of the vibrating chord, which leads to an equation of precisely the same form. * The linear velodtyis equ&l to the angular velocity mnkiplied into tlie distance, .-.at a distance represented by unity, the linear velocity =n, and since the angular velocity at all distances from the axis is the same, at a distance=v' ~z^~+p' the linear velocity = n. V 22+^1, the direction of the motion being perpendicular to the radius in order to obtain the velocity parallel to the coordinates r,y, we should multiply n. \/z»+^» into the cbsiiies of the ailgfes Vhich z and^'make with the "tangent, but these cosifles are respectively y — z /. „ ; . ~„ ' — „ ' for the motion being circular, if one of the cordinates be increased, the other will be diminished .•. v=z nz, Y=ny. t The tenns torr^pwdfng to -{ t- f > j -7- 1 , \ -r- \ .in the equation (i!/) vanish, because the time does not enter into the values of u, v, V.in like manner a and its differential ooefficients vanish, and from the values of v, V, given above, it is manifest that PAItT I.— :^00,K I. 1^3 wiiich equation is possible, because its two members are exact yariations. The equation (^K) pf tjie same nu^iber will become and it is manifest that this equation will be satisfied, if the fluid mass ,be homogeneous. The equations of the motion of fluids will therefore be satisfied, and consequently, the motion is possible. The centrifugal force at the distance 4/3/* +z* from the axis of ro- tation, is equal to the square Ti'.(_^+i/^) of the velocity, divided by this distance; therefore the function n^.(i/Si/ + z.Sz)i is the product of the II 2 /— ), ( — -},are equal respectively to nothing, consequently the only terms wliich have a finite value are V. \-f-)> '"•V j~)' which are respectively equal to zz-r-n^y, — «' «> •••the equation (H) will become ^^ SF+n^(v5^+^Sz), this equation determines the pressure e when ^ is- constant, «r. when it is a function of p, * The equation (K) is resolvable into two parts as before, (I) +«-(|)+- (|)+-■(l)+^{(|)+(|)+(£^)^ the velocity being uniform, its increment resolved parallel to the axes of x, y, z, i, e .du \ ( ^'v\ /dV . {di)'^~d^)'{oiry must be severally equal to nothing, this is evident for v, V, from their values which have been given above, with respect to the velocity u, it must be produced by the part of the velocity which is parallel to x, and if it was not uniform, the fluid would not have a uniform motion of rotation about the axis of x, zdz-{-yh/ t The centrifugal force = ?j\\/ 2-+ v^ the variation of the distance = , •■• "'•(z^^ + ySy) is = to th^ centrifugal force mi^ltiplied into the element of the distance. S44 CELESTIAL MECHANICS, centrifugal force, by the element of its direction ; thus, if we compare the preceding equation of the motion of a fluid, with the general equation of the equilibrium of fluids, which has been given in No. 1 7, we may perceive that the conditions of the motion are reduced, to those of the equilibrium of the fluid mass, solicitedby the same forces, and by the centrifugal force which arises from the motion of rotation ; which is sufficiently evident from the nature of the case. If the exterior surface of the fluid mass be free, we shall have Sp—Q, at this surface, and consequently = SV-^n^.{ySy-\-zSz) ; * Substituting for SFwe obtain -!—=^P.^x-\-QJy\-R.'^z-\-m.yly\-rfiz.tz,t]\e quantity added i, e, the centrifugalforcemultiplied into the element of distance, being an exact variation, itfollows that the expression for —will in this case be an exactvariation, n is some function of the distance of the molecules from the axis of rotation, as the tme is not involved in the preceding equation, it follows that the conditions of the motion of a fluid mass, about an axis, with a given velocity, are the same as the conditions of equilibrium of a fluid mass, the same forces as before soliciting the molecules, combined with the centrifugal force, arising from the uniform revolution about the axis. The molecules of the fluid, though they have a motion about an axis, are relatively at rest. * At the exterior free surface Sp=0, .*. 3 F+n -(?/Jy + 2J2)=0, .-. in order that the form of the fluid, may remain the same, during the entire motion, n must be constant. If die fluid was water contained in a vessel open at its upper surface, j is constant, and 3 V=g.2x the axis of rotation being supposed vertical, .•. Q.Sy, iJSz vanish, and P=g, consequently, we shall haveZ.^ — gx-j-n^.i " "^^ j+/«and at the free surface, wehave.T=w".^ ~^-^ j -| for the equation of this surface ; if m*. \/ z^ + if which expresses the centrifugal force varied at the 2r — 1 power of the of the distance from the axis of rotation i, e, as 2r— 1 2. r— I (2*+/);" ' =a ». (a »+i^ -), and/«^0^^+zSi) r ' \ ^r I' ^ "ir.g ) g PART I.— BOOK I. 24J from which it follows that the resultant of all the forces which actuate each molecule, must be perpendicular to this surface, moreover it must be directed towards the interior of the fluid mass. If these conditions be satisfied, an homogeneous fluid mass will be in equilibrio, whatever may be the figure of the solid, which it covers. The case which we have discussed, is one of those in which the variation uSx + vSy-{-YSz * is not exact ; for then this variation becomes .•. if r is positive, x is least, when (2*4-2^) =0, when r =1 all the molecules revolve in the same 2^-4- 2/ \ li time, and *= a * . i ^ J -i — which is the equation of the concave surface of the parabo- loid, of which the parameter= — — , the periodic time being equal to the force divided by the distance = — . .•. if the time of revolution, be called T, we shall have the parameter of a the generating curve rsto —T'Sc— = — ~ Zj/ - +A — ps .*. x being the same, the pressure is gi'eater at a greater distance from the axis of rotation. When r is negative, at the point where i*+y2 =0, x is infinite, and when= — h the surface of the fluid will be such, as would be generated by the revolution of aconical hyperbola, about its asymptote, the axis of x is in tliis case the as)Tnptote. The constant quantity h denotes the distance of the origin of the coordinates from the other asymptote, .*. both in this case and where the surface of the fluid is paraboloidal, the constant quantity depends on the quantity of water in the vessel. If the vessel was cylindrical, we could determine the area of the paraboloid, provided that we knew the area of the base of the cylinder, and also the points of greatest elevation and depression, for the paraboloid is half the circumscribing cylinder. This paraboloidal figure is that which is assumed by the molecules of the fluid, in the ex- periment which Newton adduces, in order to shew that the effects by which absolute and relative motions are distinguished from each other, are the forces of receding fi-om the axis of circular motion. See Princip. Math, page 10. • wJx-t-uJy+ V.J2 is not an exact variation in the preceding investigation, for substituting for V, and V, we obtain t,=_«;,V=ny, . •. wJx-f- v.ly+ V.S2=n.(!/Jz— z.Sy), consequently it appears, that though the circumstance of the preceding expression being an exact variation, would facilitate very much, our investigations, still it is not essentially necessary, that this should be the case, in order that the motion should be possible. :■ Since in the case of the sea, revolving round with the earth round its axis, and relatively quicscing with respect to the 246 CELESTIAL MECHANICS, ^^n{zh/—ySz] ; therefore in the theory of the flux and reflux of the sea, tve are not permitted to assume, that the variation concerned is exact ; since it is not so in the very simple case, in which the sea has no other motion, but that of rotation, which is common to it, and the earth. 35. Let us now determine the oscillations of a fluid mass which covers a spheroid revolving about the axis of t; and let us suppose that it is deranged from the position of equilibrium, by the action of very small forces. At the commencement of the motion, let r represent the distance of a molecule of the fluid, * from the centre of gravity of the' spheroid over which it is spread, and which we shall suppose immoveable ; let 6 be the angle which the radius r makes with the axis of a:, and zr the angle which the plane passing through the axis of x and the radius r, constitutes with the plane of x and of j/. Let us suppose that after the time t, the radius ?• is changed into r + a,s, that the angle fi is changed into 9 + aw, and finally, that the angle t3- is changed into 7it+-By + a.v; a.s, aw, and af, being very small quantities, of which the squares and products may be neglected, we shall have x = (r-\-cis). cos. (6 + Ml) } ^ = Cr4-«s). sin. (9+«m). cos. (n?+ in- + aw); 2;:=(r+a5). sin. (S + aw). sin. {nt-i-zr^-aV). eartb, u'ix-\-vii/-\-y.h is not an exact variation, we may conclude a.Jbrtiori, that it is not one, where the oscillations arise from the attractions of the sun and moan, which produce tlie flux and reflux of the sea. In order to ascertain whether an incompressible fluid solicited by accelerating forces, 'and also by a centrifugal force, may be at the surface of a given figure of revohciion, wA substitute in the equation 0=^V+n'^{i/di/+z.'iz) the forces parallel to x, y, z, which would result from this hypothesis, the resulting expression should be the differential equation of the given surface, if it is not, then we may be certain that the given curve does not satisfy the equilibrium of the fluid. See Book 3. Chap III. No. 1'8. * If a perpendicular is let fall from the extremity of r on the axis of a-, it will be equal to ''r; sin. (, and the projection of this perpendicular on the plane ofij,x, is equal to the coordinate y and its value will be r. sin. «. cos. -ar, and this perpendicular projected on the plane s x will be the coordinate •«, and it will be equal to r. sin. i. sin; «t. PART I BOOK I. 247 Substituting these values in the equation (^F) of No. 32, we sh^li obtain, the square of « being neglected, * * Since xu, »^, ««, are very small quantities, of which the squares and products may be neglected, the time t will of the same order as «, so that at is of the order a. *, consequently sin. tcu := »u — &c.= xu, COS. «a=l — - — — =1 .•. x = {r.-{-»s). qas. («+««) ^r. COS. 6. COS. au — r. sin. i, sin. au-^-»s. cos. ^.cos, t^u — tts. ^in. I. sin, »u = by neglecting quantities of the order <«*, r, cos. 6 — r. sin. 6, ccu+*s. cos. 0, r and t are independent of t, dx du . , , ds . d^x d^u . , . d*s .•.-— = — y .ar. sm. tf-f- -T- «. cos. J ; — — = — -r— xr. sm. t-\- rrr^. tt. cos. t, dt dt ' dt df dti ^ dt^ (d'x \ ■d*u id^s ^^) = ^. ,J.j/'ri-«5).sin. (9 + «mU +(^7— A rfs . . . . . (is 4- II.-T-. sm. 4. COS. «r— «$ s. sin. 4. sin, « —»nt. sm. ^. sm, «, -r- at at d*y ■ . ■ '^^'" o -A ^'^ I . d^" — i= — ar. sm. 6t sm. w.-rr- ■— Z«r. at. sm. t, cos. *n. COS. ^ sm. w.-r- + . cos. «r. -^ — 2«n. sm. t, sm. v-^j — 3y=3r. sin. «. cos. iir-|-9^, r. cos. i. cos. «t — Jw. r. sin. <. sin. -a, rejecting those quantities in the value of iy, where ■• cos. (. sin. to+Sit. r. sin. 6. cos. w, neglecting those terms which mvolve », (at as was before mentioned, in the product 350 CELESTIAL MECHANICS, (e. sm. #• cos. <• sm. *ot.— +«r'. cos. 6. sm.^u.-^ rfa . » rf^i . . ds\ +2«Kr2.cos.2*. sin.w.cos.w.-T+aj-. sin.«. cos. 6. sin.'^cj.r-j-f 2«nr. sm.«.cos.«.sm. ro. cos.w.-^^ j- f o d% o o dv a . . -2. sin.«. cos. «. cos.^n-.^-far. sin. ^(. sin. ts-. cos. sr. -*+2«rtr.sin.-<. cos.^w. ^| , d^x , . d^y , ^ d^z d^u ,dh . ,. . .„ d-v — r«. sin. 6. cos. «. -72+«- cos. \ ^— **■• ^'^•°^' sin. ^r. cos. w.^ dv d'U . . du —2nr». sinsi. cos.^w. — + ctr. sin. 6. cos. «. cos. ^a. j-^ — 2ixrn.sm. «.cos. 6. sm. tr. cos.w. ^ +«. sin. *tf. cos. *ar. -Tj — 2««. sm. *«. sm. w. cos. w. ^ rf^t) rfu , . ■ J, d'u + ra. sin. »^. sin. v. cos. cr. jj 2nrit. sin. i'^. sin.'ar.^ +«r. sm. 6. cos. «. sin. '"--^fr +2«nr.5ia.<.cos.«,8Jn.«r.cos.zj-.^'+«. eio, *«. sin. *sr. ^+2«k. sin.'«. sin. w. cos-s-. -^ J^ PARTI.— BOOK I. 251 the force of gravity, and my the elevation of a molecule of water at its surface, above the surface of equilibrium, which surface we shall con- sider as the true level of the sea. The variation (JF) in the state of motion, will in consequence of this elevation, be increased by the quan- kk2 (= by concinnating a'ir. ( Yt — 2n»-. sin- ^^- ^) ) ;+3^- -j »^«- "n. *«.-^— r«. sin. «. cos. «. ^ d^v „ . . „ dv — <»r.* sin. «. COS. (. sin. w. cos. sr. -r- — 2nr*{c. sin. l, cos. 6. cos. V.-r— 2«r*n COS. *i. sin.o. cos. a-. — +«r.Bm.*,cos.*. cos.**— w dt dt ' rft* ds . . . d^v — 2icnr. sm. ^.cos. t, sm.«r.cos.i7.-7--|-r'« sin. i, cos. i, sin. tt. cos. 17. -rs- — ^2«r*« sin. *. cos. «. sin. ^ct. r+«'"'' cos.'^. sin. "w. —. +2«n7*.cos.*^.6in.a-.c09.w.— p at at* at . a ^'s . ^ . . ds \ -j-«r. sin. (. cos. 4. $m. V. -^ — l-2«»r*sin. tf. xos.0. S1B.0. co8.n-. -r r ' dt* dt i (and by concinnating we obtain the coe£5cient of li = to (....|||| equilibrium; it is evidently a function of «andjr. y being the eccentricity, it is evident that the differential of the normal according to wliich the gravity acts, in case of equilibrium, differs from the differential of the radius, by a quantity which r: the product of the eccentricity into the differential of N, a function of 0. .-. at the surfoce of the fluid in equilibrio, {^V)—g. S, (r'+q. N), at the surface of the fluid in motion, the normal corresponding to r' -f «^, has not the same direction as when in equihbrio, its variation=S. (r'-^/jN+ai/ +«7. qN) ; the attraction of the spheroid in motion differs from the attraction of the spheroid in equilibrio by quantities of the order «^ •.• let it be equal to ai/g', then {g+^yg')- i(r'-\-qN+ui/-\-cci/ q. ]S')—(g + uyg). S[j-+jiV) + g. Uy, rejecting quantities of the order « S and remarking that ^r is of the order q.h, the first term of the second member of the preceding equationz=(S f^ . •. the second term is the quantity by which in the slate of motion (S V) is increased, as has been stated in the text. 254 CELESTIAL MECHANICS, the variations Sy, and SV, being taken relatively to the two variables 6, and w. Let us now, consider, the equation relative to the continuity of the fluid. For this purpose, let us conceive at the origin of the motion, a rectangular parallelepiped, of which the altitude is dr, the breadth r. dw. sin. fi. and the length 7-.d9. * Let r', 8', ir', represent what r, 6, ■a-, become after the time /. By following the reasoning of No. 32, we shall find that after this interval, the volume of the molecule of the fluid, is equal to a rectangular paral- (dr't lelepiped, of which the height is -j — r- .dr ; of which the breadth is dr being eliminated, by means of the equation Finally, its length is ' • r Bin. » = radius of a smaU circle, whose plane is parallel to the equator, and as the plane of the axes of x, and y, is fixed, r, sin. 6. dw= the differential of the arc of thi« circle, to wliich dr is evidently perpendicular, also, the differential of the nieridian:=r.<^«, is perpendicular both to r. «aa. i. d«r and to dr, .•. these three differentials, constitute the parallelepiped mentioned in the text. • When the fluid is in motion, this expression becomes, — J.(r+i»4 + <»y) sin. (« + »«)* .-.the part which corresponds to«y, is «*.«3y.(r-J- its density, con-esponding to the time t, we shall obtain, by putting the primitive value of its mass, equal to its value after the time t, p. e'r'*. sin. y = (p). r*. sin. 9 ; this is the equation relative to the continuity of the fluid. In the case we are at present considering, r' = r+a.s; 6' = 9 + «u; ■ar=nt+zr + aV', * r" t ■s/ are generally functions of r, t, w, and t, see page 217, notes ; the reasoning is precisely the same as in page 218, substituting the coordinates j-, t, w, in place of j;. ;/> *• 256 CELESTIAL MECHANICS, consequently, we shall have by neglecting quantities of the order «• Let us suppose that after the time /, the primitive density (p) is changed into (p) + «p' ; the preceding equation relative to the continuity of the fluid, will give 36. Let us apply these results, to the oscillations of the sea. Its mass being homogeneous, f' vanishes, consequently, » dr'=dr->r»ds, d^=d6-^»du,d^*d^J^*dv .'. (x-)-(^)-(^)=-- (dr-\-ctds\ (d6-^»du\ ^f dtn-X-adv \ , . {" ds \ , / du\ /dv \ it is plain, that if there was no motion, the differential of any coordinate 6, with respect to another coordinate, would vanish, after the time <, this differential is of the order <••. — -7— — ^ ^ ( ^T** — )'s of the order f or «■", consequently it may be neglected, from (JOT ) ^ di ' which it appears, that all the terms in expression for €' after the first may be neglected. r»{sin. «+«M.cosO \ =(5). r». sin. «) (»''+2«s). sin. <+«M. cos, i) > =(§). r*. sin. t, i, e, PART I.— BOOK I. 257 Let us suppose, conformably to what appears to be the case of nature, that the depth of the sea is very small in comparison with the radius r of the terrestrial spheroid; let this depth be represented by y, y being a very small function of 6 and -a, which depends on the law of this depth. If we inte- grate the preceding differential equation, with respect to r, from the surface of the solid which the sea covers, to the surface of the sea, * it is obvious that the value of 5 will be equal to a function of 9, w, and /, independent of LL (j). (r»-l-2«r5).(sln. ()+««cos. 0)+ («)• '■'•«n. i. J«-{§} + {^] + {£} } +aj'. r'. sin. 6-=z{() r«. sin. 6. .-. (j).r«.««cos.«+(5)2*w.sin. «+(j).r'.sin.«.|«|^|+ {^|+ {^}} -}-«§'. r'. sin, tf=0 .•. dividing by sin. 6 and », we obtain • The depth of the sea being inconsiderable, in comparison of the terrestrial radius, we may suppose, that for this depth r*, and the factor of r» in the second tema, of the second member of this equation, are constant .*. integrating we obtain , / /du\ , tdv \ u COS. (■y as the increment of the radius at the surface of the spheroid = aug. (-j-j +avg. ( j- ) see notes to page 252, .-. s' at the surface of the sea _ f (du-) Cdvl tt.cos. «1 . r)— {- } + «3^'— « — . the three first terms destroy each other .•. aiV — « — +| is equal to the first member of the equation (L), and i 1/ liace it is an exact variation, the first member of the equation(L)wiU Be so also, .*. V — dif- ftfenced With respect f f, is equal M the tentt of the first metrHbex of the equatioB (t). wWch is multipKed by Sf . — «. sin.«.cos. »= t.'^cos.<0 , in order that —2». ^ ^ ^ . sin. <. cos. » may b* *f ifienme wfder m | - - I it 16 Becessay that « | — ? should be of the order y or y, which is a function of ^.•ct, and <,=A, the other term being of the order —may be rejected. If we only consider the terms, which refer to S and -r, the first member of the equation (L) is the same as the first member of the equation (M), near the surface, the last term of the first member of the equation (L') vanishes .-. the equation (L) must in this case coincide with the equation (M), but A the member of the equation (L) does not vary .•. we have the second member of the equation (L) — the second member of the equation(M) i. e, Ja=JF' — giy '■> but Sa = 3./ V'—JL I ••• S- •[ V'— ^ ? =.lV'—g.ly, from the theory of the tides * / it appears that the 5 F' fin these two members are the same, .•. g^xf=:.— and ■p'^egy -^ a con- % slant arbitrary quantity ; when the integral is taken between the surface of spheroid, and eurface of the sea, this constant arbitrary quantity may be rejected. PART I.— BOOK I. 261 above this surface, and this pressure is equal to a-^.gy % therefore we have, in the entire of the interior of the fluid, from the surface of the spheroid covered by the sea, to the surface of the level of the sea, p' = ^gy ; conse- quently, any point of the surface of the spheroid, which is covered by the sea, is more pressed than in the state of equilibrium, by the entire weight of a column of water, contained between the surface of the sea, and the sur- face of level. This excess of pressure becomes negative, for those points, where the surface of the sea is depressed beneath the surface of level. It follows from which has been stated above, that if we only consider the variations of 9 and is ; the equation f L) will be changed into the equation (Mj, for all the interior molecules of the fluid. Consequently, the values of u, and v, relative to all molecules, * situated on the same terrestrial radius, are determined by the same differential equations ; thus, supposing, as we shall do in the theory of the flux and reflux of the sea, that at the commencement ofthemotion,the values of w,i — Vy, I — I, were the same for all the molecules of the fluid, situated on the sanie radius, these molecules will exist the same radius, during the oscillations of the fluid. Therefore the values of r, u, and v, may be supposed very nearly the same, on the small part of the radius, comprised between the solid, which the sea covers, and the surface of the sea ; thus, if we integrate with respect to r, the equation ^ cd.r*s 1 , , ^^du) , (dv} , u cos 9 } , * At the commencement of the motion u, and v, i — f i "{ t; r > ^^^ the same, for all molecules situated on the same radius, .•• after the interval dt, the corresponding values of u and V, will be the same for all molecules situated on the same radius. t r-s-(j^i)zz)%_?2.(^)4-2,-y.(i)+y2(i)for (r^)=(r—y)* y being a function of i, and ar, when these angles are increased by the quantity «m, »v, becomes y-\-aM.. \-t-\ +«d. 5 j - f this is the value of -/ con-esponding to the angle <+«u, ^■\-nt-\-an for the surface of equilibrium, ,•• where the fluid is in motion, we must add ay to this expression. COS. 6 262 CELESTIAL MECHANICS, we shall have ( crf9 J Ccra-J sm. 8 ) (r»5) being the value of Vs, at the surface of the spheroid covered by the sea. The function r^s — (r*s) is very nearly equal to r*. [s — («)} +Qry(s), (s) being what 5 becomes at the surface of the spheroid ; con- sidering, the smallness of y, and (5), in comparison of r, we may neglect the term 2ry.(s) ; therefore, we shall have ros—(r*s')=r.' [«— (5)}. Now, the depth of the sea, corresponding to the angles 0-|-ixw, ar + w?+«f, is y + a.[s — (s)]. If the origin of the angles 6, and ni + sr, be referred to a point, and a meridian, which are fixed on the surface of the earth, which we are permitted to do, as we shall see very soon ; this same depth will be y-i- au. ^-T^^+ai'. j;r-(> plus the elevation ay of the molecule of the fluid at the surface of the sea, above the surface of level ; therefore, we shall have If we make cos. 4=:^, then sin. « ' '^ ^•' 1 — u^ —dfi ^ ^. __ —fi.dft. ^ __ j^_ COS. i sin. $ consequently the equation of continuity, on the supposition that the sea is honiogeneouA becomes, — (^) +r2. r ^^ _r2. ( rf.(«-v/rV) w Book IV. Chap. 2. ^ Md^ r+ :?: — +-d;;^ j= -r- { [^)+ —T^ S.ee Book IV. Chap. 1, No. 2. PART I.— BOOK I. 263 '-»=^+-gi( + 4l}' Consequently the equation relative to the continuity of the fluid will become * cdyjO (d.yV? yii.cos.6 ,„. It may be remarked, that in this equation, the angles 8 and nt-\-t!r are reckoned from a point, and a meridian, which are respectively fixed on the surface of the earth,and in the equation(M), these angles are reckoned from the axis of a, and from a plane, which passing through this axis, revolves about it with a rotatory motion, expressed by n ; but this axis, and this plane are not fixed on the surface of the earth, since the attraction and pressure of the fluid which covers it, as well as the rotatory motion of the spheroid, disturb a little their position. However it is easy to perceive that these perturbations t are to the values of «m, and «t', in the ratio of the mass * Substituting for s — {s), its value {du "i dv di > ^' 1Z and observing that du ^ dv lu COS. * y. . sm. i we will arrive at the value of y, which is given in the text. f In the state of equilibrium, neither the pressure or attraction of the ocean, can produce any motion in the spheroid covered by the sea, and it is only the stratum of water which irv consequence of the attractions of the exterior bodies, and of the centrifugal force, is elevated above the surface, wliich can produce any effect. The effects of the pressure and attraction^ may be considered separately, with respect to the first, if the mean radius of the earth be supposed equal to unity, «y being the elevation, the action of the aqueous stratum is equal to the diiference of the attractions of two spheroids, of which the radius of the interiors 1, of the exterior — l+»i/, naming this difference »y.k. and t its direction, uyhdv will be the expression for this attraction ; multiplied into the element of its direction, t being a function of <, and -a, dr 26i CELESTIAL MECHANICS, of the sea, to the mass of the spheroid ; therefore, in order to refer the angles 9, and nt+zi; to a point and meridian, which are invariable on the surface of the spheroid, in the two equations (M) and (N) ; we should alter u, andi^, by quantities of tl:e order^ and — , which quantities we r r are permitted to neglect ; therefore we may suppose in these equations, that a.u and a.v are the motions of the fluid, in latitude and longitude.* It may also be observed, that the centre of gravity of the spheroid being supposed immoveable, we should transfer in an opposite direction to the molecules, the forces by which it is actuated, in consequence of the re- action of the sea ; but the common centre of gravity of the sea and sphe- roid being invariable in consequence of this reaction ; it is manifest that the ratio of these forces, to those by which the molecules are solicited by the action of the spheroid, is of the same order, as the ratio of the mass therefore they may be omitted in the calculation of W. of the fluid to that of the spheroid, and consequently of the order-, The attractions are of the order ay ; for if y vanished there would be no pressure or action, but y is of the order — . The exact effect which the attractions, and pressures of the aqueous stratum produce are calculated in Book V. Nos. 10 and 11. • The centre of gravity of the spheroid is considered immoveable, because we do not consider the absolute oscillations of the molecules in space, but only their oscillations reia^ live to the mass of the fluid. The common centre of gravity of the fluid and spheroid covered by the fluid is not affected by the mutual action of these molecules, see No. 20. With respect to the action of foreign bodies, their effect is not to be neglected, as in case of the action of the sea, if we consider the centre of gravity of the spheroid immoveable, we must transfer in a contrary direction to the molecule, the attraction which such bodies exert on the centre of gravity of the spheroid, the oscillations «y and the force which actuates the particles are of the order a.-~fix ».q. \ — -i- !• ,see preceding note. PART L— BOOK I. 265 37. Let us consider in the same manner, the motions of the atmos- phere. In this investigation, we shall omit the consideration of the variation of heat in different latitudes, and different elevations, as well as all anomalous causes of perturbation, and consider only the regular causes which act upon it, as upon the ocean. Consequently, we may con- sider the sea as surrounded by an elastic fluid of an uniform temperature ; we shall also suppose, that the density of this fluid is proportional to its pressure, which is conformable to experience. This supposition implies, * that the atmosphere has an infinite height ; but it is easy to be assured, that at a very small height, its density is so small, that it may be regarded as evanescent. This being premised, let s', u\ and w', denote for the molecules of the atmosphere, what s, u x\ designated, for the molecules of the sea ; the equation (L) of No. 35, will then become 2 .„ t \d*ii {m--^->--'-m\ + 1 ^ 3- ( • 2fl /■ d""^' \ . o -ft „ A fdu'\ , 2w. sin, ^9 / ds' \\ '\-o^'rJzT.\sm.^\—-^) +271. sin. 9. cos. 9, ( — 1 -| .( — ) ( dt- ' ^dt ' r ^ dt ' . <^-Sr. $ (^~)—2nr. sin. \ (^)^ = |-. <^.(^+ ccs').sm.^ + xu').Y +^F- ^P. e M M • x\ccording as the fluid is elevated above the surface of the earth, it becomes rarer, in consequence of its elasticity which dilates it more and more, as it is less compressed, and it would extend indefinitely, and eventually dissipate itself in space, if the molecules of its surface were elastic ; consequently, if there is a state of rarity, in which the molecules are devoid of elasticity, the elasticity of the atmosphere must diminish in a greater ratio than the compressing force. 266 CELESTIAL MECHANICS, At first let us consider the atmosphere in a state of equilibrium* in which case s\ zi' andt/ vanish. Then, the preceding equation, being integrated becomes, ■— .r^. sin. *9 + F— P-^ = constant. The pressure p being by hypothesis proportional to the density ; we shall make j) = I. g. p, g represents the gravity at a determined place, * which we will suppose to be the equator, and / is a constaat quantity which expresses the height of the atmosphere, of which the density is throughout the same as at the surface of the sea : this height is very small relative to the radius of the terrestrial spheroid, of wliich it is less than the 72Dth part. The integral A^ is equal to Ig. log. f ; consequently the preceding equation relative to the equilibrium of the atmosphere becomes, ig. log. p = constant + r+ — .-r*. -sin. *9. At the surface of the sea, the value of F" is -the same for a molecule of air, as for a molecule of water contiguous to it, because the forces which solicit each molecule, are the same ; but the condition of the equilibrium of the sea requires, that wc should have V-)r — . r^. sin.^S=constant ; 2 * An homogeneous atmosphere is an atmosphere, supposed to be of the same weight as that which actually surrounds the earth ; its density being uniform, and every where equal to the density of the air at the surface of the earth. Let h be the height of the mercury in the barometer at the equator, and d its density, we shall have lg=h.d:. /x— and by e substituting for A and e^ and g their rnumerical values, /comes out equal to 5;^ miles very nearly, which is somewhat less than ^he 720th part of the radius of the equator. When the temperature is given, this height is a constant quantity, whatever be the ohang«s wliich 'the pressure undergoes. PART I— BOOK I. «67 therefore p is constant at this surface, i, e, the density of the stratum of air contiguous to the sea, is every where the same, in the state of equilibrium. Let R represent, the part of the radius r, comprehended between the centte of the spheroid and the surface of the sea, and r' the part comprised between this surface and a molecule of air ele- vated above it ; r' will differ only by quantities nearly of the order — . r' 1 , * from the /leigkt of this molecule above the surface of the sea ; we may without sensible error neglect quantities of this order. The equation between p and r will give Ig. log. f = constant + l + — . RK sin. *H?^* -R/. sin. -fl : 2 the values of V, (-p^and (-r-g) being relative to the surface of the sea, where we have, constant = V+ ^'R'- sin. -9j the quantity *- l—^ \— n* R, sin. % expresses the gravity at the same M m2 * V being a function of R, 6, and vt, i{ R receive the increment /, V becomes s V "^ T I d~ \ "^ T9\ J~t\'^ ^^' ^^ the expression^ R^- sin. -e will be increased by the quantity n* R/, sin. *«^ — -/.' sin. *(, but this last term being indefinitely small, may be rejected. 268 CELESTIAL MECHANICS, surface; which we will represent by g'. The function \tll * being niul^ !;/:> ■€. Ldr 3 ^ tiphed by a very small quantity r? we may determine it on the hypothesis, that the earth i. spherical, and we may neglect the density of the atmos- phere relatively to that of the earth ; therefore, we shall have very nearly, Ur i ~ * m' m expressin g the mass of the earth ; consequently S— J = '~Jlz= — ^ ; therefore we shall have /^. log. p= constant —^'g' — ^g' ; from which may be obtained _r'g' C r) p = n,c t *If the earth was a sphere then r', would be equal to the height of the molecule of the atmos* phere above the surface of the sea, and as in the case of a spheroid the height is determined by a normal drawn to the surface from the molecule, the difference between / and the part of this normal which is exterior to the surface, depends on the ellipticity of the spheroid, which is 1 for he afterwards supposes that the earth is at the surface of the sea very nearly ^ spherical, .*. the only abberration from sphericity can arise from the greater centrifugal force of the molecule of the air, the ratio of this excess of cen- trifugal force to gravity, for a molecule elevated at the equator, above the surface of the earth r= , and the mtercept at the surface between the du-ection of r, and the direction of a normal drawn from the molecule of the air must be evidently of the order of the ellipticity t, e, of the order , and the difference between r' and this height is equal to the square of this quantity divided by R very nearly. t Sf— P3x+ Qlj/-i-Rh, and if we refer the molecules to the polar coordinates r, i, w. PART I.— BOOK I. . 269 c being the number of which the hyperbolical logarithm is equal to unity, and n being a constant quantity evidently equal to the density of the air at the surface of the sea. Let h and /?' represent the lengths of a pen- dulum, which vibrates seconds at the surface of sea, under the equator, and at the latitude of the molecule of the atmosphere, which has been is that part of the force SF, which is resolved in the direction of the radius of the earth, tf:= the complement of latitude .'. »*7l sin. '< is the part of the centrifugal force, which acts in the direction of the terrestrial radius. The force varying inversely as the square of the dis- 1 dV m tance, V-^ —, and — r: -j- see Book II. No. 12. R dr R' The earth being supposed spherical 5 ;7- >• 'S nearly the same in every parallel, and .-. equal (d'Vt to its value at the equator, where it is equal to g very nearly ; in the value of < -j-^ > we sub- stitute ^ in place of ^^ , for thus the error of the supposition that g =:^is somewhat cor- rected ; substituting for /'dV\ „ . . fd^V\ «» „, . (^-) + mR.sm.^>,^^)+-^R'.sm.'e .their values and of remarjiing that V-{- — R.* sin. '« is constant, we obtain the value of Ig. log. ^ which is given in the text. The density of the atmosphere being inconsiderable with respect to that of the earth, we may without sensible error, neglect the attraction of its molecules. The variable part of the value of § is necessarily negative, for the density decreases, ac- cording as we ascend in the atmosphere ; const ^g'-(.y\ Ig Ig V-^Rl , const r'c^.,1 , r'\ const md at the surface of the sea / :rO .••{=<: = n which is consequently the value of 5 at the surface of the sea; when the times of vibration are given, the lengths of the isochronous lendulums are proportional to the forces of gravity, .*. — :i -r-. 270 CELESTIAL MECHANICS, considered : we shall have— = -, and consequently, g h 7/ C r'\ Ih f—n. c * From this expression of the density of the air, it appears that strata of the same density, are throughout equally elevated about the surface of the sea, with the exception of the quantity -i — ~-^ j however, in the exact determination of the heights of mountains by observations of the barometer, this quantity ought not to be neglected. Let us now consider the atmosphere in a state of motion, and let the oscillations of a stratum of level, or of the same density in the state of equilibrium, be determined. Let acp represent the elevation of a mole- cule of the fluid, above the surface of level, to which it appertains in the * If we expand the value of g into a series it becomes equal to ■ V ' r'h' and neglecting higher powers of /',=!— =7- .-. in strata of equal elevation above the level h'—h of the sea, the difference of density is equal to r. (j — j ; in like manner, if the density of two strata, in latitudes of which the forces are respectively equal tog and g'; be the same, we shall have Ih 7' and r* being the heights which coiTespond to the respective latitudes, .•. neglecting quantities of the second order we shall have, when the density is given, /A'=r"A .-. r'/= — conse- h quently the difference between/ and /''/(=—)= r'. ('— ^V PARTI.— BOOK I. 271 state of equilibrium ; it is manifest that, in consequence of this eleva- tion, the value of tVvfill be increased by the differential variation —»g.S

i.g.Sip + »SV' ; (^SV) being the value of S V, which, in the state of equilibrium, corresponds to the stratum of level, and to the angles 9 + «w, and nt+zr+otv j SV being the part of iV, which is produced by the new forces, which in the state of motion, agitate the atmosphere. Let fi=:(f) + «f', f being the density of the stratum of level, in the state of equilibrium. By making -4-=y, we shall have but in the state of equilibrium we have, 0= ^J.{{r+»s). sin. ($ + «^)}»+(JD- /^'y } therefore, the general equation relative to the motion of the atmosphere will become, relatively to the strata of level, with respect to which ir very nearly vanishes, +r^.J^.|sm. .9. |-^j+2«. sin. 6. cos. 6.^-^1+ r-\rf^|| = neglecting quantities of tlie order a', Ig- 8(5) (•(s)~(g).~-«.{') 272 CELESTIAL MECHANICS, =iV'—gJ(P'^gSy' + n^r. sin. »6.$. (/ — (/)),* a («') being the variation of r, which in the state of equilibrium corre- sponds to the variations a?/, «u', of the angles 9, and zy. Let us suppose that all the molecules, which at the commencement of the motion existed on the same radius vector, remained constantly on the same radius in a state of motion, which, as appears from what pre- cedes, obtains in the oscillations of the sea ; and let us examine whether this supposition is consistent with the equations of the motion and continuity of the atmospjiere. For this purpose, it is necessary that the values of u' and of v', should be the same for all these molecules, as we shall see in the sequel, when the forces which cause this variation are de- termined ; consequently, it is necessary that the variations Sip and St/, should be the same for these molecules, and moreover that the quantities •V 2nr. S-sr. sin. 'Q. S'^> , and n'r. sln.^^ J. S , Sdu") ^ <:dt'} u.' cos. 9 % r + as' is equal to the value of r at the surface of level, which corresponds to the angles 8+«m, and sr + uv, together with the elevation of a molecule of air above this surface ; the part of as' which depends on the variation of the angles G and iB-jt being of the order — '—, may be neglected in & N N * Dividing tliis equation by r» (5) we shall obtain (?) ^~ I' ^d« ) W j sin. 6 \r' dr )' f The part of «/ which corresponds to the variations »u', av', is of the same order as the products of these quantities by the eccentricity of the spheroid, see page 258, and the ec- centricity in this case is proportional to the fraction — , consequently the variation of «/ which corresponds to the variation of the angles I and 1?,=: ; the entire variation of cts' is g made up of two parts, of which one is equal to the elevation of the molecule above the sur- face of equilibrium, on the supposition that the angles 6 and ar are not varied, and this part of the variation of us'=ttip, the other part of the variation is the part which corresponds to the variations au' and «u' of the angles 6 and w, and from what precedes it appears that this part may be neglected, consequently we have r d.r^sf \ 2s' . ds' the second term= f~- J by substituting

is supposed to be equal to y ; its derivitive function with respect to r must vanish,

(cfw'? z/cos 9 ? These oscillations of the atmosphere ought to produce corresponding oscillations,, in the heights of the barometer. In order to determine these last by means of the first, we should suppose a barometer fixed at any elevation above the level of the sea. The altitude of the mercury is pro- portional to the pressure which the surface exposed to the action of the air experiences ; therefore it amy be represented by Ig. p ; but this surface is successively exposed to the action of different strata of level, which are alternately elevated and depressed like the surface of the sea ; thus the value of p at the surface of the mercury varies, 1st, * because it appertains to a stratum of level, which in the state of equilibrium was less elevated by the quantity a.y; 2dly, because the density ofa stratum increases in the state of motion, by a/ or by— yi^ . In consequence of the first cause, the variation of f is augmented by the quantity — »y, ( -f }or ^'pi. therefore the en- tire variation of the density f at the surface of the mercury, is gtCp)- , . It follows from this, that if we represent the height of the mercury, in ^ * (rfr)~ '^' (/)• '° *^ *^'® °^ equilibrium /^\=g'. (§) see (page 223) ••. (^\ = ^^ consequently — ay, ( —~ J = ' j \-r-j 's negative because the density in- creases as we ascend in the atmosphere . The temperature of the air being supposed to remain unvaried, its specific gravity will vary as ({) its density, and this quantity varies as Jc. 276 CELESTIAL MECHANICS, the barometer, in the state of equilibrium by k j its oscillations, in the state of motion will be represented by the function — '^" ; conse- quently at all heights above the level of the sea, these oscillations are similar, and proportional to the altitudes of the barometer. It only now remains, in order to determine the oscillations of the sea, and of the atmosphere, to know the forces which act on these respective fluids, and to integrate the preceding differential equations j which will be done in the sequel of this work. END OF THE FIRST BOOK. TREATISE OF CELESTIAL MECHANICS, BY P. S. LAPLACE, MEMBER OF THE NATIONAL INSTITUTE, &C. PART THE FIRST— BOOK THE SECOND. TRANSLATED FROM THE FRENCH, AND ELUCIDATED WITH EXPLANATORY NOTES. BY THE REV. HENRY H. HARTE, F.T.CD. M.R.I.A. DUBLIN : PRINTED AT THE UNIVERSITY PRESS, FOR RICHARD MILLIKEN AND HODGES AND M'AKTHUR. 1827. R. GRAISBERRy, PRINTER Ty THE O.NIVERSITT. TABLE OF CONTENTS. BOOK II. Of the law of Universal Gravitation, and of the Motion of the centre of gravity of the Heavenly Bodies. -..-.. Page 1 CHAPTER I. Of the law of Universal Gravitation, deduced from the ■phenomena. The areas described by the radii vectores of the planets, in their motion about the sun, being proportional to the times ; the force which sollicits the planets, is directed to- wards the centre of the sun, and conversely, .... No. 1 The orbits of the planets and comets being conic sections; the force which actuates them is in the inverse ratio of the square of the distance of the centres of these stars from that of the sun. Conversely, if the force varies in this ratio, the curve described is a conic section, ...... .- No. 2 The squares of the times of the revolutions of the planets being proportional to the cubes of the major axes of their orbits, or, what comes to the «ame thing, the areas described in the same time, in different orbits, being proportional to the square roots of their parameters ; the force which sollicits the planets and comets ivill be the same for all bodies placed at the same distance from the sun, ... No. 3 The satellites in their motions about their respective primary planets, present very nearly the same phenomena, as the planets do in their motion about the sun ; therefore the satellites are soUicited towards their respective primary planets and towards the sun, by forces which vary inversely as the squares of the distances. - - No. 4 Determination of the lunar parallax, by means of experiments made on heavy bodies' and on the hypothesis of the force of gravity varying inversely as the square of the distances. The result which is thus obtained, being perfectly conformable to obser- vatiofis; the attractive force of the earth is of the same nature as that of all the hea- venly bodies ........ No. 5 IV CONTENTS. General reflections on what precedes ; they lead to this principle, namely, that all the molecules of matter attract each other directly as the masses, and inversely as the square of the distances, -------. No. 6 CHAP. II. Of the differential equations of the motion of a system of bodies subject to their mutual attraction, - • - . . . 27 Differential equations of this motion. - - - - - No. 7. Development of the integrals which we have been hitherto able to obtain, and which re- sult from the principles of the conservation of the motion of the centre of gravity, of areas, and of living forces. ....... fjo. 8 Differential equations of the motion of a system of bodies, subject to their mutual at- traction, about one of them, considered as the centre of their motions ; development of the rigorous integrals which we have been able to obtain. - - No. 9 The motion of the centre of gravity of the system of a planet and of its satellites about the sun, is very nearly the same as if all the bodies of this system were united in this point; and the system acts on the other bodies, very nearly as in this hypothesis. No. 10 Discussions on the attraction of spheroids : this attraction is given by the partial dif- ferences of the function which expresses the sum of the molecules of the spheroid, divided by their distances from the attracted point. Fundamental equation of partial differences, which this function satisfies. Different transformations of this equation. No. 11 Application to the case in which the attracting body is a spherical stratum : it may be proved, that a point situated in the interior of a spherical stratum is equally attracted in every direction ; and that a point situated without the stratum is attracted by it, as if the mass was condensed in its centre. This result likewise obtains for globes com- posed of concentrical strata, of a variable density from the centre to the circum- ference. Investigation of the laws of attraction in which those properties obtain. Among the infinite number of laws which render the attraction very small at consi- derable distances, that of nature is the only one in which spheres act on an exterior point, as if their masses were united in their centres. This is likewise the only one, in which the action of a spherical stratum on a, point situated within the stratum vanishes. ' . ^ No. 12 Application of the formulae of N°. 11, tp the case in which the attracting body is a cy- linder, of which the base is a reentrant curve, and of which the length is infinite. When this cuiTe is a circle, the action of the cylinder on an exterior point is reciprocally proportional to the distance of this point from the axis of the cylinder. A point situ- ated in the interior of a circular cylindrical stratum, of a uniform thickness, is equally attracted in every direction. ...... No. 1 3 Equation of condition relative to the motion of a body. ... No* 14 CONTENTS. V Different transformations of the differential equations of the motion of a system of bodies subject to their mutual attraction. ..... No, 15 CHAP. III. First approximation of the celettial motions, or the theory of elliptic motion. - - - - ---97 Integration of the differential equations, which determine the relative motion of two bodies, attracting each other directly as the masses, and inversely as the square of the distances. The curve which they describe in this motion is a conic section. Ex- pression of the time in a converging series of the sines and cosines of the true motion. If the masses of the planets be neglected relatively to that of the sun, the squares of the times of the revolutions are as the cubes of the greater axes of the orbits. This law obtains in the case of the motion of the satellites about their respective primary planets. No. 16 Second method of integrating the differential equations of the preceding number. No. 17 Third method of integrating the same equations ; this method has the advantage of fur- nishing the arbitrary quantities, in functions of the coordinates and of their first dif- ferences. ....... Nos. 18 and 19 Finite equations of elliptic motion : expressions of the mean anomaly, of the radius vector, and of the true anomaly, in functions of the excentric anomaly. - No. 20 General method for the reduction of functions into series ; theorems which result from it. . - - - - - . - - No. 21 Application of these theorems to elliptic motion. Expressions of the excentric anomaly, of the true anomaly, and of the radius vector of the planets into converging series of the sines and cosines of the mean anomaly. Expressions in converging series, of the longitude, latitude, and of the projection of the radius vector, on a fixed plane, a little inclined to that of the orbit. ...... No. 22 Converging expressions for the radius vector and time, in functions of the true anomaly, for an extremely excentric orbit. If the orbit is parabolic, the equation between the time and true anomaly is an equation of the third degree, wliich may be solved by means of the tables lof the motion of comets. Correction which ought to be applied to the true anomaly computed for the parabola, in order to obtain the true anomaly corres- ponding to the same time, in an extremely excentric ellipse. - . - No. 23 Theory of hyperbolic motion. ....-- No. 2t Determination of the ratio of the masses of the planets accompanied by satellites, to that of the sun. ......-- No. 25 CHAP. IV. Determination of the elements of elliptic motion. - 167 Formulae which furnish these elements, when the circumstances of the primitive motion are known. Expression for the velocity, independent of the excentricity of the orbit, b2 VI CONTENTS. In the parabola, the velocity is reciprocally proportional to the square root of the radius vector ......... No. 26 Investigation of the relation which exists between the major axis of the orbit, the chord of the arc described, the time employed to describe it, and the sum of the extreme radii vectores. ........ No. 27 The most advantageous means of determining by observations, the elements of the orbits of comets. ........ No. 28 Formula; for determining by means of any number of contiguous observations, the geo- centric longitude and latitude of a comet at a given instant, and also their first and second differences. ....... No. 29 General method for deducing from the differential equations of the motion of a system of bodies, the elements of the orbits ; the apparent longitudes and latitudes, and also their first and second differences being supposed to be known for a given instant. No. 30 Application of this method to the motion of comets, supposing them to be actuated by the sole attraction of the sun : it gives, by the solution of an equation of the seventh degree, the distance of the comet from the earth. The sole inspection of three consecutive and contiguous observations, enables us to ascertain whether the comet is nearer or farther than the earth, from the sun. ..... No. 31 Method for obtaining as accurately as we please, and by means of three observations only, tlie geocentric longitude and latitude of a comet, and also their first and second differences divided by corresponding powers of the element of the time. - No. 32 Determination of the elements of the orbit of a comet, when for any instant whatever, its distance from the earth, and the first differential of this distance, divided by the element of the time is given. Simple method of taking into account the excentricity of the earth's orbit. ........ No. 33 When the orbit is parabolic, the axis major is infinite ; this condition furnishes a new equation of the sixtli degree, for determining the distance of the comet from the earth. - ........ No. 34 Hence results a variety of methods for computing parabolic orbits. Investigation of that, from which we ought to expect the greatest accuracy in the results, and the greatest simplicity in the computation. ..... No. 35 and 36 This method consists of two parts ; in the first, the perihelion distance of the comet, and the instant of its passage through the perihelion, are determined in an approximate man- ner ; in the second, a method is given of correcting these two elements by means of three observations made at a considerable distance from each other, and from them all the others are deduced. ....... No. 37 Rigorous determination of the orbit, when the comet is observed in its two nodes. No. 38 Method of determining tlie ellipticity of the orbit, in the case of a very excentric ellipse. No. 39 CONTENTS. Vll CHAP. V. General methods Jbr determining by successive approximations, the ntoiions of the heavenly bodies. - - - - - . • 23'1 Investigations of the changes which the integrals of the differential equations ought to undergo, in order to obtain those of the same equations, increased by certain terms. No. 40 Hence we deduce a simple method of obtaining rigorous integrals of differential linear equations, when we know how to integrate these same equations deprived of their last terms. -.-. ..... No. 41 An easy method is likewise deduced of obtaining continually approaching integrals, of the differential equations. .--.... No. 42 Method of making the arches of circles which are introduced into the approaching inte- grals to disappear, when they ought not to occur in the accurate integral. No. 43 Method of approximation, founded on the variation of the arbitrary constants. No. 44 CHAP. VI. Second approximation of the Celestial Motions, or theory of their per- turbations. ........ 259 Formulae of the motion in longitude and latitude, and of the radius vector in the disturbed orbit. An extremely simple form, under which they appear when we only take into account the first power of the disturbing forces. ... No. 46 Method of obtaining the perturbations, in series arranged according to the powers and products of the excentricities and inclinations of the orbits. - - No. 47 Development in series, of the function of the mutual distances of the bodies of the system, on which the perturbations depend. Application of the calculus of finite differences in this development. Reflections on this series.* .... No. 48 Formulae for computing its several terms. , - - - . No. 49 General expressions for the perturbations of the motion in longitude and latitude, and of the radius vector, the approximation being carried as far as quantities of the order of the excentricities and inclinations. , . . '- No. 50 and 51 Recapitulation of these different results, and considerations on ulterior approximations. No. 52 CHAP. VII. 0/ the secular inequalities of the Celestial Motions. • 307 These inequalities arise from the terms which, in the expression of the perturbations, contain the time, without periodic signs. Differential equations of the elements of el- liptic motion, which makes these terms to disappear. ... No. 53 If the first power of the disturbing force be solely considered, the mean motions of the planets are uniform, and the major axes of their orbits are constant. - No. 54 VIU CONTENTS. Development of the differential equations relative to the excentricicities and position of the perihelia, in any system whatever of orbits having a small excentricity and small inclination to each other. ...... No. 55 Integration of these equations, and determination by observations, of the arbitraries of their integrals. ........ Nq. 56 The system of the orbits of the planets and satellites is stable with respect to the excen- tricities, that is to say, those excentricities remain always very small, and the system only oscillates about a mean state of ellipticity, from which it deviates very little. No. 57 Differential expressions of the secular variations of the excentricity and position of the pe- rihelion. ........ No. 08 Integration of the differential equations relative to the nodes and inclinations of the orbits. In the motion of a system of bodies very little inclined to each other, their mutual in- clinations remain always very small. ..... No. 59 Differential equations of the secular variations of the nodes and inclinations of the orbits ; 1st, with respect to a fixed plane; 2dly, with respect to the moveable orbit of one of the bodies of the system. ...... No. 60 General relations between the elliptic elements of a system of bodies, whatever may be their excentricities and respective inclinations. . , - . No. 61 Investigation of the invariable plane, or that on which the sum of the masses of the bo- dies 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 bodies, inclined to each other at any angle whatever, - . - No. 62 CHAP. VIII. Second method of approAmation of the celestial motions. - 352^ This method is founded on the variations which the elements of the motion supposed to be elliptic, experience in virtue of the secular and periodic inequalities. General me- thod for determining these variations. The finite equations of elliptic motion and their first differentials, are the same in the variable and invariable ellipse. - No. 63 Expressions of the elements of elliptic motion, in the disturbed orbit, whatever may be its excentricity and inclination to the planes of the orbits of the disturbing masses. No. 6+ Development of these expressions, in the case of orbits having a small excentricity and inconsiderable inclination to each other. First, with respect to the mean motions and the major axes ; it is proved that if the squares and products of the disturbing forces be neglected, these two elements are only subject to periodic inequalities, depending on the configuration of the bodies of the system. If the mean motions of the two planets are very neariy commensurable, there may result in their mean longitude two consi- derable inequalities, affected with contrary signs, and inversely as the products of the masses of the bodies into the square roots of the major axes of their orbits. It is CONTENTS. IX from such inequalities that the acceleration of the motion of Jupiter, and retardation of that of Saturn arise. Expressions of these inequalities, and of those which the same relation between the mean motions may render sensible, in (he terms which depend on the second power of the disturbing masses. .... No. 65 Examination of the case, in which the most sensible inequalities of mean motion occur among the terras, which are of the order of the squares of the disturbing masses ; this remarkable circumstance obtains in the system of the satellites of Jupiter, and we deduce from it the two following theorems: The mean motion of the Jirst satellite, minus three times that of the second, plus twice that of the third, is accurately and constantly equal to zero. The mean longitude of the Jirst satellite, minus three times that of the second, plus twice that of the third, is constantly equal to two right angles. These tlieorems subsist notwithstanding any change which the mean motions of the sa- tellites may undergo, "either from a cause similar to what 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 only differs for eacli of the three satellites by the magni- tude of its coefficient, and which according to observations is insensible. - No. 66 Differential equations which determine the variations of the excentricities and perihelias. No. 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 contains the periodic variations ; the other independent of this configuration, con- taining the secular variations. This second part is furnished by the differential equa- tions which we have previously considered. .... No. 63 Simple method of obtaining the variations which result from the nearly commensurable rela- tions between the excentricities and perihelias of the orbits ; they are connected with those of mean motion which correspond to them. They may produce in the secular expres- sions of the excentricities and of the longitude of the perihelia, terms extremely sensible, depending on the squares and products of the disturbing masses. Determination of these terms. ......-- No. 69 Of the variations of the nodes and of the inclinations of the orbits. Equations which determine their secular and periodic values. - - - - No. 70 A simple method of obtaining the inequalities which result in these elements, from the nearly commensurable relation which exists between the mean motions; they are connected with the analogous inequalities of mean motion. - - No. 71 Investigation of the variation which the longitude of the epoch experiences. It is on this variation that the secular equation of the moon depends. - - - No 72 Reflexions on the advantages which the preceding method, founded on the variations of the parameters of the orbits, present in several circumstances : method of inferring from them the variations of the longitude, of the latitude, and of the radius vector. No. 73 ERRATA Page Line 20, 3,./o/- This readThe. 28, 7, for (2"+z'}-, read {z"—:^)'-. 34, 6, /or mm, read mm'. 50, 19, Jbr from the M, read from M. 51, 12, /or their, jeac? its. 52, 16, Jbr z — z, read z — z'. 62, 11, for its, read these. 68, 5>forr.[-^),readr.f^—y 81, last line, for the second |, read |. 96, 8, ./or supply, read solely. 96, 19, /o,iL,,,,rf4l', dx dx 103, l.^or e, read c. 143' 17, ./or COS. en, rearf cos. tji<, 152, 19, Jbr u^, read v. 163, 3, /or tan. '^c, reafif tan. ^«. 166, 17, Jor value, read ratio. 174, 10, ybr COS. £. cos. £, rcarf cos. S. cos. S'. 174, ll.^re, read c. 174, 20, Jor sin «. sin. «'-, read sin. u. sin. u'. 216, 1,/orJS", readJS"'. 2 2 219, 1, /"or — , read . r r 224, 20, /or t/— T', read U'—V. 240, 2, ybr the second aQ, read a Q'. 244, 11, ybr these, read the. 256, 4,/or— 0, read=0. 266, l,JordR, read dR. 271, i;Jordf, readdt\ 284, 3, for a\ read »\ ERRATA. Page Line " a , a 285, 11, for a ■=. — -, read a = —-• a a 287, 13, for -__i(-_), read -. ef. 287, 16, dele — before — - . a - 299, 2, /or n + t, read nt+i. 300, 7, for e. COS. it', read e' cos. •a'. 306, 5, /or m', rearf to. 315, l,fx"dr, readfx."dR. 318, 5, yor motion, rfac/ motions. 323, \^, for m' .^^a, read m'.'/a' . 328, 20, ybr the second € J — €, readZt—Q^. 380. 2, /or in «, read (a). A TREATISE ON CELESTIAL MECHANICS, PART I.— BOOK 11. OF THE LAW OF UNIVERSAL GRAVITATION, AND OF THE MOTIONS OF THE CENTRES OF GRAVITY OF THE HEAVENLY BODIES. CHAPTER I. Of the law of universal gravitation, deduced f-om the phenomena. 1. After having developed the laws of motion, we proceed to deduce from these laws, and from those of the celestial motions, which have been given in detail in the work entitled the Exposition of the Sys- tem of the World, the general law of these motions. Of all the pheno- mena, that which seems most proper, to discover it, is the elliptic motion of the planets and of the comets round the sun, let us therefore consider what this law furnishes us with on the subject. For this purpose, let PART. I. — BOOK II. * B 2 CELESTIAL MECHANICS, X and 1/ represent the rectangular coordinates of a planet, in the plane of its orbit, their origin being at the centre of the sun ; moreover, let P and Q represent the forces with which the planet is actuated in its relative motion round the sun, parallel to the axes of ^ and of j/, these forces being supposed to tend towards the origin of the coordinates ; tinally, let dt represent the element of the time which is supposed to be constant; by the second chapter of the first bool^,* we shall have d''v . = ^ + Q. (.) If we add the first of these equations multiplied by — i/, to the se- cond multiplied by x, the following equation will be obtained : ,^ d. (xdy—ydx) , „ „ = — ^ ^^if + xQ—ijP. It is evident that xdy — ydx is equal to twice the area which the ra- dius vector of the planet describes about the sun during the instant dt; by the first law of Kepler this area is proportional to the time, conse- quently we have xdy — ydx = cdt, c being a constant quantity ; hence it appears, that the differential ot the firsi member of this equation is equal to cypher, which gives xQ—yP = 0, * These laws refer strictly to the motion of the centre of gravity of each planet ; it is therefore the motion of this point which is determined, and by the position and velocity PART I.— BOOK II. 3 it follows from this, that the forces P and Q are to each other in the ratio of cT to ^ ; and consequently their resultant must pass through the origin of the coordinates, that is, through the centre of the sun, and as the curve which the planet describes is* concave towards the sun, it is evident that the force which acts on it, must tend towards this star. The law of the areas, proportional to the times employed in theic description, leads us therefore to this first remarkable result, namely, that the force which solicits the planets and comets, is directed towards the centre of the sun. 2. Let us in the next place, determine the law according to which this force acts at different distances from this star. It is evident that as the planets and the comets alternately approach to and recede from the sun, during each revolution, the nature of the elliptic motion ought to conduct us to this law. For this purpose, let the differential equations (l) and (2) of the preceding number be resumed. If we add the first, multiplied by dx, to the second, multiplied by dy, we shall obtain dx.d''x + dy.d''u , „ , ^ , 0= -—^ — ^+Pdx -!- Qdy ; which gives by integrating of a planet, we always understand, unless the contrary be specified, the position and ve- locity of its centre of gravity ; hence it is evident, that the equations of the motion of a material point, which have been given in the second chapter, are applicable in the present case. * The areas being proportional to the times, the curve described is one of single curvature, {see Book I. page 28, Notes), therefore two coordinates [x, y) are sufficient to determine the circumstances of the planet's motion. As the curve described by the planet is con- cave to the sun, it is plain that in the equation —pr= P; -jj- must be taken nega- tively, because the force tends to diminish the coordinates. See Book I. Chapter II. page 31. 4 CELESTIAL MECHANICS, 0= -^^^^ -V2J\Pdx H- Qdy\* the arbitrary constant being indicated by the sign of integration. oodii^^^ u fix Substituting instead of dt, its value — - — - — , which is given by the lavs^ of the proportionality of the areas to the time, we shall have For greater simplicity, let us transform the coordinates x and j/, into a radius vector, and a traversed angle, conformably to the practice of astronomers. Let r represent a radius drawn from the centre of the sun to that of the planet, or its radius vector ; and let v be the angle which it makes with the axis of x, we shall have then, xz=.r. cos. v; y =.r. sin. r ; r ■=. y/a* + ij* ;t from which may be obtained, ■dx''-\-dy''-=.r''.dv^-\-d7''' ; xdy — ydx zz r'dv. If the principal force which acts on the planet be denoted by (p, we shall have by means of the preceding number, . P z= (p. COS. t; ; Q = (p. sin. i; ; 9 =:\/P*i-Q* ; which gives Pdx+Qdyz=.(pdr ; dx -4- dii' * The equation = ^^ h '^■/{Pdx + Q(/y), has been already deduced in No 8 ; by substituting for dx^ and dif- their values in terms of the polar coordinates, we obtain — — p -J — — — [- 1J . sin. u.(rfr. sin. i>+ rdv.coi.v), V Pix+Qrfy = ?irfr.(cos.^u+sin. ^d), +(prft). (r.cos. r. sin. «— r. cos. v. sin.ii) (T (dx'' + du'') = dr; therefore by substituting in the equation — ^ — —rr:: + 2f(Pdx+ Qdy) = 0, [xdy—ydxy we obtain ijl^lj^+Ii ^ g /©t/r = ; and • • l—ch"—T\ 2f^dr). dv" = c^dr" ; as the r «D^ ■ variables dv and (/)• are separated in the equation dv = . j " can r.V—c"—2r\fq)dr be integrated and constructed, the radical ought to be affected with the sign ±, when I) and r increase the same time, the sign is +, and in the contrary case the sign is — ; these circumstances depend on the initial impulse of the planet. The determination of v, or of the orbit described by a body, when the law of the force (p is given, is called the in- verse problem of central forces, the expression for dv coincides with that given by Newton in Prop, il, Lib. 1st. Princip. for it is there demonstrated that XY. XC = O T^ 0\'^ XY ' , from the construction it is evident that -rrrrr = dv, that IN = dr. Q A X. XC that Q;=c, and finally that A = r, and as Z« OC -^ , and ABTD = the square ot XY ^- ^'^ the velocity, V ABTD — Z- = v^ —fifdr— f_': -^^ = dv = ^V.^yiVT) — Z= cdr ■■ -H by r. r\i/—c'—'lr"f(pdr If the force

■"+*) — pl> = 0, by squaring this equation, we get b-r' 2 2 J . o»+i r- . r"'*"3 — p"b- = 0. When n is even, this equation may have four possible roots, when it is odd, it can only have three ; but as this equation is the square of the given equation, some of the roots are in- troduced by the operation, so that the equation to the apsides can never have more than two possible roots, consequently no orbit can have more than two apsides, i. e. there are only two different distances of the apsides, but there is no limit to the number of repetitions of these, without again falling on the same points, if ?2 = — 3 or a greater negative number, the equation can have only one possible root, and the orbit but one apsid. If in the equation — j- + -r-7-" +2/?'£?'")— be substituted in place of r, it becomes c2_ /_ 1 J\ 2f(p. —T! which is a much more convenient form, particularly when the PART I.— BOOK II. dr The orbits of the planets are ellipses, having the centre of the sua in one of the foci ; if, in the ellipse, is- represents the angle which the axis major makes with the axis of x, moreover if a re- presents the semiaxis major, and e the ratio of the excentricity to the semiaxis major, we shall have, the origin of the coordinates being in the focus, 1 +e. cos. {y — ut) ' which equation becomes that of a parabola, when t? = 1, and a is in- finite, it appertains to an hyperbola, when e is greater than unity. law of the force being given, the nature of the orbit is required; for instance the equation in page 2rf2, page 5 becomes, when — is substituted for r then differentiated, and the result divided by W^"^ ■ A f ,> , 9 /'^''^ oX , 1 1-4-e. cos. (u— sr) j.^ . . _ . d^z e.cos.(u-ro) d-z „ 1 cV differentiating twice -— = ~ — -i, '.--rir -H- = -75 Tx' "•■ ^ = rf„8 - «.(!— e«) ' • dir ^~ ~ «.(1— e»)' * ^ ~ rt.(l— e^) c'".(r*.(/'D*4.rfr''') c' c'.dr" * — ' ■ ■ = — ^ — ^-— r= — Ifiidr, • • by differentiating and dividing by r*.dv r- r^.dv by dr we obtain d.l—--A—(p. r \r*dv^/ t The greatest and least values of r correspond to v—vs^ijr, u— ot'=0, •.• they are re- spectively 0.(1 + e), «.(! — e), consequently they lie in directum; hence it is easy to per- ceive, that when and the ditferential of the se- ^ r\dv' r^ a.(l—e- r a^.{l — e*) 2 2 1 cond member divided by dr will be equal to -^ — , consequently we have r' a.(l — er)' r- ' the value of dr- \ c- c- c- 1 c' c- , f dr- \ dr PART I.— BOOK II. 9 forms an equation of condition between the two arbitrary quantities a and e, of the equation of a conic section ; therefore the three arbitrary quantities a, c, and ra-, of this equation, are reduced two distinct quantities, and as tlie differential equation between r and v, is only of the second order, the finite equation of conic sections is its complete integral.* From what precedes, it follows, that, if the curve described is a conic section, the force is in the inverse ratio of the square of the distance, and conversely, if the force be inversely as the square of the distance, the curve described is a conic section. S. The intensity of thet force ?, with respect to each planet and c* comet depends on the coefficient — r- .- : the laws of Kepler fur- ^ a{l — therefore we have c 2 * The polar equation of the parabola is r = ~ -; •.• when v — v = 0, i. e. •^ ^ r+cos. (u— cr) at the perihelium, r =. — ' =— =D, :• a(l — e»)=2D. Nowthis is the same thing, (J g2\ as if a was made infinite, and e= to unity, in the equation, rz=:a.- -, which expresses the distance of the nearest apsis from the focus of the ellipse, for substituting for the ex- centricity its value V^a^—A^, r becomes equal to a.{— filZ—l | — as (i* — af) ~ — -2.' and as v'a* — apz=a — +( }• — = when a is infinite 2a '^ 2 o a.{a—a +-^) „ £_ f — ''^ — JL and it is evident that e is equal in this case to 2 ' ~ 2a * unity. •.• If we suppose that the synchronous areas are as the square roots of the parame- ters, or c = , we will have ; . dt : ^tat ^/2D :: dtiT; :' 1 —/c* a'- k 2k \ The constant ratio which c bears to the square root of 2D, is that of 2x '.h, which is the same for all the planets; -^, or --p^-rT- is the value of the "^ «' 0.(1 — e'; 12 CELESTIAL MECHANICS, The coefficient , , being the same for all the planets and comets, it Ic' force ip at the unity of the distance of a planet from the sun. The accelerating force of the planets being the same at equal distances from the sun, it follows that the moving force will be proportional to the mass; and if all the planets descended at (he same instant, and without any initial velocities from different points of the same spheric surface, of which the centre coincided with that of the sun, they would arrive at the surface of the sun, being tupposed spheric, in the same time ; here, we may perceive, a remarkable analogy between this force and the terrestrial gravity, which also impresses the same motion, on all bodies situated at equal distances fiom its centre. If the apparent diameter of the sun be observed accurately vvith a micrometer, it will be found to vary in the subduplicate ratio of his angular velocity ; from this phenomenon the equable description of areas may be inferred ; for as the apparent diameters of the sun are inversely as the distance of the sun from the earth, the angular velocity of the sun must be inversely as the square of the distance of the sun from the earth, therefore the product of the diurnal motion into the square of the distance, i. e. the small area must be constant. If the sun's mean apparent diameter be called m, and his least apparent diameter m — n, his appa- rent diameter at any other time, will be m — n cos. z, z being the angular distance of the sun from the point where his diameter is least, lience it may be inferred, that the orbit is ellip- tic ; for as the distance is inversely as the apparent diameter, r:=i — -, when m — n cos. (i' — ■sr) r is greatest, v — ■az^O, when least v — CT=:jr, •.• viu- — nr cos. (u — -z) = j('« — n), x being the greatest distance, and mr = s (m — n) -\- nr. (cos. v. — «r), let (m — ?;). x = nx', and then 7nr = ?i(r. cos. (v — to) -|- x), :• m : « : : r. cos. (v — t!!)-\-x' : r ; now r. (cos. [v — w) is equal to a part of the axis intercepted between a perpendicular let fall from the sun's place on this axis, and the place the earth is supposed to occupy, and x' is a constant quan- tity, •.• producing the axis in an opposite direction from the sun, till the distance from the earth is equal to x', and erecting a perpendicular to the produced axis at the extremity of its production, x -\- r cos. [v — ■cr) is e(iual to the distance of the sun from this perpendicu- lar, and as it is to r the distance of the sun from the earth, in a-given ratio of major ineijua- lity, namely m : n, it follows that the curve is an ellipse of which the directrix is a perpendicu- lar, erected at the extremity of x'. This conclusion might also have been inferred fi-om th« polar equation to the ellipse r = ~^ = a(l— e'). (1-f ecos. (u— w))-'. '^ H-ecos. (y — ot) Kepler directed his observations to the planet of Mars, of which the motion appeared te be more irregular, than the motion of the other planets, and by determinmg several di^ tances of the planet from the sun, and tracing the orbit which passes through them all, it will appear that this orbit must be an ellipse, of which the sun occupies one of tlie foci, it PART I.— BOOK II. IS follows that for each of these bodies, the force ip, is inversely as the square of the distance from the centre of the sun, and that it only va- ries from one planet to another, in consequence of the change of dis- tance ; from which it follows that it is the same for all these bodies sup- posed at equal distances from the sun. We are thus conducted, by the beautiful laws of Kepler, to consider the centre of the sun as the focus of an attractive force, which, decreasing in the ratio of the square of the distance, extends indefinitely in every di- rection. The law of the proportionality of the areas to the times of their description, indicates that the principal force which solicits the planets and comets, is constantly directed towards the centre of the sun ; the ellipti- city of the planetary orbits, and the motions of the comets which are per- formed in orbits, which are very nearly parabolic, prove, that for each planet and for each comet, this force is in the inverse ratio of the square of the distance of these stars from the sun ; finally, from the law of the squares of the periodic times proportional, to the cubes of the greater axes of their orbits, i. e. from the proportionality of the areas traced in equal times by the radii vectores in ditlerent orbits, to the square roots of the parameters of these orbits, which law involves the preceding, and is applicable 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 would fall towards the sun, with equal velocities. 4. If from the planets we pass to the consideration of the satellites.! we find that the laws of Kepler being very nearly observed in their mo- tions about their respective primary planets, they must gravitate towards the centres of these planets, in the inverse ratio of the squares of their distances from these centres ; they must in like manner gravitate very nearly as their primaries towards the sun, in order that their relative mo- tions about their respective primary planets, may be very nearly the same can also be sliewn that the angular velocities are inversely as the squares of the distances from the sun, from which it fblluvvs that the areas are proportional to the times. 14 ^ CELESTIAL MECHANICS, as if these planets were at rest. Therefore the satellites are solicited to- wards their primaries and towards the sun, by forces which are inversely as the squares of the distances. The elliplicity of the orbits of the three* first satellites of Jupiter is inconsiderable ; but the ellipticity of the fourth satellite is very perceptible. From the great distance of Saturn we have not been able hitherto to recognise the ellipticity of the orbits of his satellites, with the exception of the sixth, of which the orbit appears to be sensibly elliptic. But the law of the gravitation of the satellites of Jupiter, Saturn, and Uranus is principally conspicuous in the rela- tion which exists between their mean motions, and their mean dis- tances from the ceiitre of these planets. This relation consists in this, that for each system of satellites, the squares of the times of their revo- lutions are as the cubes of their mean distances from the centre of the " planet. Therefore let us suppose that a satellite describes a circular orbit, of which the radius a is equal to its mean distance from the centre of the primary, T expressing the number of seconds contained in the duration of a sidereal revolution, and tt expressing as before the ratio of the semiperiphery to the radius, — '—— will be the small arc described by the satellite in a second of time. If, the attractive force of the pk' * The frequent recurrence of the eclipses of the satellites, enables us to determine the synodic revolution with great accuracy : and by means of this revolution, and of the motion of Jupiter, we can obtain the periodic time. The hypothesis of the orbits being very nearly circular, in the case of the first and second satellites, is confirmed by the pheno- mena, for the greatest elongations are always very nearly the same ; besides the supposition of the uniformity of the motions, satisfies very nearly the computations of the eclipses. The distances of the satellites from the centre of Jupiter, may be found, by measuring with a micrometer, their distances from this centre, at the time of their greatest elongation, and also the diameter of Jupiter at this time, by means of which, these distances may be obtained in terms of the diameter; however they cannot be determined with the same preci- sion as the periods of the satellites. As it is necessary in a comparison of a great nu nber of obsenations, to modify the laws of circular motion, in the case of the third and fourth •atellites, but especially in the case of the fourth, we conclude that the orbits of these sa- tellites are elliptical. PART I.— BOOK II. 15 net ceasing, the satellite was no longer retained in its orbit, it would recede from the centre of the planet along the tangent, by a quantity equal to the versed sine of the arc , that is by the quantity* ■ ; therefore this attractive force makes it to descend by this quantity, to- wards the primary. Relatively to another satellite, of which the mean distance from the centre of the primary is represented by «', 7" being equal to the duration of a sidereal revolution, reduced into seconds, the descent in a second will be equal to , ■; but if we name (p, (p', the attractive forces of the planet at the distances a and a', it is mani- fest, that they are proportional to the quantities by which they make the two satellites to descend towards their primary in a second ; therefore we have 0:0 •• — = — : — — — . The law of the squares of the times of the revolutions, proportional to the cubes of the mean distances of the satellites from the centre of their primary, gives v T* : r' :: a' : d* : From these two proportions, it is easy to infer 1 1 :: a* d' consequently, the forces 9 and 9' are inversely as the squares of the dis- tances a and d. • T: 1" :: 2ax : arc described in a second, on the hypothesis that the motion is uni. form, the versed sine of this arc = ^~.. As the orbits of all the satellites are notd- 2a 1^ liptic, we cannot determine from the nature of the orbits, whether the force for each satel- lite in particular, varies inversely as the square of the distance or not. l6 CELESTIAL MECHANICS, 5. The earth having but one satellite, the ellipticity of the lunar orbit is the only phenomenon, which can indicate to us the law of its attractive force ; but the elliptic motion of the moon, being very sen- sibly deranged by the* perturbating forces, some doubts may exist, whe- ther the law of the diminution of the attractive force of the earth, is in the inverse ratio of the square of the distance from its centre. Indeed, the analogy which exists between tliis force, and the attractive forces of the sun, of Jupiter, of Saturn, and of Uranus, leads us to think that it follows the same lawt of diminution ; but the experiments which have been instituted on terrestrial gravity, offer a direct means of verifying this law. Fort this purpose, we proceed to determine the lunar parallax, by • The orbit of the moon differs sensibly from the elliptic form, in consequence of the action of the disturbing forces, and the variation of its apparent diameter shews, that it de- viates more from the aVcuZar form, than the orbit of the sun. The first law of Kepler may be proved to be true, in the case of the moon, In the same manner as for the sun, namely, by a comparison of her apparent motion, with her apparent diameter. Indeed, if great accuracy is required, the observations ought to be made in the syzygies and in the quadra- tures ; for in the other points of the orbit, the disturbing force of the sun deranges the proportionality of the areas to the times employed in their description. See Princip. Math. Lil). 1. Prop. 66. and Lib. 3, Prop. Sand 29. f Newton demonstrates that th? force which retains the moon in her orbit, is inversely as the square of the distance, in the following manner : if the distance between the apsides was 180°, the force would be inversely as the square of the distance, as has been already pointed out. See Note to page 7- Now the apsides are observed to advance three degrees and three minutes every month, and the law of the force which would produce such an advance of the apsides, varies in- Tersely as some power of the distance, intermediate between the square and the cube, but which is nearly sixty times nearer to the square ; •.• on the hypothesis, that the progres- sion of the apsides, is produced by a deviation from tlie law of elliptical motion, the force must vary very nearly va the inverse ratio of the square of the distance; but if, as Newton demonstrates, the motion of the apsides arises from the disturbing force of the sun, it follows, aforliori, that the force must be inversely as the square of the distance. X The value of the constant part of the parallax is deduced on the hypothesis, that the force soliciting the moon, is the terrestrial gravity, diminished in the ratio of the square of PART I.— BOOK II. 17 means of experiments on the length of the penduUim which vibrates se- conds, and to compare it with observations made in the heavens. On the parallel of which the square* of the sine of the latitude is J, the space through which bodies fall by the action of gravity in a second, is, from observations on the length of the pendulum, equal to 3'°"[",65548, PART. I. BOOK II. D the distance ; and if this parallax agrees with the observed parallax corrected for the lunar inequalities, we are justified in inferring, that the diminished terrestrial gravity and the force solliciting the moon are identically the same. • Let unity represent the radius of a sphere equicapacious w ith a spheroid, its density being supposed to be the same with the mean density of this spheroid; if the greater semi- axis of the spheroid be = 1+g, and the lesser = 1 — s, we shall have for the oblong spheroid the following equation, -—.1^=—— (1 + ^).(1 — s)*, v P = 1+g — 2* neglecting the squares and products of s and §, which is permitted as the ellipticity of the spheroid is supposed to be inconsiderable, consequently we have {:^2«, ••• in an oblong spheroid, such as would be generated by a revolution about the greater axis, the ele- vation of the spheroid above the equicapacious sphere is double of the depression below this sphere ; and if r be the radius of the equicapacious sphere, a the greater, and b the lesser axis of the spheroid, we have a — r= 2r — 2b, :• r = — - — ; if the spheroid be oblate, i. e. such as would be generated by a revolution about the lesser axis, 4* i 43- —.1^ = — . (1 — «)(l + e)^, hence j=2j, i.e. the depression inthisca«eis equal to twice the elevation, •-• 2a — 2r=r—b, andr= — — — . ' ' 3 If a sphere be inscribed in a spheroid, the elevation of any point of the spheroid above the inscribed sphere, is to the greatest elevation of a spheroid above the inscribed sphere, i. e. to the difference between the radius of the equator and seniiaxis, as the square of the cosine of the angular distance A from the axis major, to the square of radius, •.• the elevation =z {a — b) cos. *a, and as the equicapacious sphere is elevated above the lesser axis, and •/ above the inscribed sphere by a quantity equal to r — b, the ele- vation of the spheroid above the ctjuicapacious sphere =(a-^b) cos. "t^ — r-{-b:=(_a — b). 2a4-b ., / —2a + 2b\ ,,,,•• <^ ,, cos. 'A — +A, ^= I, consequently when the elevation is 0, we have 2 1 cos. "-A = — , V sin. 'a = — , and a = 35°16'. This situation is also remarkable 3 3 for being the distance from the quadrature at which the addititious force of the sun, ig equal to that part of its ablatitious force, which acts in direction of the radius of the moon's orbit. 18 CELESTIAL MECHANICS, as we shall see in the third book : we select this parallel, because the attraction of the earth on the corresponding points of its surface, is very nearly, as at the distance of the moon, equal to the mass of the earth, divided by the square of its distance from its centre of gravity. Under this parallel, the gravity is less than the attraction of the earth, by f * of the centrifugal force which arises from the motion of rotation at the equator ; this force is the - th part of the force of gravity ; consequently we must augment the preceding space by its 432d part, in order to obtain the entire space which is due to the action of the earth, which on this parallel, is equal to its mass divided by the square of the terrestrial radius ;_ therefore this space will be equal to 3'"',66394. At the distance of the moon, it must be diminished in the ratio of the square of the radius of the spheroid of the earth, to the square of the distance of this star, to effect this, it is sufficient to multiply it by the square of the sine of the lunar parallax ; therefore X representing this sine under the parallel above mentioned, we shall have ■2'*.3"'%66394, for the height through which the moon ought to fall in a second, by the attraction of the earth. But we shall see in the theory of the moon, that the action of the sun diminishes its gravity towards the earth by a quantity, of which the constant part ist • The centrifligal force at the equator is to the efficient part of the centriftigal force at any parallel, as the square of radius to the square of the cosine of latitude, i. e. in this case, 2 1 as 1 to — , -.' as the centrifugal force at the equator is the-—— th part of the gravity, the force o 288 2 1 1 at the parallel in question, will be = -^ '"ooq" "^ 3 288 432 \ m being the mass of the sun, and d its distance from the moon, a the radius of th^ moon's orbit, the addititious force = — r— > and the part of the ablatitious force, which acts in the direction of the radius vector ==-— -. 3 sin. ^-sr, -a being the angular distance from quadrature, see Kev.ton, Princip. Prop. 66 ; ••• -^C — 3 sin. 'sr) is the part of tlie sun's PART I.— BOOK II. 19 equal to the th part of this gravity ; moreover, the moon, in its re- lative motion about the earth, is sollicited by a force equal to the sum of the masses* of the earth and moon, divided by the square of their mu- tual distance ; it is therefore necessary to dimipish the preceding space by its 358th part, and to increase it in the ratio of the sum the masses of the earth and moon, to the mass of the earth ; but we shall see in the fourth book, that the mass of the moon deduced from the pheno- raena of the tides, is a — - — th part of the mass of the earth ; therefore 5o,7 the space through which the moon descends towards the earth, in the interval of a second, is equal to -^. — ^ . ■r^3'"^66394. 358 58,7 Now a representing the mean radius of the lunar orbit, and T", the duration of a sidereal revolution of the moon, expressed in seconds ; d2 disturbing force acting in the direction of the radius, which is efficient at any point; (hence it appears that it vanishes when sin. *w — . — , see Note, page 17); in order 3 ma to obtain its mean quantity, multiply this expression by dtn and it becomes • : (rfrar — Srfar. sin. »ir) = -— - {dia — --rfar-f- — rfzircos. %b), and its integral = ——{vi~- 3,3. ..„ ., ma v . 2"^ •'"Z"' ^'^ ' ^^ entire circumference, z. c. when ■»=«-, Tr""5~' ■•' the mean disturbing force := , but — : ii' the force retaining the moon in its orbit : : J**"* T^ ' ' 2re the periods of the sun and moon) ••• -rj- r: — 7^^^ Ttq' "t"^ ' = .^Q -, and — ~qjr^^ ~ 'oTo'' ■•' ■" consequence of the diminution of her gravity by the action of the disturbing force, the moon is sustained at a greater distance from the earth, than it would be if the action of the sun was removed, and as the mean area de- scribed in a given time in the primitive and disturbed orbits is the same, the radius vector is increased by a 358th part, and the angular velocity is diminished by a l79th part. * The moon being considered as a point, if it revolved about the centre of the earth, in 20 CELESTIAL MECHANICS, 9/7 * ^,^ will be, as has been already observed, the versed sine of the arc which it describes during a second, and it expresses the quantity, by which the moon has descended towards the earth, in this interval. This value of a is equal to the radius of the earth, under the above mentioned parallel, divided by the. sine of x ; this radius is equal to 6369514"''; therefore we have 6369514""'^' a- . X but in order to obtain a value of a, independent of the inequalities of the moon, it is necessary to assume for its mean parallax of which the sine is x, the part of this parallax, which is independent of these inequalities, and which has been therefore termed the constant part of the parallax. Thus, tt representing the ratio of 355 to 113, and T' being =: 2732166" ; the mean space through which the moon de- scends towards the earth, will be 2.(355)16369514"" (113/..r.(2732lG6)'" the same time in which it revolves about the common centre of gravity of the earth and moon, the central force which should exist in the centre of the earth capable of effecting this, should be ::: to the sum of the masses of the earth and moon ; for a being the dis- tance of the earth from the moon, and m,rril their respective masses, the distance y at which the moon would revolve round the earth by itself, considered as quiescent, is I , see Prin. Math. Prop. 59, Book I. and T ' =: ^—=: — ; — , , hence if a be the distance, the central force =: m-\-m', ••• as the versed sine of the arc described in a second is the space through which the moon descends in consequence of the combined actions of the earth and moon, this must be diminished in the ratio of ?n : m-J-i«' to obtain the space described in consequence of the sole action of 7«. The two corrections, wliich are here applied to the space through which a heavy body would descend at the latitude 55' 16', diminished in the ratio of the square of the distance, are in the Systeme du Monde, applied to the versed sine of the arc described in a second, hence it appears that they must be affected with contrary signs. PART I.— BOOK II. 21 By equalling the two expressions, which we have found for this space, we shall have ,3_ '2.(355)-. 35S.5S,7.6369514< ^^ ~ (USy. 307. 59,7. 3,6ii39ii'-2732l66y ' from which we obtain 10536 ",2 for the constant part* of the lunar pa- rallax, under the parallel in question. This value differs very little from the constant quantity 10540,7 which Triesnecker collected from a great number of observations of eclipses, and oft occultations of the stars by the moon ; it is therefore certain that the principal force which retains the moon in its orbit, is the terrestrial gravity diminished in the ratio of the square of the distance ; thus, the law of the diminution of gravity, which in the planets attended by several satellites, is proved by a comparison of the times of their revolutions, and of their distances, is • In order to find the constant part of the parallax, we apply to the observed parallax, all the corrections which theory males known, and we may perceive from this how the theory of gravity, by indicating the forces which act on the moon, furnishes us with the means of determining the mean motion, and the nature of the inequalities which act on it. f If in a partial eclipse of the moon, the time be noted in which the two horns of the part which is not eclipsed, are observed to be in the same vertical line, it would be easy to shew that the height of the centre of the moon at this instant, will be the same as the height of the centre of the shadow ; •.• if at this instant the height of each of the horns be observed, the mean height,which will be the heightofthecentreof the shadow, will be the apparent height affected by the parallax ; but as the centre of the shadow is diametrically opposite to the centre of the sun, the true height will bo equal to the depression of the sun, which is known from the time of observation ; •.• the ditl'erence of these heights will be the parallax of the moon for the observed altitude, by means of which we can easily determine the greatest parallax; and if in a total and central eclipse, the height of the moon be observed at the instant that it is entirely immersed, and also when it Jint begins to emerge, the mean height will be the height of the centre of the shadow as it is affected by parallax. In an occultation of a fixed star, the star's parallax vanishes, and the difference of ap- parent altitudes is = to the difference of the true altitudes -|- parallax in altitude of the moon ; hence by the known formulae we can obtain the true parallax. A constant ratio exists between the horizontal parallax, and the moon's apparent diameter at the same terrestrial latitude. 22 CELESTIAL MECHANICS, demonstrated for the moon, by comparing its motion with that of pro- jectiles near the surface of the earth. It follows from this, that the ori- gin of the distances of the sun, and of the planets, ought^in the com- putation of their attractive forces, on bodies placed at their surface, or beyond it, to be fixed in the centre of gravity of these bodies ; since this has been demonstrated to be the case for the earth, of which the attrac- tive force is, as has been remarked, of the same nature with that of these stars. 6. The sun and the planets which are accompanied with satellites, are consequently endowed with an attractive force, which decreasing in- definitely, in the inverse ratio of the squares of the distances, comprehends all bodies in the sphere of its activity. Analogy would induce us to think, that a like force inheres generally in all the planets and in the comets ; but we may be assured of it directly in the following manner. It is a con- stant law of nature, that one body cannot act on another, without expe- riencing an equal and contrary reaction ; therefore the planets and comets being attracted towards the sun, they ought to attract this star according to the same law. For the same reason, the satellites attract their respec- tive primary planets ; consequently^'ais attractive force is common to the planets, to the comets, and to the satellites, and therefore we may con- sider the gravitation of the heavenly bodies, towards* each other, as a general property which belongs to all the bodies of the universe. We have seen, that it varies inversely as the square of the distance ; indeed, this ratio is given by the laws of elliptic motion, which do not rigorously obtain in the celestial motions ; but we should consider, that the simplest laws ought always to be preferred, unless observations com- pel us to abandon them ; it is natural for us to suppose, in the first in- stance, that the law of gravitation is inversely as some power of the dis. « Besides, it follows from tlie sphericity of these bodies that their molecules are united about their centres of gravity, by a force which at equal distances solicits them equally towards these points ; the existence of this force is also indicated by the perturbations which the planetary motions experience. PART L— BOOK II. 23 tance, and by computation it has been found, that the slightest differ- ence between this* power and the square, would be very perceptible in the position of the perihelia of the orbits of the planets, in which obser- tion has indicated motions hardly perceptible, and of which we shall hereafter develope the cause. In general, we shall see throughout this treatise, that the law of gravitation inversely as the square of the dis- tance, represents with the greatest precision all the observed inequalities of the motions of the heavenly bodies; this agreement, combined with the simplicity of this law, justifies us in assuming that it is rigorously the law of nature. The gravitation is proportional to the masses ; for it follows from No. 3, that the planets and comets being supposed at equal distances from the sun, and tlien remitted to their gravity towards this star, would fall through equal spaces, in the same time ; consequently their gravity will be proportional to their mass. The motions almost circular of the satellites about their primaries, demonstr;ife that they gravitate as their primaries towards the sun, in the ratio oi their masses ; the slightest difference in this respect, would be perceptible in the motions of that satellites, and observations have not indicated any inequality depending * See No. 58 of this book ; this also follows from Prop. 45, Book 1st, Prin. For if the force which is added to the force varying in the inverse ratio of the square of the distance be called X, the angular distance between the apsides = 1 SO. . = 180.(1 — X), the square of -/l+^X ^ X being neglected, and conversely if the distance between the apsides be given, wt can determine X. The force X is supposed to vary as the distance. f See Newton Princip. Prop. 6, Book 3, where it is shewn, that if the satellite gravitated more towards the sun than the primary at equal distances from the sun, in the ratio ofd:e, the distance of the centre of the sun from the centre of the orbit of the satellite, would be greater than the distance of the centre of the sun from the centre of the primary, in the ratio of V «/ : v/ e , ••• if the difference between d and e, was the thousandth part of the entire gi-avity, the distance of the centre of the orbit from the centre of the sun, would be greater than the distance of the centre of Jupiter from that of the sun, by a th part ^ ' " 2000 ^ of the entire distance. 24 CELESTIAL MECHANICS, on this cause. Therefore it appears that if the comets, the planets and satellites, were placed at equal distances from the sun, they would gravi- tate towards this star, in the ratio of their masses ; from which it follows, in consequence of the equality between action and reaction, that these stars must attract the sun, in the same ratio, and consequently their action on this star, is proportional to their* masses divided by the square of their distance from its centre. The same law obtains on the earth ; for from very exact experiments instituted by means of the pendulum, it has been ascertained, that if the resistance of the air was removed, all bodies would descend towards its cefitre viith equal velocities ; therefore bodies near the earth gravitate to- wards its centre, in the ratio of their masses, in the same manner as the planets gravitate towards the sun, and the satellites towards their pri- maries. This conformity of nature with itself on the earth, and in the immensity of the heavens, evinces in the most striking manner, that the * The mutual attraction does not affect the elliptic motion of any two bodies when their mutual action is considered, for the relative motion is not affected when a common velocity is impressed on the bodies, ••• if the motion which the sun has, and the action which it experiences on the part of the planet, be impressed in a contrary direction, on both the sun and the planet ; the sun may be regarded as immovable, and the planet will be sol- licited by a force ::' to the sum of the masses of the sun and planet, divided by the square of their mutual distance ; •/ the motion will be elliptic ; but the periodic time will be less than if the planet did not act on the sun, for the ratio of the cube of the greater axis of the orbit to the square of the periodic time, is proportional to the sum of the masses of the sun and planet; however as this ratio of the square of the time to the cube of the distance, is very nearly the same for all the planets, it follows that the masses of the planets must be comparatively much smaller than the mass of the sun, which is confirmed by an estimation of their volumes. See No. 25, and Prop. 8, Lib. 3. Frincip. Math. Tlie comparative smallness of the masses is also confirmed by the laws which Kepler was enabled to an- nounce, for tliese laws were deduced from observation, notwithstanding tlie various causes which disturb the elliptic motion ; hence appears the reason why, in the commencement of this chapter, the sun was supposed to be immoveable, and to exert its action on the planets as on so many points, which do not react on the sun, neither was the mutual action of the planets on each other taken into account ; the same simplifications were employed, when the motion of a satellite about its primary was considered. PART I.— BOOK II. 25 gravity observed here on earth, is only a particular case of a general law, which obtains throughout the universe. The attractive property of the heavenly bodies does not appertain to them solely in a mass, but is peculiar to each of their molecules. If the sun only acted on the centre of the earth, without attracting in particular each of its parts, there would be produced in the sea, oscillations much greater, and very different from those which we observe ; there- fore the gravity of the earth to the sun, is the result of the gravitations of all its molecules, which consequently attract the sun, in the ratio of their respective masses. Besides, each body on the earth gravitates towards its centre, proportionally to its mass ; it reacts therefore on the earth, and attracts it in the same ratio. If this was not the case, and if any part of the earth, however small, did not attract the other part, as it is attracted by this other part, the centre of gravity of the earth would have a motion in space, in consequence of the force of gra- vity, which is impossible. The celestial phenomena, compared with the laws of motion, conduct us therefore to this great principle of nature, namely, that all the molecules of matter mutually attract each other in the proportion of their masses, divided by the square of their distances. We may perceive already, in this universal gravitation, the cause of the perturbations, which the heavenly bodies experience ; for the planets and comets being subject to their reciprocal action, ought to deviate a little from the laws of elliptic motion, which they would accurately follow, if they only obeyed the action of the sun. The satellites in like manner deranged in their motions about their primaries, by their mutual attraction, and by that of the sun, deviate from these laws. We may perceive also, that the mole- cules of each of the heavenly bodies, united by their attraction, should constitute a mass nearly spherical, and that the result of their action at the surface of the body, should produce all the phenomena of gravitation. We see moreover, that the motion of rotation of the heavenly bodies, should slightly alter the sphericity of their figure, and flatten them at the poles, and that then, the resultant of their mutual action, not pass- PART I BOOK II. E 26 CELESTIAL MECHANICS, ing accurately through their centres of gravity, ought to produce in their axes of rotation, motions similar to those, which are indicated by ob- servation. Finally, we may perceive why the molecules of the ocean, unequally acted on by the sun and moon, ought to have an oscillatory motion, similar to the ebbing and flowing of the sea. But the deve- lopement of these different effects of universal gravitation, requires a profound analysis. In order to embrace them in all their generality, we proceed to give the differential equations of the motion of a system of bodies, subjected to their mutual attraction, and to investigate the exact integrals which may be derived from them. We will then take advantage of the facilities which the relations of the masses and distances of the heavenly bodies furnish us with, in order to obtain integrals more and more accurate, and thus to determine the celestial phenomena, with all the precision which the observations admit of. PART L— BOOK II. S? CHAPTER II. Of the differential equations of the motion of a si/stem of bodies, sub- jected to their mutual attraction. 7. LET m, m', m", &c. represent the masses of the different bodies of the system, considered as so many points ; let ^, i/, z, be the rectangu- lar coordinates of the body m ; a/, y', z', those of the body m', and corresponding expressions for the coordinates of the other bodies. The distance of m' from m being equal to v/ {a^-xy + (T/'—yy + (z'—z}\ its action on m, will be, by the law of universal gravitation, equal to n^ i^—^y+ii/'—^y-^iz'—zy If we resolve this action, parallel to the axes of a\ of y, and of z, the force parallel to the axis of ,r, and directed from the origin, will be m(y—x) * W-^y+{jj'—yy-^(,z'^zy)Y^ E 2 * The force parallel to the axis of x: -7 , .,,"*> ; •: ^— x) : mm' JTD -xi 1 / / ;; , , - — r^ , .. / be differenced with V(x— x)-+(y'— y)^+(2_2)- ; and if ^y_a,,i+(^'_^)i + (2'_j)i respect to x, and then divided by m.dx, it will become _ _j[__. nim'.(x ' — x).d.T: 28 CELESTIAL MECHANICS, or ^ dx ) We shall have also, 1 /* J mm" A m '< 'v/ (y'— ^)'^ + {i/'—j/y+(z"—z)^ > V dx y for the action of m" on m, resolved parallel to the axis of x, and corres- ponding expressions for the other bodies of the system. Consequently if T^m' mm'' + "^'"^'' + &c • A. representing the sum of the products of the masses m, rti, »»", &c, taken two by two, and divided by their respective distances ; — . j— 7-^f * will express the sum of the actions of the bodies rn, m", kc. on m, resolved parallel to the axis of x, and directed from the origin of « 1 f '^^ \ i^j """' 'm''\dx)~ nT X V(j'— f)'+(y— y)^+(;'— z) ' "^ dx mm" . ■> vi'.{x'—x) __.. ., . -^"\ dx ^/ (x"-^)^■f (y-^)^ +(z"_z) » ^ • 3 (C:r'-x)»+(y =_y)'-Ks'-«)*) ' " {x"-x) 4- jT-r, ,.,,,,, .,,,., r^TT-^ 4- &c. = the sum of the actions of the bodies m', m", »»'", &c. on m, resolved parallel to the axis of ir. PART I.— BOOK II. . 39 the coordinates. Therefore dt representing the element of the time, supposed constant ; we shall have by the principles of dynamics, ex- plained in the preceding book, O = m . — \ i . dt^ I dx S In like manner we shall have d^u ^ dx-) di* Idt/S dt^ I dz ]• = m.- If we consider, in the same manner, the action of the bodies m, m", &c. on m' ; that of the bodies m, m', on m", and so of the rest, we shall have the following equations, namely, dt* \dci/y ' dt* Xdy'S ' df \dz"S The determination of the motions of m, m', m", &c., depends on the integration of these differential equations ; but as yet they have not been completely integrated, except in the case in which the system is composed of only two bodies. In other cases, we have not been able to 90 • CELESTIAL MECHANICS, obtain but a small number of perfect integrals, which we proceed to develope. 8. For this purpose, let us first consider the differential equations in •r, x\ af', &c. ; if we add them together, observing at the same time, that by the nature of the function x, we have we shall obtain, = l.m. ^ • We shall have also, = S,m. -— ; etc t»* d'z 0= l.m. . Let A', Y, Z represent the three coordinates of the cen- tre of gravity of the system j we shall have by the nature of this centre l..m l.m 2,m therefore we shall have d'X ^ d^Y ^ d*Z ^ = -dT' "" = -dT'' "^ = -dT' and by integrating, we shall obtain X = a+bt ; Y = a'+ b't; Z = a"+b"t ;t • Suppose that there are only three bodies, then l.m.—— =^-j-y + \ 'TTj'r \~7T' / _ m'm.{(x '—x)—(x'—x)) ^ mm"({x"—a:)—{x"—x) ) _ , -i- , — "' '" U*^ ^-^ )—v'^ ■^); 3^ _Q ti)g sgjpg proof may be extended to any num» ber of bodies. PART I.— BOOK II. SI a, a', a", b, b', b", being constant arbitrary quantities. We may per- ceive by this, that the motion of the centre of gravity of the system is rectilinear and uniform, and that consequently, it is not deranged by the reciprocal action of the bodies composing the system ; which agrees with what has been demonstrated in the fifth chapter of the first book. Resuming the differential equations of the motion of these bodies, and multiplying the differential equations in y, y', y'', Sec, respectively by a:, of, x'\ &c., and then adding them to the differential equations in *, **, *", &c. multiplied respectively by — y, — y', — y" , &c. ; we shall obtain \ r x"d'y"—y"d'x" 1 , , but from the nature of the function x, it is evident that c rfx ) ^ dx } „ X. ■m—m-''-' grating," = a, and X=at+b, the constant quantity a depends on the velocity of the centre of gravity at the commencement of the motion, and b depends on the position of this centre, at the same instant. S2 CELESTIAL MECHANICS, consequently,* by integrating the preceding equation, we shall obtain In like manner we shall have, xdz — zdx ^ f xdz — zdx > = ^•"^•1 — dF-y^ f ydz—zdy \ . 1 'dt r c" ='E.m. c, c', (f, &c. being constant arbitrary quantities. These three integrals involve the principle of the conservation of areas, which has been ex- plained in the fifth chapter of the first book. Finally, if we multiply the differential equations in x, x', x', &c., re- spectively by dx, dx, dx", &c. ; and those in y, y', y' , &c. respectively by dy, dy', dy", &c. ; those in ;:, z', z", &c., respectively by dz, dz', dz", &c. ; and then add them together, we shall obtain _ _, (dx,d'x4-di/.d''i/4-dz.d^z) , , O — ^.m.— ' ■% , — dx, T dr * Suppose that there are only three bodies, then i/( j— ) + v(-p- l'*'-^ Y"/^)"" n".( '^'^'y"—if'd'^x' \ _ mm'.{y{,x'—x)—y' ( x'—x) ) ( [x-xy+[y'—y)'-\-[^-zy ) ^ , mm"(y{i^'—x)—y"{x"—x) ) , m"m( ) y'(x»—x')—y''{x'—x') ) , '^ {(x'-xy-\-(y"—yY+[^'-z)^)i + ( (x/z-y) - My"-y') ' + (^"-^r ) * mm'(x{y' — y) — x'(y' — ;/) ) ^ mm\x[y" —y) — x"{lj' — ^j)) , ( {x-xY+{i,—y)'^{z-zY-y —{(3!--xY^{y"-y) -^{J< -z^f _ »»^/m'(y(y"-y)-^'(y/-y')) ,_ t Bymultiplying |^^ |^ J + &c. by ^., cf^. rfx",&c.;{ A | , ^ |, J ,+ PART I.— BOOK II. 33 and by integrating, h being a new arbitrary quantity. This integral contains the principle of the conservation of living forces, which has been treated of in the fifth chapter of the first book. The* seven preceding integrals are the only exact integrals, which we have hitherto been aljle to obtain j when the system is composed of only two bodies, the determination of their motions is reduced to differ- ential equations of the first order, which can be integrated, as we will »ee in the sequel ; but when the system is composed of three or a greater number of bodies, we are then obliged to recur to the methods of approximation. 9. As we can only observe the relative motions of bodies ; we refer the motions of the planets and of the comets, to the centre of the sunj and the motions of the satellites, to the centre of their primaries. Therefore in order to compare the theory with observations, it is neces- sary to determine the relative motions of a system of bodies, about a body which is considered as the centre of their motions. Let M represent this last body, m, m, m", &c., being the other bo- dies, the relative motion of which about M, is required ; Let (, U and y be the rectangular coordinates of M, ^+x, n + t/, y+z, those of WJ ; l-i-x', n-t-j/', y-f r", those of m', &c. ; it is manifest that x, y, z, will be the coordinates of vi, with respect to M ; that /, y', z', will be those PART I. BOOK II. F 4c, by dy, dy', dxj' , &c. and then adding these quantities together, their aggregate is equal to the differential of a considered as a function of x, x', &c. i/, y, &c. z, «', &c., and •.• it is equal to dx. * Three of these ii\tegrals are furnished by the principle of the consei-vation of areas, three by the principle of the conservation of the n.otion of the centre of gravity, and one bj the conservation of living forces. 34 CELESTIAL MECHANICS, of m' referred to the same body, and so of the rest. Let r, r', &c., re- present the distances of m, m', &c. from the body M, so that and let us also suppose m'm Vi^v- xY + {jsJ—yy + (2'-z)* mm" -L — — + &c. This being premised, the action of m on M, resolved parallel to the axis of X, and tending from the origin, will be — j- ; that of vfi on M resolved in the same direction, will be — -7-, and so of the other bo- dies of the system. Therefore, to determine^, we will have the fol- lowing differential equation : dt' r' " and in like manner. d'n ""- di^ dt' mz PART I.— BOOK II. 39 The action o£ M on m, resolved parallel to the axis of .r, and directed from the origin, will be —, and the sum of the actions of the bodies m; m", &c. on m, resolved in the same direction, will be — . m f -7-;- j ; consequently, we will have and substituting in place of — -1 its value S.^, we will obtain di r^ dt^ r* f ' m I dx b in like manner, we will have d*z , Ms ^ mz 1 C d\ 1 ,_, F 2 * — . •{ -7— > is equal to the sum of the actions of the bodies m', m", &c. on m, re- »i (^ ax 3 solved parallel to the axis of x, •.• if we add to this expression the action of M oh m. which is equal to , we will have the actions of all bodies of the system on m, ana r' d^ip^x) Mx •■• hj the principles of dynamics established in thfr first book, — — \r ~^ 1_ (d\l _ m ' \dx $ ~~ ' sa CELESTIAL MECHANICS, If in the equations (I), (2), (3), we change successively the quantities m, X, 1/, z, into m', x', y', s! ; tw^ a!', y", z', &c. ; and reciprocally, we will obtain the equations of the motion of the bodies w, m", &c. about M. If we multiply the differential equation in ^, by M+S.m. ; that in x, by m ; that in «', by wi', and performing similar operations on the other differential equations ; by adding them together, and observing that by the nature of the function a, we have ''-m-m^^-^ we will obtain from which we obtain by integrating • The differential equation in ^, becomes by this multiplication, (M-j-2.m.) — _ M.2. — — 2.W.2. ^ = ; and if the differential equations in s, jf, x", &c. be multi- plied by m, m', m", &c., respectively, and then added together, their sum will b« = if this expression be added to the preceding, we will have, observing the quantities which ^j + 2. TO, |-^ 1=0, and by integrating we have (M+2.OT.)- 1 "^ | +2.m. j -^ ^ =^ t l.mij d^x , " dt* T^ ^ r^ ^ ^ \ dx S ^ M + 2.m dt'' ^ 2.my.M ^ m, ^.m.^-mj, ..J^ _ J=?f^ 2. j--j ; multiplying the differ. Af+2.m. >•» + M+2.m ' r» M+2.m I dx i ?.mx , , , '^■mx ential equations in y,i/',i/", &c. by mx-m. ^^_^^ ^^^ , « *' -»« -^^^s.,^ ' &c, we OB- d^u xu my 5 '^^ ? *" tain for the equation in ^, mx.—^+ M.m.^ + mx.2.— ^'I'^ij^ M+Tm.' i-y mM y 5,»nx ^ my 'S.-mx 5^? . ]f the same operation be performed for the equations in ^r'andy", &c. we obtam, by addmg ihes0 equations, and concinnating rf*y . ,, m.xij my „ \ dx \ d y 2.mx.2.^ + ,V.2.-^+2.;«r.2.-f -2.x.|_ \ ^ 2.m. ^^. ..mx .^ iW + 2.m PART I.— BOOK II. Sy of which equation the integral is Const"'. = £.ff». ^ ^ ,f — -— ^rp-- S.W. -jf at M-{ S.7» rf/ E.m?/ dx + — 7 — t: — . 2.WJ.- orc =: M^^.m ' "•'"• (f? ' M.l.m.^^^Jl:^^^ x.mm'. { i^'--Udy'-dy)-iy'-y-^d^-d^) | . ^.^ . 2.— ^+2.ni«.2. < — p- \ ; JV/+2.m this equation being added to the equation obtained, by taking the sum of the equations i* X, K, &C. gives f 'i.my. d's 2 m.x d^y "» S.tkx.S. I , . my.m'dx' my.m"dx" m'y'.mdx m'j/.m"ds;' (M-]rm-\.ni^m')dt "'■(M+ ra-f „;'+,«//)dy'—iifd j!^ . „ Udy—ydx-'r3^'dy'~u"dsf') mm. ________ I „,,„ _v — £ — J — J J (/< ^ dt A. ^„.' (''dy'-}i'\idy—ydx-ifx'dy—y'dJ\, , {ydJ—xdy') , Av di —i' dy) . . mm -^ — ^ ——■ — ^ ■ + mm'. -■? = — i-i -\-mm'.^ ; £.'+4c.= dt ^ dt ~ dt . {{x—T).(dy'—dy)—( u'—y).(dx'—dx)') «'«--^ -^-^ y^'y yn —"'')\ ... making the factors of n/m", m'm", *f. PART I—BOOK II. 41 at ^.mm. < ^ — -—^ <-^ > ; (5) [ydz — zdy') c"-=.MX.m.- dt ^ (6) c' and c" being two new arbitrary quantities. If we multiply the differential equation in x, by ^ , „ H.m.dx 2mdx — '2m.- the differential equation in y, by „ , - l.m.dv -~mdy-^m.—^, the differential equation in z, by li.m.dz 2mdz — 2m. and if, in like mannfer, we multiply the differential equation in x', by PART I. BOOK II. G also to coalese, and obliterating the quantities whicl> destroy each other, we have (M+2.m). Const. = c=the second member of tlie equation in the text, it is evident that the same proof is applicable to any number of bodies. 4a CELESTIAL MECHANICS, M+Hm ' the differential equation in y', by 2m'.dy'. — 2ot. ^ ■ the differential equation in 2', by 2'm!.dz'.~'2m\- and so of the other bodies ; if we then add together these different equations, observing that we will obtain A_£>v idx,d'x-\-dy.d^i/ ^-dz.d'z) Q'E.mdx d^x 2I..m.dy ^ d^y £E.m.dz ^ d'z . ^,, rwrfr * The differential equation in x, being mul(iplied by this quantity becomes = + M. — \-2mdx.l.-~- —2A—-\dx — — .m.- -rrr^— ■ dt* mx 2m . »«x , 2 , /^ dx \ .„ — —X.mdi,~—— S.mdx.'S.-—4- rr; .S.mdx.l -r- 1, if corresponduig operations be perfonned on the differential equations in x', x", &c. we will obtain by adding them toge- ther, dxd^x . ,,„ mxdx , „ , mx ^ (dxl . 2z.m. — — -- +M.22. — +22.7nrf*.2.— - —22.-? y > dx— dt^ r* r* \ OS ) PART I.— BOOK II. 4S which gives by integrating const - Z TB Jdx^+dy^-^dz^) —(•E.mdxy—(J:.mdyY-'i'£.mdz y ' ' dt* {M-\.Y..m')dt'' — 2M.S.— — 2A, r or h=MX.mX ^ •' L + dt"" Z.mm'. \ (dx'—dxy+(di/'—dyy-\-(dz'—dzy ^ * " dt^ d'x 2M , mx 2.S.m , mx 2 dt^ M+2.W , r' M+2.W r* ^ M-\-^.m. M+2.M ^t]''"^^ 2M this equation by reducing, and observing that — rrr . 2.wrf'.r.2. 22.»» , mx „ , mx , , , 2 , dx 2.>«rfx.2. — — = — 22.»«(ir.2 , and also that . 2.mdx.^, =0, M+2.m r^ r' M+2.m ' ' * [-iydy-\-'2zdz=2rdr, we shall obtain the differential equation of the text. '• ^S] ■ '^" + '- [|] • <^ + ^- [Zzl • '^'='^' seepage 28. • If there are but three bodies, we have by multiplying by (Af-f-m-j-wi'+m"); Const. .(M+ )«4-»n'+"'") = h; and if we only consider the coordinates parallel to the axis of x, we will have M (mdx^ + m'dx'^ + m"dx"^) -{- {in -}- m' + m"). (mrfx*+»i'c?x * + m"rfx"*)— (w + m'+?)i")rfx+fl'x'+rfx")]s = Af.2.n;i;^+mVx' + m'^rfx''-|-m"-rfx"^+mm'rfx"+»;m'£f^) * ~~ W^ R' "~ R' a! x'' we shall have the values of -^j- , -777, &c. by distinguishing the let- ters X, y, z, &c. by one, two accents, &c. ; but by the nature of the centre of gravity, 0=S.ff2X,; 0=:S.7m/,; 0=^7nz/y therefore we will have, neglecting quantities of the second order, T.^„ mx ^•^•— MX S.m R^ ' ((x+x,)*+(y+3/,)^+(z+2/)' ^ ^ '^^ ' ^^ ^'' ^^ ^ '> ^ by neglecting quantities very small of the second order, X.(X'+2Xx,-{-Y'+2Yi/,+ 3 '2 Z» +2Zz,r^ + iX-S^^+ Y^+Z')-^=X{X'+ Y'-+Z^)~''—~X.{2Xx,+2 Y>/,+2Zz,)R' + x,{ X» + Y'+Z^ ) ^= (by substituting R' for X*+Y'+Z^) •;p + "^ — (Xx + Yv+Zz.) ^ ,^ mx I MX.'S.m S.mx, rX2.«x,+ y2.n>.v,+Zv.>.z) ) _ _ MX ^^^ ^^^ ^^^ ,^^ ^^^^ ^j. ^j,, «econd member of this equation ranish. PART I.— BOOK II. 47 consequently, the centre of gravity of the system is sollicited by the action of the body M parallel to the axis of a:, in very nearly the same manner as if all bodies of the system were concentrated in this centre. The same conclusion evidently obtains for the axes of 7/ and of z, so that the forces by which the centre of gravity of the system is actuated parallel to these axes, by the action of M, are pT~> pT" * When we consider the relative motion of the centre of gravity of the system about M, we should transfer in an opposite direction, the force which sollicits this body. This force resulting from the action of ttz, m\ ml', &c. on Mf resolved parallel to x, and acting in a direction tending from their origin, is S. — j ; if quantities of the second order are neglected, this function is by what precedes, equal to XS.wi R' In like manner, the forces by which M is sollicited, in consequence of the action of the system, parallel to the axes oi y and of -2, in a di- rection tending from the origin, are F.E.TW . ZX.m ■, and R^ R» It appears from this, that the action of the system on the body M, is very nearly the same, as if all the bodies were condensed in their com- mon centre of gravity. By tiansferring to this centre, and with a con- trary sign, the three preceding forces ; this point will be sollicited pa- rallel to the axes of .t, of _y, and of z, in its relative motion round M, by the three following forces : - and if these {R— p. COS. ■vt)^' {R-\-g. COS. z,)*' ^ ^' R^' R- be transferred to M with a contrary sign, the entire force by which the centre is urged, is p-^ . It appears from this discussion that the centre of gravity of the earth and moon describes very nearly an ellipse about the sun ; now a comparison of this expression, with that which gives the action of M on m, disturbed by the action of m' on M and on m, shews that the curve described by the centre of gravity, approaches much nearer to an ellipse than the curve described by m, for the force on m, acting in the direction of R — p. cos. w _ M+m m '.{R— p. COS. ■a) ,f 1 R-j-q.cos.ir) \ ~ (R—p COS. -sry "^ ~" ~r' '■'"■ l{/J+y. cos. ^=)^ r^ )' cos. 6, ( being the angle at which r is inclined to a radius drawn from M to m, this ex- pression becomes by rejecting very small quantities of the second and Iiigher orders, M4-m-l-m' . w'. COS. S , , , . ., , .,. - + rm — , and the last term is evidently greater than (R — p. COS. ■a)'' [R-\-q. COS. -sr) 6Af. COS. *i '•' the accelerating force on the centre of '^ (R-^-q- cos. try C Mmp Mm'.q 1 1 f Mmp gravity parallel tor= \ ^R_pJ^_^y - ^Rj^^_,,,^^y \;^:;;^^{-Rr SMtnp' . cos. TO Mm', Mm'q SMm'fl. 'cos. ar 1 1 , / _a R^ ir+ 2_ 1 _p-,= because «p-^'y=0. 3MC0S 33- I, J. U u rr— — ,. ("iy^+w'g'); the part of this force which is perpendicular to it disturos the 50 CELESTIAL MECHANICS system ; these centres will move very nearly in the same manner, as if the bodies of the respective systems were concentrated in them ; for the ac- tion of the first system on each body of the second system, is, by what precedes, 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 on the centre of gravity of the second, will, therefore, by what has been just established, be the same as in this hypothesis, from which we may conclude generally, that the reciprocal action of different systems, on their respective centres of gravity, is the same as if the bodies of each system proportionality of the areas tlescribed by the centre of gravity to the times, and it is evi- dently less than — - — '■ ' — •, See Princip. Math. Lib. 1. Prop. 66. Cor. 3, 4, &c. '" (E+q. cos, ^Y "^ r > ' The distance of the centre of gravity from M differs from the distance of m from M re- til 1 solved parallel to R, by p. cos. ct, = , • r. cos. w. (by the nature of the centre of gravity"). In like manner the abberration m longitude =p. sm. ss- = — — — ;. r. sm. ■a, ••• it varies as the sine of the angle of elongation of M from m ; if i be the tangent of the latitude of the earth, the distance of the earth from the plane passing through M and the centre of gravity m' of »n and m', = sp = rs, r> t\ow «=tan. (f. sin. (v — $), ^ being the inclination of the orbit of the moon to the above mentioned plane, and v — 6 being = to the distance of the m' moon from her node. The distance from this plane, as seen from the M = — - — r- . '^ m-\-m ^. See Book 7, and Newton Princip. Math. Prop. 65, 66, 67, 68. What has been R stated at the commencement of this note, shews the truth of Newton's 65 and 67 Prop. Lib. 1. And it would be easy to demonstrate, as Newton states in Prop. 64, that when the force varies as the distance, the centre of gravity describes an accurate ellipse about M, for the force soUiciting m parallel to the axis of x, = — Mx, ••• the force which solicits the centre of gravity parallel to this axis, — — — — MX '■ ■', now this last terra vanishes, if we add to this force, the force 2.mx = X2.m-fS.m.j;; by which M is sollicited in a contrary direction, the entire force on the centre of gravity parallel to this axis = — (M-{-'S,.m.)X, V the centre of gravity describes an accurate ellipse, and m describes an ellipse about the common centre of gravity of Man d m' ; the periodic time in this elb'pse depends on the number of bodies composing the system, and it varies inversly as the square root of the sum of the masses. PART I.—BOOK II. 51 were concentrated in them, and that consequently those centres move, as they would do, in the case of this concentration. It is manifest, that this conclusion equally obtains, whether the bodies of each system are free, or connected together in any manner whatever, because their mu- tual action does not affect the motion of their common centre of gravity. Therefore, the system of a planet and its satellites acts very nearly in the same manner on the other bodies of the solar system, as if the planet and its satellites were united in their common centre of gravity ; and this centre is attracted by the several bodies of the solar system, as in this hypothesis. Each of the heavenly bodies, being composed of an infinite number of molecules, endowed with an attractive power, and their dimensions being very small compared with its distance from the other bodies of the system of the world; its centre of gravity is attracted very nearly in the same manner, as if the entire mass was concentrated in it, and it acts itself on the several bodies of the system, as on this hypothesis j therefore in the investigation of the motion of the centre of gravity of the heavenly bodies, we may consider these bodies as so many massive points, placed in their centres of gravity. But the sphericity of the planets, and of their satellites, render this hypothesis, already very near to the truth, still more exact. In fact, these several bodies may be conceived to be made up of strata very nearly spherical, and of a density which varies according to any given law ; and we novr proceed to show that the action of a spherical stratum on a body, which is exterior to it, is the same as if its mass was united in its centre. For this purpose, we will establish soiue general propositions, relative to the attractions of spheroids, which will be very useful in the sequel. 11. Let X, y, z, represent the three coordinates of the attracted point, which we will denote by m ; let dM represent a molecule of the spheroid, and a/, y', s^, the coordinates of this molecule, j denot- ing the density, which is a function oi'af, y, z', independent oia;,y,z-y we will have dM zz ^.djfdg.dz'. H 2 52 CELESTIAL MECHANICS, . The action of dM on m, resolved parallel to the axis of x, and tendiqg towards the origin, will be ^,dx'.dy '.dz',(x — x) {{^—'^y-viy-y'y^iz—z'rf^ ' and it will consequently be equal to ^ J ^.dx'.dy'.dz' ^ {. dx } therefore if V denote the integral p ^.daf.dy'.dz' ^^ J y/{x-xy-^iy—j/Y^^{z^^Y ' e.dx' .di/ .dz' * The action of dM on m, is expressed by , rrrr} — TvTTT ^ » ■•" ^^^ force p.di!.di/ .dJ . , paraUel to the axis of ^: ^^_^,^._^^^_^,^,_^^^_^^. : : (^-x ): ^.dif .di/ .dif •^{x-xr+{y-y'Y+^-^r . consequently it is = ^(^_y).+(_yly).+(,_^), )i . »!« p,dx' .dy' .d:! expression ,-, ,^, , ,~ yx. , , iTxTT ' differenced with respect to x, and divided by p.dx'.dy.'dz'Jx—x') .. dx. becomes— ^j ^__^,;,^^^_^,,)r )^ : V th^ express.onor r , e.dx'.dy'.dz' -i 3 a. ,, ,, , /,, ■ , ,., y , expresses the actidn of a molecule of the sphe- l ^/ix—x)^+(t/—i/y+(.z—zy 5 dx roid, on a point without the surface of the spheroid, consequently, if we take the sum of the corresponding expressions for all the molecules of the spheroid, «. e. if we take }d f ^' ^' — > = — ■( -r f I this quantity expresses the dx action of the spheroid, on the point m, resolved parallel to the axis of «; the characteristic, d refers solely to the coordinates x, y, t, it does not denote an operation the reverse of that indicated by the characteristic yt PART I.— BOOK II. 53 extended to the entire mass of the spheroid ; — < —— ?- will repre- sent the entire action of the spheroid on the point vt, resolved parallel to the axis of cT, and directed towards their origin. V is the sum of the molecules of the spheroid, divided by their respective distances from the point attracted ; in order to obtain the attraction of the spheroid on this point, we should consider F" as a function of three rectangular coor- dinates, of which one may be parallel to this line, and then take the differential of the function, with respect to this coordinate ; the coeffi- cient of this differential, affected with a contrary sign, will express the attraction of the spheroid parallel to the given line, and directed to- wards the origin of the coordinate to which it is parallel. Denoting the function ((4: — a/)*+(?/— ^')*+(^ — 2;')*)"% by S, we will have As the integration only respects the variables a/, y, z', it is manifest that we will have but we have ^-{dx'S'^idT/* S^ i^J'' • ^ — (^— x') d»e _ _i J 3(t— /)t _— (^— r')'— (v— v')*— f2— «')'4-3f*— y^' 54 CELESTIAL MECHANICS, consequently we will have also This remarkable equation will be extremely useful in the theory of the figure of the heavenly bodies ; we may make it to assume other forms, which will in different circumstances be more convenient ; for instance, let a radius be drawn from the origin of the coordinates to the point at- tracted, which radius we will represent by r, let 9 be equal to the angle, which this radius makes with the axis of x, and w the angle which the plane passing through r and this axis, makes with the plane of the co- ordinates X and 7/ ; we will have a; = r. cos. 9 ; i/ =r, sin. 6. cos. w ; z = r. sin. 6. sin. w ; consequently we shall obtain by means of these expressions, we can obtain the partial differences of IB like manner, -— , -— - , are respectively equal to d'Z d-S . d' A-t\ i\-^\ t it is only ^ idxS IdxyidsS necessary to make x the sole variable in the preceding expressions for r, cos. 9, and tan. -sr, consequently, if we difference these expressions, we will have {rfr "> . C 6?9 > sin. 9 d-sr ^|=cos.9;|^|.= —'>-^=0; by substituting '"'■ { ^ C ' { ^ } > we obtain the value o*" | ^ } » which has been given IN the text. 56 CELESTIAL MECHANICS, which gives ' m=-'-m- sin. (dV .4^1 By this means we can obtain the partial difference 1 —r- c , in partial differences of the function V, taken with respect to the variables r, 6, and V. By differencing this value of^-y— C a second time, we shall obtain the difference \ -^ ^ in terms of the partial differences of V, taken relatively to the variables r, 9, and n-. We can obtain, by a si- milar process, the values of -5 —^ >, and'S . ^ r* By the preceding operations, we can transform the equation (A) into the following : ^ Cd'F? cos.0 ( , U\rV} ,n^, sin. *( _ _ ■/y'+z' . j <^'* 1 _ 2xVJ/^"+2^_ 2.sin. 6. COS. ^ . J ^ 7 . 7* ' '■ l'rfx« j ~ r* ~ 7^ ' Ida )'* tcfx^i ~ *^ ' '■' 1 "^ ) "~ a!) « * dx* "^ dr'lx^'' di'- 'cfx'"^ d6' dx^^ d*V dV sin. »« , d^V sin. «« ». sin. ^T "5^* FTsmTTS ' 1^~T'" >in. *«. cos. *Kr ( «?0 7 »- is? (••■"•' . , . d'r 1 z» x*+2/' COS.**. +sin sm. 4. sui. «r; -; — = = ^— = dz" r r^ r^ sin. *. cos )s. *. sin.CT _ S^^\ cos. *. sin. w ^■^ ^ rf'O "^*£__ COS.* COS. *. sin. 'iff V^'-f«'.r»'^(i(»+z»)-Tr» V^»^z».r* r^ sin. « sin. «. r» 2 sin. S. COS. «. sin. 'ar fcfw'} ^ cos. w rf'w 2zv r* ' \dz J "" ^*+s« ~ r.sin. «' dz^ ^*+T^ ~" 2sin.ti7. cos.g fc?F-) ^dV y S'^f\^SdV\ WHj.f'^^l V'^'l sin.»<.r« '\1^\-\TrV\Jz\^ \di\'\Tz\'^ XdZl'lIz^ PART. I.— BOOK II. S» CELESTIAL MECHANICS, if COS. 6 be put equal to i*, this last equation will become f(^ri . CdV\ COS. fl. sin. ST . idVt cos.w id'-V\ = | — I .m...sin..+ |_} . 4.|_}.__;|_} C<^'^? ■ 2, • « , Cc^^? Ccos.««.+sin. »«. COS. »o7 , td'V\ = J^}- -=«•««'• *- + [^^.^ S- l + ilFi' COS. '*. sin. '■!? , C^^ 1 COS. « — COS. «. sin. '■KT 2sin.fi . [. \-r- > i-T-— J— COS. fi. sin. 'w • r* ^ «/« J r^sm. « r' iT. COS. w f'^'^l COS. V 5^1 2 sin, ^j. "*" l"rf^J *PTsmT« Irfari* rSsin.*« ' if the corresponding terms are made to coalesce in the values of i -— j + ^ -7-7 \ + I -^ ? , we will obtain the following expression C d''V ■) dV r sin. ^6 \ —5- ^ . (cos. ""^^-sin. *«. COS. *«r^sin. '^6. sin. *«)+ -r-.) — '■ \r cos, "fi.+sin. '<. sin, 'ct . cos. M+sia. ^L cos, 'ct \ ^d^ Vl sin. a<+cos. 'fi.cos. *g-+cos. ^L sin, ^ig) J <^^ 1 J 2 sin. «. cos. fi cos. 6 ") ^dV\ f2sin. «. c r*sin. ^ cos. (. COS. *iiT 2 sin. i. cos. fi.cos. 'w cos. cos. fi. sin. ^-a ' r». sill, fl ^ " "*" r^. sin. i r'.sin. < 2 sin.fi. cos.fi. sin, 'ct ^ j^ ( rf^T i sin. 'w cos. *sr 1 _i_ J '^^' 1 "~ r» 5 I. <^a-* J r^ sin. "fi J-', sin. »fi j i ^ 3 ( 2 sin, ig. cos. OT — 2 sin, tg. cos, zr ^ _ frf^Fi nf^^l ^j.)*^^^! ^ r'.sin. »fi )~ t'^^l''" {"dV I'T l~^J *T^ , CdF-» cos.fi ((i^F 1 1 , , . , . . -r \~jT \—7—- — -+ \ -rr- f • -r— — rr=0, V hence multiplying by r«, we obtain I rffi ) r>.sin. fi l dw^ i r».sin. *fi tr j a j ' 1 rfr» J ^ I dr i^ 1 c/fi» ) ^ i dfi 5 sin.« l^'w' i sin.ifi "•""'"'' PART I.— BOOK II. 59 12. Let us now suppose, that the spheroid ,is a spherical stratum, the origin of the coordinates being at the centre ; it is obvious that V will only depend on r, and that it will not contain /ut or w ; the equa- tion (C) will therefore be reduced to from which we obtain by integrating, r I 2 \ ~1F' S^^'-Xd^S'^^'X'dP]''^'' •'^"'S considered as constant, •.• r. | -^ J may be substituted in place of r». s ? + 2r. ■{ — f . Ifwe make COS. . = ^. then —= (_).(J^), and _ = (_). -^ +f -^ V f ^ J) and as rf« is constant, and rf^= — rfO. sin. «, d*fi=: — dS*, cos. *; d'V .d-^V. dV. ,iV. ,dV^ rf^ dV j/<^^\ cos.< _^ dV ^ '/l—f^Kft . _ /rf*FN ,rfF. cos.» _ rf*y . sin.«~ <'/**'/lZ:^ * W«'/'^('5r}' sin.« ^ = ; from this it appears, that a point situated within a spherical stratum does not experience any action, or, which is the same thing, it is equally attracted in every direction. If the point m exists without the spherical stratum ; it is mani- fest that if we suppose it at an infinite distance from its centre, the action of the stratum on this point, will be the same, as if the entire mass was collected in this centre ; therefore if M represent the mass of this stratum ; — s— :— r or — will become in this case, equal to — r-» from which we obtain B = Af, therefore we have universally,* r' * When the point is at the centre — j- = 0, when r = 0, as has been already re- marked, see preceding page ; this is also evident from other considerations, and as B must he the same, wherever the point is assumed within the surface, B in all such cases s= ; V V=A, the value of A may be easily determined. When the point is infinitely distant, the action is the same as if all the molecules were united in the centre of gravity of the sphere, see page 47, and in this • • , M idV) B M D ,T i^ ^ . -^^ case the action IS equal to , v — \ — S- or — = , ••• B=M; V=A-\ . hence when the attracted point is infinitely distant, A=0, •.• it is always =0 ; and V= r r ' If the attracted point be without the sphere, the attraction towards the convex part is equal to the attraction to the concave part of the surface : and when the point is on the surface, the attraction to the spherical stratum is only half of what it is, when the point is at a distance from the surface. This is immediately evident from the expression u*.du.d'a.d6.sm,i ^^ ,., , , „ 1 , . j j j^ • « -jr [r — u. co».«. «.(/)), which, when ip-if)OC-— becomes u*.du.dv!.di. sm, t. T'^^ti, COS 6 ' ' , and it is easy to shew that this expression is the same for two elements situ- ated on the convex and concave sides of the spherical stratum, and which lie on two lines drawn from the attracted point, and making an indefinitely small angle with each other, for u sin. tz=. a perpendicular let fall on r from the attracting element, r — «. cos. 6 = 62 CELESTIAL MECHANICS, with respect to exterior points, :dV} M CrfF7 _ M 'id? i ~ r* ' that is to say, they are attracted by the spherical stratum, in the same manner, as if the entire mass was united in its centre. A sphere being a spherical stratum, of which the radius of the in- terior surface vanishes ; it is obvious, that its attraction on a point si- tuated on its surface, or beyond it, is the same as if its mass was united in its centre.* This conclusion is equally true, for globes composed of concen- trical strata, of which the density varies from the centre to the surface according to any given law ; for this is true for each of its strata ; thus, as the sun, the planets, and the satellites may be considered, very nearly, as globes of this nature ; they attract exterior bodies almost, as if their masses were concentrated in their centres of gravity, wliich is conform- able to the result of observation, as we have seen in No. 5. Indeed, the figure of the heavenly bodies deviates a little from the spherical form ; however, the difference is very small, and the error which results part of r intercepted between attracted point and this perpendicular, and it is manifest from similar triangles that the perpendicular let fall on r, and also the intercepts between these perpendiculars and attracted point are respectively as the distances of the attracting elements from the attracted point, and udS is also in the same ratio in both cases, see Princip. Math. Book I. Prop. 72, •.• for the two elements at above mentioned, u.di.u. uu.. 6.(r — m.cos. ^) . , r ■, , . .1 .• i.- i_ ■ IS the same for both, consequently the attractions which vary as these expressions will be equal, and this being true for every two corresponding elements existing on the same right hnes, itis true for the entire stratum. Hence if the attracted point is indefinitely near to the spherical surface, its attraction to the molecule contiguous to it, is equal to its attraction to the rest of the spherical stratum ; if the attracted point ap- proaches still nearer, so as to become identified with this molecule, it will then be a part of the stratum, and its attraction will now be only half what it was previous to its contact with the stratum, * For w being the radius of the homogeneous sphere M= -rr— • "'» V — < -j- f — PART I.— BOOK II. 63 from the preceding supposition, is of the same order as this difference, relative to points contiguous to this surface ; and with respect to those points which are at a considerable distance,* the error is of the same order as the product of this difference, by the square of the ratio of the radii of the attracting bodies to their distances from the points attracted, because we have seen, in No. 10, that the sole consideration of the great distance of the attracted points, renders the error of the preced- ing supposition, of the same order as the square of this ratio ; the heavenly bodies, therefore attract one another very nearly as if their masses were concentrated in their centres of gravity, not only because they are at considerable distances from each other, relatively to their respective dimensions ; but also because their figures differ little from the spherical form. The property which spheres possess in the law of nature, of ac- tracting, as if their masses were united in their centres, is very remark- able, and it is interesting to know whether it obtains in other laws of attraction. For this purpose, it may be observed, that if the law of gravity is such, that a homogeneous sphere attracts a point placed with- out it, as if the entire mass was united in its centre ; the same result will have place for a spherical stratum of a uniform thickness ; for if we take away from a sphere, a spherical stratum of a uniform thickness, we will obtain a new sphere of a smaller radius, which will possess the property equally with the first sphere, of attracting as if the entire mass — j-= when r=:a, — — . a; for a point which is situated within the sphere, it is evident the action of the strata between the point and exterior surface vanishes, consequently this case is reduced to the former. • This ratio may be deduced from what has been established in No. 46, page 10; sec also Systeme du Monde, page 255, and Book 3, No. 9. If the force varied as the distance, a homogeneous body of any figure will attract a particle of matter placed any where, with the same force and in the same direction, as if all the matter of the body was collected in the centre of gravity. See notes to page 50. This will appear immediately if the force of each element be resolved into other forces parallel to three rectangular co- ordinates. 6* CELESTIAL MECHANICS, was united in its centre ; but it is evident, that if this property belongs to these two spheres, it must also belong to the spherical stratum which constitutes their difference. Consequently the problem reduces itself to determine the laws of attraction, according to which a sphe- rical stratum, of an uniform and indefinitely small thickness, at- tracts an exterior point, as if the entire mass was collected in its centre. Let r represent the distance of the attracted point from the centre of the spherical stratum ; u the radius of this stratum, and du itg thickness. Let 9 be the angle, which the radius u, makes with the right line r, -u the angle made by the plane which passes through the two lines /■ and m, with a fixed plane, passing through the right liner; m"c?m.c?ot.c?9. sin. 0,* will represent the element of the sphe- rical stratum. If then f denote the distance of this element, from the point attracted, we will have f^ = r*—1ru. cos. 9-f«*. Let us represent the law of the attraction, at the distance /"by .(/ ) ;• therefore if we denote J^djl (?(/), by (p,(J) ; we shall obtain the entire action of the spherical stratum on the point attracted, by means of the integral t^.du.fdvs.d^. sin. ^'p,{f), diiferenced with respect to r, and divided by dr. This integral relatively to w, should be taken from t3-=0, to n- equal to the circumference, and after this integration, it becomes ^Tt.u^.du.fd^. sin. 6. ip/y) ; TT expressing the ratio of the semi-circumference to the radius. The value ofy differenced with respect to 6, will give TU and consequently, 2,r.«Vw./d/9. sin. G. ^X/) = 27r.i^ • ffdf. ^^'^ shall have* -ffdf. i. e. the least and greatest values of/, it is evident that by makiDgJ'/df.(p,[/) = Mj')> the integral of the second member will assume the form in the text. t = 4!Tr.vrdu. ?(/•)} (D) and by integrating with respect to r, we shall have 4'(r+u) — »]^(r — u) =:Qru.Xdr. (?{r) + rU, U being a function of u, and of constant quantities, added to the in- tegral* ^u.fdr,(p{r). If we represent ^(r-\-u) — ■^(r — u), by R, we shall obtain by differentiating the preceding equation. id'R\ , , . , „ d.(p(r) (. dr-j dr K2 coefficient of the second member of this equation, with respect to r= — '— — . {—2?/) = ; iiru*dtp= the mass of the spherical stratum, for 5ra'= the area of a circle whose r' radius=«, •••■ixu' = the surface of the spherical stratum, and ^jtu. 'du= the mass of the stratum, of which the tliickness =; du. * Multiplying both sides by dr, and dividing by 2ic.udu we obtain by integrating 68 CELESTIAL MECHANICS, but by the nature of the function R, we have t~d^s - Cd^V'^ consequently, or o ^^ ^ ^ . r.d. (d"Ul 3rdrf. ?(/) = d'Mf), •■• (dr+duy. ({r-u))=d\i,{r-u); ^_ d\-^{r-^u)—d\-^{r—u) _ d^R __ d\4.(r+ii)—d'4{r~u) _ d'R dr' ~ dr' ~ dti" ~ du' ' In order to obtain the attraction to a sphere, we should integrate the expression ' W'^-i-^) — ■4'('' — u) from m=0 to u = L, L being the radius of the spliere, and then the differential of this function taken with respect to r, and divided by dr, will give the attraction of the sphere. — See Book 12, No. 2. PART L— BOOK II. 69 must be equal to a constant arbitrary quantity, which we will denote by 3 A ; therefore, we have + — -7- — — j^ , r dr from which we obtain by integrating, B (p{r)z=Ar+-^i B being a new arbitrary quantity. Consequently, all the laws of at- traction, in which a sphere acts on an exterior point, placed at the dis- tance r from its centre, as if the entire mass was collected in this centre, are comprised in the general formula Ar-\ — -. In fact, it is evident, that this value satisfies the equation (Z)),t whatever may be the values of A and B. If we suppose A zzO, we shall have the law of nature, and it is evi- dent that in the infinite number of laws which render the attraction very small at great distances, that of nature is the only one, in which * Since u does not occur in the first member, nor r in the second member of tliis equa- tion, the equality of these members can only arise from their being respectively equal to a constant quantity, independent of both u and r. Multiplying both sides by r*dr, we shall have 2r.ir.?)(r)-|-r'.rf.(p(r)=3^r'.rf>-. •.• r'. divided by df; we shall likewise denote by V(X)> ^^^ difference of ^'(f) divided by df, and so on ; we shall thus obtain by two successive differentiations of the preceding equation, with respect to r, ^"(u+r)—^\.'Xu-~r) zz 0* As this equation obtains, whatever may be the values of ti and r, it follows that ^"{J") mu^t be equal to a constant quantity, whatever may be the value ofj"; and that therefore 4'"'(y)= 0} but, we have by what precedes, from which we deduce d, — (%|.(«-|-r)— ^)fl— r)), differenced with respect to r, if ■ ■ ■ w only equBl to U, its differential with respect to r must vanish, for then the quantity to which this JO differential is equal vanishes : ue- 4wM\rfK^r=0. ^^ —. ■^[uJf-r—^u—r) S = — rf. = 0; r is B.(—u-\-r)=. —^Br: •.' d.i—. ^(u-i-r—d^lu—r) > = — rf. = 0: r is less IT than u when the point is assunied within the sphere, •>• the limits of/ must be taken ji+f, u — r. * d-M-+-)^-M—r) ^ u=,^'{u+r)-nu-r) : and r{^^r)-V(«-r) = \ 72 CELESTIAL MECHANICS, and therefore = 2. K/) +/ (/) z: — ,• and consequently the law of nature. 13. Let us resume the equation (C) of No. 11. If this equa- tion could be generally integrated in every case, we would ob- tain an expression for V, involving two arbitrary functions, which could be determined by seeking the attraction of the spheroid on a point situated in a position which facilitates this investigation, and then comparing this attraction with its general expression. But the inte- gration of the equation (C) can only be eflFected in some particular cases, such as when the attracting spheroid becomes a sphere, in which case the equation is reduced to one of ordinary differences j it is also possible, in the case in which [the spheroid is a cylinder, of which the base is a curve returning into itself, and of which the length is infinite : we shall see in the third book, that this particular case involves the theory of the rings of Saturn. Let us fix the origin of the distances r, on the axis itself of the cy- linder, which we shall suppose to be indefinitely extended on each side of the origin. Denoting the distance of the point attracted, from the axis by r', we shall have r' = r.v/l— ^*. dp - rfr ~''• lJ/'(^t+r) is always equal to \J-"{m — r), now this could nqt always be the case unless each of them was constant. • M/) =ffdf. 9JJ); :■ ^>{J) =/. ^,(/), and V{/) = (f)df + /'n/)-4f^0, •.■fK' + k'd-a) become by making ——, — — = dr dr dr dr" respectively — p, and substituting -j- s/ — \.r', — v' — i./, fori and A'; du= — + V'— 1. rfcr, dv= —j v' — 1. dvT, consequently K=log. Z-}-*^ — l.tn, i= log. r'—^ — 1. PART L— BOOK II. 75 ?(?•') and 4/(r') being arbitrary functions of r', which may be deter- I, 2 . , du 1 dt w, are particular integrals of the preceding differential equations ; let —y = — 7 = n; -r-, 1 , du ^j, = -7~" ' TT" ='^— 1. = m; —- = — V— 1. = m ; g = np' + nq', (see Collection of r/ q' Examples of differential and integral calculus, page 466,) = —■ +— y-S *" = — ""z "1" aZ—p'—o'+fl'-ffl' =0, -.• 4/=0, i.e. 4^ = 0, and F = ^' cos. ar — /. %/ — 1. sin. w), by substituting cos, w ±v — 1. sin. «r for e~ ^ , and assuming the arbitrary function

^.\/— 1. sin.TO) rf.(/. cos.TO+r'.^— 1. sin, to)'- ^'^''" ^ d.{y. cos. ar+;^ ^Hr. sin. ■^)» * dr"* rf. ^(r. cos-TO+Z-y/ — 1. sin, to ) -. cos. TO+r'.\/ — 1. sin, o-) rf (/. cos. TO-f /. ^/ — 1. sin. to) d^T^ , d'^.^{r'. cos, a- — r'.y/ — 1. sin, to) rf. (y-'. cos, to — r^.v* — 1. sin, g")^ (/.(/. cos. TO— r'. \/Iir. sin. ar)* * ^^^ rf. ■v|'(r'.cos. ar — r'.ij — 1. sin. to) d'.{r'. cos. sr — r'.^/ — 1. sin. to) rf.(/. cos. TO— r'.v/— 1. sin. to) ' '''»* PART L— BOOK II. 77 If the base of the cylinder is a circle, F will be evidently a function ofr', independent of 13- ; the preceding equation of partial differences will consequently become, which gives, by integrating, t dr'i - r' ' <^.(r'.cos.«r±/V— 1. sin. jsr) , . , , , — - — i r = — /.SUl.W±/.v' — 1.C0S.W. aw dK(r'. COS. u±r'.'y^r. sin. zr) , , ,—-.. , (tl'S -T-j = — r.cos. wrpr. v' — l.sin. w); •.• \ ^„^J rf'.«)(»^. COS. ■CT + r'.\/ — 1. sin. w) ,. , . , ^ , — - . , , = ■ ■ ; . r'».(sin. ^■a—'2V — 1 .sin. w. cos. «r — cos. 'w) d.{y, COS. «r+r'.\/ — 1. sin. ot)^ , ^.^'(r'. COS. 3-4/. \/ — l-sin. w) , ,, , ./— T • \> 4» —^ . (-^ j'.(C0S. zr-f- V — 1. sm. -a)) d:{/,cos, •sr + r'.v' — l.sin. w) ^dK4^(/. COS. ^-V^. >^.sin ^)^ ^^ ^^.^_ '^,)+2v/=T. sin. «. cos. «_cos- U). d.{r'. cos. -a — >/ — 1, /. sin. *ot) rf.il/(r'. cos. TO — /.V — l.sin. w) , , , ./ — r • v« , — ii ^= '— . (_r. (cos. -n— V— l.sin. to)). rf.(r'. cos. ar — r'.^. — 1. sin. to) substituted ; consequently this integral satisfies the given differential equation. When TO Tanishes F= ^(r')-\-^{r'), and when to=:90°, V= , ?(/), and ij'(r') may be determined. 78 CELESTIAL MECHANICS, H being a constant quantity. In order to determine it, we will sup- pose / very great with respect to the radius of the base of the cylinder, which consideration permits us to regard the cylinder as an infinite right line. Let A represent this base, and z the distance of any point of the axis of the cylinder, from the point where r' meet this axis, the action of the cylinder supposed to be concentrated in its axis, and re- solved parallel to r', will be equal to f- Ar. dz the integral being taken from ;:= — oc, to ^ = oc ; which reduces this integral to — — ; this is the value of — \—r-,>j when r' is very consi- derable. By comparing it with the preceding expression, we obtain H = 2 J, and it is evident that whatever may be the value of r, the 2A action of the cylinder on an exterior point, is — —.* * If the base of the cylinder be circular, V will be always the same, when / is given, ••• V will be a function of r', independent of w ; dividing by /, and multiplying both sides by d/, we obtain r = >/r'*-\-z'; .'. the attraction in a direction perpendicular to the base, : to the at- traction towards the assumed point = — — ; — - ) ;;/ : v^/^+z', hence as Adz is the dif- A'/dz ferential of the area of the base; — — ? is the differential of the entire force and its [r +z )'^ Az integral = . /- ,^ =, (see Lacroix, No. 192), when z = OC this integral becomes A A — , and when z = — OC, it becomes — —r ; and as we want the attraction of the pomt r' r to the cylinder between these two values of z, the difference of the expressions in these PART I.— BOOK 11.^ 79 If the attracted point lies within a circular cylindrical stratum, of an uniform thickness, and of an infinite length ; we have also — ) — -i 2A two cases, = — — > must give the attraction required. Wlien r is very considerable with respect to the radius of tlie cylinder, it is the same thing as if the mass of the cylinder was concentrated in its axis. When the point is situ- ated within the cylinder, F is of a different form from what it is, when the point is situ- ated without the cylinder ; and as it is of the same form wherever the point is assumed witliin the cylinder, whatever it is in one case, it will be the same in all. The length of the cylinder must be infinite, otherwise the point, even when situated in the axis, would not be equally attracted in the direction of the axis. When the base is circular, — ) — - i = —7- •.• — | -— i . clr — H. — —, • • V (^ but these expressions will be evidently pro- V'a^-l-r'"- Va'^+r'' portional to the reciprocals of the secants of the angles at the base of the triangle. If the attracted point exist in a perpendicular to the plane of a circle which passes through the centre, x being the distance of the attracted point from the circumference of a circle, con- centrical with thegiven circle, the distance of the centre from this point being=;-', then ■xr.fjc'^ — r'*)=the area of this circle, and ^.■xxdx is the differential of the area, and as the attraction in t' the direction of r' b as — j- ; the differential of the attraction of the point towards the circle iTT.r'.dx „,.,,. , . IW ^ , , , , =r , or which the mtegral is 1- C, and when x = r the attraction va- x'^ X ■ishes, ••• C=r 2a-, and the corrected integral = 1t.(\ I, hence the attraction of a point situated in the vertex of a cone to all circular sections of the cone is the same, and for similar cones the attraction varies as the side of the cone. If the attracted point exist in the produced axis of a finite cylinder witli a circular base, of which the radius =«, r' being as before the distance of the attracted point from any point in the axis, \/n''-f»"'* will be the distance of the circumference of the cylinder from this point, the attraction to- r wards this circumference is as I — , , and the differential of this attraction is as a, ''' ,- ^ ,^ of which the integral = r* — v'a'l-)-'^, r, and r^, being the greatest and least values of/, the attraction to the entire cylinder = — r, + /,, — v'a»+r^„ -j- ^a^-f-r/; r, — r//= the length of the cylinder. If the length be infinite r, =V^a*-f-r/', •.• the attraction is as r, — •'-y)^M^-zy)\ + ((x'-.r) •+(y-^) '+ {z-zf ) "** f — »/ %m'(z'—zY ,\4.&c rf'Q rf'Q (j'Q _ — 3(M+ffl).r^-f3(M+OT).r '' — 3m^(J'-I)'4-(/— y)H(z'— z)-) ^ ((x'— .r)- + (»/-t/)^ + (z'-^)')' -|-3ffl'. ,; , (, 7/ 3^.Vl +';--) s :=0. In the expression for -.W + ;^ + VTi PART I.— BOOK II. 83 we shall obtain, by adding them together. ' dr In like manner, if we multiply the first of the equations (0. by — y. cos, 9. sin. v ; the second, by r. cos. 6. cos. v, we shall obtain by their addition N' being supposed equal to — 2rfr. «(<. sin. 6, cos. S. COS. -v — 2dr.dv. cos. ^S. sin. v. cos. i' — d^i, r. sin. <. cos. 6. cos. 'v — (/<'■. r. COS. *«. COS. "ii-|-2c?i;.rf«. r. sin. (. cos.^. sin. r. cos. v — d^vr. cos. *^. sin. u. cos. v — rfu'. r, cos. »«. COS. "^u ; di/=: dr. cos. «. sin. v — rd6. sin. S. sin. v -\-rdv. cos. «. cos. v; :• d'!/=d^r. COS. i. sin. II — rfnrfS. sin. 6. sin. v + dr.dv. cos. «. cos. u — dr.df. sin. <. sin. v — rdS'^. cos. J. sin. v—rdLdv. sin. «. cos. v — rd'K sin. d. sin. v\-dr.dv. cos. *. cos. v — rdv*. cos. ^. sin. v— rdv.de. sin. ^. COS. V -|- rd'' v. cos. (. cos. v; v d^i/. cos. 6. sia.v = d'^r. cos. '(.Bin. 'v. 84 CELESTIAL MECHANICS, The values of ?; v, and 6, involve six arbitrary quantities, which are introduced by the integration of the preceding differential equa- — Qdr.de. sin. 6. cos. 6. sin. ^v -{■2dr.dv. cos. ^6- sin. v, cos. v — rd$^. cos. '6. sin. 'v — rd^t. sin. <. COS. (1. sin. ^T) — rrfn'.cos. ^6.s'm.^v-i-rd'v.cos. 'i. sin. r. cos. v — Src'^.rfr.sin. *. cos. *. sin. t). cos, u; dz—dr. sm.6-\-rd6. cos. fl ; •.• d'z—d^r. sin. 6. -\-2drdS. cos. 6 -\-rd^e. cos. i — rdi^. sin. S; ••• d^z. sin. 6=d'r. sin. '^6-[-2dr.dL sin. ^. cos. 6-\-rd'(. sin. L cos. ^ — rdt'^. , rf^x (i*u . , d'z . d^r ids'" sin. I 6, consequently, -j-^cos. 6. cos. v-\- -~ cos. 6. sin. v-|- —3- sin. S=-t-i —rj- ctt ctt ('i (*(• at rdv- ^ dx dy . dz . d'x ; — . cos. '6, but -r— = cos. (. cos. V ; — p- :=cos. «. sin. ii ; -7- = sin. S. •.• -—— dt^ dr dr dv dt^ , '^'i/ . • a- '^^^ • . COS. *. COS. 11 + , . cos. L sin. v-j- — -r-. sin. ^= di^ dt^ In like manner, if d'x and its value be respectively multiplied by the differential of x, on the hypothesis that v is the only variable quantity, we shall obtain ; — r.d'^x. cos. *. sin. 1)= rd'r. cos. 'i*. sin. v. cos. vf2dr.de.r. sin. d. cos. S. sin. d. cos. v-^2dr. dv.r. cos. =e. sin. ^v+d^6. r" . sin. u. cos. r. sin. ^. cos. i-\-di'^.r'^. cof, '^ sin. t). cos. v-\-d~v. r». cos. *#. sin. 'u+rfi)' r-. cos. ^<. sin. v. cos. u — Idv.di. r^. sin. «. cos. 6. sin. 'r ; and multiplying d^y and its value by the differential of y, taken on the same hypothesis, we obtain r,d''y. cos. *. cos. v—r-d'r. cos. *^. sin. v. cos. i) — 2c?r. dS. r. sin. ^. cos. i. sin. ». cos. D_j_2rfr. dv, r. cos. '^. cos. ^u — r'^di'^. cos. »*. sin. v, cos. d — r* . r.cos.e.sin..+ |^|. r. cos. .. cos. . = J-j. j— j + |^j. |^^ — i — \:=zN'. Multiplying d^j and its value, by the differential of x, taken on the supposition that 6 is the variable quantity ; — rd'^x. sin. 9. cos. v= — rd'r. sin. *. cos. I. COS. 'v+2rdr.d6. sin. '«. cos. 'ii4-2r(/r.rfu. shi. 6. cos. «. sin. v. cos.v + r*d6^. sin. «. cos. t. COS. *w ^r'di.dv. sin. '«. sin. ij. cos. v + rVu' sin. «. cos. 6. cos. ^v + r'^.d^i. sin. '«. COS. 'D+r'd'v. sin. «. cos. 6. sin. r. cos. v. ; performing a similar operation on d^y and its Talue, we obtain —d^y.r. sin. 6. sin. v^—rd^r. sin. «. cos. «. sin. ^ 5i-j-2) rfr.(^<'. sin.^i. sin. *« — 2r ' \ dr'S'ldc J "*■ Idr.dvS'ldcy ^ Idr.dri' ldc\ -~ I dc V We can obtain by means of those equations, the value of ^ , a y and if we make idv\ cd^^ cdv^ rdn "'={Tb\'{d-A-Uj'{dby' '' = uru}-{da}'U\' sin. v.cosiv -J- r'dv^. sin. 6. cos. f. sin. 't'-+- r^.d^S. sin. 'S. sin. 'v — r'd'v. sin. *. cos. i. sin. V. COS. v; and in liiie manner d'^z.r cos- 6:^rd''r. sin. 1 cos. tf-j-2rrfr. dS. cos. '< — r'^dt'. + , , , (/-.T.r . d^ii. r . . , d^z.r T^d'^L cos. ^, •.• T— sin. ^. cos. V ^^-^- sin. «. sni. t) H = — dt^ dt^ dt^ ' 2rdr.de r'^.dv^ . d') , dx . \dy cos. i— \- sin. 6. COS. 6-\-t ".— — , but — — = r. sm. 6. cos. v ; — r- = — r. sin. «. sin. v. dz , ( rf*«"l J^V ] • ■ . f '^'^ 7 — = cos. « ; and — < -r-, f . r. sni. «. cos. v — < -r-^ > . r.sin. 6. sin. n-J- < -rrr f ^-cos. «. '2r.dr.d6 d-6 . (dv"] r: j-T r-'' --I— sin. «. cos. fl+r«. -{—-!- = P. di" ~ dt^ ^ \dt^ i • The vakes of r, v and 6 are determined by the integration of equations of the second order, ••• two arbitrary quantities are invoWed in the determination of each variable. 86 CELESTIAL MECHANICS, ^' - UayidbS LJbS' Id^V' - ldc]'W'^dc^~^d'J'^dc^'^dby' '^ ldhl'\dc^'\da^~^Jbl'lTa^'\Tc^' [drl {dv\ ff/6 7 Ulrl (idvl \d^\ ^dc^'^la^'^Tb^-l'ckl'idb^'^d'J' we shall have * From the value of J — I = M' ; it is evident that M' is a function of r, v and «'; »nd as these coordinates are functions of a, b, c, and conversely, it follows that I da I 1 dr \- IdaS'^ \ dv \' ida]'^ \ dS \'\da\~ fhy substituting for M' its value -I -j- \ \ Xdr^ r \da i"^ Xdr.dvS' 1 «'« j I dr.dS ]' \daS ' by similar operations we obtain the values of j —jj- }• • \ —j- \, &c. Multiplying J — — I and its value, by m and its value, J.— jr-J and its value, by n and its value, ' dM' •% ^ —j— i and its value, by p and its value, we obtain /rf'Ql idr\,(dv\ (di\ (dv\ f"'M ^ . ^ ''"'^ PART I.— BOOK II. «7 Ib like manner if we make (.drl Sdn Sdrl S '^^ I '''= Ida^' idc^-lTc^'ld-ay' Sdrl Sdn idr} ^dn P'= lTbyiTa\-\TJ'Uby' iTc\-{dc\-U\)=''"iiir\- id-a\-id-a\'id-c\)'='n-drl Sd^\ s^lfS^'l /^\_/£!\ f 'l!.\)4. S JIB} X dr^ ]'ldcS ^ Idal'Xdb] \db]'\daP^ Idr.dvy {dv\(idv\ Cd0-) Cdvf (dCW, i dQ\ ^dil ( <:dv1 \dcS^\da]'ldb\- IdbS'Xda])^ ld7MriToy\lday id-bS~idbS'id'aS)^^'t'd^y Adding these three expressions together, and observing that the coefficients off J , I — — ) are respectively equal to cipher, and that the coefficient of ( ) = S, we will obtain the expression given in the text. We can by a similar process obtain the values •f f——\, ( -T-j )i now if we substitute these values in the equation (F), and also M' and ^'' for ■( J- f ; i -J- ( > 3"d multiply by € and cos *e, we will arrive at the equa- tion (G). 88 CELESTIAL MECHANICS, the equation N' -rzf —\ will give Finally, if we make Idby IdcS Idcy ldb)i ' ^dr-) cdv-i cdr) Cdv') Ldcy Cda^ CdaS' tdcS' P" \drn \dv) (rtr^ S"'^? ~ id^iS' (M) ~ ldh\ ' Idal' The equation P^— -s -j^ r will give Consequently, the equation (F) will become, 0=wi.r* cos.«8. 5 —7—( +n.r'^ cos. *e. < —rr- >-\-p.r^. cos. '9.3^ f ^ da ' t db J I dc y + rw". cos. 9\ \ — ^ + n". cos. ^^-{-tA + /'• cos. '9. | -^ | . + S(fZrM'. cos. ='^— P'. sin. ^. cos. ^). In the theory of the moon, we neglect the perturbations, that its action produces in the relative motion of the sun about the earth, which implies that its mass is indefinitely small. Then tlie variables a/, y', z', which are relative to the sun, are independent of j:, y, z, and the PART I.— BOOK II. 89 equation (G) obtains in this theory ; it is therefore necessary that the values found for r, v and 9, should satisfy this equation, which fur- nishes us with a means of verifying these values. If the inequalities which are observed in the motion of the moon, are the result of a mu- tual attraction between these three bodies, namely, the sun, the earth, and the moon, the observed values of r, v and 6, deduced from obser- vation, should satisfy the equation (G), which furnishes us with a means of verifying the theory of universal gravitation ; for the mean longitudes of the moon, of its perigee, and of its ascending node, occur in these values, and a, b, c, may be assumed equal to these longitudes. In like manner, if in the theory of the planets, we neglect the square of the disturbing forces, which we are almost always permitted to do; then, in the theory of the planet, of which the coordinates are ,r, 7/, z, we can suppose that the coordinates x', yf, z', x', &c. of the other planets, are relative to their elliptic motion, and consequently, independent oix^y^z; therefore the equation (G) obtains in this theory.* 15. The differential equations of the preceding No. drr rdv' — T-s-. cos. *9 — r. — -= }——i dt- • dt^ Idr^ de d.Cr^.——. cos. ^9) ,„ , ^ dt -S^Q) J.; (H) dt Xdv^ di' ^ df ^ dt' I d& S PART I> BOOK II. N * We arrived at the equation (G) on the supposition that x,j/, z were independent of a/ w', »', &c. In the case of elliptic motion x, y, z, are independent of x', y', z', and conversely, and as when the square of the perturbating force is neglected, the motion is q.p. elliptic, it follows that x, y, z, are in this case independent of x', y' , z' . See page 49, of the text. 90 CELESTIAL MECHANICS, are only a combination of the differential equations (?) of the same No. ; but they are more convenient, and better adapted to astronomical com- putations. We can assign other forms to them, which may be useful in different circumstances. Instead of the variables r and 9, let us consider u and s, u being equal to -, that is to unity divided by the projection of the ra- dius vector, on the plane of x and of y ; and s being equal to the tan- gent of 6, or to the tangent of latitude of m above the same plane, by multiplying the second of the equations (H) by rdv. cos. *9, and theu integrating, we shall obtain I ti.dtS -^ \dv \ u^ h being a constant arbitrary quantity ; consequently we have dv dt — '•V*-.^/{f}.^" If the first of the equations (H) multiplied by — cos. 6, be added to the third multiplied by — '- — , we shall obtain u 1 dx^ , idQ) , ^ ^dQ) ^ u df Idu^ IdsS dt from which we deduce There are two distinct objects, one to verify the values of r, v, 6, and the other to verify the theory of universal gravitation. dv /do:^ ,, , ^/. dv /dQ.\ PART I.— BOOK II. 91 (u'.dt) u.dt C ( du) u (ds)J If we consider dv as constant, we shall obtain by substituting for dt its value, which has been already given ~ dir . jdv } ti'dv du u" \ds\ * Cay ) vr N 2 rfV , V dividing by dv, and the radical du) ^ u\ ds i quantity we obtain the expression which is given in the text. ds , ^ d^s 2sds^ . \+s^ , d'i d^s 2s '1+s" 1+s^ (l+s'Y u' ' dt'~ u-dt' (l+s^) ds^ , . , s ^ dv'- . ^ . s dv' ^ , 2sds , (sin. 6. COS. 6 =z , ••• r^. —, — . sin. 6. cos. 6= — -— - : 2rdr= ,c-.dt' ' ^ l+s' dt~ u^ • dt' ' u' ^du.(l+s') „ , , 2s-ds^ 2du.ds , d*6 r' A^i 1 r" ^—^ — , V 2rdr.d6 =z „ ; but r^. = — .d.i — c = — ;-. M» ' {l-^s-y u^ '' dt^ dt t-dt^ dt J — '—— f ; •.• by substituting for d'0, d6 and r* their values already given, and d't for ; — its value dt^ .,-.V.^./{g}.A^Jgi.V;..+v]fj-.^ dv PART I.— BOOK II. 9S Therefore in place of the three difFereutial equations (H), we shall have the following : " - "rf^ "^ '' """ Idv /• u'dv " \du J u'Xds 5 By making these equations to assume the following form, we avoid fractions and radicals, (K) +" •-r•-;- -. V, the third equation (H) becomes = s dv* dv.'Uh^ r d's 2s.ds^ Cds_ 2u.du 1 X^TdF^ (l+«*).M^flfi* ■ ■*" i a'* dv.dt J 2.^s' 2duAs_^Uai \'t\^\^3\yi\\ = oy substitut- _— .p^ + _ . 2«rf«.p' + -,. ^. PART I. BOOK II. 98 CELESTIAL MECHANICS, =: —X +s. The area described by the projection of the radius vector, during dv the element of time dt, being equal to i. — =■ ;t the first of these IT equations indicates that this area is proportional to this element, and that consequently in a finite time, it is proportional to the time. By integrating the last equation we obtain s ■=. y. sin. (u— 8),t- * [f\ = _^_, /^l=_Z^3, <^m =0; therefore if these values ofi— >,^— ^,^-r-^ be substituted in the equations (K); tlie second of these equations becomes rf'zi , dQ s clQ d^u fi , u.s^ d'u + " - -d^l — T- ^7r= rxi +""- " + •- ' rfu* ^ du u ds — ^^dv^ "^ ^Vf+T*" k^(l+s^)^ dv' j^ u L_ 5 J and the third equation becomes • ' - h^[\+s'^y- dv" U.US uus d^s , + • — :r: — . dv. r"-. cos. '« r: tbe element of the area described in a given ti»e by the projection of the radius vector ; see page i. rf'5 d'^s.ds .,„,,,. ■ ds^ , . X + i = ; ••• — ; h sds = 0, therefore by integrating —rrr + ** =: c, it dv^ dv' dv is evident that s = sin. v. or s = cos. ti, and that •.• s = a sin. v, or « = i. cos. v, and consequently s = a. sin. v-\-b. cos. v. will satisfy the given equation, and be its com- plete integral ; as it contains two independent arbitrary quantities. Now, a sin. v + 6. COS. V. may be reduced to the form y sin (v — 6), by assuming a — -y. cos. 6, b— — y. sin i, which gives a. sin. v + b. cos. ti = y. (sin. u. cos. 6 — cos. v. sin. f) — y. sin. (v'—e), and it may be shewn that y. sin. (d — 6), Ukewise satisfies this equation. It is also PART I.— BOOK 11. 99 y and being two arbitrary quantities. Finally, the seconi equation gives by its integration « =T^7rT-5T •(^l+«' + ^' cos- (^'-^) \ = \/l+s* r .* e and isr being two new arbitrary quantities. By substituting in this o 2 evident, that s = a. sin, (d— 6) -[- o. cos. (u— ^) will satisfy the equation - + * z: 0, and may be used when convenient, but in this case a, h and i, must be selected in such a manner, that they may be reduced to two independent quantities. * In the equation -j-^ + m — ,,,,'" ,^ 3 , let P = '^ •% , and m = a. sin. (u — (l)-4-5. cos. (« — <) will be the complete integral of the equation -7-5- -|- «= 0; and a sin. (v — d) and 6. cos. (v — 6) will respectively satisfy the equation ——^ -f u — ; now if the expression a. sin. [v — 6) + b. cos. (t) — S) be regarded as the integral of the differ- ential equation — ; \- v — P = ; a and b must in this case be functions of the va- dv riables v, and as there is only one equation to verify by means of a and b, we can impose certain conditions on them whicli will facilitate their determination ; supposing them to be functions of v in the equation a = a. sin. (v — «) -f- *• cos. (u — 6), we shall have du =■ adv. COS. (y — S) — b. dv-sm. [v — 6) ■\- da. sin. (u — 6)-\- db. cos. (u — f); but as there are two quantities to be determined, and as the proposed question furnishes us with but one condition, we are at liberty to select the other condition ; for this pvir- pose let da. sin. (u — () + db. cos. (v — 6) = 0; then duz^ dv. {a. cos. (u — 6) — 5. sin. {v — 6)) ; and consequently, d'u=^ — dv'^. {a. sin. {v — f) -\-b. cos. (« — t)) -}- dv, da. cos. (v — I) — dv.db. sin. (y — 6) ; and this value of d'u being substituted in the equation — -— \- u — ,— — -3 gives, dv^ A^(l4-s^)T adv'' . (sin. (u — f) — sin. {v — 6)) +Wd^. (cos. (d — 6) — cos. (u — *) ) + da-dv. cos. (y — i) — db.dv. sin. (u — 6) — Pdv^ =0; ••• da.dv. cos*, (u — S) — db.dv. sin. (v — «). cos. (w— ') 100 CELESTIAL MECHANICS, expression for u, in place of s, its value in terms of v, and then sub- stituting this expression, in the equation dt — — — j- ; the integral of the resulting equation will give t in a function of v ; therefore we shall have V, u and s, in functions of the time. — P. COS. (v — 6). dv^ — 0; and if this equation be divided by dv, and then added to the equation da. sin*, (u — S) -\- db. sin. (u — 6). cos. (u — 6) =0, we shall have da z= P. COS. [v — 6). dv, of which the integral h a —a' -\-f P. cos. [v — 6). dv; in like manner if the same equations be respectively multiplied by cos. (v — 6), sin. (v — 6), we obtain by subtracting the second, divided by dv, from the first ; db= — P. sin. {v — 6) dv; and •.• hz=.b' — J P. sin. {v — i). dv. Therefore u = a. sin. (v — 6) -\- h. cos. {v — 6) = a'. sin. {v — 6) + sin. {v — 6) J' P. cos. (u — 6) dv. +6'. cos. (v — e) — cos. (v — i)./P. sin, (■a — i) dv ; a' and b' are the values of a and b when P — ; P = ;,,. . — r;3 = (by substituting for s^ its value) -: —. ~3, therefore 6m.{v — I) /P. cos. (v — 6)dv = ^ -.-i -. f ^ : 3 , but ^ '-^ ^ ' A' •^(l-f-y\sin. "(u— e))'' cos, {v — 6) dv _ sin. (m — i) sin. (ti — 9) •^ /i^(l-|-y=.sin.i(t)— «)^ ^ A*(l-fy\sin.^-(u— 9)t ' *"^ ' /j'(H-y'.sin.^(u— (i)i COS. (u — e\dv y^.sin ''{t' — fl). cos. (ti — ^^.dv , , . ^ : — —1. — ; — ^ = by reducmg to a cotn- ^^l+y^sm. ^(«— 0)^ A'(l +y-. sin. '(■!;— «))i ' ^ co%.{v—6).dv K.sin. (p— «) mon denommator 77-- — — 3 ; consequently A cos, (t) — 6).dv _ ^ sin, ^{y — 6) •^(l + y^sin. ^(v — «;)! ""F"* (14-y^sin. ''0^— «)^* — cos. (u — S). f P. sin. (u — 6). dv = , , .\ /. sin. (t) — d). dv , sin. (n — 6). dv -^.(cos.i.-^).fj,^^--—---,^ , ,r.if.^^_^^—-—.^, _ —1 cos. {v—6) 1 COS. (f— ~ (l+V^)* F(I+771iir>i::«)' ' 1+7'-" Ani+y^sin.a(x,_9))^ 1 sin, (v — 6). dv I ^ sin. (v—e). cos. ■'{v—e). dv l+y''h'{l-i-'y\sm.'{v—e))i i+ya-'i'- A"(l+y=, sin. =(v— «)) -^ : by reducing — —i fa'n-(t— ^) +y^ sin. (t,-0) (sin. ^(t^-Q+cos. ^(v-ll)).dv ^ + y'" A^{l+y^sin.*(«_(l)l PART I— BOOK IL loi I The calculus may be considerably simplified, by observing that the value of s indicates that the orbit exists entirely* in a plane of which y is the tangent of the inclination to a fixed plane, and of which 9 represents the longitude of the node, reckoned from the origin of the angle v. Consequently, if we refer the motion of m to this plane, we shall have s =0, and y = 0, which gives = — = -|^< 1-he. COS. (v — ;!r)>. This is the equation of an ellipse, in which the origin of the radii is at the focus : — rr- j-, is the semiaxis major, which we will repre- sent by a ; e is the ratio of the excentricity to the semiaxis major ; 1 sin. (u — (l).(l +y^). rfu sin. (v — 6).dv ' 1+yi* A'(l+ySsin. ^(u— «))4 A"(l-fy'.siu. ^(u— «))T ' ft. sin. {v — 6). fcoi. [v — 9). dv ft. cos. (v — (l)ysin. [v — S). dv /j»(l fy^ sin. '(v—6)f. h'(l+y\sm.''{v—6))i __ ftsiD.^(v—6) 1 ^ COS. "(u — e) ~ F(r+y^^'sin~HJ^— ^^ HV ' /j'.(H-y^ sin. i(v—6))^ ~ (sin. ''{v—S)+cos.^(v—e)-^y^.s\n.^v—e) _ (l +y ^. sin,^(v-«))^ (l+y)'./%»(l-|-y«.sin. «(d— 9)« " (l+y"). A« \i = f'- (i+y«)/^^ ' •'•'* = °'- sip.(v-e)+i'.cos.(.-^) + ^.-^ \;;;^,;^.^ , and as e'. d'u cos. {v — w) satisfies the equation -yT + " = 0, we may write this function instead of a', sin. {v — 6) -\- 5'.(cos. (v — C), and as e is arbitrary we can assume it equal to / — — , • e, by means of which the expression for u will assume the form given in the text. * y is evidently equal to the tangent of latitude, when v — i = 90, and consequently it is in this case equal to the inclination of the orbit ; and as sin. (v — 6) = — = s. cotan- y gent of inclination ; the orbit described must be a plane, for this equation expresses the relation between the two sides, and invariable angle of a spherical triangle. 10^ CELESTIAL MECHANICS, dv finally, tb- is the longitude of the perihelium. The equation dt zz -j—, becomesj by substituting in place of i/, \/[A. (l+e. COS. (y — •B-))^ Let us expand the second member of this equation, into a series pro- ceeding according to the cosines of the angle v — ■a-, and of its multiples. For this purpose, we will commence by expanding >—- , ^_ into a eimilar series. By makiag X = 1 + ^1-e^' we shall have l+e.cos. (t;-..) v'r=?tl+ ^. c^"— ^^-^ l+^.c-^'-'^-'^-^J '■ * — = r = -7; ; r: — — n -, — ^r- , ••• a = — • ; hence h =1 u ^(l-|-e. COS. (u — •a)) 1+e. cos.(u— ar) jit{l — e») t By reducing the coefficient cX—=^ ^ \a. the second member of this equadon to the same denominator, it becomes equal to 1— ^" (u— ar)7^/_l _(u_sr)V— 1) •1— e=.(l + x» + ^(c^ '4- c but c — c =2 COS. (v — to), •.• this second member = PART I.— BOOK II. 103 e being the number of which the hyperbolical logarithm is equal to unity. By expanding the second member of this equation, into a series ; namely, the first term relatively to the powers of c ~^' * and the second term relatively to the powers c~^~'°)'^—y^ , and then substituting in place of the imaginary exponentials their expres- sions in sines and cosines ; we shall find 1 1 \+e. cos. (w — w) y/i ^' (1 — 2x. cos. (u — c3-)+2a^ cos. 2(y — =r) —2a'. cos. 3,(u— w) + &c.) ; By representing the second member of this equation by ?>, and making q — — , we shall iiave generally, 1 x» g , ; and from the equation a = ■ , we obtain VI— ««)(14A=' + A. COS.(t)— ar)) (l-j-v/l_e^) ~ \Tl/, n? ) . and 1 + A' = S J=i-; •.■ by substituting for !->.*, andl+A« we obtain 2(1— g^+y/ l-e') ^ i 2.'/l_e"(l+^l— e").(l+e.cos.(u— w)) 1 + e. cos. (v—,,) * The expression of the first term gives the following series : the expansion of the second term gives making the factors of the same powers of a to coalesce in the two series, and observing . ^ i. ilv — a).) -' ""^'"^ '^ S'^^" '" t'le text. 104 CELESTIAL MECHANICS, e-'"-\dr^-^ ±______iJL (1-fe. cos.(w — •sr)) 1'2'3 m.dq^ in which rfg' is supposed to be constant, and the sign is + or — , ac- cording as m is even or odd. From this, it is easy to to infer, that if we make :!^ — (I gs^— I (1+e.cos. (t;— Tir)7 "~^ ^ (14-E^^\ COS. (v—sr) 4- -E^^^. cos, 2(i)— x^)+£(^). cos. 3(t)— TS-) + &c.) J we shall have, whatever may be the value of i. (l + \/l — e»>' the sign being +, if i is even, and — if i is odd ; therefore if n be ♦ Substituting — for e we obtain — 7; -. r =— '- = ip, :• ° ^r 1 "t^e. COS. (11 — st) y+cos. (d — ar) (9+COS. (u — w)) 9'" y + cos. (i; — w) ' " (^+003,(1) — w)' " 1 9 J ' iaiAd\\ — \=d- —■ -^dq= , % -, and di. 1^1 = [q) {g + cos.(v — T^y {ij-f-cos.(v — ar)-5 lyj =d.; ; rr .1- dg = ■ — ; -^ — • : hence generally we obtain d"' i — J- (y+COS. (e— sr))i • ^ (y+COS. (u— ar))+ ^ ' \ g j ^ ±. 1.2.3 w ± 1.2.3 me '^ 7W 4- 1 ?K 4- 1 (jr+C08. (v — ■a)) (1-fff. COS. (l) — to)) t Substituting — for e, in the value of ffl, we obtain— = .(1— 2>. cos. («—«;) + 2a'. ? ? •?»— 1 1 -2 f ■) cos, 2(i>— «7)— 2a 3. COS. 3(r— s-)+ &c.) v 77-; ,,, =e .d.{ — Vz^\he ^ ' ' (1 +e. cos. V — a)Y \ 9 J dq PART I.— BOOK II. 105 supposed equal to a '•v /*, we shall have ) ^ (5 . COS. (v — u7)-\-E COS. Z(y — zs) + &c.) ; (1) (2) (3 ndt = dv. (l-h- E . COS. (v — 3-)-f £ . cos. 2(t' — ■o-) + E • and by integrating nt + t=v+E . sin. (f — zr^ + ^.E . sin. 2(t; — w) + ^£ . sin. 3(v — -sr) + &c. s being a constant arbitrary quantity. This expression for nt-\-i is very converging* when the orbits have a very small excentricity, such as the orbits of the planets and of the satellites ; and we can, by the PART I. BOOK II. p preceding series differenced with respect to q, and divided by e' ; the differential of the — 2 , 1 ^ i „ — 2 2 terra = e —2 V 1 ^^—21 2e. — r^x3'— — . ±2e (y+^y^_l)H-l = by simplifying and reducing to a common denominator, — °* ■ —, ,• , which becomes, by substituting — for o, , 2-»_y^ 4c.) the value of 17, differenced with respect to x, y, 2, successively, for 1^ = - f • -0+73 -^^^ + Ti (£i/*+^«+2Gx) = _ -^. (2).(y'+z»)-%x - 116 CELESTIAL MECHANICS, consequently we can obtain, by this means, the values of U, U', U" ; and the equation V zz constant, will become const. = — 4- {Dx+EyJ^Fz+2G)^{A-\-Di/—Ex').{xy'—ya:')* ■ + {B + Dz'—Fs').(xz'—za:') -i- {C-\-Ez—F>j').{yz'—zy') This equation satisfies the equation (I), and consequently the dif- D.xy-F.y.-1Gii), ^ = - ■^- F.^. ^ . Fz^+ ^ (Dxz+Eyz+2Gz) = - fr- { J (.T*4-y*) — -Dx2 — Eyz — 2Gz), ••• if these equations be multiplied by :c', y', J , respec- tively, the sum of the terms at the left hand side will be equal to d\], and the sum of those on the right hand, will coincide with those already given. * This equation evidently satisfies the equation (I), for ir = _il. D. + — . (,D.v'+Exy+Fxz4-2Gjc)+Ay+Dy"—Ex'y'+Bz'-^Dz-—Fx'z' dx r r^ ^=—^,E + -^.(Ey''+Dxy-\-Fyzj-2Gy)—Aj^—Di/'se'^Ex''-iCz' + Ez'''—F,/:^, dy r r^ '^Z.= .-fl.F+—. (Fz^ + Dxz4-Eyz-{-2Gz)—Bx'—Dz'x' + Fx"—Cy'—Ez'y'+Fy'^. dz r r^ , dV , , dV , , dV dx dy dz _JLm)f-\-z^)—Exy—Fxz—2Gx).x'.+Ay'JJrDy'^-v'—Ex''-y'-\-Bzx^Dz'^x'-F£'z', _ iL. (£(xH2" )-Dxy-Fyz—'iGy),/-Ay'x'-Dy'^x+Ei^^y'-\- Cz'y'^Ez'iy'-Fy^'z' , — ^. {F{x^-^y^)-Bxz -Eyz->f2Gzy-Bx'z'-Dz''x>J[. Fx'^z'—Cy'z'—Ez''y'+Fy"z: = by obliterating the quantities which destroy each other _fL.(Dh/''+z')-Exy-Fxz-2Gx)x'J^E{(x^^z')-Dxy-Fyz-2Gy)y'^F{x^^y')-Dxz-Eyz -2Gz)z'; '^=—E{xy'—yz')—y{A + Dy' -Ex' ]—F{xz'-zx')-z{B+Dz'-Fx-)+2Gx, PART I.— BOOK II. 117 ferential equations (O), whatever may be the arbitrary quantities J, B, C, Z),. ^. F, G. Supposing them all to vanish first , with the exception of A ; 2dly, with the exception of B ; Sdly, with the „ „ . , . . dx dt/ dz . , c > ' , exception of C, &c., and restoruig — — , -^ , -r:, m place oix,y, z', dt dt dt we shall obtain the integrals c = xdy—1/dx ^_ xdz — zdx _ „ _ydz--zdy__ \ dt c'zz dt ; C" - dt n-f.^S'^ (dl±dz-)l ydxj.dx zdzxlx .. "-•/+'^-\7 -Of )■+ de ^ df ' f* {da^^dz^) \ xdx.dy zdz.dy \ S^ dl^ "^ df ' I a~f"j.. S ^ {dx^^df) ) xdx.dz ydy.dz^ _ "-■/ +^-|7 2f > dt^ ^ de ' a r dt- (P) / c, c\ c'', f,f',f", and a being constant arbitrary quantities. '^-L = B{xj/-t,jf)-^x{A+D^--E=if)-F{y^-Z!/)-z{C+E^-Fy')+'2Gy', ^ = D{xJ-zxf)-\rx(B^D^—Fx-)J^E{y:^-Z!/)+y{C^Ez'~Fy')-^^Gz', ' Multiplying these three equations by x, y, z, respectively, and observing that those terms, of which one factor is the product of two of the coordinates, x, y, z, destroy each other, we obtain, by concinnating —r-;X-{ — tt- V + -r-r- z^ — £(x'+z^)v' — i)(y^+3*U'' — ■' ^ dx dy' -^ dz ^ ;,y w , Fiy'' +x''y+E{yx)-\-Fxz+2Gi)x-^(Dxy+Fzy+2Gy)iJ+{pxz+Eyz-^2Gz)z', and It dV this expression, when multiplied by -^ is identical with the preceding valueofj:'-^ \- , dV , dV .V--T- + ^ dy dz * Supposing all the constant quantities but A to vanish, the preceding equation be- comes const. =^A{xy' — ^x') ; supposing them all except D to vanish, we shall have const.= 118 CELESTIAL MECHANICS, The differential equations (O) can only have six* distinct integrals of the first order, by means of which, if the differences dx, dy, dz, be eli- minated, we shall obtain the three variables %, y, z, in functions of the time t; therefore one at least of the seven preceding integrals should occur in the six others. We may perceive even, a priori, that two of these integrals must occur in the five remaining. In fact, as the sole element of the time, occurs in these integrals ; they are not sufficient to determine the variables x, y, z, in functions of the time, and conse- quently tiiey are inadequate to the complete determination of the mo- tion of m about M. We proceed to examine how it happens that these integrals are only equivalent to five distinct integrals. zdii^^v d '*' If we multiply the fourth of the equations (P) by — , » ^^^ then add it to the fifth, multiplied by — ; we shall obtain A- r i^dy—ydz) {xdz-zdx) {xdy—ydx) ^-J' It ^^ ' dt + ^' dt C|iA (dx^ + dy'^) } (xdy — ydx) f xdx.dz ydy.dz\ t (T If i"** di \~~dF"^~~df~S' yy.J— zzV which will be equal to the fourth of the equations (P), by substituting for x', y, ^, their values. Supposing G to be the only constant arbitrary quantity, we ob- lu. , , . const. u, , , . . tain, const.= G { — + (^"+i/^-\-'^-))\ ■•' makmg — - — = -i- , and substituting r for x'. y, j/, we obtain the expression given in the text. * As the differential equations (O) are of the second order, and since the complete integration of each equation furnishestwo constant arbitrary quantities, the entire number cannot exceed six. f Performing this multiplication and addition, we obtain ft. (xzdy—xydz)—xzdy^ —xzdy.dz''+xydy-.dz-\-xydz- , zydy^ .dx-\-zldxdydz PART I.—BOOK 11. 11 T, 1 ^v x- • 1 r z'hi~—udx xdz — zdx ydz — zdy , . By substituting m place of — - — -^ — » ; 1- — their ' 6 f ^; dt dt values, which have been determined by the three first of the equations (P), we shall have — f^'—f^" -L o. Sit _ C^ljh^l X . ^^f^^dz ydy.dz c '^ Ir I dt' S)'^ ~dF~'^~^t''~' This equation coincides with the sixth of the integrals (P), by making ./' = LE^ZiJL, or 0=fc"—f(f+f'c. Thus the sixth of tlie integrals (P), results from the five preceding, and the six arbitrary quantities c, d, c'', f, J", f", are connected together by the preceding equation. If we take the squares of the values o? f, f, f", which are deter- mined by the equations (P), and then add them together, we shall obtain — y}.dx.dy.dz — yz.d*z.dx fi j t f^ (xy.dz — t/zdx) — yxAx^dz — yxdz^ yz.dx^ + yz.dz'.dx x^.dx.dy.dz-\-x.zJz'.dy xz.dx'.dy z'.dx.dif.dz + ^3 + -dT^ df^ d^ =by mak- ing factors to coalesce-/c'^i/'c'+..^ iffc^ _ z. ±^ (J^l+^ ^ ,. fd_y ^ -^ ^ ^ r dt dt dt" ~ dt {dz^—dx'^)xy.dz(dy^ + dz-) xy.dz (dx^+dz^) _ y.dx {dy^~ dz^) y.dx ^^ '^'Yt d? dT dt' + ^" dt df" "^""ir * {dz^+dx') , , ,^dx.dydz , , . dxdy.dz „ . . , ., -^: ^^ i- + («'— ^') — ^ij— + (x'— «'). j^ — = after all reductions, and obli- , . , , , , , (zdy — ydz) (xdz — zdx) su terating quantities which destroy each other, j — r/ • j, 1- — ^. (xdy — ydx) (xdy—ydx) {dx^ + dy"^ ) xdy ^ ydy.dz xdxJzl ydx ~^t ^"^^ 1? ^~dr'i de + dt'' s~~dr' € xdx.dz , ydy.dz) ),.,., ... < — — — V , 1 i which IS the expression in the text. 120 CELESTIAL MECHANICS, /~K=(^- 1 a? S~lTtS \' I d? T> in \vhiclW= is, for the sake of abridging, put equal to /'' + /* 4-/"% but if we take the square of(, tlie values of c, c, c", which are given by the same equations, and then add them together, we shall have, by making c + c' +c = n ; *•>'"— ~7r- + ^ ' dt^ r ' dt' rfdy'^.dx^+z''dz-.dx^ 2yz.di/.dz.dx^ l^x (ydy.dx-\-zdz.dx) ^ (dy*+dz^) + '' di^ *■ dt* + r dt' ■^' ^i" (ydy.dx^zdz.dx ) .„_ j^V, , (.dz') {x +y +z% ■■ ~ X . ( dx^+^dx^dy^-^-^dx^'.dz^ ) , ( rfi/*4-2^■^'■rfy'+2%^f/^°) _^ (dz*i-2dz^.dx'-^2dz--d,,') dt* ^ ' dt^ " ' df- dx-dy ,^ dxdz , ^ dii.dz idx'^A-dy^-X-dz') 2/* , , , ((/j'+rfy''4-&') 2^ (■T^f/c''.4-,yVv'+zV/;:-42-r.y-c?-r-"'.y) (2xz.dx.dz-\-'2yz.dy.dz ) dp •" r M* "^ dt'' , , (du\dx''+d!/\dz') '(dx\dy'+dx\dz') , , {dz'du^+dz\dx') „ ^ literating tlie quantities which destroy each other, and observing that r''dr^ =x' this equation determines the position of the axis major of the conic section. * Substituting — xdx ior ydy + adz, and —ydy for xdx -{-zdz in the two last terms of the second member of this equation, and they will become .-. multiplying the first by Y, and the second by X, and then subtracting, we obtain the expression given in the text. / PART I.— BOOK II. 127 If by means of the last of the equations (r), -tjz be eli- df minated from the equation r . — ,:^ ro-= « > we shall '■ at' dr obtain ~di ^.r-J^-l^^^h- but dr vanishes at the extremities of the greater axis ; therefore at these points we have, The sum of the two values of r in this equation, is the axis major of the conic section, and their difference is equal to twice the excen- tricity ; thus, a is the semiaxis* major of the orbit, or the mean dis- tance of m from M ; and v 1— is the ratio of the excentricity to the semi-axis major. Let e represent this ratio j and by the pre- * The coefficient of r with its sign changed is the sum of the two values of r, and their difference is equal to twice the radical, and •.* = to 2 a. y 1 , and V-'^ \/ 1 is the ratio of the excentricity to a ; \/ »« — -^-— =s fte fi. \ ft • r = ;«'e* = i' ; c?r = ae. sin. udu, •.• rdr = a\e. sm. udu.{l — e cos. u\ 2r = a.((2 — 2e. cos. a)— ( 1 -}- e* . cos. * « — 2e. cos. «)) := a.( 1 — e » , cos. *u), and V 2r a.(l — e') = ac^.(l — cos. *u) = ae*. sin. *«, and therefore rdr , , a'-.e. sin. u.[\ — e cos. m) du a^ ; _ __ - {~dt\ = = — -. V^.\j2r-rl.-.a.{X-e^). •^^/«^sin.^« V'^ (1 — e COS. u)du. 128 CELESTIAL MECHANICS, ceding number, we have a ~ k' ' therefore f^e ~ I. Thus, we can know all the elements which deter- mine the nature of the conic section, and its position in space. 20. The three finite equations found in the preceding number, be- tween ,r, 7/, z, and r, give x, y, z, in functions of r ; thus, in order to determine these coordinates in a function of the time, it is sufficient to have the radius vector r, in a similar function, which requires a new integration. For this purpose, let us resume the equation a ar by the preceding number, we have, therefore we shall obtain rdr dt = \/ju. \jlr~- a.(l— e°) In order to integrate this equation, let r ■=. a.{\ — e. cos. w), we shall have at = — -pz^. (I — e cos. U), from which may be obtained by integrating, t -^ T =■ —p.' (w — e sin. u) ; (S) T being a constant arbitrary quantity. This equation determiuM u. PART I.— BOOK II. 129 and consequently /• in a function of ;; and as x, y, z are determined in functions of r ; we shall obtain the values of these coordinates, for any instant whatever. We have thus completely integrated the differential equations (O) of No. 17 ; this integration introduces the six arbitrary quantities a, e, I, 6, (^, f, f, &c.) } we shall have, by what precedes, ,. _5 /+"'-^""^'^".Ku'.r.&c.) I ^"■'-•""' •" 1 1.2.3... «. 1.2.3...n'. 1.2.3...n".&c.i If M is a function of t and a, only, we shall have ^'' ~ 1.2.3. .M.dr' therefore K^+.)_,(/)+— ^^+_-^^^+^— ^.^-^ + &c. 0) Let us in the next place suppose that u, instead of being given im- mediately in a and t, as in the i^receding case, is a function of x, x being given by the equation of partial differences, i-j— c — •^* )1~[ » Cdx J Cdt J in which z is any function whatever of x. In order to reduce u into a series proceeding according to the C d^u") powers of a, the value of } — -^\^ must be determined in the case in which ji=:0 ; but in consequence of the proposed equation of partial differences, we have \din _ d.fz.du ,^ (Jc) \- dt ' 134 CELESTIAL MECHANICS, fc5-terW~~'w*i dt r therefore, we shall have (dtci _ ldZ\~ This equation being differenced with respect to «, gives c d'u 1 _ d-.fz.du \'dJ'\~ do^.dt ' but the equation [k) gives, by changing u into fz.du, (.d.fz.du} _ ^ df^.du 1 id^VX dt y consequently [c?^M> d-.fz^.du ldA~ df ' This equation being differenced again with respect to a, gives id^Wi _ d^.Jz-.du w\~ dx.df ' but the equation (k) gives, by changing u into ^Vm . . du' du dx du' dx dii fz.du' * Let /.rf« =u', then -^ = -^. ^=--^-^ ^-rfT = '■ "1^- '^'^'^ '^ . d. fz.du _ d. Jz-.du substituting for du' its value, we obtain — ^j — . f As the characteristic/ indicates an operation, the reverse of that denoted byrf.we can remove the sign f, by depressing the index of d by unity. PART L— BOOK II. 135 therefore Ldx'\~ I df~y By continuing this process, it is easy to infer generally SdHi) U':fz\dzn (d''-\z\\-]\ id^^\=l—dF-r\ — r-^r Les us now suppose that by making «=0, we have x zz. T, T being a function t ; we shall substitute this value of x, in z, and in u. Let Z and u represent what these quantities then become ; we shall have on the hypothesis that azzO, ^d'tn di teS ~ dt^'~ ' and consequently, by what precedes, we shall obtain, - which gives . = u + «.Z._ + — .^.J^^+^^.c/^)^(+&c;(P) It only now remains to determine what function of t and «, x repre- sents ; which will be effected by the integration of the equation of partial differences jT-f^^'jjTf- For this purpose, we shall ob- serve, that rf^ = {§}.rf^4-{|}.^«: 136 CELESTIAL MECHANICS, and by substituting in place of < -r^t its value ^' ■< -7- r we will ob^ tain therefore, we shall have dz\ — -. <:f.(/+a3) 1 + \dxy Idt} which gives by its integration, x =^(t+a,z'), (i>(t-\-a.z) being an ar- bitrary function of t-\-o(,z; so that the quantity which we have termed T, is equal to qi^t). Consequently, as often as there exists between a. and X, an equation reducible to the form x = (p(^t+ az) ; the value of u will be determined by the formula (P) in a series proceeding according to the powers of «. dz * zd»'=dicz — a. —7-. dx, therefore, by substituting this value of zd», we obtain the dx expression for dx given in the text; now as dx is an exact differential, the member, at the dx right hand side of the equation must be also an exact differential, consequently, —r- •— I !-)-«. •—3—) > must be equal to (^(t-\-itz), ip' denoting the derivative function \ dx dt ' of (p. 2 being by hypothesis a function of x, let it equal F(x) and we shall have x = ^{t ■\-aF[x)), and it is easy to obtain from this expression the proposed differential equa- tion of partial differences, for dx_ da = ?'(<+«FW)- ^ (F(x))^»F'{x\^^ ^ -^ = «''('+«-fW) { ^+''-^<^)-^^ = and by eliminating '(t). F; -— = -7-. — , + — . -J-;, and when a, a =0, rf« c?« rfar d* dx da dx da, dx daf . , » „ , du du , , , „ _ , , , dx , ,, . dx . . ^ = ^/^. F= *i. F; ^= 4^1, ^. F=^, F, .-. by substituting z for da dx dt dt ' da' d»'- df ' dt ' ^ F we obtain — — 2. —r- = when x = ;7- f » ^^ shall have on these suppositions, and consequently, \d"'.u} \d«.\dx"''S ( dx S' dt" We shall have in like manner. m-{'-m\ df If we suppose a' to vanish after the differentiations, and if besides we suppose that in the second member of this equation, x' — »J/(/'+a'y»'); we shall obtain dr-\di" 140 CELESTIAL MECIHANCS, provided that we make a. and a' to vanish after the differentiations, and also that we suppose in the second member of this equation a; = ?.(^+«^") ; x' =^(lf-\-ai.'z"^)', which comes to supposing in the second member as well as in the first mem- and to change in the partial difference ] ;( , of this second c dx.da. J ber z into z", and z' into z''^. Thus, we shall have on those suppo- sitions, and also by changing z into Z, s/ into Z', and u into u, C d" -"'-"-. [ -^^^ } - __< l.da..da.') y ( 1.2.3 n. 1.2.3 n'.dr-\dt"^'-^ J ° By following on this reasoning, it is easy to infer, that if we have r equations, x"=n(r+o^"z")i &c. z, z', z", &c., being any functions whatever of x, x', sf', &c. j u being supposed to be a function of the same variables, we shall have generally t n+n'+n"+&c.-r^ C d'U ? ^ _-^ Xdx.da.' .d! a.' . &C.5 r" ^". »' « ' &c. - ( 1.2.3.. .w.l.2.3...?z'.1.2.3,..n".&c.c?r-'.rfr'-'.c?r.""-'' ^ c^-^M provided that in the partial difference < - — r? - y - v. o — f » ^^e change Ldx.da. .da. . &c. J z into s", e' into z'"', &c., and that afterwards we change z into Z, z' into Z', 2* into Z", &c., and m into u'. PART I.— BOOK II. 141 If there is but one variable x, we shall have CrfM? rdul therefore ■dt ^■(- 1^\ \ 9" = 1.2.3 n.dt"-'^ If there are two variables a: and x' ; we shall have this equation differenced with respect to «', gives but we have < — ; >■ = s'. j -7- f 5 ^^^ ^^ i" this equation x is sub- stituted in place of u, we have < — ? =2'. ^ ;7- c ; therefore S (Pu 1 _ i'^'^'idFsK , ^dz) idu) idZd:^^- -• I — irS ^^'\d'i;^'\di^' * By substituting 2" for z, &c. we have made the coefficient , ,, , gn,n to ae- pend on a coefficient of the second order, and the ilifFerentiations relative to t and t' will not be difficult when «, «' are = to cipher. du , du du'' , , , 4 du > ; 2v/=T j (2 j' sin i being any number whatever. If we expand the second members of these equations, and then substitute, in place of c'^"*V— • jjjjjj ^f c—rnt.^—i^ their values cos. rnt. + ^IITsin. mt.^Hl, and cos. rnt. — ^ ~y Sin. rnt. »/ —i, r'being any number whatever ; we will obtain the i powers of sin. nt and of cos. nt, evolved according to the sines and cosines of the angles nt of its multiples ; this being premised, we shall find sin. nt + ■—. sin. ^«?+ ___, sin.'«/+ 7-r^ • sin. *nt + &c. g = sin. nt — ;; ("cos. ^nt — 1 ) 1.2.2^ e* — 5 (sin. 3nt — 3 sin. nt) 1.2.3.2^ ^ ^ ef 4 3 \ H ^ ^ . ^, (cos. 4«/ — tcos. 2n/ + i. —1— ) 1.2.3.4.2' ^ ^ 1.2 y "^ 1.2.3^4.5.2^ '^^'"' ^"^"^ ^^"' ^"^■*' Ti" • ^'"' "^^ — &c. * SeeLacroix, Traite Complete, Tome 1, page 76, 95, of the Introduction. 144. CELESTIAL MECHANICS, Let P* represent this function ; if it be multiplied by ^{''("O ^^^ then if each of its terms be differenced, with respect to /, as often as there are units in the power of e, by which it is multiplied, dl being supposed constant ; and if then these differentials be divided by the corresponding power of ndt, the formula (9) will become 4/M = 4/(nO + eF). P representing the sum of these differentials thus divided. * The series P is always the same where the equation m = n/ -f. e. sin. nt obtains, whatever be the form of the function indicated by 4' ; therefore when the form of -.^ is given, the expression for ■^/(m) will be obtained by performing the operations indicated in the text. When the value of P, is multiplied by e. cos. nt, the form of the terms multiplied into the even powers of e, will be cos. ;'. 7it. sin. i. nt, and the expansion of this product is effected by the formula sin. a. cos. b = sin. ^ ^'"' •, therefore the terms multiplied by the even powers will be the sines. The form of the terms multiplied into the odd powers of e, will COS. in. cos. snt the developement of which is effected by the formula cos. a. cos. b = ' ' "*" — ^'^ ~~ i ^ consequently the terms multiplied by the odd powers of e will be the cosines. If any term of the form Ke^'. sin. snt. be differenced as often as there are units in 2r, it is evident that when this terra is divided by ndt)-'; the result- ing terms will be Ke^'. s'"'. sin. snt, for as the terms are alternately cos. snt, sin. snt, when the number of differentiations is even the last term must be sin. snt, and as « is in- troduced as a factor at each successive differentiation when the number of differentiations is 2^, i^"" will be a factor of this last term, the first term is -f cos. int, and the signs of the subsequent terms are minus and plus in pairs, .*. the signs of the successive differential coefficients including the first, are plus minus, minus plus, plus minus ; i. e. + — , — -J-, -f — , &c. ; hence it appears, that when r is an odd number, the sign of the last term will be — , and when r is an even number, the last terra will be -f- . In a term of the form of the Ke'"-*-'. cos. snt the number of differentiations being odd, the last term must be of the form Ki''+^. s^''+^. sin. snt, the signs of the terms in this case are alternately minus and plus in pairs, i. e. , -f +, , -f- -f-, and as the sign of sin. snt, is the opposite of the sign of the penultimate terra, when r is even this agn is fvidently — , and when r is odd this sign is -{-• PART I.— BOOK II. 145 It would be easy by this method to obtain the values of the angle u, and of the sines and cosines of this angle, and of its multiples. If for example, we suppose »J/(w) = sin. iu ; we shall obtain vj/'(wf) =:i cos. int. The preceding value of P, must be multiplied by i. cos. int, and the pi'oduct should be expanded into sines and cosines of the angle nt, and of its multiples. The sines will be multiplied by the even powers of e, and the cosines will be multiplied by the odd powers of e. Then any term of the form Ke . sin. snt will be changed into ± Ke^''. s^^, sin. snt, the sign + having place, if r is even, and the sign — obtain- ing, if r is odd. In like manner any term of the form Ke "^ . cos. snt. will be changed into T Ke "^^ . s^^'^^ . sin. snt, the sign — having place if r is even, and the sign -f- obtaining, if r be odd. The sum of all these terms will be the value of F, and we shall obtain sin. iu = sin. int + eF. If iJ/(m) be supposed equal to u*, ^'(nt) will be equal to unity, and we will find uzznt-\-e. sin. nt + . 2 sin. 2nt A -,.("3*. sin. Snt — 3 sin. nt) 1.2.2 ^ 1.2.3.2 V + - — -; -,.(4^.sin.4n/— 4.2'.sin.2«/) 1.2.3.4.S' ^ + ■ — -J (5*. sin. Snt — 5.3 . 5.4 sin. 37it+ —V. sin. nt). 1.2 PART I. BOOK II. V * If ^^'{a) = u, then -^(nt) = nt, and ^'(nt) =: — r- =1, the series P' becomes sin. nt ndt e , COS. (2«f— 1) e' (sin. 3nt — 3 sin, nt) 1.2.2 ndt 1.2.3.2'' ' (ridt)'- 1.2.3.4.2' 4 3 (cos. lent — 4 COS. 2nt+i. — ^ <^'. ,.^ '— + &c. which will be reduced to the expression in the text, by performing the prescribed difFerentiations. 146 CELESTIAL MECHANICS, This series is very converging for the planets, u being thus deter- mined for any instant ; the corresponding values of r and v, will be given by means of the equations (y) of N°. 20 ; but we can obtain these last quantities directly in converging sei'ies, in the following manner : Eor this purpose, it may be remarked, that by No. 20, we have r = a(l — e cos. ii) ; and if in the formula ((/), we suppose ^(u) zz 1 — e. COS. u, we shall have ^\ni) = e. sin. nt, and consequently . , " • 2.1 e^ d' sin. ^nt , "* I — e. cos. u-=zi — e. cos. nt-\-e-. sin. ^nt-r . }- 1.2 ndt 1.2.3* d"-. sin. ^/2; + &c. n\de Therefore by the preceding analysis, we shall obtain r c 6" — = 1 + e. cos. nt cQ!i.2nt* a 2 2 •———7,(3. COS. 3nt — S. COS. nt) ,.(4'. COS. 47it — 4.2". COS. 2nt) l.'i.i).'2 5.4 — : — 4.(5^ COS. Snt — 5.3 . COS. 37li-{ . 1.2.3.4.2 ^ ,1-2 COS. nt) * Since ■4'(a) = 1 — e. cos. u, -^{iii) = 1 — e. cos. ;?;; by substituting for sin. ^w<, sin. ^nt &c. their values, the expression for 1 — f. cos. ic becomes 1 — e. cos. nt-\ • (1 — it , . e^ , { — sin. 3Kf + 3 siu. ?;i) e* (cos. 4«f — 4cos. 2«* + 3) eos. Int)^ — . d. -^^ + -^^.d .- ^^-^^ ±J + &C., now wlicn the differentiations indicated by the characteristics ^-c I ,, 1 i-x.c«-^- r and consequently, log. CI — A. c ) — log. (1 — X. c" J V — 1 from which may be obtained, by reducing the logarithms into se- ries,! c'-'^-'+l 1— AC ^ 1— '^c ^ c + 1 _ i_e+/i-e^ _ ^YUe. ^^^-g+J^+ f), ... by substituting these values of l+> 1+^1— e^ ~ 1 + yi-e^ and 1 — A, we obtain t Log. c"'^^ - vV^ = log. c""^~'+ log. (1 - A.c~"'^^)- log. (1-A. .«V-X)^„.^Z:] ^ log.(l-A.c-"-v^-' )_log. (1-A. c"-^^) ; log.(l-A.c-"^^) = _A -^■'_£. -2«V-_ xi_ r^«-^-_&e.-.log.(l_.).c«-^"* 12 3 & V / = — . c ^ + ■^- <= + -^ • c + &c.; .-. log.(l— A).c — log. (1— A).c =Y. (c -c )+-2-'^* " '"^T- PART L— BOOK II. 149 V = u + 2x. sin. u+ . sin. 2u-\ . sin. 3m + . sin. 4m + &c. 2 t 4t by what goes before, we have u, sin. u, sin. 2m, &c. in a series arranged according to the powers of e, and expanded into sines and cosines of the angle nt and its multiples, therefore in order to obtain v expressed in a similar series, it is only necessary to expand the successive powers of / into a series ranged according to the powers of e. The equation u — 2— — , will give by the formula (p) of the preceding number, JL__L + i:^ 4. ML+Jl _£L + i'(i + 3Xi + 5) _£l . &, . M' ~ 2' 2'4-2 "^ 1.2 * 2' + * 1.2.3 •2^4-6 ^ « ., and as we have, u •=. 1+v 1 — e* i we shall have This being premised, we shall find by continuing the approximation to (c "' "' — c "' "') +"&c., V dividing by v' — 1, and substituting 2v^-l sin. . sin. nt; the factors of sin. 2nt are -—- 4 f- — - -|- — ^ -f- -«-+ T- + IF + -5 r - -5- ' &'=• See page 145. 1S2 CELESTIAL MECHANICS, If instead of fixing the origin of /, at the moment of the passage through the perihelium, we fix it at any instant whatever ; the angle nt will be increased by a constant quantity, which we will denote by t — Ts; and consequently the preceding expressions of — , and of v will become 1 +ie*— (e— |- e ). cos. (ntA-i—^)—{^ ^— 1 e* ). r ~a ' ' ''' ^ 8 cos. '2..(nt + £— xir) — &c. V = nt-t i +(2e— -e'). sin. (nt-\-i-^)+ (— e'— — c*), sin. 2(nt + t — -sr) + &c. j V is the true longitude of the planet, and nt + t is its mean longitude, these two longitudes being referred to the plane of the orbit. Let us now refer the motion of the planet, to a fixed plane, a little inclined to that of the orbit. Let (p represent the mutual inclination of these two planes, and 9 the longitude of the ascending node of the orbit, reckoned on the fixed plane j let 6 be this longitude reckoned on the fixed plane of the orbit, so that 6 is the projection of 6 ; also let v, be the projection of v on the fixed plane. We shall have tan. (f, — 0) =cos. (p. tan. (u— §). This equation gives v, in terms of u, and vice versa ; but we can have these two angles, each in terms of the other, in very converging series, by the following method. The series X* x' ^v = ^M+A. sin. u -f -— -. sin. 2m + — . sin. 3«+ &c. has been already deduced from the equation PART I.— BOOK II. 153 by making tang. i» = V -—- • tang. \u. ^ 1 —e ^yTTe l—e + 1 yl + e into cos ip ; we shall have COS. a> — 1 *1 ^ — TT- — — tan. -r- 9;T COS. (p+l a ^ ' the equation between i^v and i^ti, will be changed into an equation be- PART. I. — BOOK II. X (7.(1 gi) • By making e negative in the equation r = ■— i , v will be equal to cipher, 1 -re. COS. V ^ ' when r = a.{l+e), i. e. at the aphelium, •/ it is from this point that the angle v is reckoned. Since the results must be identically the same, when v is reckoned from perihelium and aphelium, and since the signs of the odd multiples are necessarily changed, in order that these expressions may remain the same as before, the sign of the factors which multiply these odd multiples, must be changed at the same time, i. e. these factors must be odd powers of e. t 1—2 sin. -— ip= cos. ip; 2 cos. -— COS.

> ^ COS. ^^v. < 1 -i tan.-^i;^ which gives, by reducing into a series, r= — rr- i 1 ~- tan. %v+ < -^ > . tan. *^f— &c. \ COS.^^W(. 2~x ^ (2— a) ^ J In order to have the ratio of v to the time t, we will observe that the expression of the arc in terms of the tangent, gives tt = 2. tan. ^u. (1 — J. tan. %u+^. tan. % u — &c.) ;t but we have tan. 1m = V J—— • tail' k"" 5 therefore we shall have uzzl.sl^^ tan. It;. 1 14(^)- tan.^i- t; + i (^)'.tan.«4i'-&c.) * «'=«* — 2<»+ 1 : vr_ i_j_2. COS. -iu— 1— 2«. COS. »iti+* «.a.(2— «) J-(2— «) 2. COS. HH^l— 2 COS. "if) 2. cos, 4u— «. cos.,*ir+«. sin.^^w _ ■^•'" "^ • dividing the numerator and denominator by 2— «, we ob- ■~ COS. ^^1.(2— «) + «. sm. ^\u ' ^ tain the expression in the text. PART I.— BOOK II. lik we have likewise 2 tan. Am "*"■ "*- l+tan.^i^" " ^' ^^"* 2"-Ci— tan. '^w+tan. ♦iw— &c.) ; from which may be obtained e. sin. M = 2(l — «)• v 5 — • **"• i'^O ^ t^"* 'i*^ + {^-^}ltan.>-&c.( . a tan. *^w+ &c.^. 2. sin. iu 2 tan. J u COS. iM 2 sin. iu. cos. |u l-f- tang. ^|u sin. ^\u sin. 'iu-j-cos. *|m ' COS. '^\u _'J~ D .. _ «T\/|M u — e. sin. u _ dI ^ 2 , if the parts which destroy i each other in this expression be obliterated, and if a^ which occurs both in the numerator and denominator, of the part which remains, be likewise obliterated, the resulting quantity will be value of t given in the text. • It appears from this value of t, that the times in which different comets moving in pa- rabolick orbits, describe equal angels about the sun placed in the focus, are in the sesqui- plicate ratio of the perihelium distance. See Newton, Prop. 37, Book 3, and also No, 27. BOOK I.— PART 11. 159 sun and earth ; in place of which, in the second case, it expresses the sum of the masses of the sun and comet ; but the masses of the earth, and of the comet, being much less than that of the sun, they may be neglected, and we may suppose that ^ is the same for all these bodies, and that it ex- presses the mass of the sun. Therefore by substituting in place of \/^ its value -— in the preceding expression of ^ ; we shall have = -7=. (tan. \v^\. tan. %v), TT.V 2 t = This equation contains no quantities which are not comparable with each other, it will easily determine t, whenever ^ will be known ; but in order to determine y, by means of t, we must solve an equation of the third degree which admits of but one real root. We may dispense with the resolution, by making a table of the values of v, correspondino- to those of t, in a parabola of which the perihelium distance is equal to unity, or equal to the mean distance of the earth from the sun. This tabic will give the time which corresponds to the anomaly v, in any parabola of which Z) represents the perehelium distance, by multiplying by JD^, the time which answers to the same anomaly, in the table. We shall obtain the anomaly v, which answers to the time, by dividing t by Z)^> and then seeking in the table, the anomaly which answers to the quo- tient of this division. Let us now suppose that the anomaly v, which corresponds to the time t, in a very eccentric ellipse, is required. If quantities of the order a- be neglected, and of 1 — e be substituted, instead of a; the preceding ex- When this equation is reduced to an original form there will be only one mutation of sign ; •.- there will be only one real and affirmative root ; when u and D are givefl, r and t may be obtained immediately by the solution of a simple equation. 160 CELESTIAL MECHANICS, pression of / in v, in the ellipse, will give _ Z)i\/2 ("tan. 4^+^. tan. %v t=: ^- |+(1— e). tan. \v.(^\-\. tan. *i)w— f tan. *!■ \v)\ We should seek, in the table of the motion of comets, the anomaly which answers to the time t, in a parabola of which Z) represents the perehelium distance ; let U represent this anomaly, TJ-\-x being the true anomaly in the ellipse, corresponding to the same time, x being a very small angle. If we substitute in the preceding equation TJ-^-x in place of V, and then reduce the second member of this equation into a series arranged according to the powers of a; ; we shall obtain by neglecting the square of x, and the product of x into I — e. ^^ D'-s/i •/> r(tan.ii7+i.tan.^xt.)+^^^£^ ]• C+ — . tan. iZ7.(l--tan. %U—t. tan. ^IT)} 2Di 2Dt ^|.-*+4-"} (neglecting the square and higher powers ^/2 of «) = '^^ — '■ — • ^ 1-j > , V the value oft becomes = -:?^".(l+-|-)-*«"g-*''[l + (^- j)-tan. '^-(^-»).«.2.-«tang.»|v | = 7^=— \ -tang.^B + -. tang. |^u+|. tang. ^v\- -— -. tau. '^«— -— . tang. '\v— V ft. ( * *•'' 3 4« -2 ) _ ■\/2.i)l —2 . tang. > I -^. {tang.i«+f tang. > + (!—«). tang. >t.(4 + (^ijj-_^). tang. °\v) — \. tang. *|. i>) ; 1 — e being substituted for «. • Substituting [/ + « for r ; this equation becomes dI - ^ = -— : v/2 . (tang. i( i;+;r)+^. tan. ^i( U-\.x)+[\-e) tan. |( 17+ *).(i _i. tang. '^ PART I.— BOOK II. 161 but by hypothesis, we have /= -^. {tan. ^U+^. tan. %Zr};* PART I. — BOOK 11. Y ,TJ^^ 1. 4,rr-i- ^^ ^^ ^? C tang. 1 1/+ tan. ^^ , f tan. ■?;+ tang, -x ^ ^ ' { U+x)—^. tan. '( U+v))=-— ] u x+^ }' U 7 t ■^f ^ I— tan. — . tang.- ^1— tang. — . tan.- j (1— e). tan. — +tan.- j J ?• tang. —4- tan.- "^ l_tan. — . tan. — ( (^ i _ tan. — . tan. — J ftan.-— +tan.-|- ") ") 2)1^2. / 17,, x , ^ .t 'U ,,, =U l_tan.— .tan.|]) ) +S tan. 1^. tan. ^ + Stan. -^ . tan. ^) +(1 _c).tan. — (^ _ x tan. — — f 1 e U ![7 3jj» xx H — . tan. — -. (1 — tan. — f tan. -— ), and since tan. — =: — , when x^, x^, 'U *U I ^f7\^ 1 &c. are rejected, and 1+2 tan. -™|-tan. -j^= (14- tan. '-^] = ttt, by sub- COS.— X x '[7 *t7\ stituting — for tan. — . (1 +2 tan, — \- tan. — j, we shall have the exprcs- cos.— sion given in the text. * Therefore the two last terms of the second member of this equation are equal to cipher, con- sequently — = ——. tan. — f^_ 1 -f tan. - — |-f- tan.-^J; v^ or sm. x = 2 COS. — - 1— c 17 / „ *17 . =t7 ^[7 *V\ —^ — tan- -g- {—2- COS. -^ + 2 sm. — . cos. -^ + | sin. —^\ , (by substituting 17 1 e / U / *U ^U *U for tan. -^its value); = ——(tan. — 1—2. cos.— -+2 cos. — 2 cos. — - +f 162 CELESTIAL MECHANICS, therefore by substituting in place of the small arc s, its sine, we shall obtain sm. x = — . (1— e). tan. 1Z7.(4— 3. cos. %U~6. cos. % U). Thus, by constructing a table of the logarithms of the expression, — . tan. 4Z7.(4— 3. cos. %U—S. cos. '*l£7') ; it will be sufficient to add to them the logarithm of 1 — e, in order to obtain that of sin. x ; consequently if this correction be made to the anomaly U, computed for the parabola, we will have the cor- responding anomaly in a very eccentric ellipse. 21. It remains for us to consider the motion in an hyperbolic orbit. For this purpose, it may be observed that in the hyperbola, the serai- axismajor a becomes negative, and the excentricity e surpasses unity. If therefore in the equation (^) of No. 20, we make « = — a, and k= / , and then substitute in place of the sines and cosines, their V 1 ^ values in imaginary exponentials ; the first of these equations will give a^ (1-2. cos— + COS. -^)) = -^- tan. — (-4. cob. — +2 cos. — + f /, ^ ''U *U\ l—e U /„ „ *U , '-U (1-2 COS. -^ -t- COS. — j= -^-. tan. — . (f _ SjO cos. g - f • cos. — + f COS. —-1^ evidently the expression given in the text. Vfc, \ Vfi • ' u! u! * nt=u — esin.M, (n in this ca8e= / — ~ — 7=- ;'•• nt=— — t= +e. sin. — ;=r^; PART I.—BOOK 11. 163 The second will become r=a'.(i.e.(c«'+c-"')-l);' finally, if we make a corresponding change in the sign of the radical of the third equation, in order that v may increase with t, and conse- quently with u' ; we shall have Let us suppose that in these formulae, y! zz log. tan. (j^r+l^j) "^ being the semicircumference of which the radius is equal to unity, and the preceding logarithm being hyperbolic ; we shall have WtJi. _ a » = e. tang, w — log. tan. {^v + |w) -,% t2 sin. ^—r >-^=- ' ♦ ^2^' * " +^' o • / u' , — tt' \ * r/=a(l— e cos. w), becomes / = — a'(l— c) f ^ +'' j« tt' «' -m' ' /■" sin -^— ^ ■ __— y I ""ivIT c-^ + o^ c — 1 c +1 *T«ng-(^ + ^;=c and— -— _=cot.(- + -)=c ; .'. tan. - - <= =tan.(_+ _) _ cot. (^ + y j= 'M^+jJjHj;^ 164 CELESTIAL MECHANICS, r=a'.\ 1 J-; (.COS. W J tan. ^v zz \ -— j- . tan. ^w. The ai'c. -^ — - is the mean angulai- motion of the body m, during the a'^ time t, supposed to move in a circle about M, at a distance equal to a. This arc may easily be determined by reducing it into parts of the radius ; the first of the preceding equations will give by trials, the value of the angle w, corresponding to the time t ; the two other equa- tions will then give the corresponding values of v and of v. 25. T expressing the sidereal revolution of a planet of which a is the mean distance from the sun ; the first of the equations (/) of No. 20, will give T = 2'7r; but by the same number we have — j^=« j there- h' — u tang, ar ; /. by substituting this expression for '^ ^ ; we obtain the value of '• '^llL given in the text. /+""'= tan. (JL+ ^) +cot.f — + ~ ) =2 sec-. ^ '§ V4'2/\4'2/ a- 2 tan. 2C0S.— . sin.— ^— „ , ,. ,, '^^+^ ,,„' -^ = -7^= . cot. — . tang. -= , (ascot. -= 1), -7=f- '^n-o- 2 sin. — . cos. -5- ^ ^ ^ PART I.— BOOK II. 165 fore we shall have V /A If the masses of the planets, relatively to that of the sun, be ne- glected ; jM. will express the mass of this star, and this quantity will be the same for all the planets ; thus, for a second planet, of which a and T" express the mean distances from the sun, and the time of the side- real revolution ; we shall have in like manner ■ T > consequently we shall have r*: T"-.\a>: a'3 that is to say, the squares of the times of the revolutions of different planets, are to each other, as the cubes of the greater axes of their or- bits ; this is one of the laws discovered by Kepler. It appears from the preceding analysis, that this law is not rigorously true, and that it only obtains when we neglect the action of the planets, on each other, and on the sun. If we assume for the measure of the time, the mean motion of the earth, and for the unit of distance, its mean distance from the sun ; T will in this case be equal to Sir, and we will havea = 1 ; therefore the preceding expression for T will give n*=:l ; from which it follows that the mass of the sun ought then to be taken for the unity of mass. We can thus, in the theory of the planets and of the comets, suppose /^ =:!, and assume for the unity of distance, the mean distance of the earth from the sun ; but then, the time t is measured by corresponding arc of the mean sidereal motion of the earth. The equation 166 CELESTIAL MECHANICS, enables us to determine, in a very simple manner, the ratios of the masses of the planets which are accompanied by satellites, to the mass of the sun. In fact, M representing this mass, if we neglect the mass m of the planet relatively to that oi M -^ we shall have ~ s/m' If we afterwards consider a satellite of any planet m' ; and if p re- present the mass of this satellite, and h its mean distance from the centre of m, and T, the time of its sidereal revolution, we shall have therefore, m'-\-p __ h^ f ^Y M -^ a^'\T: ) ' This equation gives the ratio of the sum of the masses of the planet m' and of its satellite, to the mass M of the sun ; if therefore the mass of the satellite be neglected in comparison with that of its primary, or if we suppose that the ratio of these masses is known ; we will ob- tain the value of the mass of the planet, to that of the sun. We will give, in tlie theory of the planets, the values of the masses of the planets about which satellites have been observed to revolve. PART I— BOOK II. i&7 CHAPTER IV. Determination of the elements of Elliptic Motion. 26. After having treated of the general theory of elliptic motion, and of the mode of computing it by converging series, in the two cases of* nature, namely, in that of orbits very nearly circular, and in the case of very eccentric orbits ; it now remains for us to determine the ele- ments of of these orbits. If the circumstances of the primitive mo- tions of the heavenly bodies were given, we could easily deduce the elements from them. In fact, if we name V the velocity of m, in its relative motion about M j we shall have dt' ' and the last of the equations (p) of No. 18, will give In order to make ju. to disappear from this expression ; let U denote the velocity which m would have, if it described about M, a circle of which the radius is equal to the unity of distance. In this hypothesis, we have r = a = 1, and consequently £/"*=/* ; therefore V'=U\ . r ay This equation will give the semiaxis major a, of the orbit, by means of the primitive velocity of m, and of its primitive distance from M. a is positive in the ellipse ; it is infinite in the parabola, and negative in 168 CELESTIAL MECHANICS, the hyperbola ; therefore the orbit described by m, is an ellipse, a para- bola, or an hyperbola, according as V is less, equal to or greater than U. ^ — . It is remarkable that the direction of the primitive mo- tion, does not at all influence the species of conic section.* In order to determine the excentricity of the orbit, it may be ob- served, that if i represent the angle which the direction of the relative dv'' motion of m, makes with the radius vector r ; we have — r-j- = V^. cos. *£. By substituting in place of V", its value u. \ C, we shall L r a S have dr"- C 2 1 J ^ ^ — -— - =u. i. \ . COS. £ ;T df- t r a S I 2 V- * From the equation — = — , it appears that when V and r are given, the axis major and therefore the periodic time are constantly the same. Hence since U. Y — = the velocity in a circle at the same distance, it follows that in the ellipse the velocity at any point is to that in a circle at the same distance in a less ratio than that of \/'2 : I, in a parabola, it is in the ratio of 'V^ 2 : 1 ; and in the hyperbola it is in a greater ratio than that of V^ : 1. See Princip. Math. Prop. 16. In the ellipse when the velocity of projection diminishes, the distance increases, and when F vanishes, r becomes equal to 2a, in this case the excentricity e becomes equal to unity. In the hyperbola, the limit of the velocity, when r is infinite, is 17^ — = the velocity in a circle, at the distance of a transverse semiaxis from focus. It is also manifest that when the distance is equal to the semiaxis major, the velocity is equal to that in a circle at the same distance, and that in general the velocity in an el- lipse, is to the velocity in a circle at the same distance in the subduplicate ratio of the distance from the other focus to the semiaxis ; for it is as V 2a — r : Va. t -/ = the velocity resolved in the direction of the radius, .*. it is equal to V, mul- dt tiplied into the cosine of the angle which the radius vector makes with the curve or tan- gent, t. e. it is equal to F. cos c. PART L— BOOK IL 169 but by No. 19, we have therefore we shall have c 2 1 ) a (I — e' )=:>'''. sin. ^s. } C; ( r a ) by means of this equation, we can determine ae the excentricity of the orbit. From the polar equation of a conic section, namely «.(!—£-) /• ^ i i — 1-f-e. COS. V ' we obtain a. (I — e') — r COS. v= — ^ . er PART 1. BOOK II. Z dr^ , *er^ / 2 1 \ Substituting for — —r its value, we shall have 2«.r— -^ — — m i ]r'. cos. -i= «<■ ■ a V r a / ixa.{l — e"), .'.(2r I. ( 1 — cos. -s) = n(l — e") = the parameter ; hence it appears that when the distance and axis major are given, the parameter varies as the square of the sine of projection, since k * = "^^ , see page 4, a(l — e') = >-. — — - , •_• the parameter depends on that part of the velocity which acts perpendicularly to the radius vector, it is termed the paracentrick velocity, and it is evidently a maximum at the ex- tremity of the focal ordinate. (2 1 \ . I, it follows that sin. "s varies in- (2(1 r \ ), but the sum of the two factors is given, being equal to 2«, •/ the product is a maximum, and consequently the sine of projection is the least possible, when the distance from the focus is equal to the seraiaxis major. 170 CELESTIAL MECHANICS, We shall thus obtain the angle v, which the radius vector r constitutes with the perihelion distance, consequently we have the position of the perihelion. The equations (f) of No. 20, will make known the angle ii, and by means of it, the instant of the passage through the perihelion. In order to determine the position of the orbit, with respect to a fixed plane passing through the centre of M, supposed immoveable ; let (p represent the inclination of the orbit on this plane, and S the angle which the radius r constitutes with the line of the nodes ; moreover let z be the primitive elevation of m, above the fixed plane, which elevation we suppose to be known ; we shall have r. sin. e. sin. (pzz z ; so that the inclination (p of the orbit will be known, when we shall have determined S. For this purpose, let \ represent the angle, which the primitive direction of the relative motion of m, makes with the fixed plane, which angle we suppose to be known ; if we consider the triangle formed by this direction produced to meet the line of the nodes, by this last line, and by the radius r ; I representing the side of the tri- angle which is opposed to the angle Q, we shall have r. sin. £ sin. (S+0 we have also -~- zz. sin. a ; therefore we shall have V , z. sin. £ tan. C= r. sin. A — z. COS. £ • r. sin. e=fl perpendicular let fall from the extremity of r, on the line of the aodcs, and z = this perpendicular multiplied into the sine of '.'. sin. £ : sin. (S -fi); . , r. sin. S ?-. tan. € ~ •/„:„, » .•. /— — : , — = -: , . . (r. sin, A — z. co«. «;. sin. £. cos. £-{-sin. S. cos. J sin, e-J*^''"' *• '-''^' ' sm. a tan £ = z. sin. i. PART L— BOOK II. 171 The elements of the orbit of the planet being determined by these fornuilfB, in functions of the coordinates r and z, of the velocity of the planet and of the direction of its motion ; the variations of these ele- ments, corresponding to the variations which are supposed to take place in its velocity and in its direction may be obtained ; it will be easy, by the methods which will be given in the sequel, to infer the differential variations of these elements, arising from the action of disturbing forces. Let us resume the equation F*= V\ \— ^ L r a In the circle a=r, and consequently V ■= U.\ — ; from which it appears, that the velocities of the planets in different circles are reci- procally as the square roots of their radii. In the parabola, a = oc, .-. y ~ Z/M — ; therefore the velocities in different points of the orbit, are in this case reciprocally as the square roots of the radii vectores, and the velocity in each point is to that which the planet would have, if it described a circle whose radius was equal to the radius vector r, as \/ 2 : 1. An ellipse, of which the minor axis is indefinitely small, is changed into a right line ; and in this case, V expresses the velocity of di, if it descended in a right line towards M. Let us suppose that m sets out from a state of repose, and that its primitive distance from M is r ; let us moreover suppose, that having attained the distance r, it has ac- quired the velocity V ; the preceding expression for the velocity, will give the two following equations : r a ( r a S Z2 172 CELESTIAL MECHANICS, from whicli we obtain rr ' this is the expression of the relative velocity acquired by wa, in depart- ing from the distance r, and in falling towards M, through the height r — /, We can easily determine by means of this formula, from what height a body m, which moves in a conic section, ought to fall towards M, in order to acquire, in departing from the extremity of the radius vector r, a relative velocity equal to that which it has at this extremity ; for V expressing this last velocity, we have i S ~ ^ } but the square of the velocity acquired in falling through the height r — r , IS ^ — ; by equating these two expressions, we shall have 4a — r „ . , . , 2a— r ^r—^r" * By equating these expressions we have = ; — , • • (2a — r)r' = 2a.(r — r*); and consequently {4a — r) r = lar ; .'. r' = , andr — >'= Sar ia- , in the ellipse 4a — r is greater than twice 2a — r, .'. r — r' is less than -— ; in the — r « o i^ parabola a being infinite, r — r'z:.—, in the hjfperbola r — )'= — — , and as in this case ^a-\-r is less than twice 2a -)- >•, r — r' is greater than ~-. In order to determine the space through which a body must fall externally, so tliat it may acquire the velocity which it has in a conic section, r' — r must be substituted lor r — r*, and then we equate — — ~ to , from which we obtain fiflr* — Sor = rr ar PART I— BOOK II. 173 In the circle «=r, and then r — T'-=i\r ; in the ellipse, we have r — r'Z-\r; a being infinite in the parabola, we have r — r'-=.\r% and in the hyperbola, in which a is negative, we have r — r'>^r. •27. The equation is remarkable, in that it determines the velocity independently of the eccentricity of the orbit. It is contained in a more general equation, which exists between the axis major of the orbit, the chord of the el- liptic arc, the sum of its extreme radii vectores, and the time em- ployed to describe this arc. In order to arrive at this last equation, we will resume the equations of elliptic motion, which|have been given in No. 20 ; /* being supposed for the sake of simplicity equal to unity. These equations will consequently become ~ 1-f-e. cos. V ' r z=. a.{\ — e, cos. U) ; t •=. a^.(u — e. sin. u). Let us suppose that r, v, and t correspond to the first extremity of the elliptic arc, and that r', v', and f correspond to the other ex- tremity ; we will have ,_ a.(l—e^) ~ 1-f e. cos. t/ ' r'zz a.(l — e. cos. u') ; f= a^. {ill — e. sin. m'). Let/'_/=Tj i^±ii=ej '^±^^^; r'+r=R, 2fli- — r'r, •/ ia an ellipse /' is = to the axis major, in a circle it 18= to tbe diameter, it is infinite in the parabola ; and in the hyperbola r' becomes = — 2a. 174 CELESTIAL MECHANICS, subtracting the expression of /, from that of t', and observing at the same time that sin. u' — sin. u -=.2. sin. €. cos. 6' ; we shall have T n 2a"5. (€ — e. sin. 6. cos S'). If we add together the two expressions of r and of r' in terms of u and i/, and if we observe that COS. ti -f cos. M = 2 COS. E. COS. S' ; we shall have R zz 2c.(l — e. COS. €. cos, f ). Now, let e represent the chord of the elliptic arc, we have c''=:r''-{-r"' — 2rr'. cos. (v — v') ; but from the two equations a.(l — e*) ., >. r = — -^ — ; rzzaJl — e. cos. u\ 1-^e. cos. V ^ we obtain a. cos. w — e) . a.V I — e^'.sm.u COS. I' zz — ^ : sm. vzz — • r r In like manner we have a.Ccos. XL — e) . , a.\ 1 — e*. sin. «' cos. V — — =^ ; sm. v — — — ; r r therefore we shall have rr. cos. (w — t'')=o*.(e — cos. v).{e — cos. w')+a".(l — e*). sin. u. sin. tC' and consequently c*n:2a*.(l — e'^).(l — sin. «. sin. 11 — cos. ?<. cos. iC^ +a*e*,(cos. u — cos. liy ; > PART I.— BOOK II. i75 but we have sin. u. sin. i/^-cos. u'. cos. m = 2 cos. *£ — 1 ; COS. u — COS. m' = 2. sin. g. sin. 6' ; therefore c*zr4a*. sin. *e.(l--e^ cos.*?');* consequently we will by this means obtain the three following equations, R = 2«.(1 — e. COS. e. COS. S') ; T = 2a^. (£ — e. sin. £. cos. C) ; - e= 4>a\ sin. *e.(l — e*. cos. ^e'). » «'=£' + £, «=€' — £; .'. sin. u' = sin. £'. cos.€ + sin. g. cos. €' ; sin. uzz%m.Z'. cos. C — sin. S. COS. S' ; .*. sin. v! — sin. «= 2. sin. S. cos. £', hence Tzr/' — <= ai. (u u e. (sin. ii' — sin. u) = the expression in the text. Cos. u'= cos. £'. cos. £ — sin. S. sin. £' ; COS. i/=cos. €. COS. €'+ sin. S. sin. S', .'. cos. «+cos. u=i2. cos. S'. cos. £, and r'-\-r=R=L (7.(2 — e. (cos. m'+cos. a) := the expression given in the text, t — v' is evidently equal to the angle contained between r and r'. a.(I — e") . «.(1 — e") — a.(l — e. cos. u) „ 1-fc. cos.v= ; ..e. cos. "j= ; :, (by substituting for , . ae.fcos. « — e) . fi.(cos. m — e) r Its value) = ^^ ; . . cos v= — )- : • sin. -v = r r n-.{\ — 2e. cos, it+e'. cos. ';<) — cos. ';f+2g. cos. i<^ e') a^.{\ — cos. -!«— (-'.(sin. -u) — ^ — p; T j~ tt'.(l — e"). sin. '(( , , ,, = 5 ' consequently cos. |t) — v )=cos. v . cos. i^ + sin. v. sin. v'~ a'.(cos. ?'-e).(cos. !i'-e) + (7^.(l-e") sin.u. sin.M* _i LI IZ^^ ; r~ + r^=n"-.(2 _2e.(cos. ?< -f cob. «')-[- r".(cos. "u+cos. -u')), .*. r* + r'-— 2?-/. COS. {i—v') =c-.(2 — 2e.(cos. «+co6. u')+e-. (cos.'u +cos."«') — 2o^(e' — e.(cos. k + cos. ;(')4.cos. u. cos. ;/) — 2a". sin. u. sin. M'+2aW. sin. u. sin. ii' = by reduction 2fl^(l — t?). (1 — sin. «. sin. u') — cos. u. cos. u' -)- a'.e'. (cos. "k -j- cos. "«') — 2o.-.e°. cos. «. cos. ;/. Cos. n. cos. !(' + sin. ii. sin. k' = cos. 2£= cos. £■ — sin. €^= 2. cos. °S — 1 . .-. & = 2a-(l— «'').( 1 + 1. — 2. cos. =S)+ n".e=.(4.. sin. =£. sin. '£'=40^(1— e'),(l— cos. 'S) 4-«a'.e°. sin. "«. sin. =£' =4a-.(l— e*). sin. -£+ 4a'.e^ sin. % — 4a^e^ sin. €"-. cos. V = 4c"'. sin. ■€.(! — e". cos. -o'). 176 CELESTIAL MECHANICS, The first of these equations gives e. COS. fe zr — 1 2a, COS. G by substituting this value of e. cos. C', in the two others, we shall have c zzi^a . tan. e. ■{ cos. £ — { ) {• I \ '2a J S These two equations do not involve the excentricity e ; and if in the first we substitute in place of £, its value given by the second, we shall obtain T in a function of c, R and a. It appears from this, that the time T depends only on the axis major, the chord c, and the sum R of the extreme radii vectores. If we make ^ _ Qa —R-\-c , _ 2a—R—c ^ ~ 2a ' ~ ~ 2a ' the last of the preceding equations will give cos. ae = x^'+\/(i— ^^).(T^'*) ; Irom which may be obtained, 2 £ := arc. cos. z' — arc. cos. z ;* c' . ,. /2n — Ry sin. *£ , c la — R , . .. * = sin. o — I ) . -. , let =11, ■ =: )", and as sin. -lozr 4n- \ 2a / COS. ■£ 4a- 2a 1 — COS. -£, .'. 71. COS. '^b=cos. ^£ — COS. *£ — to^+jb". cos. % .'. COS. ■"*-!-(" — '"'" — !)• *"*• "^ fn m'^ — 1) \^(7i — nr — 1)- — im'^ — — m', and solving this equation cos. °£ = ^ ± j, , and as COS. 2?= 2 COS. ■£ — 1, we have cos. 2»=— n+nj'± V(n— m" — If — 4?)!*, .and substi- tuting for n and m, we obtain ; PART I.— BOOK II. 177 arc. COS. z denotes here the arc, of which the cosine is z ; consequently we have sin. (arc. cos. z'^ — sin. (arc. cos. s) tan. ^■=. ^^ ^-— > — ^= ; we have likewise z -\- z' ■=. ; therefore the expression of T will become, (by observing, that if T is the duration of the sidereal revolu- tion of the earth, of which the mean distance from the sun is taken for unity, we have by No. 16, T — S'tt) ; T= — — . (arc. cos.z — arc.cos.2: — sin. (arc.cos.2;')-f sin.(arc.cos.;s)). («) As the same cosine may appertain to several arcs, this expression of PART. I. BOOK II. A A 2a—RV^-c'- ^ d c* + {2a—Ry—2^.(2a—Rf 2c°-— 2.(2a— /?)' , i^i.^os 2g- ia' ~~ \ (2a}* 4a- •" / ' "" ' the part of this radical of which the denominator (2a)'' = s°.2'° ; „ „ (2a— /?)--!- c=+2.c.(2a—iJ) ^ ,„ {2a— ff)'+c^— 2.c.(2a— iJ) , . for 2- = i — — -!- — i ' and z" = -i -■ ,„ ,. — ^ ; and the (2a)- (2«)- part of this radical of which the denominator is 4a-= — ~- — :''= — (2a — RY — C- — (2a — RY — c'+2c.(2a — R) — 2c.(2a — R), the part without the radical is evidently equal to 22/, .'.by substituting we shall find the cosine of 2% ^^ zz' ■{•V z- z"^ — z'^ — =''+1, which is evidently equal to the expression given in the text. Let z,z' represent the cosines of two arcs, and the cosine of the difference of these arcs will be = 22' 4- '/l — 2''.(1 — 2*') =cos. 2£; .'. 2S = the difference of two arcs of which the cosines are 2 and zi. o- • , n («+*) • (a—h) , , _ («+A) {a—b ) bm. a — sm. 6zz 2. cos. - — - — . sm. , cos. a 4- cos. 6 = 2. cos. — - — . cos. — -— 2 2 2 2 sin. sin. a — sin. b 2 __ (n — b) cos. a+cos. 6 la — b) °' 2 cos. -i 2 value of tan. S, which is given in the text. = tang. — , from this formula may be inferred the 178 CELESTIAL MECHANICS, T is ambiguous, and it is necessary carefully to distinguish the arcs to which the cosines z and z' belong. In the parabola, the seraiaxis major a is infinite, and we have arc. COS. z — sni. (arc. cos. z) =: -—. I — ~ ]' * o \ a / By making c negative, we obtain the value of the arc. cos. ^ — sin. (arc. COS. z) ; the formula (a) will therefore give for the time T, employed to describe the arc subtended by the chord c. The sign — having place, when the two extremities of the parabolic * Arc. COS. z' — sin. arc. COS. :'=arc. sin. V'l — z" — V'l — ^'^ —by expressing the /i *'2^■^ arc. in terms of the sine, v'l — z" -{. ~ + &c. — Vl — z'^ _ AW—iia—R)" + 2c.{2a—R)—c') \ i _ {{iRa—R^ + 4.ac—2cR—c'' ))i — when n is oo, ——-■ -, — * ' (2.a.4a') In the expression for arc. sin. VI — ;:", the approximation is not continued beyond the second term, because the subsequent terms disappear in the value of T, when a is supposed to be infinite. The second term of the value of T vanishes when c passes through the focus, and T is less when the angle fomied by r,r' is turned towards the perehelion, than when the second term vanishes, it is manifest that the sign of the second term must be in this case negative, and positive in every other case. The second term of the second member of this equation vanishes when the extremi- ties of the arc described, are bounded by the focal ordiaates, .'. the time of describing the parabolic arc intercepted between vertex and focal ordinate varies in the sesquiplicate ratio of the parameter. See Newton, Princip. Vol. 3, Lem. 9, 10. Indeed it appears from the value of T, that the time of describing any parabolic arc, of which the chord passes through the focus, varies in the sesquiplicate ratio of the chord. PART L— BOOK II. 179 arc are situated on the same side of the axis of the parabola, or when one of them being situated below, the angle formed by the two radii vec- tores, is turned towards the perehelion, it is necessary to make use of the sign -j- in every other case. T being equal to 365''^'", 25638, we have — = 9""% 688754.. In the hyperbola, a is negative ; z and z become greater than unity ; the arcs, arc. cos. z, and arc. cos. z' are imaginary, and their hyperbolic logarithms are, 1 . arc. cos. 2;= __ . log. (^z-\-s/z'—\ ; 1 / arc. COS. s'= / — — . \og. {z' + \/ z'- — 1 ; consequently the formula {a) becomes by changing a into —a, 3 rp T=^ .(v/^"— 1 +v/;s^— 1— log.(2r'+v/2'*— i)±log.(zf-\/5^IIi. The formula ( • • ^s x — arc. sin.— \/ , and arc. a ^ a- "" V a* /lac c' sin. = ^ T have the same sine, T varies in nn ellipse as the arc — sin. ; whicli agrees with Newton's conclusion; Princip. Math. Lib, 1. Prop. 37- See Prony Mecha- nique Analytique, Tom. 2. No, 914, and Euler's Mechanics, No. 272, 672. If c=2a the time of falling to the centre will be equal to a?T PART I.— BOOK II. 181 in the actual state of astronomy, the elements* of these orbits require only very slight corrections ; and as the variations of the distances of the planets from the earth are not at any time sufticiently great to render them invisible to us, we can observe them perpetually, and by a comparison of a great number of observations, correct the elements of their orbits, and also the errors tliemselves to which the observations are liable. This method cannot be applied in the case of the comets, as they are only visible near their perehelion ; and if the observations which are made on them during the time of their appearance, are ina- dequate to the determination of their elements, we have not then any means of following these stars in imagination, through the immensity of space ; so that when the lapse of ages brings them back towards the sun, it is impossible for us to recognise them ; it is therefore of the greatest consequence to be able to determine by observations made during the time of the appearance of a comet, the elements of its orbits; but the rigorous solution of this problem surpasses the powers of analysis, and we are obliged to recur to methods of approximation, in order to obtain the first values of these elements, which we can afterwards correct with all the precision which the observations admit of. If we employ observations which are at a considerable distance from each other, the eliminations would lead to impracticable computations ; it is therefoie necessary to restrict ourselves to the consideration of near observations ; and even with this restriction, the problem presents con- siderable difficulties. It has appeared to me, after mature reflection, that instead of employing directly the observations themselves, it would be more advantageous to deduce from them data, which offer a simple and exact result ; and I am satisfied that the geocentric latitude and longitude of the comet, at a given moment, and their first and second * In the present state of Astronomy, the motions of the planets may be considered as very accurately known, and the object of these observations is to determine them with still greater accuracy. And when the elements have been determined uuder the most favourable circumstances, i. e. in those in which they have the greatest influence, they should be afterwards corrected simultaneously, by the method of the equations of con- dition. 182 CELESTIAL MECHANICS, differences divided by corresponding powers of the element of the time, are those which best satisfy this condition ; for by means of these data, we can determine rigorously, and with facility, the elements, without having recourse to any integration, and by the sole consideration of the differ- ential equations of the orbit. Tiiis mode of considering the problem permits us also to employ a great number of neighbouring observations, and by this means, to embrace a considerable interval between the ex- treme observations, which is very useful in diminishing the influence of the errors, to which these observations are always liable, in consequence of the neb:;losity which surrounds the comets. I proceed now to pre- sent the formulas, by means of which the first differences of the longitude and latitude may be deduced from any number of neighbouring observa- tions ; I will afterwards determine the elements of the orbit of the comet by means of these differences, finally, I will point out the means which have appeared to me the simplest, for correcting these elements, by three observations, made at a considerable distance from each other. 29. Let at any given epoch, a. be the geocentric longitude of a comet, and 6 its northern geocentric latitude, the southern latitudes being supposed negative. If we denote by s, the num- ber of days which have elapsed since this epoch ; the geocentric longitude and latitude of the comet, after this interval, will be ex- pressed in consequence of the formula (i) of No. 21, by the two fol- lowing series, The values of a, ( y ) , \ 'T'^) ■> ^^ '> ^' ( ^ ) ' ^^' '"^^ ^^ ^^' termined by means of several observed geocentric longitudes and la- titudes. PART I— BOOK 11. !83 In order to obtain them in the simplest manner, let us consider the infinite series which expresses the geocentric longitudes. The coef- ficients of the powers of 5, in this series, may be determined by the condition that it ought to represent each observed longitude, when we substitute for s, the number of days which corresponds to it ; we shall by tliis means obtain as many equations as there are observa- tions ; and if the number of these last be n, we cannot determine by their means, in the infinite series, but n quantities a, f — j , &c. However, it ought to be observed, that s being supposed very small, we can neglect the terms* multiplied by s", s""*"*, &c., so that the infinite series is reduced to the n first terms, which we are able to deter- mine by the ?i observations. These determinations are only approxi- mative, and their accuracy will depend on the smallness of the terms which we have neglected ; they will be always more exact, in proporron to the smallness of 5, and to the greater number of obser- vations employed. Therefore the question is reduced to a problem in the theory of interpolations, namely to find an entire and rational function of 5, of such a nature, that when we substitute for s, the number of days which correspond to each observation, this function is changed into the observed longitude. Let £, S', 6", represent the observed longitudes of the comet, and * As the values of the differential coefficients in the series expanded according to the formula of No. '2 1 , are independent of the value of the increments, these values will remain, when the increment varies; and there are as many series of the form « -f — ( -r) "4" . f -— ) -[-(~j"— )> ^c., the quantities e, e', Q'\ &c. into y, y'y y\ he. These expressions will be more exact, according as the number of observations is increased, and as the intervals which separate them, are less ; we could therefore* employ all the neighbouring obser- vations of the selected epoch, provided that they were exact ; but the errors to which they are always liable, would lead to an erroneous result ; therefore in order to diminish the influence of these errors, the interval between the extreme observations should be increased, in proportion as a greater number of observations is employed. We are able in this manner, with five observations, to embrace an interval of thirty-five or forty degrees, which ought to lead to very approximate values of the geocentric longitudes and latitudes, and of their first and second diflPerences. If the epoch which we select, is such that there exists an equal number of observations before and after it, so that each longitude which follows, has a corresponding longitude which precedes it by the same interval ; this condition will render the PART I. BOOK ir. B B * The number of observations will of itself produce an increase in the error, .'. in order that the error may be distributed over a greater number of degrees, we must increase the interval between the extreme observations. 186 CELESTIAL MECHANICS, values of a. I t; ) , ( yi ) more accurate,* and it is easy to perceive * When the obsenrations are assumed at different sides of the epoch which is selected j', i"', i""', &c. are negative when i, i", i"", &e. are positive, and vice versa. In the values of a, which are given above, the terms after the first, are negative and positive in pair* and in the values of —;-, — r^. the coefficients of c?£, rf'S, &c. are less than when all the as as- observations are made at the same side of the selected epoch, .*. the convergence of the terms will be more rapid, and the terms which are omitted are of less consequence. Let the number of observations be odd, and =:2r + l, and let i be the number of days between each observation, and let the epoch from which we count be the instant of the mean observation when «=?'''' , then we have d»__ J_ Is ~ 2i u — . ^ A^e^'--^) + A=e('-3) \ ^ 1.2.3 A.5 I ^ S =-^^ -J A ■'£''■-3) + A '£"•-♦) 1.2.3.4.5.6.7^ ^ (P«. a'€('-i> 1 di« ~ 2.?^ 2.3.4..r A *.£('•-*) ^ - A^eC-S) f_± r^. A^e'-^'+Ac. 1.2.3.4..5.6.Z' 2.3.4..S.6.7.8«7 A is the characteristic of finite differences, so that a .EM = €<'■+') — €('>. If the number of observations be even, and equal to 3r, we should assume for the epoch, the mean time between the first and last observation, and then we shall have g(')+£(r-i)_ _L. a'csc- D+et'-s') + 2Z6¥-^*(^''-"+«<'-'') A«.(C('--3)+e'-* ) 2.4.6.8.10.12 d» A C''"^' 1 •> ft, o\ , 3 * ^fr. t a -T-= -^ T-?^- ^-^^ '+TT-5-r7r^' a'.SC-' — &C.; dt t 4.6.« 4.6.8.10.t PART I.—BOOK II. 187 that new observations assumed at equal intervals, and at opposite sides of this epoch, will cause quantities to be added to those values, which will be, with respect to their last terms, of the same order, as the ratio of 5*. [ -TT J to «. This symmetric disposition obtains when all the observations being equidistant, we fix the epoch in the middle of the interval contained between them ; it is therefore advantageous to em- ploy corresponding observations. In general it will be always useful to fix the epoch towards the middle of this interval ; because that the number of days which separates the ex- treme observations, being less considerable, the approximations are more convergent. The calculus will be likewise simplified by fixing the epoch at the very instant of one of the observations ; for then the values of a. and of 6 will be immediately given. When, by the preceding process, we have determined, ( ~ J , f -j-^ ) ' ( Z ) ' ( rf? ) ' ^^ ^^'^ deduce in this manner the first and second differences of a. and 9, divided by the corresponding powers of the element of the time. If the masses of the planets and of the comets, are neglected in comparison with that of the sun assumed to re- present the unity of the mass ; if, moreover, we assume for the unity of distance, its mean distance from the earth ; the mean motion of the earth round the sun, will be, by No. 23, the measure of the time/; let, therefore x represent the number of seconds which the earth de- BB 2 \.1.ds^ 4.4^ ^ ^ ' 4.6.8.21 ^ ^ ' 4..6.8.10.1'i.j» • '^ •^* ^ I °"-^ It is easy to prove these theorems from the theory of finite differences. See Lacroix, Tom. 3. 188 CELESTIAL MECHANICS, scribes in a day, in consequence of its mean sidereal motion ; the time t corresponding to the number s of days, will be xs ; therefore we shall have \dt)- x' \Js)' \l?)~—{l?)' Observations give in logarithms of the tables, log. Azr 4,0394622 ; more- over, log.A"=log. A + log. ^, R being the radius of the circle, re- duced into seconds; from this it appears that log. a- = 2,2750444 ; therefore, if the values of ^ -^ j , and of (~\, be reduced into se- conds ; the logarithms of ( y j and of ( ~ J will be obtained, by subducting from the logarithms of these values, the logarithms, 4,0394622, and 2,2750444. We shall obtain in like manner, the the logarithm of { 7^ ) » and of ( -77 1 ; by subtracting respectively the same logarithms, from the logarithms of their values reduced into seconds. ■ As the precision of the following results depends on the accuracy of the values of ., (^), (^^:), 6, (i^) , and (^^') , and as their formation is very simple, the observations ought to be selected and multiplied in such a manner, as to obtain them with the greatest pos- sible precision. We now proceed to the determination of the elements of the orbit of the comet by means of these values, and in order to generalize these results, we will consider the motion of a system of bodies actuated by any forces whatever. 30. Let X, y, z, be the rectangular coordinates of the first body ; x', y', z', those of the second body, and so on of the rest. Let us conceive that the first body is sollicited parallel to the axis of a; of y, and of z, by the forces X, Y, and Z, which forces we will suppose PART L— BOOK II. 189 to tend to diminish these variables. Let us conceive, in like manner, that the second body is sollicited parallel to the same axes, by the forces X', Y', Z, and so of the rest. The motions of all these bodies will be given by differential equations of the second order, &c. If the number of bodies is n, the number of these equations will be 2>n, and their finite integrals will involve &n arbitrary quan- tities, which will be the elements of the orbits of these different bodies.* In order to determine these elements by means of observations, we should transform the coordinates of each body into others, of which the origin will be at the observer. Therefore supposing a plane, of which the position may remain always parallel to itself, to pass through the eye of the observer, while the observer moves on a given curve, let f, f', ^' , represent the distances of the observer from the different bodies, projected on this plane ; and «, a', a.", &c., the ap- parent longitudes of these bodies, referred to the same plane, and 6, 6', %", their apparent latitudes. The variables x, y, z, will be given in a function of f, a, 6, and of the coordinates of the observer. In like manner, a/, y', z, will be given in functions of f', a', 6', and of the coordinates of the observer, and so of the rest. Moreover, if we sup- pose that the forces X, F, Z, X', Y', Z, &c., arise from the re- ciprocal action of the bodies of the system, and from the action of foreign bodies j they will be given in functions of f, p', {, &c. ; «, «, * Each body furnishes three equations, and consequently tlie n bodies furnish 3« equa- tions, and as in the integration of each differential equation of the second order, two ar- bitrary quantities are introduced, the total number of arbitrary quantities must be 6«. 190 CELESTIAL MECHANICS, 01." y &c. ; 9, 0', %" , &c. ; and of known quantities ; consequently the preceding differential equations will be between these new variables, and their first and second differences ; now observations make known, for a given time instant, the values of a, ( -^ ) » ( -^ ) , 6, { -7- ) . I — j ; o''* { 7/7 ) ' ( ~j^ ) > ^c. ; therefore, the quantities which remain unknown, are f, f', ^' , &c., their first and second differences. These unknown quantities are Zn in number, and as we have 3?* differen- tial equations, we can determine them. There is also this advantage connected with this method, that the first and second differences of j, f', ^', &c. occur in these equations, in a linear form. The quantities «, 6, j, «', 6', §', &c., and their first differentials di- vided by dt, being known ; we shall have for any given instant, the the values of x, y, z, a/, r/, z', Sec, and of their first differentials di- vided by dt. These values being* substituted in the 3n integrals of the preceding differential equations, and in the first differences of these integrals will give 6?z equations, by means of which we can determine the 6n arbitrary quantities of these integrals, or the elements of the orbits of these different bodies. 31. Let us apply this method to the motion of the comets. For this purpose it may be observed, that the principal force which ac- tuates them, being the attraction of the sun, we may abstract from the consideration of every other force. However, if the comet passes sufficiently near to any large planet, to experience a sensible per- turbation, the preceding method would still make known its velo- city, and its distance from the earth ; but this case being of rare oc- currence, we shall only consider, in the subsequent researches, the action of the sun. * The number of unknown quantities for each body is tliree, namely 5, --^, -r-f- , therefore there are 3« unknown quantities in the system of n bodies. PART I.~BOOK II. J 91 Assuming the mass of the sun to repi'esent the unity of mass, and its mean distance from the earth, the unity of distance, and moreover placing the origin of the coordinates x, y, z, of a comet of which the radius is r, at the sun; the differential equations (O) of No. 17, will become, (the mass of the comet, in comparison with that of the sun being neglected) 0= '^'^ • ^ Let us now suppose that the plane of x and of ?/, is the plane of the ecliptic ; tliat the axis of or is the line drawn from the centre of the sun to the first point of Aries, at a given epoch ; that the axis of z/ is the line drawn from the centre of the sun to the first point of Cancer, at the same epoch ; that the positive z' are on the same side with the north pole of the ecliptic ; and finally, that a/ and y are the coordinates of the earth, and R its radius vector ; this being premised, Let the coordinates a; y, z, be transformed into others relative to the observer ; and for this purpose let a. represent the geocentric longitude of the comet, 9 its geocentric latitude, and f its distance from the earth projected on the ecliptic ; we shall have x-=.3!-\-^. COS. a, ; y=.y'-\-f. sin. a. ; zzz^. tan. 6. If from the first of the equations (k), multiplied by sin. «, be sub- tracted the second multiplied by cos. a, we shall have n— o-„ ^°* '^V I ^- sin* <^—y- cos. a. „ 0= sin. «. -^ - COS. a.-^ + -^ * ^ ds _dx' di . d» , d^s . dV . rf«e 192 CELESTIAL MECHANICS, Iience we deduce, by substituting for x and y their preceding values c?V d°y' x' . sin. a — y' . cos. a. sin. *. -^ - COS. «. -^ + y, . The earth being retained in its orbit, as the comet, by the attraction of the sun, we have df ^ R' ' dt' ' R' consequently, dV d^u' 1)'. cos. a — x'. sin. a therefore, we shall have 0=:(^.cos.a-y.sm.«).J-^— ^J-2.J-^5.J^j-f. ^^|. Let A be the longitude of the earth, as seen from the sun j we shall have x'=R. COS. A ; y'=R. sin. A; therefore y'. COS. a — x\ sin. x=zR. sin. (^ — «) ; the preceding equation will consequently become, ^ dp da . „ dx^ . „ d°-tc dy du' sin. «. COS. « — 2. — ^. -T— . sm. -« — g. sin. «. cos. «. -.- e. sin. -«. -t-t- ; — j- = -3— ri< dt ^ dr dr dt dt de . , (l» . '^'y dW , d-p + -^. sm. «+£. COS. «. — r- ; . . -~ . COS. » — ,„ . COS. » + -rr- sin. «. cos. a ^ Ozitan. 9. \ cos. a.. ----+sin. «. ^ ^ 4-tan. 9. ^ TA. ( IF r^' This equation will become, by substituting for x, y, z, their values, = tan.9.|)^+^^.cos.«+|^'+f(.sin..j_* ^ 5^^? ^^/i cos. *9 PART I. BOOK II. C C * -7?T-= — rr T — rr- cos. « — 2. -p-. —7- . sin. « — ». cos. «. — — £. sin. «. — — * if this valae of -^ be subtracted from the first, and if we suppose W ^. sin. 9. COS. fi. cos.(yf — »)+ j-ji . sin. (.4 — «) t tan. e. [ —7-- . COS. « J fi- . sin. «. -^ . cos. « + •^. sm. «. 1 4- -^ — . tan.«_ 2.—. — _^_ __._____.._. sm. 6- —. tan.^ ^. tan. 6 p.= * This value of ~ is derived immediately from the preceding equations, by multiply- ing the entire expression, by cos. "l. and dividing by — , and observing that tan. «= '—, f If the two values of —, be multiplied by -^'—r > and if the second be then sub- dt "^ •' dl dt we shall obtain PART I.— BOOK 11. 195 The projected distance ^, of the comet from the earth, being always positive ; this equation shews that the distance r of the comet from the sun is greater or less than the distance R of the earth from the sun, according as /u.' is positive or negative ; these two distances are equal, if i^' = 0. We can, by the sole inspection of the celestial globe, determine the sign of/ ; and consequently, whether the comet is nearer or farther than the earth from the sun. For this purpose, let us conceive a great circle, which passes through two geocentric positions of the comet, indefinitely near to each other. Let y represent the inclination of this circle to the ecliptic, and ^, the longitude of its ascending node •, we shall have tan. y. sin. (« — x)— tang. 9 ; fiom which may be obtained rfO. sin. (a — x):=.doi.. sin. 6. cos. 6. cos. (a — a) ; differentiating a second time, we shall have c c 2 traded from the first, the quantity by which g is multiplied is the numerator of the value of fi', and the quantity independent of 5, is its denominator. If r be less than R,— ^ is positive, .". in this case ft' must be positive ; if r is greater than R, then — — is negative, .', /*' must in this case be negative ; when 196 CELESTIAL MECHANICS, + <.-r, c • sm. 9. COS. 6: (at ) f/*9, being the value of c?^9, which it would have, if the apparent mo- tion of the comet continued in the great circle. Consequently the value of j!*' becomes, by substituting for d0, its value dx. sin. S' cos. G. cos. (a — x) ^ sin. (« — x) ' ^ sin. 0. cos. 6. sin. {A — x) j The function ^■^"' ; { ■ is constantly positive : therefore the value sin. 9. cos. G •' "^ ^ .. . . . T W9) Sd'^J u .1- 01 ju.' is positive or negative, according as j-jjit — ) -ttt c has the same or a contrary sign, to sin. (^4— x) ; A — x is equal to two right an- gles, plus the distance of the sun from the ascending node of the ^ , , , di ... COS. (« — a), tan, 6 di . * d*. COS. let — a), tan. y— j- , . . a«. ■ — 7—^ — r = r- , . . rf«. cos. ^ ' COS. ^« ' sin. (a— x) cos.'« (« — x). sin. 6. COS. 6 = dS. sin. (« — x) ; hence dH,. sin. (« — a)+6?«. d6. cos. (« — x)=:d'<». sin. 6. cos. «. cos. (a — A)+c? A the sign of sin. (^A — a) is negative, and if —r~, -r-^ be also negative, the comet in the third position must deviate from the great circle from the very direction in which the sun appears as seen from the earth, if a be Z A, then sin. {A — a) is positive, .'. if d^i d^6 — - -fibe also positive, it is evident that the comet must be nearer than the earth to the sun, and .'. that in the third position, the comet must deviate from the great circle, from the direc- tion in which the sun appears from the earth ; on the contrary, if sin. (^-a) be negative, and d't d'e ,. . —jY -rf positive, in order that this may obtain, in this situation of the bodies, it is necessary that in the third position the comet should deviate from the'great circle, from the opposite side to that in which the sun appears, as seen from the earth. See Memoirs of the Academy of Berlin, for the years 1772, and 1778. t **=x'*+2§.j'.cos. « + 5*. cos. '« ; 7/' = y + 2ry. sin. « + g^ sin."*; s'=5» 198 CELESTIAL MECHANICS, but we liave a/zzR. cos. A ; y'zzB. sin. A ; therefore r' = — ^—-- + 2R.(. cos, (A—c,)i-R'. COS. -0 ^ By squaring the members of the equation (3), when arranged under the following form, we shall obtain, by substituting in place of r^, its value, ■ (h^+^^?- '^'- (^—n^y- (y-'-R'?+iy'=R' ; (4) In this* equation, ^ is the only unknown quantity, and it ascends to the seventh degree, because the term which is entirely known in the first member being equal to 72*, the entire equation is divisible by f. Hav- ing by this means determined f, we will obtain \-4.{ by means of the equations ( 1 ) and (2). By substituting, for example in the equation (1), instead of —^ ^^ , its value -^, which is given by the equa- tion (3) ; we shall have Y4l= "/ $5??? + />^'. sin. (A-«) I. The equation (4) is frequently susceptible of several real and positive roots ; by making its second member to coalesce with the first, and tlien dividing by f, its last term will be tan.'fl. .-.ar^+^H^'^ •'^''-+j/" + 2e.(j'. cos. a+y. sin. a)+5^(l-ftan. '^)=['^^^. Multiplying both sides of equation (3) by fi'. R'.r^, and we obtain ft'.R\^.r^ = R* ~ R.i", :. r\lfi'.R'i-\.l)=R'; .'. substituting for r^ its value we obtain f ~^ + 2i??. COS. (A—u) + R^)r. ifc'Rr-.^+l)=R\ * Ti*^ occurs on both sides of this equation with the same sign, therefore it may be omitted, and as the remaining quantity is divisible by g, it may be depressed to the seventh degree. PART I.—BOOK 11. 199 2. R'. COS. «9. (ix'.R^+3. COS. (A—<^)) ;* Thus the equation in ^, being of the seventh degree, it will have at least two roots which are real and positive, if /^'. iJ'+3. cos. (A — a) is positive jt for by the nature of the problem, it must always have a positive root, and it is evident from the nature of equations that when this is the case the number of its positive roots cannot be odd. Each real and positive value off, gives a different conick section for the orbit of the comet ; therefore we will have as many curves which satisfy three neighbouring observations, as ^ will have real and positive values, and in order to determine the true orbit of the comet, we must have recourse to a new observation. 32. The value of f, deduced from the equation (4) would be rigor- ously exact, if «, ('^),(^), 6, (|),^^^ , were exactly known ; but these are only approximate values. Indeed, we can by the method already laid down approach to them nearer and nearer, by employing a considerable number of observations, which has also the advantage, of enabling us to consider intervals sufficiently great, and thus to compensate by each other, the errors of observations. But this method is liable to the analytic inconvenience of employing more than three observations, in aprobleni in which three is sufficient. We can obviate this inconvenience in the following manner, which at the same * This equation when expanded becomes ^]i^.cos.(A—»)).R*-^R%{ft'°.R*f+2ftR''i + l)=RS when ii-^ is obliterated, and this expression is multiplied by cos. ''6, and divided by {, the absolute quantity is evidently equal to (2i?. cos. ( 4—»). SR^ + ^fiR"). cos. H. f This equation being of the seventh dimension, when the absolute quantity is positive it must have one real negative root, -and from the nature of the problem it has one real affirmative root, .'. as impossible roots enter questions by pairs, the number of those in the proposed ' equation cannot exceed four ; consequently, in order that the sign of the absolute quantity may be positive, the remaining real root must be pos itive. 200 CELESTIAL MECHANICS, time that it only employs three observations, will render our solution as accurate as we please. , For this purpose let us suppose that a and 6 represent the geocen- tric longitude and latitude of the intermediate observations ; by sub- stituting in place of a-, y, z, their values a^+f . cos. « ; y +f • sin. « ; and J. tang. 6; they will g'^^)^^?') ^^(^"'^ LdfY ^" functions of f, «, and e, of their first differences and of known quantities. By differen- tiating these functions, we will obtain, -l --| oi -;^ f and -{^Yy in functions of f, «, 9, and of their first and second differences. We can eliminate the second difference of f, by means of its value, and its first difference, by means of the equation (2) of the preceding number. By continuing to difference successively, the values of <-^>-, J — t, and then by eliminating the differences of a. and 9, superior to the second, and all the differences of f, we will obtain the values of j^ll' ®' {SKIS} ' *^"' ^""^ premised, let «, a', a'", be the three observed geocentric longitudes of the comet ; 6, 9', 9''', its three corresponding geocentric latitudes ; let i be the number of days which intervene between the first and second observation, and i', the number which separates the second observation from the third j finally, let a be the arc which the earth describes in a day by its mean sidereal motion j by No. 29, we shall have PART I— BOOK II. 201 By substituting in these series, tor ■< -T-f r > < -jji r > ^^- i ^ j > -7^ >-, &c., their values obtained by the preceding method; we shall have four equations between the five unknown quantities f, {l}.{|-"}'{l!}'{S}- These equations wUlbe alwa,sn,ore exact, according as we consider a greater number of terms in the preced- ^ .. , „ . • ff/»l (d"i^\ idH . ing series. By this means we shall obtain, < j^f, iTTai'i^j ^ -3 T-y > , in functions of f and of known quantities ; and by substitut- ing them in the equation (1) of the preceding number, it will only involve the unknown quantity ^. In fine, this method which has been detailed here, merely in order to shew how by means of three observations only we can obtain continually approaching values of f, would require in practice, very troublesome computations, and it is at the same time more exact and more simple, to consider a greater number, by the method explained in No. 29. 33. When the values of ? and of 4 -r^ > shall have been determined, we can obtain those of J-, t/, ^A^)>\-Jt) ^^^(~77) ^y "1^^"^ of the equations x — R. cos. A-{-^. cos. a ; j/ =r 72. sin. A + ^. sin. a. ; z zr. ^. tang. 6 ; and of their differentials divided by dt. PART. I. BOOK II. U D '202 CELESTIAL MECHANICS, (§)= ©• «'»• ^ + ^- ©• ^««- ^4+ (|)- «'"• -+e- ©• ^•'>^- - ; (^)=(|).tan.H.O. COS. ^e The values of f-^] and of (-77) are given by the theory of the mo- tion of the earth : in order to facilitate their computation, let E repre- sent the eccentricity of the earth's orbit, and H the longitude of its perihelion ; by the nature of the elliptic motion we have, •dA\ ^^\—E^ „ l—E'' fdA\_ Vl-E\ dti- R^ ' l+E,cos.{A—H) These two equations give ulR\ _ E. sin. (A— //) . let R' represent the radius vector of the earth corresponding to A, the longitude of this planet, increased by a right angle ; we shall have R'= '-^' l—E. sin. (A—H) ' from which may be obtained R'—l + i- E. sin. (A—H)=- M' dA , . * , being equal to the angular velocity of the earth, it is equal to the square root of the dt parameter divide by the square of the distance, .'. it is equal to — ™ — • PART I.— BOOK II. 205 consequently (dR\_ R' + E'—l ^ \dtf- R'.k/x—E'' If we neglect the square of the excentricity of the terrestrial orbit, which is very small, we shall have the preceding values of i—jj) ^^^\^) ^^^^^ consequently become (§)=(^'-''- ™^- ^ -■ Tr-+ (*)• ""'■ "-<• (^)- ''"• "• (f )= (^'-l).si„. A + SStA+ ©, sin, .+, (§). COS..; R, R' and ^ being gi/en immediately by the tables of the sun, the com- putation of the six quantities x, y, z, (^j)' (~^)' ^^^ \^)^^^^ be easy, when f, and f;^) will be known. The elements of the orbit of the comet can be deduced from them, in the following manner. D D 2 dt dt {\-\-L.cos. A — Hf (1 — L-f ^ ^ ' • (\—E%E. sin. (A—H) ^ E. sin. {A—H) _ 1— £' {\-irE.coi.{A—Hf VT^E" ' ~ l+E.cos. {A + 90—H) =(1 — £').(! — E. sin. (^ — //))-'= (when the square of E is ne- 1— jE. sin. (^-//) glected) l+£. sin. (A—H), :. R'—\(=E. sin. (A—H)) isequal(^) , when £> dected. '204 CELESTIAL MECHANICS, The indefinitely small sector, which the projection of the radius vector of the comet on the plane of the ecliptic, describes during the element of time dt, is - — ^^-^- — ; and it is manifest that this sector is positive or negative according as the motion of the comet is direct or retrograde} thus, the sign of the quantity x. x-jA — 2/'(^f)> ^^''' indicate the direction of the motion of the comet. In order to determine the position of the orbit, let us name which determines sin. /, and consequently the angle 7, which expresses always the longitude of the ascending node. a and ea representing the semiaxis major, and excentricity of the orbit, we have, by N°'. IS and 19, f* being supposed =1, I dz rfy \ By substituting we obtain ^ — sin./.(..^_^.^) dy , dx . ^ du dii _, \ y-^- cos- '—y'~^- *"*' ^—y- -^- cos./+x. -^. sm. /). tan. ?.j (dy dx\ . ^ '• -df-y-df )■''''■ ^■ t dy dx\ . ^ = / du dx \ T ■ '="'^ f- 906 CELESTIAL MECHANICS, a~ r \dt J \dt ] \~di ) ' The first of these equations determines the semiaxis major of the orbit, and the second determines its excentricity. The sign of the timctioncr. ( y, J -\-y- \Zjf) +**{"^) '"^kes known whether tlie comet has ah-eady passed through its perihelion ; for if this function is negative, it approaches towards it ; in the contrary case, it has already passed this point. Let T represent the interval of time comprised between the epoch which we have selected, and the passage of the comet through the pe- rehelion ; the two first of the equations (f) of No. 20, will give, by observing that /^ being supposed equal to unity, we have n = a~^, r— 0.(1 — e. COS. ii) ; Tzza^.(u — e. cos. u). The first of these equations gives the angle u, and the second makes known the time T. This time added or subtracted from the epoch, according as the comet approaches or departs from the perihelion, will give the instant of its passage through this point. The values of a; and of i/, determine the angle which the projection of the radius vector r makes with the axis of x, and as we know the angle / made by this axis, with the line of the nodes, we shall have the angle which this last line constitutes with the projection of ?• ; from which may be obtained, by means of the inclination (p of the orbit, the angle which the line of the nodes makes with the radius /•. But the angle u being known, we shall have by means of the third of tlie equations (J"), of No. 20, the angle v, which this radius forms, with the line of the apsides ; therefore we will have the angle comprised between the two lines, of the apsides and the nodes, and, consequently, the position of the peri- PART I.— BOOK II 207 lielion. All the elements of the orbit will be thus determined. 34. These elements are given, by what precedes, in functions of c ( -£ ) , and of known quantities ; and as ( -4.) is given in f , by No. 31 ; the elements of the orbit vvill be functions of f, and of known quantities. If one of them was given, we would have a new equation, by means of which we could determine j ; this equation will have a common divisor with the equation (4) of No. 31; and seeking this divisor by the ordinary method we will arrive at an equation of the first degree in ^, we shall have besides, an equation of condition be- tween the data of the observations, and this equation will be that which should have place, in order that the given element might belong to the orbit of the comet. Let us now Jpply this consideration to nature. For this purpose, we may observe that the orbits of the comets are very elongated el- lipses, which are sensibly confounded with a parabola, in that part of their orbit in which these stars are visible ; therefore we may suppose without sensible error, that a = oo, and — = ; consequently the expression for — of the preceding No. will give, ^- T de If we afterwards substitute, instead of{^)>(-hj)f^] their va- lues, which are found in the same No. ; we shall have, after all reduc- tions, and by neglecting the square of R' — 1, -a)V(^:)v{(i)— ^my t cos. *G y 208 CELESTIAL MECHANICS, ^dx\ C ,^ • r ^ N COS. (^ — a) 7 1 2 by substituting in this equation, instead of ( -^^ ) its value which has been found in No. 3 1 ; and then by making + f tan... (g?) +.■. .a,.. .. si„. (..-.)-., {^,).{§M' ^ ^ cos.^0 ) |(g)+p-.si.,.(^-.) ^ .^ * By making this substitution, the equation (.5) becomes ^ ( -© "^/ V — ^^~+;.c'.sin. (.J— «)y(iJ' — l).cos.(^ — a) — sin.-i;^!^^) COS. *(i J dt + .<.(S){,.._„.s..C.-,,+^i^}+i-f It is evident from an inspection of this expression, that B is equal to the quantity by ^vhich 5^ is multiplied, and that C is equal to the corresponding factor of {. *) ) . tan. (. PART L— BOOK II. 209 _ /'da\ C .„, . . , . . COS. [A — at) ) R we shall have and consequently r\|5.f^+C.e + -i^r=l; this equation is only of the sixth degree, and in* this respect it is simpler than the equation (4) of No. 31 ; but it is peculiar to the pa- rabola, on the contrary, the equation (4) is applicable to every species of conic section. 35. We may perceive by the preceding analysis, that the determi- nation of the parabolic orbits of comets, leads to more equationst than unknown quantities, we can, by different combinations of these equations, form as many different methods of calculating these orbits. Let us investigate those from which we ought to expect the greatest precision in the results, or which participate the least in the errors of observations. It is principally on the values of the second differences \-f^) and i-fjij, that these errors have a sensible influence ; in fact, it is neces- sary, in order to determine them, to take the finite differences of the geocentric longitudes and latitudes of the comet, observed during a PART I. BOOK II. E E * This equation is of the sixth degree for {* and r^ occurs in it, and if we substitute for r^ its value, in terms of j ; f will be the liighest dimension of 5 which occurs in it. t The reason why there are more equations than unknown quantities in this case, is because the axis major is supposed to be infinite. o 2W CELESTIAL MECHANICS, very short interval of time ; but these differences being less than the first differences, the errors of observation are a greater aliquot part of them ; besides, the formulae of No. 29, wliich determine, by the com- parison of observations, the values of «, 9, \7>)'(;t,)'Vj7i)» ^^^ \7jp') ' determine with greater precision the four first of these quantities, than the two last ; it is therefore advantageous to rely as little as pos- sible on the second differences of a. and of 9 ; and as we cannot reject them all at once, the method which only employs the greatest ought to lead to the most exact results; this being premised. Let the equations which have been found in the N"'. 31 and 34, be lesumed >_ ? » ^*= 7^-^ + 222f. cos. (^-^)+72* ; cos. *9 (d^'\_ R- sin. (^— g) (J_ 1 5 /rf^«\ ^dtf- ^~7^Y~V/i^""^i _^;^5 (L) ■ \dt) ^ r(U\ "' \dt/ , C(^) ^^ /^")\sin.^.cos.O \7ff)— — aP- •< ,„ , + 2. {-j-J. tan. 64- ^^ jr — — — , -r, sin. 6. cos. 6, cos. (A — a) (I 1 > ^dtJ »=(S)"+e*.{|)Vi©--+!j)r t COS. ■'9 ) (dp\ ( „ V , .. N sin. (A — a) > ^p. j(/i'— 1). COS. {A-») 5^ ^^ PART I.— BOOK II. 211 COS. (J — a) ? . 1 2 +2e.(^).[(i?'-l).sin.(^-.)+^ "•"ie* rd'h If we wish to reject (tjs)* 'f is only necessary to consider the first, the second, and the fourth of these equations ; eliminating (-J\ , from the last, by means of the second, we will obtain an equation which freed from fractions will contain a term multiplied by r*. f'', and other terms affected with even and odd powers of ^ and of r. If all the terms affected with the even powers of r, be reduced|into one member, and likewise all the terms affected with the odd powers of r ;* the term multiplied by r*. f* will produce one multiplied by r"'. f* ; there- fore by substituting instead of r*, its value given by the first of the equations (L), we shall have a final equation of the sixteenth degree in f. But instead of forming this equation, in order afterwards to resolve it, it will be simpler to satisfy by trials, the three preceding- equations. If we wish to reject ( / ; we must consider the first, the third and the fourth of the equations (L). These three equations would also lead us to a final equation of the sixteenth degree in f, which can be easily satisfied by trials. The two preceding methods appear to me the most exact which can be employed in the determination of the parabolic orbits of the comets ; it is even indispensably requisite to recur to them, if the mo- tion of the comet in longitude or in latitude is insensible, or too small for the errors of the observations not to alter sensibly its second differ- ence ; in this case we should reject that one of the equations (L), which contains this difference. But although in these methods, we G£ 2 • By squaring each member, we get rid of the odd powers of r, and the value of any even power will be obtained by means of the first of the equations (L ). 212 CELESTIAL MECHANICS, only employ three of the preceding equations ; yet the fourth is useful, in order to determine amongst all the real and positive values of ^ , which satisfy the system of the three other equations, that which ought to be admitted. 36. The elements of the orbit of a comet, determined by what precedes, would be exact, if the values of a, 6, and of their first and second differences, were rigorously correct ; because we have taken into account 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 are only permitted to neglect the square of this excentricity, as being too small a fraction for its omission to influence sensibly the results. But 6, a, and their differences, are always liable to some inaccuracy, as well on account of the errors of observation, as because these differences are collected from the obser- vations in an approximate manner. It is therefore necessary to correct these elements by means of three observations at considerable intervals from each other, which may be effected in an indefinite number of ways ; for if we know very nearly two quantities relative to the motion of a comet, such as the radii vectores corresponding to the two ob- servations, or the position of the node, and the inclination of its orbit, by computing the observations, at first with these quantities, and then with other quantities which differ very little from them ; the law of the differences between these results, will easily make known the cor- rections which those quantities ought to undergo. But among the binary combinations of quantities relative to the motion of the comets, there is one of which the calculation is the simplest, and which on this account deserves to be preferred ; and in a problem so compli- cated, it is a matter of importance, to spare the computer every superfluous operation. The two elements which have appeared to me to afford this advantage, are the perihelion distance, and the instant of the passage of the comet through this point ; for they not only may be readily deduced from the values off and of(^-r.)i but it also PART I.— BOOK II. ' 2J3 is very easy to correct them by observations, without being obliged, at each variation which they are made to undergo, to determine the other corresponding elements of tlie orbit. Let us resume the equation which has been found in No. 19, a.n—e^')=2r—~l- r\dr' dV a.{\ — e*) is the semiparameter of the conic sections of which a is the semiaxis major, and ea the excentricity ; in the parabola, where a is infinite, and ea equal to unity, a.(l — e*) is equal to twice the peri- helion distance ; let Z) equal this distance, the preceding equation becomes, relatively to this curve, -=^-4- m- ■ is equal to '^ ,' ; by substituting in place of r*, its value dt ^ dt + 2/?f . COS. {A — a)+i2*, and instead of ] -^ ^ and of j -t- [ , COS. *9 their values found in No. 33, we shall have rdr p C C dttl . ( d^ dt =^•lls^e■li^-■«^«•li•-(^-.)• V/T^. ,\ ,A ^ sin. M — a) 7 + ?• ^(^—1)- cos. (/f— «) ^-^ -\^ + f.i2. &?. sin. (^— «)+iJ.(i2'— 1). rdr -iff. sin. (^ - «) { (^)- (J*) } + iJ-C^); and by substituting R'-\ for (^) , 214 CELESTIAL MECHANICS, Let P represent this quantity ; if it is negative the radius vector r goes on diminishing, and consequently the comet* tends towards its perihelion ; but it moves from it, if P is positive. We have then tiie angular distance i; of the comet from the perihelion will be de- termined by the polar equation of the parabola cos. i» = -^; finally, the time employed to describe the angle v will be obtained, by the table of the motion of comets. This time added or subtracted from that of the epoch, according as P is negative or positive, will give themoment of the passage through the perihelion. 37. These different results being collected together, will give the following method, for determining the parabolick orbits of co_ mats. A general method for determining the Orbits of the Comets. This method will be divided into tdvo parts-; iii the first, we will give the means of obtaining very nearly the perihelion distance of the comet, and the instant of its passage through the perihelion ; in the second, we will determine exactly all the elements of the orbit, these quantities being supposed to be very nearly known. and _L for f4) we shall have ^ = -1- j /|) +,.i!^.g) I + cos. (A-,). R' \dt/ dt cos.^i \\dtJ COS. 6 \dt/ } PART L— BOOK II. 215 An approximate determination of the perihelion distance of a Comet, and of the instant of its passage through perihelion. i Let three, four, or five, &c. observations of the comet be selected as nearly as possible* equi-distant from each other ; with four observa- tions we can embrace an interval of 30° ; with five observations, an interval of 36°, or 40°, and so on of the rest ; but it is necessary always that the interval comprised between the observations should be more considerable, as they are more numerous, in order to dimi- nish the influence of their errors ; this being premised. Let €, €', C^', &c. be the successive geocentrick longitudes of the comet ; y, y', y'', the corresponding latitudes, these latitudes being supposed positive or negative, according as they are north or south. Let the difference S' — E be divided by the number of days which se- parates the first from the second observation ; in like manner, the difference C^' — S' be divided by the number of days which sepa- rates the second from the third observation ; we will also divide the difference C'''" — S ', by the number of days which separates the third from the fourth observation, and so of the rest. Let J£, J6', i^'\ be these quotients ; let the difference ,* 0=3/^+«\.r^+ Ji/. tan. e+ -^^j-^l +2y. ^ ^^^ L-^R'-l). COS. (E—=c)) (4) • All tlie observations made in the interval between the extreme observations may be made use of in determining « a, i, «, '■, and / ; for if each observaiion'be expressed in a linear function of these data, there will be more equations than unknown quantities ; the first final equation will be obtained if each equation be multiplied by the coefficient the first un- known quantity, the second final equation will be obtained by a similar process, and so on ; and the data will be given by a resolution of these equations with a precision which will be greater, as moie observations are made use of. This advantage is peculiar to this mcr thcd. (See Connaissance ties Temps, Annee 1824.) PART I.— BOOK 11. 219 a rr-D, in • r t? \, COS. (£ — a) 12 — 2ff^.((/2'— 1). sin. (£— «)H ^ V-^^ +— . In order to deduce from these equations, the values of the unknown quantities x, y, and r ; we must consider, in the first place, whether, abstracting from the sign, b is greater or less than /. In the first case, we employ the equations (l), (2) and (4). We make a first supposi- tion for a;, by supposing it, for example, equal to unity ; and from this we conclude, by means of the equations (IJ and (2), the values of r »nd of ^. We substitute then, these values in the equation (4), and if the remainder vanishes, it shews that the value of x has been rightly assumed ; but if this remainder be negative, the value of x must be increased, and it must be diminished, if this remainder be positive. By this means, we shall obtain by a small number of trials, the values of X, y, and r. But as these unknown quantities are susceptible of several real and positive values, it is necessary to select that value which satisfies exactly or very nearly the equation (3). F F 2 Since the publication of this book M. Laplace has ascertained that the best means of diminishing the influence which the errors of observation have on their results, consists in combining the equations (2) and (3), by multiplying the first by a', and the second by A', and then adding the products together, by means of which the following equation will be_obtained, _asin. (£ — a) — ^. sin. ^. cos. fl. cos. (£ — a).R /_! 1_ -\ X. A^. tang , i + ^ al. + \. h. l-\-\. a°h. sin. 6. cos. 6 i This equation combined with the equations (1), (4), will give the values of x, y, r. By making a first hypothesis for x, the equations (a) will give the corresponding values of r. and then the equation (5) will givey. Now if the value of x has been properly assumed, these values, when substituted in the equation (4) ought to satisfy it ; if this equation is not satisfied, a second value of x should be taken, and so on. Hence the perehelion distance D, and the instant of the passage through the perehelion, may be determined, 220 CELESTIAL MECHANICS, In the second case, i. e., if we have / > 6, we must employ the equations (1), (3), and (4), and then the equation (2) will serve to veri y the values deduced from these equations. Having by this means obtained the values of .r, y, and r j let P be assumed = ^'/^ + ''• *"• *^°' n — ^^' ^^^' ^-^ — *^ +^. f '"' '^f ~''^ — (^'—0. cos. (E—cc) l—R.ajr. sin. (E— a) + E.(R'—1). The perihelion distance D of the comet will be determined by the equation the cosine of its anomaly v will be given by the equation COS. ^.V := — } and from this we infer, by the table of the motion of the comets, the time employed to describe the angle v. In order to obtain the in- stant of the passage through the perihelion, this time sh.ould be added to the epoch, if P is negative, and subtracted from it, if P is positive, because, in the first case, the comet approaches the perihelion, and in the second case, it moves from it. Having thus determined very nearly the perihelion distance of the comet, and the instant of its passage through the perihelion, we can correct them by the following method, which has the advantage of being independent of an approximative knowledge of the other ele- ments of the orbit. PART I.— BOOK II. 221 An exact determination of the elements of the orbit, when ive knoxv very nearly the 'perihelion distance of the Comet, and the instant of its passage through the 'perihelion. In the first place, three observations of the comet, at a considerable distance from each other, should be selected, and then from the peri- helion distance of the comet, and from the instant of its passage through the perihelion, as data which are determined by what pre- cedes, we compute three anomalies of the comet, and the radii vectores which correspond to the instants of the three observations. Let V, xf , x/', represent these anomalies, (those which precede the passage through the perihelion being supposed negative) j moreover, let r, r', r", represent the corresponding radii vectores of the comet ; f' — V, x/' — V, will be the angles contained between r and r', and be- tween r, and r^' ; let U be the first of these angles, and U the se« cond. Likewise let a, a', a.", represent the three observed geocentrick longi- tudes of tlie comet, referred to a fixed equinox ; 0, 6', fi', its three geocen- trick latitudes, the southern latitudes being supposed to be negative ; let e, S', %", be its three corresponding heliocentrick longitudes j and sr, ■zs', -m", its three heliocentrick latitudes, finally, let E, E', E\ be the three corresponding longitudes of the sun ; and i?, il', jB' , its three distances from the centre of the earth. Let us suppose that the letter S indicates the centre of the sun ; 222 CELESTIAL MECHANICS, T that of the earth ; C the centre of the comet, and C, its projection on the plane of the ecHptic. Tlic angle STC\ is the difference of the geocentrick longitudes of the sun and of the comet ; by adding the logarithm of the cosine of this angle, to the logarithm of the co- .sine of the geocentrick latitude of the comet,* we will obtain the lo- garithm of the cosine of the angle STC; therefore in the trian^^le STC there will be given the side 52' or R ; the side SC or ?; and the angle STC; we can thus by trigonometry obtain the angle CST. The heliocentrick latitude of the comet will then be obtained by means of the equation sin. S. sin. C^r . sm. ro-n -. — „^p . T sin. CTS The angle TSC is the side of a right angled spherical triangle, of which the hypothenuse is the angle TSC, and of which one of the sides is the angle zj ; from which we can easily obtain the angle TSC', and consequently, the heliocentrick longitude S of the comet. In like manner, z/, Q', zr", ^" ; and the values of §, ^■,^", will deter- mine whether the motion of the comet is direct or retrograde. If we conceive the two arcs of latitude w, w', to meet in the pole of the ecliptic, they will make an angle equal to C — S ; and in the spherical triangle formed by this angle, and by the sides -— - — w, and -^ — w'. * If £ be the longitude of the sun, STC'=:x — E, and in the right angled spherical tri- angle, of which one side is the measure of « — E, and the other side about the right angle the measure of «, the hypothenuse will be equal to the measure of the angle at the earth between the sun and comet i. e. equal to S TC, .'.by Napier's rules we have cos. («—£). cos. «=cos. STC. t Sin. CST : CTS : distance of comet from earth : r'.: sin. w : sin. I, .'. sin. «7 = sin. i. sin. CST sin. CTS PART I.— BOOK II. 223 7r being the semiciicumference, the side opposite to the angle €' — g, will be the angle at the sun, contained between the radii vectores r and r'. It may be easily determined, by spherical trigonometry, or by the following formula : sin. *iF. rz COS. *i-. (zj-f-T;/) — COS. *^ (%' E). cos. zr. COS. -a/,* in which V represents this angle ; so that if we name A, the angle of which the square of the sine is cos. ^tV.(S — £). cos. -sr. cos. -s/, and which can be readily derived from the tables, we shall obtain sin. 'ir=COS. (iTir+iar' + J). COS. (i^+|n-' — A). Naming in like manner V the angle constituted by the two radii vec- tores r and ?■', we will have sin. 'iP = cos. (iur+i7i/+^0- COS. (^^ + ^t;/ — ^'), A' being what A becomes, when T3-' and C are changed into is" and ^". Now, if the perihelion distance of the comet, and the moment of its passage through the perihelion were accurately determined, and if the * This expression may be easily derived from the known formulae of spherical trigono- metry, for if we assume B = (S' — S) ; C = — ro ; C'= — — ■a' ; we shall have, cos. B * cos. V — cos. C. cos. C , _ 1 _ 1— COS. B— 2 sin. «--. B sm. C. sin. C 2 _ sin. C. sin. C— cos. F+cos. C. cos. C _ cos. f C — C) — cos V ~ sin C sin. C sin. C. sin. C" ' .-. 2 sin. ^i B. sin C. sin. C =cos. (C—C)— cos. F =2 sin.'f F— ? sin. ^^ (C— C), and since sin. "\B = 1 — cos."^B; and sin. -^ {C — C) = sin. 'A (C+C) — sin. C. sin. C, we shall have (2—2 cos. '^. 7?). sin. C. sin. C'= 2 sin. ^ F— 2 sin. 'J (C+ C)+ 2. sin. C. sin. C". .'. sin. =iF= sin. i (C+ C')— cos. ^iR sin. C. sin. C; which will give the expression in the text when their values are substituted for B, C, C. 224 CELESTIAL MECHANICS, observations were rigorous, we would have V=U; F'=U i but as this can never be the case, we will suppose m= U—V; m'= U'—V. (It is to be observed here, that the computation of the triangle STC, gives for theangle CST two different* values. Most frequently, the nature of the cometary motion will make known which of them ought to be employed, especially if these two valufes are very different ; for then one of them will place the comet farther than the other from the earth, and it will be easy to determine, by the apparent motion of the comet at the instant of observation, which ought to be selected. But any uncertainty which remains on this account may be removed, by taking care to select that value which renders Fand V very little different from U and Z7'.) Then we will make a second hypothesis, in which the instant of the transit through the perihelion remaining the same as before the perihelion distance varies by a small quantity; .e g. by a live hun- dreth part of its value, and then we seek in this hypothesis the values of U— V, and of U'—V ; let then n = Cf— V; n' zz U—V. Finally, we make a third supposition in which, the distance of the perihelicm remaining the same as in the first hypothesis, we make to vary by half of a day, more or less, the instant of the passage through the perihelion. And then let the values of U — V, and of U' — V be investigated on this new hypothesis. Let in this case p- U—V; f'- U'—V. This being premised, if u represents the number by which the sup- • The values of C5r, are CST, and 180— 2 STC— C'S 7'. PART I— BOOK II. 225 posed variation in the perihelion distance should be multiplied, in order to obtain the true distance, and t the number by which the supposed variation in the instant of the passage through the perihelion should be multiplied, in order to obtain the true instant ; we shall have the two following equations, {m — n). u-\-(7n — p). tz=i7n; (m' — ?/). tt-\- {m' — p'). t = m! ; by means of which equations we obtain the values of wand of?, and consequently the corrected flistance of the perihelion, and true instant of the passage of the comet through the perihelion. The preceding corrections suppose that the elements determined by the first approximation, are sufficiently accurate to enable us to treat their errors as indefinitely small. Both if the second approximation does not appear to be sufficient, we must recur to a third, by ope- rating on the elements already corrected, as we have done on their first values ; it is solely necessary in addition to secure that they un- dergo small variations. It will also suffice to compute by these cor- rected elements the values of U — V, and of Z7' — P j by represent- ing them by iVf and N, and substituting them in place oi m and m', in the second members of the two preceding equations ; we shall have by this means two new equations which will give the values of u and of t, relative to the corrections of these new elements.* Having by this method obtained the accurate distance of the peri- PART. I. BOOK II. G G If in place of computing 17, U' , V, V, on the three hypothesis mentioned in the text, they were computed on the five following hypotheses, 1st, with the elements found in the first approximation ; 2dly, by making the perihelion distance to vary by a very small quan- tity ; 3dly, by making it to vary by twice the same quantity ; 4thly, the same perihelion distance as in the first hypothesis being preserved, by mab'ng the instant of the passage 226 CELESTIAL MECHANICS, helion, and the true instant of the passage of the comet through the perihelion ; the other elements of the orbit may be inferred in the fol- lowing manner. Let J be the longitude of the node which will be the ascending one, if the motion of the comet be direct, and (p the inclination of the orbit ; we shall obtain by a comparison of the first and last ob- servation, . tan. ■ST. sin. Q'^ — tan. zj''. sin. g ^^ ''^~ tan. ■nr. cos. S'' — tan. z/^ cos. Q ' tan. ■sr''' tan. ' -r^ ^f^ ian.^vzz — ; D = Therefore we shall have v the anomaly of the comet at the instant of the first observation, and its perihelion distance D, hence it is easy to infer the position of the perihelion, and the instant of the passage of the comet through this point. Thus, of the five elements of the orbit of the comet, four are known, nauiely, the pei'ihelion distance, the position of the perihelion, the instant of the transit of the comet through this point, and the position of the node. It only remains to find out the inclination of the orbit ; but for this purpose it will be necessary to recur to a third observation, which will also be useful in indicating amongst the different real and positive roots of the equation in f, that of which we ought to make use. 38. The hypothesis of the parabolick motion of the comets, is not rigorously true, it is even very improbable, considering the infinite num- ber of cases which give an elliptic or a hyperbolic motion, relatively to those which determine a parabolic motion. Besides, a comet which moves r sin. -A u ♦ Dividing r and its value by r'and its value respectively, we have -t = — " — n — , — -, and also we nave \- —r— — - — = -^rr • D r ' r' rr D PART I.—BOOK II. 231 in either a parabolic or an hyperbolick orbit, would be only visible once ; therefore we may with great appearance of probability suppose, that the comets which describe these curves, if any such ever existed, have long- since disappeared, so that at the present day, we only observe those, which moving in orbits returning into themselves, are perpetually brought back, after greater or less intervals, into the regions of space, near to the sun. We can by the following method, determine nearly within an interval of some years, the duration of their revolutions, when we shall have made a great number of very accurate observations before and after the passage through the perihelion. For this purpose, let us suppose that we had four or a greater number of accurate observations, which may embrace all the visible part of the orbit, and that we have determined by the preceding method, the parabola, which satisfies very nearly these observations. Let V, v', i/^, if', &c., be the corresponding anomalies, r, /•', -Z^, r"', &c., the corresponding radii vectores. Let also i/—v=U; i/'—v=U'; 'i/''—v=U"; &c. this being agreed upon, we compute by the preceding method, with the parabola already found, the values of U, U, U'\ &c., V, V, V\ &c.; let m=U—F; m'=U'—V'; m"=iU"—V"; m'"=U"'—V'"; &c. Afterwards, suppose the perihelion distance in this parabola, to vary by a very small quantity ; and let in this hypothesis, n=U—V; n'zzU'—V; n"zzU"—V"; n'"zzU"'—V'" ; &c. We then make a third hypothesis, in which the same distance of the perihelion being preserved, as in the first, the instant of the passage through the perihelion is varied by a very small quantity ; let then p= U—V; p'=zU'—V' ; p"=U"—V''; f'=U'"—V'"; &c. Finally, we will compute with the perihelion distance, and the instant 232 CELESTIAL MECHANICS, of the passage of the comet through the perihelion of the first hy- pothesis, the angle v, and the radius vector r, on the hypothesis that the orbit is elliptic, and that the difference 1 — e between its excen- tricity, and unity, is equal to a very small quantity, for example, to a 50th part. In order to obtain the value of the angle v on this hypothesis, it will suffice, by No. 23, to add to the anomaly v, com- puted in the parabola of the first hypothesis, a small angle of which the sine is -j^^. (1 — e). tang. \v. (i— 3. cos. *^t;— 6. cos. "^^v).* By substituting then in the equation r-=z iT-.il — ^ . tan.^lyC; in place of v, this anomaly thus computed in the ellipse ; we will obtain the corresponding radius vector. In a similar manner we can compute, v', r', %>", r", 'o", r'", &c. ; by means of which we can obtain the values of U, V, U", U'\ &c., and by No. 37, those of V, V, V\ &c. Let in this case q=zU—V; q'=U'—V'', q'zzU' — V"; q"' = U"—V"; &c. Lastly, let u denote the number by which we must multiply the supposed variation in the distance of the perihelion, in order to obtain the true distance ; and t the number by which the supposed variation in the instant of the transit through the perihelion must be multiplied, in order to obtain the true instant ; and s the number by which the * When the orbit is supposed elliptic, we must have at least four observations ; and then if the arc obsei-ved be considerable, and particularly if it is greater than 90°, the ellipticity will be very sensible, and the periodic time may be determined with tolerable pre- cision, if the four observations be made with all the precision of modern observations. If the square of a be neglected, the expression for r will be — ^-57 — II — ^-— . tan. i I which becomes the expression in the text when 1 — e is eubstituted for a. PART L— BOOK 11. - 235 supposed value of 1 — e must be multiplied, in order to obtain the ac- curate value, we will thus form the following equations, (m — n). u-\-(7)i — p). t-\-(m — q). s=m (in — n'). u-\-{in' — p'^. t-{-(iii' — q'). s:zzni (^m"—n"). u+(m"—p"). t-\-{vi"—q''). s=m" (ot'"— n'"). u-^(ni"—p"). t-\-(rn"'—q"'), s=m"' ; &c. The values of m, t, s, may be determined by means of these equations, from which we can infer the true distance of the perihelion, the true instant of the transit of the comet through the perihelion, and the true value of 1 — e. Let D be the perihelion distance, and a the semiaxis major of the orbit ; we shall have a =: — ; the time of the comets sidereal revolution will be expressed by a number of sidereal years, equal to a~, or to { ) ", the mean distance of the sun from the earth being taken for unity. Afterwards by N°. 37, we shall get the inclination of the orbit, and the position of the node. Whatever be the precision of the observations, they will always leave some uncertainty as to the duration of the comets revolution. The most exact method to determine it, consists in comparing the observa- tions of a comet, in two consecutive revolutions; but this means is not practicable, except when the lapse of time brings the comet back to- wards its perihelion.* PART I. — BOOK II. H H 234 CELESTIAL MECHANICS, CHAPTER V, General methods for determining, by successive approximations, tlie motions of the lieavenli] bodies. 40. In the first approximation of the motions of the heavenly bodies, we have only considered the principal forces which actuate them, and from thence the laws of the elliptic motion have been deduced. We will consider, in the following investigations, the forces which disturb this motion. In consequence of the action of these forces, it is only requisite to add small terms to the differential equations of the elliptic motion, of which we have previously detei'mined the finite integrals : it is necessary now to determine, by successive approximations, the inte- grals of the same equations, increased by the terms which arise from the action of the disturbing forces. For this object, we here subjoin a general method, which is applicable whatever be the number and the degree of the differential equations, of which it is proposed to find the perpetually approaching integrals. Let us suppose thai we have between the n variables y, i/', y", &c. and the variable /, of which the element dt may be considered as con- stant, the n differerential equations &c. P, Q, F, Q', &c. being functions of t, y, y', &c. ; and of their dif- ferences continued to the order i — 1 inclusively, and a being a very PART I.— BOOK II. 235 small constant coefficient, which, in the theory of the celestial motions, is of the order of the disturbing forces. Let us in the next place sup- pose that we have obtained the finite integrals of these equations, when Q, Q, &c. vanish ; by differencing each, i — 1 times in succession, they will constitute with their differentials, in equations by means of which we can determine by elimination, the arbitrary quantities c, c', c", Sic. in functions of t, i/, y, y", &:c. and of their differentials to the order? — 1. Therefore, if T, V, V", &c. represent these functions, we shall have* c=V; c = V'; c"=V" ; &c. These equations are the in integrals of the order i — 1, which the dif- ferential equations ought to have, and which their finite integrals fur- nish by the elimination of the differences of these variables. Now, by differentiating the preceding integrals of the order i — 1, we shall have 0-dV; O-dV; = dV" ; he. but it is evident that these last equations being differentials of the order i, without involving arbitrary quantities ; they can be no other than the sums of the equations H H 2 • In every differential equation of tlie order i, the number oi Jirst integrals is equal to f, these integrals are of the order i — 1, and therefore they only contain the i — 1 differen- tial coefficients — > — — — —4; and if these could be eliminated we would have dt dt~ d'~h the 1'* integral, or the primitive equation, which corresponds to the proposed differential equation ; consequently, if we have n differential equations, the number of first integrals, or of integrals of the order i — 1, must be in, from which if the differential coefficisnts of the variables y, y, y", &c. could be eliminated we would obtain the n finite integrals of the proposed equations. 2S6 CELESTIAL MECHANICS, multiplied respectively by suitable factors, in order that these sums may be exact differences ; therefore representing the factors which ought to multiply these equations respectively in order to form the equation dV = 0, by Fdt, F'dt, &c., and in like manner, representing by Hdt, H'dt, Sec. the factors which ought respectively to multiply the same equations, in order to constitute the equation O^dF' ; and so of the rest, we shall have dV-F.dt. I ^ + -P ( + -P"'-^^- 1 "^' + -P' ( + ^^' ' dV = HAL 1^^ + P^+//'.rf/. j^' + P'( 4- &c. 5 F, F', he. Hf H', &c. are functions of t, y, y', y" , &c. and of their differences to the order i— 1 : it is easy to determine them, when V, d'y V, &c. are known ; for F is evidently the coefficient of — ^ , in the d'y' differential of F; jF" is the coefficient of-^ , in the same differential, and so on of the rest. In like manner, H, H', &c. are the coefficients of -^ , ^, &c. in the differential of V ; consequently, as the tit Ut functions of V, V, &c. are supposed to be known, by differencing d'~^ 11 d'~^ v' them solely with respect to -^^ , -r^ , &c. we will obtain the factors by which the differential equations should be multiplied in order to obtain the exact differences ; this being- premised, let us resume the differential equations = -g- + P^.,Q, ; = -^ + F^«.Q; &c. PART I— BOOK II. 237 The first being multiplied by Fdt, the second by F'dt, and so of the rest, and then added together, will give - dV'+cc.dt.(FQ+F'Q+ &c.) ; in like manner will have, =z dr'+o^,dt.(HQ+H'Q+ &c.) ; &c. hence we obtain by integrating, c — a^.fdt.{FQ-\-FQ+kc.)-V; c'—oc.fdt(HQ+H'Q+kc.) = V ; &c. ; we will have by this means i?i differential equations which will be of the same form as when Q, Q', &c. are equal to nothing, with this sole dif- ference, that the arbitrary quantities, c, d , d', ho., must be changed into c_a./^?/.(J'Q+FQ'+&c.)j c—o,.fdt.{HQ.-\-H'Q+ &c.) } &c. Now if, on the hypothesis of Q, Q', &c. equal to zero, we eliminate from the m integrals of the order i — 1, the differences of the variables y, y, &c. ; we shall have the n finite integrals of the proposed equa- tions ; consequently, the integrals of the same equations, when Q, Q', &o. do not vanish, will be had, by changing in tlie first integrals c, d, &c. into • c—^.fdt.{FQ + F'Q+ he); c—c..fdL(HQ + H'Qi- &c.) 4(1. If the differentials dt.(iFQ+FQ:-\- &c.), dt.{HQ-\-H'Q[-\- &c.) &c. were exact, we could obtain by the preceding method the finite integrals of the proposed differential equations ; but this does not obtain except in some particular cases, of which the most extensive and the most in- teresting, is that in which these equations are linear. Let us, there- 233 CELESTIAL MECHANICS, fore, suppose that P, P', &c. are linear functions of y, y', &c. and of their differences to the order i — J, without any term independent of these variables, and at first let us consider the case, iu which Q, Q, &c. vanish. The differential equations being linear, their successive inte- grals will be also linear, so that c— V, d — V, being the in integrals of the order i — 1, of the differential linear equations V, V, &c. may be considered as linear functions of ?/, y, he. and of their differences, to the order i — 1. In order to demonstrate this, let us suppose, that in the expressions of r/, y, &c. the constant arbitrary quantity c is equal to a determinate quantity, added to an indeterminate Sc ; the constant quantity c', is equal to a determinate quantity added to the indeterminate, Sc, &c. ; these expressions being reduced into series, arranged with respect to the powers and products of Sc, Sd, &c., we will have by the formula of No. 21, &c. Y, Y, { -J- ) , &c. being functions of t, without arbitrary quan- tities. By substituting these values in the proposed differential equations, it is manifest that Sc, Sc, &c. being indeterminate, the coefficients of the first powers of each of them, must vanish in those different equations ; but these equations being linear, we shall have PART I.— BOOK II. 239 evidently the terms affected with tlie first powers of Sc, Sc', &c., by sub- stituting ( -7— ) ■ oC-\- { —p- !. which being comprized in the two 4a COS. ^ ' ■' or terms — . sin. at-\ . cos. at, may be neglected ; 2dly, the term a 7^f COS ± . . \at-\-t), the sign + obtaining, if the term of the expression ^Cl sill* of Q is a sine, and the sign — having place,* if this term is a cosine. It appears from what has been stated above, how the arc t is produced without the signs of sine or cosine, in the values of 2/, y, &c. by the "/ adding these two expressions, and observing that a-°-[-a-*.m'-|-a-^.w*-f &c. = we shall arrive at the expression given in the text. * The parts under the sign of integration in this case are respectively /. sm. ( at+t). COS. at.dt,/. sin. (a« + e). sin. at.dt) —f. sin. at. cos. at. cos. i.dt -\- f. cos. ^at.dt. sin. 1, f. sm, ''at. cos. i.dt +_/. sin. at. cos. at sin. i.dt, and these expressions are = \f. sin. 2,at. cos. i.dt + ^f. cos. 2at, sin. udt -|-^ /sin. i.dt, and — ^f. cos. 2at. cos. i.dt-\-^/. cos. i.dt ■\-\f. sin. 2.at. sin. i.dt, and by integrating these expressions become — — . cos. 2at. cos. c J . sin. 2at. sin. i + A. sin. e. t, — — — . sin. 9.at. cos. s +* cos. ut. ; — . cos. 2.at.' ^ 4a ''4a 4a sin. t, and if the three first terms be multiplied by '■ — '■ — , and the three last by a a. cos. at ^ . . , they become respectively + -^. sin. at. cos.2a<. cos. 1 — ; . sin. 2a*. sin. at. sin. 1 — -. sin. at. sin. t.t — —-r. ' 4a* 40*^ 2a 4a' sin. 2a^ cos. at. cos. i A . cos. at. cos. i.t — -—,. cos. at. cos. 2ai. sin. t = ^ 2a 'ia' 4. JL . sin. at. (cos. 2a^ cos. e — sin. 2at. sin. e) _ -^ . cos. at. (sin. 2af. cos 1 + ' 4a" 40' COS. 2o^ sin. A + ~ . (cos. a«. cos. «— sin. at. sin. t) = — ^. sin. a^ cos." (2a< + t) — cos.af. sin. (2at + 1)) =^^- s'". (ai+i)+-^ ■ cos.(af+e). PART I.— BOOK II. 243 successive integrations, although the differential equations do not con- tain it under this form. It is evident that this will be the case as often as the functions FQ, F Of, &c. HQ, H'Q\ &c. contain con- stant terms. 42. If the diflFerences rfA(2^Q+ &c.), dt.(HQ-\- &c., are not exact, the preceding analysis will not give their rigorous integrals; but it suggests a simple means of obtaining integrals more and more approach- ing, when a is very small, and when the values of i/, y' , &c. on the hypothesis of a. being equal to cypher, are known. By differentiating these values, i — 1 times in succession, we will obtain the following differential equations of the order i — 1, cz^V; c'-V; &c. d'li d'li' The colficients of —-jjj- , —~ > i" the differentials of V, V, &c. being the values of F, F, &c. H, H', &c. ; we will substitute them in the differential functions dt.iFQ+FQfi- &c.); dt.(HQ-{-H'Q+8cc.)* Afterwards, we must substitute, in place of y, y', &c,, their first ap- proximate values ; which will give their differences in functions of t, and of the arbitrary quantities c, d, &c. Let T.dt, T'dt, &c., be I I 2 * Let y = (p.(<, c, c', c", &c.) be the value of ;/, when «=0, which being substituted in place of^, in the function dt.{FQ+F'Q.'), dt.(HQ+H'Q.')+ &c. these functions will de- pend on t, and c„ c/, c,", &c. •.• i/ = .(t, e,)=zX) the expression in the text. See an example of this method m Chapter 7, Article 53. PART I.— BOOK II. 249 &c. hence it is easy to infer by eliminating R, and its differentials, from the preceding expression of J/, that X is a function of t, and of the constant quantities c, d, d', &c. ; and as these constants are functions of 9,'X is a function of t and of 9, which we can represent by (p.{t, S). The expression of y is, by the formula (0 of N°. 21, the expansion of the function (f.{t, Q + t—d), according to the powers of t — d ; therefore we have J/ zz (p.(t, t) ; it follows from this that the value of ^ will be had by changing 9 into t, in A'. The proposed problem is by this means reduced to the determination of X, in a function of /, and of 9, and consequently to the determination of c, d, d', &c., in functions of 9. For this purpose, let the equation ^ + X+(^— 9). F+(f— 9)*. Z+(/— 9)'- 8+ &c. be resumed. The constant quantity 9 being supposed to disappear from this value of y, we have the identical equation By applying to this equation the same reasoning as in the case of the equation 0=A. + A''^+ ¥t'' + &c., it will appear that the coefficients of the successive powers of {t — 9), must of themselves be equal to zero. The functions X, Y, Z, &c., do not involve 9, except as far as it is PART I. BOOK II. K K i250 CELESTIAL MECHANICS, contained in c, d, &c. ; so that in order to constitute the partial dif- ferences I -37- ) > ( —j^ j , I —jT- j , &c., it is sufficient merely to make c, d, &c., vary in these functions, which gives (dX\_(dX\ ^,(dX\dd_ (dX\ dd' \d^ )~ \dc )' d^^Kddj' M "^ \dd')'~dr'^ ^^' ' \d^ )~ \dc )'d^^ \dd )' d^^ \ dd'J ' d^ "^ ^^' ' &c. Now it may happen that some of the arbitrary quantities c, d, d', &c., multiply the arc t in the periodic functions X, Y, Z, &c. ; the dif- ferentiation of these functions relatively to 0, or which comes to the same thing, relatively to these arbitrary quantities, will develope this arc, and make it issue from without the signs of the periodic functions ; the diflFerences (jfl-j'f-^j'f"^)' '^^•■> '^'^ ^'^^'^ ^^ °^ ^^ following form : i: in which X\ X", F', Y\ Z, Z", &c., are periodic functions of U and moreover involve the arbitrary quantities c, d, d', &c., and their first differences divided by d^, which differences do not occur in these functions, except under a linear form ; we shall therefore, have PART I.— BOOK II. 251 &c. This value being substituted in the equation (a), will give ^ + (i^9). ( F + 6 Y' + X"—<2Z) + (t—^)\ {Z' + ^Z'+Y"—SS)+ &c. hence we deduce, by putting the coefficients of the powers of / — 9, separately equal to nothing, = X'+ 6. X"—Y-y = r'+ 0. Y"+x"—^z-y O = Z' + 0.Z''-{-Y"—3S; &c. The first of these equations, being differenced i — 1 times in succession, with respect to t, will give a corresponding number of equations be- tween the arbitrary quantities c, c, d', &c., and their first differences divided by dt ; the resulting equations being afterwards integrated, with respect to 0, will give these constant quantities in functions of 0. The sole inspection of the first of the preceding equations will almost always suffice to determine the differential equations in c, c', d'. Sec, by comparing separately the coefficients of the sines and of the cosines which it contains ; because it is manifest that the values of c, d , 8cc. being independent of ^, the differential equations which determine them ought to be equally independent of this quantity. The simplicity which this consideration produces in the computation, is one of the principal ad- vantages of this method. Most frequently these equations can only be integrated by successive approximation, which may introduce the arc 9, without the periodic signs, in the values of c, d, &c., even when this arc does not occur in the rigorous integrals ; but we can make it to disappear by the method which we have laid down. KK 2 252 CELESTIAL MECHANICS, It may happen that the first of the preceding equations, and its i — 1 diflPerentials in /, do not give a number i of distinct equations, between the quantities c, c', c", &c., and their differences. In this case, we should recur to the second and subsequent equations. When the values of c, d, d', &c., shall have been determined in functions of 9 ; we can substitute them in X, and by changing after- wards 9 into t, we will have the value of y, in which no function of the arcs of a circle occur, which are not affected by periodic signs, when this is possible. If this value still preserves them, it will be a proof that they existed in the exact integrals. 44. Let us now consider any number n of differential equations P, Q, F, Q', &c., being functions of y, y', &c., and of their dif- ferentials, continued to the order i — 1, and of the sines and cosines of angles increasing proportionably to the variable t, of which the dif- ference is supposed to be constant. Let us suppose that the approxi- mate integrals of these equations are y-X-\-t.Y-\- t\Z + t'S + &c. ; y'=X,-\- 1. Y,+ t\Z,-\- t'S, + &c. ; X, Y, Z, &c., X,, Y,, Z,, &c., being periodic functions of t, and containing the in arbitrary quantities c, d, d', &c., we will have, as in the preceding number, = X'+^.X"— Y; = Y' + e. Y" + X'^^Z; = Z'+t Z" -h Y" — 35 ; &c. The value of y will in like manner give equations of the following form : PART I— BOOK II. 25S = X: + 0. X/+ Y,', = y;+ 9. 17+ X/— 2Z,; &c. The values oi y", y''', &c., will furnish similar equations. By means of these different equations we can determine the values of c, c\ d', &c., in functions of 9, those equations being selected, which are the sim- plest and most approximative : by substituting these values in X, X,, &c,, and afterwards changing 9 into t, we will have the values of 3/, y, &c., not containing the arcs of a circle without the periodic signs, when this is possible. 45. Let us resume the method which has been explained in N". 40. It follows from it, that if in place of supposing the parameters c, d , d', &c., constant, we make them to vary, so that we may have dc = ^ cidt. (FQ + FQa- &c.) ; dc = — aclt. {HO. + H'Q + &c.) , &c. we will have always the in integrals of the order { — 1, namely c = V; d = V'j d'z=V''; kc. as in the case of a equal to zero ; hence it follows, that not only the finite integrals, but also all the equations in which only differences of an order inferior to i, enter, preserve the same form in the case of « equal to nothing, and of a being any finite value whatever ; because these equa- tions can result solely from a comparison of the preceding integrals of the order i — 1. Consequently, we can equally, in these two cases, difference i — 1 times in succession the finite integrals, without making c, c, &c., to vary ; and as we are at liberty to make all, vary at once, there results an equation of condition between the parameters c, d, &c., and their differences. In the two cases namely, of « equal to nothing, and of* being any quan- tity whatever, the values o^y,y', r/' , &c., and of their differences to the 254 CELESTIAL MECHANICS, order i — 1 inclusively, are the same functions of t, and of the para- meters c', d', &c. J let, therefore, Y be any function of the variables, Vi y't y"y &c., and of their differentials inferior to the order i — 1, and let us name T, the function of t, into which it is changed, when we substitute in place of those variables and of their differences, their values in t. We can difference the equation Y-=.T, by con- sidering the parameters c, d, d\ he, as constant ; we can even assume the partial difference of F, relatively to one only, or to several of the variables y, y, &c., provided that we only make to vary that part of T, which varies with them. In all these differentiations the parameters c, d, d', &c., may be always regarded as constant ; because, by substituting for y, y', &c., and their differences, their va- lues in /, we will have equations identically nothing, in the two cases of « equal to nothing, of a, having any finite value whatever. When the differential equations are of the order i — ], it is no longer permitted, in differentiating them, to treat the parameters c, c', d\ &c., as if they were constant. In order to difference those equations, let us consider the equation — dt.(SP + S'F-i- &c.) becomes identically equal to nothing, in consequence of the equations c ■= V, c'z=. V, &c. ; and as the same equations also obtain when the parameters c, c', c\ &c., are variable, it is manifest, that in this case also, the preceding function is identically equal to nothing, the equation (f) will consequently become " = (§)•* -(^) •*'+*«• — ».dt.CSQ + S'Q-\- &c.) : (^) It appears from this, that in order to difference the equation ^ = 0, it is sufficient to make the parameters c, c', — 0; d^=:0 rf^'4. = 0; therefore, by changing successively in the equation (x), the function ip into 4-, i^, =0; S'' ip z=0; S'lp = 0; thus, as these last equations obtain even when q, g', &c., are supposed to be variable, the equation /p,.rt(l — e'' ), ae being the excentricity of the orbit of »i- If we substitute in the expression of rir, in place of a: and of i/, their values r. cos. v, and r. sin. v, and instead of — , , the quanti- ty v/|W«.(l — e*) ; and if finally we observe that by N°. 20, we have p. = w*.a' ; we will obtain Sr = {'dR-i i — a. sin. v.fndt. r. cos. v.\'2,. fdR-{-r. ) ;t7 ( i «. cos. v.fndt. r. sin. v. \'2- j dR-\-r.\'—j-, ^ the equation (T) gives by integrating and neglecting the square of the perturbating forces ; PART. I. BOOK II. M M plied into y. be added to the second multiplied into — x, we shall obtain obliterating the quantities which destroy each other rl>r.y.^ + yfrlr. (^)_^ -2yfdt..AR-yf..rR _ r^r... 4 - xfr.lr. ('^) + f^t^ ^ix fdt.yd.R^xf y.r.R, at -^ \dx/ at .', neglecting quantities of the same order as the square of the disturbing force, we have r.Jr. X±ZfJL = X. /ydt.(2dR + rR')— y.fx.dt.{2dR -f rR'), which becomes the expression in the text, when r. (-r) is substituted for r.R' ; and by substituting for x xdy — ydx a.nAy their respective values r. cos. v, r sin. v, this equation is divisible by r; now ^^ . 3 an 1- =V(«.a.(l_e;) ; and Vfc = n.a' : consequently y^^' -'na'Vr^^ ~ 1 Vf<,a.(\—e'- 266 CELESTIAL MECHANICS, or.dJr+drJr 3a ^^ ,, ,„ , 2a „ ., URl ,v=-^ . -^7l^F~ ' (Y)* By means of this equation the perturbations of the motions of w in lon- gitude can be easily determined, when those of the radius vector shall have been determined. It now remains to determine the perturbations of the motion in la- titude. For this purpose, we shall resume the third of the equations (P), and by integrating it as we have integrated the equation (S), and making z zz ris, we shall have „ . . ^dRl . „ . cdRn a. cos. V. fndt. r. sm. vA-^- X — a. sm. v. J nat. r. cos. i'. J-r- C Sszz 7 — ? K^J jtA.V i — e' Ss is the latitude of m above the plane of the primitive orbit : if we dr.'Sr +2rd.^r , /./. ,,o „ , >d , „ /(^^\ * 3i)= — +ffdt-.^dR + 2r. (-p) = V'^.a.(l— r) V'^'-/ the expression in the text ; iZ'Sr is omitted as being' of the order of the squares of the disturbing forces. + Multiplying the third of the equations (P) by x, and subtracting it from the first multiplied by z, and then integrating, we shall obtain neglecting quantities of the order of the square of the distiu-bing forces dx dz , , /dR\ z. — X. — r- =r — I xAt. \—T-]t dt dt V ''2' in like manner subtracting the second of the equations (P) multiplied into z from the third multiplied into y, we shall obtain by integrating. dz dy . ,, fdR\ and multiplying the first of these equations by y, and the second by x, we obtain by adding them together PART I.— BOOK 11. 267 would wish to refer the motion of m, on a plane a Httle incHned to this orbit ; by naming s its latitude, when it is supposed to exist on this plane, s-\- Ss will be very nearly the latitude of m above the proposed plane. 47. The formula (X), (Y) and (Z), have the advantage of exhibit- ing the perturbations under a finite form ; which is very useful in the theory of the comets, in which these perturbations can only be deter- mined by quadratures. But in consequence of the little excentricity and inclination of the respective orbits of the planets, we are permitted to expand their perturbations, in converging series of the sines and cosines of angles increasing proportionably to the time, and to form ta- bles of them which may serve for an indefinite time. Then, instead of the preceding expressions of Sr and Ss, it is more convenient to make use of differential equations which determine these variables. By arranging these equations with respect to the powers and to the products of the excentricities and inclinations of the orbits, we can always reduce the determination of the values of Sr and Ss, to tiie inte- gration of equations of the form the integration of this species of differential equation has been given in N". 42. But we can immediately give this very simple form, to the preceding differential equations, by the following method. Resuming the equation (R) of the preceding number, and in order to abridge making, M M 2 -(-^-'•l)=V..*(f)-./-.(f). and by substituting for — — - — — its value V^f«.a.(l — e-), and for x and y their values r, COS. V, r. sin. v, we obtain the expression which is given in the text. 268 CELESTIAL MECHANICS, it thus becomes, In the case of elliptic motion, in which Q = 0, r* is by the N°. 22, a function of e. cos. (?it+c — zr}, ae being the excentricity of the orbit, and nt + i — sr being the mean anomaly of the planet m. Let e. cos. (nt -\- £ — ■s!-') = u; and let us suppose that ?'*= ''\(u) being equal to '^^ ■ ^ , in like manner as ?'.(w) is equal to ' ' ^ — . This being premised, if we make 7( ZZ e. COS. (?lt-\-i — ■sr)-^iu, the differential equation in u will become = ^ + "'•'"- ^^- fQ-^u. ,'.(„) + ^ .t and if we neglect the square of the disturbing force, u may be supposed to be equal to e. cos. («/+ 1 — zb-), in the terms depending on Q. The value of — found in N°. 22, gives, by carrying the precision to quantities of the order e* inclusively, r ■=■ a, (1+e*— M.(l — |e«) — a* — |w^) j hence we deduce r^ =z a*.(I + 2e*— 2m.(J— le*)— M*— M^) = ?..(z0.t If this value of <^{u) be substituted in the differential equation in hi, f Substituting in place of u its value, the part which involves the cosine will be equal to notliing, as is evident from the preceding page ; the other part is what is given in the text. X — = (as powers of e higher than the third are rejected) 1 +^e^ — (e— -|s'). cos. (?i/+ 2m- s — in) — \^. COS. 2(nt^t — sr) — ^e'. COS. 3(wi+s — w); cos. 2{«f+£ — PART I.— BOOK II. 271 and if we then restore in place of Q, its value Q.J'd.R + r.j -j- > , and e. COS. (nt-\-i — ra-), instead of tc, we shll have, as far as quantities of the order e', = — z H n.Su dt ^ g-. (1 + \^ — e. cos. {nt-\- i — Ts) — \ e^ cos. (2?j^+2£ — 2w)). |(2./di?+r.g]^- (X') . J ndt.\_^vs\. {nt-\-i — ■ss'). [1 + e. cos. (wf + t — w)]. a* 2./die+r.)f(]. When Jm shall have been determinined, by means of this differential equation ; into 1 -J- > ; hence it follows, that in order to obtain Ss, it is sufficient to effect this change in the differential equation of Su, and afterwards to substitute the value of Su given by this equation, and which we will denote by Su', in the expression of Sr. Thus, we shall have, = -^ +n^M — J-. (l+^e* — e. cos. (nt-\-( — w) — |e*. cos. (J2.nt-{-2i — So-)). [ ^ \ ■dR\ dz) 2e . , .... , .. .. fdR\ 'e f (Z) values are substituted in place of u, hence substituting for u and i^ their values, namely, ..cos.(»^+^.),-f..cos.2(„^+.-w)+ i-. ^=(2/diJ+r.(f)).(-^ A + -i-^ — -^ —e.cos. (n<+£— sr) — -^. COS. (2«<+2£— 2w)); ),= ^.(l+3„).(l_3.) ■=. (omitting terms which would by their multiplication produce powers of e higher than the third) -— j-; du ■=. — e. ndl. sin. (nl + £—•=>•)■, '-(u), their values just given we obtain the last term of the equation (X'). Jr = — a.(Ml— |e' + 2"+f. u% §"' = f- cos. 2(n< + t— r7) + f r, .•. dr=z— ahi.(l+^. e^+2e. cos. (ni-f e4-c7)+|. cos. (iiw/+2{— 2w). PART I.—BOOK 11, 273 is = — « — iw) ; cos. iixi. sin. p 1= 2 cos. i'w. sin.p = sin. (p+»'w)+sin. (p — i'to) ; i. sin. iw, cos. p=2t'. sin. i'ui.cos. /)=j'. sin. (^+»'i«)— «*. sin. (p — i'vi) ; i. sin. (fj«. sln.p=2i. sin. i'tu. sin. p= — V . cos. (p+i'iu) .(-i. cos. {p — iixi) ; in the se- cond member of these equations, the first term is changed into the second when i has a negative value, •.* if i is indifferently positive or negative, the second member is con- tained in the first ; hence we have cos. jw. cos. />=i. cos. (iio+p). &c. See note page 290. PART L— BOOK II. 2T5 This property will also enable us to express in a very commodious manner the perturbations of the motions of the planets. Let, in like manner (a* — lad. COS. {rJt—nt-\- i—C) +0'*)"* —\. E. B^. COS. t.{ril-^t->r t'—t) ; B^~'' being equal to JB<'\ This being premised, we shall have by the theorems of N°. 21, Tnf i? = — . 2. A^'\ COS. i.(n't—nt+ 1 — t)* , m' ^ fdA^'^ \ ., , , , + — . «,. S.flr. ( — T — 1 . cos.i.(n7 — nt-\-i—t) , m' . ^ , ( dA-'^ \ . , , , ^ + — . w,.E.a . ( —y-f \ . COS. t.(nt — 7tt+ i — t) —. (vl — U/).S.?. A^'\ sin. t.(n't-~^t+t — t) m' fd''.A^'^\ + — -. M,*.S.a*. I —fT~ ) • cos, i.(n't — 7it-\-c' — t) nn2 * Substituting for r, r', v, v', their values, the constant part of the value o K w , m'.aa'. COS. (n't — nt-i-t' — t)+zz^ m' become ^ I '— (o'^+2')l (a2_2(za', COS. (nV— «!!+£' — i) -j- n" -i- (z'_s)2)i which becomes (by reducing, and observing that terms higher than of the order of the square of the disturbing forces are neglected) =:m'.(aa'. cos.((jj'< — nt-\-i' 8)+«2'. (a'— ^ 5a'— V) i+ "^■^'-^^' (a* — 2aa'. cos. [n't — nt 4. i'_e ) + a'*)^ t^ („= — 2aa'. cos. {n't — nt + 1' — i) -J- a" ) 2 ' - Hi. 2.^W. cos, i.{n't-nt + ^'-^) 4. i^i^fl - i'"'-"'"- <^o ti!l'^"^ + ''-') + —-. (a'— s)^2. i5W. cos. i.(n7— H«+i'— s) ; now if a, a', n/-f s, ?i'/4-«'> be supposed to be increased by u', «/, v^, vj respectively, the value of R will be given by (he formula of N". 21, in the manner expressed in the text. 276 CELESTIAL MECHANICS, + -g-' («'/<• 2««'- [^ ^f^'J • COS. j.(n7— «/+.'— .; + — . u:\-Z.al\ { -^^ j . COS. i.{rJt—nt-\-i'—{) ^. (t'/— O' "/• S««'«" ( -J~ ) • (sin. ?. («'/—«/+ f' — t) —. {v' — t/). u/. "ZAa'. I -j-j- J . sin.«.(n7— «?+t'— . {yl—vy-. Y..i\A^'\ COS. ».(m7— «f + £'—0 rri.zz Sm'.az" , . , , , . + -^3 5^^f— • COS. («/— «f+£'— + m'.-fc^. X.B'". COS. i.(n't—nt+t—0 + &c. If in this expression of i?, the values relative to the elliptic motion, are substituted in place of u^, u', v^, w ', z and z', which values are functions of the sines and cosines of the angles tit+i, n't-{-i', and of their multiples ; k will be expressed by an infinite series of cosines of the form vi'k. cos. (i'n't — int+A),* i and i' being entire numbers. It is evident that the action of the bodies m", m"', &c., on m, will produce in R, terms analogous to those which result from the action of m', and that we shall obtain them, by changing in the preceding expression of R, all that which is relative to m', into the same quanti- ties relative to m", m'", &c. Let any term m'k. cos, {iln'l — int + A") of the expression for i?, be considered. If the orbits were circular, and existed in the same * The form of this function is always that of a cosine, for the values of ii', u[, are ex- pressed by series of the cosines of nf+i, n'Z+t', and of their multiples, which are multi- plied into a function of the form 2. cos. u{n't — nt-\-i' — i), the value of v— r/ is expressed by a series involving sin. (ntJfi) ; sin. (»'<-{- 1') ; and their multiples, and this is multiplied into a function of the form S. sin. >.(»'t — nt-f-i'— •). PART I.— BOOK II. 277 plane, we would have i— i, therefore i' cannot surpass i, or be sur- passed by it, but by means of the sines and cosines of the expressions ofu^, v^, z, uf, vf, z which by combining with the sines and cosines of the angle n't— fit +i' — i, and of its multiples, would produce sines and cosines of angles in which i' is different from i. If we consider the excentricities and inclinations of the orbits, as very small quantities of the first order, it results from the formulas of N"- 22, that in the expressions of u^, r , z or rs, s being the tangent of the latitude of 7?i, the coefficient of the sine or of the cosine of an angle, such as J.(nt-\ e), is expressed by a series, of which the first term is of the order /, the second term of the order J'+ 2, the third of the order* / + 4 ; and so of the rest. The same obtains for the coefficient of the sine and cosine of the angle ^'(nV+Z), in the ex- pressions of w/, u/. z'. It follows from this, that «and j' being supposed positive, and i' greater than i : the coefficient k in the term of m'k. cos. (i'vlt — int+A), is of the order if — i, and that in the series which ex- presses it, the first term is of the order i' — i, the second term is of the order i' — «+2, and so of the rest, so that this series is very converging. If i be greater than i', the terms of the series will be successively of the orders i — i', i — i' + 2, &c. ♦ It is evident from inspection of the series in pages 150, 152, that when all the coeffici- ents of the function cos./(n<-|-i) are collected together, they will constitute a series of the form e/±e/+* ±e-f+*±eJ'-*-^, &c., hence multiplying cos- y:(nt-|-i) into cos. i.{n't — 7ii-\-^ — i) the product will be of the form of cos. i.(n't—nt + t'—s) +f.(n't + 1)) . ^ , t= by making y -J- i = i' cos. {i'n't — ini -}- A) _ ; which is to be multiplied into the series e^, f^+^, tf+*,&c., =(asy=i' — «'= ), <""', «"-'+«, &c. 278 CELESTIAL MECHANICS, Let zr denote the longitude of the perihelion of the orbit of m, and 6 the longitude ofits node ; and in like manner let zr' denote the longi- tude of the perihelion of the orbit of m, and 9' that of its node ; these longitudes being reckoned on a plane very little inclined to that of its orbit. It follows from the formulge of N°. 22, that in the expressions of u„ v„ and z, the angle iit-\-B is always accompanied by — w, or by — 9 ; and that in the expressions of u', t\, and z', the angle n't+ 1 is always accompanied by — w', or by — 9', hence it follows that the term vnlk. cos. {i'lit — int+A) is of the following form 7nfk. cos. (i'7i't—mt+i'e'—u—gur^g'u'—g''^—f^'), g, g', g', ^", being entire numbers, positive or negative, and such that we have O^i'-i^g—g'-g'^^g". This also follows from considering, that the value of i?, and its dif- ferent terms are independent of the position of the right line, from which we reckon the longitudes. Moreover, in the formula; of N". 22, the coefficient of the sine and cosine of the angle w, has always for factor the excentricity e of the orbit of m, the coefficient of the sine and cosine of the angle Sur, has for factor the square of this excen- tricity, and so of the rest. In like manner, the coefficient of the sine and cosine of the angle 9, has for factor tang. \ip,

PART I.— BOOK 11 275 but if one of them, such as g, be negative and equal to — g, this factor will be of the order i' — i + ^2g.* Therefore, if we only preserve, among the terms of k, those which depending on the angle i'n't — int, are of the order « — i, and neglect all those which depending on the same angle, are of the orders i' — i + '2, i'—i + 4, &c. ; the expression of k will be constituted in the following manner : H.e^.e'^. tang. ^".(4?). tang. ^'".(4?0- cos. (i'n't— int -^i'/-h-g. ^-g'. ^'-^'. ^-g". r) H being a coefficient independent of the excentricities and of the inclinations of the orbits, and the numbers g, g, g", g", being all positive, and such that their sum is equal to i — u If we substitute in R, a.(l +m,,), in place of r, we shall have fdR\ (dR\ If in this same function, we substitute in place of «,, v, and c, their values given by the formulae of N". 22, we shall have fdR\_ fdR\ \dv)-[dj' provided we suppose that t — xs-, and e — 6 are constant, in the differential of R, taken relatively to i ; for then u,, v^, and z are constant in this differential, and as we have v ::: nt + t + v^, it is evident that the preceding equation has place. We can therefore easily obtain the values of r. ( -^ j and of | -7- J , which occur m * For in this case i' — i+^g=g-{-g'-\-g" +g"'- /dR\ . rfr dR dR dr dr ,, ^ . /dR\ dR\ Q 280 CELESTIAL MECHANICS, the differential equations of the preceding numbers, when we shall have obtained the value of R expanded into a series of the cosines of angles increasing proportionally to the time. It will also be very easy to determine the differential d/?, by taking care that the angle nt, solely varies in the expression of R, the angle n't being supposed to be constant ; because d22 is the difference of R, taking on the supposition, that the coordinates of m', which are functions of n't, are constant. 49. The difficulty of the expansion of R into a series, is reduced to the determination of the quantities J'*', i?''\ and their differences, taken relatively to a and a'. For this purpose, let us consider generally the function (a' — 2aa'. cos. 6 + a'*)"', and let us expand it according to the cosines of the angle 6, and of its multiples. By making — = «, it will become a'~^.(l— a. cos. 6 + a)~'. Let fl— 2«. cos. e-|-«*)-' =^. bi'^+bi'K cos, O+^f . cos. 26. +3f. C0S.39+&C. bf', U}\ bf\ &c., being functions of a, and s. If we take the loga- rithmic differences of the two members of this equation, with respect to the variable 0, we shall have —25. a. sin. 9 —bi'\ sin. 9— 26f . sin. 29— &c. 1— 2«. cos. e-|-«* ~ i.6f'+6i". cos.9-i-*i->. COS. 29 + &C. ' by multiplying transversely, and comparing together like cosines, we find generally m _ 0-1 ).(i+.-).z>r"-(i + s-ou.br'' ,. by this means the values of bf\ bf\ &c., will be given, when bf\ 6^'^, are known. PART I.— BOOK II. 581 s being changed into 5-f 1 in the preceding expression of (1 — 2a. COS. ^)+«^)~'~\ we shall have (1— 2«. COS. 9+«^)-'-'zr }2'l^fli. + b%i COS. 6 + bfl^. cos. 29. + 6fj,. C0S.39+&C. By multiplying the two members of this equation, by 1 — 2a;. cos. 9 + a*, and by substituting in place of (1 — 2a. cos. 94-a*)~', its value in a se- ries, we shall have i. 6('»+^,w. cos. 9-(-if . COS. 29+ &c. =(1— 2«. COS. 9+a*).(4.i.fj,+Z.^'].i. cos.9-t-6fj,. COS. 29+6fji. cos. 39+&c.) ; from which may be obtained, by a comparison of similar cosines i« = (l + «^). b^Ji,- cc. b^-l^ — «. iii+/).* The formula (a) gives the preceding expression of bi'\ will therefore become ' i — s PART. I, BOOK II. O O When this transverse multiplication is performed we must substitute for cos. t. sm.'(l), sm. 6. COS. (•'«, their values m terms of ^— ^ — ■ 1 > ; hence we obtain '2 2 _ f«e('-2J. sin. (i— 1). «— ««e«. sin. (f— 1). 6— (I + ««).(i_l). €j-i, sin. (j— 1). e + «?£('). sin. (i— 1). *. +«.(j— 2). €('-2). sin. (i— 1). «=0. .". Sj. «.(»—«) = (1 + «=).(?—!). Ij>-i) _ a.{i— 2+ji). gj-2. * To obtain this value of S<'', it is to be remarked that cos. t. cos, it cos. (i'+l). 9+cos. (j — 1). « , ,. , • , = ' 2 ' "6"ce multiplying the two factors of the second member of this equation, the coefficient of cos. id is (!+«') £'"' , ab^'+^^ «4(*-\'. 282 CELESTIAL MECHANICS, By changing i into i+1, in this equation, we shall have and If we substitute in place of ^il^i', its preceding value, we will have ''• ~ {i—s).{i—s+\).o<. These two expressions o? b'l\ and of ^i'"*"", give to -_L___ , f ; (b) (1-.^) by substituting for 6^'+^', its value deduced from the equation (a), we shall have e:=-i TT—^ ' (^)* (1— a ; which expression might have been inferred from the preceding by changing ? into — ?", and by remarking that i,''= Ur'^. We shall con- sequently obtain by means of this formula, the values of 5f+i, 6f|.,, bfl^i, &c., when the values of bf\ b[^\ bi'\ &c., will have been known. In order to abridge, let x denote the function 1 — 2a. cos. 6 + a', by differentiating with respect to «, the equation X-' = 1. bm. + 6i'> cos. i-bi'K cos. 29 + &c. ; we will obtain -2s.(«-cos. 0). A-'-^=i. -^ + ^^ . COS. e + -^. cos. 29+ &c.; ^ "^ ^ da, dtt. da. * Hence if we know the coefficients of the multiple cosines in the series which is equi- valent to(l — 2«. COS. S+a^)-*, we know the coefficients of the multiple cosine's in the series which is equivalent to (1 — 2«. cos. i-\-a-)-^~'^. PART I.— BOOK II. 283 but we have 1 — a* — X — X -\- COS. 9 zz ; therefore we shall have — i ^ . A '-' = J . ^-j-^ -| ^. COS. 9 + &c. ; lience we deduce generally db

, •—J • t's-t-l • M« a a By substituting in place of bi'li its value given by the formula (i), we will obtain dblf_ _ ( i+(i+^s).x' I _ 2.(i—s + l) dx ~ I «.(i— «^) S ' ' 1— «* ' ' ■ This equation being differentiated, will give dx* ~\ «.(1— a*) V dx '^\ (1—*')* «* - 1-a* • dx "*• (1— «*r ' ' oo 2 * Substituting for a-»-i, a-«, their values given in the preceding page, the coefficient of COS. U, in the value of x-'-^ is Cj^, , and the coefficient of the same quantity in the value of A—* is £'''. \ Differencing the coefficient of €^'' with respect to a it becomes — 2.(1— g^H-a^^z (i+25).(l— <»')-f2«"-.(i+2^) _ «^(1— »')"- ■*■ (!—«')' ~ _ J£— 3£)_ (z + 25). «M1— g^ ) 2«''.(i+2i) _ _ f£— 2«_l+£i a°-.(i— «')«+ *^(l_«7 + (1— «^)^ ~ ««.(i— ««)2 + (1— «*)2 ' . 284 CELESTIAL MECHANICS, By differentiating again, we will obtain -t- I {l—a'Y "^ «^ V ' 1— a* • dx' _ ^.{i-S+\).x dU:^'' 4.0— ^+l).(l+3a^) ,^„ {\—x^y ' dx (1—**)' • * ' • It appears from this that in order to determine the values of b['\ and of its successive differentials, ' it is sufficient to know those of Up, and of up. These two quantities may be determined in the following manner : Let c represent the number of which the hyperbolical logarithm is unity ; the expression of >r% may be made to assume the following form : By expanding the second member of this equation, with respect to the powers of c^V^, and of c—''v=ij it is evident that the two exponen- tial quantities eW^, and c— '"V-i will have the same coefficients ■which we will denote by k. The sum of the two terms /cc'V^T, and k.c-^W=i is 2A-. cos, i6 ; this will be the value of b['K cos. ?6 j there- fore we will obtain Up ■=. 2A-. Now tiie expression of a~' is equal to the product of the two series 1 + sx.c-W-^ + !A!±^ . ^\c-'"^-^ + &c. 1 .2 these two series being multiplied, the one by the other, will give, in PART I.— BOOK II. 285 the case of i = 0,* and in the case of « zr 1 , /, _ «. ^ *+«• -TTi-- " +—7:2 r.2.3 • * + &c. , ^ consequently, I In order that this series may converge, it is necessary that a should be less than unity ; this may be always effected by assuming « equal to the ratio of the smaller of the distances a and d to the greater, and as we have already supposed a= —r, we will assume that a is less than a! . In the theory of the motions of the bodies m, m', m", &c., it is ne- cessary to know the values of U^\ and of h[^\ when sittV, and inf. In these two cases these values do not converge rapidly unless a is a very small fraction. These series converge with greater rapidity when 5ZZ — \, and we have 4.-4 = '+ «)■■«■ + (^:)*- «'+(a^)'- '' + (^)' -'+ ^^- ,1, _ C 1.1 , 1 1.1.3 ^ 1^ 1.1.3.5 g 1.3.5 6_i_— a.^1 — — . a-— -.g^^^y.a — ^•2_4_6,8-'^ ~ 4.6.8 • 1.1.3.5.7 o ? ,. „ . a^ — &C. \ • 2.4.6.8.10 3 * i = when equal powers of 4 ^~', and c"~ ', are multiplied together and, ;=I, when powers of J)'^~^, are multiplied into powers of c"""^""', which are less by unit than these. This is evident from the value of/-. 286 CELESTIAL MECHANICS, In the theory of the planets and of the satellites, it will be sufficient to assume the sum of the first eleven or twelve terms, the subsequent being neglected, or more accurately, by summing* them as a geometric progression of which the ratio is 1 — a*. When ^>"'J, i<'i, shall have been thus determined we will obtain 6'^i by making ?=0, and s— — \, in the formula (6), and we will find h'V _(l+^')-^L°l+6-«.6"l If in the formula (c), we suppose ? = 1, and s — — \, we will have J^V= 2.«5f5+3.(l+«*). 6l!i By means of these values of h^V, and of j}t\ we will obtain by the preceding formula, the values of U'l, and of its partial differences, whatever may be the number i, from which we may we may conclude the values of lip, and of its differences. The values of Vl\ and of Vl'' 2 'It may be determined very simply, by the following formulae; (0) ^(1) I -(1—*?)^ ' 1 - ^- (1— «*)* -^ • For if (1 — «2)— 1 be expanded to a series, the sum of the remaining terms will be very nearly equal to this series multiplied into the eleventh term. (l+«^).i(0)— 2«ii^ f By formula (6) 6^ = ^w ''5 substituting for IP , i'D , their values we (1 +«2)2.60 1 +6«.(1 +«2). hi\ _4,Z.4W _6«.(1 ^.«2). m (1— «2)2i«») obtain h\ = (I — « a\2 rf'^C) 1 rfa^ ~ a"' ; in a similar manner we obtain the value of 6<'). PART I.— BOOK II. 287 Now, in order to obtain the quantities A^°', A^^\ &c., and their dif- ferences, it may be remarked that by the preceding number, the series l.A^"' + J<'\ COS. 6 + ^(2'. COS. 29+ &c. results from the expansion of the function a. COS. fl . , , . /,v_i -,i («"■ — 2aa'. COS. fl-fa*)-*, in a series ranged according to the cosines of the angle 9 and of its multi- ples ; by making —j- zz a., this same function is reduced to a' rr- b?^ + (-^ — V- *"' ) ' cos. 6—4-. bf\ cos. 29— &c. la which gives generally ^"^ = -V- *»'- when i is zero, or greater than unity, abstracting from the sign. In the case of ? = 1, we have ^(1) we have then 1_ dhl" fdo. \ ~ ~d* da. • \da J ' d\"> da ■ = — 1 db[" a!' da. ' [da J ' but we have [jaj- 1 a' ' therefore rfA^'^j 1 db^ doc ~ • d. ' and in the case of i= i, Wtt have ( dA'^\ _ -- ' ( ' dbf 288 CELESTIAL MECHANICS, Finally, even in the case of i=:l, we have, \ da" )~ a'^' dx^ ' \ da' J - a'*' dx' ' &c. In order to obtaia the differences of A'-''' relative to a' it may be ob- served, that A^'^ being an homogeneous function of a and a', of the dimension — I, we have by the nature of this kind of functions, hence we deduce , ( dA'^\ .,, ( dA^'>\ **• \da.da')- "^-y da ) ' \ da" ) ' &c. J5''' and its differences will be obtained by observing that by the preced- ing number, the series 1. B^''> -|-5<". cos. 9-f i?'=>. COS. 6+ &c. is the expansion of the function a'^. (1 — 2a. cosJ-l-a'')"^ according to PART I.— BOOK II. 289 the cosines of the angle 6 and of its multiples ; but this function thus expanded, is equal to a'-', a. ¥i'-¥¥i\ COS. + bT- cos. 29+ &c.) ; therefore we have generally hence we obtain (dB''\ _J_ dVr (<£B^\ _J_ d'bl' \-d^ ) - a'^- -ir ' \da^ )-a-'-d^' ""'' Moreover, B''^ being an homogeneous function of a and of a, of the dimension — 3, we have C dB^" •) C dB" •> from which it is easy to infer the partial differences of B'^'> teken re- latively to a', by means of its partial differences relatively to «. In the theory of the perturbations of m' by the action of m, the values of A^'^ and of fi' ' are the same as above, with the exception of /i<", which in this theory becomes -^ r. ^i"- Thus the compu- •^ a a tation of the values of ^<'', B'^'\ and of their differences, serves at once for the theories of the two bodies m and m'. 50. After this digression on the expansion of R into a series, let us resume the differential equations (A''), (F') and [Z') ofN"'. 46 and 47 ; and let us determine by their means, the values of Sr, Sv, and Ss, the approximation being extended to quantities of the order of the ex- centricities and of the inclinations of the orbits. If in the elliptic orbits, we suppose r =: a.(l + u^ ; r = a'.{i + <) ; vzznt + t-^v,; 1/ =. n't-{-i'-\-'^l ; PART 1. BOOK II. P P !^90 CELESTIAL MECHANICS, by N". 22 we shall have Ui = — e, COS. {ni-\-i — 73-) ; «/ = — d . cos. (n7+e'— z/) f, = 2e. sin. (w?+f"~^) > ""' , — 2e'. sin. (nV+i' — v) ; n^+£, n7+«' being the mean longitudes of m and m' ; a and a' being the greater semiaxes of their orbits ; e and e' being the ratios of the excentricities to the gueater semiaxes ; finally, ■s- and ■nr' being the lon- gitudes of then- perihelions. All these longitudes, may be referred in- differently to the planes themselves of the orbits, or to a plane which is very little inclined to them ; because quantities of the order of the squares and products of the excentricities and of the inclinations are ne- glected. The preceding values being substituted, in the expression of R of N". 48, will give R = -^- 2. ^'".(cos. i. (n't^nt-\-i'—i)* * As the approximation is carried only as far as terms involving the first power of the excentricity, the only terms in the general expression for R which are to be considered, are , . ^ VT .,-, ., : , /rfA('')\ /c?A(-')\ , . , ■ ^ the four first. Now as A(') = A(-') and I— — j = I —- — I, and cos. i.ui =: cos. ( — im) to representing {n't — nt-\-i' — i), and sin. ( — i.ui) = — sin, iw, we shall have generally (i' representing the positive values of i, and n representing (nt + s — w)), cos. i.w. cos. n := 2 COS. Hm. cos. n= cos. (i'.w-\-n)-{-cos. (i'M—n), and i. sin. i.tu. sin. n = 2t'. sin. i'uj. sin. ?i = + i. cos. {{i'M -\- n) — i'. cos. {n — i'.iv) See page 274, Notes). Hence substituting for n, its value, (nt + s — ar), and observing that cos. i, (n't — nt + t' — t.) cos. (nt+i — w) = cos.i'.(7i't—vt + i'—i)+vt-^t—-iir)+cos.i'.(n't — H< + e'— <) — (nt+t—ia)), and also that when 2e. sin. (nt + t — a-) is substituted for v/, sin. i.(n't — nt + t' — s). &m.(nt-^i — ot) = COS. i.(n't — nt+i' — i)J{.nt+i—a) — cos. i'.(n't — nt + i'~i)—nt—i—rv), we obtain the second term in the expression ; in like manner the third term is obtained, by taking the index i — 1 ; in the third term the circular part is PART I.— BOOK II. 291 — -. 'ZAa. \ — — V + 2i.^<''f . e. cos.(«.(«7 — nt-\-i-i)-^nt + 1 — n-) the sign S of finite integrals, extending to all integral values positive and negative of i, the value izzO being comprehended among them. From which we obtain ° 2 i da S ^ c (aa^ w — 7i 5 cos. j.(n7 — nt-{-B — i) m ~2 e'. cos. (nt + £— w') ^ i..in — n) — n c ( da ) c. COS. (?.(n7 — nt+ 1 — i) 4- w^+ * — bt) ; pp 2 made to assume a more symmetrical form, for it becomes by performing the prescribed operations, cos. (j — \).{v!t—nt ■\- i — e)4.n'<+i' — <«/), which is evidently identical with the expression cos.c.(n't — ntA^i — ej+n^ + t — t?'), besides the values when j=0, are com- prized in this expression. 292 CELESTIAL MECHANICS, 2 i da.da' ) '^ L da S t.(n — «) — n L i da S e'. COS. (J,.[n't — nt-\-i' — i) + nt + i — w') ; The sign S extending in this and the following formulas to all the inte- gral values of i, positive and negative, the sole value ?' — being * When the value of i = 0, is excepted out of the positive and negative values of i, we shall have dR m' dA'S» m rf.2AW ., , m' d^A<-'» — = , — ; \- — — • — ; . COS. t.in't — nt+^ — r--. a. , „ . e. cos. (nt+f— ar) dr 2 da ^ 2 da ^ ^ ' 2 da" v t / m' I d"A<-') dAf.') \ — — . 2.( a. — r-^ + 2?. — ; — }e. cos. ?.(n'f — nt4-i' — e) + nt — s— ar) 2 \ da^ ^ da f ^ -r / ■ 7r'\a'-^-r-i — J — je'- cos.(ni+s— ar')— 77- ^'O- , , , — 2(t— 1 . 2 V da'.da ^ da ' *■ ^ '2 rfa.t/a' ^ J d. COS. t.(n'i — wi-j-s' — e) + ni + s — ■cr'), .'. substituting for r its value, a.(l — e.), COS. («< + .-«), we shall have r. (— j = -. a. ^- . ^ • m' rfsAW ., , , , , Hi' a!2A''' c. COS. (n/ + t — 11) + -^ . a. — -5 — . cos. t.(n't — nt-f-i — — -. a. —z — e. cos. «. ^ da ^ ct(z (n't — nt 4. ^ — i) + nt-\r i — -a) (Sa-. , „ Ajiia. -t- J. e. cos. Urit — n<' + e' — f)+n< + ' — w) 2 \ aa- ^ da I m' „ d^AO) .0-. e. cos. (ni+' — '°) 2 ofa^ w' /d'^AW 2dAm\ -. a'.a. { -y-r-, k O. ; 1. C'. COS. (««+ £ 1:/) 2 \da'.da ^ da J ^ ' m' I rfAt'~'' Af>~)\ — —. 2.{a^a' 2(i— 1) d.f~- )e'. cos. ((»'<_«<+«'— .)+nt+i—sr> 2 V da da I (the remaining terms are omitted because e* occurs) PART I.—BOOK II. 293 excepted, because the terms in which i — 0, are extricated from this sign : mg is a, constant quantity added to the integral fdR. There- fore by making ^ , 3 Cc?*AW7 , ^ , C .'. 2/cliJ =2»»'ff • 2. AW. COS. i.(n't—nt-^i' — i\- — . 2a. -j—. e. cos. (nt+s — w) — ° 2 n' — n 2 aa 2 ' ^•^"~'"^ . f 2a.-^ +2iA(').)e. cos. i.(n't—nt+e—i) + nf +«-,=) — -J . 2a'. »n' — 2re+« \ da . / a dR el. cos.i.(n't — nt-\-i — i)+nt-J[-i — v'), •.• by reducing we obtain 2yd/? + r. -j-= the expression which is given in the text. 294 CELESTIAL MECHANICS, t.(7i — ?7) — n C (da) 3 the sum of the masses M-\-m being assumed equal to unity, and — ^y — being supposed equal to «* ; the equation (X') will become 6?/* ' ^2 C da ) * The equation (X') becomes by neglecting the square of the excentricity, —^ ^- n'^u—n-a.{l — t. cos. (nt+i-''sr).(2/dR + r. -j— j — 2ean^.fndU (sin. {nt + t — w). l^/dR + ''• — 7- ) ; ("''^ being substituted for -^ and M+fl» bemg by hypothesis =1). By substituting for 2/diJ+ 7- -^, its value, this equation becomes = t.(n't—nt + 1*— e) + "2- • I <* • -Jl- + 3a --^ — ) • «• COS. (n(-f !— -a) + ({n^a).(2m'g+Y) -"'-^ H2an^{2mV-j-^. a.-j—Xj. e. cos.(wf+i— or); (=7:V.Cc. cos.(««+i-:x))+.i=a.— ( aa'. .^-^+2a. -^+2a'.— -^+4.A0);«'. (cos. 7,t + ,_B^) =: (n*m'. Def. cos. (si+t— a/). ^ + (2^1,..^ + ^-'«^-^ 2.r Jn.n , .AC.) [ \e.cos.ii^n't-nt+^-,) + nt+^) J, ^. jjifl. (sa. — T— A ^. AC'-*. ) c.cos. ».(«'<— 71/ +t— «)+««+« — o) ^2 \ da n — n' I _2an«.e./«(f«.(sin. i.(n'/~«/ +t'-f) + «< + i-^\ 2m' dA^^ ing vt are Sam'gnt and 2 — . a'nt. i — ; — i, hence we will have 3m'g= -— . a. — j — . 2 \ da ' J aa PART I.— BOOK 11. 297 been di!«pensed with, by supposing them to be comprised in the elements e and s- of elliptic motion ; but then the expression for h\ would have involved terms depending on the mean anomaly, which would not have been included in those which are given by the elliptic motion : now it is more convenient to make those terms to disappear from the expression for the longitude, in order to introduce them into the ex- pression for the radius vector ; J] and J] will be so determined as to satisfy this condition. This being premised, by substituting in place of a' . \ — -r-r t Its value — A' ' — a. \ — -, — f , we sljaii obtaui i da ) I da S 11 — f.(« — n) n — t.{ii — n) \ da J moreover let JE'*' =r 3^ „ A (0^ i\{n—n').{n^i.{n^n'))—3rn n — n t .{n — n) — n PART. I. — BOOK ir. Q Q 298 CELESTIAL MECHANICS, i^«zzi^=l2^. «A«+|-. (n+i(n-n'))-3n^) n — n S , /c?AW\ 2n \ 2w^Jgw r'\ da J'^ 71— n'' S n^— (71—1.(71—71'))" ' (i—^).{Qi—l).7ia.A'-'> + (i—l).7ia\ { ^^^ ) \ da J , G«= 2.(n — t.(7i — 71')) 2n\D'''^ n'' — [71 — i.(7i — 7i')y ' we shall have -a'=^-''\-^)'r-^'^i--\^rLl}^n'' ^ ^ i\(n—7i'y—n' a A'''-'] . COS. i.(n't—nt-\-i'—i) — 7n'.fe. COS. (7it-\-t — w) — m'.f'e'. cos. (tiI-^-i — w') i-^m'. C.nt e. sin. («/+£ — iir)+iwi'. Z). w/. e'. sin. («?+£ — w') r — 7 7z T^xir-e. cos.(i.{n't-nt+B~()-i-nt+ t-sr) , , jTl (71 1(71—71)) ^ ^ ' J ' -f -T—} —, KTi- e'' COS. (i.(7it-7lt-\-i-i)-]r7lt + i — -a-') 7f -(71-1.(71 — ?z))* ^ ^ -^ 2n5.5a*.3-7— f H r- «A"'S / ^ I da i n — 71 ) > . 7«' „ f 71 ,,., 2 li.[n—7i'y ^ t, sin. ?.(n'^ — ??/+£' — i), +jn'. C.w/. e. cos. (n/+£ — ■sy) + rn'D.7it. d. cos. (w^+t — w') — ^!^ ). -. aA('\.± / n — n' PART I.— BOOK II. 299 -rr. e. sin. (iJn't — nt+ i' — t) + nt+i — ■n-) , n— 1.(71 — n') ^ ^ ' ^' H 7-7 77. e'. sin. (i.{n't — nt+i — t) + n-{-i — u) ' n — t.{n — 71) ^ ' \ in these expressions the integral sign X extends to tlie whole values of i both positive and negative, the sole value 'f=.0 being excepted. It may be observed here, that in the very case in which the series represented by S.A'''. cos. i.(nt — nt-\-^ — i) converges slowly, the ex- pressions of — , and of Sv, may be rendered converging by means of the divisors which they acquire. This observation is extremely important, QQ2 r-r — • 7- aA(') It will become =zi.^rT, — ?- (a-. — ;— t.(n — n') — n n — n' ^ da- ' 2 \ da 2n „r\ /2«+I\ 2n ^., i.in—n')~3n U {^ + J^. „ A..) + -p}^^ . (aK ^ + _i!L, . axA- £:l!=i>' V \d»f^n — n' /^ i-(n—n')—n'\ d* ' n—n' I n—n' nA('' ; now by reducing the two terms which constitute the factor off. cos. i.{n't — /i/ + f — i)J^nt-\-i — w) in page 296, to a common denominator, it will become = to {1^.{n—n'f—iin.(n—ri)—2i.(p.[n—n'f + 2i.n-—iK(n—n'f + n'—v.(n—n'f-\-2in.(n—n') (rfA'*' '■In \ a"-—! f- 7.«A(''» (divided by (a-. — ; 1 ^. a A" A. divided by 2j'.(7i— «')*—«')+ -:; — r- aM*\ ; which is \ da n — n! I 2.(?i — n') evidently equal to jE'*'. 300 CELESTIAL MECHANICS, because without it it would be impossible to express analytically the reciprocal perturbations of the planets, the ratio of whose distances from the sun, differ little from unity. These expressions may be made to assume the following form, which will be extremely useful in the sequel ; let h=: e. sin. w ; li z=. ef sin. •ar' ; / :^e. cos. 73- ; I'zz: e. cos. -a/ ; we shall have a - 6''''{ da J+ 2 ' ""■ (. K(JiII^fr^-^' ^ cos. i.(n't — nt-\-i' — t) —m'.{hf-^}if). cos.7?/+£)-m'.(//+/'/0- sin. («H0 ^^JL, (l.C + l'D').nt.{sm.(nt+s) — ^.\ih.C+h'D).nt.cos. («/+0 -T. — -r-^, ;ttt . sm. (i.(rLt—nt+ t'— 0+«^+ ?« — {ii — .(n — zi;) m \ ?r >' iXn—7i'\.(i\[n—ny—n^) ) sin. e.(?z7 — nt+i — O* -\-nL{h.C+h'.D). tit. sin. (nt+e) + m'.(l.C+l'.£>). nt. cos. (n/+0 ^ — .;' .. . sin. (i.(n't—nt + t'— + «/ + n — t.{n- — n) ^ ^ n — t,{n — n) ^ PART I.— BOOK II 301 these expressions of ir and h> being added to the values of r and of v, relative to the elliptic motion, will give the entire values of the radius vector ofm, and of its motion in longitude. 51. Let us at present, consider the motion of m, in latitude. For this purpose let the formula (Z') of N°. 47, be resumed ; and if the product of the inclinations, by the excentricities of the orbits, be ne- glected it becomes the expression for i? of N°. 48, gives, by assuming for the fixed plane, the plane of the primitive orbit of m, I ^- ) = —7^ . E. Jjw. cos. t.(nt-r-nt+£ — i); the value of i comprehending all whole numbers both positive and negative, including i ■=. 0. Let y represent the tangent of the in- clination of the orbit of ?«', to the primitive orbit of m, and n the longitude of the ascending node of the first of these orbits, on the second ; we shall have very nearly. z' = a'.y. sin. {7i't+i'—n) ;t which gives • When the primitive orbit of m is assumed as the fixed plane, the differential of the two last terms in the value of R (which is given in page 276) with respect to z, becomes (when quantities of the order m''- are neglected) the expression which is given in the text. f When quantities of the higher orders of the inclinations are neglected, we may sub- stitute for sin. {n't-\-i' — n), the longitude on the fixed plane, and we can also assume the distance of the planet from the centre of its orbit, equal to the mean distance a', ; under these restrictions it will readily appear that the tangent of latitude of m' above the fixed plane = y. sin. (n't+i — n), and v s' = a'.y. sin. (n't + i' — n). 302 CELESTIAL MECHANICS, —. a'.S.i5(*-".y. sin. (i.(7i't—nt+t'—i) + nt+i—n), the value of i, in this and the following expressions extending to all whole numbers, as well positive as negative, the sole value / = 0* being excepted. The differential equation in $u', will consequently become, by multiplying the value of ( -r^ j , by n*a^. which is equal to unity, = — ^-; — + ifJu' — m'.n.—Tz. y. sin. (n't+i' — n) ar a , m'n'' ,_,,, . , H -— . aa'. B'-'K y. sin. (nt+i—n) ^ g— . ad. Y..B'-''\y. sin.(J.(nV— n/+ /— £)+n^+£-n); from which we obtain, by integrating, and by remarking that by N". 47, ^5= — a.hi, ^'= — ^f—;f-' -^' y- «'"• (n'^+f '— n) * When this value of s' is multiplied into 2. Sl*>. cos. i.^n't — nt 4-'t'— t), it becomes, when»=l, equal to Bd). sin. («'« — ?!<+ s'— O+w'^ + f"— n) + 5('>. sin. (n<+i— n), and whenj = it becomes =5^"'. sin. (/('Z + t' — n) ; now had this product been expressed generally .a'.S.BW.y. sin. (j.(w'i — nt-\-i' — e)+n<-|.j — 11), it would not answer to the two cases in which jrz:l, and in which i=0; hence we see the reason why this product is resolved into parts in the expression for \—rj^ and also why the value »=0, is ex- cepted out of the values of /. ■f This difiFerential equation is integrated in the manner prescribed in N°. il. PART I.— BOOK II. 303 2 w — (7i — i.{ii-n )) In order to obtain the latitude of m, above a fixed plane, a little inclined to the plane of its primitive orbit, naming

which occur in the expressions of r, r, and s, are approximate as far as quantities of the third order, that is to say, the approximation in which we only consider the squares and products of the excentricities and of the inclinations of their orbits, will add nothing to their values; therefore they have all the required accuracy ; this observation is the more important, in as much as the secular variations of the orbits depend on these coefficients. The different terms of the perturbations of r, v, s, are comprised in the form k. ^^^^'ii.(nt—nt+B—i) + r7it+ri) I, r being either a positive integral number, or equal to cypher, and k being a function of the excentricities and of the inclinations of the orbits, of the order r, or of a superior order : we are enabled by means of this, to determine of what order a term depending on a given angle is. It is manifest that the action of tiie bodies nf, m"', &c., only cause to be added to the preceding values of r, v and s, terms analogous to those which result from the action of iri, and that if we neglect the square of the perturbating force, the sum of all these terms will give the complete values of r, v and *•. This fol- PART I. BOOK II. U R 306 CELESTIAL MECHANICS, lows from the nature of the formula; (X'), (Y') and (Z'),* which are linear with respect to quantities which depend on the perturbat- ing force. Finally, we shall obtain the perturbation of m', produced by the action of m', by changing in the preceding formulae, a, n, h, I, i, nr, p, q, and m', into a', n', h', I', i', txt', p, q, and m, and vice versa. * When quantities of the order of the square of the perturbating forces are neglected, the formulae X', Y', Z, are linear with respect to the perturbating force, from which it follows, that the variation of the sura is equal to the sum of the variations. PART I.— BOOK II. 307 CHAPTER VII. Of the secular inequalities of the celestial motions. 53. The perturbating forces which disturb the elliptic motion introduce into the expressions of r, —t— and of s, which are given in the preceding chapter, the time without the signs of the sine and cosine, or under the form of arcs of circles, which increasing indefinitely, must at length render these expressions erroneous ; it is therefore es- sentially necessary to make these arcs to disappear, and to obtain the functions which produce them by their expansion into a series. There has been given for this object, in the fifth chapter, a general method, from which it follows, that these arcs arise from the variations of the elliptic motion, which are then functions of the time. These varia- tions being performed with extreme slowness, have been termed secular inequalities. Their theory is one of the most interesting points in the system of the world : we proceed to present it here, in all the detail which its importance requires. By the preceding chapter we have 1 — h. sin. (nt+i) — /. cos (nt-\-i) — &c." r -^z a. 7H + -y-. (/. C-\-l.D).nt. sin. {nt-\-i) f 771 ——. Qu C-\-h'.D).nt. cos. (7it+e)i-m'S. -^ = n+2n/?. sin. {nt-\-i')-\-2nL cos. (nZ-j-O+^c- R R 2 308 CELESTIAL MECHANICS, — 7n'. (LC-\-t'D).n*t. sin. (nt+i) + m'.(;?.C+A'Z)).n*/. COS. (nt+i)+m'.T; s •=. q. sin. {nt-^i) — p. cos. {nt-\-i)-\- &c. ——. a*,fl'.(/— _p).JB'". nt. sin. (w^+ i) m' _I!L , a\a'.{q'—q-).B^^\nL cos. (nt-\-t)+m'.x ; 'V, T, %, being periodic functions of the time t. Let us at first consider the expression of —rr, and compare it with the expres- sion of y of No. 43. As the arbitrary quantity n multiplies the arc tt under the periodic signs, in the expression for —y- ; we must em- ploy the following equations, which have been found in N°. 43, 0=X' + 9. X^-'— F; = F + 9. Y"-irX" — 2Z ; let us consider what X, X', X", Y, &c. become in this case ; the fit) expression of — j— , being compared with that of ij of the above cited N". gives X zz n-\-^nh. sin. (nt + + 2^^- cos. (nt+i) + m'. T ; Y = m'.n\(h. C+h'D). cos. (nt+s')—m'?i\(l.C+l'.D). sin. (nt+i). The product of the partial differences of the constant quantities, into the disturbing masses being neglected,* which we are permitted * Since the product of the partial differences of the constants into the disturbing masses are neglected, it will not be necessary to take into account the periodic function m'.T; the second and third terms of the value of A' involve 7if under the periodic signs, .'. differencing the arbitraries contained under the signs with respect to n, we obtain the value of X", which is given in the text. PART I.— BOOK II. 'i09 to do, because these differences are of the order of the masses, we shall have bv N°. 43, X'= (^y (l+2A.sin.(«^+0 + 2/. cos.(«/+i)) +2w. {-k)'(h. cos. {nt-\-t) — /. sin. {nt-\-t)) ->r^n.(~y sin. (72/+0 + "^n-l^jA . cos. (n/+0; X'^'in. (^). (A. cos. {nt+i) — l.sm.{nt+i)). The equation O = X' + 0. X'' — F, will consequently become - ( ^^) . (1 + 2^. sin. (w^+0+2Z. cos. (jit+t)) 4-2W. [ — j . sin. (wf+f) + 2n. ( ;jq ) • cos. (n/+0 +^''-^- {%)'^ ^^)|.(/^cos.(«^+0-/.sin.N+0) —m'.n\{h. C+h'.D). cos. (?z^+ H- m'.w*.(/. C+/'.Z)). sin. w/+ f), . The coefficients of the corresponding sines and cosines, being put se- parately equal to nothing, we shall have If these equations be integrated, and if in their integrals, 6 be changed into t, we shall have by N°. 4o, the value of the arbitrary 310 CELESTIAL MECHANICS. quantities, in functions of t, and we can efface the arcs of the circic from the expressions for -5— and for r, but instead of this change we can all at once change 6 into t, in these differential equations. Tlie first of these equations indicates that n is constant, and as the arbitrary- quantity a of the expressions for r depends upon it, in consequence of the equations ?z^ rz -y ; a is likewise constant. The two other equa- tions are not sufficient to determine h, /, i. We shall have a new equation by observing, that the expression for -j— , gives by integrat- ing, fndt, for the value of the mean longitude of m ; but we have supposed that this longitude is equal to nt-\-i ; therefore we have nt+i =.fndt, which gives dn , dt ^ and as — ^ = ; we shall have also -^ = O. Thus the two arbitrary dt dt quantities n and i are constant ; the arbitrary quantities h and / will be consequently determined by means of the differential equations, ^^' "^'-^ il C+l'.D); (1) dt 2 dl m'.n dt~ "1 (Ji.C+h'.D); (2) dxi The consideration of the expression of —7— being sufficient to deter- mine the values of n, a, h, /and i; we may perceive a priori, that the differential equation between the same quantities, which results from the expression for r, must coincide with the preceding. We may be easily assured of this a posteriori^ by applying to this expression the method of N°. 43. PART I.— BOOK II. 311 Let us now consider the expression of s. By comparing it with the expression of 1/ in the N°. already cited ; we shall have X — q. sin. (nt+i) — p. cos. (nt+ e)+W. x Y = -^ . a\a'.B^^\{p—p'). sin. (nf + «2 71 + — f-. a^.a'.B^'KCq—q'). COS. (nt+t). 4 n and i being constant, as is evident from what precedes ; by N". 43, we have X^' = O. The equation = X'+ 6. X'^ — Y consequently becomes, "*'"-. a*a'. £'''. (p—p'). sin. (wM- ^ . a\a'.B^'\ (?—?')• cos. (wf+t) ; 4 from this we deduce, by comparing the coefficients of corresponding sines and cosines, and by changing fi into /, in order to obtain p and q directly in functions of t, ±.^-.I!^.a^.a'.B^Kiq^qr, (3) ^ =_^. a^a'. JS<'> (;,-/); (4) After that p and q shall have been determined by these equations, if we substitute them in the preceding expression of s, by obliterating the terms which contain the arcs of a circle, we will have 4 m'.n 312 CELESTIAL MECHANICS, s := q. sin. (w^+O — 'i'-cos. {nl-\-i)+m'. ^. dn 54. The equation —rr "=■ 0, to which we have arrived, is of great importance in the theory of the system of the world, in that it indi- cates that tlie mean motions of the heavenly bodies, and the greater axes of their orbits are invariable ; but this equation is only accurate as far as quantities of the order in .li, inclusively. If quantities of the order m! Jf , or of the superior orders, would produce in dv --TT, ii term of the form* ^kt, k being a function of the elements of the orbits of m and of rn' ; a term of the order kf would be produced in the expression of v, which by changing the longitudes of m, pro- portionably to the square of the time, would at length become ex- tremely sensible. The equation —7— zz 0, would no longer obtain, but in place of this equation there would be obtained by the preceding dn number —j- zz 9,k ; it is therefore of importance to ascertain whether there exists in the expressions for v terms of the form A/*. We proceed to demonstrate that if we only consider the first power of the disturbing masses, however far we extend the approximations relative to the powers of the excentricities and the inclinations ot the orbits ; the ex- pression of w will not involve terms of this kind. For this purpose let the formula (X) of No. 4-6 be resumed. Sr= a. cos. v.fndt. r. sin. v. j2jdli-^ r. );t^( f f* Vi- * If the value of n contained a term of the order kt°, there would exist in the exjn''esfion dv ~dt of --3-, the term 2kt, and consequently this term would exist in A', so that in comparing coefficients of corresponding terms, we would have -7- = 2i. PART I.— BOOK II. 313 — a. sin. v.fndt. r. cos. v. \9,fAR-\-r. \ --r-\\ Let us consider the part of Sr which involves terms multiplied by t*, or for greater generality, let us consider the terms, which being mul- tiplied by the sine or cosine* of the angle a,t+ €, in which « is very small, have at the same time a* for a divisor. It is evident that n being supposed =0, there will result a term multiplied by f*, so that the first case is contained in the second. The terms which have a* for a divisor can only be produced by a double integration j therefore they must be produced by the part of Sr, which involves the double integral signy^ Let us first examine the term 2a, cos. v.Jndt.{r. sin. vfAK) The origin of the angle i being fixed at the perihelion, we have in the elliptic orbit, by No. 20, a.n—e*) rzz ^^ ^—. 1 -\-e. cos. V and consequently COS. V=. — ^ — : er hence we deduce by differencing r''.dv. Sin. v ■=. — i: — dr ;t PART I. — BOOK II. S S • K must be very small, because the sine is supposed to increase with great slowness ; it is evident that if a be supposed equal to nothing, the double integrations would produce a term proportional to the square of the time. , . — enrfr— e.a.((l— «')+r). = j-^ i-i— i = -■ — ' . dr. 314 CELESTIAL MECHANICS, but by No. 19, we have y-^.dv = dt's/iJ^a.(l — e*) = a* .ndt.^ \—e* ; consequently, andt. r. sin. v rdr ■ x/l— e' e „, ^ 2a. COS. v.fndUr. sin. v.fdR) ... ,, - , The term =^ — . ^ ^^- — -, will therefore become 2. COS. ^ ^y(^^^,y^^)^ or -^^^^. (rS/d«— /r*.d/J). It is evident that as this last function does not contain any double integrals, there cannot arise any term which h^s as* for a divisor. Let us now consider the term gfl. sin, vfndt. (r. cos. v/diR) w.v/l of the expression of Sr. By substituting for cos. u, its preceding value in r, this terra becomes g. sin. v.fndt.{r — a.(l — g*))./'dj? By N°. 22, we have ;^' being an infinite series of the cosines of the angle nt + t, and its multiples j therefore we shall have Jj^. (r—a.(-e')).fdR= a.fndt.Qe-i-x:).fdR.* PART I.— BOOK II. 3J5 Denoting by x '^^ integral fxndtt we will have a. ffidt.Qe + x')f^R= h^'fndt. f d/^+ax^'/d^ — «• / x"- dh. As these two last terms do not involve the double sign of integration, no term which has a* for a denominator can arise from it ; therefore if we only consider terras of this kind, we will have 2a. s in. v./ndt.{r. cos. v.fdR _ 3rt^.e. sin. v.fndt.fdR = -^.—'fndt./dRi - nai f* and the radius r will become dr w+(i^)-v-^"*-^'"'^ (dr \ dr —f- ) being the values of r and —-rr in the case of elliptic motion. Thus, in order to consider in the expression of the radius vector, the part of the perturbations, which is divided by a*, it will be sufficient to increase the mean longitude 7it -\- t, by the quantity — ./ndt.fdR, in the expression for the mean longitude in the case of the elliptic motion. Let us now examine whether this part of the perturbations should be taken into account in the expression for the longitude v. The formula ( F) of N°.46, gives by substituting — . — ^. J ndt.fdR in place of ir, when the terms divided by a* are only considered ( 2rd'r- \-ar' ^ LZE^LA.^. /ndt./dK; ss2 316 CELESTIAL MECHANICS, but by what goes before, we have ae.ndt. sin. v $r = / — : r*dv — a^ndU s/ \ — e* ; hence it is easy to conclude, by substituting for cos. v^ its value, which has been already given* in terms of r rd'r-{-dr* i ^rd'r+dr'' —1-15. dv v/l— e* ~ ndt ' therefore if we only consider the part of the perturbations, of which the divisor is a*, the longitude v will become (y) and ( -^ j being the parts of v and of —-r- which are relative to - r.ae.n.dt. COS. v.dv ,, , . . - (a.(o.(l — (r) — r).ndt.dv r,a^,r = . — — equal by substituting for cos. v ; ■y== , vl — e- VI — er 2rd"r ^a'.ndtVl—e". dv 2ar.ndt.dv , „ a"e'n^dt' aVdtWl—e'- a"n'dtWT^~ (\—e').ahi'dt- ' 1- (aVn-'rft'. {a'.{l-e Y—2ar^ l~^) + r-) . dr^ _f_ aWT- 2a 1 , 2rd°r+dr^ _ 2dv 2rd v e' "^ rVuI? (1— e')i ' * ' ~~^iW~ - l[dt ~ {\—r)andt "^ (1— e*)! aWT^ .2a 1 1 2rdv la now--— -^j-— J- + _ 2r^dv+2a^ndt.Vu^ « , «' . ^ 1 „ . — — =0, and _=r s + — r=^=- s =0. ••since o«Vl — er dv dv -J = -J-, the preceding expression becomes equal to — -j- . PART I— BOOK II. 317 the elliptic motion. Therefore in order to consider this part of the perturbations in the expression for the longitude of m, we should follow the same rule as we have given, when considering the expression of the radius vector, that is to say, it is necessary to increase in the elliptic expression of the true longitude, the mean longitude w^+ £ by the quan- tity .fndt.fdiR. The constant part of the expression for ( —j- j , being expanded into a series of the cosines of the angle nt-\-i and of its multiples, is reduced 3fl to unity, as we have seen in N". 22 ; hence arises the term — . fndU /d-R in the expression for the longitude. If d/? contains the con- stant term km'.ndt, this term would produce f. . k.v^fy in the expression for the longitude v. Therefore in order to ascertain whe- ther such terms exist in this expression, we must consider whether &.R contains a constant term. When the excentricities of the orbits and their mutual inclinations to each other are small, R can be reduced always into an infinite series of the sines and cosines of angles proportional to the time /. They can be generally represented by the term km, cos. {int-\- int -f A), i and i' being integral numbers, either positive or negative, or equal to cypher. The differential of this term taken solely with respect to the mean motion of m, is -^ikni.ndL sin. [i' >i' t + i7it + A) ; this is the part of dR, which is relative to this term : it cannot be constant unless we have O =1 i'nf + in; but this supposes that the mean motions of the bodies m and m' are commensurable with each other ; and as this is not the case in the solar system, we ought to infer from it, that the value of dR does not contain constant terms ; and that consequently if we only consider the first power of the perturbating masses, the mean motions of the celestial bodies are uniform, or what comes to the 318 CELESTIAL MECHANICS, same thing, -j- = 0. The value of a being connected with that of n, by means of the equation w* = — ; it follows that if we do not take into account periodic quantities, the greater axes of the orbits are constant. If the mean motions of the bodies m and m', though not exactly commensurable are very nearly so ; there will exist in the theory of their motions, inequalities of a very long period, and which may be- come very sensible, on account of the smallness of the divisor a*. We will see in the sequel that this obtains in the case of Jupiter and Saturn. The preceding analysis will give in a very simple manner, the part of the perturbations which depend on this divisor. It follows from it, that then it is sufficient to make the mean longitude nt+i or fndt vary by the quantity . fndtAR ; which comes to make n, in the integral fndt, increase by the quantity . J'dB ; the orbit /* of m being considered as a variable ellipse, we have n* zr -^; therefore the preceding variation of n must introduce in the semiaxis major of the orbit, the variation* — . If in the value of --^— we carry the approximation as far as quantities of the order of the squares of the perturbating masses, terms proportional to the times will arise ; but by attentively considering the differ- ential equations of the motion of the bodies m, m', m!', &c. j it will readily appear that these terms are at the same time of the order of the squares and of the products of the excentricities and of _ . . o A* , . 2nn*dn , . . San , , r> r • From the equation «' = -V we have da = substituting . Jan for a^ -V ft dn, and we have da = = /\ , , /d'A^'^\ ) By N°. 49, we have we will readily obtain by the same N". — r^> - , ^ , in functions of bf\ d(x, da, and of 6^"; and these quantities are given in linear functions of b'^^, and of 62', ; this being premised we shall find S«0 CELESTIAL MECHANICS, therefore (0,1)= S,m' .n.oi.\h%^ 4.( !_«»)» ' let (a*~2aaf. cos fl f flf'*)* = (o, a') + (a, a )'. cos. e+(a. a'")*, cos. 2fi+&c. by No. 49, vve shall have (aa') = ifl'. Z.l!i ; » (a,ay = {a'.b%), &c. therefore we shall have consequently by N". 49, we obtain by substituting in place of b'^^ and of its differences, their values in b'^, and b!^, the preceding function will be found equal to ((l+c.^).b<2lH-'b^V . (1— a ) therefore L2LJ.J-— 2.(1'^— «^)* or we shall obtain by this means very simple expressions for (0, 1) and for [0. '1, and it is easy to conclude by the values in a series for b'l\, and for b''2\, which are given in N°. 49, that these expressions are positive, if w be positive, and negative, if « be negative. Naming (0, 2) and [oTi] what (O, 1) and [±J] become, when a' and m' are changed into a'" and w", and in like manner let (0, 3} and [oTs"] represent what these same quantities become when a' and m' are changed into of" and mf^' ; and so on. Moreover let A'-', f, h'', I", denote what PART I.— BOOK 11. 321 h and / become relative to the bodies m'', m"', &c. ; we shall obtain in consequence of the combined actions of the different bodies m', m", vr\!", &c. on m, ^ = ((0, 1) + (0, 2) + (0. 3) + &c.)). ^-[orr].r- [aj]./'"- &c. J ^=—((0, l) + (0, 2)+(0, 3)+&c.). A+[orT].A'+[al].;/''+&c.. .^ ^ dK dl' dM dl' , -u u i . • au It IS manifest that — j-, — j-; —rr>-7r> &C'> ^"^ be determined by G^ dt dt dt expressions similar to those of -^ and of -5-, and that it is easy to infer them from the preceding by changing successively, that which is relative to m, into that which refers to rnf, m'', &c., and vice versa. Let therefore (1,0), [O]; (1,2), [O]; &c. be what (0. 1), [O] ; (0, 2), [o;j] ; &c. become when we change in them that which is relative to m, into that which is relative to m', and conversely ; let also (2, 0), [ro] ; (2, 0, CEI] ; &c. be what (0,2), [oTT]; (0,1), [2:1] become when that which is relative to m, is changed into that which is relative to m', and conversely, and so of the rest. The preceding PART I. BOOK II. XT • In this case (1— 2a. cos. l+a^)— = (1— 2a. cos. e+a^)K •.• s = —i; see page 278; •.• the first terra in the expansion of a'-^.(l — 2a. cos. S+o*)-' becomes (when 4 = — ^.) a'.i'^l, and the coe£Scient of cos. i = o'.Jt".. 322 CELESTIAL MECHANICS, diflPerential equations referred successively to the bodies m, m', n^', &c. will give for the determination of h, I, h, I', h'\ I' , &c. the follow- ing system of equations, -^=((0, 1 )+(0, 2)+(0, 3)+&c.).^[0]./'-[ai].f-[0].r-&c. ^ -^=-((0, 1 )+(o,2)+(o,s)+&cO.A+[§3]A'+[^l.r+[or|].r'+&c. -^=((l,0)+(l,2)+(l,S)+&c.)/'-[k£].KiII].^HlII].^"'-&c. M' — /;(A) ^=-{(i,o)+(i,2)+(i,3)+&c.)A'+[iZo].H[i^].A'''+[r3]-^i"+&c. r CIS f-=((2,0)+(2,l)+(2,3)+&c.)^H!;i]-K!li]-KIi]-^"-&c. — =-((2,0)+(2,l)+(2, 2)-^kc.)¥-{{J^'\ .H[5rT].A'+[2r|].A''"+&c. The quantities (0,1) and (1,0), foTT] and [ITo] have remarkable relations, which will very much facilitate the computation, and which will be useful in the sequel. By what precedes we have, - _ Srn! .na^ .a' .{a, a')' If in this expression for (0, l), m' be changed intonj, n into n, a into a', and vice versa; we shall have the expression of (1,0) which will be consequently 3' m.n'a"'.a.{a'. a)' 4(a'*— a*)* but we have (d, a)' = (a| a')', because each of these quantities results from the expansion of the function (a*— 2aa'. cos. fl+a'*)* into a series arranged according to the cosines of the angle 6 and of its multiples j therefore we will have (0, 1) m.n'a' = (1,0). m\na ; (1,0) = ~ ,,...,. , PART I— BOOK II. 323 but, when the masses m, and m', &c., are neglected with respect to M, therefore (0, \).m.^a =(1,0). m'.\/^j by means of this equation we can easily obtain (1,0) when (0, 1) will be determined. In like manner we have [ oTTl m.s/a = [iTo] m'.k/a!. These two equations will also subsist when n and «' have contrary signs ; that is to say, when tlie two bodies m and m' revolve in contrary directions ; but then we must give the sign of n to the radical \/a, and the sign of n' to the radical i/«'. The following equations result evidently from the two preceding : (0,2) m.\/a — (2,0) m'.y/a!' \ [oTa"] OT.\/a — [JTo"]. m".\/a" ; &c. (l,2)TO'.\/a = (2, \)m".K/a!'; \TJ]m'.s/a' z=\TT]. m\s/7^' ; &c. 56. Now in order to integrate the equations (A) of the preceding number, let h = N. sin. (gt-i-^) i I z=N. cos. (gt+Q) ; h' = N'. sin. (gt+S) ; /' = N\ cos. (gt+S) ; these values being substituted in the equation (A), will give Ng = ((0, l)+(0, 2)-h&c.).A^ — [oTT]. iV'_[or^] N''^ &c.\ N'g= ((1, 0)+(l, 2)+&c.).A^'— [ITo]. N'—lT7r\. A''"— &c. V; (B)* A^'g-=((2,o)+(2, i)+&c.).N'—iro]. N—\jrr]' A'— &c.) &c. T T 2 * In general, the number of these algebraic equations is equal to that of the coefficient! 324 CELESTIAL MECHANICS, The number of bodies m, m', fvi', &c., being equal to t, the number of these equations will be also i, and by eliminating the constant quantities N, N' &c., we will have a final equation in g, of the degree i, which can easily be obtained in the following manner : Naming p the function N\ m.\/a.(g—(0, l)-.(0, 2)— &c.) -[.N"m'.\/^'.(g—{l, 0)— (1, 2)~&c.) -f-&c. ■J-2N. »J.v/a.([orT]. iV'+[on;]. N^'^kc.) + &C. * In consequence of the relations which are given in the preceding number, the equations (B) are reduced to the following f -r^ j =r ; (^,)=0j (^^A =0, &c,; therefore N, N', N\ &c. being considered as so many variables, (p will be a maximum. Moreover, (p being an homogeneous function of these variables of the second di- mension ; we have therefore in consequence of the preceding equations,

, in place of N its value deduced from the equation N, N', &c. ; by means of the operations performed on the function gi, the ratio of these coefficients is obtained ; one of them remains undetermined. PART I.— BOOK II. 325 ( -TTv 1=0; this value will be a linear function of the quantities N', N'\ &c. ; in this manner we shall obtain a rational function^ which is both integral and homogeneous, of the second dimension in A^', N", &c., let ?i''' be this function. By differencing ® be this func- tion. By continuing this operation, we will arrive at a function (p*'~" of the second dimension, in iV~", and which will consequently be of the form (A'^ ""'')*. /i ; A- being a function of ^, and of constant quantities. If the diiferential of ip''~'^ taken with respect to A''~", be put equal to cypher, we shall have A: = ; this will give an equation in g of the degree ?', of which the different roots will give so many different systems for the indeterminate quantities N, N', N", &c. ; the inde- terminate N^'~\ will be the arbitrary quantity of each system, we shall obtain immediately, the ratio of the other indeterminate quantities N, N', &c, of the same system to this, by means of the preceding equations taken in an reverse order, namely Let^, g^, gi, be the i roots of the equation in g\ let N, N', N\ &c. be the system of indeterminate quantities relative to the root g ; let N, N,', Nl', &c. be the system of indeterminate quantities relative to the root g,, and so on of the rest : by the known theory of differ- ential linear equations we will have hzzN. sin. {gt-\-^)+Ni. sin. (^z+ej+iVj. sin. {git-\-^d + &c- ; h'=zN'. sin. {gt-\-V}+N^. sin.C^-^+ej-hiV^s'. sin. {g^t-ir^i) + &c. ; ;/''=iV'''.sin.(^?-fe)4-iV/.sin.(^/-fe,)+A7.sin. (^2^+60)+ &c. ; &c. 326 CELESTIAL MECHANICS, ?, Si, ^2, being constant arbitrary quantities. The values of /, t, I", &c. will be obtained by changing in the expressions for h, h!, h", &c. the sines into the cosines. These different values contain twice as many arbitrary quantities, as there are roots g, gi, gi, &c. ; for each sj'stem of indeterminate quantities contains one arbitrary quantity, and besides, there are i arbitrary quantities S, Si, 62, &c. ; these values are consequently the complete integrals of the equations (A) of the preceding number. It is only now required to determine the constant quantities N, iV, &c. N', Nf. &c. €, C',, &c. These constant quantities are not given immediately by observation ; but they make known at a given epoch, the excentricities e, e. &c. of the orbits, and the longitudes -sr, ts', &c. of their perihelions, and consequently the values of A, h\ &c. /, /, &c ; thus we shall derive from them the values of the preceding constant quantities. For this purpose it may be observed, that if we multiply the first, third, and fifth, &c. of the differential equations (A) of the preceding number, by Nm.\/a, N'rd,\^ d, &c. respectively, we will have in consequence of the equations (B), and of the relations found in the preceding number, between (0, l)and (1, 0), (0, 2) and (2, 0), &c. ^^ dh ,. ^ dh' ,- d¥ ,- N. -^. mVa +N'. -^. rd.^d +N". -g^. id'.^/a!' + &c.)* — g. (N.l. mVa + N'J.77i.\/7+N''J'.m"V7' + &c.) * Multiplying the first of the equations (A) by N.m.Va, and the third by N'.m'.'t/a', we shall obtain by adding them together, _, dh ,_ dh' _ - ~dt- "'■'^'^ '^^'•'dT- "''•'V^«' = (0- 1) + (0, 2) + (0, 3) + &c.) l.N.m.Va — [ori].Z'.Mj«.Va — &c. +{(l,0) + (l,2)+(l,3) + &c.) I'.N'M'Wd — [iTo]. /.JV'.m'. Vl' — &c. = (as [i5n].m.v/5 = [M] m'.Va',) UiW^'a. ((0, 1) + (0, 2) + (0, 3) -f- &c.) iV_ [oTT]. iV' _ &c.) + V.rri- VI'. ((1, 0) + (1,2) + (1, 3) + &c. M PART I.— BOOK II. 3!27 By substituting in this equation, in place of //, fi, K', &c. /, t, I", &c. their preceding values ; we will have by comparing the coefficients of the same cosines, 0=JV.2V,.mVa+iV^'.iV;.m'.v/7+iV",iV;'.w"Va^'+ &c. ; 0=N.N^.m-y'~a + N'. No'.m'V^ + N".NJ'.m"Vd' + &c. &c. This being premised, if the preceding values of h, h', &c., be multiplied by N.m.^/a, N'.m'.^/a', Scd, respectively, we will have in consequence of these last equations, N.mh.^/a + N'.m'h'.s/'^ + N".m"h".^^'d' + &c. zz{N\m.\/a + N'\rn.\/a + N"\m".y/'a" H-&c.). sin. {gt-\-%). we shall have in like manner, N.ml.\/a-\-N'm:i'V^ + N".m"r.s/'^ + &c. ={N\m.K/a-{-N'\m'.^/7-{- N"\m!'.s/7' + &c.) cos. {gt-\-^). The commencement of the time being fixed at an epoch, for which the values of h, I, h', I', &c. are supposed to be known ; the two pre- ceding equations give — [liO]. JV— &c.) = (iV./m. Va. + N'.l'm'.>/a'. + &c.) g; now by substituting for dh dh' •^+-^+&c. /, /', &c, we obtain; m.Va. (m.g. cos. (gt-\-Z) + NNg,. cos.(g/-|-e,)-|- NN^g;,. cos. {g,tJf.Z^))+ &c. +nt'.v/a'. (N'^.g. cos. (gt+Z) + N'N.'.g,. cos. (g,t+ Z,) + NN\.g,. cos, (gj +e,)+&c.) =g)N~. mVa. cos. {gt + €) + iVA^, cos. (g^ + g,) + NN'2- cos. (g,<+e,) + m'Va'.N'^. cos. (gt + <^) + N'N,'. cos.{g,t + €) + N'N',. cos. {gtt+ ^2)+ &c.) From hence it follows, that in order for this equation always to obtain, we must have N N,mVa+ N' N;.m'.^/a' + &c. — 0. 328 CELESTIAL MECHANICS, tan. e= N.hn.\/a + WM.m'.^/d'-'r N".k"m''.s/a:'+8ic. * NJm,K/a + N'.l'.m'.x/7+N".l"m"Va"+ &c. This expression of tan. 6 does not contain any indeterminate quantity ; for although the constant quantities N, N', N", depend on the inde- terminate quantity N^'-^'> ; yet, as, their ratio to this indeterminate quantity is known by what precedes, it must disappear from the tan. e. e being thus determined, we shall obtain iV^'~", by means of one of the two equations which determine tan. €, and from it we infer the system of indeterminates N, N', N", &c., relative to the root g. And if in the preceding expressions, this root be successively changed into gi, gs, gs, &c., the values of the arbitrary quantities relative to each of these roots will be obtained. These values being substituted in the expressions for h, I, K, I', &c. the excentricities e, e', &c. of the orbits may be deduced from them, as also the longitudes w, z/, &c., of their perihelions, by means of the equations e* = /«* + l" ; e" = h*'-\-l" ; &c. h , h' tan. w r: -J ; tan. sr — —p- ; &c. thus we shall have e\= N' + iSTj* + N^'' + kc+QNNi. cos. ({gi—g).t+€i~-t)* - +2NN,. cos. ((g,—g).i+t,^Q-)+^N,N,. cos.(5-2— gi).^+e2— e) + &c. This quantity is always less than (^N + A^, + N„+ &c.)*, when ♦ By fixing the origin at the epoch when h, h', I, I', &c. are known, gt vanishes, there- fore the coefficients of N.m.Va ■\- N'^.m'.*/a' -{- &c are sin. 6, cos. S. • The coefficients by which 2NN is multiplied in the values of A'+f are sin. (gi+€). sin. (g,t + €,), COS. (gt + S). cos. {gf +€ ,), and the sura of these two = cos. {g, — g) . * + £,—£). PART I.— BOOK II. 329 the roots g, g„ &c., are all real and unequal, the quantities N, N„ &c., being supposed to be positive. In like manner we shall have N. sin. (g-/.+g)+AV sin. (g-tif+gi)+ JV„. sin. (g.^+ e„).f &c. N. COS. (^gt-\-^)-^Ni. COS. (^git-\-^i)+N2' COS. (g2^+fo)+ &c. ' tan. -azz hence it is easy to infer AVsin.((g.— g)./-fg.-€))+.Vo.sin.((gc-ff).^+e,-g))+&c. im.^-^:-^ f-6;_ ^^^^^ cos.((^'-i-5-). ^+e,-e)) +A^o. cos. ((-2--). ^+e,-e))+&c.* When the sum A^, +iV"j, +&c. of the coefficients of the cosines of tiiis denominator, taken positively, is less than N ; tang, (w — gt — S) can never become infinite ; therefore the angle ra- — gt — g can never attain the fourth part of a circumference ; so that in this case, the true mean motion of the perihelion is equal to gt. 57. From what precedes it follows, that the excentricities of the orbits, and the positions of the greater axes are subject to considerable variations, which change at length the nature of these orbits, and as their periods depend on the roots g, g , gt, &c., they embrace relatively to the planets, a great number of ages. The excentricities may therefore be considered as of variable ellipticities, and the motions of the peri- helions as not altogether uniform. These variations are very consider- PART 1. BOOK II. V U ^ r^ , , »v> tan. ar — tan. (ff< + S) h , ., * Tan. izr—{st-\-i)) = — ; l^-i-;s,= -, tan. (gtJ^^) V \& -r Ji l-f-tan. a-. tan.(^4-€) J_ - l+y.tan.(^/+e) A. COS. (f<+£) — /. sin. (rf+?) , ,../.,.,.. , ,, , ,; — r— ^ — ^-^5r> no«' by substitutine for h and / their values, and observ- /. cos.(^+£)+A. sin. (^<+e)' •' ^ ing that sin. (g«+S). cos. (^;+g,)— sin.(g/+Q. cos.(g<+€)=sin.((g,— g). t + (?,—£)), the numerator of this fraction becomes A', sin. {gt + £). cos. (gi + S) + -^^z- sin. (gf + £,). cos. {gt-\-Z)Jr &c — iV. sin. (gt + £). cos. {gl + /a^+ &c. consequently P=0, Q—0, P'z=.0, Q'=0, &c. It follows therefore that the expressions of h, I, h', I', &c., do not contain either exponen- tial quantities, or arcs of circles, and that consequently all the roots of the equation in g are real and unequal. The system of the orbits of ?«, in, m", he, is therefore perfectly stable, relatively to their excentricities ; these orbits only oscillate about a mean state of ellipticity, from which they deviate a little, the greater axes remaining the same : their excentricities are always sub- ject to this condition, namely, that the sum of their squares multiplied respectively by the masses of the bodies, and by the square roots of their greater axes is constantly the same. 58. When, by what precedes, the values of e, and of ts- shall have been determined ; let them be substituted in all the terms of the expressions for r, and —7—, which are given in the preceding numbers, the terms which contain the time t, without the signs sine and cosine, * See Lacroix, torn, 2, No. 613, for the truth of the assertion will be immediately ap- parent, in the first case, if in place of the sines and cosines their imaginary exponentials be substituted, or if in the second, the equal roots be supposed to diflFer by very small in- determinate quantities. 384 CELESTIAL MECHANICS, being effaced. The elliptic part of these expressions will be the same as in the case of the undisturbed orbit, with this sole difference, that the excentricity and the position of the perihelion will be variable ; but the period of these variations being very long, on account of the smallness of the masses m, m', m", relatively to M; we can suppose these variations proportional to the time, for a long interval, which for the planets may be extended to several ages, before and after the epoch which we select for the origin of the time. It is useful, for astronomical purposes, to have under this form the secular variations of the excentricities of the perihelions of their orbits ; they can be easily inferred from the preceding formulae. In fact, the equation e*=A*-j-f, gives edezzhdh+ldl ; and if we only consider the action of m', we have, by N". 55, §=(o,,)./-[on:.f; therefore -^=-(0,l).A + [oQ].A'> but we have h'U—hl'—e.ef. sin. (z/—sr) ; therefore we shall have -J- =[^]' ^- sin. (tj — bt) ; consequently, if we only take into account the reciprocal action of the bodies m', m", &c. we shall have -^ = \El\- e'-sin. (^— T!r)+ [ol] ^'. sin. (z^'—z,)-^ &c. -^ = [iT], e. sin. (w— i!r')+[ili]. e^'- sin. {z!"—z/)^&cc. • h—t. sin. ■zr, A' = e'. sin. -s/ ; l^ e. cos. ar ; i' = e' cos. w', •/ hi — hi = te'. sin. PART I.— BOOK II. 335 ^^1 '—r- = [iTol. e. sin. {-a — g^04-[g' Q. C sin. (w— tst^-J- &c. ; &c. The equation tang, w =: -j-, gives by difFerenceing it e^.dts zz l.dh — • h,dl. If the action of m', be only considered, by substituting for dh and dl their values, we shall have i^=(0, 1). (A» + /»)-[^]' {^A' + ^0 } * which gives -£- = (0, 1)— [oTT]. -^. COS. (Tsr'—Zir) j therefore we shall have, in consequence of the reciprocal actions of the bodies, m, m', m", Sec. ; ^"^ ==(0, 1)+ (0, 2)+&C.— [oT]. — . COS. (-sZ—Tir) — dt e" [o. g"}. . COS. (w"— w)— &C. J e -^=(l,0) + (l,2),4-&c.— [i^].-^. cos. (w— w')— . e" (" i, z"! . — r. cos. (w"— w*) — &c. } cos — — = ; but from the equation tang, -a — -j- , we have cos. 'w s ,^ — ^ — -^, V by substituting we have the expression in the text. 336 CELESTIAL MECHANICS, -^= {% 0;+(2, 0+&C. - [E^]. -^. COS. (^-;;r")- [2^].-7r. COS. (■=/ — ■et'O — &c. &c. These values of -rr, -7-, &c. : —7-. — Tr-» &c., being multiplied by at at at at the time t, the differential expressions of the secular variations of the excentricities and of the perihelions will be had ; and these expres- sions, which are only rigorously true, when t is indefinitely small, can however serve for a long interval, relatively to the planets. Their comparison with accurate observations, which are made at consider- able intervals from each other, is the most exact means of determining the masses of the planets, which have no satellites. For any time t, the 1. . . (de\ , f d'-e ,„ de d^'e ^ excentncity e is equal to e-\-tA-r j + —3-. ■•+«c; e,—r-, ,^ , &c. being relative to the origin of the time t, or to the epoch. The preced- es ing value of — ^ will give by differencing it, and by observing that a, etc a', &C.J are constant, the values of -j-^t '"ITr^ *^^" ^^ ^'^^ therefore continue as far as we please the preceding series, and by a similar process, the series relative to to- ; but in the case of the planets, it will be sufficient, in the comparison of the most ancient observations of which we are in possession, to take into account the square of the time, in the expressions in series of e, e', &c., a-, -a', &c. 59. Let us now consider the equations relative to the position of the orbits, and for this purpose let the equations (3) and (4) of N°. 5S, be resumed, 4-=:^.aV.B....(;^y). PART I.— BOOK II. 337 By No. 49, we have, fl'a'.jB<"= «*. e^^' ; and by the same number we have '}■' -_ ^-^-i (1—*)* ' therefore we shall have mn 4 •' 3*a' 4.(1— a The second member of this equation is that which we have de- signated by (0, O in N°. 55 ; consequently we shall have ^=(0,l).(y'-j); -^ = (0, 1). ip-^p) J hence it' is easy to infer, that the values of q, p, q\ ^Z, &c., will be determined by the following system of differential equations, -^=((0,l)+(0,2)+&c.).i>-(0, l).y-(0,!2).i>"-&c. \ -^=-((0, l)+(0,2)+&c.).y+(0, 1 ).?'+(0, 2). /+&C. d(i _ dt =((l,0)+(l,2)+&c.)./— (1, 0).^)— (1, 2). y— &c. ^=-((l,0)+(l,2)+&c.)g'+(1.0)- ?+(!' 2)- /"+&«. rf/ (C; ^=((2, 0) +(2, 1 )+&c.). /"-(S.O). p— (2, 1 ). p'— &C. rf^ -^= - ((2,0)+(2,l)+&c.). /+(2,0;,9+(2, 1). 9'+&c. &c. PART I. — BOOK II. X. X 338 CELESTIAL MECHANICS, This system of equations is similar to that of the equations (A) of N°. 55 ; it would coincide altogether with it, if in the equations A, h, I, h', I', &c., be changed into y, p, q, p', &c., and if we suppose [ oTT'] . = (0, 1) ; [iTo] = (1, O), &c. ; consequently, the analysis which we have employed in N". 56, in order to integrate the equa- tion (A), is applicable to the equations (C). Therefore let us suppose q z=N. cos. (gt+^)-\-Ni. COS. (git-\-^i)-^N2. cos. (gJ+Q+Scc. ; p = N. sin. (gt+&)-\-N'i. sin. (git+^O+N^. sin. (gst+Q2)+ &<=• i q' = ivr'.cos. f^f+e)+iVi'. COS. (§i?+e,)+A7. cos. (^2f+g2)+ &c. ; / = iV'. sin. (gt-^%)^Ni. sin. {g /+ei)+iV'2. sin. {git^%^-if &c. ; &c. and by the method given in N°. 5^, an equation in g of the degree i, may be obtained, of which the different roots will be g, g,, g^, &c. It is easy to see that one of these roots vanishes, because the equations (C) vvill be satisfied by supposing p, p', p", &c., equal and constant, and also q, q', (^', &c., but this requires that one of the roots of the equa- tion in g should vanish, and thus the equation is depressed to the degree i — 1. The arbitrary quantities N, Ni, N^, &c., €, fi, Sj, &c., may be determined by the method detailed in N°. 56. Finally, by an analysis similar to that of No, 57, we shall find const. = {p^-\-q'). m.\/a-\-(p'^ + q'^). m'.^+ &c. ; from which may be inferred, as in the above cited N". that the ex- pressions of p, q, p', q', &c., do not contain either arcs of a circle, or exponential quantities, when the bodies m, m', m", &c., revolve in the same direction : and that consequently all the roots of the equation in g, are real and unequal. Two other integrals of the equations in C may be obtained. In fact, if the first of these equations be multiplied by m. y/a, the third by m.\/d^ the fifth by m".ija", &c , we shall have in consequence of the relations found m N°. 55, PART I.— BOOK II. 339 0= -^. m.\/a + -^. w'VZ + &c. which being integrated, gives constant = ^.»2.v/a + 9'.»j'.\/a'+ &c. (1) In the same manner we shall find constant =. f.m.^ a -\- p .rn.sf a' -f &c. (2) Naming p the inclination of the orbit of m, on the fixed plane, and 9 the longitude of the ascending node of this orbit on the same plane ; the latitude of m will be very nearly, tang. ip. sin. (n/+£ — 6). By comparing* this value, with the following, ^. sin. («/ + i) — ^. cos. {nt-\-i) we will have p ■=. tang. (p. sin. 6 ; ^ i= tang. (p. cos, 6\ hence we deduce tang. 9 = \//)*+y* ; tang. 6 = ^^ , -JL 1 therefore the inclination of the orbit of m, and the position of its node, may be obtained by means of the value of ^ and of q. If we denote successively by one stroke, two strokes, &c., relatively to m', m", &c., the values of tang, (p, and of tang. &, the inclinations of the orbits of xxSJ » I / d'^ \ idh — hdl _ , „ , . . , J „ , w * d. tan. TT.Izz ;— 1= = , • • as P— e-. cos. 'a-, we obtain eHin^idh — hdl; \ COS. ^w/ P as hh'+ll= ed. cos. (a-'— n-) ; if -^7^= (0, 1). (A' + V)— [M]. i}iK-\-U), be divided by e*=A*4-^ we shall have the value of —j- which is given in the text. * This is the value of i, or of the latitude very nearly, when periodic quantities are ne- glected, in fact the values of ip and i, which are derived from a comparison of the two values of*, are tho mean values, only affected with secular inequalities ; see N°. 53. 340 CELESTIAL MECHANICS, m', val'., &c., and the positions of their nodes, will be had by means of the quantities f', q. f", q", &c. The quantity s/p^ + q*, is less than the sum N, Nu N^, + &c. of the coefficients of the sines of the expression for q ; therefore these co- efficients being very small, because by hypothesis, the orbit is inclined by a very small angle to the fixed plane, its inclination to this plane will be always inconsiderable ; hence it follows, that the system of the orbits is always stable relative to their inclinations, as well as relative to their excentricities. The inclinations of the orbits may therefore be considered as variable quantities comprised between determinate limits, and the motions of the nodes as not being altogether uniform. These variations are very sensible in the satellites of Jupiter, and we shall see in the sequel that they explain the singular phenomena, which are observed in the inclination of the orbit of the fourth satellite. From the preceding expressions for p and q, results the following theorem : That if a circle be conceived, of which the inclination to the fixed plane is N, and of which gt-{-^ is the longitude of its ascending node ; and if on this first circle a second be conceived inclined to it by an angle- equal to N„ g,t+^, being the longitude of its intersection with the second circle, and so of the rest ; the position of the last circle will be that of the orbit of m. The same construction being applied to the expressions of /z and of /of N". 56 ; it will appear that the tangent of the inclination of the last circle on the fixed plane, is equal to the excentricity of the orbit of m, and that the longitude of the intersection of this circle with the same plane, is equal to that of tlie perihelion of the orbit of in. 60.* It is useful for astronomical purposes to obtain the differential variations of the nodes and of the inclinations of the orbits. For this purpose let the equations of the preceding N°. be resumed, namely, ♦ It should be observed, that tlie differential^expressions which are gJvcii in this N"., are relative to the secular variations uf Uit- uoJes ana ot the inclinations of the orbits. PART I.— BOOK II. 341 tang.

'■ sin. «' cos. '6 + (0, 2). tan. (p". cos. e + &c. ; hence, obliterating the terms which destroy each other, and making corresponding factors of tan. ip", tan. (p"', &c., to coalesce, we obtain the expressions which are giveen in the text. Since dp dq dp/ . p and qzzO, the coefficients of these terms are neglected in the vhlue of "^j ''jr'~Jt' 344 CELESTIAL MECHANICS, ((1, 3)— (0, 3)). tang. • In like manner we have x'dy'—y'dx' _ / a'.(\J:^ dt ^ 1-ftang. >'' &c. These values of xdy—ydx, x'dy'—y'dx', &c., may be employed, when we do not take into account the inequalities of the motion of the planets, provided that the elements e, e', &c., (p, (p', &c., are consi- dered as variable, in consequence of the secular inequalities ; therefore the equation (4) of N". (9). will then give ' l-f-tang. *(P 1+tang.?)* +E.7nm' , ^ (a/— t). {di/—dy)—{y'^y).{dx'-dx) 7 'I dl 5* This last term, which is always of the order mm', being neglected, we shall have c = m.y — !^ /- + 7n'. V — H^ r^ + &c. ^ 1-l-tang. > ^ ^ 1-l-tang. *9' ^ Therefore, whatever changes may be produced in the progress of time, in the values of e, e, Sec, tp, ') + &c. ; on the supposition that the orbits are very nearly circular, and inclined to each other at small angles, the secular variations of the ex- centricities of the orbits, are by N°. 56, determined by means of dif- ferential equations independent of the inclinations, and which are therefore the same as if the orbits existed in the same plane ; but on this hypothesis, ?> ir 0, ?>' = 0, &c. ; consequently, the preceding equation becomes const. = e'-.niy/'a + e"'.vi'.\/'d^->re"''.m".s/ a-'-if &c. this equation has been already obtained in N°. 57. In like manner, the secular variations of the inclinations of the orbits, are by N". 59, determined by means of diflFerential equations independent of the excentricities, and which are therefore the same as if the orbits were circular ; but on this hypothesis, ezzO, elzz. O, &c. ; therefore const. z=m.\/a. tang, (p* -\-n^ .\/ a! . tang. (p'*-{-rnf'-\/a". tang. ?**+ &c. which equation has been obtained in N°. 59- If we suppose, as in this last number, that p zz tang, (p. sin. ; qzz tang. ip. cos. 6 ; It is easy to be assured, when the inclination of the orbit of m, on the plane of a; and _y, is '-■ " '"^ expression — = — — [ — j'i ) ^^ multiplied by m, it will give when quantities of the order m* are neglected, — = — — m. ^ ^'' ' , •/ by making similar substitutions for the bodies rri, m", &c., we obtain the expression which is given in the text. PART L— BOOK II. 349 which preserves always a parallel position, and which the following con- dition enable us to find easily, at all times, namely, that the sum of the masses of the system respectively multiplied by the projections of the areas described by the radii vectores in a given time, is a maximum. It is principally in the theory of the solar system, that the investigation of this plane is important, in consequence of the proper motions of the stars, and of the ecliptic, which render the exact determination of the celestial motions a matter of great difficulty to astronomers. Naming y the inclination of this invariable plane, to the plane of x and of ^, and n the longitude of its ascending node, it follows, from what has been demonstrated in N". 21 and '^2, of the first book, that we will have tan, y. sm. n — — ; tan. y. cos. n n — ; consequently. 7n.v/a.(l-e*. sin. a, sin. 6-fm'.v/a'.('l-e"'). sin. ■ ■ ■ — 7n.v fl.(l — e*). cos. <(-\-m' y d .{\ — e*). cos. '. Therefore de- noting this mutual inclination by w, we shall have m',\^a'.(l — e'*). sin. tir tan. (p= — ■ — ^^ , ^ .t m.\/a.{\ — e*)+m .%/«'.! 1 — e'*). cos.w • Multiplying both sides of this equation by 2m. Va.{^\ — e^. cos. ip, we obtain 2mc. cos. (p. •/a.(l — e* = 2ni^a(l — e"). cos 'ip + 2jn.m'. cos. ip. cos. (p'Va.(\ — ^). V a'{\ — e^), which will coincide with the second member of this equation, if we substitute for c^ its value, and observe that m?a.[\ — e'). sin. ''(p=jn"a'.(l — e^). sin. *(?'. f When q> and ~yr> <^y ^» &c., which, in the case of the invariable* ellipse, is equal to cypher ; this function is therefore likewise nothing in * When in V" — V, the values of — —, • . „ » — rr- > due to the elUptic motion are substi- tuted, the terms of the resulting equation must be identically equal to cypher ; but in the case d'^x d^v d^- of F;-F, the values of-—, — |-, -—1. must be increased by the quantities due to the aaion of the disturbmg forces ; so that after substitution, the resulting expression may be resolved into two distinct equations, one of which would obtain, if there were no dis- PART I— BOOK II. 357 the case of the variable ellipse. We have evidently in this last case, V, — F'=0 ; for this equation is the differential of V'zzO ; by subtracting from it the equations P'' — V' = 0, we shall have V' — V'zzO'y conse- quently we can in this case difference the equation V'=:0, dx, di/, dz, c, c', &c., being solely made to vary, provided that for d''x, d^'y, d^z, be substituted the parts of their values, relative to the disturbing forces. These results are precisely the same as those which we obtained in N°. 45, from considerations purely analytic \ but considering their great importance, it was deemed right to deduce them here from the consi- deration of elliptic motion. This being premised, 64. Let the equations (P) of N°. 46 be resumed. 0=4^+ dV Oz= dtf" d^z -73- de fdR\ {dxj' dR\ dyj' ( (p) If we suppose R=:0, we shall have the equations of elliptic motion, which were integrated in the third chapter. In N". 18, the seven following integrals were obtained, -zdz „ ydz — zdy xdy — ydx , xdz — z^^ „ - It ' It ' 0=/+^;. t^[^ ■\-dz^ dt (dx^'+dz' \ ) , xdx.dy + n _ /^ 2/^ _L f dx''-^d y'-+dz^\ dt' xdx.dz de dt zdz.dx ~~df zdz.dy ~dF~ ydy. dz dF~ {P) turbing forces, and by means of the other the variations af the parameter, may be ob- tained, these equations are respectively equal to V" and V, — F". 358 CELESTIAL MECHANICS, These integrals give the arbitrary quantities, in functions of the coor- dinates and of their first differences ; their form is extremely commo- dious for determining the variations of these arbitrary quantities. The three first integrals give, by differencing them, and by making the pa- rameters c, (/, d', &c., and the first differences of the coordinates solely to vary, _ xdP'y — yd^x , , __ xdi^z — zd^'x , «_ yd^z—zd^y '^'- di ' "^^ - Jt ' '^''- di ' By substituting in place of d^x, d^y, d^'z, the parts of their values whicl* are due to the actions of the disturbing forces, and which in virtue the differential equations CP), are — dt"", ( -f-;] » — dt^. ( ;t- ) > — dt^' ( -T- J ; vye shall have / ,, ^ fdR\ fdR\l We have seen in N°'. 18, and 19, that the parameters c, d, d', &c., determine the three elements of the elliptic orbit, namely, (p the incli- nation of the orbit on the plane of x and 3/, and 9 the longitude of its node, by means of the equations tan. (B ■=. -T ; tan. 6 = — j- j c d and the semipararaeter a.(l — e'") of the ellipse, by means of the equation iua.(l— e*) =z e-Vd"- + d"- ; These same equations obtain also in the case of the variable ellipse, PART L— BOOK II. S5d provided that c, c', d , are determined by means of the preceding differential equations. In this manner the parameter of the variable ellipse, its inclination to the fixed plane of x and y, and the position of its node may be obtained. By means of the three first equations (p), we have deduced in N°. 19 the finite integral Q — d'x — dy + cz ; this equation, and also its first differential, = d'dx — ddy + cdz, taken on the supposition that c, c', d', are constant, obtain in case of the disturbed ellipse. If the fourth, the fifth, and sixth of the integrals {'p) be differenced, the parameters f, /', f", and the differences dx, dy, dz, being con- sidered as the sole variables, and if then we substitute, in place of d''x„ d^'y, d^z, the quantities — dt''. { -r- ) > — dt^' 17") — '^^^ I y- J , we shall have + (jydx—xdy). ^-^j +(zdx — xdz). (^ j 5 * Differentiating under these restrictions we have „ (du.(Py+dz.dh) , d'^y ^ d'-x , , d^x . d°-z •/ by ordering the terms we have df= dy. ( ^. — - X. ^) ^dz. (..__.._)+ (ydx-xdy). ^, + d^Z , ^. fdR\ /'^^\ /<^^\ (:rfx — xdz). —„, wliich becomes the expression m the text, wheni -^1, i~7~)' Vd')' , . , ^ d''x d-ii d°z are substituted for -j-r-, -;^) -r-;. dt- d-y dt^ 360 CELESTIAL MECHANICS, + {xdy-ydx). g| + izdy-ydz). |^^ j ^^{xdz-zdx). Jg ^ +{ydz^zdy). \^\ . Finally, the seventh of the integrals (^), when differenced with the same restrictions, will give the variations of the semiaxis major a, by- means of the equation d. ^ = 2. diJ, the differential di2 being* re- ferred solely to the coordinates x, y, z, of the body m. The values oif,f\ f, determine the longitude of the projection of the perihelion of the orbit, on the fixed plane, and the ratio of the excentricity to the semiaxis major ; for / being the longitude of this projection, we have by N". 19, tan. /= ^, * Differentiating the seventh equation under the same restrictions, we obtain d. — = By means of this expression, Lagrange ascertained that the mean motions were invari- able, if the first power of the disturbing masses be only considered, the approximation being extended to any power of the excentricities and inclinations. From the extreme simplicity of this expression of the differential of the major axis, the determination of the longitude is a very easy problem. In the supplement to the third book, Laplace investi- gated the simplest form of which the other elements were susceptible, and he has suc- ceeded in assigning such a form to them, that they only depend on partial differences of the same function, taken with respect to these elements, and what is particularly remark- able, the coefficients of these differences do not involve the time, and are solely functions of the elements themselves. PART I.— BOOK II. s6i and e being the ratio of the excentricity to the semiaxis major, we have by the same number This ratio may also be determined, by dividing the semiparameter a.(l — e*), by the semiaxis major a, and by taking the quotient from unity, the value of e^ will be obtained. The integrals (p) have given by elimination, in N°. 19, the finite integral, O^j^r — h*-\-fx-\-f'y-\-f"z\ this equation obtains also in the case of the disturbed ellipse, and it determines at each instant the nature of the variable ellipse, we can difference it, f, /', /'''', being considered as constant quantities, which gives = ft.dr+fdx+f'dy-y'dz. The semiaxis major a determines the mean motion of m, or more accurately, that which in the troubled orbit, corresponds to the mean motion in the invariable orbit ; for by N". 20, we have nz=.a~*'\/y. ; moreover, if we denote by ^ the mean motion of m, we have in the invariable elliptic orbit d^ = ndt j this equation obtains equally for the variable ellipse, since it is a differential of the first order. By differ- encing, we shall have c?'^ =r dn.dt; but we have , San J «* SanAR , dnzz —- — . a. — = , therefore San.dtAR d^^ = PART I. BOOK II. S A 362 CELESTIAL MECHANICS, and by integrating ^ = — . JXayi.dtAR . Finally, it has been observed in N°. 18, that the integrals (p) are only equivalent to five distinct integrals, and that they furnish between the seven parameters c, d, d , f,f',f'\ and e the two following equa- tions of condition, rr;;, il O^fd'-fd^f'c; these equations obtain also in the case of the variable ellipse, provided that the parameters be determined by what precedes. We can likewise be assured of this a posteriori. We have thus determined five elements of the disturbed orbit, namely its inclination, the position of its nodes, the semiaxis major (which gives the mean motion), its excentricity, and the position of the perihelion. It now remains for us to determine the sixth element of the elliptic motion, namely, that which in the undisturbed ellipse corresponds to the position of m, at a given epoch. For this purpose, let the expression for dt of N°. 1 8 be resumed. J (1+e. cos.(t;— ^))* This equation being expanded into a series, gave in the number already cited, ndt =dv.(l+ E™. cos. (r— zr) + £(^'. cos. ±(v—zr) + &c.) : which being integrated on the supposition that e and w are constant, will give fndt+izzv-\- J2"'. sin. (v — v) -\ ^— . sin. 2.{v — or) + &c. PART I.— BOOK II. 363 t being an arbitrary quantity. This integral is relative to the inva- riable ellipse ; in order to extend it to the disturbed ellipse, it is ne- cessary when we make ail the terms to vary, even to the arbitrary quantities e, e and ■nr, which it contains, that its differential should coincide with the preceding ; which gives di—de. \ 1—7— \' (sin. (v — w) -f- \, \—j— \ • sm* ^'iy — ■=^)+ &c. | —d^.iWK cos.(i>-3r) + i:%cos,.2i" + Szc.)t+ A, so that the similar equa. tion of mean motion would suppose in + i'n'+i'n" + &c. =0, which is even more improba- ble than the equation i'n' — iti=0 ; besides, if this last equation obtained, when there were only three bodies, it would cease to exist when the action of the other planets was taken into account. * As i'=5, and? = 2, i'—i = 3, and consequently, the periodic function is multi- plied by quantities of the third order, with respect to the excentricities and inclinations- If the axis major is subject to an inequality increasing proportionally to the time, the mean longitude has one increasing proportionally to the square of the time. See N". 5i- 368 CELESTIAL MECHANICS, centricities and inclinations of the orbits, as has been observed in N*. 48. The preceding analysis gives the most sensible part of these inequalities ; for the variation of the mean longitude depends on tw^o integrations ; while the variations of the other elements of elliptic motion depend only on one integration ; consequently, the terms of the expression for the mean longitude are those solely, which can have the square of (?V — in) for a divisor ; therefore taking into account these terms solely, which considering the smallness of this divisor, must be the most considerable, it will be sufficient in the expressions for the radius vector, the longitude, and the latitude, to increase by these terms the mean longitude. When we have inequalities of this kind, which the action of m produces in the mean motion ofm; it is easy to infer the correspond- ing inequalities produced by the action of m on the mean motion of m'. In fact, if we only consider the mutual action of the three bodies M, m and m! ; the formula (7) of N". 9. gives const. = ™. V^+^^J:) h. „,,«fe-+y+-^^") ((mdx-{-in'da/y + {mdy 4- rri'dr/y + (jndz + midzy ) , . 2Mm QMm' 2mm' The last of the integrals {p) of the preceding number, gives by substituting for — , the integral S/dii a dx*->rdty-\-dz* _ 2.(M + 7n) __^ .^^ If we then call; R' what R becomes, when the action of m on mf i« considered, we shall have' PART I.— BOOK II. 369 ^,_ m.{xafAry2f-\-z^^ m df -^/^^q^N^* the differential characteristic d' only referring to the coordinates x't y, zl, of the body w!. By substituting in the equation (a) in place of TIT and of —-t:^-^ 1 these values, we ^ df- dt* shall have mfdR+m'J'd'R = const. C(m.dx+m'dx')*-\-(m.dy + n/.dy'y + {m.dz-{-m'.dz'y) 2.(M + m+m')dt* It is evident that the second member of this equation does not contain terms of the order of the squares and of the products of the masses m and m', which have for a divisior ifn' — in j therefore if we only consider such terms, we shall have mfdR+m'/d'H' = ; hence if we only take into account those terms, of which the divisor is (i'n' — iny, we shall have Sffa'n'dtd'R' _ __ m.{M+m).a'n' Sffandt.dR ^ M-\-m' " m'.(^M+m').an* M+m / but we have . _ S.f/andtdR __ S.ffdn'dt.d'R! ^ consequently PART 1. BOOK II. 3 B V S70 CELESTIAL MECHANICS, 7n'.(jM +m').an^' ±in.(M+m).a'n'))=. ; moreover \/M+ m , \/M4-m' , • 77' — I . S } "■ ,5 therefore m and m' being neglected in comparison with M, we shall have _ or m.y/a Thus the inequalities of ^, which have for a divisor (i'n' — iny will make known those of ^', which have the same divisor. These ine- qualities are, as we have seen, affected with contrary signs, if n and n' have the same sign, or what comes to the same thing, if the two bodies m and m' revolve in the same direction ; they are besides in a constant ratio to each other ; hence it follows, that if they appear to accelerate the mean motion of m, 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 m'.\/a' to m.\/a. The acce- leration of the mean motion of Jupiter, and the retardation of the mean motion of Saturn, which the comparison of ancient with modern observations made known to Halley, being very nearly in this ratio ; I have inferred from the preceding theorem, that they are owing to the mutual action of these two planets ; and since it has been demonstrated, that this action cannot produce any change in the mean motions, independent of the configuration of the planets, I did not hesitate to admit that there exists in the theory of Jupiter and Saturn, a great periodic inequality of a very long period. And observ- ing then that five times the mean motion of Saturn, minus twice that of Jupiter, is very nearly equal to cypher, it appeared to me very probable that the cause of the phenomena observed by Halley, was PART I.— BOOK II. 371 an inequality depending on this argument. The determination of this inequality verified my conjecture. The period of the argument {i'n't — ini), being supposed very long, the elements of the orbits m' and m experience in this interval sensible variations, which it is essentially necessary to consider in the double mtegvsX ff akn* .dt^ . sin. (i'n't — int-^A). For this purpose, we shall make the function k. sin. [i'n't — int+A) assume the form Q. sin. (i'n't — int+i'e' — ie)+Q. cos. (i'n't — int-\-i'e'—ie) ; Q and Q' being functions of the elements of the orbits, we shall have consequently ffakn*.di\ sin. (i'n't—int-^A) = • n*a. sin, (i'n't— int-\-i't— it) ^ _ 2c?Q' Sd^Q (i'ri—iny V^ [i'n'^in).dt (i'n'—in)\dt^ ^ (i'n'—inydt^^ S 3b 2 • Substituting for A its value iV — u — gzr — g'w — g"t — g"'i; sin. [in't — inf+A) = sin. (in't — intJ^iW — ii). cos.(g'o+g'a'-\-g"i-{-g"'i') — cos.{in't — int+i't — it), sin. (g-w+^'w'-t- g"H"g"'*)» hence the value of k. sin. {iri't — inf+A) will be given; calling in't — int + i't' . — ii-/t + b, the quantity to be integrated becomes fdt. fdt. siB.(^Ji+b)Q+/dt. /dt. cos. (./l+b).Q, now one integration gives fdt. sin. (Ji-\-b). Q= rr- cos. {Ji+b)+-jrf. COS. (fi + b). dt. -^.(= j^'^- sin. (/^+J)+ jx-f- sin. {.fi-\-h)dt. ^+ 4c. O \ dO 1 d^Q 1 d'Q = ^.cos.(fi + A)-yF. ^. sin. (/<+&)+ _. ^.cos. (fi + b)^-^.-^, sin.(y]! + i)^&c. ; in like manner we can obtain by partial integration, yc?^ cos.(./?+A).Q' = -^. sin (/<+6)+ -i . ^. cos. (yi + i) - jr. ^- sin- (^2+*)— *c., in order to obuin the second integrals, i. e. fdt. fdt. sin. (ft + b), each of the terms of the preceding series into which the first integrals may be resolved, should be multiplied by dt, and then integrated in the same manner as fdt. sin. (ft-\-b). Q, and if all the fac- tors of sin. (Jl + b), and cos. {ft-\-b) be respectively collected, we shall obtain by sub. stituting for y and b, the expression given in the text. 372 CELESTIAL MECHANICS, n'-a. COS. [i'n't—int-\-i't—u) C 2rfQ 3rf*Q' \Q! + (i'n'—iny I ' {i'n'—m).dt {i'n'—inydt (i'n'—iny.dfi^ In consequence of the slowness of the secular variations of the elliptic elements, the terms of these two series decrease with great rapidity. We may therefore only consider the two first terms in each series. If then we substitute in place of the elements of the orbits, their values arranged according to the powers of the time, the first power being the only one which is retained ; the preceding double integral may be transformed into one sole term of the form {F-\-E.t'). sin. (i'n't—mt-\-A-\-H.t). Relatively to Jupiter and Saturn, this expression will serve for several centuries before and after the instant, which may have been selected for the epoch. The great inequalities of which we have been speaking, produce some sensible terms among those which depend on the second power of the disturbing masses. In fact, if in the formula ^ _ 3w^^ ffakn\dt\ sin. {i'^—i?^ + A) j we substitute for | and ^ their values 2i.m'.an''k nt- -TTj-, — ^t; • sin- i^t^t — int+A) ; fA.(tn — my t j> there will result among the terms of the order ra*, the following • Assuming p = - . , and / = , , ^rr. -rrr , the value of ^ = £. PART I.— BOOK II. 37s There will result in the value of |' a corresponding term, which is to the preceding in the ratio oi m.\/a to — m'.>/d , it is therefore 3.i*.w»'*a*.n*.F C .— ,— 1 m.\/~a 66, It may happen, that the most sensible inequalities of mean motion occur among the terras of the order of the squares of the dis- turbing masses. If we suppose three bodies m, m', rr^', to revolve about M, the expression for di2 relative to terms of this order, will contain inequalities of the form A. sin. (?n/ — ?'n7+«V^+^)> now if the mean motions of m, m\ rnf, &c., are such that in — ?n' + i"n!', may be supposed a very small fraction of n, there will result a very sensible inequality in the value of ^. This inequality may even render rigorously equal to cypher, the quantity in — i'n' + i'n", and thus establish an equation of condition between the mean motions and the mean longitudes of the three bodies m, ni, rnf'-, this remarkable case obtains in the system of the satellites of Jupiter. We proceed to deve- lope the analysis of it. If we suppose M to represent the unity of mass, and if m, m', m", be neglected in comparison with M, we shall have ffdi^. sin. {irlt — int-\-(jl — p). sin. {irit — int — A) + A) (a), now ify,p be supposed to be very small, we shall have sin. ((p'— jj). sin. {in't — intJf A)) = (y— ;?). sin. {i'n't — in/-f- A)» and the cosine of the same quantity =1, in each case these expressions are true, for the first power of the disturbing force ; •/ in the expression (a) a term occurs =£. cos.(i'n't int + A) (i'pf+ip). sin. (i'n't — inf— A)= — . (i'p'+ip)' sin.2.(in't — int + A), now i'p/ + ip _ 3ian^k.(i'mVa + im'.^a' , „ Sim'^aM , _ . — : , and E =r , "the coefficient of sin. 2.(n't — nt fi.{i'n'—inf.V a! f*-"* , .. 9i\m'°-.aV.k^ {i'mVa +i'm'V7 . ^ , ,_ ^ ,., . . . + A)=: ■ , , ., :— — • 7= ), and when the double integration is per- formed, there will result the expression given in the text. S74 CELESTIAL MECHANICS, we have also dlzz ndt', d^ = n'dt; di"= nf'dt; consequently d»i 3 ^ da d^' 3 ,i da! d^?" 3 A da" dt\- r^' a* ' dt - I'd'' dt - 2*" v»- It has been observed in N°. 6l, that if we only consider inequalities which have very long periods, we have , , m , m' , m" constant = 1 r H t ; a a a' which gives da , da' . ,, da" a* a* ff'* It has been also observed in the same number, that if the squares of the excentricities and of the inclinations of the orbits be neglected, we have constant = m.\/a + vd.y/d -\-m".\/a" j which gives _ mda ni.dd . m\da" = — ;= + — 7^=- + x/a x/a' n/o* * From these different equations it is easy to infer d'^l _ 3 \ da * ~dr-'~2-'^' d^ d^_3_ m.n's / n—n" \ da \fit ~ -2' m'.n • \n'—n")' a« d*^" _ 3 m.n"i / n—ri \ da dt ~ 2 m".n' \n-^'' ) ' a** , I 1 2 dn da ^ d^K , i I da . ... • n^ss — , •• — -x* — r=— ^. and— T^=:rf«=:— — .R . — r; '« '»" manner PART I.— BOOK II. 375 da Finally, the equation — = 2/di?, of N°. 64, gives zz 2diJ. It is therefore only requisite to determine AR. By N°. 46 we have, m.r ,-N~J R = —jr- COS. (y'—v) — rri.(r* — 2rr'. cos. (r/ — 5y)+r'«) r" yJI nt'.r + ^. cos. (i/"— v)— m^(r»— Srr^''. cos. (5:;"_t;)+r"*) the squares and the products of the inclinations of the orbits being ne- glected. If this function be developed into a series arranged accord- ing to the powers of the cosines of »' — v, of i/' — v, and of their multiples : we shall have an expression of the following form, /2 = -^. ir,ry^^m:{r,ry\ cos. (v'~v')^m'.(r,rT\ cos. 2.(t/— r^) -j-w'.(r,r')'". cos. 3.(w'— w)+&c. + -^' (,r,r'y^^rri'.{ryY\co%.{v''^v)^rri'.{ryT- cos. 1.\x/'^v) 4 2 maa , maa _ "" = f as n= — 1- 1 7^=— , therefore multiplying -I -t , „ . m.nn'.da m'nri.da' mn'n".da both sides by »in'=a ''. a' *=, we shall obtain 1 1 — = -^ 1- m'rm'Kda' ^ da' _ m.n'.{n—n") da_ ..J^^_^ „,i _^__i.M.n't J? ' '•* a« ~ ?«'.«.(«'—«")' a« ' '•' dt 2' ■ a'^ ^''i^' (M_n^ -JL the expression for — r- may be obtained in a similar manner. («"_«') a* ' ^ dt ^ 376 CELESTIAL MECHANICS, heuce we obtain dR = dr. m < 2 "2~ ]d.(r,rT dr \ +'^'l dr \' ^os.{r/-v) + mf. d.(r,ry^ dr . COS. 2.(5/ — f)+&c. \ d.{r/'y^ ] : dr " +m\^ d.{ryy^\ dr ' ,008.(5/ — !:;)-|-»i''. (^ j \^y. (-008.2.(1/^— t;)+&c. I +^t, S ^'-(r.r^Tl sin. (»'-t;)+2ffz'.(r,r0''^ sin.'2.(t;*-t;)+&c. L ' W'.(r,r'/)a).sin.(r;^-t;)+2OT".(r,r"p.sin.2.(t;"-y)+&c. Suppose agreeably to what is indicated by observations, in the system of the three first satellites of Jupiter, that n — '^n', and n' — 2«". are very small fractions of n, and that their difference (n'—9,n')—(n' — ^2w"), or n — 3n'-\-2n!' is incomparably less than each of them. It results Sr from the expressions of — , and oi Sv, of N°.50, that the action of wt' produces in the radius vector, and in the longitude of m, a very sen- sible inequality, depending on the argument 2.(n7 — nt-{-t — 1). The terms relative to this inequality ,have for a divisor 4.(n'— n)* — w*, or (n — ^n').(3n — 2m'). and this divisor is extremely small in consequence of the smallness of the factor n — 2n', It appears also from a conside- ration of 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 (n't — nt-\-t — e), and which as it has for a divisor («' — n)* — n'' or n.(n— 2«') is extremely sensible. It appears in like manner, that the action of m" on m produces in the same quantities a consi- derable inequality, depending on the argument 2.(w"/^ — n't-\-i" — «'). Finally, we may perceive that the action of m', produces in the longi- tude and radius vector of m" a considerable inequality, depending on the argument n''t — n't-\-('' — t'. These inequalities have been recog- nized by observations, we shall devclope them in detail in the theoi*y PART L— BOOK II. 377 of the satellites of Jupiter ; their magnitude relative to the other ine- qualities permits us to neglect the latter in the present question. Let us therefore suppose ir-=.m'.E'. COS. 2.(wV— n^+t'— t) ; ivzzm.F. sin. 2.(n't — nt+ 1 — i) ; iT^=m".E". COS. '2.{nl't—nt-]rt"--i')+m.G. cos. (j^f—nt+t'—i) ; Sv'=m".F". sin. 2.{n't—n't+i"—i')+m.H. sin. (n't—nti-t'—i) ; irJ'-m".G". COS. {n"t—n't-\-e"—t') ; iv"=m".H'. sin. (n"t—n't + i"—i')- It is necessary now to substitute in the preceding expression for d^, instead of r, v, r', v', r", v", the values of a-tSr, ni-\-i+h', a' +3/, n't+t'+Sv', a"-\-ir'', n"t + ("-\-Sv", and only to retain the terms de- pending on the argument nt—3n't-{-2n"t+ i — Ss'+Sf", it is easy to see that the substitution of the values of ir, Sv, Sr", Sv", cannot produce any such term. Therefore it can only arise from the substitution of the values of Sr^, and of Sv' ; the term m'.{r,r'y^\ dv. sin. (y' — v) of the expression of d7?, produces the following quantity : 2 \E\ — ^V F .(a,a'y^).sir\.nt-3n'l-if2n't+(-3f]-2i"). And it is the only quantity of this kind, which the expression of dR PART I. BOOK II. " 3 C * (r/)(«) = (a,ffl')W+ '^•^''f^^'\ ^r+ ^6^. 3/ + &c. ; sin. (v'—v) = sin. («'<+»' da ■da +3i/ — nt — e — 3w) = sin. (n't — nt + s' — f) + cos. {n't — ni + s' — »). dv' &c. ; by substi- tuting for V, we shall have ^i^^^, J/ = d.(a'a')W ^ ^^„^,^ ^^^ {2.{n"t—n't+ •"— s'), da da which when multiplied into sin. {n't — n^-j-s' — i) gives a term of the form — m".E". sin. (nt — Sn't-\-2n"t-\.i — 3£'+2e"), in like manner by substituting for 3d', we obtain cos. (»V — »if +j — s'). ivf = — m".P. sin. (nt — 3n't + 2n"t + s— 3e' + 2s"), hence if we substitute for dv its value, there will result in the term m', (r,r')('). dv. sin. (i/ — r), the expression which is given in the text. 3T8 CELESTIAL MECHANICS, ir contains. The expressions of — and of J^ of N°. 50, being applied to the action of m" on m', give, when the terms which have n* — 2m' for a divisor are retained, and observing that n" is very nearly equal to UL 2 ' E' _ ^ L t dd S ' ri — n ^3 ~d~ ~ (w^2n"). (3w'— 2w") therefore we shall have ^^_ m'.rri'.ndt ^, ■\ %\a,a!T {d.(,a,a'y^) i 2 ' 'I a' da' S X sin. (nt—3n't+2n"t+('—3s' + 2s")=— 4- -r. This value of -^- being substituted in the values of — ~-, — ^ , and a* ° dt dt d''l" ■^ •, will give, because n is very nearly equal to 2n', and n is very nearly equal to 2«"; * In page 296, if we substitute for — 5 — , A'-^\ their values, the coefficient of cos. da! 2inH-n't+ ."_0 becomes = ^ . n'^.a'^ '-^^ + -^, «'.(a;«"P = t!^' ; ^ ' ' 2 da! n — n" a' (n'— 2n").(3«'— 2n") in like manner the coefficient of sin. 2.(«"i — n't+s" — t') in the expression for Jv', given b page 298, = In'', a-. -^J- + -,-^, . a'.(a!a'T ^E" -=F" = ' ^. («'— 2n").(3n'— 27i") PART I— BOOK 11. 379 d^ _ ^^+ ^S-= e^*- sin. Ciit—^nf-t^n'r+i—Si + Se") ;* at dJ' atr (6 being made, in order to abridge, equal to Or more accurately, SO that if we assume V = ^—3^' + 2|"+ £— S5' + 2£", we shall have — ■=. €.«*. sin. F. dt^" As the mean distances a, a', a", and also the quantity n vary very little, we can in this equation consider €«* as a constant quantity. By integrating it, we obtain 3c2 dr?, 3 I da * —4- = . n^ therefore multiplying by n, we obtain the coeflScient ar 2 a* of — r = — ^. n" = — -^ . n=a, therefore by substituting for — j- , we obtain —^ = j„ / 2.y ) _ '^•("^«0"' y „"-.„'^"o. sin.(«f_2n7+3?j"«+s~3»'-2e"); in like „ da . , ^d°t S mn'i 3? 9 wn'^a' manner the coefficient of—- m the value of -rr = -^' — ; — • — ^='7;* — ; — ' "- dt- 2 m'.n * 2 »»'.?< a- „ 2 which being multiplied into m"m'ndt, gives (by substituting for n'-, its value /— ) 1 —- ^^=1. E„(Wfl_dJ^Y_^ ^,^^_,^^,, sin.(«.-3«.+2«"/+.-3.'+2.«). rfr 2 \ a' da' I '^ 1d-l" The value of - - may be obtained in a similar mamier. 380 CELESTIAL MECHANICS, ±dV dt = \/c — 2S«*. COS. V c being a constant arbitrary quantity. From tlie different values of which this constant is susceptible, the three following cases arise. If c be positive and greater than ±2S«*, the angle F will increase continually ; and this will be the case, if, at the commencement of the motion (« — 3n'-j-2n"y is greater* than ±2Qn*.(l +cos. V), the su- perior, or lower signs having place according as S is positive or negative. It is easy to be assured, and shall we point it out particularly in the theory of the satellites of Jupiter, that 6 is a positive quantity relative to the three first satellites; therefore, supposing T ■a- = tt — V,f (n being the seraicircumference) we shall have dt= ^" \/c-f-2S«*. COS. ra- in the interval from zr = 0, to la ir — ; the radical v/c+^^??2*. cos.ra^ is greater than ^2Sn*, when c is equal to or greater than 2Qn' ; there- fore, the time t in which the angle u passes from zero to a right angles is less than 7=^* '^^^ value of S depends on the masses m, m', fn". The inequalities which have been observed in the motions of the three first satellites of .Jupiter, and which we have already adverted to, assign relations between their masses and that of Jupiter, from which it * If c be positive and greater than ± 2£n', the angle V must always increase, for the quantity under the radical sign can never be equal to cypher ; c — 2?n'. cos. ^ = I -77-) = (n— 3/i' + 2n"f, if this quantity be greater than ± 2£n'.(l :+. cos. V), c — 2Zv?. cos. Fmust be greater than ±2. S«^.(l rpcos. V) ; i. e. c must be greater than 2Sn'. f By making q: -srz: ir- — V, we get rid of the ambiguity of sign in the value oidU PART I.— BOOK II. 381 follows that >= is less than* two years, as we shall see in the theory of these satellites. Therefore the angle ra- passes from zero to a right angle in less than two years ; now from observations made on Jupiter's satellites, it appears that since their discovery, the angle w has been either equal to cypher, or insensible, consequently the case which we have examined, is not that of the three first satellites of Jupiter. If the constant c is less than ± 2Sw*, the angle Fwill only oscillate, it will never attain to two right angles, if € be negative, since then the radical v'^c — 2g?z*. cos. V will become imaginary ;t it will be never equal to cypher, if € is positive. In the first case, its value will be al- ternately greater or less than cypher ; in the second case, it will be alternately greater or less than two right angles. From all observations made on the three first satellites of Jupiter, it appears that this second case, is that of these stars, therefore the value of £ ought to be positive relatively to them, and as the theory of gravity assigns a positive value to €, we ought to consider this phenomenon as an additional confir- mation of this theory. Since according to observation, the angle a- in the equation * As 7) =: -— - , P being the time of revolution of the first satellite, we have t Z P ; the value of £ depends on the masses m, m', m", and also on n, ri, n", these last are had by knowing the periodic times of the three first satellites, and the first are deter- mined by their effects in producing certain inequalities, and are obtained in the same manner as the masses of Venus, Mercury, and Mars, are determined from certain effects which they produce on the earth's orbit. -|- When c is negative and less than q:2Sn', the radical is evidently imaginary when F=?r ; •/ V can never be = to «-, and it must be alternately positive and negative, its mean value being equal to cypher. If € is positive, the radical is evidently imaginary when r=0; v '" t^'® case V can never be =0, its value is therefore periodic, and in its mean state is equal (o ir. 3«2 CELESTIAL MECHANICS, dtz= . ^" ,* V C+2g?l*. COS. TS must be always very small, we can suppose cos. w = 1 — ^t^-*, and the preceding equation will give by integrating it, w = A. sin. Qiit.\/1, + y), X and y being two constant arbitrary quantities, which can be deter- mined by observation alone. Hitherto, it has not indicated this ine- quality, which proves that it is extremely small. From the preceding analysis the following consequences may be inferred. Since the angle nt — 3n7-(-2n'^?+£ — 3/ — 2/ only oscillates on one side or other of two right angles, its mean value is equal to two right angles ; therefore we shall have, if we only consider mean quantities, n — 3n' + 2Tf zz ; that is to say, the mean motion of the Jirst satellite plus tmce that of the third, minus three times that of the second, is exactly and constantly equal to cypher. It is not ne- cessary that this equality should accurately obtain at the commence- ment, which would be extremely improbable, it is sufficient that it should be nearly the case, and that n — 3}i'+2n''^, should be, abstracting from the sign, less thant x.n.^^; and then the mutual attraction of these three satellites would have rendered this relation rigorously exact. We have therefore n — 3n' + In" equal to two right angles ; hence, the mean lon- gitude of the first satellite, minus three times the mean longitude of the second, plus twice tliat of the third is exactly and constantly equal to * Tlie equation ■ , is that of a pendulum whose length is -^ , i. being VC + 'iSn". COS. ar **" the number of seconds in a revolution of the first satellite, the amplitude of the arc of vi- bration being -— • * Or in other words, at the origin of the motion, it should be comprized within the limits ± x.n.VZ, PART J.— BOOK II. 383 two right angles. In consequence of this theorem, the preceding va- lues of Sr' and Sv are reduced to the following, Sr' = (m.G—mfW)- cos. (n't—nt+^—i) ;* iv'= [m.H—m'F'). sin. {n't—nt-\-i'—i). The two inequalities in the motion of m!, arising from the action of m and oim". are consequently confounded into one, and will be always combined. It follows also, that the three first satellites can never be eclipsed together ; they cannot be seen together from Jupiter, neither in opposition nor in conjunetion with the sun ; for it is easy to perceive that the preceding theorems obtain equally for the mean sy- nodic motions, and the mean synodic longitudes of the three satellites. These two theorems likewise obtain, notwithstanding the changes which the mean motions of the satellites may experience, either from a cause similar to that which alters the mean motion of the moon, or from the re- sistance of a very rare medium. It is evident that if these different causes d''V operated it would be merely requisite to add to the value of —jpr > ^ quantity of the form j" , which can only become sensible by inte- grations ; supposing therefore Vznir — -nr, and ts very small, the differ- ential equation in V will become 0=——- + e«*.:D- + ^ df ' df As the period of the angle w/.v^S embraces but a very few number of years, while the quantities containedt in — rv ^^^ either constant or * For2n"<.J.2i"— 2n'i— 26' = a-+M'i— 2 hence ar. Vxt+^i' ' dx t^ ' di/' ' r' ' dy r^ dy /dR\ fdR\ dR (r. cos. u r^.s^. cos. v\ dR . dR manner x.l -r- ) — z. ( -— I = -—— i 1 • 1 — r. cos. v.s. + — -• \dz) \dxl ds \ r ^ 7^ I dr * ^ dv rs. sin. v , . i ■ • = the expression in the text, and by a similar process the remaming terms may r be obtained 386 CELESTIAL MECHANICS, xdy—ydxzzcdt'y xdz — zdxz=.c!dt, ydz—zdy—d'dt; these differential equations in f,f\ f", will consequently become df=-dy.\-^^\-dz.p + s^).co..v. j-^-r.. COS. r. \-^[ + \dR}-> ..sm... J^J^ ,^ S ■ UlR) , cos.w (.dR) 5. sin, w {dR) } c'-di ^dRf C car) r (Cf) r ( ds )) r ( dsy df'zzdx. ] -=- i — dz. ] (1 + 5*). sin. v. ]^-l — rs. sin. v. . , Cdv ) i^ ■' t ds S id? SdR-)} ( 6?y 3 ) + Cfl/. SCOS. V. <-z-> dR) — s. COS. sin.f ^dR} 5. COS. t' ^dR}} d\dt ( dv S r i ds $dRi I dsS' dj' z=dx. J(l-|-s»). COS. '"']--j-\ — r.s. cos. vA-j->-^s.sm.vA--r^'> dR) UR} -r- (■ — S. COS. r. <^- s- dr ) (dv ) ■j-dy.<(l-\-s').sin.v.\-j-i — r.5, sin, v. V-^[ — s. cos, t C Cus ) t dr ) , , T, ^ {dRf sin. vidRl 5, cos. f idRf) c ( dr y r (dv ^ r i ds j) , ,, ,. S • SdR} , cos. w ^dR) 5. sin. t' {dRl 1 ^ L dri r L dv ) r ( ds^} The quantities c, c" depend, as we have seen in N". 64, on the inclination of the orbit of m to the fixed plane, so that these quan- tities become equal to zero, if this inclination is nothing ; besides it is easy to perceive, from the nature of i?, tiiat j -j^i is of the order of the \_ do \ inclinations of the orbits ; tlierefore the products and the squares of the inclinations of tlie orbits being neglected, the preceding expres- sions for djl and df', will become PART I.— BOOK II. 387 „, , /dR\ , ,, C fdR\ COS. r fdR\^ but we have dx zz. d.[r. COS. v); dyzz d.{r. sin. v) ; cdt ■=. xdy — ydx n r*dv ; therefore we shall have df-=. — (dr.%\x\. v-\-2rdv. cos. v). ( -7- ) — r*.c?r. sin. r. ( "j" ) >* df'zz(dr. COS. t; — 2r//''+ J"', its value f*e, we shall have (ue. sin. uT r: /' ; jj-e. cos. -a- rr^ ; these two equations will determine the excentricity and the position of the perihelion, and we can easily infer . PART L— BOOK II. 389 t.\edezzf.df-\-f'df ; y.\e\d^ -fdf-fdf.^ By assuming for the plane of ^' and of j/, that of the orbit of 7n ; we shall have by N*". 19 and 20, in tlie case of invariable ellipses, a.{\ — e*) _ r^.dv. e. sin. {y — tt) ~l+e.cos. (w — 37) ' ~ fl.(l — e*) ' r^dv—a^ndt. \/l — e* ; and by N". 63, these equations subsist also in the case of variable el- lipses ; the expressions of ^and of df consequently become df-= ■/ -. (2. cos. v-\-^e. cos. •za'+^ff. cos. (St; — sr)). { -^rr ) ^ — a^'ndLVl — e*. sjn. vA^j ; df = ^^£=-^.(2. sin. r+f.e. sin. ^+\.e. sin. (2v—^). (-^ j -\-a*ndt.s/\ — e*. cos. v. ( ^ i > therefore /' rfar fdf'—f'df * fC-.e=r^r; •:fc\ede=fdfJrfdf: tan. ,, = ^, ■ ; -—^ =-LJ—,J-J- *.• substituting for f, we have fi^^dra ■=-fdJ' — fdf. . . ,„ r^. f/u.e. sin- (u — w). sin.u f Substituting for dr and t^dv, we obtain dfz=L a.(l— e') -/l— e' -J-2(r. cos. V. cos. (u — w) ; e. sin. u. sin. (u — ■et) =ze. sin. ^v. cos.w — e. sin. v. cos. r. sin. a-. '2e. cos. V. cos. (u — a-)=2e. cos. °i). cos. ar-)-2e. sin. v. cos. i). sin. ^=6. (sin. ''i)-[-cos. -ti). cos. ti- I . ■ ■ X , ^ cos. (2u — w) , cos. 3- -l-e.cos.D.(cos.u.cos.a-+sin. ■u.sin.'CT)=e.cos.v cos.(u — iij) = e. ^^r f-e. — - — , ',' by making similar terms to coalesce we obtain the expression given in the text, we can in a similar manner derive the other expressions. 390 CELESTIAL MECHANICS, andt . . N ,^ , , ^^ fdR\ ed-m zz J - - . sin. {y — rs). (2+e. cos. (w — ra-)). I j— ] + ^ COS. (t;-^). (^ ^j , de = ^ ■ ^ ■ (2. COS. {y — vs)-\-e-\-e. cos. *(y — •nr)). ( -jZ ) a^ndt , , fdR\ ^/\^e\ sin. (v—nr). ^ ^ j • I" This expression for de may be made to assume a form which in se- veral circumstances is more commodious. For this purpose, it may be observed, that drJ-r; J = di2 — ^'"•{'J~) > ^y substituting in place of r, and dr their preceding values, we shall have r\dv. e. sin. (v — ■=■)•{ ^) =^-C^ — ^')- di? — a.(l — e*). dv.( ^ )» but we have r"- .dv =a''ndt.\/ \ — e" ; dv= ^.j 7idt. ( 1 +e. cos. (f — zr) y (1— e*)i • fdf'—— ^^ ^^-- . 2. sin. V. cos. arH . e. sin.V. cos. w+^e. sin. (2«-^w). cos. «».( ^i- ) ldR\ -\-ltxa\ndt.V\—e'. cos. v.cos.w. ^^J f'.df= — !^±^^, 2. cos. V. sin. =r + -|-. e. sin. o-. cos. cr+Je. sin. ,j. cos. (2"— ^)-( ^j .^1 — e° , . /iiJ\ — ^e.a'.wc?i.v 1— e*. sin. v. sm. w. y-j-f \ •: fc-e\dz, =fdf'--f'df= — ~=- 2. sin. {v—a)+^.e. sin. (2t)— 2»).^^ j. , . /dR\ -\'fie. arndt.y I — e*. cos. (v — d-). f —I, PART L— BOOK II. 391 therefore, , . fdR\ a'ndt.vl— e*. sill, (y — sr). [~r:) = — !^ -. dR ; . ri+e. COS. (v—zr))\ f -^ 1 ; therefore the preceding expression for dv will give ede = <^rtdW'^\ ( ^ ^ _ * ^ 2 (. t da S C da ^ } +aM. e. sin. .•.{a.-+ J„.{f^}+4„..{^}+i.a-.{^^,}} :'«rf^.S.|f.A«+ ^a. {^^}}- sin. (i.(n7— w^+Z— 0+n^+O J — am PART I. BOOK II. 3 E * Sin. u= sin. {nt — i + v) = sin. (n<-J-t). cos. v,-\-coi.[nt-\-i).im. v„ now sin. v;=v, — -^-{- &c., COS. t!;=:l ~--\- &c., hence substituting for r, its value 2.e. sin. («<+« — w), and neglecting the square of e, we obtain sin. (nf-f-e-J-uJ = sin.(?i«-}-t) + cos. (nt + «). 394 CELESTIAL MECHANICS, The integral sign S extending in this expression, as in the value of R of N°. 48, to all entire values of i, as well positive as negative, the value t = being included. 2e.sia.{nt-\-i — w), differencing 72 with respect to a, and retaining those terms only in which the first power of the excentricity or inclination can occur, and from which we may obtain constant quantities, the first term of the expression for -(diiferenced under these restrictions) will give — ; — , the second term will give — ; — , and also a. — pr— , •/ (1 — da aa da* + /rf«\ „ , ,/(dl^''\ //rfACA /(^ACu 6— w), now this quantity should be multiplied into sin. u, or into sin. (nt-\-i)-{- cos. (nf+s). 2e. (sin.(ni+s — vr) ; hence, performing this operation, neglecting the square of e, and we 'rfA'"' shall have the coefficient of — -j — =:2.e. cos.(ni+e). &\a.(nt-\-t—'a)-{-e. cos. {nt-\-t — w). sin. (jit-\-i) — e. cos. {iit-\-i — -a), sin. (^nf +e)=2e. cos. [nt +«). sin. (nf +£ — w) = e. sin. (2) f/iA («t+s) — ct) — e. sin, a- ; in like manner, the coefficient of — a. — r-?— . is e. sin. (nt+s). cos. da (»f+i — ■Kr)=— . sin.2.(«<-}-t) — ar)+ — . sin. w ; hence, by multiplying by — , see N°. 48, the constant part of the second term of the value oi dfzz — - — . dt. e. sin. w. //rfAloiN a /d''Am\ \ . ,., manner, to obtain the coefficient of e'. sin. -a-', let the third and fourth terms of the value of R be differenced with respect to a, [i being equal to - 1. e'. cos. {n't — tii + ^ — i). cos. (n't 4" t' — =^) ; — A<''. 2.e'. sin. (w'< + s' — jt'). sin. [ti't^ — 7it-\.t' — i) ; sin. (,nt-{-t) is the only part of the value of sin. V, into which these quantities can be multiplied without introducing powers of e greater than the first, •/ when for these quantities equivalent expressions are substituted, deter- • J u .u »• r.i r ; COS. (a + 6) 4- cos. (fl — b) mmed by the equations ot the form cos. a. cos. b =■ — ;-^^ — ' — ^-^ ^ . sm. a. sm. 6 = cos. — "^ cos. ; we shall obtain the second and fourth terms of the second line of the value of df; in order to obtain the first and third terms, let the third and fourth (J n » —7-/ ■= ( -r- ) , and then these terms become — . uj. a'. — ; — . sin. (n't — nt + s' — t) ; + -—. w,' Af >. \di/ 2 da' ^ 1 / 1 2 ' PART I.— BOOK II. 395 By the preceding number, the value of df will be obtained, if the angles i, i , id-, and -r', be diminished by a right angle in that of f; hence we deduce df'= —. e. COS. ^. |..|-^j+la .|-^^|| - «'««^^- '■ ^°«- - 1^"+ 4^- {-d^]+ ^^- {-^j + ^l^^ll +tt7»'«c?/.E.|iAc^+ia.|-^||. COS. {i.{n't—nt-^i'—i)-\-7it + X. Let us name, in order to abridge, X the part of the expression of df, contained under the sign S, and Y the part of the expression of df' contained under the same sign. Moreover, let us make as in N". 55, (0, 1):. ^.|a^|— j+ i.«3. |-^^j| ; It should be then observed, that the coefficient of e'dt. sin. ts', in the expression of df, is reduced to foTTl, when we substitute in it,, in place of the partial differences of A'" in a', their values in partial differences relative to a ; finally, let, as in N". 50, 3e 2 COS. [n't — nt+e' — i), when we substitute for «/ and t)/ their values, and proceed as before, we shall obtain, after the resulting quantities are multiplied by 2. cos. («<+«). (the only part of the value of cos. v which can be taken into account) ; the first and third terms of the coefficient of d. sin. sr' ; in order to obtain the variable part of the value of df, e, d, do not occur ; the first terra of the value of B, must be differentiated with respect to v, or, what is the same thing with respect to e, and then multiplied into 2. cos. («/ + •), this same term should be also differenced with respect to a, and then multiplied into sin. (ni-{-s). 39^ CELESTIAL MECHANICS, e. sin. zT ■= h; d . sin. is ■=. h', e. COS. Ts ■=! I; e'. COS. z/ ■=. I' ; which by the preceding number, gives yzzt^-l ; J''= fj-h ; or simply, f-=.l;f'z=. h, the mass of M being assumed as unity, and the mass 7)1 being neglected relatively to M ; we shall have -^= (0, 1)./— [oriy+ffm'.TZ.F; -7r-= — (0, 1 ).A+[orT]./«'. — am'.n.X. Hence it is easy to infer, that if the sum of the terms analogous to am'n Y be named ( F), which terms arise from the action of each of the bodies ?b', m", &c. on m ; if in like manner, the sum of the terms analogous to — ani'nX, arising from the same action, be called (X), finally, if we denote by one, two, &c., strokes, what the quantities (X), ( F), h and / become relatively to the bodies wz', »/'', &c, ; we shall obtain the following system of differential equations : = ((0, l) + (0, 2) + &c.).Z-[0]./'— [O].^'— &c.+(F); =—((0, l)+(0,2)-i-&c.)./t+[o^]./i' + [ori].//''+&c.+(A') ; ^/ = ((1 . 0) -K J . ^) + &c.)/'-[ro]./— [17^]/— &c.+( F ') ; dt dl dt dh' ^^1' =—((!, 0)+(l, 2)4 &cOA' + [irr)]./i+[Tr^].//+&c. + (A'0. &c. In order to integrate these equations, let it be observed that each of the quantities h, I, //, /', &c., is made up of two parts, the one depend- ing on tlie mutual configuration of the bodies 77i, m', &c., the other independent of this configuration, cor.taining the secular variations of these quantities. We shall obtain the first part, if we consider that when we have regard to it solely, h, I, h', /', &c., are of the order of the dis- PART I.— BOOK II. 397 turbing masses, and consequently (0, !)./«, (0, l)./, are of the order of the squares of these masses. Neglecting quantities of this order, we shall have dh' dV therefore, h-f{Y).dt; l = f{X).dt; h'=f(Y').dt; l'=f{X').dt; &c. If these integrals be taken, the elements of the orbits being considered constant; and if Q be what /(Q).f// then becomes, and if SQ be the variation of Q, arising from that of the elements, we shall have fiY).dt-Q—fSQ; But as Q is of the order of the perturbating masses, and as the varia- tions of the elements are of the same order, SQ is of the order of the squares of these masses, therefore, if quantities of this order be ne- glected, we shall have f{Y).dt= Q. We can therefore take the integrals /(y).^^/, /(A^.^//, /(F). provided* that in this last partial diflFerence we make e — sr equal to a constant quantity, therefore the term of ■! -5- >- which corresponds to the preceding term of R is m.{i+g). k. sin. [i'7i't—int+i'i'—if(—gz7—g'zy'—g''''&'). The corresponding term of dR is to', ink.dt. sin. (e'nV — int-\-i'^ — u — g-m—g'Ts/ — •^'^S'), if therefore we only take such terms into account, neglecting the square of e in comparison to unity, the preceding expression for ede will give - m'.andt gk •,.,,,.,..,,. , , ,,,,s ae= — . sin. (int — tnt-{-t'^ — n — gTs — gs/ — g'^^). fj- e but we have ^ = g^^ . e'^. (tang. yy\ Q= ^l therefore we shall obtain by integrating m'a7i (dk^ ,., , ■ -i , ■ , 1 «„,^ Now, if the sum of all the terras of R, which depend on the angles i'n't — int be represented by the following quantity, vi'P. sin. (i'n't — i?it+i'i-^ii) + m'.P'. cos. (i'n't — int+i'c — it); the corresponding part of e will be * Hence — ^ = 6 — 6, \' —gta =:Z — gi, therefore if we substitute this quantity for gvr, and then take the value of -7- , we shall obtain the expression for I— J , corres- ponding to the value of R. 400 CELESTIAL MECHANICS, COS. (i'n't — int-{-i'i' — ii)). This inequality may become extremely sensible if" the coefficient in' — in is very small, as is the case in the theory of Jupiter and of Saturn. Indeed, it has for a divisor only the first power i'n' — in, while the corresponding inequality of the mean motion has for a divisor the second power of this quantity, as has been observed in N°. Q5 ; but J — - > and -! —T- t being of an order inferior to P and to P", the ine- \de) \de) ^ quality of the excentricity may be considerable, and even surpass that of mean motion, if the excentricities e and c' be very small ; we shall see examples of this, in the theory of the satellites of Jupiter. Let us now determine the corresponding inequality of the motion of the perihelion. For this purpose, let us resume the two equations, which were obtained in N". 67. These equations give df'^ l*de. cos. -a-^iAcdTsy. sin. w ; hence, if we only consider the angle i'n't — int-\-i'i — ii — g-o! — g'z/ — g''^', we shall have ( dk "i dfzzm'.andt. -J-j- |- cos. -ar. sin. (i'n't— int+i'i' — it — g-sr — g'-s/ — g'^d') — fxcdu. sin. -o-. Let — m'.andt. \\-^}+ ^'^'•l- cos. [i'n't— int+h — it^g--a-g^-s/-g'^')* • By multiplying by sin. ■a, we shall have dfz^ —m'andt.f — \, (cos. w. sin. (irii — ini-f-jV— it — g-a — ^w' — g" <') + sin.w. cos. (i' n't — int + PART I.—BOOK II. 401 represent the part of fj-edzr, which depends on the same angle, we shall have df:= m'.andt. \\-f\-\- h,^' (• sin.(«V# — int-\-t't'-u-{g-\').7ir-g'-sr-g''^'') — '"'■ ~~¥' ' ^' ^^"" (*'"'^— ^"^ + «'«'—««— (5"+ 1 )• ■^—g'-^—g'V). It is easy to perceive from the last of the expressions of ^ given in N°. 67, that the coefficient of this last sine, has for a factor e^+'.e'^.(tan.^ip)«"; r dk ~i k' is therefore of an order superior by two units, to that of -j -7- > ; consequently, if it be neglected in comparison to ^;7->-i we shall have m'.andt f dk) ,-i >. • . •■ ■ , , _//n/N . ) -J- >-• COS. [in t — mt-{-i i — ii — gsT — gsf' — g"W), for the terra of erfar, which corresponds to the term m'.k. cos. (i'rit — int — int-\-ii' — it — gzr — g'-^'—^'^'\ of the expression of R. It follows from this, that the part of bt which corresponds to the part of R expressed by vti.P. sin. {irit — int-\-'H — ?£)-}-m'.P'. cos. {i'r^t—int-^i't — it), is equal to — Tv-i — T-T — < -J -7- v. COS. (i n t — int + ti' — it) — ■{ -^— t f*.{i n' — tn),e 'Wde ) ^ ' (_ de } sin. (i'n't — int-\-i'c — u)] , we shall by this means obtain, in a very simple manner, the variations of PART I. BOOK II. 3 F iW—u—gm—g''a'—g'6').) — triandt.k. sin. to. cos. {in't — intJ[-i'i — it — gw — ^■a' — g"K) = dk — m'andt, —7- . sin. (int — int — ?V — U—ie — !)• w — g''a' — g"lf), de ^ and the two terms into the value of the coefficient of are obtained from the e , . , sm. (a4-i)4-sin. (a — b) formula sm. a. cos. b = —i — — > '-. 2 402 CELESTIAL MECHANICS, the excentricity and of the perihelion, which depend on the angle i'lH — int-\-i'i' — ft. They are connected with the variation ^ of the mean motion, which corresponds to it, in such a manner, that the va- riation of the excentricity is Sin '\de.dlj ' and the variation of the longitude of the perihelion is (i'n'-in)fda Sin.e \ dej' The corresponding variation of the excentricity of the orbit of w', due to the action of ot, will be 1 \^l i'n'.d'lde'.dty 3i'. and the variation of the longitude of its perihelion will be (rn'—in) fdn Si'n'.e' 'Xde')' And as by N°. 65, we have ^'zz . ^ these variations will be m'.\/a' m.Va _ f_^l_\ and ("'-^>0-^^V^ |^l Si'n'.m'.'\/a'Xde'.dty si'n'.e'.m'.y/a' KdC)' When the quantity i'n' — in is very small, the inequality depending on the angle i'n't — int produces a sensible one in the expression of the ^ 5 _ Sm'anH - Tfj^ZIinf — ■ ''■^' "^*'^' *''"'' ~ *'"* + «'s'— ") — P'- sin. i'n't— int + iV— is^) differ- encing ^, first with respect to e and then with respect to t, the coefficient become* im'an^.i (i'n'—^in)~'' ^^ '^^ variable part is the same as the variable part of the expression for de, bence the ratio of de to (j-^j is that of 1 to Sin ; in like manner it may be shewn, that the ratio of d^ to (§), is that of 1. to -^ \de^ e (in'— in) PART I— BOOK II. 403 mean motion, among the terms depending on tiie squares of" the dis- turbing masses ; the analysis of them has been given in N°. 6d. This same inequality produces in the expressions of de and d-u, terms of the order of the squares of those masses, which being solely functions of the elements of the orbits, have a sensible influence on the secular variations of these elements. Let us consider for instance, the expression of de depending on the angle i'7i't — int. By what precedes, we have . m'.an.dt ( (dP^ ,., „ . ^ , ., . .. ( CdP'^ de= ^r-\{d^]' '''' (' «'^-"^^+^'^ -^0 - {{^ }. sin. (i'n't — int+i'i' — u)]. By N*. 65, the mean motion 7it ought to be increased by > V , ' • L ' ) P' cos.f i'n't — int-\-i'e — ie — P'. sin. (i'n't — int+i'i' — ii) u (in — tn) .f*. (. ^ - J and the mean motion n't ought to be increased by 3m'.an''i' m.\/a ,_ ,.,,,. '— T^TT— -:-T5 — •— — 7=7. [P' cos. Ci'nt — mt+t^ — u) — P'. sin. (i'71't — int-\-i'^ — ii)} . In consequence of these increments, the value of de will be increased by the function 3m'.a''.in^.dt ,_ ,-. r rdP'^ fdP^'^ 2iJ.\'s/a.{t'n—iny '\ \de) Idejj' and the value of du will be increased by the function 3m'.a\in\dt _ cdP) „, {dP'} ■ 2^W~a'.(t'n'-iny.e<'''^''^"'-^''''"-'^'']^- ll^r^'lwl' we shall find in like manner, that the value of de' will be increased by the function 3f2 404 CELESTIAL MECHANICS, Sma'' .s/ a.in^ .dt and that the value of dz/, will be increased by the function 3ma\\/7i.in\dt ^. , ._ . .-^ (' fdP^ „ fc/P'll o ^ '7-> , — '^^' • (.tm'Va' + t'.m. ^a). i P. i -t-1+ P'-i -rr > >• 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', CT and zr', relative to the angle i'n't — int may also introduce some constant terras of the order of the squares of the disturbing masses, into the differentials de, de', disy, and dz/, and depending on the variations of e, e', zr and ta-' relative to the same angle ; it will be easy by means of the preceding analysis to take them into account. Finally, it will be easy by our analysis to determine the terms of the expressions of e, -a; s' and «/, which depending on the angle i'n'i — int-\-i'i — it have * Let the increment of nt = d.(nt') = j>.{^P. cos. A — F'. sin. A), and the increment of d.[n't) = '——: . p.(P. COS. A — P, sin. A), then we have d.(i'n't — int) = _ m'.Va' (i'TfltV d \ i\. p-(P. COS. A — /*. sin. A), calling this quantity a, and substitute m'.Va' I ing it for d.{in't — int) in the value of de, given in this page, we shall have the dP dP' factor — T— . cos. ( A+ a) — . sin. (A-|- a ), then by developing and remarking that dP dP sm. A= A, and cos. a=1 q-p., the preceding expression becomes— 7— . cos. A —. de de dP . ^ f I'm.Va ,\ ,_ A r>, ■ AS , '^P' A / sin. A ;— . sm. A. ( = — i] . P.( "• cos. A — P'. sin. A)+ -r- . cos. A.{ — de \ m'Va' ' de \ i'fw.v fli \ T^ — i). p.(P- COS. A — P'. sin. A), as sin. A. cos. A, contain only periodic functions, m'.Va' ' the quantities multiplied by them, or any powers of them, need not be considered st pre- sent ; but as sin. -A = i — i. cos. 2A ; cos. "A = A + ,^. cos. 2A ; we shall obtain (by substituting for sin. *A, cos. *A), two terms which do not involve periodic functions, and which when concinnatcd, become the quantity by which de is said in text to be aug" ment PART I.— BOOK 11. 405 not in' — in for a divisor, and those, which depending on the same angle and on double of this angle, are of the order of the square of the disturbing forces. These terms are sufficiently considerable in the theory of Jupiter and of Saturn to induce us to have regard to them ; we shall develope them in the requisite detail, when this theory will be more particularly discussed in the 8th Book. 70. In order to determine the variations of the nodes and of the inclinations of the orbits, let the equations of N°. 64, be resumed --'•{4f}-{f}}' If the action of m be solely considered, the value of R, of N°. 46, gives 1 ((a,'_x)H(y-^)*+(5r'-^)')ii (dR^ (dR\ ^ ^ ( 1 < —r t — ^- \ -^r- ?■= m'.(x'z — z'x). \ - __i I ((x'— .r)»+(y— ^)* + (z'—zyp - ?__ I ((•f"-^)*+(i/'-y)*+(^'-*)*)'^ 406 CELESTIAL MECHANICS, Let now by N°. 64, the two variables p and q will determine the tangent of the inclination (p of the orbit of m, and the longitude 9 of its node, by means of the equations tan. —p).X3/+(q — q')'3:y' —Y—^ |-.r;-'.(cos. (»'+») + cos. (u'— r)). -J. -j^llll I r/.(sin.(t;' + r) + sin. i}J — r)). The excentricities and inclinations of the orbits being neglected, we have r-=za; v zznt-\- i\ r' = c j v' ■=. ri t -\- i \ which gives 1 I J_ 1 (a" — 2aa'.cos,(?z7— n^-j-t' — £+a*)J Moreovei-, by N°. 48, we have J = \. SB"\ cos. i/n'(—nt+ 1'— 0; (y—lad. COS. (n't—nt + e'— e) + a'*)t * rr". COS. XI. COS. u' = — COS. (w^-i)') -}- cos. (v — v), rr\ sin. v. cos. »;'— -^ • (•'''• (u+w') + sin. (v — ti)). 408 CELESTIAL MECHANICS. the integral sign E extending to all entire values of ?, positive as well as negative, the value i=0 being included ; by this means, we shall have, neglecting the terms of the order of the squares and products of the excentricities and inclinations of the orbits, dp {a' — q) m'a, , , , -^- -^-^.-^(cos.Im'^-I-w^+e'+O— COS. (ivt—nt-^i'—i)) (p — p) ma , . , + ^^-^;^.-^- (sin.(n'i+n^+£' + j)— sin. (nU—nt->ri'—i))* + ~f. • TO'.aa'E5«.(cos. [(i+l).(n7-7if + t'— «)]— cos. («+l). + (fc^. m.aa'.EB").(sin.[(j + l).(«7— w^+f'— 0]— sin.(H- 1). {n't—nt-\-i — t) + Sw^-^Se]) ; dq (p' — p) m'.a , , , di ~ 2r^'~^' ^^°^' («^+«^+«'+0+cos. (n't^nt+ I'— 0) + -^-^ • —^' si"- (w'^ + «^+£' + 0+ sin. (n't—nt-\-(-—i)) -I- \ElZP}. wj.a'a.SJ5"'.(cos. [(i + l).(n7— w^+e'— 0] + cos. (i+l). 4c {iVt—nt + £' — £)+2«f +20] * The value of the third term in the expression for -j- will be had by observing that COS. (tj±i/) „,, , 2.BW ^ ^. ^. 2.B('^. cos.j.(n'<— nf+s'— e)= — - — . (cos. (i.(«'t_n< + f'— i) +«'<+«<+ t'+f)+cos. i.(n'<— 7!<4. 6'_ E)_?i7_„<_E'_£))_cos. j.(n7 — ni+i'_,)+nV— n<+i'— i)— COS. («.(n'< — nt-^i' — i) — n't — r^i-f s+'i); therefore, if we concinnate the terms of this ex- pression, we shall obtain by observing that cos. i.{n't—nt-\-i—t)-{-n't^ n<+t' + i) = cos. (i+1) n't—nt + f'—s)+ 2)1/4- Se), and' also that cos. i.{n't—nt-^i—i) f «';—«< + I'—i) — COS. (j-|- !).(«'« — r>t-\-i' — i), the expression given in the text. PART I—BOOK II. 409 + ^^-T^- m'.fla'.2Bc\(sin.[(?+ l).(n7— w^+f— 0]+sin. («+l). (n'i—nt-^ «'— + 2n^+ 2i]). The value ^" z: — 1 gives in the expression of —^ , the constant quantity-^ — ^.m'.aa'.B^~^\ all the other terms of the expression of —^ are periodic, if P represents their sum, we shall have by N°. 48, at 4c (£('> being equal to 5'-'0. By the same method, we shall find, that if we denote by Q the sum of all the periodic terms of the expression of -^, we shall have at 4c If the squares of the excentricities and of the inclinations of the orbits, be neglected, we shall have by N°. 64, c=:\/[*a. If then /x be sup- posed = 1, we have n*a^ = 1, which gives c = ; the quantity , thus becomes -, which by N . 5\) is equal to (0, l); hence we shall have |- = (0. l).{q'-qHP ; -g--(0, l).(^-p) + Q. It follows from this, that if (P) and (Q) denote the sum of all the functions P and Q, relative to the action of the different bodies PART I. — BOOK II. 3 G 410 CELESTIAL MECHANICS, m', m\ &c., on m^, and if in like manner (P), (Q), [P'), (Q), &c., denote what (P) and (Q) become, when the quantities relative to m are changed into those which refer to m', m", &c., and conversely ; we shall have for the determination of the variables p, q, p', q, j/, q", &c., the following system of differential equations, ^ =-((0, l) + (0,2)+&c.)5'+(O, \).q' + (0, 2)./^-&c.-^-(P); -^=((0, l)-i-(0, 2)+&c.;.p— (0, \).p—(0, 2)./"— &c.+(Q) ; ^=z— ((1, 0) + (l, 2)+&c.).?'+(l, 0).9 + (l, S^Z+Scc+lP-) ^zr r(l, 0)+(l, 2) + &c.).y-(l, 0).p-(l, 2).|/-&c.+ (Q'), &c. From the analysis of N°. 68, it appears that the periodic parts of p, q, p', q, &c., are p=f(P).dti q=f(Q).dt p^z=f(P').dt; q'=f(Q'-).dt, we shall afterwards obtain the secular parts of the same quantities, by integrating the preceding differential equations, deprived of their last terms (P), (Q), (P'), &c. ; and then we shall light on the equa- tions (C) of 1S°. 59, which have been already discussed with sufficient detail to dispense with our reverting to this object. 71. Let the equations of N°. 64 be resumed, namely, \/?* + C^ (/'' tan. c= ; tan. 6zz— 7-; c c from them may be deduced (/ c' — ■=. tan. (p. cos. 6 ; — = tan. a». sin. d : c c- PART L— BOOK II. 411 differentiating, we shall have d. tan. (p=.—. (dc'. cos. 9 + dc'\ sin. 9 — dc. tan. ! \d. tan.^= ^ -^^ r. ^^5. sin. (t;-9)+ 1-^ j. cos. M)| _0±^'.e«.(.-.,{f}> 3g2 fi/ c" &'.c'+t;c".c" * Hence cos. « = -■ sin. <=— ■ •/ d. tan. ip= 7 ^- „ " — o'^' = by substituting cos. 6, sin. i, for their respective values, the expression which is given in the text. di dc".c'—dc'c" d6V(^+if') dc" /_^L_._ '^•'^- ' = -^E^i= — 7- — ''•■ — 7 — =-r-U?M=^ dd.c" \ c'Vif' + c"''' •j- Multiplying the value of rfc', given in page 385, by cos. 6, and that of rfc" by sin. 6, we shall obtain by adding them together (1 +«*). cos. ("— *)-(-;|j) — ''*• ^°^- (*'~*^(^) "^ s. sin.(v— «).^— W — . tan. a =d tan. qi, hence by substituting for s its value, we shall \dv/ dv _ obtain the expression given in the text. 412 CELESTIAL MECHANICS, These two differential equations will determine directly the inclination of the orbit and the motion of the nodes. They give sin. (v — 9). d. tan. p — dL cos. (v — 9). tan. (p=0 ;* this equation may be also deduced from the equation 5=tan. (p. sin.(t'-6); in fact, as this last equation is finite, we may by N". 63, difference it, either by considering (p and 9 as constant, or by treating them as if they were variable ; so that its differential, taken by making p and 6 the sole variables, vanishes ; hence results the preceding differential equation. Suppose now, that the inclination of the fixed plane to that of the orbit should be extremely small ; so that the square of s and of the tan. p, may be neglected, we shall have dt , ,. CdR^ a. tan. (p=: . cos. (v — 6). -j -7- f 5 ,« dt . , ,,. (dill «9. tan. (p=: . sin. (v — 9). -J -3- r 5 by making, as before, p ^ tan. (p. sin. 0. ; y zi tan. (p. cos. 9 ; we shall have, in place of the two preceding differential equations, the following dt CdRl dg=--. COS. v.^-^j; * By multiplying the first equation by sin. {v — [6), and the second by cos. {v — 6), their second members become indentically equal to each other, therefore the first members will be also equal to each other. PART I.— BOOK II. 413 , dt . CdR^ but s z= q. sin. v — p, cos. v, hence \ds )~ sin. v'Xdq) ' \ds )~ cos. v'\dp)' therefore , dt {dRi dt ^dR) '^P=-^-ld^\' We have seen in N". 48, that the function R is independent of the fixed plane of x and of j/ ; supposing, tlierefore, that all the angles of this function are referred to the orbit of m, it is evident that R will be a function of these angles, and of the respective inclination of these two orbits, which inclination we shall denote by (pf. Let 6/ be the longitude of the node of the orbit of m' on the orbit of m, and let us suppose that m'k tan. ((p'y. cos. (i'7i't — int+ A — ^'.G/) is a term in the value of R, depending on the angle i'?i't — int : by N°. 60 we shall have tan. (p/. sin. 6/ zz j/ — p ; tan. ip/. cos. ^'=q' — q, hence we deduce (ta„. ,>. »in. gr= w-<,)+(p--pW='i-w-,y(P'-^W=ry. 2.v/— 1 * dq =z COS. t.d. tan.

-'. rdh. %va.{j!n't—int-\-k—[g—\\ fi)/) ; I ^ I =— ^.(tan. -'- sin- (i'«'^-^«^+ A-(5^-l).6;). * sin. ^; = ' ' , COS. 6; = ^^^^^ • • COS. «; + V_l. sin. «A« = COS. gi',. + -/= 1 . sin. -« =\-;j^^^^^^) > '^e°«=^ we obtain by multiplying by tan. ?>;, and its values, the expressions for tan. (p,«. sin. g«;, tan. ip;s. COS. g«/, which are given' in the text ; now m'h (tan. ip/)e. cos. (t'n'i— »«/+ A— ^<;) = m'i. (tan. ip/)«. (cos. .^«. cos.(f'n7 — inl + A) + sin. gL sin. [in't — mt + A), if we substitute for (tan.

. sin. gf, and their difference with respect to ;, and q, we will obtain the expressions for — , -^, which are given in the text. PART I.— BOOK II 415 This expression of s, is the variation of the latitude corresponding to the preceding term of R, it is evident that it is the same whatever may be the fixed plane to which the motions of m and of m' may be referred, provided that its inclination to the plane of the orbits be inconsiderable ; therefore we shall by this means obtain that part of the expression for the latitude, which becomes sensible from the smallness of the divisor i'n' — in. Indeed this inequality of the latitude involves only the first power of this divisor, and in this respect it is less sensible than the corresponding inequality of the mean longitude, which contains the square of this divisor ; but on the other hand, tan. (p' occurs affected with a power which is less by unity ; which remark corresponds to that made in 69, on the corresponding inequality of the excentricities of the orbits. It thus appears that all these inequalities are connected with each other, and the corresponding part of R, by very simple expressions. If the preceding expressions of p and of q be differenced, and if in the value of —~ and of —7^ , which result, the angles nt and ?i't be in- creased by the inequalities of the mean motions, depending on the angle i'71't — hit ; there will result in these differentials ; quantities which are solely functions of the elements of the orbits, and which may sensibly influence the secular variations of the inclinations and of the nodes, although being of the order of the squares of the disturbing masses ; which is analogous to what has been stated in N°. (>9, relative to the secular variations of the excentricities and aphelias. 72. It remains for us to consider the variation of the longitude i of the epoch. By N". 64, we have di =zae.j< — - — C V. sin. (v — w^-j-^. j — ^ — V . sin. 2.(y — u) + &c. —d'sr.({E''K cos. (u— T!r)-f £®. cos. 2.(t;— ■5r)+&c.); If for £'", E^'\ &c. be substituted their values in series arranged ac- 416 CELESTIAL MECHANICS, cording to the powers of e, which series it is easy to infer from the general expression for £''', given in N". l6, we shall have -sr)* — e''/ 1 — e . (2 — fe. cos. [v-w)-\-e*. cos. 2.(v--sr).< -^— > e. sm. {v—s:).{\ + -. cos. (i;— lir)). J — ^. The general expression for di contains terms of the form m'.k.ndt. cos. (i'«7 — int-[-A'), and consequently the expression for £ contains terms of the form ., / ' . . sin. (i'n't—int+A.) ; but it is easy to be as- t'n — m sured that the coeflScient k in these terms is of the order i' — i, and that consequently, these terms are of the same order as those of the mean longitude which depend on the same angle. The latter have for divisor the square of i'n — in ; we have seen that we can neglect in respect to them, the corresponding terms of i when i'?i' — in is a very small quantity. If in the terms of the expressions of di, which are solely functions * ByNo.ie, £(')=:± ^e'.(\+eVl— £)^ • • log. £("=2/. log. e+log. (l+eVlH?) (1 + V^l— e)' _i.log. (l + ^^r^T), then by differentiating and substituting for £i*' its value, we can obtain the expression which is given in the text. PART I.— BOOK II. 417 of the elements of the orbits, we substitute in place of these elements the secular parts of their values, it is evident that there will result in them constant terms, and other terms affected with the sines and cosines of the angles on which the secular variations of the ex- centricities and of the inclinations of the orbits depend. The constant terras will produce in the expression of c, terms proportional to the time, which will be confounded with the mean motion of m. As to the terms aflFected by the sine and cosine, they will acquire by integration, in the expression of t, very small divisors of the same order as the disturbing forces ; so that these terms being at once multiplied and divided by these disturbing forces, they may be- come sensible, although of the order of the squares and products of the excentricities and inclinations. We shall see in the theory of the planets that these terms are insensible, but they are extremely sensible in the theory of the moon and of the satellites of Jupiter, indeed it is on these terms that their secular equations depend. We have seen in N". 65, that the mean motion of m has for 3 expression — Jfandt. di?, and that if we only consider the first power of the disturbing masses, di2 involves only periodic quantities ; but if we take into account the squares and products of these masses, this differential may contain terms which are solely functions of the elements of the orbits. By substituting in place of these elements the secular parts of their values, there will result terms affected with the sines and cosines of angles, on which the secular variations of the orbits depend. These terms will acquire in the expression of the mean motion, by the double integration, very small divisors, which will be of the order of the squares and products of the disturbing masses ; so that being simultaneously multiplied and di- vided by the squares and products of these masses, they may be- come sensible, although being of the order of the squares and PART I. BOOK II. 3 H 418 CELESTIAL MECHANICS, products of the excentricities and inclinations of the orbits. "We shall see that these terras are likewise insensible in the theory of the planets. 73. The elements of the orbit of m, being determined by what precedes, they should be substituted in the expressions for the radius vector, for the longitude, and latitude, which have been given in N°. 22 ; the values of these three variables will thus be obtained by means of which astronomers determine the position of the hea- venly bodies. By reducing these values into a series 'of sines and cosines, we shall obtain a series of inequalities, from which ta- bles may be formed, and thus the position of m at any instant may be computed with great facility. This method, founded on the variation of parameters, is extremely useful in the investigation of those inequalities, which from their relations with the mean motions of the bodies of the system, acquire great divisors, and by this means become very sensible. This species of inequalities affects principally the elliptic elements of the orbits ; therefore by determining the va- riations which result from them in these elements, and by substituting them in the expressions of elliptic motion, we shall obtain in the simplest manner possible, all the inequalities which those divisors render sensible. The preceding method is likewise useful in the theory of comets ; these stars are only perceived for a very small part of their course, and observations solely make known the part of the ellipse, which may be confounded with the arc of the orbit which they describe during their apparition. Therefore, if the nature of the orbit, consi- dered as a variable ellipse, be determined, we shall have the changes which this ellipse undergoes in the interval between two conse- cutive appearances of the same comet ; we can therefore, an- nounce its return, and when it reappears, compare the theory with ob- servations. After having given the methods and formulae for determining by successive approximations, the motions of the centres of gravity of PART L— BOOK II. 419 the heavenly bodies, it remains for us to apply them to the dif- ferent bodies of' the solar system ; but as the ellipticity of these bodies influences in a sensible manner, the motions of several of them among each other, it is requisite, previously to proceeding to the numerical applications, to treat of the figure of the heavenly bodies, of which the investigation is equally interesting, on its own account, as that of their motions. END OF THE SECOND BOOK.