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The following ]\[omoir, on the " General Integrals of Planetary Motion," was submitted to Prof II. A. Newton, of Yale College, and Mr. (J. ^\. Hill, of Nyuck, N". Y., and has received their approval for publication in the " Smithsonian Con- tributions to Knowledge." WASiriNOTON, ^^. 0., Deccnilx'i', 1874. JOSEPH HENRY, Secretary Smithsonian Institution. (iii) •"-''• •tiillii Vfi' '"iTl.J^ IM{K1' ACK. Thk |)n'S('iit memoir may he fumsidcnd iis. in pnrt, nn rxtciision aiul jj;mprnliza- tiou of two foniicv i)ai)ci-.s liy the aiitiior: tiic first Ixuiig T/n'orir i/rsjirr/iir/iii/iniin paper. To the fornu-r class belong (1) tJK! analogy Ix'tween the expressions for tlu- rectangular co-ordinates ,»• and //, wliieh differ only hi that tlie latter is comiiosed of jn-oducts of sines, wliile tlie former is composed of similar ])rodncts of cosines ; and (•,') the omission of all consid(«rations of the modifications growing out of tlu- fact tinit in ecpiation (1) one value of /i vanishes. To the latter class belong the omission of all considera- tions r(>specting the convergence of the series encountered, respecting terms of long period, and respecting the occurrence of relations among tlie arguments, such as that known to subsist betwi'cu tlu; mean motions of three of the sati'llites of Jupiter. These subjects will naturally come up for consideration whi'U tiie process of actually integrating tlie differential ecpiations of planetary motion in the most general way is midert.dveu. Xo method for the actual execution of this integration is given at present, partly l)ccause tlie papcn- may be considered com- plete without it, partly because the author has not succeeded in working out any method satisfactory to himself It is true that a large part of the paper is devote(l out. It would be extremely agreeable to the author to learu that abler hands than his were success- fully working to effect the actual solution of this noble problem in its most general form. ( V) CONTENTS. § 1. Introduction . . . . , . . • . . § 2. Canonical Transformntioii of tlic Equations of Motion ..... § 3. Approximation to tlic Kcquirt'il Solution Ijy tlio Viiriutions of tlio Arbitrary Constants in ft First Approximate Solution ....... § 4. Formation of the Lagraiigian ('uelDcionts (n„ at), and lloduetiou of tlio Equations to a Canonical Form § 5. Fundamental Relations between t..o Coellieicuts of tliu time, b„ b,, etc., considered as Functions of c,, c,, etc. •.•..... §• G. Development of n, iij, and Q; . . . § 1. Form of Second Approximation . . , , , ... § S. General Tlieoreni ••....... § i). Summary of Llesultd . . , . . , . PAUU 1 11 10 10 24 20 2H ( vii ) ON THE GENEKAL INTEGRALS OF PLANETARY MOTION. § 1. Introilitcdon. If wc examine wliiit lius been done liy fjfeomefers towards developinjjf tlie co ordinate^ of tlie pliUiets in terms of tlie time, we sliail .see that tlie iiio.st >,'eiieral expressions yet found are those f-tv tlie development of the secular variations of the elements in a periodic form. It, i^ well known that if we negleet (piantities of tlie third order with respect to th*? eccentricities and inclinations, the intei^ia- tion of the equations which «iv(> liie secular variations of those elements, and of the loii<,'itudes of the periheh' iid of llie nodes, h-ads to the conclusion that tlie genend expressions of those elenumts in terms of the time are of the form C sin 7T (1) c COS 71 = S, Xi COS (r/,Y -f- li^) ^ sin = i\ JUi sill (/// -j- y^) ^ cos = Vj Mi cos (/// -|- y,) 71 beiiif,' tlie number of planets-, A', M;, (/,, aiid //, being functions of the eccentrici- ties at a given epoch and of the mean distances, whih; ,5, and y^ are angles depend- ing also on the positions of the perihelia and nodes at a given epoch. It is to be nmiarkinl that one of the values of hi is zero, the corresponding (piantities J/ and y depending on the position of tlie plane of reference. The num(>rical values of these constants for the solar system have been fouiui liy several geometers. 'I'he latest and most complete determiualions are thos(,' of Jje Verrier and of Stockwell.' A\ hen we consider the terms commonly called ])eriodic, that is, those wliich depiMid on the mean longitudes of the i>lan(>ts, we shall find that their determina- tion depends on the integration of differentials of the form '"'^ ,i„ ('V + // +/71' +;ri + ^-v + /^?o. ■tvher(> we put m' the mass of the disturbing planet. ' Smithsonian CuuU'ibutioiis to KMowli'dj,^', Xo. •23-2. \'ol. XVJII. October, 1874. j G K S K U A L IN T E G U A L S O V V L A \ K T A R Y M O T I O N . // ii function of the eccentricities, inclinations, and mean distances of the two phinets, devekjpable in powers of the two former quantities. /, /' the mean ionyitm' vs of tlie phniets. ;t, 7i' the longitudes of their periiielia. 0, 0' the longitudes of their nodes. (■,y, /•, numerical integer coefficients, and in which /' + / +/ -f- / -f- /,•' -f /,• — 0. Tile coefficient 7i is of the form * AtA'Jyci^"'' (1 + A,c'-\- A.,'" -f etc.), while the circular function of whidi it is a coefficient may be put in the form COS sin '•^''^ +-'"^' + /'■^' +''•'"') cos (/7 + (7) ± c!" (.^"^ +->"^' + ''■•' + ^•'''') ^i» (''' + '0. As these equations have hitlierto been integrated the different elements arc developed in powers of tlie time, and w( arc tints led to expressions of the form (.l + .17 + .l'r + ...,);'!;^^(;7'+//). But it is clear, that we sliall get more general expressions if, instead of using developnrents in pow(>rs of tlie time, wo substitute tiie genenU values of the ele- ments given by equations (1). Tlio substitution will be most readily made by reducing the circular to exponential functions, i utting in (1) for brevity n = f "N^^ the equations (1) may be put in the form ell-' =z= v.A^A -^ Tn the preceding differential to be integrated the coefficient of *"^, (/'/' -f- //) is of the form (1 -f A," + A,.'^ + etc.) Aeie'J' 0* ,?/" '^^ (^Vr +fn' + m + IM). If hi the last factor we sul)stitut(: the preceding ex])oneiitials for the circular functions, its product bv ,V^',^V^' in the case of a cosine reduces to half of the sum (di)> (ciir («)* (W' +(;,/ (,';,/(x; (^y. Substituting the values of tliese expressions in t(>rnis of the exponentials just given, developing by the polynomial tlieorem, and then substituting for the expo- GRNERAL INTEGRALS OF T L A N E T A II Y MOTION. 3 nentials their expressions in circular functions, wc find tlmt this sum reduces to a scries of terms, eacli of the form COS ''sin ('•^^' + '^^-=+ • ■ • +'A.+i,?-'.+y2>.'2+ . . . +./X), in eacli of which we liavc 'i + '^+ • • • + '■„=./+/ Ji+J.+ - ■ ■ +./„ = /■• + /.■'. The expressions A^e^ + A^e'^ -\- etc., comprising products and powers of the sfpiarcs of r, /■', (ji and ^,' by constant coefficients by the substitutions of the values (1) reduce tlieniselves to a series of terms of tiie form Ji cos (/,Xi + /A+ . . . + a„ + . . .j\>.\ +J2A+ • . . +i„>.'„), in wliich /, + /,+ . . .+;,+^;+. . . =0. By tlicse operations and by corresponding ones in the case of sines the expres- sions to be integrated iiually reduce themselves to tlie form m'A SU) cos ('"'' + '■' + ''^-^ + '""-^-^ + • • • +ii^'i + • • • +i«?.',), in each of which the sum of the integral coefficients of the variable angles van- ishes, wliile ^1' is a function of the mean distances and of tlie '2n quantities A; and Mi. By integration this expression will remain of the same form, so that we may regard it as a general form for the i)erturbutiun due to the mutual action of two jJanets, the elements of each being corrected for secular variations. If we con- sider the action of all the planets in succession, we shall introduce no new variable angles except their mean longitudes, wliich will make » mean longitudes in all. We siiall therefore have, at the utmost, not more than Sii variabk; angles. We may thus conclude inductively that by the ordinary methods of apjjroxima- tion, the co-ordinates of eacli of -in planets, moving around the sun in nearly cir- cular orbits, and subjected to their mutual attractions, may be expressed by an infi- nite series of terms eacli of tlie form /.■ CCS • ,i,; ('l>-. + '2?-= + . . . + /3„?.3„) . ;i2„ beinf »'„ ii . . . /;,„ being integer co(>ffici(Mits, different in each term ; Xi, ?vs . each of the form li-[-h,t /i, /.J . . . /;,„ b(>ing iiii arbitrary constants, and /;,, h, . . . h^Jc, being functions of Su other arbitrary constants. We sliiill fnrtiier assume that the inclination of the orbit of each jdanet to tiie plane of .r y is so small that the co-ordinates may be dev(>loped in a convergent series, arranged according to the powers of this inclination. wiiil(> it may be siiown tiiat the general expressions for tiie nvtangular co-ordinates will be of tiie form .1 ^ SI- cos ( /,>., + /,>., -j- , . . ^ ,;^,?..,j y = A7,' sin ( /,>., + /,,X., 4- . . . -f /,„>.,„) z = Sv sin (y,>.i +.y>j + . . . +/,„>..„) (=i) G K N K UAL I N T E G U A L S OF P L A N B T A II Y AI O T I N . Tlio letter •, i/, and z are expressed in this form, any entire func- tion of these quantities will reduce itself to the same form. W(> shall now proceed to show that the form (8) is a general one : that is to say, that having an approximate solution of this form, if we make further approxima- tions, developed in powers of the errors of this first solution, every approximation can be expressed in the form (3). We can make no general determination of the limits within which these approxi- mations will be convergent, we are therefore obliged to assume their convergency. § '2. Canonical Trdusformatton of the EQttationn of Motion. If we i)ut il, the potential of the « -|- 1 bodies, that is, the sum of the products of every pair of musses divided by their nuitual distance, the ditt'erential equations of motion will be ;?(/t -\- 1) in number, each of the form <1-X: oil ill vx. If we substitute for the co-ordinates themselves their products by the square roots of their masses, putting .\i=W(-.r,; Vj =/»;//„ etc., tlie ditt'erential equations will assinne the canonical form A\'e suppose the index / to assume for each of the three co-ordinates all values from to //, tile valiu; referring to the sun, and W(> thus have 3(// -j- 1) etinations of tlie form (4) the integration of which will give the co-ordinates in terms of the time, and (!(// -f" ') arbitrary constants. \\ V sliidl now (liniinish the numlier of variables to I)e determined in the follow- ing general manner: Su])pi)S(> that w(> have ii> differential ecpiations of the first order, l)etw(>en ;// variables and the time f, each being of the form (It X Suppose also that we have found /• integrals of these equations, each of the form ,/'(,»',, 3'., .... ;'■„,./)= constant. Let us assume at pleasure )ii — /• other independent functions of the variables, each of tlie form ' £, = f/;, ■{..•,, a-,,, r,J\ GRXEUAL i:\TEGllALS OF I> L A X E T A 1! Y MOTION 5 so thiit the m variablo-s x can be oxpirs.siHl as a function ot /.■ arbitrary constants tiic tinio I, and t\ui iii—k variables ' si!^2> • . . . i;„i—k- Differentiating the above expression for c., and substituting for ^^' its value X, we sliall have (U vl ex, ^ -ex, ^ ^ '"ox,; By substituting for the .-'s in the right liand side of this equation their expres- sions ni terms ot ^1, £,„_.^., ^, ,i„(l the arbitrarv constants, we sliall luive the problem reduced to tlie integration of m~k equations between that number of variables. In tile special problem now under consid(u-ation, the m variables are the co- ordinates X, //, ;, and tiieir first dcu'ivatives witli respect to tiic time. Tiie integrals by winch we shall seek to reduce the number of the variables are those of tiie con- servation of the centre of gravity. Wo sliall take for £„ c^, etc., lin.-ar functions ot X,, r,, etc., so chosen tliat tlie reduced ecpiations shall maintain the canonical form. Let us take tln' // -\- 1 linear functions of tlie co-ordinates .<:— ^0 = « + fj( = a„oa-o + fXnXi + -f «;,^a;„ L =«„oJo+ "„i'''i + +«,„i''*'ni where wo have put for symmetry w, = ca„„ or Ho, = , (6) c being an arbitrary coefficient, while the other coefRrlents urc to bn chosen, so that tlie resnltin;. differential equations shall be of the canonical form Let us represent tiic values of x which we obtain from these equations by Differentiating any one of the preceding expressions for ^, and substituting for jji Its value, we ii;jve i«n , «j2«,-2 , aj„a,„ VI., 1)1, )llo 111,, should vanish whenever / is different from y, and should reduce to unity wlienevcr /=/. In otlun- words, it is necessary and sufficient that the coefficients a sliould be so chosen that tiie (m +1/^ quantities ]/ m^ (8) sliould f(U-ni an ortliogonal system. The first line of coefficients is already deter- mined by the equation (G), the coefficiint r excepted, which is to be determined bv the eoiulition + :'+ 7)1,, ' ')» +;:=^' W'o + »'l +....+ })!„ = C", or, from (6) which gives c = y m, putting m for the sum of the nuisses of the entire system of bodies. Having thus 'Aw = 1)1, becomes y'o V''"i v/»'» \/ 1)1 ' / 1 ^z 1)1 \/ ))i «10 «ll «i» • l/»«« "■n\ ««,. l/'"o' V< ' * ' • l/»»«' O K X K UAL I N r ]•: U 11 A L S O P P [. A N E T A R V -M O T 1 O X . Tlio lumihcr of coefficients to be (letevniiiied is now ii(ii -\- 1). The total niiiii- ber of conditions nhicb the system nmst satisfy is ^'*- - /"J~-'\butoneoftliese beini,' ahrady satisfied by the quantities in tiie first line, there remain onlv - '^" "^ '^^ Z conditions to be satistied !)y ii{n + 1) quantities, we have therefore quantities which Jiiay be chosen at pleasure. The f>("iieral theory of tiie substitution Avhich we have been oonsidering, and tlu! various modes in which the orthogonal system just found may be- formed, have been developed very fully by lladau in a paper in AmiakH ,h V Kmlc Xoniia/v Supcniun; TomeV. (18(58).' We shall, ther(>foro, at i)resent confine ourselves to a brief indi- cation of the special form of the substituticm whicli has been found useful in Celestial Mechanics. AVe first remtirk that if we form the (/* -j- 1) equations by giving / in succession all values from to n, we shall have by the theory of orthogonal substitutions the (/< -|- 1) equations If we sui)pose in the first e(piatioiis we shall have from (.5) whence, l)y substituting tliese valiK's of z, and //, in the second equation, we shall have for the expression of ,r, in terms of f„, £„ etc. to replace equation (7) \/ 111 - 1 "-21 - siH- IV: Tlie first t(>rm of this expression is common to all the values of .,•„ represent iiiir, as it does, the co-ordinates of tlu! c<'ntr(! of gravity of the system. It may, there- fore, be omitted entiivly, wlieii we seek only llie relative co-onliiuites of the vari- ous bodies, and, in any case, it will disappear from the differential ecpiatious of motion. The most simple way of forming tlie coefficients a,j is to suppose"^"" of them equal to zero. L(>t us first supimse a,^ = wlienevery> /, the first line, in which / = (I, being, of course, excepted. The orthogonal system will tlien be of the form ' Siw lino Tniii>i;iniuilioii ik's Kiiiiittimi.- DiircivmifUus ilu In DviiaiiiNniL'. 8 GENKllAL IXTEOllALS OK I'LAXKTAUY iMOTlON. §/»'o \/>lli l/wio <^2() «2I «; l/Wo' V"',' 'i/ -^, 0,, (10) ""» "'II "«2 a„„ Then a„„ will be determined by the condition ?»„ ?/; ' while all the other coefficients in the bottom line will be determuied by the condi- tion ' / 4- ~ — - = y ?», «/„ ' w< Takin- the line next the bottom the diagonal coefficient will be detennined by the equation ^ a?,, „_, -\- af,_,, „_, m„_i _ ' while the remaining coefficients of the form «„_,,, will be given hf the equ. itions The gcMieral values of the coefficients to which w.- are thus led may be expressed ui tlie loUuwnig way : put .".■ = »'o + «'i + . by which m will become ,«„. Also, suppose m m,. V, = AVe shall then have Vnj .«^i • ^,2 _ »^ "/_! "i , It is easy to prove that the coefficients thus formed f.dfil the required eoncl.tions. m. f l/vr. y VI., / UKNIJUAL l.NTKUUALS OF 1' L A i\ KT A 11 V MOTION. 9 Wo sec that, suDposing .i„ to ix>i)resi,.iit tlic co-ordinates of the sini or other cen- tnil bo.ly,f. is e.iuul to tlie co-ordinate of the Urst phmet, which may hi. auv one at pkuisure, rehitively to the sun, multiplied by u func^tion of the masses, while c is efjual to tlic co-ordinate of tlie second phmet rehitively to the centre of .'ravity'of the sun and first planet multiplied by another function of tlie masses, and so on 1 lit-se functions t,, when divided by the functions of the masses just alluded to, will (lifter ironi the co-ordinates of the several planets relatively to the sun only by quantities of the order of magnitude of the masses of the planets divided by that 01 tiie sun. ' In what precedes wc have considered only the co-ordiiiat(-s av Of course the other co-ordinates are to be subjected to the same transformation. If we ivpresent by ri and ^ the corresponding functions of // and ^, and if in the expressions for ^ r and s we substitute for x, v/, ami c, the expressions (3), those quantities will tlu'in- selves reduce to expressions of this same form. \ 3. Approximation to the Rccpdred Solutions hy the Variations of the Arbitrary Constants in a First Approximate Solution. By the transformation in question W(- have for the determination of the relative motion ot the n + 1 bodies, 3« differential equations, of the canonical form vi-r d('- or:,' Jt"--c>^,- ^^^^ Let us now suppose that we have found approximate solutions of these equa- tions 111 the form (:3), the quantities x, //, z being there replaced by £, ^, ami r. that IS, solutions which possess the property that, if, on the one hand, eacli expiv.- si.m IS twice differentiated, and if, on the other hand, the values (=}) are substi- tuted m the second members of (11), the two expressions shall differ only by terms multiplied by small numerieal coeffieients. We have to show that when' w^ make a further approximation to quantities o£ the first order relative to these coefiicients the solution will still admit of being expressed in the form (3). To do this we sliall make the further approximation by the method of the variation of arbitrary constants, remarking, however, that the usual formula, of this method cannot l.^^ appli.>d, because they presuppose that the first approximation is a ri.jorous solution of an approximate dynamical problem, while, in the present cas,>, we are not enti- tled to assume that our approximate solution (3) possesses this quahtv • in oth.r >yords, we are not entitled to assume that any function ri„of the quantities s ■,, and „ can be formed, such that we shall find the -.in equations of the form " ' rigorously and identically satisfied by the approximate expressions, both with respect to the time, and the G.* constants which the solution contains. C'onse- qn.-ntly, we cannot assume tin, (>xistence of a p,u-turbative function, and must emi)loy other expressious iu place of the derivatives of that function. We set out, then, with the three sets of equations, having n in each set 2 November, 1874, 10 OEiNEllAL INTKGUALS OF PLANET AH Y MOTIUX. I, = ^7.;, cos (/,;i, + i,?., + + /,„a,„) )7, = *'/.•( sin (;,X, -f- ijXj + + '.m>.i,„) (l^) ^, = si/i sin (y,Xi +p.a + +j,,:k,), in wliich all the quantities are supposed to be given in terms of G» arbitrary con- stants and the time, each ?. being of the form /( being an arbitrary constant, wliich each b, k, and /••' is given as a function of 3?i other arbitrary constants, whieli we may represent in the most general way by So long as no distinction between a and I is necessary, we may represent the entire 6m arbitrary constants by Let us now take the complete second derivatives of (12) with respect to the time, supposing all (in constants variable. We shall suppose the variable constants to fulfil Lagrange's conditions, now 3« in number:^ ^r, 'scij dt~^' jC, ouj at ~ ' }t, Siij dt ~ "' ^ ' which will give dt ~ dt ~ ^ " '"^*'- From the second derivatives, combined with the differential equations (11), wc shall have 'Sn equations of the form i=i vuj at O^i iO-'' which it is required to satisfy. The expression in the right-hand member of this ec^uation corresponds to „ . ni the usual theory, when R is the perturbative function. Let us multiply tiiis equation by ^^^^ , and add up the 3« equations which we may form in this way by substituting for t^ all the values of -', ■,;, and ;' in succession. We may thus obtain *v ' v"" '^^/ '':'* da^ ^d^ _ '-? o^ ^f, j=i /^i da^Oiij dt ddf, i~i r/2 iia,; the sign S' indicating that all values of y; and C as wc-ll as of P. are to be included. The right-hand member of this equation corresponds to '.' in the usual theory. Let us now multiply the equations (13), the first by '}', the second by ^y', and the third by ^^^^' , and add together the 3h equations which may be thus formed by giving I ail its values. If we subtract their sum from the last ecpiation, putting UKiNKUAL IXTKGRALS OF PLANETARY MOTION. H (==i \ou,, (jiij dUj da J ^ we shall have (".",)'';;'+(«.«.)'';;' + etc_f _ 'v -^.^i,, (i5) the sign ^ uicludiiig, as before, not only all values of / from 1 to >t, but the cor- rcsijonding terms in r, and 'C,. ]})• giving h all values in succession from 1 to 6/t, we sliall have a system of ()/* differential eiinations, the integration of which will give the values of the Km quantities iu terms of the time. IJy tile fundamental assumption with wliich we set out, the expressions for £, >;, and 'C, are sueli that tiie rigiit hand members of tiu'se equations are small quanti- ties of which we neglect tlie powers and products. We may, therefore, after solv- ing these equations so as to get the derivatives iu the form integrate by a simple quadrature, sujjposing r/„ two classes of constants represented by a and /, we shall have three classes of the functions souglit, included in the forms Let us now differentiate the equations (Li) with respect to the time, puttiu" for brevity iA + '"A 4- + !Jhn = h '.^. + '2^. + + i„:A,., = N JA+JA + +./;a„ = 6' y.>.i +,/2>... + +;,„?,,„ = iV'; we sliall then have, omitting the index / of f>, k, luid iV, £', = — Shk sin .V >7',= SJ>h cos A'' n,'-,') ^', = Sh'k' cos N\ To form the combination («j., aj) we must differentiate the equations (1'2) and (15') witli respect to r», and a^, and substitute the results in (14). Li fbrininT these quantities, two series of terms represented by the sign S of summation are to be I 1 12 U K N K II A L I N 'I' E a II A T, S O 1' P L A N K T A U Y M () T ION. intiltipliod tof^ctlu"!', wliicli rciulcrs it lu'ccssiiiy to be iiion; cxitlicit in vciJVcsciitiiifif tlio (ioiiblc Kiiinination wc tiius cncouiitcr. lliiviiifj; ii. of ciich of tiic (luimtitit's x, y;, iiiid ^' (listiiij,Miislic(l Ity writing' tlu; viirious values of tlie index /, wiiicli takes all iutej^er values from 1 to ii, tlie quantities h, /.•, and X siiould all bo affeeted with this same index. Ihit it is not necessary to write it after iV or /<, because each N is common to all the £'s and >;'s, or to all the i,"s, respectively. Again, we have as many values of .Vas there are comltinations of the coefficients /„ /._,, /,„ etc., which enter into it, while each .V has its corresponding coefficients /.', / in number. "NVo must, therefore, consider /.■ to be written fu {'.■> '-i-, ''a 'V.Ji while h and N arc affected with the same indices, the first excepted. In other words, we have /' ('l, i-l-. '"a '■.„) = ''/'l + '7'3 + + iJhn ^Vl^ 'V 'a 'ii..) = '"l''-l + ''i^-i + + '■M''-Su- Then, in the sense in which wi> have hitherto used the sign of summation • and A' are common to ^ and y;, while; the corre- sponding (pnuitities for if, being of a diffen-nt form, must be written separately, ^^'e have, in fact, distinguished them by an accent. The co-ordinates and their derivatives which enter into the expressions (d^, Uj) will then assume the following form, the index i being understood after k and A''. £i = '%/\u eos X^ ra = , to quanti- ties ot the first (l(>frr,.,> of approximation, A = and A' = by the condition that t, r„ and '; satisfy the oriyinal differential equations, and the coefficient Au,v must vanish, unless wo have ^u — iV;= constant. The co(>fficicnts /*„ /,, ],,,„, l)eing supposed incommensurable, this can only happen when we have in (:{)' 'ly = 'V ; V = 'V, etc., and hcnco N^^N,, - when sin {S\ — X^ will itself vanish. Hence, (,,„ ,/.) containinjr no constant term whatever, we must liave ("*, "j) = 0. (19) A!,Min, differentiating the equations (Ki), the first three with respect to ?, and the last three with respect to Jj, we find "'= Sf^ {JJ.% cos N'f, = _ >% (i/>/,l cos N, = — S, (JJblc), sin N, = — S, iJjh'/i'), sin iV,. I ! fl ii 14 OKNKIIAL INTKOHALH OF I'LANKTAllY MOTION. From tlicsc expressions it may be shown timt in tlic same wiiy that \vv tbund {,i^, «J =(). We liiive next to consider tiie combiimtions of tiic form (u„ Ij), for which tho expression is i I I (!u^ dij i.lj Oil, ^ A. Tlie only non-periodic terms in this expression will be those in which /< = v, mid these terms reduce to or, by puttnig we have («,,/,)=_f^ ^2) These (), wliich will then divide into two classes aecordiuy as the derivative- of n is taken with respect to /„ /, or /,„, or with respect to «„ a, or /t '^ ^"f9«,,„ (/< -^^■'■' or '^ = il, (23) U K >' !•; It A I, I N T !•; (1 U A I, S O V I' I, A N K T A II V M O l' 1 O N . U Uy ffiviiijf y all viiliicH in Niuccssion from 1 to -hi, we sliiiU liiivc ;)n ('(luiitious to dctciiiiiiif tho viiriiitioiis of c,, c.,, c,,,, IVom wliich the viiriiitioiis of «,, «.,, .... .»<„, lire to l)c ohtiiiucd by tlu> iin ('(illations ('^1). Hut, for our prcscut pur- poses, it will be more convenient to consider the c's as the fundamental elements, and to consider «„ n.,, « „, to be replaced by c,, r.,, c.,„ in the orijjinal ecpiations. The second class of differential e(iuations (15) will, by (IJ)), be represented by ^ " (ft ^ rd, (-il ol' (:((,~ (.I- iki,~ <:i-(ii,\ Substitutiuf,' for the coetHcients in the first member their values (W), we sliall huvo iiti ecpiations represented by dih ill ^ i;i,,/t ^ /.„, "^ ,"i \ ft' in, "^ "-"'• } Putting /•■ successively equal to 1, "2 ;}uted by f„ /■„ and i!,'„ possessed the property that a function lio of £> 'C, and : could be found such tiiat for all values of i t t f 10 (JKNKRAL INTEGRALS OF I' L A >' i: T A 11 Y MOTION, we should have in (2:3) and (24) by puttinju: li = il — Liu, "^ c/j § T). Fiiiuhniuntal liilafloii hctu-ccn the Ciufficioitx of the tiiiiv, />„/>,.,,< Ye. , vouxhUicI ax J'Hiirliaiix "/' ('i, <'j, intc^ifvals we liave been investigating is that tiiey do not contain the time, exc(>i»t as multiplied by the '>]h factors /(, wliicii are functions of the 3/t constants o. This ciuiracteristic will enable tis to deduce a fundamental relation between tiie differential coefficients of It witli respect to c. In tlie first phice, we remark that each c has a /> to which it stands in a ))eculiar rela- tion, ni that the latter, multiplied by the time, is added to t!ie /, whicii is conjugate to c to form the corresponding '/.. Tiie theorem in (pu-stion is this; each /> l)eing supposed to be niar!s to liave comjiletely expressed the 'Sii co-ordinates g, y;, i,', as functions of the Lvi quantities «,. II., and we have just shown how to rejiiace the first Wn (pmntities liy the quantities t\,c., fj„. W we add to these the first derivatives of the co-ordinates (1(5) G E N E UAL 1 X T E (J II ALU O E 1' 1. A N E T A U Y M O T I O N . n wc shall have im variables, roprt'sentcd by c„ >;„ f, ^\ rU, Tf, expressed as functions of the G/t (|aantities Let us now suppose those equations solved with respect to these last (luantities. \Ve shall tlien have 6/t equations of the form Ci = ^i\ K = 'I'i. whence /, = i|', — ?*/, (06) ^ and 'P being functions of £, r.. if, etc. Tlie first and tliird of th(\se rxi)r<'ssions arc tlie (i/* first intef;;rals of tlie j;iven equations, or, wliat we may (ail the int(f,'nd functions, bein<,' tliose functions of the co-ordinates, and the time, which rcuiai' etjual to arbitrary constants durnig the entiro niovenu-nt. JiCt us now, for generality, once more represent the ijn arbitrary constants by "n "•> '«,„„ and let us consider the (()«)" quantities of I'oisson formed from the "•eneral ex- pression' the symbol ^\ including, as in (14), the 'in values of £, r,, and ^ in succession. Put- ting tiu' giMieral expression (14) in tiie form (..,.„,)=v;['.'-"-';----':=-"-]. forming by multiplication tiie jiroduct of tiiis expression l)y (27), then puttinj. V =y, and forming the sunuuation • noticing also that tlie expression ^ -, j 1 ( itjiii is ecpial to unity whenever x- and y represent the same symbol, and to zero in the opposite case, we find an expression which is itself ecpial to unity wh(-n u = !, and which vanishes in all otlier ( uses. Now »»„ Oj, and »»„ may here be any of the (>/( arbitrary constants. ],(-t us then suppose ((„ a^ to represent /j and /^ respectively, and Uj to represent Cj. This equa- tion will then become (',> '•>) [(«, '■>] \- Oi, '•.) [/,., '•..] f (/„ '■,)V^, r,] 4- ( (■ i f r 18 GEX.ERAL IXTEGRALS OF PLAXETARY MOTIOxV. accoi-chng as l and ^i repivsent the same or different indices. But we liave already tound that tlie .>xpressi„n (/, c,) vanish(-s whenever / is different from /, and reduces becom7 ' "" '^'"'' "''^'''■' "'" ''^""^" '^^'' "^"'''""' '"" "'" "'"^"•^•""S thus [A' f,] =^ 1, (28) wliile all other combinations [/,, cj, [/,, /.] and [c,, <■,] vanish. Let us now return to the integral equations (2(5), and first form the combination 'llw conditions (28) therefore give and t"''"^^^^^ (29) the first equation applying whenevery is different from /, the second when they arc the same. Let us next consider the combination [f„ /.] wliich we know must vanish for all vdue^ot I and j. forming the general expression (27) from the integrals (2G), we [/,, y = [7, and ^ respectively. By tliis substitution we sliall introduce no terms of tlie form' 0)7, jtC, or 'Ct. Hence, wlion wc substitute for £, >r, and C, their expressions in infinite periodic series, tiie reduced expressions will contain cosines only. In fact, usiu"- the forms ° ^i = SK'i cos iV r,i = Sl'f sin ^V <:,=: A7/,sin.V', we shall have from (12) when we put for brevity C"-"'0^'+("'~"-^>' + «tc. XWij IHj I \?H,. 9»^. / - ' •»•,• — Xj := .S7",j cos iV ; 2/, — .'/, =-- 'S'Z-,^ sin iV; = * (jf (32) \t>\2 2, — 2,'- z= A'/.',., sin iV. Each d<>nominator in H will therefore assume tlie form ^/ ( A7- cos .V)- + (,S'A- sin A')- -f (A'A-' sin .V' Wlien W(< form these three scjuares we find that every term of th(- form // cos (-^u + ^^) 1" till' fii'st s(iuare is destroyed by a corresponding term — // cos ( .V« -I- K.) in tii(> second square. Hence the sum of th(-se two squares will only contain ternis of the form ;, cos {N^ — X,). Since in each value (IT)) of .V we have '\ + !: + h + -\-L = h we sludl have in X^ — iV, Also, since in .V the sum of these coefficients is zero, it follows that the smn(< thins,' will liol.l tru(> of th(> third of the precedinj,' s(iuares. The denominator in question may therefore be expressed in the form in wiiich each A'^ is of the form where «. + '■■ + (, 4- + 'a„ = 0. mmmmt^^- J±:J..M:^.^d^y,j0^ f y f r 20 GENERAL INTEGRALS OP PLANETARY MOTION. The possibility of developing the reciprocal of this denominator in the usual way depends upon the condition tluit the constant term of *S1- cos N is larger than the sum of the coefficients of all the other terms, a condition which, so far as we yet know, is fulfilled by all the planets and satellites of our system. Representing tliis constant term by 1,-^ and the quotient of the dum of all the other terms divided by k^ by A, so that SI; cos iV = ^•„( 1 + A) the developed expression for fl will be n=s'';;^(i-iA+i-|A»-etc.). When we develop the powers of A this equation will reduce itself to the form n = Sh cos (/,;i, + L?., + 1,?.3 + -(- i3,.x,„), (33) each X being, as before, of the form ,., . ^ ^i = h + ht, while in each term h + h + h-\- + hn = 0. To form the seco id part of ilj and of il) in (23) and (24) we have to differen- tiate the expressions (12) twice with respect to the time, and once with respect to the arbitrary constants wliich enter into them. Putting, as before, for brevity, ^=*l^l + *'2^3+ +hn^3n b = iA -\-iA-\- + HnK, we have ^•2 = — Sh^Jci cos N ^< = -SIH; sin N (34) ^^,*= — Sh"k',smN'. or For the other derivatives which enter into il'j we have ^1 ^ — SI: k, sinN ^f = Sijki cos N (34)' .>,'= SJjl:'iCos N'. Forming tlie sum of the products which enter into 11^, in the manner represented in § 4, it becomes ^r S^S, ^( !jl-:) , {h%,^ sin (N. — JV;) + UjA'i\{f>''l'''<)^{^^n(N', — N'^)-sin(N',-\.N'^))\. (35) OENKllAL IXTEGKALS OF PLANETARY MOTION. 21 This expression reduces to the form S II cos N, where in each value of iVwe have li = 0. In tliis expression it may be worth while to give the complete value of ^corre- sponding to any value of N. The value of the latter is comphstely determined by the indices /„ i,, etc., which multiply X,, ?.„ etc., in its expression. Let then represent the value of N for whicli we wish to find the corresponding value of mii'J.i /,„) by means of (35). The required term will be found by taking in (35) all combinations of )• and /i for which we have N,-N^= N, N\. — .v; = N, or .v; -f iV; = K I-ct us represent the combination of indices v in iV" by k„ L, vtc, and those in N\ byy,,y^, etc., so tliat we have iV, =i,?., +y,?., + +y:,„X3„. Then, in order that tlie sum or difference of these angles and of iV^ may make .V, according to tlie formula; just written, we nmst have -X'^ = (,"1 - 'l)'<. + (,«2 — «.)>.2 + + («,,„ — i3„)?.3«, and or ^V = ('l - i,)>-. 4- ('2 — .?;)>... + + ihn —J,J>.,n- For the corresponding coefficients of the time h, we have ^v = (,". — h)^ + (ih - i^h, + + («,„ _ ,;„)7>,„ ?-; ± (.A — A)^. ± a - Q\ ± ± (i„. — i.u,)K. Affecting k and I' with the proper indices, as explained in § 4, tlie part of the coefficient IIj{!,, /., /,„) corresponding to any one value of tlie angle N,, will be i ^ n ^Xjljki{lii,^l„_, ) A-j(,M,— «i,/<2— ^- + J f^jj^^'U^ ./=' Wm I ^■',(./.-'.,.y;.-'i, )—/.•',('■,—;„ 'o— i, ) I where the values ofh^ anu /^ are those just given. The complete value of //;(/„ !.,, ) will be found by taking the sum of all the terms which we can form by giving to ;/i,|(*o, ctc.,ji,j.,, y„„ in these expressions, all admissible combinations of values, that is, the complete expression will be given by writing before the first line the symbols |Ui=OC A«2=0C |Uj„=OC 2 2 2 i"i = — OC !«, =— oc |U3„ = — OC j' 22 GENERAL INTEGRALS OF PLANETARY MOTION. V CM) and before the second one y,=oc y,=oc .2 2 ... _ i, = -cx: ,72=— oc .;3,> = -oc Differentiating {US) with respect to /,, wo have By the substitution of these expressions (23) now assumes tlie form (ft V/jSm.V, (;57) putting for brevity h' = ij7i -)- If., By the fundamental Iiypothesis tliat tlae adopted expressions for £, ., and C are (. -.) and (3(,) all the terms wh.eh are not of the order of those neglected in the n le^eHlT r" ' """'' ''^"'' ^° '^''' ^' '' '' '''"^ -•^"- «^' ^'-' ^--'ities neglected m that approximation. To form the equations (24) we differentiate (12) with respect to ., wlu-rel)y omit- Un,^ mdex . wUh which ., „ ^, Z, and , are always to be considered as :^:d, — A cos A + / ,V^ . sin A^ dc f'C: cc 'i „ OK . ,, /:f) , = aS" . sm X-{-tS/,- . cosiV ?: r,f, ' : ,, vc (iny The sum of the products of these expressions by (:3-t) which enter into (24) is -\ l'[ S-^,„ I (i=/.)^ J^ ,0, (^; - ^) - ^ ilrA% ^^'>' «in ( a; _ A^) + l(i''A-% ^J- (cos ( at; - A^;) _ cos ( A^; + A^;) ) i / (^'^A04^' (sin (A;- a;) _ sin (A-; + A^;) I , while by differentiating (;33) we find .;^=A(.^cosA^_/^.^^si„x). (07)" Taking the difference of these two expressions, the equations (24) will assume the \ji =—Sh" cos .V+ 1 Sir sin .V. (38) the qmmtities J, and r being formed by a process similar to that used in forming n. We have now to mtegrate the expressions (37) and (3S), and substitute th^ O E N K 11 \ L INT K G K A L S OP P L A N E 'I' A U Y AI OTIO X . 23 resulting values of <■, and /^ in the expressions (12). Representing the perturha- tions of eacli quantity by the sign h, we shall have to increase each value of X by the quantity ^?., = ,M, + thbi. We here have the time t outside the signs s!n or cos in both (V,, from the integra- tion of (;38), and in (hlji. We must next find the sum of tlie terms thus introduced into b?.i. Dilfereutiuting this expression we have We have now to form the sum of the terms in the seeond member of this ecpiation wiiich are multiplied by /. iJeginniug with the second, we have, omitting the in- dex of I) dh ch dc, . c/> dc. dt ~dc,dt '^ dc.dt '^ ^^^' Substituting for ' their values in (31), this equation becomes dh ( dh ch . , ^ ih \ . .. which, after multii)lying by t, is to be added to the last member of (38). But it will be more convenient, instead of using h and It" in these expressions, to retain T^C /- 7-''* the expressions ,!?,',,', and ', !? in their present analytical form, lleprcsentina' them, at dt- dl' ^ for brevity, by t", y;", and ^", the equations (23) and (24) become dcj _ eq .■ = » i ^„ cJi, „ fV* , .„ o^s ,• 1 dt ~ eij ,r, t ^ *^ "^ '^ 'eij ^^'efjl (4«) dt dcj ^ .• , . 1 ^ ' dcj ^ '^ '■ c-;^ ^ ^ '• 6cj i " If in the first of these equations we substitute for the derivatives their values in (34)' and (3(>i, it becomes 'J = - ^' { '^'' - - (^'"'^''■') } ^'" ^^ + - (^"^'J ^'^ ^"'^ "^' + - (^'".A /'•'.) cos ^"• Substituting in the first of the above expressions for , we have dt '^'' — V / . Sh oh ' ^ . rlh ) , . „ ^ dt—'n 'v.,+'^,.c,+ +'-,.e: [^^'"^ j i 1 34 GENERAL INTEGRALS OF P L A N t i .\ R Y MOTION. We have next, in tl>e second of equations (40) to substitute tlie expressions for the derivatives in (37)' nud (37)", nitaininy only the terms multiplied by I. This gives by substituting for b its developed expression b — ij)i -\-LJ)i+ + '■;.,> K + ,\^,{:,^ + J^+ + ,.«.■)} e„»^ + .{.r^',(^^^ + ^*+ +i,*.)} .^K. Adding this expression to (41), we find that the sum reduces to a series ot .ms eacli of which has a factor of the form 6b,_dhj By (31) these factors arc all zero. Hence the terms of (39) multiplied by < destroy each other, and we have the parenthesis around ^ ' indicating that all the terms multiplied by the time in that expression are to be omitted ; in other words, that, in taking tin; derivatives of n, ^, y;, and ^ with respect to o„ we are only to consider the coeffici* iits h, k, and k' as functions of these quantities, and are not to vary i,, b,, etc. § 7. Form of (he Second Approximation. The rest of our process is now as follows : By iutegratii.j,' (37) and (38), the last member of (38) being omitted, we have Scj = S^'-'cosiV (,y.) = _ S^''j sin N. The co-ordinates £, y;, and s in (12) being expressed as functions of the quanti- ties Cj and /j, we are to suppose these quantities increased by their perturbations, that is, we are to lind ^^ = 24^^ + 24^^' or, since we have replaced /, by ?.<, K N E UAL 1 N T K (J U A L S O V V L A X 10 T A U Y M U T J O N . 25 111 (4;J) wc liuve and, integrating, J = :\n hb,=^'p ,'.cj=,S X 7': /('; (b. oc, COS iV, >-i ov. ^^^< = ('V,)+J'Vv/^ f/' j - 3/1 /(', (7/ \ 6 ;- I 0- cr, I iiu iV, which, for brevity, wo may represont by h'/.i = «S',7v, sin iV. (44) l)iitting Tn adding tho ofroct of tho ix-rtuvbiUioiis hci to £, >;, and ^, wl- iiir to vary only /•, tile fxpressions for h^, etc., being .;,„) [ h = '% I '^^- .sin N+ k cos .V(/,,^>., -f Uv,., + + 4„,S>.,„) [ W( = S, I a-'sinXf //cosXcy,,^?.. +^V^X2+ +iJ>.J I We are to put in these expressions and the values of (^?. in (44). We tiius find + ^ «^<. { i:. ( ^' ^J| ), - Z^. (/,/., + /,A, + + /,„A,„x I cos ( a; _ ^;) ;i, = 1 s\,, I V, (^' ;:J' ), + k^ {i,L, + ;,/., + .;...+ /,„z,„), I sin (X + X.) + ^ ^n,.. I :^. (^^ ;;'■ ). - A-, {lA -V u. + + /:,„A„),. I sin ( a; - X,) 5C = i '^%|:t.(^'^|), + ^<„(.^Z,+^A,+ +./.Z.J.|sin (X', + N.) + h ^^. [ S< (^'j; ^),-l/^J,^ +.^A, + +i<„Z,„X I sin (A-;, - N,) Since, in iV„ we have 'ii= 1, while in ^V, " " ii = 0, 4 November. 1874. 'f it 26 GENKllAL IXTEGllALS OF PLANETAUY MOTION'. it follows tlmt all these terms will be of the same form with those already contained in t, y;, and s (12). In the preceding inte<,'ration we have tacitly siipposed the coefficient of the time, b, never to vanish in any case, lint some of the values of A' will necessarily he zero, and in this case, instead of having JA(//cos jy'= .' sin iV, we must put fi (It cos N = Id. The only terms of this form are found in hi. If, in (3S), we represent the coeffi- cient of the vanishing term by /("„, we shall have for the terms in question -V = - rj. This adds to '/. the same expression, and is equivalent to diminishing h by the quantity /("„. We make this chang(> not only in the original terms of £, r,, and 'C, but also in tiie terms of (S^, (V, and (^^', because the change will only affect them by quantities of tlie second order, which we have rejected throughout. Making these changes, the expressions i + Ki r -\- 'Vi "i»l if + K, will now satisfy the differential equations (11) to quaniities of the second order, while their Ibrni will still be in all respects the sani(! as in (12). As we have made this one approximation without changing tlie form of tlie original integrals, so may we make any number of successive approximations. We may, therefore, regard the form e = ,S7.- cos (;,?., + ?j?.2 4- + /;,„?.:,„) y; = aS7.- sin (;,?.i + *,?., + + l,„X,„) (,' 3= S/i sin (./,Xi -\-jnX., + +j,n>-jJ, where each '/>. is of the form ?..• = '.-{- bA 7, being an arbitrary constant, and /.', Jc, and b^ being each functions of 3)t other arbitrary constants, while '■| + h+ + '■),. = 1, fini)r<-sn,t the li „• ..l,.n,..Mts ..f tl.. systr... l.y r^. .,,, vtc. Sinro u-, y, 2, a.ul «, >:, s, ill-'" Mil liiuiir .o-..i(liiiutc.s, »-,. Imvr in ili,. (.xpirssioiis (1(1) of tlit" latter Kv.iy tin,.- Nv.. .lifH.rcutiiite th.-se ••xpicssions with resp.Tt to the time we miiltiply the eoifficieuts by h, u linear liiucticm of />„ I,,, cte. Ik'uce : = [""\6'^'j. 'I'lic form of the potential il shows tlmt :/;i;:'\:':::;':.."'';r,™ ""■ '""■ '"■ '""■""'"■■ "'°'-"""»' *° "- ■"-■™ -i- In order that the diftWeutial e,,uation ;^f = ^^ n..y he satisfied identically we must have or K", 6'-"] = [„'--■)], The expression (-^1) for ,•, /.• hrinfr Unrar in a, is of the form Ileue.., wh.M, wt> express A, in terms „f r„ <•„ cte., we must have Tiie fundamental property of homo, neons functions now gives Substituting in (4G), we find t/c, which is the tlieorem enunciated. '* This theoren. camiot b,. .lirectly employe,. ,„ ohtuin the values of /,, for the eason tl.at 1 cannot be .Ictern.ined as a InnCou c.f the canonical constants until the equations ot motion are completely iiitei.'r .te.i. §9. Snmmnri/ of UrmUK The following is a brief summary of some of the results which follow from the preceding nivestigation. We first suppose that we have found expressions for ,", ,, and ^ of the forn, (!■>), .uch as ulent.cally satisfy the .lifferential equations (11). We also conceive the i (J K X !•; u A li 1 N r !•; o ii a i, s o i i- 1, a > k r a ii y m o t i o n . 29 (luiuifitifs /.■ and /* as cxpn ssed in terms of '-in t'anonical coustauts t„ f^, f, t.,„, so chosiin tliat tlio fxpriNsion I I I (Cj f/j. fc^ i/j ffj ell, J Ninill rcilncc to unity wlicii k—J, tinil sliall vanish whenever any ot'er of the Cii (iniintities '•, r„„ /, /,„ is substituted tor /j. 'I'iien: — Tlici'.itn I, — If, taking the entire si'ries of ',iit co-ordinates represented by £, i',,, r^i >:.4. s'l Cfu ^ve multiply the square of each toetticient k l)y the coefficient of the time in the correspond injj angle /jXi -j- I'a?., + etc. (that is, by the corresponding (piantity !',<», -\- ih^ -j- etc., ory,/*, -\-JJ>i -\- etc.), and by the coefficient !j or j) of any one of the ?.'s, as >.j, which '/. is to lie the same tiiroughout, then all tiie constants c, except r^, will identically disappear from the sum of all these products, which sum will reduce identically to '-.V^. I'his theorem is expressed in eipiatiou ('21). T/i'DiTiii II. — The 'An coefficients of the time, 6,, tj, etc., considered as functions of ci, t'a, etc., fulfil the ' ,, — conditions expressed by vh, /'■''> where / and ./ may have any values at pleasure from 1 to 3n. They arc therefore all till' partial derivatives of some one function of c,, v., c,„. 'J7wuiT)ti 111. — This function is the negative of the constant term of the expres- sion for the living force in terms of c,, c,.,, etc., as shown in the last section. 'f/iioniii IV. — The sum of the canonical elements c,, Cj c,,, is equal to the " constant of areas." this constant being cither the sum of the canonical areolar velocities on the plane of A""!', or, which is the same, the sum of the products ob- tained by multiplying the actual areolar velo( ity of each body around any point, fixed with reference to tlu> centre of gravity of the system, by the mass of the body. This theorem is demonstrated as follows : The sum I 2 w,(av/, — .r'i2/i) 1=0 is known to be a constant by t!ie principle of conservation of areas. From the ex- j)ressi()ii ({)) for :»•,, and the corresponding (expression for y„ introducing the quantity Uoi as ill ((S), we have (•'i.'/i •' illi) — *- „ y.'.j', k i^j'.kJt J =0* ^ wr,- multiplying by «;„ and then summing with respect to /, we have j =0 V I 1 - })li I By the condition of the orthogonal system (S) the sum in brackets vanishes when- ever/ is different from /•, and beconi(>s unity when these indices are equal. More- over in (5) c'y and „ vanish whenever the origin of co-ordinates is fixed relatively 30 G E N K II A L r X T K G H A L S O F PLAN E T A R Y M O T I X. I > SuLstitutins for ^, ,, C', and ,' their oxpressioas (16), the constant term of this expression heconies WW. But if we a/^, neon,,, V.-Thc constant part of the living force, whicli is itself eqnal to the constant //.n the integral of living forces, nsually expressed in the forin IS represented hy kV,+6,c, + +&,,„<■,,„), as already shown in § }). The constant part of H itself is therefore equal to V,+ft,r-.+ +h,„c,„. The equality of // to the constant part of T may be shown by the preccdii..- ^l.eorv or U may I. easily deduced directly trom the theorem of lini.g Lr. as Jl.own W Jacohi. ( \or/rsunf/('ii iihrr Dumimll,; p. •>[) ) The conditions that the Lagrangiau coefficients ,.„/,), the sum of the-canonical auH, lar velwcitu-s and the .l.tterence between ,he potential an.l living force are al constant, give r,se to a nun.bcn- of relations b..twe.-ii the quantities I /, ad th. nvat.ves w,th respec. to c, which I have not jet t;.„n,l of any use \n the o e ^ turns ot integration. I theirtore omit to c.ite them, especially as tlu-ir ,.ompl..,e expressions are rather complex. i«<«mpi(i(. l^e i^nns whi..h we have been considering are those in which it vould be cce.ss,u> to develop flu. ..xpress.ons for co-ordinates of th,. planets, if we wished these expressions to hoi, true for all tim,>. The usual expivssious are su«ici,.utlv -•••cct tor a few centun,-s, but fail .ititvlv wh,-n w,- exten.l the tim,- l.von,! c-i- tmn bmits But, in the case of th,- plaiu.fary system, we are ol.li,..., ,„ :„„,.„, , , h,.m tor th,^ r..as„n that formulas dev..i,.ped in multipl,. of the •>;5» in,I..p,.n.l..„t a guments o that system w,udd b,. uun.anag,.abl,- iu p,.a,.ti,.e. Hut, i„ ,L ,.,J ot the subsuhary systems, as the Tellurian ami Jovian for instanc,-, the scdar wUiLh UHaiccs the ium.b..r ul milly imlqu.n.ient arpiim.nts t.. ;)/,-!. '.'■.■^.■f" a K X 10 15 A L 1 N T K (J U A L S O V V L A X K T A K Y M O T I O N . 31 vaiiatious of tlic orbits are so rapid that the apimiximatioii in pow(>rs of tlu< tiino fails wen for \n-csvn\ uses, llnicc, tiu- lunar theory, eonsidered as a problem of three bodies oidy, is always treated in a manner analogous to that in which the general theory of planetary motion has been eonsidered in the present paper, the three arjrumeiits introdue( living force as contains /., Cr, and 7/, by differ- entiating which with respect to the latter (pmutities, lie obtains the expressimis for the motions of the thriM- arguments. Tlie theory of Jupiter's satellites has been treated by ^l. Souillart in such a niam.-r that the <\)-ordinates may contain, instead of the longitudes of the ju-ri- ioves, the varying angles on which these longitudes de|)end. His analytical theory IS given in the Aiiiialrfi s. One of the special i)rol)Iems growing out of this geiu'ral theorv will !)(< the determination of the coeffici(-nts of the time, A„ /<.,, etc., eitlier in terms of the canoni- cal constants c„ c,, etc., m- of th(> largest of the coefJici,-nts /•. in the expressions for the co-(mlinafes of the several planets. These coefHcients are, approximately, the mean distances of the planets. The (pianlities /> ought, perhaj.s, to api)ear as the re.;;s of an e(piation of the '.]u//i degree, but the writer h.is not yet succeeded in forn.'iig any expression fitted to give ris.> to such an cpiation, excei)t one in which only the .ijiiares of the (luantities in nuestiou appear. ril|)i.,.,aK|l 11V TIIK .SMITIISOMAN 1 N Ml T li T U) N , W A « a 1 N U I' 1) \ (■ 1 ■!• V _ l> K !■ K V M K II . i S 7 4 .