GENERAL INTEGRALS OP PLANETARY MOTION. 3 



nentials their expressions in circular functions, we find that this sum reduces to 

 a series of terms, each of the form 



in each of which we have 



3.+h+. . .+/„ = /. + //. 

 The expressions A-fi^ -\- J-ge'^ + etc., comprising products and powers of the 

 squares of e, e', ^ and ^' by constant coefficients by the substitutions of the values 

 (1) reduce themselves to a series of terms of the form 



h cos (*A + *2^2+ • • ■ + inK + • • -ii^'i +y2^'2 + • • • +y«^'«), 

 in which 



H + H+. . ■+h+h+. . -=0. 



By these operations and by corresponding ones in the case of sines the expres- 

 sions to be integrated finally reduce themselves to the form 



*^'^' cos ^*'^' + ^^ + ^'^^ + *''^^=^ + • • • +ii'^'i + • • • +i«^'«)' 

 in each of which the sum of the integral coefficients of the variable angles van- 

 ishes, while A is a function of the mean distances and of the 2n quantities iV^ and 

 M^. By integration this expression will remain of the same form, so that we may 

 regard it as a general form for the perturbation due to the mutual action of two 

 planets, 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, which will make n mean longitudes in all. 

 We shall therefore have, at the utmost, not more than 3?i variable angles. 



We may thus conclude inductively that by the ordinary methods of approxima- 

 tion, the co-ordinates of each of 3re 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 each of the form 



1^ sin ^^'^'^ + ^^^'^ + • • • + hnKi) (2) 



^l, io . . . %n being integer coefficients, different in each term ; ^l^, ^la . . . /ls„ being 

 each of the form 



li + ht 



hi h ■ • ■ hn being 3n arbitrary constants, and 6,, b^ . . . \n^, being functions of Zn 

 other arbitrary constants. 



We shall further assume that the inclination of the orbit of each planet to the 

 plane of xy h so small that the co-ordinates may be developed in a convergent 

 series, arranged according to the powers of this inclination, while it may be shown 

 that the general expressions for the rectangular co-ordinates will be of the form 



OC = Sk cos ( ■tiXi + i'aXa + . . . + Hn^sn) 



y = Sh sin ( ?;iX,i + 4^2 + • • • + hn^3n) (3) 



z — Sc sin (yiXi +^2^2+- • • -\rhnK,) 



