262 Variations of the Arbitrary Constants in Elliptic Motion. 



m'r [cos. (v' — v) -{-ss'] 

 (93), or R = ^ -,—^ ~ 



m' 



75 (94) ; we must now ex- 



j-^2 _2rpcos. (ij' — ■y)4-p^+(ps^ — ^5)-]^ 



press r, u, 5, p, v' ^ s', in terms of fndt, a, E, e, hc.,f,ndt, a', E', 

 e', Stc, on the supposition that the disturbing force =0, in which 

 case the elhpse described by m would be invariable, and the ele- 

 ments a, E, he. are constant, a.hofndt=nt since n= const., in the 

 same way a% E', &;c. the elements of the motion of m' correspond- 

 ing to those of m, are to be considered as constant, d\sof^ndt=pt, 

 as before ; for the method of expressing r, v, s, p, v', s', as directed, 

 we shall refer to Vol. I of the Mec. Cel., p. 181, &c. then for R, 

 as given in (93) or (94), we shall refer to p. 263, he. of the same 

 volume, where it is expressed in a function of nt, a, E, &;c. ^nt, n', 

 E', he. Then observing that the characteristic d of differentials in 

 the above formulae refers only to the t or nt in the invariable ellipse 

 described by m, but that the integral sign /refers to t, whether it is 

 introduced into R by the values of r, v, s, or those of p, v', s', we 

 shall readily find 6r, Sv, Ss ; and in the same way we might find the 

 variations of r, v, s arising from the action of another body m", re- 

 volving around M', and so on for any number of bodies whatever, 

 then by adding all the variations of r, v, s, according to their alge- 

 braic signs we shall get the total variations of r, v, s arising from the 

 disturbing bodies m', m", he. which being applied to the values of 

 r, V, s, in the invariable ellipse at the time t, will give the correct 

 values of r, v, s, at the same time by neglecting quantities which de- 

 pend on m'2, m'^, he. m"^, m"^, he., and so on ; it may be ob- 

 served that as the invariable ellipse is taken for the plane of a;, y, the 

 complete value of os will be the latitude of m, but if the invariable 

 ellipse makes a very small angle with the plane x, y, we must add 

 the complete value oi hs to the value of 5 in the invariable ellipse at 

 the time t, as stated above. 



The application of what has been done to the solar system is 

 easy, for in the case of a primary planet or a comet disturbed by the 

 attractions of the other planets, we are to consider M' as denoting 

 the sun's mass, m that of the disturbed planet or comet, and m', m", 

 he. as the masses of the disturbing planets ; but in the case of a 

 secondary planet revolving around its primary, and disturbed in its 

 motion by other secondaries revolving around the same primary, 



