262 Variations of the Arbitrary Constants in Elliptic Motion. 



(93), or R 



m'r [cos. (v' — v) +s$'] 



) 



3 



2 



m! 



[r 2 -2r P cos.(^-fl)4- P 2 + (ps'-rs) 2 ]* 



press r, y, 5, p, v\ s', in terms of fndt, o, E, e, See, f flit, a', E', 

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

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

 ments «, E, &c. are constant, d\sofndt=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, also f \ndt = t nt , 

 as before ; for the method of expressing r, v, s, p, v' 9 $', 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, &c. of the same 

 volume, where it is expressed in a function of nt, a, E, &c. ,nt, a\ 

 E', &c. 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, &v, 8s ; 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", &c. which being applied to the values of 

 r, t>, 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' 3 , &c. m" 2 , m" 3 , &c, and so on ; it may be ob- 

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

 complete value of 8s will be the latitude of m, but if the invariable 

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

 the complete value of 8s to the value of s 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", 

 &c. 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, 



