Variations of the Arbitrary Constants in Elliptic Motion. 263 



M' will denote the mass of the primary, m that of the secondary, 

 which is disturbed, and m', m" y &c. will denote the masses of the 

 disturbing secondaries ; but as the method of finding the integrals 

 which are indicated in (B), and in the values of Sr, &v, 6s, is too long 

 to be inserted here, we shall refer to p. 362, Vol. I of Pontecou- 

 lant's Systeme du Monde, where the value of F which he has giv- 

 en, denotes the value of R, that is to be used in computing the sec- 

 ular variations of E, •#, e,p,q; and for finding the periodical varia- 

 tions, we shall refer to pp. 346, 463, where the value of R that is 

 to be used in finding the periodical variations of the above quanti- 

 ties together with those of a and n' is given ; the value given at p. 

 463, will enable us to find the variations which involve the first pow- 

 ers of e, e\ p, q; then for finding <5r, 5v, by (C) and (E), we shall 

 refer to pp. 474, 475, and for finding Ss, to p. 483, of the same vol- 

 ume, or Mrs. Somerville's Mechanism of the Heavens may be con- 

 suited, where the above subjects are treated after the manner of 

 Pontecoulant. 



We will now proceed to obtain other formulae for finding <5r, &v, 

 fa, which will be useful in many cases. 



► . xd 2 x+yd 2 y+zd 2 z M 



Put "-^f + - =T, then from r*=x*+y*+z\ 



d*(r 2 ) m M dx 2 +dy 2 + dz 2 

 we get -2^2- =T-- + fa > or by (40), we shall 



rf2 ( r2 ) m, M < M l • • • d2 ( rSr ) Mr8 ' 



have 2 1.2 =T+ — "~T' whose variation gives — 772~ + — 



3 



Mb a . MSa 



*T + —r> by (39) —5- =-2/dR, also by (19) <5T=_rR', for 



in the invariable ellipse T=0, and in the variable ellipse it = — rR', 

 but since the first power of the disturbing force is only considered, 



dR d 2 (rSr) MrS 



we have by (66) rR'=r -r- ; hence we shall have — jri — + - 



r 3 



dR 



+ 2/ dR -f- r -i— =0, (H), which agrees with the equation given at 



p. 257, Vol. I of the Mecanique Celeste. 



It is evident by (52), that if we take the differential of <5r as ex- 

 pressed in the first and second forms of (90), relative to 5a, 5e, Su, 



ddr dr 

 only, we shall get ^ = ^ (SafdR), (95) ; substituting the value 



dv /7R //R 



°^ ndi ^ rom (®®) * n ^)> l ^ en s * nce a ~d~ =r HP ky comparing 



