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", &tc. 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 ^r, Sv, 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, ■ztf, 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, g ; then for finding Sr, Sv, 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- 

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

 Pontecoulant. 



We will now proceed to obtain other formulae for finding lir, h'^ 

 6s, which will be useful in many cases. 



xd'^x+ycPy+zdr-z M 

 Put -^ — +- =T, then from r^ =x^ ■\-y^ -{-z'', 



d^{r^) M dx^+dy^ + dz^ 

 we get -^dt^ ^'^~7 + — dt^ ~' °^ ^^ (^^)' '^^^ ^^^^^' 



d^r"") ^ M M d^{r8r) 'MrSr 



have -^^p- =T+ 7--' whose vanation gives —^p— + -^ = 



Mha M^a 



5T+^^. by (39) -^ =-2/£ZR, also by (19) <5T=-rR', for 



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



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



, „ c?R , d^{r6r) Mr6r 



we have by {<oQ) rR'=r-7- ; hence we shall have ,2 — + — — 



dR 



-f 2/fZR+r-^=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 hr as ex- 

 pressed in the first and second forms of (90), relative to 6a, 6e, 6Uf 



d6r dr 

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



^ dv ^ ,„ ., . ,^^ dR dR 



of ^ from (88) m (F), then smce a-i^—r-^^ by comparing 



