xliv THE THEORY OF THE LONG INEQUALITY 



mu , e na r\ d (^) t d R* « dR A J J- ,.• * *r, • 



Ihe value of / te — ; — + 0,e--— +d„e— —} depending on this part of dR is 



2 J t [ de de 2 de J r r 



very small, and may therefore be neglected. 



51. Let F be the constant term of R which is independent of the eccentricities and 



clinations ; and let $R = — — ^a + — — $.d, 



da da 



then * na *f t ~V - - ;7*n» *? (P sm x + * cos x) 



8mnria ! d* d?F .^. . 



+ » ■ ,,, j-j3 ( Q sm X + Q cos x )' 

 w sin 1 dada 



aF = -—abjl\ 



2 



d*F m 8 «PV»> , r 



a 3 -=-r = a 3 — f— = - m aC, suppose, 



da 2 2 d a a rr 



in- 



dada \ aa aa I 



2ri, 



= + — a 2a — - — + a 2 " = m aC 2 suppose 

 2 V da da 2 J 



then log aC,. = T'76899, log aC 2 = 0-02293. 



Hence ^e = - o"-431 sin X + o"'063 cos X. 



The formula for Neptune is, 



, „ r d$R!) ^mnn'a'd 2 d*F a . _ 



J« ! / \, - , • ,„ v-A (-P sin X + P' cos X), 



•/ t da w sin 1 daaa 



8»iV 2 «' 4 dlF , . ^ 



+ < , • ," ^ W sin X + Q cos X), 

 m w 4 sin 1 od ! 



4mm nri ad Cx . 



«= s - ^ — v — \P sin X + F cos X), 



to sin 1 



8mW 2 C 2 ' try , 



o . ,„ (Q sm X + Q cos X), 



to sin l 



where a'C 2 ' = 1 (#,«) + 4a ^ + a 2 ^) ; .-. log a'C 2 ' = 0' 



.-. 3V= + 0"-78I sinX - o"-114 cosX. 



66762 ; 



52. The above terms are of very little importance : but there are terms of the same kind 

 arising from the action of the constant part of the disturbing force of Saturn and Jupiter upon 

 Uranus and Neptune which are much more considerable. Let F x be the constant part of the 

 disturbing function for Saturn and Uranus, m t the mass of Saturn, a t the \ major axis of its 



orbit d - — 

 a 



