OP URANUS AND NEPTUNE. xlv 



then aFi = b ', and $R = — — ' S,a, 



2 da 



. rd($R) im'nW ctF t , n . 



2na* /-i- — — 77 —^(Psin\ + P'cos\), 



J t da or sin 1 da* 



= - — I 2o + 4a — - + a" 1 — ? I = - «i 1 aC suppose ; 



2wa' 



da' 

 -d (&ff) 4m,»reVa ! C 



f , = + — , . ,„ (PsmX + P cosX). 

 . da w i sin 1 y 



According to Pontecoulant (Liv. VI. p. 375), 



log 6/= 0-33131, log a -S- = T-53098, log a' 2 ~° ? = T'76910, 



da ° da 2 



ntl = ^oT^R ; •"• lo S aC - °" 4 9 3 79, log l . ,, m 1-46410; 



3501-0 ° a» 2 sin 1 



.-. #« = - 3" -431 sin \ + 0"-517 cos X. 

 Similarly, if F 2 be the constant term of the disturbing function for Jupiter and Uranus, 



772 



m 2 = mass of Jupiter, then aF 2 = b ", 



2 



,d 2 P 8 «i. /.. „ „db" „„dV> 





2rca 2 



- m 2 afl suppose. 



dP 25 



— M; 

 da 



d (3i2) 4m 2 mVa 2 Z) 



55 = — — ^a; 

 da 



/•a id« ) ^m.jtnn'a'-U , n . 



/ -V-* = + jT- ,» ( P sin * + P cos *)> 



^, da <o 2 sin 1 v ' 



log 6 " = 0-30928, log a" -% = 2-90360, log a" 2 — 2- = 2'97473, 



do da 



1 , _, , 4m.m'»' 



, log aZ> = 0-35133, log -j-i — 9 = 1.98805 ; 



m* 



1047'9 



.-. S'e = - 8"-259 sin X + l"-244 cos X. 



For the action of Saturn on Neptune, the formula is 



tf« = 2n a' 2 / -A_£ * . ., (Q sin X + Q'cos X), 



•>t da w sin 1 J 



where o'C = 6,' + 2a' ^ + l a ' 2 ^ , 



da " da~ 



and log a' = 1-50179, log b ' - 0-31249, 



^ = 105534 log a 2 -,, 



