OF URANUS AND NEPTUNE. XXXV 



dR 3 dR> dR* , dR, 



* R= ^d7& + -dr&+-£-b + - d ji» 



dR, . dR., s dR 3 , dff 2 , 



de de d-ar «ar 



dR 3 . , dR 2 * , di? 3 » , ^^2 « , 

 de de dsr d-w 



41. We shall first calculate the terms which will be divided by to 



dR, «. dR, , 



^ = to'(P cos X - P'sin X) , (Art. 1 6.) 



.-. Si? = - «'(£, - *£') (P sin X + P'cosX) 



- - !»' fl + ~ a 2 ) £(P sin X + P'cos X), 



<a 2TO „ , . . . 



since %- = r a" nearly, see (Art. 9-) 



& » 



!<*-?> = _ TO '(i + ^a 2 )r,(P cosX-P' sin X), 

 de to 



- - ^-^ (1 + ^a 2 ) (P sinX + P cosX) (P cos X - I* sinX), 



to* TO 



„ _ Sm ^ (1 + ?% a .) | ( P *_ p'*) sin 2X + 2PP cos 2X} ; 



2t0 2 TO 



-d(&R) 9TO' 2 w 4 a 2 4m 



rra^^^n^ . „ 2 ) { (P 2 -P 2 ) sin 2X + 2PP' cos 2X}. 



V, cfe 8w 4 sin l v to' 7 l 



4to ,_ , to' 2 w 4 



1( >g (1 + — 7 « ) = 0-39076, log -— : — -, - 3-49819 ; 



to to* sin 1 



TO 



therefore by the values of log aP and log aP" in Art. (16), this part of 



$£- + 12"-169 sin2X - 3"-750 cos2X, 



and the corresponding part of 8£= - a 2 . $£ nearly, 



.-. $£m - 8"-877 sin 2X + 2"'736 cos 2X. 



42. Next let 



^ dR ly dR lyl dR 2 y dR t y, 



m = -d7^ + -dT^ + -dV^ + -d7^' 



where R 2 = to' (P 2 cos 2X - P 8 ' sin 2\), Art. (28). 



if 



