IN PHYSICAL ASTRONOMY. 
297 
-~b, 
u* 
4 a t - " 3,i — 1 2 a, 3 3>l 
+ 2 + - n / 2 ( r* r A — ( - 
+ -i, ,»-»,) + »! V 
+ £?**■•+ 0 + p. (7- »;r f *w } e sin (*(•“-*.') + »'- «) 
„ n fo* /3a 2 . a, 
r2 i(* -*,)+»,{. ^ ^ 1 •» V 3 ’ i - 1 2«, 3>i 
- Ai s 3,i+i) e . sin (*(»*-», o + »,<-»,) 
If « > a ( , and 
1 
/ 1 — ^ cos 9 + ^-1 2 = i 6, o + 6, , cos 9 + 2 cos 2 9 + &c. 
La a 2 J ’ 
3 
/ 1 — — cos 9 + = i 6 3 0 + 5 3 ! cos 9 4- b 3 2 cos 2 9 + &c. 
La a- J ’ ‘ ’ 
the value of R may be easily inferred from the value which it has in the former 
case. Considering only the terms multiplied by the eccentricities 
( d R\ 3 m. a , , s , m. a . . , 
- — ) — 1 — e cos (nt — rar ) + — i — e cos (2 nt — n.t — w) 
dr J 2 a* v ^ 2 a* y 1 ’ 
+ V — e , cos (nt — 2n,t + &,) 
fli 
+ w , 2 (- — ^-K • T + - 1 + 2i) b„ ■ 
J L 4 a- 3,i — 1 dr 2 a 3,! 
- T Jr 6 s,i + i} ecos (* + 
^ / 3 (1 + i) e, , , ia ; 2 , 
+ 4 — ^5 1 + ^r 6 3,i 
+ (1 J ?) 5 b 3,i+ 1 } e / cos (*(«<“ V) + »i * - 
All these expressions are to a certain extent arbitrary, on account of the 
equation which connects b 3>i _ 13 b 3>i , and b 3>i + 1 
(2 i + 1) a ^ i (a 2 + a, 2 ) ^ (2 i — 1 ) a 
2 a, 3 >* + 1 
2 a, 3 ’ ? “ 1 
t r* being the coefficient of the cosine of the same argument in the expression for and excluding 
the case of i = 0. 
2 Q 2 
