IN PHYSICAL ASTRONOMY. 
291 
, J o 3m,an R.t, 3 \ nee* . , n . - . , _ . 
+ r,„ --- (2 '„_ 5a|) ) s,n (2 „ < - 5 »,< + w + 2 -,) 
fo ■ 3m,an R 0 .^\ Kes ' n 2 . , 0 , s , , 0 . 
+ 1 2 r 273 + r i33 — — i f 777 t — r sin (2 n t — 5 n,t + ® + 2 v ( ) 
L i^(2 n— on,)J (2 re — 5re ( ) 
r o p -v re e.sin 2 -4- 
+ < 2 r 3 i -2 ~ r ” ,gW - 31 s > To e — 7 sin (2 n « - 5 n/ + + 2 v ( ) 
L ft(2n- 5n ( )J (2 re — 5n ( ) 
The coefficients of the terms in the development of R multiplied by the cubes 
of the eccentricities, as regards the quantities b 3 and b 7 , (they also contain the 
quantities b 3 ,) may be found by changing b 3 into b 5 , in the terms in R multi- 
plied by the eccentricities, and multiplying the result by 
_ 9 (a ? e ? + a* e *) 3 a* e 2 cos2x _l ± ( e * + e* + 2 sin 2 -h-^cos t + .£ JL e *cos (t + 2x) 
8 a* 8 a* 4 a t \ 2 / 16 a t 
[0] [50] [1] [61] 
9 a 
3 a 
3 a 
27 a 
— — c — e - cos (t — 2z) + — — e 2 cos (t — 2x) + -r — e^cos {t + 2z) + — — ee,cos (t - x + z) 
1 o cl. 1 o 4 o a t 
[111] [51] [121] [91]' 
— ~ — ee. cos (t + x + z) — — ee,cos {t — x — z) + — ee,cos (t + x — z) 
8 a. 8 a, 8 a t 
[81] [71] [101] 
+ YT sin * Y cos (* + 2 ^) + f - e / 2cos 2 2 
[141] [110] 
and changing b h into b 7 , in the terms in R multiplied by the squares and pro- 
ducts of the eccentricities, and multiplying the result by 
— and — — e cos x + — e cos (t — x) + — e cos (t + 2) — — e cos (t + x) 
6 a 2 a, a i a i 
[10] [11] [41] [21] 
— — e ( cos (t — z) —2e t cos z 
[31] [30] 
and changing b 3 into b 5 in the terms in R multiplied by the squares and pro- 
ducts of the eccentricities, and multiplying the result by — — and the same 
quantity. 
Thus R 155 results from the combination of the arguments 
51 x 14, 50 x 15, 61 x 16, 10 x 55, and 11 X 54. 
