46 
MR. LUBBOCK’S RESEARCHES 
| cos (t — z) + A a a t e- e t cos (t + 2 x — z) 
- a a t e ef cos (< — x — 2 2 ) + a a i e e t - cos (t + x — 2 2 ) + e, 3 cos (( - 3 z) 
8 8 3 
— ^ e e t - cos (f — x + 2 2 ) + ~ cos (t + x + 2 2 ) + ^ cos (f + 3 z) 
— 3aa t e sin 2 cos (t +x — 2 y) + a«,e sin 2 -h- cos (t — x — 2y) 
— w 
— 3 aa t e l sin 2 -k- cos (< + 2 — 2 ?/) +aa ( e j sin 2 -X cos (t — 2 — 2 y) 
w ^ 
a 2 c 3 
e / 
2/. 3 
+ -—p- cos 2 — a< e ‘ cos 3 2 
4 4 
} 
{ 3,0 + 6 3> , cos t + b 3)2 cos 2 £ + &c. 
+ terms independent of b. 
Multiplying out, the coefficient of each term may be put in terms of bs,i _ 2 , 
bs,i - 1, b 5 t i, bs,i 4.1 and bs,i + 2. 
The quantities h, 0, bi,i, from which all the other quantities b 3 , b 5 , &c. 
depend, may be obtained at once from Table IX. in the Exercices de Calc. In¬ 
tegral, by M. Legendre, vol. iii. See also vol. i. p. 171 . of the same work. 
the integrals being taken from <p = 0 to <p = t. 
A = ^ (1 — c- sin 2 <p) 
c2 = , 4 . fla/ - v , = 7 . a * beir) §' = — as in the notation of the Mec. C6L 
(a + aM (1 + a) 2 a, 
6m = 
it (1 + a) 
F> 
i, , = --- { — (F 1 — E>) - F> 1 
1,1 (l + a) 1 c 2 v ; / 
In the theory of Jupiter disturbed by Saturn, u, = *54531725 ; and hence in 
this instance if c = sin 6, 0 = 72° 53' 17". 
By interpolation, I find from Table IX. p. 424, 
F(72° 53' 18") = 2-6460986 
* p in the notation of Woodhouse’s Astronomy, vol. iii. p. 287. 
