46 
MR. LUBBOCK’S RESEARCHES 
— a a t | e$ + e ~ t j- cos (f — z) + a a t e- e t cos (t -f 2 x — z ) 
a a t e e t - cos (< — x — 2 z) + — a a t ee t - cos (t + x — 2 z) + — e ; 3 cos (( - 3 z) 
8 8 3 
— e e, 2 cos (t — x -f 2 z) + cos (t + x -f 2 2 ) + cos (i + 3 z) 
8 8 12 
— 3 aa t e sin 2 -fi- cos (i + x — 2 y) -f a a ; e sin 2 -h- cos (t — x — ‘l y) 
A a 
— 3 a a, e, sin 2 -k- cos (< + 2 — 2 z/) + a a ; e i sin 2 -k cos (t — z — 2y) 
A A 
+ 
a? e, 
cos z — 
are: 
cos 3 2 
f b 
1 W 3,0 + £>3,1 COSf + b 3i2 cos 2 1 + &c. 
+ terms independent of b. 
Multiplying- out, the coefficient of each term may be put in terms of 65,1 _ 2 , 
bs,i — l) b^ t i, b$ } i -j_ 1 and b^,i -f- 2 - 
The quantities fa, 0 , fa,\, 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. 
( 1 + €*«■ “ 4 / 4 * 
the integrals being taken from <p = 0 to <p = or. 
A = v' (1 — c 2 sin 2 ^) 
^ — 4 a a ‘ ~ 4 a a * bein §' = — as in the notation of the M£c. C£L 
* 1.0 = 
(a + a y ) 2 (1 + a)® 
4 
(1 + a) 
2 F‘ b ltl = 
it (1 + a) 
{|(F'-E')-p} 
In the theory of Jupiter disturbed by Saturn, a = -54531725 ; and hence in 
this instance if c = sin 0, 0 = 72° 53' 17 k 
By interpolation, I find from Table IX. p. 424, 
F (72° 53' 18") = 2-6460986 
* p in the notation of Woodiiouse’s Astronomy, vol. iii. p. 287. 
