IN PHYSICAL ASTRONOMY. 
373 
These examples will serve for the present to show how the development may 
be obtained from Table II. 
M. Damoiseau has given (Mem. sur la Theorie de la Lime, p. 348,) the ex¬ 
pression for a l -4 in terms of the true longitude. In order to obtain a com¬ 
parison of his results with those which may be obtained by the preceding 
method, it is necessary to transform his expressions, which may be done by 
Lagrange’s theorem, into series containing explicitly the mean longitude. 
If we suppose 
% = A 0 + A ! cos (2 A' — 2mA') + e A a cos (c A' — rar) + e A 3 cos (2 A' — 2 m A' — c A' + &) + &c. 
r 
s = B Ii6 y sin (g A' — v) + B ul y sin (2 A'— 2 m A — g A' + v) + B 14S ysin (2 A' — 2m A' +gA' — v) + &e. 
n t — A' + C x sin (2 A' — 2 m A') + e C a sin (c A' — m) + e C 3 sin (2 A — 2 m A' — c A' + ot) -f- &c. 
in which expressions A, B, c are the same quantities as in M. Damoiseau’s 
notation, the indices only being changed according to the remark, Phil. Trans. 
1830, p. 246, in order that Table II. may be applicable to the transformation 
required; A.' is called v, and & . -j, h u in the notation of M. Daivioiseau. 
~~A 0 + i (2—2 to) Ay C l + ~e°~A„C, + 1 (2 - 2 m - c) e*A s C 3 + 1 (2 - 2 m + c) e*A 4 C 4 
T h Z> L £ 
+ 1 e? A b C 5 + &c. 
+ | A x — -i- ce 2 ^ 2 C 3 + 1 ce^A^Ci — 1 (2 — 2m — c) e'-A^C^ + 1 (2 — 2m + c) e-A 4 C a 
— 1 me?A b C 6 + ~ m e"- A b C 7 — -I (2 - 3 m) e*A 6 C b + 1 (2 - m) e t °-A 7 C 5 J cos 21 
[ 1 ] 
+ |^ 9 + 1(2 - 2m) ,4, C 4 +l (2-2 m)A l C 3 + -1(2-2 m-c)A i C l 
-J- 1 (2 — 2 m + c) /4 4 C, | e cos x 
[2] 
+ { + 1 (2 — 2 m) C 2 + Cj} e cos (2 < — x) 
[3] 
+ | A,- 1 (2 — 2 m) A X C 0 _ - L C, } e cos (2 t + x) 
[43 
3 c 
MDCCCXXXII. 
