246 
MR. LUBBOCK’S RESEARCHES 
Table II. may be used in forming the developments required in the method 
employed by MM. Laplace and Damoiseau ; for this purpose it is only ne- 
cessary to make t = instead of n t — nf 
X — C X' — w . . . cnt — n r 
Z = C ; X, — ST, . . . 7ST l 
and y — g X' — v . . . gnt — v 
The notation throughout is the same as that used Phil. Trans. 1830, p. 328, 
with the exception of the indices of the arguments. 
In the elliptic movement ; 
i 5 r 5 = 1 + 5 e c - 
(j + ^ye 2 ^ + 5e^l + -jj- e 2 ^ cos x + 10e°-^l + ^ e 2 ^ 
cos2x 
1 At. 74s , 
+ — - e n cos 3 x 4- — - e 4 cos 4 x 
~ 8 48 
a 4 r~ 4 = 1 4 - 3 e 2 4 - 4 ecosx 4 - 7 e 2 cos 2 x 
a 3 r~ 3 = 1 + + -|e 2 ^ + 3e^l + |r e 2 ^ cos x + ~ e 2 ^1 + ^ e 2 ^cos 2 x 
r.o 77 
-l _ e 3 cos 3 x + — e 4 cos 4 x 
8 8 
a 2 r~ 2 = 1 + (l + | e 2 ) + 2 e (l + y e 2 ) cos * + e 2 (l + ^ e 2 ) cos2 x 
1 o i 
+ _ e 3 cos 3 x + — - e 4 cos 4 x 
^ 4 24 
( e 9 \ / e 2 \ 9 4 
1 — -g j cos x + e- ( 1 — -g j cos 2 x + g- e 3 cos 3 x+ -g-« cos 4 x 
_L — 1 cosx — ~ (\ — cos 2 x — cos 3 x — L C os 4 x 
a 2 \ 8 / l \ S / 8 3 
_ 1 _j_ — 2 e ^1 — cos x — ^ cos 2 x — L cos 3 x — e - cos 4 x 
= 1 + 3 e 2 ^ 1 + - 3 e ^ 1 + A cos x - A e* cos 2 x + e i cos 3 x + L-cos 4 x 
l! = l 4 . 5 e 2 — 4 e cos x + e- cos 2 x 
a 4 
a 
- = r 0 
r 
+ r, cos 2 < 
+ e t\j cos x 
4- CTj cos (2 2 — x) 
4 - <? r, cos (2 t 4 - x) 
4 - e, r, cos z 
4- e t r 0 cos (2 / — 2 ) 4- &c. &c* 
