510 A THIRD MEMOIR ON THE PROBLEM OF DISTURBED ELLIPTIC MOTION. [218 
or, neglecting the periodic quantity cos (2v — 2v), and writing y for sin y cos y; also 
putting as usual ra = n' 2 a' 3 = m 2 n 2 a 3 , and r = a', we have 
and thence 
or 
d£l 3 
~^y ~ ~ m ‘ri r 2 . | y, 
8 = — m 2 n 2 r 2 .18y 
1 * d£l 
— o — = — f m 2 n- oy. 
r- ay i ° 
Substituting this value of - 8 and putting also r = a, — 0, ^ = 0, the 
v~“ ctdj at 
equation for 8y becomes 
dt 
d~8y 
dt 
y + n- (1 +| m-) 8y = — 2Bn + 
<BA 
dt 
III. 
To deduce the formula, seventh book of the Mécanique Céleste, I proceed as follows : 
Putting 
1 da 
we have 
d6' 
dt cos cf)' dt ’ 
A cos cf)' = ~ sin cf)' cos (v — a) — ~ cos cf>' sin (v — a), 
,/ da' 
, d(f)' 
B cos cf>' = sin cf)' sin (v — a')+ --A cos (fy' cos ( v _ 
which may be written 
/ • ct d 
A cos cf)' = — sin v ( s i n <f>' cos a ) + cos v (sin cf>' sin a'), 
dt 
d d 
B COS cf)' = cos V ^ (sin cf)' cos a') + sin v (sin cf)' sin a). 
Laplace in effect assumes that the variations of the ecliptic are given in the form 
sin cf)' sin a' = — %k sin (int + e), 
sin cf)' cos a' = 2& cos (int + e), 
(it + e is there written for the argument, n being assumed =1) where i, k, e are 
absolute constants, the quantities i being all very small in comparison with m 2 . Sub 
stituting these values, and putting cos cf>' equal to unity, we have 
A = — Sik cos (v + int + e), 
B = — 'lik sin (v + int + e);