508 A THIRD MEMOIR ON THE PROBLEM OF DISTURBED ELLIPTIC MOTION. [218 
II. 
Neglecting the terms which involve A 2 , AB, B 2 , we have 
21 = v ^— 2A — 2B sin y cos y , 
23 = ^(r 2 B sin y cos y) + r 2 (^B^ — A sin y cos y 
— r 2 (cos 2 y — sin 2 y) B 
dv 
dt 
we have 
and if we then neglect also the products of A and B by y and C ~ 
21 = 0, 
23 = 0, 
where it may be noticed that, in order to obtain this last value of (5, the only 
neglected term is a term containing B sin 2 y. 
Now, attending to the values of A and B, we have 
dA _ dA dv d'A 
dt dv dt dt 
jj dv d'A 
dt+ dt ’ 
where here and in the sequel ^ denotes differentiation in regard to t in so far only 
as it enters through the quantities a, 6', <j>, which determine the position of the 
variable ecliptic. 
Hence 
6 — + r 
dt 
„ f „ dv d'A\ „ „ 
2 - B -T7 + —T7- )-r 2 B 
dt 
dt J 
dv 
dv 
dt 
n . dr d A a oT> 
= 2Ar -j- + r 2 —j—— 2r 2 B 7 , , 
dt dt dt 
and, as above, 21 = 0, 23 = 0. 
Let r, v, y be the values obtained on the supposition that (5 = 0, and 
r + Sr, v + Sv, y + 8y, 
the accurate values; the first and second equations show that, neglecting the products 
d V 
dhy 
, we have 8r = 0, 8v = 0; so that the values of r and v 
of y and into 8y and ^ 
are not affected by the variation of the ecliptic. And then, substituting in the third 
equation y + 8y in the place of y, and for 
cos (y + 8y) sin (y + 8y), = cos y sin y +- (cos 2 y — sin 2 y) 8y,