288 
[212 
A MEMOIR ON THE PROBLEM OF DISTURBED ELLIPTIC MOTION. 
da 
de 
and we thence obtain for the variations the new system of formulae, 
+ 
2 ein 
na dg 
dt, 
l—e- dD. Vl-e-dCl , 
H — dt r— -7— dt, 
na 2 e dg na 2 e civ.r 
dq =-^ d ^dt- 
" na da 
l-e-dO. 
na 2 e de ’ 
dvr — 
Vl — e 2 dfl 7 , 
+ o 7T 
na 2 e de 
d<$> = 
— cot<I> dfl , cosec <f> ein ,, 
na 2 Vl—e 2 dt m 2 Vl — e 1 d® 
(cos + sin S' sin fidd'), 
dt = 
d0 = 
cot <I> dCl 7j 
_j =-=- cti 
wa 2 Vl-e 2 
+ cosec <f> (sin S'clxp' — cos S' sin fidd'), 
cosec d> dfl 
H . 7 t ui 
?ia 2 V1 —e~ d<v 
+ cot ( h (sin S'dxf)' — cos S' sin field'), 
where, as before, il = il (a, e, g, vr, t, 0, <f>). This is the second form of the expressions 
for the variations of the elements. It is hardly necessary to remark, that if in either 
system of formulae we omit the terms involving eld' and def)', and in the place of 
2, ©, ( 1>, write cr, d, </>, we have the formulae for the variation of the elements when 
the orbit of the planet is referred to a fixed plane, and the disturbing function is 
given under the form il = H (r, p, cr, d, </>), or O = il (a, e, g, vr, a, d, </>). 
The demonstration of the two preceding systems forms, as before remarked, the 
object of the present Memoir. But it is proper to give also the systems for the 
variations of the elements in the form in which they would have been obtained, if 
the notion of the departure had not been introduced into the investigation. To do this 
I revert to a preceding system of equations, which may be written 
dr = 0, 
dd 
cosec (/> 
fa 
d$ ’ 
na 2 V l — e 2 
dz 
— COt (f) 
dfl j 
# dt - 
na 2 Vl — e 2 
7 nae sin f 
d J 
Vl -e 2 
= 
dn 7. 
7 - dt, 
dr