212] A MEMOIR ON THE PROBLEM OF DISTURBED ELLIPTIC MOTION. 
289 
/ dCl 7, 
d na 2 v 1 — e 2 = ^7 dt, 
,, — cosec 6 £¿12 7 , cot <6 £¿12 7 / 
dip = ;— r ^ at + ——-j - dt = 
rca 2 vl - e 2 ?ia 2 VI - e 2 dz \ 
na 
cot z £¿12 7 . 
-rrdt), 
,2 Vl - e 2 d(p 
where 12 = 12 (r, 2, 0, 0). 
a — e 2 ) 
Substituting in these equations for r, 0, the values — y and <D +/ we 
obtain 
da = 
tie — 
2e sin/ £¿12 2 (1 + e cos/) 2 £¿12 7 
—-j—dt + — V- -y- ¿i, 
w v 1 - e 2 m (1 - e 2 ) # 
Vl 
e 2 sin f dil e + 2 cos /+ e cos 2 / cZO ^ 
dr no? Vl -e 2 ^ ’ 
^ _ (1 — e 2 ) (— 2e 4- cos/4- e cos 2 /) £¿12 ^ (2 + e cos/) sin/ (¿12 ^ 
nae (1 4- e cos /) c?r %a 2 e dz ’ 
Vl — e 2 cos/ <ifl ^ i (2 + e cos/) sin/ £¿12 
~!T—T, ~dz 
riE = - - * v waj ~ dt + K ~ ' " ~:J-L ~ dt ^¿=~dt, 
dr m 2 V 1-e 2 " 
wae 
na 
2 Vl - e 2 d(f) 
dcf) = 
£¿6» = 
cot (f) dil 7 cosec <f> £¿12 
?ia 2 Vl — e 2 ^ wa 2 Vl — e 2 d6 
cosec <jb c?il ^ 
na 
2 Vl — e 2 £¿0 
where 12 = 12 (r, 2, 0, </>), as before; and which is the first system for the variations 
of the six elements, a, e, g, E, 6, <f). 
But if in the disturbing function we replace r, z by their values, then if on 
the left-hand side 12 = 12 (r, z, 6, $), as before, we find 
1 — e 2 £¿12 
£¿0 
dr ’ 
1 + e cos / dr 
r dil (2 4- e cos/) sin f di2 _ £¿12 
~ acos f + rVe 2 
(1 4- e cos/) 2 £¿12 _ £¿12 
ae sin / £¿12 
Vl -Z 2 dr 
di2 
dcf) 
di2 
dz 
(¿12 
dd 
(1 - e 2 / dz dg’ 
_dil 
d(f) 
_ £¿12 
£¿^3 ’ 
_ £¿12 
“(¿0 ’ 
where on the right-hand side 12 = 11 (a, e, g, E, 0, </>). 
C. III. 
37