276 A MEMOIR ON THE PROBLEM OF DISTURBED ELLIPTIC MOTION, 
and attending to the equations cos 2 <£ sec' 2 y + sin 2 cf> cos 2 x — 1 we deduce at once 
d na? V1 — e 2 = 
1 
[212 
. dil 7 . . dCl j 
cos © sec 2 y , a£ 4- sm © cos a? -y- at, 
^ J dv dy 
dcf) 
na 
2 Vl - e 2 
— sin cf> cos 2 ¿c ^ di + cos </> cos x dt). 
Now the position of the planet may be determined by the quantities r, z, 6, cf>, or 
we may consider Cl as a function of the last-mentioned quantities. And if on the 
right-hand side fl = d (r, v, y) as before, the formulae of transformation are 
dCl 7 dCl 7 dCl 7 dCl-. dClj dCL 1Q dCl , 
-f- dr + - 7 - dv + — 7 — a?/ = —- dr+-r-dz+ dd + yy- a© 
ar aw ay J dr dz dd dcp 
where 
and we have 
dv — cos cf> sec 2 y dz — tan z cos 2 x sin cf> dcf> + dd, 
dy — sin cf) cos x dz + tan 2 cos x cos cf> d<f>, 
dCl 
dCl 
dr 
~ dr ’ 
dCl 
dCl 
dd 
~ dv ’ 
dCl 
/ 
— sin (f) COS 2 X 
dCl 
+ cos </> 
dCl\ 
dcf) 
= tan z 1 
dv 
cos x 7 — 
dy) 
dCl 
^ cos cf) sec 2 y 
dCl 
+ sin (f) 
dCl\ 
dz 
dv 
COS X y— 
dy) 
where on the left-hand side il = if (r, z, 6, cf>); and these equations give 
dCl ^ dQ. ,dCl 
co t^ = co t^- c --c°s ec « 3?> 
an equation which is satisfied by O = il (r, z, 0, cf>). We have thus 
dr = 0, 
dv = 0, 
dy = 0, 
7 nae sin f 
d .... ./ = 
V1 — e 2 
d na 2 V1 — e 2 = 
d<f> = 
na‘ 
m 
dr 
<m 
dr 
cin 
Vl - e 2 # 
cot z 
dt, 
clt, 
dt,