i 
284 
A MEMOIR ON THE PROBLEM OF DISTURBED ELLIPTIC MOTION. 
[212 
d na? V1 — e 2 = dt, 
a\> 
dd> = t= (sin >8 ~rj — cos >8 (cot cf) ( 7 +coseccf) i Jn ) )dt— (sin S'd(f>'+sin S'sin cf)'dd'), 
no 2 V1 — e 2 \ # v do- ddjj 
dH = — C °( ^ (cos 4-sin $ ["cot </> ^-fcosec </> dt+cosec <i> (sin S'dcf>'—cos S'sin ft dd'), 
d© = 
na? V1 — e 2 
cosec d> 
— ^cos >8 ^ +sin $ ^cot <£ + cosec c ^ + cot^sinß'd^'—cos$'sin0 , d0 / ), 
m* Vl — e : 
where il = il (r, ]?, a, d, 0), as before. 
But il may be expressed in the form il = il (r, ]>, 2, 0, d>, a', 6', </>'), or dis 
regarding a', 6', (f)', in the form il = il (r, ]>, 2, ©, d>), and to effect the transformation 
of the differential coefficients we must write, 
dd> = cos S d<j) + sin S sin $ dd, 
d® = — cosec 4> (sin S d<f> — cos S sin <fi dd), 
d2 = (da — cos $ dd) — cot d> (sin Sdcf) — cos S sin (f> dd), 
or, what is the same thing, 
d(f> = cos S d(f) — sin S sin d? d®, 
dd = cosec (f) (sin S d<f> + cos S sin d> d®), 
da = d2 — cos d 5 d® + cot <£ (sin S dd> + cos S sin d> d®), 
and substituting in 
dfl 7 dil 7 dil 7 dil 7/ , dil 7 . 
dr + -dp d * + <& daJr Cie de+ cl4 d4, 
dr 
dil 
dr 
dil 7 dil 7l dil 7N , dil 7 ~ dil 7 * 
= ^ + f g®+m dS + 
l the 
right-hand side 
il = il(r, J?, cr, 
d, </>) as 
before, then we 
have 
dil _ 
dil 
dr 
dr ’ 
dil _ 
dil 
d\> 
dj? ’ 
dil 
dil 
d2 ~ 
da ’ 
dil 
(— cos 4> + cot </> 
^ . dil 
+ cosec cf> 
. dil 
. ~ . , dil 
d® ” 
cos $ sin d>) ^ 
cos o sm d> — 
dd 
sm o sm d> j— , 
Clip 
dil 
, , ■ n dil 
. . ~dil 
c< dil 
d2 — 
cot <6 sm o 
da 
+ cosec (p sm o 4- 
COS S -=-r , 
d<£ 
where on the left-hand side il = il (r, J?, 2, ®, d>).