212] 
A MEMOIR ON THE PROBLEM OF DISTURBED ELLIPTIC MOTION. 287 
(cos S' d<f>' + sin S' sin d6'), 
7 , — cot c70 7 , cosec <ï> dil 7 , 
c№ = 7 tv, dt , . -TT^dt- 
,2 Vi - e 2 dZ 
na 
d Ç - cot<1> * 
na' 2 V1 — e 2 
7 „ cosec <1> rfil 7l 
d© = — : jxdi 
wa 2 Vl — e 2 
na 
,2 VI - e 2 d© 
+ cosec $ (sin >8' d<f>' — cos /S' sin <// ¿20'), 
+ cot <3> (sin $' dcf)' — cos S' sin <f>' d6'), 
where, as before, il = il (r, p>, X, ©, <T). This is the first form of the expressions for 
the variations of the elements. 
But we may in the disturbing function il replace r, }), by their values in terms 
of a, e, g, and if on the right-hand side il has the last preceding value, we have 
m 
da 
di1 
di1 
dt1 
dil 
diI- 
da ' + U de+ + + + + 
cZct 
d© 
dil 
cZil 7 dil 7l dil 7V , dil 7 „ ¿il 7 , 
= —T- dr + “37 db + 7V dX -f 373 d© + 7 , d<J>, 
dr d]> J dX d© d4> 
where on the left-hand side il = il (a, e, g, w, X, ©, <1>); and the expressions for the 
differentials dr and d]?, are 
7 1— e 2 7 j., , aesin/, 
dr = — >aa—acos/ae+ ^ r■ dg, 
1 + e cos / 
and we have therefore 
(2 + e cos/) sin/ 
1 — e 2 
dil 
1 - e 2 dil 
da 
1 + e cos/ dr ’ 
dil _ 
de 
,dil 
— a cos/ + 
II 
G 
•ts re 
ae sin/ dil 
Vl _ e 2 dr 
dil 
dil 
cZ-zzr 
d]> ’ 
dil 
dil 
d®~ 
d<S>’ 
di1 
dil 
dX ” 
dX ’ 
tZil 
dil 
d©~ 
d© 5 
Vl— 
+ ecos f) 
(i-O* 
1 — e 2 iZ}? 5 
(1 + e cos/) 2 dil 
~(Ï - df ty ’ 
where on the left-hand side il = il (a, e, î/, w, X, ©, < T),