514 A THIRD MEMOIR ON THE PROBLEM OF DISTURBED ELLIPTIC MOTION. [218 
But 
s = sin cf)' sin (v — a) 4- s t 
ds 
dv 
sin (fV cos (v — a) + ~ sin cf)' sin (v — a) + ^ , 
fjz* rV fj rf' 
dv , = - sin <t>' sin ( v - o') + ^ sin <f>' cos (v -<r’) + dv dv sin f sin (v - a') + 
d' 
where ^ denotes differentiation in respect to v, in so far only as it enters through 
cf)' and a ^these are functions of t, which is = ^ . Hence 
d 2 s d 2 s / d' . d d' .... , 
i= i7 + ! ' + S sm * cos( ’-" ) + iS s 
but 
dv 
dv 
sin <6' sin (v — a') = - fsin v sin 6' cos a — cos v ~ sin ci>' sin <r' 
dv T n\dt dt T 
If. d 
d . 
— A cos <£' 
n 
and thence 
= -- A 
n 
d d' . „ 
1 dA 1 d'A 1 n 1 d'A 
dv dv 
sin <6' sin (v — a') = 7 7 = B — „ , , 
T n dv n dv n n 2 dt 
and similarly 
df . 
1 
dv 
d 
d . 
j- f sin cf>' cos (v — a')— - -jcos v ^ (sin cf>' cos a') + sin v sin <f>' sin a 
dt 
- B cos 6' 
n 
Hence 
-B. 
n 
d 2 s d 2 s 2 1 d'A 
„ + « = ;ri'+*, + " ~JT\ 
'* dv 2 n n 2 dt 
dv 
d 2 s _ 1 d‘ 2 s / 2 1 d’A 
dv 2 + S n 2 dt 2 +S '+n n 2 dt 5 
or, putting v = nt, 
SL %