218] A THIRD MEMOIR ON THE PROBLEM OF DISTURBED ELLIPTIC MOTION. 
507 
To reduce these, we have in the first place 
dT . dT 
dr 
dT 
dr r 
dT 
dv 
dT 
= r(Q 2 + R 2 ), 
dQ . n dR\ 
=r- R cos y, ^ = r>(Q d £ + R d “)=^ (- QB - RA sin y), 
2 n dT n -pdR 
= - r 2Q = r 2 R -j- 
dy dy dy 
The equations are thus reduced to 
d 2 r ~ n 2 a? 
r( Q. +Ä)+ 
= r 2 R(— sin y .V — B cos y). 
dQ 
dr ’ 
(r 2 cos y. R) +r 2 (QB + RA sin y) — ^ , 
dQ 
) +^R{Any^ + B eoB y) = 
and then substituting for Q, R their values, viz. 
dy 
Q = 
dt 
dv 
+ A, 
R = cosy ^ -Bsin y, 
we 
find 
<Pr 
dt 2 
d 
dt 
dv''' 
r B, Hs 
+ 
n 2 a 3 dQ „ 
+ r> =* +ä ’ 
(r 2 
dv 
r 2 C0Vy Jt 
dQ 
dv 
+ 33, 
'(-S) 
dt 
where 
+ r 2 cos y sin y 
{dv A 2 
\dt) 
dQ ~ 
=W +6 ' 
2f = r (— ZA — 2B sin y cos + A 2 + B 2 sin 2 y) , 
dt 
dt 
23 = (r 2 B sin y cos y) + r 2 — A sin y cos y — — AB cos 2 y ^ , 
dv 
+ r 2 (— (cos 2 y — sin 2 y) B^+B 2 sin y cos y j, 
in which 
a / r\ • # / dO . . #. dcp 
A = cos (v — <7 ) sin cp + sm (v — cr ) , 
-r. • / • , / dO' . dtp' 
B = sm (v — a) sm (p + cos (v — a ) ; 
O', a, <p' being given functions of t such that da — cos cp' dO' = 0. 
The foregoing equations of motion are rigorously accurate. 
64—2