274 
• A MEMOIR ON THE PROBLEM OF DISTURBED ELLIPTIC MOTION. 
[212 
The equations 
r = a elqr (e, g), 
f= elta (e, g), 
treating all the quantities as variable, give 
1 — e" 
oesin/ 
J H*«/ 
da — a cos f de, 
(l-e\ 
to which is to be joined 
d/= (l-Mc°s/)» dg + sm/(2 + eeos/) de _ 
1 — e 2 
— e- 
= 0^(1-6») sin/ , . 1 
(1 + e cos/) 2 J 1 + e cos/ 
, a(2e — 1 + e 2 cos f) , 
da H—-Tz jrrz-~ de, 
(1 + e cos / ) 2 
all which formula? will be useful. 
If we treat the elements as constant, then in the foregoing expressions for dr 
and df, we must attend only to the part involving dg, and must put this equal to 
ndt; the values first obtained for dec, dg, dz, dv, correspond to this assumption, and 
we have 
dr _ nae sin/ 
dt ~ Vi _ e 2 ’ 
df _ na 2 VI — e 2 
dt r 2 ’ 
dz _ na 2 Vl — e 2 
dt f 2 
dec 
dt 
cos cf> sec 2 y 
and we then deduce 
dv 
= cos (f) sec- y 
dy . , 
— = sm © cos x 
dt 
d dr 
dt d,t 
no? V1 — e 2 
r 2 
3 
na? Vl — 
62 
ry* 2 
na? Vl — 
e 2 
r 2 
) 
na 3 e cos / 
r* 
d 
dt 
cl_ 
dt 
r 2 cos 2 y 
dy 
dt 
dv 
dt 
)=°, 
ii at 
dt \dt, 
= — cos 2 c/> sin y sec 3 y 
2n 2 a 3 e cos f 
rihd (1 — e 2 ) 
r* 
values which satisfy the undisturbed equations.