THE ELEMENTS OP THE ORBIT OF A SATELLITE. 
761 
or 
dy 
dt 
Now let 
fkn 
cos 2 i 
t' 
\ 
\T 
cos y+i 
n C0S J 
r 
hrt 
t' 
«i = 
= T 
n 
h - 
tC 
b — 
r£ 
n 
u 2 — 
n 
And we have 
J 
( a i cos cosyd-a 3 cos j)y-\-cti cos j cos 2j.r/ ''j 
cJy 
= — (cq cos 2 i cos j-\-ci. 2 cos j)z —oq cos j cos 2 jX 
and by symmetry from the two latter of (110) 
dK 
dt 
dr] 
dt 
= (b 1 cos 2j cos i-\-b. 2 cos b x cos i cos 2 i.y 
= — (b 1 cos 2 j cos i-\-b 2 cos i)£ — b l cos i cos 2 i.z 
J 
( 112 ) 
y • • 
J 
i 
y • • 
(113) 
(114) 
These four simultaneous differential equations have to be solved. 
The a’s and b’s are constant, and if it were not for the cosines on the right the 
equations would be linear and easily soluble. 
It has already been assumed that i and j are not very large, hence it would require 
large variations of i and j to make considerable variations in the coefficients, I shall 
therefore substitute for i and j, as they occur explicitly, mean values i 0 and j 0 ; and 
this procedure will be justifiable unless it be found subsequently that i and j vary 
largely. 
Then let 
a=cq cos 2 i Q cos j 0 -\-a 2 Cos j 0 
a=cq cosj 0 cos 2j Q 
fi=b 1 cos 2 j 0 cos i 0 J rb 2 cos i 0 
b = 6 1 cos i 0 cos 2 i 0 
(Hereafter i andy will be treated as small and the cosines as unity.) 
Then 
dz_ 
dt 
dy 
dt 
% 
dt 
dr] 
dt 
«*/ + &V 
— az— a^ 
^+b y 
— fiC—hz 
> . 
(115) 
( 116 ) 
