THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
841 
® 1 =^cos ( k0-\-/3) = —k cos (k—l)-\-k cos (/c+1) 
® 3 =yqcos (k0-\-/3) = (k 2 —%k) cos (k— 2)— 2k 2 cos /j+(^ 2 +|^) cos (k-\-2 ) 
r 73 
0 3 =z^ cos (k9+/3) = -(F-^+^k) cos (£-3) + 3(F-fP+i&) cos (k- 1) 
— 3(A. ,J -f- cos (k 1 )d - {k" + —jrk - -±-k) cos (&fi-3) > 
@ 1 =A cos (M+/3) = (^-^+|f^+13F-^) cos (k- 4) 
-(4^-15F+16^ 2 -^) cos (£—2) 
+ 3(2& 4 -^P+2F) cos (&) — (4^+15P+16F+^) cos (£+2) 
+(^+¥^ 3 +^ 2 + 1 3F+^^) cos (£+4) 
(268) 
where the 0’s are merely introduced as an abbreviation. 
Then by Maclaurin’s theorem 
cos (k9-\-/3) = cos (/M2-f /3)-f e© 1 +^e 2 ® 2 +^e 3 © 3 + 2 i 4 e 4 ©, i . • • • (‘269) 
In order to obtain further rules of approximation we will now run through the future 
developments, merely paying attention to the order of the coefficients and to the factors 
by which S2t-\-e will be multiplied in the results. From this point of view w T e may 
write 
<j>(a) = (e°) cos (2d) + (e)[cos (3d) + cos (d)]+(e 2 )[cos (4 0)-\- cos (0)] 
+ (e 3 )[cos (5d) + cos (d)] 
^■(a) = R=(e°) cos (0) + (e) cos (d) + (e 2 ) cos (2d) + (e 3 ) cos (3d) 
The cosines of the multiples of d have now to be found by the theorem (269) and 
substituted in the above equations. 
In making the developments the following abridged notation is adopted ; a term of 
the form cos [(^+s)/2+/3—Sar] is written {&+«}. 
Consider the series for ffi(a) first. 
We have by successive applications of (269) with k— 1, 2, 3, 4, 5. 
