THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
735 
Then 
<f>(N)wi=% p-^^A n P s ~^Q^p 7 ~ %l q^~ l e~ 2tl 
^( e )w I ==-2^^A w P 8 -^Q>7-^+J e -2 f 1 
Therefore by addition 
/>—2f x 
(/j(A r , e)Wj= 2[_n(pr -f- rf) — 4g 2 ]P 8-2K Q 2H » 7_2? 'o' 2 "~ 1 -—7== 
3\/ —1 
Now when 
= 0, A* 
- 1 
1 — 4> 
?i(y> 3 +r/ 3 ) — 4y 3 = — 4y 3 
= 1 , 
— 4, 
CO 
1 
Cl 
II 
= 2, 
= 9, 
= 2 (y> 3 —y 3 ) 
= 3, 
— 4, 
= 3 p 2 -q 2 
— 4, 
— 1 
4> 
CQ 
II 
If we had tahen the second teim of "W*j we should have ha>d the same coefficients 
hut multiplied by — e 3fl /2\/ — 1 instead of by e -3fl /2\/ — 1. Therefore, since 
(«a-^)/ 2 y“l= sin 2t; 
<£(IV) e)Wj= — F : sin 2fj[— P B q) 7 q -f- 4 P Q Q 2 p 5 q ( pr — 3 g 3 ) -}-1 8 P’Q l jrq } (yr— q~) 
Then let 
+ 4 P-C/'pxf^yp 1 — (f) + Q s pp] 
4? x =^[P 8 ^ 6 — 4P 6 () 3 p 4 (y> 3 —3y 3 ) — 1 8P i 'Q i p‘ 2 q 2 (p 2 —cf) — 4P 2 t/ J g 4 (3p 2 —y 3 ) — 0 8 y G ] .(41) 
and remembering that 2 pq— sin^, we have 
<£(iV, e)W I =2dF’ 1 F 1 sin 2f x siny.(42) 
Sidereal semi-diurnal terms (2n). 
W n = F[ra 3 K 2 zrr / 'V 3 C“ 31 -j- cAc 3 ®/ 3 ^ 3 ^]] 
Here the epoch is wanting, so that <£(2V, e) = <£(iV). 
5 b 2 
( 43 ) 
