THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
739 
It will be found that <£(iV) performed on the first term is zero, as it ought to be 
according to the general principles of energy—for the system is a conservative one as 
far as regards these terms. 
Let 
W Q = {zSKVt' k'Y^ 
7! TK=PQp z +pq(P 2 -Q z )e N -PQq z e zir 
nW=pzQy+ 2 PQ(P*-Q z )p 3 qe N +[(P 2 -Q 2 ) 2 - 2 P 2 Q 2 ~]phfe ZN 
- 2PQ{P 2 -Q 2 )pq^ N +P 2 QYe iN 
ct ,2 k / 3 = the same with — N’ for N 
w 0 =r P^ P s + 4 P 2 Q\P 2 ~Q 2 Yfq 2 e N - N '+ [(P 3 - Q z ) 2 -2P*Q*]pYeW- N,) 
+ 4P 3 £ 3 (P 3 - Q 3 ) 2 pVe 3 ^“ if ' ) +P 4 ^ 8e4(M ' J > 2(e '~ £)+2h 
4 P 3 Q 3 (P 3 - Q 2 )Yq+2[(P 2 -Q 2 y-2P 0 ~Q z JpY 
+ 12P Z Q Z (P 2 -QYpqY 4P 4 Q 4 ^1 
„2h 
^W w « = V=! 
„2k 
#K=-27nl 4 - pi W2+ 16 - p2 « a (- pa —ew+c-f”-<?y-2P^]w 
+ 1GP 3 Q 3 (P 3 — Q 2 ) 2 pq 7 -{- iP^Q 4 - 
Adding and arranging the terms 
P 2h 
^(AT, e )w 0 =-^ ri ^{4P^ 4 (P 6 -2 6 )-4P 2 ^(P 3 -^) 3 {i> 3 -2 2 ) 3 
~W(p 3 -g 3 )[(P 3 -Q 3 ) 3 ~2P 3 Q 3 ] 3 } 
Then let 
&=l{2P*Q\p*-q*)-2P z Q z {P z -Q z )\p 2 -q z y 
~p 2 q 2 (p 2 -q 2 )\_(P 2 -Q z ) z -2P 2 Q 2 J} . . (59) 
and we have 
<}>(N, e)W 0 = — 2^H sin 2h sinji .(60) 
This is the last of the seven sets of terms. 
Then collecting results from (42-5-8, 51-4-7, 60), we have 
^tjf t = ^ 2 ^i F i sin 2f i+ 2 #F sin 2f—2# a F a sin 2f a +2<S* 1 G 1 sin gj 
+ 20G sin g~2^oGo sin g 2 —2$|H sin 2h} . (61) 
