38 
MR. LUBBOCK’S RESEARCHES 
cn 
= n / 1 - - m .i— b 3 , j 
l 4 [a, a, 9 ’ / 
= n { ] nearl y* 
This coincides with the first term of the expression, Math. Tracts, p. 59. 
i 
f , 2 m, a 3 1 
a = a < 1 —- • — > 
l 3[x, a, 3 J 
j , 2 ?», a 3 _ m, a 3 j . m l a 3 
1 + r 0 _ 3 jn a, 3 2 a, 3 _ 6 a a, 3 
a a a 
The equation for determining z gives 
<Uj 
dr ' r 3 
^!i + =o 
dr- r 3 ^ V dz j 
If s= ysin (gni + e — v) 
_g2 + i + 3 rQ + |!i.^ 6 30 — o 
r 0 = 3 ( g 0 - — b \ 
0 [A, 12a, 3 3 ’° 2 a* 3,1 J 
„ , , m, f 3 a 3 7 3 a 9 , . a 3 , 1 A 
s p 12 a, 3 3,0 2a, 2 3,1 2a, 3 3,0 / 
, m, f a 3 . 3 a 2 , "1 
g = 1 + — i — 3 ^3,o ~-r -i 6 S,I > 
[A* L a, 3 4 a,- ’ J 
« = n (1 — 2r 0 } 
*" = "{* + ir {$ Sj -° “ t $ b ’-'}} { 1 - ? {§ 5 ” - «7 s> -‘)} 
=-{ i+ f; $*»•,} 
= n (l + nearly. 
This also coincides with the first term of the expression, Math. Tracts, p. 59; 
and it appears that when the square of the disturbing force is neglected, the 
mean motion of the perihelium of a planet is retrograde and equal to the mean 
motion of its node taken with a contrary sign. 
The equations 
d v + 
r' 2 sin (A' — v) f ,, 0 , /d R\ \ /dR\ /dfi\ /ds\l n 
w^r- { (1 + s >(di) ~ r 8 (d?) - (<u-) (s?)) d * = 0 
