336 
MR. LUBBOCK’S RESEARCHES 
<z r 3 /d , .a 2 e . \ a t> a 
+ T \dT/ d y + T sin (u “ a ) d R = ° 
i 2a 2 
da = d it 
V- 
d e' = {2 e' — 2 cos (y — a) — e' sin (y — a) 2 } d R — sin (y — a) (^~) d y 
e d a = — sin (y — a) {e 1 cos (v — a) — - 2} d i? + ~ cos (v — a) d y 
J * ti d f + e - vs — v — a — e' sin (y — a) 
d g— d ns = — d a — sin (y — a) d e' + e' cos (y — a) d a. 
The equation of condition which obtains between the constants a, e, ns, v, i 
and h may be found from the equation 
dc 
_ la + JL = 0 
r a 
dr 2 h 2 2 u. u, 
d tf 2 1 r 2 cos r r ' a 
which gives 
^ cos_»_ I p ^ g . n2 ^ ^ { i q_ e cos („_sr)} {1 _ e CO s (i> — w )}| , + ~ = 0 
hr 
Equating the values of r which have been found, 
= a { 1 — e 1 cos (y — a) } 
7 i 2 v / 1 + s 2 
/x cos » 2 { 1 + s 2 4 - e cos (A — ct) } 
since the origin of the angle v is arbitrary, we may suppose X — nr = 0, and 
v — a = 0 at the same time, 
so that 
1? */ 1 + tan j 2 sin (-cr — v ) 2 
= a (1 — e') 
fj. cos » 2 { y'' 1 + tan » 2 sin (ot — v) 2 + e} 
All the equations which have hitherto been proved are rigorously true, 
whatever powers of the disturbing function be retained. They are susceptible 
of simplification when the square and higher powers of the disturbing function 
are neglected : in this case, if the orbit be supposed to coincide with the 
plane x ?/, tan / = 0, and if the longitude be reckoned from the perihelion of 
the planet P, 
h 2 d e + {2 cos X + e + e cos X 2 } r 2 ( jyj d X d — — \~^) d X = 0 
