330 
MR. LUBBOCK’S RESEARCHES 
x . l 
CO.. (X, ) ^ -{- cos 3 » tan 9 (A ( — v))i 
r r ( ' sin ( X — X]) = r r' t {sin X cos X] — cos X sin X y } 
cos X' = cos (X ; v — v) cos v — sin ( X , — v) sin v 
\ (Ay — v) cos v — sin (Ay — v) sin v ^1 — 2 sin 3 
COS I 
sm v 
(1 — sin 3 1 sin 3 (Ay — v)^ 
cos Ay + 2 sin 3 — sin (Ay — v) sin v 
JU 
(l — sin 3 1 sin 3 (A ; — v))i 
sin Xy' = sin (X ; ' — v) cos v + cos (X y ' — v) sin v 
sin (Ay — v) cos v ^1 — 2 sin 3 + cos (Ay — v) si 
(l — sin 3 « sin 3 (Ay — v))+ 
sin Py S P' = sin i sin (X ; — v), r] = r l cos Py S P,' = r { ( 1 — sin 2 1 sin 2 (X ; — p)) 1 
therefore, 
r r ; ' sin (X — X]) = r r t | sin (X — X y ) + 2 sin 2 ~ sin (X ; — v) cos (X — v) J 
= r Ty | cos 2 y — + sin 2 s* 11 (*• + X, — 2 v) j- 
similarly it may be shown that 
r r] cos (X — Xy') = r r l | cos 2 •— cos (X — X y ) + sin 2 cos (X + X ; — 2 v) j- 
s p _ g/(l - e ?) _ a i( l - g / 9 ) 
1 1 + Cy COS (PySN+N8L — OTy) 1 + Cy COS (Ay — OTy) 
d 3 or ftx /(LR\ /d^\ _ d 3 * pz /d.RA _ 
T? + r* + \dx) ~ °’ r 3 + Vdj// “ d^ 3+ ^ + Vdlj “ 0 
r * d a' * + d r' 9 ( 1 +S 3 ) + 2 r' s d ?•' d s -f r' 3 d s 3 2 jtt 
+ ^- + 2/dft = 0 
d^ r'(l + s 3 )’ 
d /£ being the differential of Ii with regard only to the co-ordinates of the 
planet m. 
+ ( d a?)=° 
d* r ( 1 + s ; ) — r d a' 3 + 2 s d r d s -f r s d 3 s p 
d/ 3 
d. 
r' 3 d A' 
r'(l + s 3 )* 
/ d R\ 
+ U) = ° 
_At_ 
d t 
r s d' r + 2 r d r d s + r' 3 d 3 s 
d (* 
+ - 
i H~ 
(1 + s'J-f V ds / 
