16 
ON HANSENS LUNAR THEORY. 
[163 
Now NP = L — d, or (supposing, as before, that the differentiations relate to t, only in 
so far as it enters through the variable elements) d'P = — dd, and thence dd = 5* di; 
smi 
we have also d<£> = — cos i dd. The equations containing and ~ give 
ctJu aA 
cos i d na 2 V(1 — e 2 ) — na 2 V(1 — e 2 ) sin i di 
„ dCl 1 
= n 2 a 3 -jf dt, 
dJu 
dd 
cos 'P sin i d na?\f( 1 — e 2 ) + na 2 V(1 — (?) (cos \P cos i di + sin 'P sin i dd) = n 2 a 3 ^ dt; 
or, expressing dd by means of di and reducing, the second of these equations becomes 
cos i 
sin i d na 2 V(1 — (?) + na 2 \/(l — e 2 ) 
di 
n 2 a z jx 
cos 2 A cos 2 'P cos ~ dA 
and combining this with the first of the two equations, and observing that 
cos 2 i 
cos 2 A cos 2 Mf 
i rv 2 n — 
+ sin 2 1 
we find 
d na 2 yYl — (?) = n 2 a z ( 
V 
di 
cos i 
cos 2 A 
cos 2 'P ’ 
dLl . T di1 
-ttf + sin i cos t 7 . 
dL dA 
na* 
V( 1 - e 2 ) 
. . T di1 . dil 
— sin % COS 2 V + COS l COS T -J— 
dL dA 
dt, 
dt. 
Now, considering il as a function of r, d, i, d>, then A, L are given as functions of 
d, i, <f> by the equations sin A = sin i sin <f>, tan 'P = cos i tan <1>, 'P = L — d, and after 
some simple reductions, 
<m 
dr 
<m 
dd 
dil 
dH 
dr’ 
dil 
dL’ 
dll , , / . . „ , T , dil . , 
-rr = tan $ -smi cos 2 t —¡^ + cos i cos \P 
di \ dL 
dil\ 
dA) , 
dil 
d<P 
cos i dil . . _ dil\ 
TT Tf d” Sin I COS x -j-jT ) j 
cos 2 A dL dA r 
whence also