[163 
163] 
ON HANSENS LUNAR THEORY. 
15 
theory of 
the base, 
Hence also 
d fdr 
dt \dt 
d „ A dL 
-77 r 2 COS“ A -77 
dt \ dt 
d ( „dA\ 
n 2 a 3 e cos / 
= 0, 
dt dt 
d fdf 
dt \dt 
cos 2 i sin A n 2 a 4 (1 — e 2 ) 
cos 3 A r 2 
2 n 2 a 3 . „ 
—e sin /. 
q-><j ** 
The foregoing values show that the equations of motion, neglecting the terms which 
involve the disturbing functions, are satisfied by the elliptic values of r, L, A: and 
in order to satisfy the actual equations of motion, we have only to consider the 
elements as variable and to write 
dr 
dL 
dA 
7 dr 
d jt 
d (r 1 cos 2 A 
dA 
= 0, 
= 0, 
= 0, 
rrcr 
dñ 
dr 
dt) n a dL 
dL 
n-a° 
dCl 
d ( r ' dt) — " dA 
dt, 
dt, 
dt, 
where the differentiations relate only to the elements, or, what is the same thing, to t 
in so far only as it enters through the variable elements: the system is at once 
transformed into 
dr 
= 0, 
dL 
= o, 
dA 
= 0, 
j nae sin f 
V(i-e 2 ) 
= n 2 a 3 ^ dt, 
dr 
d na 2 \/(l — e 2 ) cos i 
— n 2 a 3 dt, 
dL 
dil 
d na? V(1 — e 2 ) sin i cos 'F = n 2 a 3 
dt.