483] 
EQUATIONS IN THE LUNAR THEORY. 
539 
Q 1 = km 2 (|- + f cos 2D), 
Q. 2 = km? {3^ sin 2D + pi (| + f cos 2D)}, 
Q s = km 2 {— Sv 2 sin 2D — 3wj 2 cos 2D 
4- PtV x . 3 sin 2D 
+ PÁ2+ f cos 2D)}, 
&c. 
P 1 —jm 2 (— f sin 2D), 
Po =jm 2 (— 3v x cos D — 3p x sin 2D), 
P 3 =jm? {— 3v 2 cos 2D + 3y x 2 sin 2D 
— QpiV x cos 2D 
+ (2/o 3 + /Op). - f sin 2D}, 
&c. 
In particular attending to the values of P 15 Q x the equations for p x , v x are in their 
original form 
it d ir^ d iD hm ^ + §™ m - 
cl (dv 
dt \dt 
+ 2 p ; 
—jm 2 ( — I sin 2D), 
whence in the transformed form they are 
and 
dv x 
dt 
d~pi 
dt 2 
3jm? 
4 (1 — m) 
+ 2p x = Hj 4- t jdj ' "' :z\ cos 2D, 
+ Pi = 2fli + km?{\ + f cos 2D) + cos 2D. 
dv, 
Thus the constant term of p x is 2Q. x + \km?, giving in —^ a constant term — 3ilj — km 2 
this must vanish, or we have ilq = — km 2 ; and the equations thus become 
~ +2= — 1 km 2 + cos 2D, 
4 (1 — m) 
+ pi = — £&m 2 + km 2 + cos 2D, 
and then completing the integration 
— #&m 2 
Pi = ~i km 2 + 
+ 
— fjm 2 
3 — 8m + 4m 2 (1 — m) (3 — 8m + 4m 2 ) j 
cos 2D, 
Vi = 
\km? 
+ 
§ jm 2 (7 - 8m + 4m 2 ) j gin w 
((1 — m) (3 — 8m + 4m 2 ) (1 — m) 2 (3 — 8m + 4m 2 ) j 
which are the accurate values of p x and v x . 
Expanding as far as m 6 we have 
p 1 = k(— ^ m?) + cos 2D { k (— m? — f m 3 — - 2 ^- m 4 — m 5 — m 6 ) 
+i (— | m 2 — ^ m? — f| m 4 — Afi 5- m5 ~ tW- w 6 )}, 
which for j = & is =fc(- m 2 -J^m 3 -^m 4 - ^ m 5 - ^ m 6 ) , 
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