540 
ON A PAIR OF DIFFERENTIAL EQUATIONS IN THE LUNAR THEORY. [483 
and 
v 3 = sin 2D { k( I m 2 + ^-m 3 + f| m 4 + m 5 + $£££■ m 6 ) 
+ j ( | m 2 + m 3 + to 4 + m 5 + ^ff 1 m 6 )}, 
which for j = A; is = A; ( m 2 + ff m 3 + m 4 + m 5 + 4 ¿4^- m 6 ) . 
I have, not in general, but for the value j = k, calculated p 2 and v 2 as far as m 6 : 
I have not made the calculation for p 3 and v 3 , but their values may be deduced from 
the foregoing values of p, v; the final expressions (when j=k) of p, = 1 + p 2 + p 2 + p 3 + ... 
and v, = t + v 1 +v 2 +v 3 ... are 
+ A; (— £ m 2 
) 
+ k 2 ( 
3 31 770 4 1 
288 7)1 + 
83 
IF 
m 5 + 
5113 
288 
m 6 ) 
+ A; 3 ( 
- 
1621 
12 96 
m 6 ) 
4- cos 2D { k (— m 2 - 
- ig-m 3 - 
- J^-m 4 — 
895 
54 
m 5 — 
5 5 9 7 
16 2 
m 6 ) 
+ A; 2 ( 
| m 4 + 
31 
"TT 
m 5 + 
329 
'2 V 
m 6 ) 
+ k 3 ( 
- 
2381 
2304 
m 6 )} 
+ cos 4D { A; 2 ( 
- | m 4 - 
1217 
"4F0" 
m 5 — 
76589 
"7200 
m 6 ) 
+ k 3 ( 
+ 
7 
27) 
m 6 )} 
+ cos 6D { k s ( 
— 
59_ 
2 56 
w 6 )}, 
+ sin 2D { k (^ m 2 + 
f§ m 3 + 
m 4 + 
896 
"2V 
TO 5 + 
419 7 5 
F48 
m 6 ) 
+ k 2 ( 
- 
\ m 4 - 
ft 
m 5 — 
43 
3 
m 6 ) 
+ k 3 ( 
- 
783 
2048 
to 6 )} 
+ sin 4D { k 2 ( 
2 01 /vv}4 1 
25^ m + 
649 
T2“0 
m 5 + j 
365263 
^8800 
to 6 ) 
+ k 3 ( 
- 
u 
to 6 )} 
+ sin 6D { k 3 ( 
+ 
37 15 
6 144 
to 6 )}; 
which for k = l agree with the foregoing formulae (verifying them as far as m 5 ); the 
present formulae exhibit the manner in which the expressions depend on the several 
powers of the disturbing force.