578 
MR. G. H. DARWIN ON PROBLEMS CONNECTED 
Also we have the equations 
sin 4e a 
sin 4e 
= l-\, 
sin 2e\ 
sin 4e 
=1(1 -2X), 
sin 4e 
sin 4e 
sin 4e 2 
sin 4e 
= l+\ 
sin 2e'_ 1 
sin 4e 2 ’ 
sin 2e'o 
sin 4e 
a(l + 2X), 
sin 4e" 
sin 4e 
> . 
( 64 ) 
Now we shall obtain a sufficiently accurate result, if the corrections be only applied 
to those terms in the differential equations which do not involve powers of q (or sin \i), 
higher than the first. Then for the purpose of correction the differential equations to 
be corrected are by (77), (78), and (79) of Section 17 of “Precession,” viz. :— 
d% m a 
clt 
dN„ 
clt 
Nw 
=i—sin 4e,=/x 
9”o 
y/t 0 
o 
T a 
sin 4e 1 +-|p 5 g(p 2 +3g' 2 ) sin 2e\ —^pq{p z — S' 2 ) 3 sin 2e'] 
_ rff 
clt 
> . (65) 
As we are treating the obliquity as small, we may put 
2 p 1 q— : kp hc Ai^^^ < f)—\p c Ap~~ c ff—\PQ and p 8 =P, 
when P —cos i, Q= sin i. 
Then for the purpose of correction, the terms depending on the moon’s influence are 
cli 
clt 
__r/AA = ry psi n 4 
—^’= 4r~ii :> ^[sin dciffi sin 2e\ — sin 2e 
A g/; 0 4 ^ 1 11 1 
* 
T 
clt 
’5”o 
i=P 
clt 
( 66 ) 
And by symmetry (or by (81) “ Precession”) we have for the solar terms 
^=4-—4e, 
clt A r g« 0 4 ^ 
dN mr - .T/p . 
— - = 1 —P sm 4e 
clt 
’S«o 
(67) 
For the terms depending on the joint action of the sun and moon we have, by (82) 
and (33) “ Precession,” when the obliquity is treated as small, 
'1 PQ s i n -V 
clt ’ 
S"0 
clN„, 
. . . . ( 68 ) 
= 0 
