758 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
and 
M 3 3 =^ sin 3 7(1 —\ sin 3 /)-j-| sin 2 /(l —sin 3 7) 
+ 5 sin 27 sin 2 j cos (W—i//)—^ sin 3 7 sin 2 / cos 2(N—xp) 
Now 
ijr(sin 3 7+sin 2 j) —f sin 3 7 sin 3 /—■§■= —-§-(1 —f sin 3 7)(l —f sin 3 /) 
Wherefore 
W = tC {^(1 —f sin 3 7)(l —f sin 2 /)—5 sin 27 sin 2 j cos ( N — xp) 
+ \ sin 3 7 sin 3 / cos 2 (N —xp)} . (105) 
This is the disturbing function. 
Before applying it, we w T ill assume that 7 and / are sufficiently small to permit us to 
neglect sin 3 7 sin 3 / compared with unity. 
Then 
^(1 — f sin 3 7)(1 — -f sin 2 /)=3 1 2+i — i sin 3 7— | sin 3 / + sin 3 7 sin 3 /— d sin 3 7 sin 3 / 
=-^ 2 + 4 cos 27 cos 2 /—\ sin 3 7 sin 3 / 
Hence, when we neglect the terms in sin 3 7 sin 3 / 
W=^tC{-|-+cos 27 cos 2/— sin 27 sin 2j cos (IV— xp)} .... (106) 
Then since this disturbing function' does not involve the epoch or y, we have by 
(13), (14), and (18) 
P . dj dW P . dN dW . di dW . dylr dW 
k J dt clN k J dt dj dt d^’ dt di 
Thus as far as concerns the influence of the oblateness on the moon, and the reaction 
of the moon on the earth, 
p . ,dj 
j sin/— = —/r£ sin 27 sin 2 / sin (TV — xp) 
j smj~= — {cos 27 sin 2 /-{-sin 27 cos 2 / cos ( N—xp )} 
n sin 7— = \tZ sin 27 sin 2 / sin (N—xp) 
n sin i —-|r£{sin 27 cos 2 /-{-cos 27 sin 2 / cos (N—xp)} 
(107) 
If there be no other disturbing body, and if we refer the motion to the invariable 
plane of the system, we must always have N=xp. 
In this case the first and third of (107) become 
7 = 4=0 
dt dt 
and the second and fourth become 
