77 8 
MR, Gr. H. DARWIN ON THE SECULAR CHANGES IN 
n sin i~=^j sin (JV— \fi) 
T" ... TT 
— (sin 4f : + sin 2g x —sin 2g) — — sin 2g 
(158) 
This gives the additional terms due to bodily tides in the equation for dxjjjdt, viz. : 
the last of (133). 
If the viscosity be small 
where 
sin 4fj4- sin 2g, — sin 2g= sin 4f(l — 2\)~' 1 
sin 2g=^ sin 4f 
X= 
n 
■ . (159) 
Next consider the change in the obliquity of the ecliptic; for this purpose we must 
find (1— ^i 2 )dW/idf — dW/idxjj 'and may drop terms involving j sin ( N—xjj ). 
First take the case (iii.), where the moon is both tide-raiser and disturber. 
Then from (129) 
~ g = — Fj {(1 — i~—j~) sin 2fj+ ij sin (N-xjj- 2f : ) - ij sin (A r —i//+2f x )} 
(/A j /t g * n 2^-\-ij sin (N— xfj— 2^) — \ij sin {N— ^+2^)} 
df g 
—W'-t-t /-=F 1 -|i 2 sin 2fj 
2 dx l g 12 1 
(160) 
Therefore 
i ' 2 ’ df i df _ 
/ g =Pi{i sin 2 f x +j sin (N-xjj+2^)} 
~^(i-\-j cos (N — x(j)) sin 4f : . . . 
From (130) 
dW 1 /r° 
(] x / g 
= sin g Y -ij sin (N-xfj-gJ+ij sin (N—xft+g^+f sin gg 
cTW !t~ 
- xp 1 / ^={ 2i ~ sin gi— 2 # sin ( N ~ -g'i )+V sin { N ~ +gi) +/ sin g x } 
, ..cm, /t 2 
0 
d x / g 
(161) 
(162) 
Therefore 
i ' 2 ’ df i df_ 
!r~ 
= \G{i sin g x —j sin (N—xfi—g^} 
=i(^+? cos (N—xjj)) sin 2g x . . 
(163) 
