776 
MR. Gr. H. DARWIN ON THE SECULAR CHANGES IN 
Collecting results from the six equations (141-6), we have for the whole perturbation 
of the moon by the lunar tides 
(7 JW'+W d?)/g = iO'+i cos (A r -^))(sin 47- sin 2 gl + sin 2g) . (147) 
Next take the case (ii.), and suppose that the sun is the tide-raiser. Here we need 
only consider W 3 . Then noting that dW 2 /de'=0 absolutely, we have from (131) 
(I (V'-f-g)-y Sin (N-N'+g)} 
Accented symbols here refer to the moon (as perturbed), unaccented to the sun (as 
tide-raiser). Therefore j= 0. Then reverting to the usual notation by shifting accents 
and dropping useless terms, this expression becomes 
cos (N— xfi) sin 2g.(148) 
Then collecting results from (147-8), we have by (13) 
f f = —iU+i cos ( N ~'I J )) ^ ( sin 4f i“ siu 2 g’i+ sin 2g)—cos (N—xfi) ~ sin 2g (149) 
This gives the additional terms due to bodily tides in the equation for dj/dt, viz.: the 
first of (133). 
If the viscosity be small 
sin 4fj— sin 2g 1 -{- sin 2g= sin 4f 
sin 2 g=-§ sin 4f 
Before proceeding further it may be remarked that to the present order of approxi¬ 
mation in case (i.) 
dW 
de’ 
\ sin 4f x 
and in case (ii.) it is zero ; thus by (11) 
.( 151 ) 
We now turn to the perturbations of the earth's rotation. 
Here we have to find dW/di and cot i dW/df—dW/ sin i dxjj' or 
(1 — \i~)dW/idx — dW/idf, and hi the former may drop terms in i and / cos (N — xjj), 
and in the latter terms in/ sin (i\ r — xjj). 
