774 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
In each case, after differentiation, the transition will be made to the case of viscosity 
of the planet, and the proper terms will be dropped out, without further comment. 
First take the perturbations of the moon. 
For this purpose we have to find dWjdf and c/W/sin/ cZiV'-j-tan \j' dW/de or 
dW/fdN'+^j'dW/de. 
By the above principle, in finding dW/dj' we may drop terms involving 
j and i cos (N — xfj), and in finding dW/j'dN' dW/de, we may drop terms involv¬ 
ing i sin (A— xfj). 
We may now suppose x~X> { I J=X P'- 
Take the case (i.), where the tide-raiser is the moon. Then as the perturbed body 
is also the moon, after differentiation we may drop the accents to all the symbols. 
From (129) 
From (130) 
^f=iFA~joos 2f 1 -icos (V-^+2f,)} 
sin {N — xfj) sin 4f x .(134) 
cos (w-i//+gi)+y cos gl } 
= —\i sin (N—xp) sin 2g 1 .( 135 ) 
From (131) and symmetry with (135) 
^ ft- 1 
df / 9 
= sin (N — xp) sin 2g 
(136) 
Adding these three (134—6) together, we have for the whole effect of the lunar tides 
on the moon 
2 
wi\ = ^ sin ^ v_ ^ ^ sin 4fj ~ sin 2gi + sin 2g ^ ■ • * • ( 1 y 
Now take the case (ii.) where the tide-raiser is the sun. 
Here we need only consider W 2 , but although we may put x = X> X I J=X I J > we 
must not puty=y v , N=N', because the tide-raiser is distinct from the moon. 
From (131) 
nw / tt ' 
v/T = * G[7<cos ( 7V -^-g)+i cos (tf-^+g)} 
Here accented symbols refer to the moon (as perturbed), and unaccented to the sun 
(as tide-raiser). As we refer the motion to the ecliptic j— 0, and the last term 
disappears. Also we want accented symbols to refer to the sun and unaccented to 
