782 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
We easily find 
f=-r*-Gi ' 
|=_ry_G, 
... ! 
1 = A£+D, 
1= a ^+ d A 
(178) 
These equations contain the additional terms due to tides, which are to be added 
to the equations (116), in order to find the secular displacements of the proper planes. 
The first application, which will be made hereafter, will be to the case where the 
viscosity is small, and it will be more convenient to make the transition to that hypo¬ 
thesis at present, although the greater part of what follows in this part will be equally 
applicable whatever may be the viscosity. In the case of small viscosity the 
functions T, A, G, D will be indicated by the corresponding small letters y, 8, g, d. 
Then by (140), (150), (159), (172) we shall have • 
_ it sm 4 V -s i i TT q 
g — 4£ ^”L T J 
. k sin 4f r 
+ d=T^V-(l-2X)-irr'] )■ 
where X.=^ 
n 
. (179) 
And in the present case where i and j are small, we have by (112) and (121) 
t + t' 
+ /3: 
k , tC 
a— - tC, b=- 
t n 
where , the permanent ellipticity of the earth 
• (180) 
These equations (180) are the same whether the viscosity be supposed small or not. 
Then the complete equations are 
