WITH THE TIDES OF A VISCOUS SPHEROID. 
567 
The tidal disturbance is supposed to be sufficiently slow to enable us to obtain a first 
approximation by the neglect of the inertia terms. 
In proceeding to the second approximation, the inertia terms depending on the squares 
and products of the velocities, that is to say, iv(^a ^-\-/3 may be neglected com¬ 
pared with iv-j. A typical case will be considered in which W=Y cos ( vt J r e ), where 
CLu 
Y is a solid harmonic of the i a degree, and the e will be omitted throughout the 
analysis for brevity. Then if we write I=2(t-f-l) 3 +l, the first approximation, when 
the inertia terms are neglected, is 
a; 
1 
lv 
'i(i + 2) c; ,_ (i + l)(2i + 3) j ,; 
[_2(7 —1) 
2(2i+l) 
dY 
dx 2 i +1 
I _„,2i+3_ r L^. 
dx 
X Y) l cos vt* 
(31) 
Hence for the second approximation we must put 
da. wv , 
— w~=—1 1 Sill vt. 
And the equations to be solved are 
dp , , dY' , , iov 
— — +i»V cc —— COS Vt -p 
dx 
dx 
i(id- 2) (7+1)(27 + 3)^ j2 
ffi [ [_2(7—!) 
2(2<+l) 
i ,., „ d 
dY 
dx 
27+1 
r 2 ' +3 ~(i v X Y) jsin vt=Q 
— ( j >J r &c. =0, — &c. =0 
dy dz 
(32) 
These equations are to be satisfied throughout a sphere subject to no surface stress. 
It will be observed that in the term due directly to the impressed forces, we write Y' 
instead of Y ; this is because the effective potential due to gravitation will be different 
in the second approximation from what it was in the first, on account of the different 
form which must now be attributed to the tidal protuberance. 
The problem is now reduced to one strictly analogous to that solved in the paper on 
“ Tides;” for we may suppose that the terms introduced by w~ &c., are components of 
bodily force acting on the viscous spheroid, and that inertia is neglected. 
‘The equations being linear, we consider the effects of the several terms separately, 
and indicate the partial values of a, /3, y, p by suffixes and accents. 
First , then, we have 
— j -j-uV j cos rt=0 } &c., &c. 
* “ Tides,” Section 3, equation (8), or Thomson and Tait, ‘ Nat. Phil.,’ § 834 (8-). 
MDCCCLXXIX. 4 D 
