WITH THE TIDES OF A VISCOUS SPHEROID. 
587 
5 wa? (~ a\ 
—- ( S COS Vt— g - 
19u \ a' 
Hence the total radial displacement is 
5 wa? /- , <r 5 «?« 2 79 r 2 r 0 \ 
—— (S cos vt — 0 ;-+— 7 — .-7777 -o cos vt) 
19y V 19v loO r + q / 
150 lA q 
But the total radial displacement is itself equal to cr. 
Therefore 
and 
a 0 cr 79r 2 ~ 
1'- = S COS Vt — + t—} - :b COS Vt, 
a ■*«, 150(r + g) 
cr S / 79 y 2 \ 
-=-COS vt[ 1 + 77 -^- 
a i + q V lo 0 (v + q)/ 
This is the second approximation to the form of the tidal spheroid, and from it we 
see that inertia has the effect of increasing the ellipticity of the spheroid in the 
79v 2 
proportion 
Analogy with (76) would lead one to believe that the period of the gravest vibration 
/ 79 
of an elastic sphere is 2 77- —7— 
r \150r 
If g be put equal to zero, the sphere is devoid of gravitation, and if X he put 
equal to zero the sphere becomes perfectly fluid ; but the solution is then open to 
objections similar to those considered, when viscosity graduates into fluidity. 
It is obvious that the whole of this present part might be easily adapted to that 
hypothesis of elastico-viscosity which was considered in the paper on ‘“Tides,” but it 
does not at present seem worth while to do so. 
By substituting these second approximations in the equations of motion again, we 
might proceed to a third approximation, aucl so on ; but the analytical labour of the 
process would become very great. 
this result might be tested experimentally. 
IY. Discussion of the applicability of the results to the history of the earth. 
The first paper of this series was devoted to the consideration of inequalities of short 
period, in the state of flow of the interior, and in the form of surface, produced in a 
rotating viscous sphere by the attraction of an external disturbing body : this was the 
theory of tides. The investigation was admitted to be approximate from two causes 
—(i) the neglect of the inertia of the relative motion of the parts of the spheroid ; 
(ii) the neglect of tangential action between the surface of the mean sphere and the 
tidal protuberances. 
