58G 
MR. a. H. DARWIN ON PROBLEMS CONNECTED 
]t has already been remarked that the approximate solution of the problem of 
determining the state of internal flow of a viscous spheroid when inertia is neglected, 
is identical in form with that which gives the state of internal strain of an elastic 
sphere ; the velocities a, (3, y have merely to be read as displacements, and the 
coefficient of viscosity v as that of rigidity. 
The effects of mutual gravitation may also be introduced in both problems by the 
same artifice ; for in both cases we may take, instead of the external disturbing 
potential icffiS cos vt, an effective potential wr 2 (S cos vt — £J^, and then deem the 
sphere free of gravitational power. 
Now Sir William Thomson’s solution shows that the surface radial displacement 
(which is of course equal to <x) is equal to 
5 weft (^ , <r\ 
^-(Scosrt—jQ.( 82 ) 
19i 
If therefore we put (with Sir William Thomson) V = ryw, we have cos vt. 
r+ct 
This expression gives the equilibrium elastic tide, the suffix being added to the 
cr to indicate that it is only a first approximation. 
Before going further we may remark that 
S cos vt — (t—=-S cos vt 
r + q 
(83) 
When we wish to proceed to a second approximation, including the effects of 
inertia, it must be noticed that the equations of motion in the two problems only 
differ in the fact that in that relating to viscosity the terms introduced by inertia are 
-wy, —w-j-, —~r, whilst in the case of elasticity they are —w—~, — —w~. 
COZ CLC CIO CIO ccz ctz w 
Hence a very slight alteration will make the whole of the above investigation 
applicable to the case of elasticity ; we have, in fact, merely to differentiate the 
approximate values for a, (3. y twice with regard to the time instead of once. 
Then just as before, we find the surface radial displacement, as far as it is due to 
inertia, to be (compare (55)) 
ivv-a? 79 Y 
A- 2.3.19 3 C0S Vt ’ 
and—, cos vt must be put equal to (the first approximation) S cos vt — Hence by 
i A 79 r 
due to inertia is —— —— -o cos vt. 
V 2.3.19 2 r + 
To this we must add the displacement due directly to the effective disturbing 
potential wn( S cos — where cr is now the second approximation. This we 
know from (82) is equal to 
(57) and (83) the surface radial displacement 
