SPHEROIDS, AND ON THE OCEAN TIDES UPON A YIELDING NUCLEUS. 27 
one gives high water, the other gives low water. The result is applicable to any kind 
of supposed yielding of the earth’s mass; and in the special case of viscosity, the table 
of results for the fortnightly tide at the end of Part I. is applicable. 
III. 
SUMMARY AND CONCLUSIONS. 
In § 1 an analogy is shown between problems about the state of strain of in¬ 
compressible elastic solids, and the flow of incompressible viscous fluids, when inertia is 
neglected ; so that the solutions of the one class of problems may be made applicable to 
the other. Sir W. Thomson’s problem of the bodily tides of an elastic sphere is then 
adapted so as to give the bodily tides of a viscous spheroid. The adaptation is ren¬ 
dered somewhat complex by the necessity of introducing the effects of the mutual 
gravitation of the parts of the spheroid. 
The solution is only applicable where the disturbing potential is capable of expansion 
as a series of solid harmonics, and it appears that each harmonic term in the potential 
then acts as though all the others did not exist ; in consequence of this it is only 
necessary to consider a typical term in the potential. 
In § 3 an equation is found which gives the form of the free surface of the spheroid 
at any time, under the action of any disturbing potential, which satisfies the condition 
of expansibility. By putting the disturbing potential equal to zero, the law is found 
which governs the subsidence of inequalities on the surface of the spheroid, under the 
influence of mutual gravitation alone. If the form of the surface be expressed as a 
series of surface harmonics, it appears that any harmonic diminishes in geometrical 
progression as the time increases in arithmetical progression, and harmonics of higher 
orders subside much more slowly than those of lower orders. Common sense, indeed, 
would tell us that wide-spread inequalities must subside much more quickly than 
wrinkles, but only analysis could give the law connecting the rapidity of the sub¬ 
sidence with the magnitude of the inequality.* 
* On this Lord Rayleigh remarks, that if we consider the problem in two dimensions, and imagine a 
number of parallel ridges, the distance between which is X, then inertia being neglected, the elements on 
which the time of subsidence depends are gw (force per unit mass due to weight), v the coefficient of 
viscosity, and X. Thus the time T must have the form 
T—(giu) x v'J XL 
The dimensions of giv, v, A are respectively ML~ 2 T -2 , ML - 1 T -1 , L; hence 
x + y — 0 
— 2x — y + z—0 
— 2x—y—l, 
• v 
And x= —1, y — 1, z— —1, so that T varies as- 
gw\ 
If we take the case on the sphere, then when i, the order of harmonics, is great, X compares with 7 ; 
i 
vi 
so that T varies as-• 
Qiua 
E 2 
