28 MR. DARWIN ON THE BODILY TIDES OF VISCOUS AND SEMLELASTIC 
I hope at some future time to try whether it will not be possible to throw some 
light on the formation of parallel mountain chains and the direction of faults, by 
means of this equation. Probably the best way of doing’ this will be to transform the 
surface harmonics, which occur here, into Bessel’s functions. 
In § 4 the rate is considered at which a spheroid would adjust itself to a new form of 
equilibrium, when its axis of rotation had separated from that of figure ; and the law is 
established which was assumed in a previous paper.* 
In § 5 I pass to the case where the disturbing potential is a solid harmonic of the 
second degree, multiplied by a simple time harmonic. This is the case to be considered 
for the problem of a tidally distorted spheroid. A remarkably simple law is found 
connecting the viscosity, the height of tide, and the amount of lagging of tide ; it is 
shown that if v be the speed of the tide, and if tan e varies jointly as the coeffi¬ 
cient of viscosity and v, then the height of bodily tide is equal to that of the equi- 
librium tide of a perfectly fluid spheroid multiplied by cos e, and the tide lags by 
6 
a time equal to 
It is then shown (§ 6) that in the equilibrium theory the ocean tides on the yielding 
nucleus will be equal in height to the ocean tides on a rigid nucleus multiplied by 
sin e, and that there will be an acceleration of the time of high water equal to ——- 
The tables in § 7 give the results of the application of the preceding theories to the 
lunar semidiurnal and fortnightly tides for various degrees of viscosity. A comparison 
of the numbers in the first columns with the viscosity of pitch at near the freezing 
temperature (viz., about 1'3X 10 8 , as found by me), when it is hard, apparently solid 
and brittle, shows how enormously stiff the earth must be to resist the tidally deform¬ 
ing influence of the moon. For unless the viscosity were very much larger than that 
of pitch, the viscous sphere would comport itself sensibly like a perfect fluid, and the 
ocean tides would be quite insignificant. It follows, therefore, that no very consider¬ 
able portion of the interior of the earth can even distantly approach the fluid state. 
This does not, however, seem to be conclusive against the existence of bodily tides in 
the earth of the v kind here considered; for although (as remarked by Sir W. Thomson) 
a very great hydrostatic pressure probably has a tendency to impart rigidity to a 
substance, yet the very high temperature which must exist in the earth at a small 
depth would tend to induce a sort of viscosity—at least if we judge by the behaviour 
of materials at the earth’s surface. 
In § 8 the theory of the tides of an imperfectly elastic spheroid is developed. The 
kind of imperfection of elasticity considered is where the forces requisite to maintain 
the body in any strained configuration diminish in geometrical progression as the time 
increases in arithmetical progression. There can be no doubt that all bodies do 
possess an imperfection in their elasticity of this general nature, but the exact law 
* Phil. Trans., Vol. 167, Part I., sec. 5 of my paper. 
