The results of the theory of the elastic yielding of the earth would of course be more interesting, if it were possible fully to introduce the effects of the want of homogeneity of elasticity and density of the interior of the earth; but besides the mathematical difficulties of the case, the complete absence of data as to the nature of the deep-seated matter makes it impossible to do so. It is, however, possible to make a more or less probable estimate of the extent to which a given yielding of the surface will affect the ocean tide-wave, when the earth is treated as heterogeneous. And as we can only judge of the amount of the bodily tide in the earth by observations on the ocean tides, this estimate may be of some value.

The heterogeneity of the interior must of course be accompanied by heterogeneity of elasticity, and under the influence of a given tide-generating force, this will affect the internal distribution of strain, and the form of the surface to an unknown extent. The diminution of ocean tide which arises from the yielding of the nucleus is entirely due to the alteration in the form of the level surfaces outside the nucleus. But it is by no means obvious how far the potential of the earth, when its surface is distorted to a given amount, may differ from that of the homogeneous spheroid considered by Sir W. Thomson; and in face of our ignorance of the law of internal elasticity, the problem does not admit of a precise solution.

In a series of papers read from time to time during the past two years before the Royal Society, I have investigated the theory of the tides raised in a rotating viscous spheroid, or planet, by an attendant satellite, and have also considered the secular changes in the rotation of the planet, and in the revolution of the satellite. Those investigations were intended to be especially applicable to the case of the earth and moon, but the friction of the solar tides was found to be a factor of importance, so that in a large part of those papers it became necessary to conceive the planet as attended by two satellites.

The differential equations which gave the secular changes in the system were rendered very complex by the introduction of solar disturbance, and I was unable to integrate them analytically; the equations were accordingly treated by a method of numerical quadratures, in which all the data were taken from the earth, moon, and sun. This numerical treatment did not permit an insight into all the various effects which might result from frictional tides, and an analytical solution, applicable to any planet and satellite, is desirable.

In the present paper such an analytical solution is found, and is interpreted graphically. But the problem is considered from a point of view which is at once more special and more general than that of the previous papers.

In a well-known investigation Sir William Thomson has discussed the problem of the bodily tides of a homogeneous elastic sphere, and has drawn therefrom very important conclusions as to the great rigidity of the earth.

Now it appears improbable that the earth should be perfectly elastic; for the contortions of geological strata show that the matter constituting the earth is somewhat plastic, at least near the surface. We know also that even the most refractory metals can be made to flow under the action of sufficiently great forces.

Although Sir W. Thomson's investigation has gone far to overthrow the old idea of a semi-fluid interior to the earth, yet geologists are so strongly impressed by the fact that enormous masses of rock are being, and have been, poured out of volcanic vents in the earth's surface, that the belief is not yet extinct that we live on a thin shell over a sea of molten lava. Under these circumstances it appears to be of interest to investigate the consequences which would arise from the supposition that the matter constituting the earth is of a viscous or imperfectly elastic nature; for if the interior is constituted in this way, then the solid crust, unless very thick, cannot possess rigidity enough to repress the tidal surgings, and these hypotheses must give results fairly conformable to the reality. The hypothesis of imperfect elasticity will be principally interesting as showing how far Sir W. Thomson's results are modified by the supposition that the elasticity breaks down under continued stress.

The papers contained in the present volume form in effect a single investigation in speculative astronomy. The tidal oscillations of the mobile parts of a planet must be subject to frictional resistance, and this simple cause gives rise to a diversity of astronomical effects worthy of examination.

The earlier portion of the investigation was undertaken with the object of explaining, if possible, the obliquity of the earth's equator to the ecliptic, and the results attained were so fruitful and promising that it seemed well to examine the whole subject with the closest attention, and to discuss the various collateral points which arose in the course of the work.

It is the experience of every investigator that he reaches his goal by a devious path, and this, at least, has been the case in the present group of papers. If then the whole field were now to be retraversed, it is almost certain that the results might be obtained more shortly. Then, again, there is another cause which precludes brevity, for when an entirely new subject is under consideration every branch road must be examined with care. By far the greater number of the forks in the road lead only to blind alleys; and it is often impossible to foresee, at the cross roads, which will be the main highway, and which a blind alley. Clearness of view is only reached by the pioneer after much labour, and as he first passes along his path he has to grope his way in the dark without the help of any sign-post.