320 
MR. S. S. HOUGH ON THE ROTATION OF AN ELASTIC SPHEROID. 
to be well founded ; and it is with a view to examining these assumptions that I have 
attempted to exhibit the solution of the problem in an analytical form. 
The analysis in the present paper is confined to the case of a homogeneous spheroid 
of revolution, composed of isotropic, incompressible, gravitating material, while no 
account is taken of the surface waters. For further simplification we have also sup¬ 
posed that the figure conforms to that required for hydrostatic equilibrium, so that 
when the body is undisturbed we may suppose it free from strain in its interior. We 
have reason to suppose that this condition is approximately realized in the case of the 
Earth. 
In §§ 1-2, I have obtained the rigorous dynamical equations for the oscillations of 
such a system. The form of these equations is, however, such as to render an 
approximate solution necessary. Hence it has been assumed in the subsequent 
sections that the ellipticity of the spheroid and, consequently, the angular velocity 
of rotation, are small quantities. This assumption leads to a considerable reduction 
in the differential equations which express the elastic displacements, and, in fact, 
reduces them to the familiar equations for the equilibrium of a strained elastic body 
The boundary conditions are also similar in form to the surface equations which are 
obtained in treating of the problem of the deformations of an elastic sphere, and as 
the solution of this problem is well known, we are in a position to obtain a solution of 
our differential equations applicable to the problem in hand. §§ 3-4 are devoted to 
the transformation of the equations into a form convenient for solution, while the 
actual solution is given in § 5. 
The method of approximation followed up to this point fails to lead to a determi¬ 
nation of the period. When, however, the body is supposed perfectly rigid, we are 
able to determine the period accurately by means of the equations of angular 
momentum for the whole system. This suggests the use of an analogous process, 
which is employed in § 6, to determine the period when elastic deformations are 
taken into account. 
The principal results of this paper will be found in § 8, where they are compared 
with the hypotheses made by Newcomb. It is found that the general character of 
the motion agrees with that assumed by Newcomb, but that his quantitative law as 
to the displacement of the pole, due to elastic distortion, is slightly in error. The 
bearing of these results on the theory of the Earth’s constitution is discussed in the 
final section (§8). 
§ 1. Differential Equations of Motion of Isotropic, Incompressible, Elastic Solid, 
referred to axes rotating uniformly. 
Take as axis of z the axis of rotation, and let the system be rotating with angular 
velocity about this axis. Let x Q + u, y Q + v, z 0 + w be the coordinates at the time 
