GEOLOGICAL CHANGES ON THE EAETH’S AXIS OF ROTATION. 287 
II. THE PRINCIPAL AXES OF THE EARTH. 
9. Preliminary Assumptions. 
It is assumed at first that, in consequence of some internal causes, the earth is 
undergoing a deformation, but that there is no disturbance of the strata of equal density, 
and that there is no local dilatation or contraction in any part of the body. The cases 
at present excluded will be considered later. 
The result of this assumption is, that the volume of the body remains constant, and 
that the parts elevated or depressed above or below the mean surface of the ellipsoid 
have the same density as the rest of the surface. Such changes of form must, of course, 
be produced by a very small flow of the solid matter of the earth. Since the whole 
volume remains the same, this hypothesis may be conveniently called that of incom- 
pressibility ; although, if the matter of the earth flowed quite incompressibly, there 
would be some slight dislocation of the strata of equal density. 
It is immaterial for the present purpose what may be the forces which produce, and 
the nature of, this internal flow ; but it was assumed in the dynamical investigation 
that the forces were internal, and that the flow proceeded with uniform velocity. 
After deformation the body may be considered as composed of the original ellipsoid, 
together with a superposed layer of matter, which is positive in some parts and negative 
in others. The condition of constancy of volume necessitates that the total mass of this 
layer should be zero. If we take axes with the origin at the centre of the ellipsoid and 
symmetrical thereto, and let h F(0, <p ) represent the depth of the layer at the point B, <p, 
the condition of incompressibility is expressed by the integral of F(0, <p ) over the surface 
of the ellipsoid being zero. Then by varying h , elevations and depressions of various 
magnitudes may be represented. 
10. Moments and Products of Inertia after Deformation. 
Before the deformation : — 
Let A, C be the principal moments of inertia of the earth; a , b its semiaxes; M its 
mass ; 2B its mean density ; § its surface density ; and c its mean radius, so that 
3c=2a + 6; and let the earth’s centre of inertia be at the origin. 
After the deformation : — 
Let a, b, c, D, E, F be the moments and products of inertia of the above ideal shell of 
matter about the axes ; x, y , z the coordinates of the earth’s centre of inertia. 
Then, since the ellipticity of the earth is small, the integrals may be taken over the 
surface of a sphere of radius c , instead of over the ellipsoid. Therefore, 
a=/^c 4 jj F(0, <p) sin B (sin 2 B sin 2 <p +cos 2 B) dB dp, 
Ma’=A§c 3 j) F(0, <p) sin 2 B cos p dB dip, 
M=|t tBc 3 , 
and other integrals of a like nature for b, c, D, E, F ,y, z. 
2s 
MDCCCLXXVII. 
