GRAVITATIONAL STABILITY OF THE EARTH. 239 



the geoid, is expressible, at least roughly and approximately, by means of harmonics 

 of the first, second, and third degrees. Now, if the shape of the lithosphere is at all 

 close to that in which it may be presumed to have consolidated, the inference would 

 seem to be that, in respect of general features, as distinguished from local irregularities, 

 the positions of the continental blocks and oceanic regions have not changed much 

 since the date of consolidation. This view has in recent times met with considerable 

 support among geologists. 



The theory also enables us to make some attempt to indicate the general nature 

 of those changes which could be expected to take place. In estimating the value of 

 such an attempt some allowance must be made for the fact that the theory of an 

 elastic body in a state of initial stress is very far from complete. We try to follow 

 out certain clues drawn from the scanty knowledge we possess of bodies in states of 

 initial stress. Among these the behaviour of cast iron under tensile tests is perhaps 

 important. It is well known that cast iron which has not previously been tested 

 exhibits a stress-strain curve which is essentially different from that of mild steel, but 

 that, after several tests, its behaviour approaches to that of steel. It has been 

 conjectured that the tests have the effect of gradually removing a state of initial 

 stress, and thus reducing the substance to a "state of ease." That state of a rotating 

 gravitating planet which would correspond to a state of ease in solid bodies at its 

 surface would seem to be a state in which the' material would be arranged in 

 concentric spheroidal layers of equal density, and the external surface would be an 

 oblate spheroid, the ellipticity being determined by the speed of rotation and the 

 distribution of density ; the state of stress in the planet, when in this state of ease, 

 would be one of hydrostatic pressure, and the surface would be an equipotential 

 surface under gravity modified by the rotation. The partial reduction of the body to 

 the state of ease would be effected by gradual stages, prol>ably of the nature of local 

 fractures. Now the wrinkling of the surface, expressed by harmonics of the third 

 degree, arose as a consequence of the displacement of the centre of gravity and of the 

 ellipsoidal configuration. It would at first IKJ small in comparison with the deviations 

 from spherical symmetry which are expressed by harmonics of the first and second 

 degrees. We should therefore exj>ect that the tendency of secular change in the 

 shape of the lithosphere would lie to diminish the coefficients of the harmonics of the 

 first and second degrees. An exception must be made in the case of the coefficient e 

 of the zonal harmonic of the second degree ; for this coefficient represents a difference 

 of ellipticity of the meridians of lithosphere and geoid, and these ellipticities depend 

 upon the speed of rotation. When this coefficient is left out of account, the harmonic 

 inequality of the second degree represents ellipticity of the equator* and obliquity 

 of the principal planes ; the harmonic inequality of the first degree represents 

 displacement of the centre of gravity from the centre of figure. If the coefficients of 



* G. II. DARWIN concluded from his theory of the tidal deformation of a viscous spheroid that an initial 

 ellipticity of the equator would tend to be obliterated. ' Phil. Trans. Roy. Soc.,' vol. 170 (1879), p. 30. 



