GRAVITATIONAL STABILITY OF THE EARTH. 225 



Effect of certain External Forces. 



52. The effect of forces such as the attraction of the Moon at the time when its 

 period of orbital revolution did not differ much from the period of rotation of the 

 Earth would l>e to draw the planet out into a shape more nearly ellipsoidal, with 

 three unequal axes, than spheroidal. If the planet could have had a symmetrical 

 shape it would have l)een practically ellipsoidal, and the surfaces of equal density 

 would have been ellipsoids. Whereas the effect of rotation is the same as that of 

 forces derived from a potential of the second degree, symmetrical ivlxmt the axis of 

 rotation ; such forces as we are now considering are derived from a potential, the 

 most important terms of which would lx? of the second degree, but not symmetrical 

 about the axis of rotation. If the elasticity was too small for an ellipsoidal figure to 

 l)e stable, the planet would have been in a disturbed state, the nature of which can 

 1)6 inferred from the preceding investigation. We have only to replace in 50 the 

 initial potential, modified by the rotation, by a general expression of the second degree 

 in the co-ordinates. The only change that would be made in the result would be that 

 those terms in the radial displacement which are expreased by harmonics of the third 

 degree would not be of the specialised type introduced by the rotation, but would be 

 of general type. The figure of the planet would be derived from the ellipsoidal 

 figure appropriate to the rotation, and to the external forces, by a radial inequality 

 expressed by sin-face harmonics of the first and third degrees. The equipotential 

 surfaces would IK? obtained from the ellipsoidal equipotentials appropriate to gravity, 

 modified by the rotation and the external forces, by surface harmonics of the same 

 degrees. The result would be that the shape of the planet, as determined by 

 difference of level above or below a certain equipotential, would be a wrinkled 

 ellipsoid, displaced towards one side ; and the wrinkle would be expressible by means 

 of a spherical surface harmonic of the third degree. 



Tlie Problem of the Shape- of the Lithosphere. 



53. The problem of determining the form of the equipotentials near the surface of 

 the Earth includes the problem of determining the figure of the surface of the 

 ocean (the " hydrosphere "). The equipotentials which lie outside the nucleus (or 

 " lithosphere ") on one side, and sufficiently near to it, cut the surface of the 

 lithosphere towards the other side. Among these equipotential surfaces that one 

 which, outside the lithosphere, coincides with the surface of the ocean is known as 

 the " geoid." The surface of that part of the lithosphere which lies outside the geoid 

 is occupied by land, and can be observed directly ; the surface of that part which lies 

 within the geoid can only be observed indirectly by means of soundings. We have 

 no means of investigating the form of the surface of this part of the lithosphere 

 except by estimating its depth at a point below the geoid. The most important 

 deviations from sphericity both of the lithosphere and of the geoid are of such 



VOL. CCVII. A, 2 O 



