GRAVITATIONAL STABILITY OF THE EARTH. 221 



The variation of the excess density along the axis of the harmonic is illustrated in 

 fig. 1. The surface r = a(l + ecos0) can be an equipotential surface if 



ne = - 



and thus a sphere of radius a with its centre at the displaced centre of gravity is an 

 equipotential surface. The relative situation of the bounding surface and of this 

 equipotential is illustrated in fig. 2, in which O denotes the undisplaced centre, C the 

 centre of the displaced surface, and G the centre of gravity of the strained sphere. 

 The figures are drawn for the case in which s*a* =10. 



The type of disturbance which has been called above a lateral disturbance with a 

 hemispherical distribution of density would be the same in a body possessing some 

 degree of rigidity, but the numerical details would be different. 



49. If the equipotential surfaces of a nearly spherical body, with a nearly 

 symmetrical distribution of density, are referred to the centre of gravity of the body 

 as origin, their equations take such forms as 



r = 



in which a , ... denote small coefficients, and S_.. . . . denote spherical surface harmonics 

 of degrees indicated by the suffixes. There is no term of the form e,S,. In the case 

 of the Earth, the coefficients c 2 , ... can be determined by means of pendulum 

 experiments. If we referred to a different origin, near the centre of gravity, a term 

 of the form ,8! would be introduced, but the coefficient e, could not be determined by 

 means of pendulum experiments, for it does not affect the formula for the variation 

 of gravity over the surface.* If we choose an origin in accordance with geometrical 

 considerations, e.g., as the centre of that oblate spheroid which most nearly coincides 

 with the surface of the ocean, the results of pendulum experiments cannot tell us 

 whether this origin coincides with the centre of gravity or not. 



Effect of Rotation upon a Planet with a Hemispherical Distribution of Density 



50. In all the preceding work the rotation of the Earth has been neglected. We 

 have now to consider the effect of rotation upon a nearly spherical planet which, in 

 the absence of rotation, would have a hemispherical distribution of density. To 

 simplify the analysis, we shall disregard the concentration of mass towards the 

 centre and also the rigidity of the body. We shall take as the " initial " state of the 

 body a state in which the density is uniform and the stress is hydrostatic pressure, 



* The result may be inferred from STOKES' investigation of the " Variation of Gravity over the 

 Surface of the Earth," Cambridge, ' Trans. Phil. Soc.,' 8 (1849), or ' Math, and Phys. Papers,' vol 2, 

 Cambridge, 1883. It is easy to prove it independently. 



