si, 82] General Theory 81 



and in the new configuration is V" . The new potential energy will be 

 jSwF", so that we may write 



The first term jSw (V V) represents the work which would-be^ done in 

 effecting the displacement if the equipotentials remained fixed in space. The 

 boundary originally was an equipotential, so that the work done will be that 

 of moving certain matter from positions inside this equipotential to new 

 positions outside. It is therefore necessarily positive, and the ellipsoidal 

 configuration will be unstable if a displacement can be found such that 

 ^m(V V") is negative and numerically greater than the first term. 



In the displacement just considered let 8n denote an average normal 

 depth of matter which may be supposed excavated from one part of the surface 

 and piled up on other parts, so that p&n is the average mass removed per unit 

 area of surface. The mean change of potential V V" for such matter will 

 have an average value comparable with (dV/dn)Sn, so that the work done 

 will be of the order of magnitude of 



IS (210). 



The integral is only taken over those parts of the surface where the dis- 

 placement consists of a depression ; this may be supposed to be half of the 

 entire surface. Remembering that 



dn 



when the integral is taken over the whole surface we readily find that ex- 

 pression (210) is of the order of magnitude of 



-27rpM(Sn)* (211). 



This is the negative value of the first term on the right of equation (209). 

 We now proceed to consider the value of the second term. 



Values of P which are of degrees 0, 1, 2 in x, y, z result in displacements 

 which give rise only to new ellipsoids, so that we need only concern ourselves 

 with values of P which are of degrees 3 and higher. Displacements in which 

 P is of degree higher than 2 produce a furrow or system of furrows in the 

 original ellipsoid. When there are a great number of furrows, either the- 

 ellipsoid must be very long or the furrows very close together. In the latter 

 case the second term on the right of equation (209) becomes very small, 

 through the gravitational effects of successive elevations and depressions 

 neutralising one another. No corresponding effect occurs in the first term 

 on the right of equation (209), of which the value is represented by expression 

 (211). Hence it is seen that the ellipsoid can only become unstable through 

 a many-furrowed distortion when it is itself very long. It is easily seen that 

 the more furrows there are in the distortion the longer the ellipsoid has to 

 j.c. 6 



