GRAVITATIONAL STABILITY OP THE EAKTH. 177 



on the surface r = a. By the method of spherical harmonics we can, when W is 

 known, write down the expression for the function W (u) which is the potential at 

 external points of the same distribution. The surface characteristic equation gives 



This is one of the conditions which must be satisfied at the surface r = a. To 

 obtain the other conditions which must be satisfied at this surface, we observe that 

 the disturbed surface r = a + U is free from traction. If /, m, n denote the direction 

 cosines of the outward drawn normal to this surface we have three equations of the 



type 



, = 0, 



which hold at the surface r = a + U a . If in this equation we substitute for X T , ... 

 from equations (l), we see that in the terms containing u, ... we may replace I, ... by 

 the approximate values x/r, yfr, z/r. The only term which does not contain u,... is 

 the term Ip arising from IX^ Now p vanishes at r = , and therefore at 

 r = a + U a we have 



to the first order in u, v, w. Hence in this term also we may replace / by x/r. On 

 substituting for^> rt from (5) we find that the equation 



v , cu 



must hold at the surface r = a. By an easy transformation this equation becomes 

 the first of the three equations written in (14) below. The remaining two of these 

 equations are obtained in the same way. The equations which must hold at the 

 surface r = a are therefore equation (13) and the equations 



J_(Ur) +,- |H -u +| 3!fiLxUr = 0, 

 ox or p 



JL(Ur)+r - r +|E3ieLUr = 0, }- (14) 



oy < r p 



A C/ /TT \ Utv A TfrfJt\ TT 



-?A + =-(Un+r;r tp+J ir z\Jr = 0. 



ft 01 O1' fJ. 



These e<juations can be interpreted in the statement that the traction on the mean 

 sphere is a pressure equal to the weight per unit of area of the material heaped up to 

 form the inequality U a . 



VOL. ocvn. A. 2 A 



