GRAVITATIONAL STABILITY OF THE EARTH. 189 



a generally valid expression of F, in terms of CD. and <. By means of equations (35) 

 and (59) the equation becomes 





15. The three remaining conditions which hold at the surface r = a are expressed 

 by equations of the type 



Every term of the left-hand member can be expressed in terms of the spherical 



solid harmonics 



Sw. ,+, 3 / a). \ 9<ft, *,+ 3 3 



aF' 3*1^7' ax' ai 



We have 





Also 

 rU = 



+ 3^2^T)P [{ ^~" P ' +( " + l)alQ " } ' = a -^ 

 and therefore 



<rU) = [nP.- 



.(*r)J3*. ,+i 3 / *i M 



M-^ 5 -' :r- (~ ^7 r 



rfr ISx axVr*** 1 /] 



