194 PROFESSOR A. E. H. LOVE ON THE 



where A is an arbitrary constant. The second solution of the differential equation 

 for U becomes infinite at r = 0, and thus the above is the most general form for U. 



20. The condition that the surface ? = a + \J a is free from traction can easily be 

 shown to be the condition that 



= 

 r 



at r = a. Hence we have 



(77) 



where v = /u/(A.+2/i), so that 2A/(\+2/*) = 2 4i/. The frequency equation is 

 therefore 



The condition of gravitational instability is obtained by putting A 2 =0. It is 



) ... = o. (79) 



21. The coefficient of s^a 2 " in the left-hand member of (79) is 



or 



(-)'ff 1 , 1 _ 1 



2 Ll3.5...2K-l 3.5...(2c+l)j 



2 l3.5...(2K-l) 



v -- -- -- 



(2f-l) 3.5...(2ic + l) 3.5...(2K+3)JJ' 



and the equation (79) can be written 



where x is written for sa. Now we have 



(I I ~3 5 



o e'^ = e .-(,-| + ^ 



and therefore 



1 _ ^ + ^!_ _ . . . = x -* e -v (' 

 33.5 Jo 



