188 PROFESSOR A. E. H. LOVE ON THE 



P'. Q'. 'A*- Various relations between these functions have been noted incidentally. 

 It will be convenient hereafter to have noted the following properties of /,, (kr) : 



1 

 1.3.5. ..(2/1+1) I 2(2n + 3) 2.4 (2n + 3)(2n+5) 



^ j _ ; .^ ; 



A-r (#?) \ 



kr)}. . . . (59) 



Adjustment of the Harmonics to Satisfy the Boundary Conditions. 



14. In order to express F n in terms of &> and <f> n we use the condition that 

 3 (E n co B + F n ) is the potential at points within the sphere r = a of a distribution 

 of volume density within the sphere and of surface density on the sphere. The 

 corresponding external potential is 



where E B (a) is the value of E B at r = a. The surface density is /5 U a , where U a is 

 the value at r = a of the radial displacement U. Hence we have the equation 



which holds at the surface r = a ; it gives 



Now 



U = P n - 



2n- 



.pl =U , 

 J r = 



It follows that the equation 



Q 



holds at the surface = a. Since this equation connects the values at r = a of three 

 spherical solid harmonics of the same degree n, it holds for all values of r, and gives 



