-47- 293 



Introduce the quantity 

 A 



(8) a = ^ 



The quantities d can be expressed in terms of oc 



We have 



A _ A 1 



n+1 ~ m or, TTn' ^S T 



Thus 



(9) d^ = 06,..., d^_^^ = ^ , ... , 



so that the d's are convergents to a continued fraction. 



The quantities D can be obtained as functions of 

 Oi from the definition (6). They can also be obtained by 

 solving; a second order linear difference equation with 

 constant coefficients for the quantity =?— , obtained from 

 the recursion formula (9). " 



It is convenient to expand <^ in the neighborhood 

 of the surface of the sphere in terms of spherical harmonics. 

 The expansion of a unit point source lying outside the sphere 

 at a distance X from the center Is 



2 



(10) I * ^ P-l(cos e) + ^ P2(cos e) + ... , 



X X 



while a point souTce lying Inside the sphere has an expansion 



2 

 (11) ^ + -^P^ (cos e) + ^ P2(cos «■) + ... , 

 R R 



