294 _48- 



where |P (cos 9)j are the Legendre polynomials of cos 0© 

 (See diagram on page4X'» Integration with respect to X 

 yields the expansion of a uniform line source. The ex- 

 pansion of <$ is then easily obtained by using the distri- 

 bution of sources and sinks given in (7). 



For the present purpose it suffices to know merely 

 the value of ?r / <i>^dS, the mean value of ^ over the 



surface K of the sphere. The integral of a unit point 

 source over the surface of the sphere is, by (10) and (11), 



(12) 1 or ;^ 



according as the source lies outside or inside the sphere. 



The first of these is to be expected from the mean value 



theorem for potential functions, while the second is a 



constant, independent of the position of the sources inside 



the sphere© 



The value of — =-s- / 4>„(3S can now be obtained by 

 4TrA^ -/k ° 



using the distribution of sources and line sinks (7). The 



contribution of the distribution ins ide the sphere reduces 



P -I 2 



to A J due to the source of strength A at the center. 



This follows from the second result in (12) and the fact 

 that the successive images, each consisting of a source and 

 a line sink, have a total strength zero. The contribution 

 due to the source A^D at Q^+i ^'^'^ *^® ^^^® sink of 

 strength ^ per unit length along Q^Qn+l ^^®® equation 



n-1 

 (V) ) is 



2 1 A "^ 



OQ . °n-l OQn+i 

 n+1 



A D„ . , - T^— log 



Vl 



n-1 - ^ZT ^"^ -^ 



