250 ELECTROSTATICS. [PT. II. CH. VI. 



so that 



we have 



(6) R = 2-TraV J- f "<I> (r) sin 



do J o 



We may now change the variable from 6 to r by differentiating 

 the relation 



r 2 = a 2 + 6 2 - 2a6 cos 0, 



7-^r = ab sin #c#, 

 rdr 



(7) s 



For = 0, r = a 6, and for = TT, r = a + b, so that 



. 



b ab 

 Calling 

 we have 



ra+ 



.(8) 



^ a- 



and 



(9) R = ^aa- 1 JJ (^(a + 6) - (a - 6))J . 



By the conditions of the problem this must vanish, so that we 

 have the differential equation for "", 



(10) 



which being integrated gives 



(IT) i 



(a + 6) -(-&) =(76, 



a functional equation to determine . Differentiating twice with 

 respect to b, 



(12) 



