972 Dr. Gr. Breit on the Effective Capacity of 



relation of (r, 0) to (#, y) is shown in fig. 7. By virtue 

 of (3), when r = a, 



V o = V=- L [0+sin0cos0;^ if -l<e<+~] 



u 7r L at 2 2 



and }> (4) 



V o = V=--[9r-0-sin0cos0]^ if |<tf<+**J 



7r L J at 2 2 J 



for if r = «, then V = V. 



Thus V is known over the circumference of a circle. 

 It vanishes at infinity. Therefore, if the value of V over 

 the circumference could be written as a Fourier series in 6, 

 say as 



(V ) r=ra = -^ + 2 (b m cos md + a m sin md), . . (5) 



then the value at any point (r, 0) (r>a) is 



h / a m d m \ 



Yo=^+X{b m —cosm0 + a m ^smm0y . . (6) 



because this satisfies (1), vanishes at infinity, and satisfies 

 Laplace's equation 



a 2 Vo , a 2 Vo _ o 



It remains, therefore, to express the two equations (4) in 

 the form (5) . 



This is done by defining two functions f{&) and (f)(0) in 

 the following manner : — 



f(6) = e if -|<0<+|, 

 ftp) =>ir-e if f <0<+ 3 f, 



/(0) is periodic of period 2ir. 



0(0) = sin cos if —~<0< + £, 



0(0) = -sin0cos0 if ^<6><+ — 

 0(0) is periodic of period 2tt. 



