﻿Capacity of a Pancake Coil, 733 



It is advantageous to transform this by 

 fju = sin v\ 



v = j si nil ?/, 

 which reduces (6) to 



(?) 



vv=- a Ua-»->£K{(.-<}.. 



(8) 



Now the electrostatic problem to be solved is that of 

 finding for V a solution which tooether with its first 

 derivatives is finite and continuous, which is independent 

 of 0, which satisfies the Lnplacian 



V 2 V =0 (9) 



which vanishes at infinity at least to the first order and 

 which at the disk becomes 



V = V »- L 5I ^ 



But the equation of the disk is 



u ^ ; 

 in which case (3) reduces to 



r = a cos v. 

 Hence, using (7), equation (10) becomes 



V = V -L(l-^)| (11) 



Now the expression 



[a H P n {v) + /3 n Q n (v)'] [a»P»00 + WJ,(rt], 



where F n , Q n are Legendre functions of the first and second 

 kind respectively, when substituted in (9) satisfies (9) in 

 virtue of (8). If, then, one should be able to find such 

 values of «„, /3„, a m 6 n , and such values of n that 



V = 2 [« B P„Cv) + /3 B Q„W][a„P B (/ A ) + ft„Q w (iu)] 



n 



should vanish at infinity to the first order and should 

 degenerate into (11) when v approaches zero along the axis 

 of pure imaginaries, then, in virtue of the uniqueness of the 

 solution of (9) for given boundary conditions, the summation 

 written gives the value of V. 



If the summation written is an infinite series it also 

 gives V, provided it is universally convergent as to /jl 

 and v. 



