422 Dr Bromwicli, Symbolical methods 



Putting s = \ the symbolical formula for v at r = a becomes 

 V _ tanh oa , 1 /, ^ „ ^ . 



Thus the first approximation (analogous to § 2) is given by 



^ = 1-1 = 1-2 /^=i-^ h 



% qa \l ira^ 77 \ tt ' 



where the value of 1/^ is given by § 4 (vi). The value of the next 

 term in this series is 



2 ,-2.a 4 r ...,d,V 



qa V'^ J ai^(Kt) w^' 



which requires the error-function integral to obtain a formula 

 suitable for actual numerical calculation. However, the numerical 

 value is less than 



V -na^ 



or ( - e— =/ia> ^ __ 



\77" / 7T V 77- 



The following terms in the series will be neghgible if a'^Ud 

 exceeds 2. 



It is easy to test the accuracy of our results by taking say 

 co-i with a7/c^ = 7r2- 9-87; so that {Ktla"^) e-'^'M is of order 

 ■| X 10-5, and so is negligible. 



The Fourier-expansion is then 



— (e . + ^e-t + _j_g--i- +...)_ __^ (.7789 _^ .^^^^ ^ .q^q^^ 



= -J (-7907) = V, (-6409). 

 The symbolical formula gives (for co = i) 



^ = ^0 (1 - ^) = vo (1 - -3592) = V, (-6408) 



and thus the agreement is as close as could be hoped for. 



Even for oj = 1 when {Ktja^) e-'^V«« is of the order 1/700, the 

 two formulae agree to three significant figures ; the Fourier-expan- 



sion IS 



-» (e-.- + ^er. + ...) = ^0 ^.g^gg _^ .^q^^) = v„ (-4926) 



,2 



while the other gives 



^ == ^0 (1 - ^) = ^'0 (1 - -5079) = V, (-4921). 

 It is therefore clear that, under proper conditions, the Fourier- 



