420 Dr Bromwich, Symbolical methods 



It is easy to confirm tlie Fourier-expansion from the symbolical 

 formula above, by using Heaviside's general rule*. 



However, in the actual problem we want the value of v at 

 r= a; and for this purpose, from the considerations already given 

 in § 2, it will be sufficient to replace coth qa by unity to obtain 

 the series 



^ = 1 - i_s + sqa =-''' + I^ (^^^ + ^'^'«' + ^V«^ + •••) 

 where n = s/{l — s). 



Then (as in § 2) we obtain the asymptotic series 



^ = n\-l+ ^ f 1 _ ^ ^' I L- 3 n^a^ 



Vn I (1 — s)^/(TTK-t^\ 9. u-f. 9 i ,.2/2 



(1-5) Vi-rTKt) \ 2 Kt ~^ 2 74 k¥ 



As explained in § 2, the formula is valid only if e-«-A< is negH- 

 gible; and to obtain four-figure accuracy, this requires a^/Kt to be 

 not less than about 8-5. Hence to obtain good results from the 

 asymptotic series, n must be small, of about the order 1/15 to 1/20. 



In the Fourier-expansion, to avoid the labour of the actual 

 calculation of the roots of the equation for d, I decided to adopt a 

 simple value for 6^ (which is the root requiring the greatest 

 accuracy), and to deduce the corresponding value of s 



The value 6'i = ITOWISO = 2-9671 was selected ; and this gave 

 l/s = 17-827. Then d^ was not very different from 29^, and the 

 value ^2 = 5-944 was comparatively easy to calculate. A further 

 simplification in the arithmetic was made by taking 



a^Kt = 6^^ = 8-8 (nearly). 



Then the first and second terms in the Fourier-expansion for v 

 were found to be 



^0 (-04248 -f -00191) = (-04439) v^. 

 The corresponding asymptotic series is found to be 



1 - -01546 + -000725 - -000075 + -000008 = -98520. 

 Then on substitution we find 



^^^Vo (-04441); 



* This rule may be written in the form 



liP}^ FJO) F{a)^ 



A{p) A(O)"^^ aA'(a) ' 

 where the summation refers to all the roots p = aoiA(p) = 

 Ph,?M °^ ^^0.^"^!^^^°" (^^°^ different points of view) have been given in 



Paifr;, vo^ 2fpp 226;3^r'''' ' ' °'^'' discussion will be found in his Electrical 



iromtt^Yhfr'l^^'^'i^T'^^^ !'™ ^(0)/A(0) reduces to 1 and cancels a term 

 wrSng^ai^^). °^ *^' equation; and the values of a are - .e^a^ (found by 



