132 Mr. 0. Heaviside. [June 15, 



bj (84). Using these in (123) we obtain 



(126) 



The first series is the ordinary expression for e~ x x~ l with the terms 

 inverted, whilst the latter contains a reminiscence of the companion 

 to the Fourier cylinder function. 



60. To see whether there is a notable convergency for calculation, 

 take x = 2. Then 



e - 2 = 0-1353, 





This is evidently about J by the look of it, especially when diagram- 

 matically represented. Also 



---' =9-7479. 



JB 



So (126) gives 



log 2 = 0-1353X9-7479 0-5772 0-375X0-1353 

 = 0-6909. 



By common logarithmic tables we find log 2 = O6923. The differ- 

 ence is 0'0014. Doing it another way, we may prove by multiplication 

 that 



which is an interesting transformation. This, with x = 2, gives 

 1*3203, and produces a much closer agreement. It is probably for- 

 tuitous. 



'>!. 



Independent Establishment of the Last. 

 61. We can establish (126) independently thus : We have 



1 c x i _ f x 



-=~ -f -1_ (128) 



a? a; a? 



-* 



..... (129) 



