﻿46 Dr. J. G. Leathern : Effect of a Long 



and there is an extra term 



i r d\ 



tt(0 ( 



It is not necessary to pursue the modifications for the 

 remaining cases. The essential point is that the only factors 

 occurring in the denominator of the integral are those of 

 J(X), so that the statement made above as to the general 

 character of the integrals is applicable in all cases. 



V. A Simplification of the Mathematical Discussion of the 

 Effect of a Long Straight Wire on Electric Waves. By 

 J. GL Leathem, M.A., D.Sc* 



1. npHE use of Bessel's functions in the discussion of 

 JL problems relating to the transmission of electric 

 waves along wires is of course inevitable when precise 

 mathematical formulation is required, but it has the dis- 

 advantage of presenting formidable difficulties to the student 

 of rather limited mathematical attainments. Even one who 

 has had considerable training in mathematical analysis may 

 perhaps feel discouraged when, at the very outset of his 

 study of a physical subject of great practical interest, he is 

 faced with a function K denned by the relation 



K (» = (-5772157 ... + log 2- log«)J (a?) 



-2{J 2 (^)-iJ 4 (^)+-iJ 6 (^ }, 



and finds that upon his appreciation of the values of this 

 function for different complex values of x must depend his 

 right understanding of the subsequent investigation. To 

 find even approximate formulae for the values of this and of 

 the J functions when x is very large or very small, re- 

 course must be had to advanced treatises such as Heine's 

 Kugelfunctionen. 



Now while the definitions of the J functions as power 

 series, and of the K functions as series of J functions, are 

 from the mathematical point of view absolutely precise, yet 

 the formulae are of so difficult a character that they utterly 

 fail to convey to the mind of the average reader a corre- 

 spondingly precise idea, and so the precision theoretically 

 attained is to a considerable extent ineffective. Even if this 

 were not so, the precision of the mathematical formulation is 

 ultimately spoiled by the substitution of values which are 



* Communicated by the Author. 



