﻿Straight Wire on Electric, Waves. 49 



other infinite for x = co ; the one which vanishes must he the 

 approximation to h(x), the other the approximation to #(«£')• 

 Thus if the real part of ix is negative, g(x) is a , e~ ix and h(x) 

 is b'e lx . When x is real neither becomes infinite, and so 

 there is nothing to indicate which solution or combination of 

 the two solutions is g(x) ; but when that case arises we shall 

 find a method of picking out the particular approximation 

 which we require. 



An exponential infinity or zero is an infinity or zero of 

 very high order, and might possibly be associated with a less 

 important infinity or zero represented by a power of x ; this 

 power factor would escape notice in the first approximation, 

 so in order to detect it if it exists in the present instance we 

 assume 



y = a"x K e~ ", 



and try whether this will give a better approximation to 

 solution of the differential equation (1). Substituting and 

 neglecting x K ~ 2 we get 



which is satisfied by X=— |. The other solution maybe 

 treated in the same way. Thus we have the better approxi- 

 mations 



g(x) = a" e-^/^/x, h(x) = b" e*/ sj~7, 



and, by differentiation, 



g\x) = - ia" e-™l x /x~~, h'{x) = ib" e ix J V^7 



a 7 'and b" being constants. 



It is to be noticed that though a and a" are arbitrary, the 

 ratio a" \a is not arbitrary; it is a definite complex constant 

 whose value we do not need to ascertain, and may be denoted 

 by A. Similarly we denote b" \b by B. We assign to a and 

 to b the value unity, and have finally the following table of 

 approximate forms : — 



(2) 



It is readily verified that g(nx) and h(nx) are solutions 

 Phil. Mag. S. 6. Vol. 12. No. 67. July 1906. E 





x small. 



x large. 



9 (•») 



1 —±r 2 



Ae~ ix / \/x 



•(*) 



2 x 



— ike- ix j \/x 



h {x) 



logx 



B&fy/Z 



*'(«) 



1/x 



iBe*/ \ f x 



