﻿Straight Wire on Electric Waves. 47 



only approximately correct for the physical constants 

 involved. 



It seems desirable, therefore, to try to make the study of 

 this part of the theory of Electric Waves easier and more 

 attractive to the average student by simplifying the mathe- 

 matical formulation ; and while this must necessarily involve 

 a sacrifice of that appearance of definiteness which attaches 

 to the use of functions that have been at least partially 

 tabulated, it will be shown that all those results of the 

 mathematical theory which are of interest because of their 

 admitting of direct experimental test can be obtained as well 

 by quite elementary reasoning as by the elaborate and rather 

 troublesome discussion which depends upon the use of 

 standard functions. 



The simplified treatment will be applied to the transmission 

 of oscillatory currents by a wire, and to the scattering of 

 waves by a perfectly conducting wire. The main idea of the 

 method is that we take as specification of a solution of a 

 differential equation, not an infinite series, but the differential 

 equation itself, together with the limitations to which the 

 solution is subject for indefinitely small or indefinitely great 

 values of the argument. This specification leaves undone 

 much that has to be done before the corresponding function 

 can be evaluated for any assigned value of the argument ; 

 nevertheless it is a precise specification, and proves to be 

 well adapted to the purposes to which we shall apply it. 



2. Solutions of the differential equation 



We shall assume that there is a solution g(x) which is 

 finite for x = but possibly infinite for ,r = co; and that 

 there is another solution h(x) which is zero for # = oo, but 

 possibly infinite for ,x = 0. The equation being linear, each 

 of these solutions will contain an arbitrary constant factor. 



Case of x very small. 



In considering the case of x very small we take the equation 

 in the form 



°^y dy , a 



and in attempting to get approximate solutions we note that 

 the last term has the appearance of being unimportant in 

 comparison with the middle term. We therefore, as an 



