398 Mr. 0. Heaviside on Electromagnetic Waves, and the 

 which, when t\ is infinitely small, becomes 



H =-t^ < 182 > 



First of all, at a point distant r from the centre, comes the 

 primary disturbance or head, 



H=^-, (183) 



when vt = r—a, lasting for the time t x . It is followed 

 by the diffused negative disturbance, or tail, repre- 

 sented by (182), lasting for the time 2a/v. At its end comes 

 the companion to (182), its negative, when vt=r + a, lasting 

 for time t h after which it is all over. This description applies 

 when r > a. If r < a, the interval between the beginning and 

 end of the H disturbance is only 2r/v. From the above 

 follows the integral solution expressing the effect of f x varying 

 in any manner with the time. 



27. Alternating Impressed Forces. — If the impressed force 

 in the sphere, or wherever it may be, be a sinusoidal function 

 of the time, making p 2 =— n 2 , if n = 2-7T x frequency, the 

 complete solutions arise from (132) to (135) so immediately 

 that we can almost call them the complete solutions. Of 

 course in any case in which we have developed the connexion 

 between the impressed force and the flux, say e = ZC, or 

 C = Z _1 e, when Z is the resistance operator, we may call this 

 equation the solution in the sinusoidal case, if we state that 

 p 2 is to mean — n 2 . But there is usually a lot of work needed 

 to bring the solution to a practical form. In the present 

 instance, however, there is scarcely any required, because u 

 and w are simple functions of qa, and g 2 is real. The sub- 

 stitution p 2 =z—n 2 in u results in a real function of nr/v, and 

 in wina real function x ( — 1)*. Thus : — 



nr v . nr 



:COS Sill 



v nr v 



i, .... (184) 



./ . nr v nr\ 



w 1 =i[sm — + — cos — ) 



\ v nr v J 



( A 3r 2 N nr 3 



U 2 = ( 1 5~2 C0S 



V rrrv v n, 



U- 3t ,2 \ . nr '6v nr") 

 1 — 2~r sm •" —cos — }■ 

 rfr) v nr v ) 



In the case m = l, if (f i) cos nt is the form of f h so that 



i. . (185) 

 6v^\ . nr . ov nr^ I 



