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

 so that, finally, 



(f~ )V")=Jo(<rtO- • • (225) 



It is also worth notice that, integrating in a similar manner, 



\ P 2 ) K) 2 |2_ 2 2 J2 |4 + ' * * ' 



=J (<rtf) (226) 



These theorems present themselves naturally in problems re- 

 lating to a telegraph-circuit, when treated by the method of 

 resistance-operators. A special case of (225) is 



p*(l)=(rt)-i, (227) 



which presents itself in the electrostatic theory of a submarine* 

 cable. 



We have now to generalize (225) to meet the case (221). 

 The left member of (221) satisfies the partial differential 

 equation 



v 2 S7 2 =p 2 -(r 2 , (228) 



so we have to find the solution of (228) which becomes 

 J (crti) when r=a. Physical considerations show that it must 

 be an even function of (r— a), so that it is suggested that the 

 t in J (ati) has to become, not t—(r—a)/v or t + (r — a)/v, 

 but that t 2 has to become their product. In any case, the right 

 member of (221) does satisfy (228) and the further prescribed 

 condition, so that (221) is correct. 



* Thus, let an infinitely long circuit, with constants R, S, K, L, be 

 operated upon by impressed force at the place s=0, producing the potential- 

 difference V there, which may be any function of the time. Let C be 

 the current and V the potential -difference at time t at distance x. Then 



where g=(R+Lp)*(K + S#)*. Take K=0, andL^ then, if V be zero 

 before and constant after £=0, the current at z = is given by 



C = V (S/R)^ (1), 



and (227) gives the solution. Prove thus : let b be any constant, to be 

 finally made infinite j then 



p*(l)=^(l+&p- 1 )-* 



= b*J (±bti)e- bt/2 



by the investigation in the text. Now put b=ao , and (227) results. 



In the similar treatment of cylindrical waves in a conductor, pi, p$, &c. 

 occur. We may express these results in terms of Gamma functions. 



