46 BELL SYSTEM TECHNICAL JOURNAL 



It follows as an immediate corollary that the Heaviside operational 

 equation 



h = l/H(p) (8) 



is merely a short-hand or symbolic equivalent of the integral equation 



^ = /"e-'/,W*. (9) 



The significance of the operational equation and the rules of the Heaviside 

 operational calculus are therejore deducible from the latter equation. 

 The whole problem is thus reduced to the purely mathematical problem 

 of solving the integral equation. 



It should be remarked in passing that, while the Heaviside opera- 

 tional calculus has been elucidated in connection with the solution 

 of a set of differential equations involving a finite number of variables, 

 it is not so limited in its applications. It is applicable also when the 

 number of variables is infinite and to such partial differential equ- 

 tions as the telegraph equation. The foregoing theorem applies also 

 to all such physical problems where an operational formula A = \/H{p) 

 is derivable. 



Before discussing the solution of the integral equation (9) and de- 

 ducing therefrom some of the rules of the operational calculus, a 

 simple but interesting and instructive example of the way the oper- 

 tional formula is set up will be given. 



Consider a transmission line of infinite length along the positive 

 X axis and let it have a distributed inductance L and capacity C per 

 unit length. Let a unit voltage be applied to the line at the origin 

 re = at time / = 0; required the line current / and voltage V at 

 any point x at any subsequent time /. 



The differential equations of the problems are 



L^I = - — F 



cgv= -i-i. 



pi 

 Replacing — by p, we get 



ol 



px 



V=e-^Vo, 

 where v = 1/ V LC and Vo is the line voltage at jc = 0. 



