704 BELL SYSTEM TECHNICAL JOURNAL 



Heaviside's first step in attacking this problem was to start with the 

 differential equations, and replace the differential operator d'dt by 

 the symbol p, and the operation | dl by l/p, thus reducing the equa- 

 tions to an algebraic form. He then wrote the impressed e.m.f. as 

 1 (unity), thus limiting the validity of the equations to values-of t = o. 

 The formal solution of the algebraic equations is straightforward and 

 will ln' written as 



/; = 1 IKp) (33) 



where /; is the "generalized indicial admittance," or Heaxiside func- 

 tion (denoting current, charge, potential or any variable with which 

 we are concerned) and II{p) is the corresponding generalized im- 

 pedance. Thus, if we are concerned with the current in any part 

 of the network, we write 



A=l'Zip). (34) 



The more general notation is desirable, however, as indicating the 

 wider applicability of the equation. 

 The equations 



h = l/II{p) 



A = \:'Z(p) 



are the Ileaviside Operational Equations. They are, as yet, purely 

 symbolic and we ha\e still the problem of determining their explicit 

 meaning and in particular the significance of the operator p. 



C"om[iarison of the Hea\iside Operational Equations with the 

 integral e(|uations (29) and (30) of the preceding ch;ipter leads to 

 the following fundamental theorem. 



The Ileaviside Operational Equations 



A=i:Z(p) 



h = \:ii(p) 



are merely the symbolic or short-hand equivalents of the corresponding 

 integral equations 



pikprr"^"'-""- 



The integral equations, therefore, supply us n'ith the meaning and sig- 

 nificance of the operational equations, and from them the rules of the Oper- 

 ational Calculus are deduciblc. 



