ilNClIT llll:Ot<y .-/A'/i (ifHR.ll lOX.Il. C.ll.CCI.rS 7(U 



Thr fonmilas aiul nu-tlKxIs (iodiirfd aluno apply not only to finik- 

 networks, in\<>Kins» a finite system of linear ecjiiations, but to infuiile 

 networks and to transmission lines. invol\inj{ infinite s\slems of equa- 

 tions, and partial dilTeri'ntia! eciualions: in fact to all I'leetrical and 

 ilynamieal systems in wliieli liie connet'lions .md constants aie lini'ar 

 and invariable. 



.Secondly the \arial)le determined 1>\' formula (20) and (2',)) need 

 not, of course, be the current. It may e(|u.dly well be the charjje, 

 potential drop, or any of the \ariables with which we may ha()[)en 

 to be concerned. This fact may be explicitly recoj;ni/ed by writing 

 the formulas as: 



' = fhine-f'dt, (30) 



pII(P) 



xit)=j^£h{t-T)E{r)dr. (31) 



Here E(t) is the applied e.m.f., .y(/) is the variable which we desire to 

 det»Tmine ("ch.irije. current, potential drop, etc.), and 



x = E.II{p) (32) 



is the operational equation. H{p) therefore corresponds to and is 

 determined in precisely the same way as the impedance Z{p), but it 

 may not have the physical significance or the dimensions of an im- 

 pedance. Similarly in character and function, /;(/) corresponds to the 

 indicial admittance, though it may not have the same physical sig- 

 nificance. It is a generalization of the indicial admittance and may be 

 appropriately termed the Ileaviside Function. SiniilarK- //(/>) may 

 be termed the generalized impedance function. 



CH.MTER 111 



TiiH Heavisidi-: Prof^le.m .\nd the Oi'er.\tion.\l Equatio.n 



The physical problem which Heaviside attacked and which led to 

 his Operational Calculus was the determination of the response of a 

 network or electrical s\stem to a "unit e.m.f." (zero before, unity after 

 time t = o) with, of course, the understanding that the system is in 

 equilibrium when the electromoti\e force is applied. His problem 

 is therefore, essentially that of the determination of the indicial 

 admittance. In our exposition and critique of Heaviside's method of 

 dealing with this problem we shall accompany an account of his own 

 method of solution with a parallel solution fnjm the corresponding 

 integral equation of the problem. 



