THE HEAVISIDE OPERATIONAL CALCULUS 45 



iary variables h\ . . . h„, — 



a,i/'i + . ■ • + a^„h„ = 1 



(/>0) (4) 



a„i}h + . . . + a„„h„ = 0. 



The function on the right hand side, written, in accordance with 

 the Heaviside notation, as unity is identically zero for /<0 and 

 unity for />0 and hi . . . h„ are identically zero for /<0. 



It may then be shown that ' 



xj = j^ £f (t - y) hAy) dy, (j=l,2,...n) (5) 



so that the solution of (3) depends entirely on (4). 



H^quations (4) formulate the problem actually dealt with by Heavi- 

 side who did not explicitly consider the more general equations (3). 

 His method of attack was as follows; Writing p" for the differential 

 operator d"/dt" equations (4) become formally algebraic and yield a 

 purely symbloic solution 



^' = Wpy ^^^ 



Equation (6) is the Heaviside operational formula; as it stands, 

 however, it is purely symbolic and the problem remains to find the 

 significance of the equation and to deduce therefrom the value of 

 h = h(t) as a. function of /. 



Heaviside's method from this point on was one of pure induction. 

 From the known solution of specific problems he inferred general 

 rules for expanding and interpreting the operational formula: the 

 body of rules thus developed for solving the operational equation 

 may be appropriately termed the Heaviside Operational Calculus. 



The contribution of the present paper to the theory of the Heavi- 

 side operational calculus depends on the following proposition and 

 its immediate corollary.^ 



The differential equations (4), subject to the prescribed boundary con- 

 ditions, may be written as: 



hj = 0, for /<0 and 7 = 1, . . . n, 



PH^P) 



e-P%{t)dt, for/>0. 



The integral equation is an identity for all positive real values of p and 

 consequently determines hj(t) uniquely. 



' This formula has been established in previous papers. It is briefly discussed 

 in .Appendix I. 



^ See Appendix I. 



