6 BELL SYSTEM TECHNICAL JOURNAL 



These expressions may be interpret ed i n either of two ways. (1) 

 \{ p = iu where co = 2x/ and i = V— 1, then the formulas are the 

 usual complex steady state expressions. On the other hand if p is 

 regarded as djdt, they are operational formulas. It is worth while to 

 explain the latter briefly on account of its own interest and its bearing 

 on the operational calculus. 



The differential equations of A and $ are 



1 ^2 \ 



where 



Now it will be recalled that the differential equation of the electro- 

 static potential V is 



V^F = 47rp 

 and that its solution is 



V= f^dv. 



Operationally 



and the corresponding solution is 



,/ r 



V2 - ^ ) $ = 47rp 



-(pic) 



$ = I ^ dv 



-I 



pe- 



Now this is an operational equation in which p is an arbitrary time 

 function. Its solution depends on the following general operational 

 theorem.^ 



// X is defined by the operational equation 



X = me-"\ 



then 



X =f{t -X). 



Consequently, the solution of the operational equation for $ is ® 



6 See "The Heaviside Operational Calculus," Bull. Amer. Math. Soc, Jan., 1926. 

 * A proof of this theorem by operational methods was privately communicated to 

 the author several years ago by Stuart Ballantine. 



