348 HRI.L SYSTEM TECHNICAL JOURNAL 



Performing the indicated differentiations 



V{t) = \-RoA {o) V{t) - Ro£ V{t)A \t - T)dT. 



Now A{o)= J^ and 



A'{t)=pe-P'{h{pt)-Io{pt)) ^\^ 



i 



and Ro = \^L/C; therefore the equation becomes 



V{t)=\ + I jrV(/ - r) [lo {pr) - h{pr) Y-P^dr. 

 As a matter of convenience we change the time scale to pt, and get 



F(0=o + o i"v{t-r)\Io{T)-h{T)]e~Hr (263) 



where it is understood that / is actually pt. This is the integral equa- 

 tion of the problem and is in the canonical form of Poisson's integral 

 equation. 



Before solving this equation numerically I shall show how a simple 

 approximate solution is obtainable immediately; an advantage often 

 attaching to this type of integral equation. 



The function -T- e^'/oW is equal to -1 for t = and converges 

 dt 



rapidly to zero. V{t) has, as we know from the operational equation, 



the initial value 1/2 and the final value 1. Neither function changes 



sign. It follows from the mean value theorem that the equation 



can be written as 



where a<\. Integrating 



and 



V{t)=\-\v{t){e-^h{at)-l\ 



^^« = r+^^TO- ^'''^ 



