366 BULL SYSTLM ILCHKiClL JOUR.X.IL 



If the solution is carried out as indicated by (291) below, and if the 

 notation at = x is introduced, we get without difficulty 



I{t)=Ioil-r(l-e^(x)e-^)-\-r-{l-e,ix)e--^)-r'{l-e3{x)e--^)+ . . . (- . 



where the function e„{x) is defined as: 



e„Cx-) = l+.v/l!+-^-V2!+.TV3!+ . • . +x''~W(n-l)l 



= first n terms of the exponential series. 



For all finite values of the resistance increment r the series can be 

 summed by aid of the identity 



l-en(x)e-'^= I dxe-'^x"-^/{n-l)l 

 Jo 



Substitution of this identity gives 



I(t)=Io(l-r / V(i+'-)-^rfjcj 



l_|_^g-(l+r).x- 



lo- 



1 + r 



Equation (289) is an integral equation of the Volterra type, which 

 includes the Poisson integral equation as a special case. Its formal 

 series solution is obtained as follows: — Assume a series solution of the 

 form 



L(t)=h(t)-Ii{t)+Io{t)-Is{t)^ . . . (291) 



and define the terms of the series by the scheme 



h(t)=~j\{r)Io{r)A{t-T)dT 



(292) 



h+i(t)=~j\(r)h{r)A(t-r)dr. 



Direct substitution shows that this series satisfies the integral equa- 

 tion. Furthermore, it is easily shown that it is absolutely con- 

 vergent. 



While this series solution is not, in general, well adapted f(jr numer- 

 ical calculations, it throws a good deal of \aluable light on the ulti- 

 mate character of the oscillations in the important case where E(t) 

 and rU) both vary sinusoidally with time. In this case, if the fre- 

 quency of the applied e.m.f. be denoted by F and that of the resistance 

 variation by/, it is easy to show that the cm-rent Iu(t) in the un\aried 



