(7) 



(8) 



516 BELL SYSTEM TECHNICAL JOURNAL 



F{t — t) — exp i{t — T)ui -}- i \ iidri \ 



= exp i(t — t)(jo -\- i \ fj-dri — i | ixdri 



L Jq J l-T 



= exp [^S2(/)]-exp — tcor — i \ lJ-{t — T\)dTx 



Substituting this expression in (5) for F{t — t) and writing 

 exp ( - j r ix{t - Ti)dTi j = AI{t, t), 



we have for the current in the network 



/ = e'-^^'^'- r Af(t, T)e-''''A'(T)dr. (9) 



We now split the integral into two parts, thus: 



J ft /^OO /•(» 



Jo J t 



The second integral on the right represents an initial transient which 

 dies away for sufficiently large values of time, t, while the infinite 

 integral represents the total current, /, for sufficiently large values of /. 

 We have therefore 



/ = e'^^dt. r M{t, T)e-'^'A'{T)dT + It (10) 



Jo 



= F(*co, /)e'^«''' + It, 

 where 



M(t, T)e-'-'A'(r)dr. (11) 







The transient current,^ It, is then given by 



/j, = g'-y^-i^ r M{t, r)e-''^'A'{T)dT. 



(12) 



The foregoing formulas correspond precisely with the formulas for 

 a constant frequency impressed e.m.f. ; these are 



I.s = e'"' r e-'-^A'{T)dr, 

 Jo 



(10a) 



^Hereafter the transient term It of (10) will be consistently neglected and the 

 symbol / will refer only to the quasi-stationary current. 



