352 BELL SYSTEM TECHNICAL JOURNAL 



These two equations are simultaneous integral equations of the Poisson 

 type in Vi and V2, which completely determine these voltages provided 

 the admittances and the impressed voltages are known. They therefore 

 represent the application of a new type of integral equation to the problem 

 of electric circuit theory. 



The numercial solution of the general case, either by (248) or 

 (272-273) is necessarily laborious when the terminal impedances are 

 complicated and is only justified when the technical importance of the 

 problem is considerable. I wish, however, to emphasize two points 

 in this connection : the numerical solution is always entirely possible 

 and, compared with other and older forms of solution, enormously 

 simpler. One has only to inspect the classical forms of solution of 

 problems of the type to realize the truth of this last statement. 



I shall now give two applications of equations (272-273) to specific 

 problems, in one of which an approximate solution of the integral 

 equation can be gotten, and in the other of which numerical integra- 

 tion is applied. 



Problem I. Given a non-inductive cable of distributed constants 

 C and R and length s, with unit e.m.f. applied at .r = 0, while at x = s 

 the cable is closed by a condenser Co- Required the terminal voltage 

 V{t) across the condenser Co. 



We first write down the short-circuit indicial admittances of the 

 cable; from equation (168) of a preceding section and equation (267) 

 they are : — 



S,,{t)^S2o{t) 



\~C' { -1/3 16/i 36/J ^ 



= \ VRt \ l + 2^""^+2^^~~+2^'~^+ • • } ' ^^^^^ 



Svi{t)=S2^{t) 



where /3=5-i?C/4. 

 Now the current at x = s is equal to 



It is also the condenser current due to the voltage V{t); that is 



^4 ^w- 



