340 Bf.J.J- SVSTP.M ■/T.CtfXlC.II. JOURNJL 



sultanl voltage across the impedance K. Then, operationally, 



so that Vo, thus defined in physical terms, is the Vo of equations (247). 

 Now let this voltage be impressed on a circuit consisting of an 

 impedance 2Z2 in series with an impedance K — Zi so that the total 

 impedance is X+Z2. Let the resultant voltage drop across the im- 

 pedance element K — Zo be denoted by Vu then operationally 



K K-Z. ^ 



"^'^ Ki^r K^zr^^' 



which agrees with Vi as given by equation (247). 



Similarly if voltage Vi is applied to a circuit consisting of an im- 

 pedance 2Zi in series with an impedance K — Zi and if Vo denote the 

 voltage drop across impedance K — Zi, then 



V2 =\/J.llJ.9. 



We can thus see physically what the voltages Vo,Vi,V2 . . mean in terms 

 of simple circuits consisting of K and Zi in series and K and Zj in 

 series respectively. 



I shall now work out a specific problem exemplifying the preceding 

 theory. The example is made as simple as possible for two reasons. 

 First because its simplicity makes it more instructive than when the 

 phenomena depicted and the essentials of the mathematical methods 

 are obscured by complicated formulas and extensive computations. 

 Secondly while the general method of solution illustrated is thor- 

 oughly practical we cannot hope to arrive at the numerical solutions 

 of the complicated problems without a large amount of laborious 

 computations. Problems involving transmission lines with com- 

 plicated terminal impedances are among the most difficult, as regards 

 actual numerical solution, of any which present themselves in mathe- 

 matical physics. On the other hand, the formal solution (248) gives 

 at a glance the essential character of the phenomena involved. 



The specific problem we shall deal with may be stated as follows: 

 A unit e.m.f. is directly impressed on the terminals of a transmission 

 line of length s, the distant end of which is closed by a condenser Co. 

 The line is supposed to be non-dissipative, its constants being in- 

 ductance L and capacity C per unit length. Required the current at 

 any point .i;(x<5) of the line. 



We write VL/C = k, l/\/LC = v: then by virtue of the preceding 



