76 Scientific Proceedings, Royal Dublin Society. 



9. We may now presume that we know the size of conductor to be used. We 

 seek to determine its electrical performance. 



The equations connecting the voltage v and current c at any point distant x 

 from the sending end of the line are 



_ (^ _ /'„ J d\ 

 dx \ cltj 



dc I d \ 



dx \ dt) 



We are mainly concerned with the space variations of v and c along the wire. 

 We suppose a sine wave of voltage to be impressed on the line at the sending end. 



Differentiating the first of these equations with respect to x and using the 

 second, we obtain 



dx' V dt \ dtj 



We may write i]} for 3- in these equations, changing v, the instantaneous 



Cvo 



voltage, into %, the maximum voltage at the position considered, so that 

 -5-I = {R + ipL) [K + ipS) v^ = m-Vq. 



The solution of this equation is 



Wo = ^ofi'""^ + B.c-'"'^. 



If V and C be the effective values of voltage and current at the position r, we 

 have F=^e'- + ^e-- 



where A and B are constants independent of x. 



Also - ^ = (« + WL) Co and ^^ = m (^„e»- - £,e-"'-). 

 Hence «» = ^^ (-I0.- + ^o.-)- 



But ^^i^ = n, .: nc, = - ^/"- + ^o«— . 



m 



and nC = - Ae'"^ + Be-""". 



At the sending end of the line x = 0, and therefore 

 Vi = A + B, nC, = - A + B; 

 .-. 2A= Vi- nCu 

 2B= V, + nC,. 

 Hence 



Y ^XYi (e'"" + c-'"") - \nCi (c""' - e""'*) = F, cosh mx - nCi sinh mx. 

 Likewise 



Q = 1(7 igmx + c"'"^) - FT-Vc""" - e""") = Ci cosh mx 'sinh mx. 



