680 Prof. J. Perry on 



B , &c. Between A and B we have the standard cable. 

 Let the currents in the line at A, A , and B be c, c , and C. 

 Let the voltages at these points be i\ v , and Y. Let 

 V/C = v/c = p. Knowing the nature of the contrivance we 

 can calculate r and c from v and c. It is also known that 



V = v cosh mn -\ c sinii mn, 



C = c cosh mn -\ -,—. r sinh mn, 



r + Iqi 



where n= >/(r + lqi)(s + kqi). 



Putting V/O or p eqnal to v/c, we have a quadratic to 

 calculate p and therefore V and C. 



I am in the habit of writing c = l, meaning c = sin qt, and 

 then v = p. Whatever the contrivance may be, 



V = "+/3p 



C P a + bp> 



where a, /8, a and b are given in value ; they are usually 

 complex unreal quantities. Solving for p and finding we 

 have two answers, 0, and C 2 say. In general C i C 2 =l, 

 andi£i(a + £) be called P, 



C = P ± s/W^-T. 



Now P is very easily evaluated, and therefore C. 



Example : — (1) If our contrivauce is a resistance i\ in 

 series with the line c = c=l, v = r 1 -\-p, Thus if i\ is lan 

 Ohmic resistance II in series with a capacity K and an 

 inductance L, 



r, = B + L^+^j. 



This is a very simple case, the simplest form of which is that 

 of Mr. Pupin, a mere inductance coil in the line. The trouble 

 in such a case is that the whole telephonic current passes 

 through the coil and there is considerable Ohmic loss of energy. 

 It is to be remembered that the value of R to be taken in any 

 such case may be many times the ordinary resistance, because 

 c 2 R is taken to be the total loss of power, and this is due to 

 hysteresis, eddy-currents, &c, as well as mere Ohmic loss. 



(2) Again taking a case already mentioned ; the con- 

 trivance consists of two equal resistances i\ in series with 

 the line and a resistance leak to earth ?' 3 from the point of 

 junction of the other two. 



