18 Mr. 0. Heaviside on the 



their influence on the current-amplitude, whilst the resolution 

 into independent factors is no longer possible. 



The only serious attempt to formulate the effect of the 

 terminal appaiatus with which I am acquainted is that of the 

 late M: . C. Hockiu (Journal S. T. E. and E., vol. v. p. 432). 

 His apparatus arrangement resembled that usually occurring 

 then in connection with long submarine cables, including, of 

 course, many derived simpler arrangements; ard from his 

 results much interesting information is obtainab 7 e. But the 

 results f re only applicable to long submarine cables, on ac- 

 count of the omission of the influence of the self-induction of 

 the line. The work must, therefore, be done again in a more 

 general manner. It is, besides, independently of this, not 

 easy to adapi his formulas, in so far as they show the in- 

 fluence of terminal apparatus, to cases that cannot be derived 

 from his. For instance, the effect of electromagnetic in- 

 duction in the terminal arrangements was omitted. I have 

 therefore thought it worth while to take a far more general 

 case as regards the line, and at the same time have endeavoured 

 to put it in such a form that it can be readily reduced to 

 simpler cases, whilst at the same time the results apply to any 

 terminal arrangements we choose to use. 



The general statement of the problem is this. A homo- 

 geneous line, of length £, whose steady-flow resistance is R, 

 inductance L, electrostatic capacity S, and conductance of 

 insulator K, all per unit length of line, is acted upon by an 

 impressed force V sin nt at one end, or in the wire attached 

 to it ; whilst any terminal arrangements exist. Find the 

 effect produced ; in particular, the amplitude of the current 

 at the end remote from the impressed force. If the line con- 

 sist of two parallel wires, R must be the sum of their resist- 

 ances per unit length. 



Let be the current in the line and V the potential dif- 

 ference at distance z from the end where the impressed force 

 is situated. Then 



-§ =( K+S !) V > -f = R " C > • • w 



are our fundamental line equations. Here R'^R + L (d/dt) 

 to a first approximation, and = Bf + U (d/dt) in the periodic 

 case, where R / and 1/ are what R and L become at the given 

 frequency. Let the terminal conditions be 



Y = Z l C at z = l end,l ,„** 



-V sinw* + V = Z C at ^=0 end, J ' * ' [Z0) 

 so that V=Z C would be the ^=0 terminal condition if there 

 were no impressed force. 



