ee 
XXV. A Model illustrating the Propagation of a Periodic 
Electric Current in a Telephone Cable, and the Simple 
Theory of its Operation. By J. A. Furmine, M.A., D.Se., 
F.R.S., Professor of Electrical Engineering in University 
College, London *. 
[Plate IV.] 
se propagation of alternating currents of electricity 
along linear conductors has been the subject of much 
mathematical investigation. Its importance in reference to 
the practical operation of submarine cables, telephone-lines, 
and more recently of the radiating antennee in Hertzian wave 
telegraphy, is well understood. Nevertheless, practical tele- 
graph and telephone engineers have not always shown 
readiness to assimilate the ideas elaborated by theorists, and 
in some cases unsound theories of the phenomena have been 
promulgated. The assistance which practical men derive 
from considering the operation of a working model when 
appropriating physical ideas is considerable, and the author 
has found that for teaching purposes the model here described 
is of great use in making plain the meaning of mathematical 
expressions. 
The simple theory of it may first be given. The object is 
to explain the propagation of a periodic or alternating electric 
current in a telegraph or telephone cable having resistance, 
inductance, capacity, and leakage, or insulation conductance 
of a certain value per mile or knot. 
The discussion necessarily proceeds on the assumption 
that the currents and electromotive forces with which we 
are concerned are of simple harmonic form, and accordingly 
the only mathematical symbolisation necessary is that required 
for representing a simple harmonic motion or simple sine 
function and its time-rate of change. This is most readily 
achieved by the use of the complex quantity to denote the 
maximum value and phase of the periodic quantities with 
which we are concerned. ‘The representation of a vector 
quantity by the symbol a+ b, where a and 0 are lengths of 
lines and 7 is the segn of perpendicularity denoting that b is at 
right-angles to @ is now so universally understood that no 
further explanation of it is necessary. io 
Consider, then, in the first place, an electric cable in- 
definitely extended in one direction, and at the terminated 
end a simple periodic electromotive force applied. Let the 
cable have a capacity C farads, an inductance L henrys, a 
* Communicated by the Physical Society: read June 10, 1904. 
