Periodic Electric Current in a Telephone Cable. 229 
3 to 4 ohms, the capacity per knot 0°3 microfarad, the insu- 
lation resistance 300 megohms per knot, and the inductance 
say 300 henrys. We should then have L/R=C/K=100. 
Hence we should have «=0:0001 and @=10? corresponding 
to p=1000. Accordingly the distance for decay to half 
voltage would be some 7000 knots. Such an achievement 
may be quite beyond the reach of practical work, but an 
approximation to it is perhaps not impossible. It cannot be 
done, however, merely by increasing the iron armour or 
bringing it nearer to the copper, because in that case the 
capacity Is increased very considerably. The annulment of 
distortion by creating lateral leakage either through inductive 
or non-inductive leaks always leads to large values of a. 
In other words, to great attenuation. Hence, although 
the distortion of the complex potential or current wave 
may be cured, the decay of amplitude would extinguish 
audibility as far as telephonic transmission is concerned 
within quite moderate distances, and far less than that 
which could be obtained by the right application of 
inductance. 
As a small contribution to this practical consideration of 
the problem, the model here described and method of 
discussing the theory may perhaps be useful. 
Note added July Yth, 1904. 
In reference to the above method of discussing the problem, 
it has been represented to the author that it would assist some 
mathematical readers if the steps by which the usual alge- 
braical expressions for the potential and current at any point 
in the cable can be deduced from equations (17) and (18), 
were set out more at length. 
The equations (17) and (18) are expressions showing the 
manner in which the potential and current at any distance w 
along the cable, considered as vectors, are related to the 
vectors representing the potential and current at the origin 
at the same instant. 
If we desire to ascertain the state of affairs at any time ¢ 
we have to consider these vectors as rotating round their 
extremities with a periodic time T. Accordingly the proper 
expression for thevector representing the potential at the origin 
at any time ¢ from the beginning of the epoch will then be 
Ee’, where p=27/T. Hence the corresponding expression 
for the vector representing the potential at any distance « 
along the cable at the same instant will be obtained by 
multiplying equation (17) by ©’, and will therefore be 
