Periodic Electric Current in a Telephone Cable. 225 
Take a line OX (see fig. 1, Pl. IV.) to indicate the cable, and 
set up a perpendicular OH to represent in magnitude and 
direction, the voltage at the generator end. Then at equidis- 
tant points draw other lines decreasing in length in geometrical 
progression, and each shifted backwards or forwards in direc- 
tion relatively to the preceding line by an equal angle. If we 
suppose tnese lines to revolve with equal angular velocities 
round their ends as centres situated at equidistant intervals 
on the line OX ; then their projections at the same instant on 
vertical lines drawn through their centres will represent at 
that instant the actual voltage at these points in the cable. 
The periodic change with time and distance may be repre- 
sented by a working model made in the following manner :— 
On a long steel axle AA’ are fastened a number of eccentric 
pulleys Hy, Hy, H,, &c. (see fig. 2, Pl. IV.). The eccentricities 
of these wheels decrease in geometric progression, and each 
eccentric is set in phase backward behind its preceding 
neighbour by an equal angle. These wheels are embraced 
by cords C,, Cy, C3, &e. attached to balls or blocks of metal 
P,, P,, P3, &e. sliding on vertical rods R,, Rz, Rs, &e. placed 
below each eccentric wheel. These cords are all of the same 
length. 
When the axle carrying all the eccentrics is revolved by a 
handle W, the blocks P, &c. will rise and fall with a nearly 
simple harmonic motion, and at any instant all the blocks 
will be situated on a sinuous curve of continually decreasing 
amplitude. As the eccentric axle revolves the motion of 
the balls will depict the progression of a wave of potential 
along a cable having capacity, inductance, resistance, and 
leakance. 
Returning, then, to the expressions (13) and (14) for the 
value of 2 and 8, we may give these more convenient forms. 
Taking the formule (13) and (14), add to, and subtract 
from, the quantity under the radical the term 2p?CLKR, and 
after rearranging terms we obtain 
Qa? =/(KR+ p*LC)? + p( LK—CR)?+(KR—p?LC) (20) 
26? =./(KR+p?LC)?+ p?(LK—CR)?— (KR—p*LC) (21) 
If the cable constants have such relation that LK -CR=0 
or L/R=C/K, we have Mr. Oliver Heaviside’s distortionless. 
cable, and then obviously when this is the case 
a= /KR, B=p eee eS (22), 
Phil. Mag. 8. 6. Vol. 8. No. 44. Aug. 1904. Q 
