1012 ME. F. JENEIN’S EXPEEIMENTAL EESEAECHES ON THE TEANSmSSION 
We can pass fi:om this case to that of a finite cable of the length I with one end in 
connexion with the earth by the device of electrical images. 
Superimpose two potential curves satis:fying equation (2.) with equal but opposite 
potentials +V and — V at their origins O and Oj. Let these points be situated at a 
distance 21 apart, and let the curves extend from their origin to meet and cross one 
another. The resultant potential will necessarily be zero halfway between O and Oj, 
and the resultant curve between this point and each origin will represent the variation 
of potentials in a cable of the length I with one end in connexion with the earth and 
the origin at a certain positive or negative potential. 
By putting these results into a mathematical form, we obtain from equations (1.) and(2.), 
v,=v(£- V t) (4.) 
Q,=Q(s-Vf+g-<==^--)'/T) (5.) 
But V and Q do not represent the potential and current at the origin of the finite cable, 
but the potential and current at the origin of the hypothetical infinite curves supeiim- 
posed. In order to obtain and in function of any given potential Vo at the origin 
of the finite cable, we must obtain the value of V in function of Vo by putting 5^=0 in 
equation (4.), and substitute the value of V in function of Vo in the general equations 
(4.) and (5.). 
Then we have 
and hence 
and 
V —V 
▼ a*— » 0 
(6-) 
Q. 
/m / tn 
(im)® 
/ m 
1 -2l\/ -r 
1 — g V t 
(7.) 
To obtain the current flowing into the cable at its origin, make w=0; then 
Qo=7T 
Vo 1+e- 
/ m 
«V T 
and 
Q.=Qo 
( m )* ^_-21s/y 
V m / m 
/m » 
l + e“«VT 
or, for brevity, writing and 
and 
n - Xo !(!!±0 
(s.) 
Qz=Qo 
g9/_|- 
(9.) 
