222 Prof. J. A. Fleming on Propagation of a 
copper resistance RK ohms, and a dielectric or insulation 
resistance 7 ohms, all per unit of length. 
If we take any element of length in the run of the cable, 
then there is a difference between the potential and the 
current at the two ends of this element. Let dv and & be 
this difference, v and i being the potential and current at the 
generator end of the element. Then the symbolical expression 
for the manner in which this difference of potential and of 
current is created, is given by the familiar equations : 
di 
Lox — 
+Roci=dv, . . +50 
dt 
Cie +Kirv=di,. . . . - (2) 
¢ 
where K is written for 1/r. Hence 
La + R= Ay? . . ° e ° e (3) 
ra (4) 
i + AV= dass. aie é 
If ¢ and v both vary in a simple harmonic manner, and if 
I and V are their maximum values during the period ; then, 
since the maximum values of d/dt and dv/dt considered as 
vectors are at right angles to the maximum values of 7 and v, 
we may write (3) and (4) in the form 
dV B) ¢ 
a =(R+jph)l, . a ee (5) 
K+70)V,, . 
where j is the sign of perpendicularity, and p=27 times the 
frequency n of the current. 
Separating the variables by differentiation, we have 
ay ' Mest 
qa? = (R+yph)(K+)p0)V, ~ hha 
Ao ae 
aa (R+jph)( +jpC) 1.) 2 ae 
The above are the well-known vector equations for the 
propagation of a simple periodic current in a cable having 
the four constants L, R, C, K. Bi ss 
If we write P for ./R+jpL . ./K+jpC, we have the 
