224 Prof. J. A. Fleming on Propagation of a 
It should be clearly understood that in speaking of the 
potential or current at any point in the cable we mean the 
maximum value of the potential or current regarded as a 
vector. 
Since P=a+j@ and e+” = cos Bv+jsin Bx, we can 
write (15) and (16) in the form 
V=He “(cos Br—7sin Bz). . |. . See 
ue Pi con B«e—jsnPx). . (18) 
VR+jph 
Let us notice the physical signification of the above ex- 
pressions. Equation 17 shows us that the potential V in 
magnitude and direction at any point at a distance w in the 
cable, reckoned from the generator end, is obtained by multi- 
plying the magnitude of the terminal applied voltage E by 
two factors, (i.) by the attenuation factor «“’, and (ii.) by 
the phase factor (cos Ba—) sin Bx). At any distance w from 
the origin, V is less than E in magnitude in the ratio of e~™” 
to 1 and is shifted backwards through an angle Bz relatively 
to i as regards phase. 
This is easily seen to be the case if it is remembered that 
cos 8+) sin @ is a rotating operator, and when it operates on 
a vector a+ jb it gives as a result a vector of the same size, 
viz. Va*+l?, but shifted forwards through an angle 0. The 
size of the vector (a+ b)(cos@+/ sin 0) is Ya?+b?, but its 
slope @ is such that g=tan-*~ +0 where tan“! is the 
ISB 
slope of the original vectora+ jb. Hence the expression (16) 
merely means that the potential at any point in the cable 
whose distance from the origin is # is less than E in magni- 
tude in an assigned ratio, and is shifted backward in phase 
by an assigned angle. 
If 2’ is the distance from the generator end at which the 
maximum voltage has fallen to half the numerical value at 
the origin, then 2’ is given by the equation 
= ean! 
dole 
y) 
, 0°7033 
Te = ee : 
a 
or (19) 
Again, since cos B2= cos (82+ 27) it is obvious that the 
wave-length of the potential wave is 27/8, and therefore the 
velocity of propagation of the potential wave is p/B. 
These facts may be presented graphically as follows :— 
