1919 .] 
Departmental Reports. 
129 
radians per second and / is the number of alternations per second, 
combining the two equations we have 
^ = (E + *pL) (K + ipS) ve'P‘ 
dt? 
the solution of which is 
v = (Ae mx + Re~ mx )eM 
where 
m = V(R ip L) (K + ipS) 
which is called the “ attenuation constant ” of the circuit. 
For the present purpose the variation with time is not required, and 
substituting for v and c the effective values Y and I of the potential 
difference and current respectively we have 
V = Ae mx + Re~ mx 
If V s and I s be the potential difference and current respectively at the 
sending end (i.e., when x — o), the following hold : — 
V s == A + B and — n\ = A — B 
where 
n 
= a/ b + *P L 
K + ipS 
called the “ surge impedance ” of the circuit. Substituting for the values 
of A and B we have for the difference of potential at any distance x along 
the line 
Y = V s cosh mx — wl s sinh mx 
Let the length of the line be /. Then when x = Het V = Y r , then 
Y r = Y s cosh Im — wl s sinh hn .(1). 
Likewise, if Y r and I r be the difference of potential and current at the 
receiving end, we have for the potential Y at any point distant x from 
the sending end the following equation :— 
Y = Y r cosh mx + nl t sinh mx 
when x = l, V becomes V s , and 
Y s = Y r cosh Im + nl r sinh Im .(2). 
Similar equations are derivable for the value of the current at any 
point on the line from the differential equation 
dx 2 
= (R + ipLi) (K + ip$) I 
The solution for any point distance x from the sending or generating 
end of the line :— 
I = I s cosh mx — — sinh mx 
n 
and for any point at a distance x from the receiving end of the line:— 
I = I r cosh mx + — sinh mx 
n 
When x = l we have 
y 
I r = I s cosh Im -* sinh Im ...(3) 
n 
and 
Y 
I s = I r cosh Im + — 1 sinh Im 
n . 
,(4). 
9—Science. 
