1919.] Anderson.—Long-distance Electric Transmission-lines. 
97 
Referring to fig. 1, the new axis of x is shown passing through O', the 
distance 00' being equal to c. 0' now becomes the origin of the co-ordinates. 
The relationship established in equation (2) is shown in fig. 1 by the 
triangle PTM, obtained by the following construction : Produce the tangent 
PQ to T, and make PT equal in length to the arc PO ; drop a perpendicular 
from P to intercept the axis of x at M, and draw a perpendicular MT from 
M on to PT ; then PM 2 = PT 2 + MT 2 ; but as PM = y and PT = s, TM 
must be equal to c in accordance with equation (2). 
Again, PM, or the ordinate of any point P on the curve, represents the 
tension at that point for y = Vs 2 -j- c 2 and wy = V / o) 2 s 2 -f- w 2 c 2 , but 
o ) 2 c 2 — T 2 , the tension in the wire at 0 and Vw 2 s 2 -f- w 2 c 2 is the resultant 
tension at P previously denoted by r. Therefore (»y = t ; that is to say, 
the tension at any point along the wire is equal to the weight of wire equal 
in length to the ordinate y or PM. When x = o, y = c, and the tension 
= cjC = T. 
Let the half-span be denoted by l and the dip or sag by d. Then the 
co-ordinates of the point of suspension B are l and c -f- d. The tension 
at B is to (c -f d), in accordance with the foregoing principles, and as o x 
is the tension at 0 the tension at B is T -f cod ; likewise the tension at any 
other point is T -f- tod' where d' is the vertical distance from the point in 
question to the horizontal. 
It is now desired to express y in terms of x so as to get the relation 
between the co-ordinates, and from that the relation between the dip and 
the span. To obtain this, substitute in equation (1) the value of s in terms 
of y ; then 
dy _ s _ . Vy 2 — c 2 
dx c r 
whence 
when x 
x 
i — 
dy 
= c log (y + Vy 2 — c 2 ) + B 
Vy 2 — 
o, y = c ; then B = — log c, consequently 
X = c log 
y 
Vy 2 — c 2 
= c cosh' 
1 1 
c 
(3). 
X X 
By inversion we have y — c cosh -, and on expanding cosh - we obtain 
c c 
1 /v» 2 1 4 
y = c ^+ 7 _ 2 ^ + / 4 ^ + &C ' ) 
X 
Under ordinary conditions of tension and span - is a small fraction, and 
c 
the third term and higher terms of the series become negligible, and on 
rejecting these terms we have 
1 x 2 
y ~ c \ l + 2T* 
which is identical with the equation between the co-ordinates of a parabola. 
In order to obtain a relation between s and x differentiate equation (3) 
and substitute in equation (1), and we have 
/y» 2 ry\ 4 
S=ltl + 67* + 1207 + k 
(4) 
7 —Science. 
