1919.] Anderson.—Long-distance Electric Transmission-lines. 103 
Sag-deflection is therefore dependent on x and c only ; and is independent 
of slope. 
Where there is considerable difference in level between anchorages the 
tensile stress at the higher end will be greater than at the lower end, and 
since wy = r the difference in tension is equal to the weight of wire equal 
in length to the difference of level. 
Thus, if one end of a strain is 255 ft. higher than the other, taking 
copper as weighing 3-92 lb. for 1 ft. X 1 sq. in., there will be a difference 
in tension of 1,0001b. per square inch between the two ends. The tension 
at the top cannot be increased or under wind load it will exceed the elastic 
limit; therefore the tension at the lower end will be less and the sag cor¬ 
respondingly increased. This is true whether the drop is all in one span 
or distributed over several spans in a strain. For this reason it is advis¬ 
able to have a strain at the foot of any considerable drop. A more definite 
reason for placing a strain at the foot of a hill will be discussed when con¬ 
sidering stability. 
In the above statement it is assumed that the wire passes over free 
rollers. This is not strictly true, the wire being attached to insulators 
which are free to swing about the point of suspension. As the tension 
downhill increases, the insulators will swing over in that direction, and in 
so doing will relieve to some extent the disparity of tension. 
Stability. 
So long as the line passes over level country or country with an even 
slope the weight of the wire and insulators is sufficient to ensure stability. 
Even under high wind the insulators will always be in tension, though 
inclined at a considerable angle. But where the middle one of three poles 
lies considerably below the line of the other two there may be no 
weight on the insulators at all, or the tension in the wire may even lift and 
buckle the string of insulators. When this occurs the wires must be held 
down in some way. This can be done by tying down, using a second set 
of insulators below the wire, or by making that pole a straining-point. 
A method sometimes employed is attaching a weight under the wire. This 
is objectionable, tending to accentuate swinging. 
Fig. 5 shows the limiting slopes which may be allowed without special 
provision being made. It also shows the direction and extent of vertical 
pull under all conditions. 
We need only consider the condition of datum temperature in still air, 
since, if the insulators are in tension then, they will be more so under any 
other condition. 
In the parabola ABCD (fig. 6) take any chord BC. From the apex A 
draw a chord AD parallel to BC. Let the middle points of these two 
chords be distant 2 from the axis. 
Since in the parabola x = 2c tan <9, AF = 2z = 2c tan DAF, or z = c tan 6 
where 6 = angle of slope. 
Now, the distance of the apex from the lower end of the chord 
■ - AG = {z — x) = c tan 0 — x. . . . ... (7). 
Similarly, AH = (z + x) = c tan <9 -J- x. 
Values of apex distance (z — x) for various values of x and 0 are 
shown in fig. 5. 
