282 



ELEMENTS OF ELECTRICAL ENGINEERING. 



length of any element, ab, of the wire is very nearly equal to the horizontal pro- 

 jection, dx, of the element. Therefore the weight of the element is very nearly 

 equal to w dx, w being the weight of the wire per unit length. Furthermore, the 

 horizontal component of the tension of the wire has necessarily the same value, T, all 

 along the span of wire. 



Consider the element, ab, of the wire of which the coordinates of the end, a, are x 

 and y, and the coordinates of the end, b, are x -f- dx and y -f- dy. Let dy\dx be the 

 value of the first differential coefficient of y at the end, a, then dy\dx -\- a z yjdx* dx is 

 its value at the end, b. The force, T a , pulling at the end, a, of the element is the 

 tension of the wire at a, its horizontal component is T, and its component vertically 

 downwards is T tan B or T dy\dx. The force, T^, pulling at the end, b, of the element 

 is the tension of the wire at b, its horizontal component is T, and its component verti- 

 cally upwards is T tan Q f or T{dy\dx -\- d 2 yldx 2 dx} . Therefore the unbalanced force 

 pulling upwards on the element, ab, is T d 2 yldx 2 dx, and this unbalanced force is 

 equal to the weight of the element, w dx, so that : 



whence 



Ty 



but, since y = o and dy\dx = o when x = o, the constants, c and */, must be each 

 equal to zero, so that : 



>= () 



From Fig. 161 it is evident thatj , when x = l\2 ; therefore, substituting these 

 values in equation (ii), we have equation (39). 



The second member of equation (40) consists of the first two terms of the infinite 

 series which expresses the length of the arc of a parabola in terms of its chord, /, and 

 the distance, h, of the middle of the arc from the chord. 



W/W/////W////////////^^^ 



Fig. 162. 



Equations (41) and (42) are derived from equations (39) and (40). Consider a 

 given span of wire between two poles, A and B, Fig. 162, at a horizontal distance, 



