1 . . . H = wL²/8y,
or, calling x the distance from the vertex to the point of support,
H = wx²/2y,
The value of H is equal to the maximum tension on the bottom flange, or compression on the top flange, of a girder of equal span, equally and similarly loaded, and having a depth equal to the dip of the suspension bridge.
Consider any other point F of the curve, fig. 70, at a distance x from the vertex, the horizontal component of the resultant (tangent to the curve) will be unaltered; the vertical component V will be simply the sum of the loads between O and F, or wx. In the triangle FDC, let FD be tangent to the curve, FC vertical, and DC horizontal; these three sides will necessarily be proportional respectively to the resultant tension along the chain at F, the vertical force V passing through the point D, and the horizontal tension at O; hence
H : V = DC : FC = wx²/2y : wx = x/2 : y,
hence DC is the half of OC, proving the curve to be a parabola.
The value of R, the tension at any point at a distance x from the vertex, is obtained from the equation
R² = H²+V² = w²x4/4y²+w²x²,