or,
2 . . . R = wx√(1+x²/4y²).
Let i be the angle between the tangent at any point having the co-ordinates x and y measured from the vertex, then
3 . . . tan i = 2y/x.
Let the length of half the parabolic chain be called s, then
4 . . . s = x+2y²/3x.
The following is the approximate expression for the relation between a change ∆s in the length of the half chain and the corresponding change ∆y in the dip:—
s+∆s = x+(2/3x) {y²+2yΔy+(∆y)²} = x+2y²/3x+4yΔy/3x+2∆y²/3x,
or, neglecting the last term,
5 . . . ∆s = 4y∆y/3x,