J. C. NAGLE — VERTICAL CURVES EOR RAILWAYS. 
25 
E 
V 
i 
In Fig. 2 let HK=z be the correction at a distance x from A; m = CF, 
the correction at vertex; d the algebraic difference of gradients, and 21 the 
length of curve in stations; then from the first property of the parabola, 
referred to above 
■XT 2 
z = m— (4) 
Prolong AC to E to meet a vertical through the end of curve at B, then 
EB — Id, and, since CF=JCG- by the second property, similar triangles 
give CF= JEB, or 
m — \ld (5) 
The length of curve, 21, may be fixed by local circumstances, or the 
average rate of change of gradient per station may be similarly fixed. 
Call this rate of change r, then for l in stations, 
If we substitute the value of d from (6) in (5) and the resulting value 
for m in (4), we shall have 
2 = 
rl 2 x : 
0.5rx 2 
( 7 ) 
When x= 1 station, z t = .5r, when x=%, Zy 2 = 0. 125r, when x=2 , z 2 = 
2 r, when £C=3, 2 3 =4.5r, etc. 
In The Engineering Hews of November 26, Yol. XXX YI, was pub- 
lished a short table of corrections for a few values of d and assumed values 
of l, which I computed by formula (7). Similar tables may be computed 
for other values and field computations lessened by their use. The cor- 
rections are to be added when d is minus, and subtracted when d is plus. 
The table is reproduced below: 
