J. 0. NAGLE — VERTICAL CURVES FOR RAILWAYS. 
23 
Whence az — z 2 = 2 ay. 
Substituting value of z from (2) we get 
nJ2ax — 2x -J- 2y 
Squaring and arranging terms 
4x 2 -j- 8 xy + 4 y 2 — 2 ax = 0 (3) 
— the equation to a parabola, which is tangent to the axes at a distance 
of -Ju from the origin. 
If such a curve were substituted for the two intersecting grade lines, 
the center of gravity of the train would now be found to follow a curve 
lying somewhat above the parabola given by equation (3). A second 
approximation might be made by finding the locus of the center of grav- 
ity of an arc of the parabola of constant length, which would yield a 
curve differing slightly from the above parabola, but sufficiently close to 
the desired curve to answer all purposes in an actual case. 
However, it is unnecessary to pursue this line any further, for no 
matter how many approximations we make, the final length will be a 
function of the length of train only, and entirely independent of the 
angle between grade lines, though the actual form of the curve will not. 
For every different length of train, a different curve would result, and the 
best that could be done would be to construct the curve for the longest 
train likely to pass over the road. Evidently the length of curve should 
bear some relation to the angle between grade lines, increasing with that 
angle. 
It would seem possible, at summits at least, to so adjust the curve 
that for any given length of train and speed the pressure upon the rails 
should be constant; but the pressure upon the rails is a matter of no 
special moment in any practical case. 
It would seem, then, that the only other consideration affecting the 
question in a material way would be the crowding forward of the rear 
of train in sags, with the consequent jamming of drawheads and sudden 
tensile stress developed in the couplings as the effect of the engine begins 
to be felt after passing on to the second grade. At the summits the in- 
crease in tensile stress as the engine and forward part of the train pass 
over and begin to descend might be in danger of rupturing a draw-bar — 
in either case breaking the train in two. 
Wellington, in his book on the “Economic Theory of Railway Loca- 
tion,” lays down the proposition that if it be desirable that all danger 
of slack is to be avoided in sags, the difference in the rate of grade of 
track at the points occupied by the front and rear of train should never 
exceed the so-called “grade of repose” of last car — a quantity dependent 
upon the velocity as well as rolling friction. 
Taking the dynamometer resistance of last car as six pounds per 
