54 C. J. MERFIELD. 
Where differential or integral calculus is employed the usual 
notation. 
THE Cusic PARABOLA. 
To overcome a train’s centrifugal force, the outer rail of all © 
railway curves should be elevated, Every change of curvature — 
should have a corresponding alteration in the cant of the rails. 
_As this alteration can be made only gradually, changes of curva- ) 
ture should be gradual. The cubic parabola, owing to its easy : 
application, seems to be especially suited for this purpose. In | 
the following pages, data necessary for their location are deduced. 
Let the gradual superelevation of the outer rail be taken 
uniform, for distances from 0 along the axis X, or let + be the 
rate of rise, then the rise at any distance « will be “. (See fig. !) 
If p be the radius of curvature at the corresponding point om 
the curve, we have the superelevation equal to 
gauge x velocity” 
32:17 x radius ” | 
the gauge being expressed in inches, the velocity in feet per second, 
and the radius in feet. 
2 o 
Therefore 
Now for a given velocity and rate of rise +, the quantity a7 a 
becomes a constant, let it be represented a Be 
te a eee Ee eer 
Therefore ¢ = ie, and 
a ! 
41+(5%)" 53 
a oe ee a d s* 
But 5 s dae dy 
Let us for simplicity ne d s=d « then we have 
vhs ey ee = rr 
“. = = integrating twice we ar 
= 
-. eae 
This is an eciiadion to the cubic parabola, which may be written 
y= m o* 
