90 ON A SIMPLE PLAN OF EASING RAILWAY CURVES. 



The problem that has to be solved, is to so locate the circle and 

 the cubic parabola, that at the point where they have an equal 

 radius of curvature, they may have a common tangent. 



Theoretical Investigation. 



For the following investigation of the properties of the cubic 

 parabola, as applied to transition curves, the author is indebted 

 to Professor James Thompson, of the Glasgow University. 



Definition : — The cubic parabola is a plain curve, which deviates 

 from the point of zero curvature by a perfectly gradual increase 

 of curvature, and for a small portion of the curve the curvature 

 is small, and is approximately proportional to the distance from 

 the point of zero curvature. 



The equation of the cubic parabola is y = mx s where m is a 

 constant numeric, where the axis of x is tangential at the point 

 of inflection. 



Fig. 1. 



Definition : — The rise per unit of length = steepness. 



To find the steepness of a curve at any point : — Let be the 

 point of inflection, and YY the axis of Y, and XX the axis of X. 



Fig. 2. 

 A B = dx 

 BP 2 - AP X = dy 



(Vi + dy) = y % 

 To express the steepness of the curve at any point it is expressed 

 as the tangent of the inclination at that point. 



Let = angle of inclination to OX at P 1} then tan — 



steepness of the curve at P u but steepness at the point P lt = -^ 



. • . — d. = tan 6 where -JL is very small. 

 dx dx 



Fig. 3. 



Let AE be a unit of length, and let the angle DAE be 0, then 

 DE = tan 6. 



Let AB be very small and be represented by dx, and let BG 

 be very small and be represented by dy. 



In the cubic parabola we have : — 

 y = mx 3 , (in general). 

 y x = mxl, when y u and x lt are particular values of x and y. 



