92 ON A SIMPLE PLAN OF EASING RAILWAY CURVES. 



But when dx is very small, 3 mdx vanishes and -=- = 6 mx x 

 for curvature at x t . x 



In general -=- = 6 mx 



But -=- is equal to curvature = y. 

 . •. y = 6 mx. 



To apply the cubic parabola to easing the transition of circular 

 curves. 



y = mx 3 

 y = 6 mx 



b mx 



-~ = tan = 3 mx 2 = steepness at the point x. 



To adjust the circle to the curve. 



Fig. 5. 

 Let R = radius of curvature of circle. 

 X and Y particular values of x and y. 

 Let P be the point of contact. 



r\m_ y> = values of x and y at point of contact P. 

 Radius of curvature at P = R = 



6 mX 

 X = length of curve of adjustment. 



6 raX = -^ m is a constant. 



1 

 .*. m 



6 Xi? 



. *. the curvature is determined. 

 Proposition :— To find the value of (Y = TP) in terms of 

 (X = OT) and also in terms of R. 

 Y=mX\ 

 1 



. Y = 

 . Y = 



6 Xi2 



X s 

 W~XR' 



x*_ 



6R 



6 V2/ i? 



