ON A SIMPLE PLAN OF EASING RAILWAY CURVES. 91 



Also 2/ 2 = mx\ when x 2 and y 2 are particular values of x 

 and y. 



If 2/2 — 2/1 = dy, and x 2 — x x = dx 



Then y 3 = m (aii + c&c) 3 . 



.•. 2/ 2 = m (a? 3 -f 3 pal dx + 3 a^ (c&c) 2 + (cfe) 3 j = m# 3 



+ 3 mas? dx + 3 mx x (dx) 2 + ra (e&c) 3 

 But y x = ma;?, and dy = y 2 - y x , 



.' . dy = 3 mx\ dx + 3 ma^ (cfe) 2 + m (c&c) 3 



.'.-,- = 3 ma? 2 . + 3 mx x dx + m (dx) 2 . 

 dx x ' 



di/ 

 When j- are very small they become equal to 3 mxl, but 



/i dy o 

 ton = -7^ = o mail. 

 dx 



The tangents of small angles are (approximately equal) == to 

 the angles in radian (circular) measure. . •. === 3 mx\. 



Fig. 4. 



ace 



tan 6 = 3 mx\ 



== 3 race 2 (when # is small). 



Let efe = a small increase of length on curve, for length dx on 

 ordinate. 



When the angle is small we may use ds, or dx, at discretion, 

 for they are = to each other. 



Let the curvature at P X P 2 be denoted by y (gamma). 



Curvature = y = change of angle of inclination =—=-=- 

 Jn dx ds 



Required the value of -=- which will give the curvature y. 



1 = 3 mx\ 



6 2 = 3 m (x x + dx) 2 , 



2 = 3 ra [cc 2 + 2 x x dx + (<£c) 2 j 



2 = 3 race 2 + 6 mccx dx + 3m (dx)' 



(0 a _ 0, = dO) .'.dO = 6 mx x dx + 3 m (e&c) 2 



.*. -7- = 6 mcCi + 3 m dx 

 dx 



