1911 - 12 .] 
The Railway Transition Curve. 
333 
XXIV.— 1 The Railway Transition Curve. By E. M. Horsburgh. 
(MS. received November 21, 1911. Read December 4, 1911.) 
If a railway circular curve spring directly from the tangent, the curvature 
at the point of contact is discontinuous, since it is zero for the straight line, 
and finite for the circle. This causes shock whenever rolling stock enters 
the curve, and at high speeds this is a matter of danger. Further, the 
outer rail is elevated on the curve, while the rails are at the same level 
on the straight. This indicates a further discontinuity. The transition 
curve is introduced to give both a gradual change of curvature and a 
gradual cant or super-elevation of the outer rail. 
The curve to be found is one in which the curvature shall be a known 
function of some selected variable, and whose equation or equations contain 
a sufficient number of arbitrary constants to allow it to be fitted to the 
requirements of the case. Under the most general circumstances the 
equation should involve six arbitrary constants, since the curve should 
pass through two arbitrarily chosen points, and have at each of these points 
the required gradients and curvatures. This may be modified in practice, 
as the most general case is reducible to that of a transition curve springing 
from a straight line. In order to avoid difficulties of setting out, the 
equations obtained should be of the simplest nature. 
If the curvature changes uniformly in passing along the transition 
curve its equation will be and as l/p,— y"/(l-\-y' 2 )% where p is 
the radius of curvature, the equation of the curve may be written 
(1 + y' 2 )y'" — 3 y'y" 2 — k (1 + y' 2 y = 0 , where k is a known constant, and y', y", y"' 
represent ~ . If the deflection angle be small, so that y /2 is a 
negligible quantity, this simplifies to y'" — Sy'y" 2 = k. 
The complete primitive, even if it could be obtained, would be of little 
practical use, and it would only contain three arbitrary constants. What 
is chiefly important is an approximation to the shape when the deflection 
angle is small, i.e. in the neighbourhood of the origin. Assuming that it 
may be obtained, and that it is expressible in the form of the convergent 
power series y = '2a n x n , and taking the tangent and normal as axes of x 
and y respectively, and considering the curve as springing from a piece of 
straight track, the following conditions are satisfied : — y 0 = 0, y' 0 — 0, y" 0 = 0, 
