335 
1911—12.] The Railway Transition Curve. 
A, point o£ transition curve (P.T.C.) taken as origin of reference. 
Q, point of compounding (P.C.C.). 
D, point of circular curve (P.C.). The co-ordinates of D relatively to 
A are AB = a, BD = /3. 
R, radius of circular curve. 
QM (rf) is the perpendicular on DM (£), the tangent at D. 
AY, a line perpendicular to AX, and taken as 7/-axis. 
A',B',D' denote corresponding points for the other tangent. 
S and S', points where the original and the parallel tangents intersect 
(points of tangent). 
Let P he any point on the transition curve. Let the co-ordinates of 
P be (oj, y) in cartesians and (r, 0) in polars. Let TP be the tangent at 
Fig. 2. 
P, XTP is the deflection angle \fr, and tan \fr — ^ = y' is the gradient at P. 
When there is any danger of ambiguity the suffix 1 will he used to dis- 
tinguish the elements at Q, the point of compounding. Though polar 
equations, freedom (or parametric) equations, or ordinary constraint 
equations may he used, the methods deduced must he of sufficient simplicity 
to be suitable for setting out in the field. 
Although it is possible to fit a transition curve to a circle without shift 
or offset, yet a curve so calculated would be unsatisfactory. A simple 
method of procedure would be to select from practical considerations the 
approximate position of the P.C.C., by selecting an approximate value for 
£ and deducing an approximate value for the deflection angle \Js at the 
point of compounding. A convenient value of \jr is then chosen to locate 
accurately the point of compounding, Q. From this value of \Js, the radius of 
the circular curve or the degree of curve being known, the exact values of 
(• and rj are obtained, also AN, NQ (fig. 3), and l, and hence /3 (the shift), and 
a, and finally SB. The intermediate pegs on the circular and transition 
