TRANSITION CURVES FOR RAILWAYS. 593 



The transition curve is set out by offsets from the original tangent 



line. Any ordinate at a distance x^ from the point A :=^~- . .(6) 



X being the length of the transition curve, and the denominator 

 6 R a: becoming a constant for each curve. The offset H F 

 must from equations (5) and (B), be =: ^ shift and ^ E C. 



From the equation referring to the ordinates (6) the cubic 

 parabola received its name because the offsets vary as the cube of 

 the distance from the bej>inning. 



In addition to the above, which give all the data required, the 

 new intersection point I' should be found and marked for future 

 use. The internal angle is halved and I I' set off == H G X sec. 

 of ^ intersection angle. From this new intersection the crown peg 

 S' can be located. 



If the curve is cut up by a sub-tangent the intersection 1'" (see 

 Fig. 2) can be found thus : — Set off at right angles to T I" the 

 length 1" V" equal to I I'. The crown peg S' can now be found in 

 the ordinary way. 



On investigation it will be found that the curve as set out above 

 begins at A with so large a radius that it can be practically con- 

 sidered a;» straight, and that the radius diminishes till it becomes 

 at C equal to that of the original circular arc. This form of 

 transition curve should, for these I'easons and on account of the 

 simplicity of the setting out, commend itself to engineers who are 

 not conservative. 



Rankine sets out the original curve first, and then transfers it 

 nearer to the centre by the amount of the shift (hence the term). 

 The transition curve is then set out by ordinates fi'om the new 

 position of the arc, beginning at C instead of at A. As Rankine 

 uses for an example the case of reverse curves of different radii 

 meeting without any straight betw^een them, it is difficult to imder- 

 stand what is proposed for the junction of a curved and straight 

 portion ; in fact, the writer has after careful study failed to find 

 the method for reverse curves possible. However, half of the 

 transition curve can be set out by ordinates from the circular arc 

 starting at C, using the formula (6), and if the arc be continued 

 beyond the tangent point to K the other half can be set out. 

 Strange as it may seem, and a fact worth noting, this method gives 

 a curve identical with that fixed by ordinates from the straight. 



It has been foimd that a tangent to the arc at C (Fig. 1) meets 

 the straight (produced) at F and A F = f of the length A E of 



the transition curve. Tan. angle a = (or, more exactly, twice 



D 2R ^ ^ 



the tangential angle for ^ x with radius R). From this a new 



tangent line F N could be located for setting out the circular arc, 



but would not prove so simple as the parallel tangent lines 



suggested previously. Should a exceed 24° 5' (which is not likely) 



the cubic parabola instead of increasing in curvature as it gets 



p2 



