PROCEEDINGS OF SECTION H. 567 



In Fig. 1, denotes the origin of co-ordinates, OFC the curve, and 

 0:2' the initial line is tangent to the curve at 0. Then if P is any 

 point on the curve, PT tlie tangent at P, OP — r, angle POT r: it 

 ie readily proved that 



angle OPT = 20 (2). 



PTN^ = dd. 



If p denotes the radius of curvature at P we have 



^ = t ^»)' 



go that p varies inversely as r. 



If this curve joins on at the point C to a circular curve of radius 

 JR, we have 



p = -^ . ; (4). 



3-OC ■ ^^ 



The length of OC will not be very different from the length of the 

 transition curve itself^ and we may generally select a suitable value to 

 satisfy the conditions of the problem. " Thus this equation determines 

 a' when we know P and OC. 



Denoting the angle CON by «, we have 



0C2 = rt2 gin 2a = 3R.0C sin 2«, 



.'. 0C= 3/? sin 2a . . . (5). 



Having fixed the value of a^ from equation (1), the values of r cor- 

 responding to different values of 6 may be tabulated and the curve may 

 be set out by polar co-ordinates from 0, or the values of PA'' and ON 

 (r sin d and r cos d) may be computed for different values of 6 and the 

 curve set out by offsets from O-v. 



Denoting OJ^ by x and PJV by y, the equation of the curve may be 

 put in the form r* = 2a'a;?/ or y =-:— 5 — This is very nearly the same 

 as the cubic parabola if r is nearly = x, as will be true in most cases of 

 transition curves. The ^^-^ is thus equivalent to the m of the equa- 

 tion to the cubic parabola y =.mx^. 



The oalue of the constant o?. 



There is considerable difference in practice in the length of 

 transition curve adopted on railways. In some cases a fixed length is 



