TRANSITION CURVE. 149 



between the tangent to the curve and the main tangent 

 increases as the square of the length of the curve. 



(C) The curvature increases directly as the distance 

 along the curve, this being as already shown the basic prin- 

 ciple of the transition curve. 



(D) The spiral angles to various points of the curve, 

 being limited by the corresponding normals to the curve, 

 increase as the square of the distance, as is evident from 

 equation (4). 



(E) The deflection angles at various points of the curve 

 are approximately equal to one-third the corresponding spiral 

 angles and therefore increase as the square of the distance. 

 This can be shown thus : — 



BR klV kl\. 



sm BAR = — — = — — • =^ (approximately) 



BA 6lc 6 



sin BNR^smSc^=Sc^ — — from (4), (approximately) 



dl 2 



Sc 



.-. BAR = — (approximately) (8) 



A similar argument applies to all points of the curve. 



(F) The circular arc if produced backwards from B 

 until it becomes parallel to AX will have a length equal to 

 one-half the transition curve. Thus, from (4) and (6) there 

 results for the angle of the tangent to the transition curve 

 at B, approximately, 



dv . kPc re le 



--- ^= sin Sc ^= Sc = 



dl .2 2lcRc 2Rc 



Again, the angle of the tangent to the circular arc at B in 

 terms of distance Lc from D is 



s - ^ 



