Mr Sang m the Effects of the Curvature of BaUways. 335 



One thing is certain, that the change of curvature must 

 never be abrupt, and that the junction of circular arcs is in- 

 admissible. 



Viewing the distance measured along the rail as the ab- 

 sciss, the curvature may be regarded as a function of that 

 distance ; and this function must be of such a nature that 

 the curvature may be zero at the point where the deviation 

 from the straight line commences. Putting / for the length 

 reckoned from this point, and ^ for the curvature, the simplest 

 function which satisfies this condition is ^—nL That is, the 

 curvature is proportional to the distance from the said point. 

 But this function is insufficient for the purpose, since it would 

 give a perpetually increasing curvature, while the general ob- 

 ject of a railway curve is to lead us from one direction to an- 

 other — to join two straight parts of the line. For this purpose 

 the curvature must increase, reach a maximum, and again 

 diminish to be zero at the place where the second straight 

 part is reached. The simplest function which possesses the 

 requisite properties for this is ^ = nl (L — /), where L is the 

 whole length from the one straight part to the other. A 

 curve possessing this characteristic would be entirely free from 

 the fault to which I have pointed ; yet, pushing our exami- 

 nation still more narrowly, it would not be altogether free 

 from defects, since the vertical projection of the outer rail 

 (supposed stretched in a straight line) would be parabolic. 



S 



Thus, A and B representing the two ends of the curved part, 

 and AB the level of the inner rail, the parabola ACB would 

 represent the level of the outer rail ; in the case of the circular 

 sweep, the line a /3 would represent it wdth a sudden stop at 

 each end. The curve ACB must undoubtedly be preferable to 

 the line A a /3B (which, in fact, never can be adopted in prac- 

 tice), yet even it must give a hai'shness at the points A and 

 B ; which harshness must augment as the second power of the 

 velocity. 



The curve ACB ought to have touched the straight line at 



