566 PROCEEDINGS OF SECTION H. 



1. THE LEMNISCATE AS A TRANSITION CURVE, 

 By Professor R. W. Chapman, M.A., B.C.E. 



The cubic parabola is now very generally adopted by British 

 engineers for transition curves; but although a very convenient, it is by 

 no means an ideal curve for the purpose. The georaetricHl construc- 

 tions by means of Avhich it ie set out and fitted in between the straight 

 and the circular curve are usually approximately correct only, and, 

 although the error made under ordinary circumstances is not enough to 

 be a practical defect, in special cases, particularly when the circular 

 curve is of small radius, the common methods may easily lead to errors 

 of appreciable magnitude. It would certainly be preferable to have a 

 curve that could be fitted in perfectly under all circumstances, and 

 such that the method of setting out should be exact and not 

 approximate. 



In searching for a curve to satisfy this condition, the writer tried 

 the curve whose radius of curvature varies inversely as the radius 

 vector. This turned out to be the Lemniscate of Bernoulli, and it was 

 soon seen that, whilst quite as easily set out as the cubic parabola, 

 it may be fitted in with perfect exactness for any length of transition 

 curve, and for any radius of circular curve desired. Further 

 investigation showed that the writer was not the first to make use of 

 this particular curve for the purpose, and that the employment of the 

 Lemniscate of Bernoulli for transition curves was advocated by Paul 

 Adams in the Annnles des Pants et Chaussees, October, 1895. 



Properties of the Curve. 



The form of the complete lemniscate is that of a figure of 8, but 

 only a portion of the complete curve would be used for transition 

 curves, and for this purpose its equation can be most conveniently 

 written in the form 



r«=:rt.»sin2fl (0. 



