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While?!|| theory of the transition curve as here developed is bases upon the methods given by M. Nordling (see Annales desF onts etC haussto 1867), the equations for both the cubic parabola and the spiral appear in Professor Rankine sC ivil Engineering (E d. 1863), and are accredited toD r. William Froude, undoubtedly ante-dating those of Nordling. Translations of a portion of Nordling sdemonstration have appeared from time to time, but only so far as related to connecting a straight line with a circular curve. That portion relating to connecting circular curves of different radii, by means of the cubic parabola, has not appeared in the form given by Nordling, so far as I am aware. The formuke deduced in the latter case are of general application and equally true for connecting curve with curve, or curve with tangent, when proper values are introduced into the equations. That of joining a tangent with a circular curve, by means of the transition curve, is a special case where one of the radii becomes infinitely great. Some of the recent spirals, adopted as curves of adjustment in railroad location, easily develop from the equations of the cubic parabola by making the proper substitutions in them. Following the supposition indicated by M. Nordling, i.e., regarding xL and substituting Lfor xin theN ordling equations, the formuke (42-46) were obtained by the writer during the summer, of 1884, as were also (2-9) inclusive, by an independent process, resulting in the verification of the formuke invented by Froude in 1841 and published in 1861.
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