The Railroad Spiral: The Theory of the Compound Transition Curve Reduced to Practical Formulæ Rules for Application in Field Work; Multiform Complete ... for Five Hundred Spirals (Classic Reprint)

On a straight line a railway track should be level transversely; on a curve the outer rail should be raised an amount proportional to the degree of curve. At the tangent point of a circular curve both of these conditions cannot be realized, and some compromise is usually adopted, by which the rail is gradually elevated for some distance on the tangent, so as to gain at the tangent point either the full elevation required for the curve, or else three-quarters or a half of it, as the case may be. The consequence of this, and of the abrupt change of direction at the point of curve, is to give the car a sudden shock and unsteadiness of motion, as it passes from the tangent to the curve. The railroad spiral obviates these difficulties entirely, since it not only blends insensibly with the tangent on the one side, and with the circle on the other, but also affords sufficient space between the two for the proper elevation of the outer rail. Moreover, since the curvature of the spiral increases regularly from the tangent to the circle, and the elevation of the outer rail does the same, the one is everywhere exactly proportional to the other, as it should be.
(Typographical errors above are due to OCR software and don't occur in the book.)

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