The points of concourse of the salient angles, are called the nosinys of the steps.
The line drawn through all the nosings of the steps, is called the line of the nosinys.
Now let the portion of the cylinder before uncovered, be again enveloped, the development in this state becomes an envelope, and the line of nosings becomes a uniform helix, which would be the form of the rail for such a stair.
In this ease, it would be easy to execute the rail to any length, in equal portions succeeding each other ; for as the curvature of the helical line is everywhere the same, the same moulds which are used in the formation of one piece, would serve for every succeeding piece.
The steps around the circular part are termed winders; in these the risers tend to the axis of the ey tinder.
Steps with their treads of the same breadth, are termed flyers ; in these the risers are all parallel.
Very few staircases are entirely circular ; but those of the semi-circular form, with winders in the semicircle, and flyers below and above, are very numerous ; in such, the line of nosings would be crooked, and would form an angle at the junction of the flyers and the winders, and that round the semicircle would be an helix, consisting of half a revolution.
In the development of the steps, the line of nosings would consist of three straight lines ; the two straight lines through the nosings of the flyers, would be parallel to each other, and each extremity of the middle one would join one extremity of each of the other two ; but the angles are commonly taken away, by introducing a curve in their places.
A hand-rail, however, is not a mere helical line, but a solid, which may be contained between two concentric cylin dric surfaces, or concentric prismatic surfaces. The prin ciples are the same, whatever be the form of the plan. A solid erected upon any plan, is called a prism; a cylinder is therefore a round prism, and a eylindroid an elliptic prism. A hand-rail may stand upon a circular base, or partly circular and partly straight, or upon an entire elliptic base. In the construction of hand-rails, all prisms are excluded, which consist of plain surfaces ; or, which is the same thing, where the sides of the plan consist entirely of straight lines ; as in such cases, the rails themselves are either straight, or partly curved and partly straight upon the top and lower sides only, the sides being in vertical planes.
Let us therefbre confine ourselves to prisms upon a circular or an elliptic base, or upon a base partly circular and partly straight ; or lastly, upon a base partly elliptical and partly straight. The two last may be said to have compound bases
or plans, and the two former simple bases or plans. Such a prism may be denominated a curved prism. The plan of any curved prism is understood to be of the Sallie breadth, and consequently the solid erected thereon will be every where of the same thickness. The prism may therefore be a hollow cylinder, or a hollow clyindroid, or a concave body, partly cylindric and partly straight ; the latter may be open on one side, and may have the fbur planes which join the curved surfaces parallel to each other, and tangent to each of the cylindric sin faces.
Let us therefore suppose such a prism as that last men tioned, to be cut entirely through its vertical surfaces, in such a manner that any point in the surface of division may coincide with a straight line perpendicular to the external prismatic surfhee, then, every such line will be parallel to the plane of its base, and those lines in the cylin drieal part of the prism will tend to the axis. Now it is evident, that the cut, or dividing surface, will not be a plane, but will wind or twist between the cylindric surfaces. It is also evident, that the cut may pass through a line drawn in any manner we please, in one of the prismatic surfaces; or, that the development of this line may have any degree of curvature in the whole length, or in any portion of the length, or may even be a straight line. One of these being supposed to be the case, let the upper part of the prism be taken away, then the upper surface of its remaining part will be brought to view ; let a line be drawn on the exterior surface parallel to the arris, and another on the con cave side parallel to its arris; and let another cut or dividing surthee be made to pass through the two lines thus drawn, and let the upper part be removed by this division ; then the part thus removed will form a solid helix, or kind of half screw, which may be either uniform in its upper and lower surthces, or have any degree of curvature in any part that may be required, according to the development before men tioned. This is the form of the rail for such a stair ; but to form the solid helix, without cutting it from a hollow curved prism, is what is required in hand-railing.