Now, as two of its sides are actually cylindrical, and would be vertical if placed in position, and the other two winding surfaces may be formed to any development desired ; take any determinate portion of the helical solid, as a quarter of a revolution, or perhaps something more, as occasion may require, and endeavour to form such a portion, or wreath, out of a thin plank, instead of cutting it from a solid curved prism. Before this can be done, it is necessary to understand the principle of cutting a prism through any three fixed points in space, by a plane passing through those points; the points may be in the surface of the prism itself, and may be either all in the concave side, or all in the convex side ; or partly in the concave side, and partly in the convex side;— that such a supposition is possible will readily appear, since any three points are always in the same plane; and, there fore, the plane may cut the prism through any three given points.
The three points through which the section is cut, are said to be given, when the seats are given on the plane of the base of the prism, which plane is understood to be at right angles to the axis of the prism, and when the distances or heights from the scats to the points themselves are given.
It is always to be understood, that the three seats are not in a straight line, and consequently the three points them selves not a straight line.
The seat of a point in space on any plane, is that point in the plane where a perpendicular drawn through the point in space cuts the plane.
In the helical solid, the winding surface connecting the two prismatic surfaces, has been defined to be of such a property as to coincide with a straight line perpendicular to the exte rior prismatic surface, and, consequently, if the axis of the curved prism be perpendicular to the horizon, every such line will be parallel to the base ; now, let the seats of three such lines be given on the plan, viz., let each extreme boun dary be one, and let another be taken in the convex side passing through the point, which would give the middle of the development of the said side of the plan; the three seats would be terminated by the convex and concave sides of the plan, and will always be perpendicular to the convex side, and equal in length to each other. Call the three level lines, of which their seats are given, the lines of support ; let a plane be laid on the three lines of support, and it will rest either upon three points, or upon one of the said lines and two points; hence the points which come in contact with the plane, will be at one extremity of each line of support ; let each of the points, which come in contact with the plane thus posited, be called a resting point. The three resting
points are the three points in space, through which the plane is supposed to pass that cuts the curved prism.
Now because each line of support has two extremities, there will be six extreme points in all, but as only three can be resting points, unless the plane coincides with one of the lilies of support, it will be proper to show, which three of the six are the resting points. Let the plane, thus laid upon sonic three extremities of the lines of support, be continued to intersect the base of the curved prism, then the nearest extremity of the seat of any line of support, to the intersect ing line, is the seat of the resting point of that line. For this purpose, let a development of the convex side of the rail be made according to the plan and rise of the steps. The part of this development that is made to bend round the concave or convex eylindric surface of the helical portion or wreath, is called which is supposed to be brought to an equal breadth throughout its length. Only one filling-mould is used in the construction of hand rails. Let therefore the for the convex side be con strneted, and let two straight lines be drawn from the ends of the upper edge of that part of the falling-mould corresponding to the ends of the wreath perpendicular to the base of the whole development.; also let another intermediate line be drawn parallel to the other two. so as to bisect the part of the base intercepted by the said two parallels, the three parallels will give us the heights of the three resting points, the shortest height is at one extreme, and the longest at the other. Suppose now the shortest of these three heights taken from each of the three, and the remainders taken as heights, instead of the whole, then the height of the first resting point ill be nothing and will therefore coincide with its seat ; and if the middle height be less than half the length of the remaining height. the seats of the resting points will be the first and second extremities of the first and second lines of support taken on the convex side, and the extremity of the third on the concave side. The first resting point is a point in the intersection of the plane of the base with the inclined plane.