The curves of the face-mould for the hand-railing of any stairs of circular plan are elliptical, and may be drawn by a trammel, or in any other convenient manner. The trouble, however, attending the process of obtaining the axes, so as to be able to employ the trammel in describing the curves, is, in many cases, greater than it would be to obtain the curves through points found by ordinates, in the usual manner. But as this method for certain reasons may be preferred by some, an example is here given in which the curves are drawn by a trammel, and in which the method of obtaining the axes is shown.

Let Fig. 168 represent the plan of a hand-rail around part of a cylinder and with the heights set up, the intersection line obtained, the measuring base-line drawn, the rake-line established, and the tangents on the face-mould located - all in the usual manner as hereinbefore described. Then, to prepare for the trammel, from o, the centre of the cylinder, draw o b' parallel with the intersecting line, and b' o' perpendicular to b'f' the rake-line; make b'o' equal to bo, and o'a' equal to oa; through o' draw o'h parallel with b'f' From 0 draw oe perpendicular to ob'; continue the central circular line of the rail around to e; parallel with o b' draw c f, and from f' the point of intersection of ef with b'f', and perpendicular to b'f' draw f'e'; make f'e' equal to fe; then o' is the centre of the ellipse, and o' a' the semi-conjugate diameter and o' e' the semi-transverse diameter of an ellipse drawn through the centre of the face-mould. To get the diameters for the edges of the face-mould, make a' c' and a'd' each equal to half the width of the rail, as at cad; parallel to a line drawn from a' to e' and through c' draw the line c'g; also, parallel with a line drawn from a, to e' draw d'h (see Art. 559); then for the curve at the inner edge of the face-mould, o'g is the semi-transverse diameter, and o'c' the semi-conjugate; while for the curve at the outer edge o'h is the semi-transverse diameter, and o'd' the semi-conjugate. So much of the edges of the face-mould as are straight are parallel with the tangent. Now, placing the trammel at the centre, as shown in the figure, and making the distance on the rod from the pencil to the first pin equal to the semi-conjugate diameter, and the distance to the second pin equal to the semi-transverse diameter, each curve may be drawn as shown. (See Art. 549.)

285 Face Mould Curves Are Elliptical 204

Fig. 168.