At Fig. 163 so much of the plan at Fig. 162 is repeated as is required for the face-moulds, but for perspicuity at twice the size. The horizontal projection of the tangents for the first wreath are ad and db drawn at right angles to each other, tangent to the circle at a and b. Let those tangents be extended beyond d; through m, the lower end of the wreath, draw md', making an angle with md equal to that in Fig. 162, between the line af and a,f; or let the angle dmd' equal a fa' of Fig. 162. Make dd" equal to dd' Make bb" equal to b"'b" of Fig. 162; join d" and b" and extend the line to e"; make b" bv equal to b"b"" of Fig. 162, and draw bve" parallel with de. From e" draw e"e parallel with b"b; through e and f draw ef tangent to the circle at f; then b e and ef are the horizontal projections of the tangents for the upper wreath. Then if the plane B be revolved on ad, the plane C on de, and the plane D on ef until they each stand vertical to the plane A, the lines md" d"e"' and e"'f' will constitute the tangents of the two wreaths in position. This arrangement locates the upper joint of the upper wreath at /, leaving fc, a part of the circle, to be worked as a part of the long level rail on the landing. As the tangent over ef is level, the raking part of the rail will all be included in the wreath bf, so that at the joint f the rail terminates on the level.

281 Face Moulds For Winding Stairs 199

Fig. 103.

The portion fc, therefore, is a level rail requiring no canting, and it requires no other face-mould than that afforded by the plan from f to c.

For the face-mould for the rail over m a b, let the line e" d" be extended to mv, a point in the base-line b mv; then mv is a point in the base-plane A, as well as in the cutting plane E; therefore the line mv m is the intersecting line parallel to which all the ordinates on plane A are to be drawn. Perpendicular to this intersecting line mv m, at any convenient place draw m' b'; make b' b"' parallel to mv m and equal to b b"; connect b"' with m', a point at the intersection of the lines mv m and b' m'; then the angle b b"' m' is the plumb-bevil. Through d, parallel to m' m' draw d d"'; from the three points m" d"' and b"' draw lines perpendicular to m' b"'; make m' m" equal to m' m; make b"' b"" equal to b' b. Since the measuring base-line m' b' passes through d, the point of the angle formed by the two tangents, d"' is the point of this angle in the cutting plane E; therefore join m" and d"' also d"' and b""; then b"" d"' and d"' m" are the two tangents at right angles to which the joints at m" and b"" are drawn. The curves of the face-mould are now found as usual, by transferring the distances by ordinates, as shown, from the plane A to the plane E, making the distance from the rake-line m' b"' to each point in plane E equal to the distance from the corresponding point in the plane A to the measuring base-line m' b' Now, to obtain the sliding distance and the vertical line upon the butt-joints, make b"' bv equal to half the thickness of the plank; parallel with m' b"' draw bv bvi; also, b"" bvii and m" m"'; make b"" bvii and m" m"' each equal to bv bvi; through bvii and m"', and parallel to the respective tangents, draw bvii bx and m"' m""; then bx and m"" are the points from which, through the centre of the butt-joints, a line is to be drawn which will be vertical when the wreath is in position. (See Art. 284.)

For the face-mould for the upper quarter, through b, Fig.

163, draw b e, parallel with d" e"; make e e"' equal to ee'; draw e"' f' parallel with e f. Now, since e"' f' is a horizontal line and is in the cutting plane F, therefore, parallel with e"'f' and through b', draw b n; then b n is the required intersecting line. Extend e f to f"; make f f" equal to ff; join f" and n; then the angle ff" n is the plumb-bevil. Perpendicular to nf" draw f"f" and n n" and make these lines respectively equal to e f and b n; join f" and f"'; also f"' and n'; then f" f"' and f"' n' are the required tangents. The butt-joints at f' and n' are drawn perpendicular to their respective tangents. To get the slide distance and vertical lines on the butt-joints, make f"fv equal to half the thickness of the plank; parallel with n f", through fv draw fvf"";also, through n, draw n'n"; make n' n" equal to fvf""; through n", parallel with n ' f"', draw n" n"'; then n"' is the point through which a line is to be drawn to the centre of the butt-joint, and this line will be in the vertical plane containing the tangent. So, also, parallel with the tangent /„ /„,, and through f"", draw f""fvi then fvi is the point through which a line is to be drawn to the centre of the butt-joint (see Art. 284). The curve is now to be obtained by the ordinates, as before explained.