Let j k and l m, Fig. 142, represent the central or axial lines of the hand-rails of the two flights, one above, the other below the platform; and let the semicircle j d l be the central line of the rail around the cylinder at the platform, the risers at the platform being located at j and l. Vertically over the platform risers draw gg'; make gr' equal to a riser of the lower flight, and r'g' and s s' each equal to a riser of the upper flight. Draw g's and gk' horizontal and equal each to a tread of each flight respectively. Through r' draw k' a" and through g' draw s' t'. Vertically over d draw a, t' Horizontally draw a" a"" and t' t".

It is usual to extend the wreath of the cylinder so as to include a part of the straight rail - such a part as convenience may require. Let the straight part here to be included extend from l to b on the plan. Vertically over b draw b' c"" and horizontally draw b' w"; at any point on b' w" locate w„, and make w"w' equal to j l, and bisect it in w; erect the perpendiculars w' a"' w dv"' and w" v; join t" and a""; from dv" horizontally draw dv" dv'; parallel with r' k' draw dv' c"". We now have the plan and elevations of the prism, containing at its angles the tangents required for the wreath extending from b to d on the plan. The elevation F is a view of the cylinder looking: in the direction dc.

Fig. 142.

Comparing Fig. 142 with Fig. 141, the line b/W// is the trace, upon a vertical plane, of the horizontal plane abcd of Fig. 141, or is the ground-line from which the heights of the prism are to be taken.

The triangle a' b' a" is represented in Fig. 141 at abg, and the inclined line b' a" is the tangent of the rail of the lower flight, and is represented in Fig. 141 at bg; while a"" t" is the tangent of the railing around the cylinder, and the half of it is represented in Fig. 141 at ge. The height b'c"" is shown in Fig. 141 at c f, while the height w dv" or a' dv is shown in Fig. 141 at de.

The vertical planes B E C may now be constructed about the prism as in Fig. 140, proceeding thus: Make c c' equal to b' c"" and dd' equal to a' dv'; through c' draw d' h; through b draw h b"; perpendicular to h b" through a draw b" dv; from d parallel with b" dvdraw d d"'; on d as centre describe the arc d'd""'; draw d"" dv, also dd"" parallel with hb"; on d" as centre describe the arc dvd" join d"' to b" Parallel with b" h draw from each important point of the plan, as shown, an ordinate extending to the line b" d"' and thence across plane D draw ordinates perpendicular to b" d"' and make them respectively equal to the corresponding ordinates of the plane A, measured from the line b" dv; join e to f, a"' to b" a"' to e, and b' to f; also join l,to r' Then a"' b' is the tangent standing over a b, and a"' e is the tangent standing over a d. The line b'l, is the part of the tangent which stands over bl" the portion of the wreath which is straight. The curve en'p'l' is the trace upon the cutting plane of the quarter circle dnpl, traced through the points n'p" and as many more as desirable, found by ordinates as any other point in the plane A. Thus we have complete the line b' l' n' e, the central line of the wreath extending from b to d in the plan. This is the essential part of the face-mould, which is now to be drawn as follows: At Fig. 143 repeat the parallelogram a"' b'fe of Fig. 142, and, with a radius equal to half the diameter of the rail, describe, from centres taken on the central line, the several circles shown; and tangent to these circles draw the outer and inner edges of the rail. The joint at b' is to be drawn perpendicular to the tangent b, a"', while that at e is to be perpendicular to the tangent ca"'. This completes the face-mould for the wreath over blnd of the plan. If the pitch-board of the upper flight be the same as that of the lower flight, the face-mould at Fig. 143 will, reversed, serve also for the wreath over the other half of the cylinder.

In using this face-mould, place it upon a plank equal in thickness to the diameter of the rail, mark its form upon the plank, and saw square through; then chamfer the wreath to an octagonal form, after which carefully remove the angles so as to produce the required round form. The joints, as well as the curved edges, are to be cut square through the plank.

Many more lines have been used in obtaining this face-mould than were really necessary for so simple a case, but no more than was deemed advisable in order properly to elucidate the general principles involved. A very simple method for face-moulds of platform stairs with small cylinders will now be shown.

Fig. 143.