To draw the curve, the points or foci where the pins are to be fixed must be found on the major axis. To find these points, take the length of b k (which is, as previously found, the exact length of the semi-major for the inside curve) on the dividers; fix one leg at 2, and describe the arc r, cutting the major where the pins arc shown fixed, at o and o. Now take a piece of string long enough to form a loop around the two and extending, when tight, to 2, where the pencil is placed ; and, keeping the string tight, sweep the curve from b to b.
The same method, for finding the major and foci for the outside curve, is shown in the diagram. The line drawn from b on the outside of the joint at n, to w, is the semi-major for the outside curve; and the points where the outside pins are shown on the major will he the foci. To draw the curves of the mould according to this method, which is a scientific one, may seem a complicated problem; but once it is understood, it becomes very simple. A simpler way to draw them, however, is shown in Fig. 120.
The width on the minor and at each end will have to be determined by the method just explained in connection with Fig. 119. In Fig 120, the points b at the ends, and the points in which the circumference of the circle cuts the minor axis, will be points contained in the curves, as already explained. Now take a flexible lath; bend it to touch b, z, and b for the inside curve, and b, w, and b for the outside curve. This method is handy where the curve is comparatively flat, as in the example here shown; but where the mould has a sharp curva ture, as in case of the one shown in Fig. 101, the method shown in Fig. 119 must be adhered to.
With a clear knowledge of the above two methods, the student will be able to put curves on any mould.
The mould shown in these two diagrams, Figs. 119 and 120, is for the upper wreath, extending from h to 71, in Fig. 94 A practical handrailer would draw only what is shown in Fig. 120. He would take the lengths of tangents from Fig. 94, and place them as shown at it in and in n. By comparing Fig. 120 with the tangents of the upper wreath in Fig. 94, it will be easy for the student to understand the remaining lines shown in Fig. 120. The bevels are'shown applied to the mould in Fig. 105, to give it the twist. In Fig. 106, is shown how, after the rail is twisted and placed in position over and above the quadrant c d in Fig. 94, its sides will be plumb.
In Fig. 121 are shown the tangents taken from the bottom wreath in Fig.
95 It was shown how to develop the section and find the angle for the tan gents in the face-mould, /n Fig. 113. The method shown in Fig. 119 for putting on the curves, would be the most suitable.
Fig. 121 is presented more for the purposes of study than as a method of construction. 7 t contains all the lines made use of to find the developed section of a plane inclining unequally in two different directions, as shown in Fig. 122.
Arrangement of Risers in and around An important matter in wreath construction is to have a knowledge of how to arrange the risers in and around a well-hoie. A great deal of labor and material is saved through it; also a far better appearance to the finished rail may be secured.
In level-landing stairways, the easiest example is the one shown in Fig. 123, in which the radius of the central line of rail is made equal to one-half the width of a tread. In the diagram the radius is shown to be 5 inches, and the treads 10 inches. The risers are placed in the springing, as at a and a. The elevation of the tangents by this arrangement will be, as shown, one level and one inclined, for each piece of wreath. When in this position, there is no trouble in finding the angle of the tangent as required on the face-mould, owing to that angle, as in every such case, being a right angle, as shown at w; also no special bevel will have to be found, because the upper bevel of the pitch-board contains the angle required.
The same results are obtained in the example shown in Fig. 124, in which the radius of the well-hole is larger than half the width of a tread, by placing the riser a at a distance from c equal to half the width of a tread, instead of at the springing as in the preceding example.
In Fig. 125 is shown a case where the risers are placed at a dis tance from c equal to a full tread, the effect in respect to the tangents of the face-mould and bevel being the same as in the two preceding examples. In Fig. 126 is shown the plan of Fig. 123; in Fig. 127, the plan of Fig. 124; and in Fig. 12S, the plan of Fig. 125. For the wreaths shown in all these figures, there will be no necessity of spring ing the plank, which is a term used in handrailing to denote the twisting of the wreath; and no other bevel than the one at the upper end of the pitch-board will be required. This type of wreath, also, is the one that is required at the top of a landing when the rail of the flight intersects with a level-landing rail.