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°f which their seats are given, the lines of support; leta _ plane be laid on the three lines of support, and it will rest^ther upon three points, or upon one of the said lines and*Wo points ; hence the points which come in contact withplane, will be at one extremity of each line of support;tet each of these points, which come in contact with the planethus posited, be called a resting point. The three restingPoints are the three points in space, through which the plane18 supposed to pass that cuts the curved prism.

Now, because each line of support has two extremities,there will be si.' extreme points in all, but as only three canhe resting points, unless the plane coincides with one of thehues of support, it will be proper to show, which three ofthe six are the resting ooints. Let the plane, thus laid upons ome three extremities of the lines of support, be continuedto intersect the base of the. curved prism, then the nearestExtremity of the seat of any line of support, to the intersect-In g line, is the seat of the resting point of that line. Forthis purpose, let a development of the convex side of thefail be made according to the plan and rise of the steps.The part of this development that is made to bend roundthe concave or convex cylindric surface of the helical portion°r wreath, is called a falling-mould, which is supposed to bebrought to an equal breadth throughout its length. Only°ne falling-mould is used in the construction of hand-rails.Let therefore the falling-mould for the convex side be con-ducted, and let two straight lines be drawn from the ends ofthe upper edge of that part of the falling-mould correspondingto the ends of the wreath perpendicular to the base of the"'hole development; also let another intermediate line bedrawn parallel to the other two, so as to bisect the part of'•he base intercepted by the said two parallels, the threeParallels will give us the heights of the three resting points,the shortest height is at one extreme, and the longest at the°ther. Suppose now the shortest of these three heights takenfrom each of the three, and the remainders taken as heights,'Ostead of the whole, then the height of the first restingPoint will be nothing, and will therefore coincide with itsSft at ; and if the middle height be less than half the lengththe remaining height, the seats of the resting points willbe the first and second extremities of the first and secondones of support taken on the convex side, and the extremitythe third on the concave side. The first resting point is? Point in the intersection of the plane of the base with thernclined plane.

The process is now completely reduced to that of finding*be section of a prism through tiller given points, which sup-pose to be done, and the plane ,)f section will touch the sup-posed wreath at the resting j jints of each line of supportjy*thout cutting the wreath at any such line; then the threeones of support will be on the same side of the plane, viz.,° n the under side. Suppose now another section taken below,a nd parallel to the former, so that the wreath may be just con-fined between these parallel sections or planes, and theQ, stance between the two sections will represent the thicknessthe plank. The section of the prism through its verticalSurfaces, is called the rake or the rake of the plan ; and a mouldbeing C ut to the rake is called the face-mould.

The manner of forming a helical line or screw is as

follows:—

Plate VIII. Figure 1. Divide the circumference of the®hter circle of the base into equal parts ; draw a line through“ e centre to represent the axis of the cylinder, parallel toV e axis, and through the points of division in the outera, fcumference draw lines ; divide the line of heights or the, ne representing the axis, into as many equal parts as thefcumference of the base is divided into, and through the

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points of division draw lines at right angles to the axis, inter-secting, as in the figure ; through the points of intersection,draw a curve which will represent the helix, or one of thearris lines of the rail.

Figure 2.—The projection of the solid helix coiling roundthe cylinder ; which helix represents the hand-rail before itis moulded.

Figured. —No. 1.—A solid section of a cylinder containedbetween two parallel planes, a part of the side of this solidcontained between two planes passing through the axis is theform of a piece from which the rail is made after it is cutout of the plank. No. 2, half the solid section, No. 3, theinclination of the cutting planes.

Such large portions as these, however, are by no meansproper to be employed in hand-railing, as the size would neces-sarily occasion the fibres of the wood to run in a transversedirection to the length of the rail, and consequently weaken it,but they are useful in this place to convey clear ideas of theprinciple upon which the art is founded.

Figure 5. —No. 1.—The proper form of a part from whicha portion of the rail is to be made after it has been cut out ofthe plank ; exhibiting the convex side of the same, and theupper plain surface, which is that of the plank. No. 2. Theconcave side of the same, with the joints and lower surface ofthe plank.

Figure 4.—A portion of the rail completely squared withthe concave side, the joints, the bottom part of the upperwinding surface, and the upper part of the lower windingsurface, brought into view. No. 2. The convex side of thesame, showing the upper part of the upper surface, and thelower part of the lower surface.

The business of hand-railing is to find the mould for cuttinga rail out of planks,

Though hand-railing is only treated of here, as connectedwith cylindrical well-holes : it is equally applicable to railserected upon any seat whatever.

The mould, which applies to the two faces of the plank,regulated by a line drawn on its edge, so as to be verticalwhen the plank is elevated to its natural position, is called theface-mould ; or sometimes the raking-mould.

A parallel mould, applied and bent to the side of the rail-piece, for the purpose of drawing the back and lower surface(which are to be so formed that every level straight line,directed to the axis of the well-hole, from every point of theside of the rail formed by the edges of the falling-mould, shallcoincide with the surface) is called a falling-mould.

When the upper surface of the plank is not at right anglesto a vertical plane passing through the chord of the plan, inorder to cut the corresponding portion of the rail out of theleast thickness of wood, the plank is said to be sprung.

A right-angled triangular board, made to the rise and treadof a step, is called the pitch-hoard.

In a stair-case, where there are both winders and flyers,two pitch-boards will be concerned, of different treads, but ofthe same heights, as the height of the steps must be equal.

The bevel by which the edge of the plank is reduced fromthe right angle, when the plank is sprung, in order to applythe face-mould, is called the spring of the plank; and the edgeor narrow side thus reduced, is called the sprung edge.

The bevel by which the face-mould is regulated to each sideof the plank, is called the pitch.

The formation of the upper and lower surface of a rail iscalled the falling of the rail.

The upper surface of the rail is called the back.

The first thing in the practice is to spring the plank, thento cut away the superfluous wood, as directed by the draughtsformed by the face-mould. This may be cut so very exactly

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