The intersection of the plane F G with the cone is projected horizontally in a circle described from the centre C', with the diameter F' G'. The arcs I' F' A' and H' G' B' are the only parts of this circle which require to be drawn.

Figs. 7 and 8.-To describe the curves formed by the intersection of a cylinder with the frustum of a cone, the axes of the two solids cutting each other at right angles.

The axes of the solids and their projections are laid down in the figures precisely as in the preceding example. The intersections of the outlines of the cone in the elevation with those of the cylinder, furnish, obviously, four points in the curves of penetration; these points are all projected horizontally upon the line A' B'. Now, suppose a plane, as a b (fig. 7), to pass horizontally through both solids; its intersection with the cone will be a circle of the diameter c d, while the cylinder will be cut in two parallel straight lines, represented in the elevation by a b, and whose horizontal projection may be determined in the following manner :-Conceive a vertical plane ƒ g, cutting the cylinder at right angles to its axis, and let the circle g ef thereby formed be described from the intersection of the axes of the two solids; the linej h will now represent, in this position of the section, the distance of one of the lines sought from the axis of the cylinder. Now set off this distance on both sides of the point A', and through the points k and a' thus obtained, draw straight lines parallel to A' B'; the intersections of these lines with the circle drawn from the centre C' of the diameter c d will give four points m', p', n, and O2 which being projected vertically upon a b, determine two points m and p in the curves required.

In order to obtain the vertices or adjacent limiting points of the curves, draw from the vertex of the cone a straight line te, touching the circle gef, and let a horizontal plane be supposed to pass through the point of contact e. Proceed according to the method given above to determine the intersections of this plane with each of the solids in question, the four points ¿', r', q, and s, which being projected vertically upon the line e r, determine the vertices i and r required.

OF THE HELIX.

Plate VII.—The Helix is the curve described upon the surface of a cylinder by a point revolving round it, and at the same time moving parallel to its axis by a certain invariable distance during each revolution. This distance is called the pitch of the screw.

Figs. 1 and 2.-Required to construct the helical curve described by the point ▲ upon a cylinder projected horizontally in the circle A' C' F', the pitch being represented by the line A' A3.

Divide the pitch A' A3 into any number of equal parts, say eight; and through each point of division, 1, 2, 3, &c., draw straight lines parallel to the ground line. Then divide the circumference A' C' F' into the same number of parts; the points of division B', C', E', F', &c., will be the horizontal projections of the different positions of the given point during its motion round the cylinder. Thus, when the point is at B' in the plan, its vertical projection will be the point of intersection B of the perpendicular drawn through B' and the horizontal drawn through the first point of division. Also, when the point arrives at C' in the plan, its vertical projection is the point C, where the perpendicular drawn from C' cuts the horizontal passing through the second point of division, and so on for all the remaining points. The curve A B C F A3 drawn through all the points thus obtained, is the helix required.

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Figs. 1 and 2.-To draw the vertical elevation of the solid contained between two helical surfaces and two concentric cylinders.

A helical surface is generated by the revolution of a straight line round the axis of a cylinder; its outer end moving in a helix, and the line itself forming with the axis a constant and invariable angle.

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Let A' C' F' and K' M' O' represent the concentric bases of the cylinders, whose common axis S T is vertical; the curve of the exterior helix A CF A3 is the first to be drawn according to the method above shown. Then having set off from A to A2 the thickness of the required solid, draw through A2 another helix equal and similar to the former. Now construct, as above, another helix, K C O, of the same pitch as the last, but on the interior cylinder; as also another, K2 C2 O2, equal and parallel to the former. The lines A' K', B' L', C' M', &c., represent the horizontal projections of the various positions of the generating straight line, which, in the present example, has been supposed to be horizontal; and these lines are projected vertically at A K, B L, &c.

It will be observed, that in the position A K the generating line is projected in its actual length, and that at the position C'M' its vertical projection is the point C. The same remark applies to the generatrix of the second helix. The parts of both curves which are visible in the elevation may be easily determined by inspection.

Figs. 3 and 4.-To determine the vertical projection of the solid formed by a sphere moving in a helical curve.

Let A' C'E' be the base of a cylinder, upon which the centre point C'