point of contrary-flexure, is found as follows. Draw HZ at right angles to HO and equal thereto. Draw Z L parallel to OH, and from O as a centre, with O) A as radius, draw an arc cutting ZL in the point L; then L O is the tangent at O.

The tangent at Y, where the curve is at its greatest distance from the axis, is parallel to OD; and the tangent at D is perpendicular to O D.

169. To apply the lemniscate to form an ogival arch.Let A B (fig. 136) be the given span, C Y the height of the

arch, the point C bisecting A B. Bisect A C, BC, in the points X, and draw XO D parallel to C Y, and D Y parallel to A B. Bisect X D in O, and draw the lemniscate Y O (167) having DO for its axis, DY for its greatest distance from the axis, and O for its point of contrary-flexure. Repeat the curve Y O in A O, and Y O A will form one side

of the required arch, having its tangent at A perpendicular to A B. The tangent at O should be found (168) before drawing the curve.

The curve in the figure has the radii of the generating circles in the ratio of 4 to 5.

170. To apply the lemniscate to mouldings and other architectural ornament.-Mouldings of contrary-flexure, as shown on figs. 137, 138, 139, can be drawn when the

points Y, Y' are given, and the vertical or horizontal axis O D.

Draw Y'A parallel to OD, and Y X A, Y' X', perpendicular thereto. Bisect X X' in O, which will be the point of contrary-flexure of the curve. Draw OE perpendicular to O X, and bisect the right-angle EOX by the line OZ. From O as a centre, with OY as radius, describe a circle cutting OZ in Z. Drop the perpendicular Z H upon OX,

and OH will be the radius of the smaller generating circle. Make HD equal to OH, and D is the point where the curve cuts the axis. We can now find the length of the other radius in the method described above (167), and

draw the curve DYO. The other portion, OY'D' is a repetition of the portion OY D. In fig. 137 the radii of the generating circles are in the ratio of 4 to 5, XY being

two-fifths of O D; in fig. 138 they are in the ratio of 8 to 9, XY being four-ninths of OD; and in figs. 139 and 135 the radii are as 2 to 3, the height XY being one-third of O D.

This curve may be also made to form the contour of a

leaf (fig. 140), in which O is the point of contrary-flexure, OD the axis, X Y the greatest distance from the axis. In

the example here given, the same curve as in fig. 135 is used.

There are many other curves having points of contraryflexure, such as the Trochoid and Companion to the Cyclvid, which are treated upon in Chapter VI.