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tangent. Other forms of leaf can be obtained in the same way, as shown in figs. 124, 125.

Fig. 124.
Fig. 123.


If the axis of a parabola is placed vertical so as to have the vertex A (fig. 117) for its highest

Fig. 125. point, this curve will represent the path which a projectile would take when discharged at any angle to the horizon less than a right-angle, the resistance of the air being entirely ignored. The line of discharge is a tangent to the parabola, and the greatest range is obtained on a horizontal plane when that line makes half a right-angle with the horizon.

If the axis of the parabola is placed vertical, with the vertex A (fig. 117) for its lowest point, the curve will be that of the chain of a suspension bridge in which the vertical load is uniformly distributed along a


horizontal roadway, forming a tangent to the curve at its vertex.


152. DEFINITION.—When a chain or flexible cord is susFig. 126.

pended from two points, A and B
(fig. 126), in the same horizontal
line, the curve which it takes is called
a catenary. This curve has its greatest
curvature at the lowest or middle
point C, and the tangent C Tis parallel
to A B.

153. To apply the catenary to form the entasis of a column.—Bisect A B (fig. 126) in D, and draw DC at right-angles to A B. Let DC represent the difference between the top and bottom radii of the column, BD its vertical height. Take A DB, equal to twice BD, in a horizontal position on a vertical board ; and let a chain be suspended from the points A and B so as to hang down a distance equal to D C from A D B. Then by marking the outline of the chain CFD on the board, the entasis or curvature of the column is obtained.

154. To apply the catenary to form the entasis of a spire.—Let AB (fig. 126) represent the slant side of the spire if straight, or the

chord of the proposed curve; DC the amount of deviation from the straight line which is re


quired, and which is always very small in comparison with the length of A B. Place A B horizontal upon a vertical board, and allow a chain to hang vertically from A and B to the distance DC, and the outline of the curvature required can be obtained as before.




155. DEFINITIONS._ When a curve, after bending in one direction, nakes a bend in an opposite direction, like the letter 8, it is called a curve of flexure ; and the point at which the change of direction takes place is called the point of contrary-flexure. In all true curves of flexure the curvature decreases as we approach the point of flexure, or the length of the radius of curvature (101) increases and becomes infinite at that point, the centre of curvature passing over to the opposite side and its distance gradually decreasing as we recede from the point of flexure. The Harmonic-curve, or curve of sines, consists of a series of equal undulations or waves, and is the curve in which a musical string vibrates when sounded. It is called the curve of sines from the fact of its ordinates being proportional to the trigonometrical sines of the abscissæ. .

The Lemniscate is a curve having the form of a figure oo, the point of contrary-flexure being where the curve crosses itself.

The HARMONIC-CURVE. 156. To find points on the contour of the harmonic-curve. -Draw a horizontal line or axis 0 A B (fig. 127), and draw OD at right angles to 0 A. Take OD equal to the given height B C, to which the curve is to rise above the axis, and

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