§26.
Higher Pairs of Elements.—Equilateral Curve.
Triangle and Rhombus.
Figures of constant breadth can easily be constructed of circular arcs. If from the corners of an equilateral triangle P Q R, Fig. 101, arcs be drawn with radii equal to the length of one of the sides, we obtain a figure which we may call an equilateral curve- triangle. This has everywhere a breadth equal to the side P Q, so that it can be constrained in a square or rhombus A B C D, the distance between whose opposite sides is P Q. In the square the normals intersect at right angles, in the rhombus obliquely. . If we
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Fig. 101.
take these figures as cross sections for cylinders, and give to the latter such profiles as will prevent end- long sliding, we have constructed a higher pair of elements.
We may examine the centroids of these figures, taking first that of the square. In Fig. 102 the instantaneous centre of motion is the point 0, which in the position shown in the figure falls upon the vertical bisector R 0, of the square, and is also the centre of one side of the triangle. Let the curve-triangle make L. H. rotation about this point. Its corner P then slides downwards along the side A D, while R moves to the right along D C. The normals from P and R always intersect at right angles, so that the locus of the instantaneous centre is that of the vertex of a right-angled triangle, of which the ends of the hypothenuse slide upon the arms (D A and D C) of a right angle. is thus always the corner of a rectangle PDR 0, of which the
