first revolution its corner 1 comes to the point 4, after the second to 3, after the third to 2, and after the fourth back again to 1.
§27.
Paths of Points of the Curve-Triangle relatively to
the Square.
(Plates V. and VI.)
The point-paths of this pair have, as follows from the nature of their centroids, a close relationship to those of the pair shown in Plates I. to IV. All paths of points of the curve-triangle relatively to the square consist of arcs of hypocycloids or hypotrochoids, which are in this case ellipses, while all point-paths of the square relatively to the triangle consist of arcs of peri-trochoids (including the special case of cardioids, as on p. 123). Let us first suppose the square fixed and the triangle in motion.
Plate V. shows a series of point-paths for which the describing points lie upon a line drawn from the centre M of the triangle perpendicular to the chord P Q. Point 1 gives a four-sided figure composed of elliptic arcs, its corners being elliptically rounded off by a pair of similar arcs having a common tangent; at 1 for instance one of these is given by the rolling of m 1 m 2 on 0) v and the other by the rolling of m^m z on 0^0 3 . When m 2 reaches O v w 2 w 3 begins to roll on 0^0 y The point 1 is however so chosen that m-^l is equal to the radius of the curves of the centroid m^i^m^ and being therefore upon the circumference of the smaller Cardanic circle, it describes a straight line. Thus the portions of the point-paths passing through A and B } the centres of the greater Cardanic circles 2 B arid 0^0 l; are straight lines, or more strictly, are elliptic arcs which have become straight lines. The continuation of the curve can easily be understood. It is completed when each side of the centroid m 1 m 2 m 3 has rolled on each side of the centroid 1 (9 2 3 4 , and consists therefore of twelve (four times three) separate arcs.
The point 2 describes a four-cornered figure with slightly concave sides, and the point 3 a similar figure in which the concavity is more distinct; in both cases all the curves are elliptic. The end
