diagonal (OD) is constant and is equal to P R. The locus of 0, or centroid, is therefore a circular arc having D for its centre and PR = PQ = AB= the length of the side of the square, for its radius. The centre continues in this curve until R has arrived at the same distance from C at which P is shown from A in the figure i.e., up to the point 2. The chord P Q then slides in the same way on A B and A D, giving the arc 2, 3, similar to the former one, as the continuation of the centroid, in the same way the arc 3, 4 is obtained, and lastly the arc 4, 1. The centroid for the square is therefore a curve-square having for its sides four circular arcs drawn from the four corners of the square with radii equal to its sides.
To find the centroid of the triangle we invert the pair, that is, imagine the triangle stationary and the square moving upon it. The
centroid is then the locus of the B vertex of a right angled triangle having its hypothenuse =P R, i.e. a circle with the diameter PR described from its middle point 3'. The arc of this circle which forms part of the centroid ends at 2', the middle point of the side Q R. Then follows the similar curve 2' 3', and lastly, returning to 0, the similar curve 3' 1. Hence the centroid of the curve- triangle is itself a curve- triangle, and is equilateral,
its sides being arcs described from the centres of the sides P Q, Q R, and R P, and having radii equal to half their length.
Fig. 102.
As the one figure rolls relatively to the other, the curve 1.2' rolls on 1.2, then 2'.3' on 2.3, 3'.1 on 3.4, and so on. In order again to reach its initial position the instantaneous centre must traverse equal distances on both centroids, must therefore traverse the three sides of the curve-triangle four times, and the four sides of the curve square three times. For each completed rolling of the former on its three sides the element to which it belongs turns through an angle of 90 relatively to the square, so that after the.
