the centroids here differs very greatly from that of the centroids in Plates V. to VIII. The centroid for the curve-triangle is a three- rayed figure, huilt up of three circular arcs of a radius C Q = C R = half the length of the side of the rhombus; the centroid of the rhombus is an equilateral duangle, its sides arcs described with the radius B A = B C = the side of the rhombus; or twice the magnitude of the radius with which the sides of the first figure were described, The centroids are therefore again arcs of Cardanic circles, and the point-paths built up of trochoidal arcs.
Some of the paths described by points of the triangle relatively to the rhombus are given. Point I, on the centre of the perpen- dicular A Q to one of the sides of the rhombus, gives a figure symmetrical about two axes, and resembling in profile a double- headed rail; the centre, II, of the triangle describes its concen- tral point-path, which is nothing else than the diameter E F, of the duangle (in the direction D B) this line being traversed three times in each whole period. This concentral point-path coincides with the homocentral, and is at the same time (as the path of a point on the centroid) a common form of the curve. The point- path I' is a curtate roulette of the rhombus; the point-path II" is a prolate roulette for the same figure. Every point in the diameter E F describes a homocentral curve in the curve-triangle; one of these that corresponding to the points E and F is given. The variety of the forms here taken by the roulettes shows that it is impossible to draw conclusions from analogy alone as to the general character of the forms of any series of point-paths.
Plate X. 1. Equilateral curve-pentagon in Square. The curve pentagon is constructed by describing arcs of circles having a radius equal to the diagonal about each of the corners of a regular pentagon, and a figure of constant breadth is thus obtained. The centroids are: for the square A B C D the four-cornered figure 1' 2' 3' 4', consisting of arcs of circles having the corners of the square for their centres and the side length P Q of the pentagon for radius; for the pentagon another equilateral curve-pentagon, described with radii equal to the half side-length of the pentagon from the centres m v m 2 , m s , m, m 5 of its sides. The centroid of the square rolls within that of the pentagon, and in every period each side of the one centroid must roll upoti every side of the other, so that the instantaneous centre traverses the one five times
