with the radius T 0. To find the magnitude of this radius in terms of quantities already known, suppose P Q to slide until it stands per- pendicular to either of the arms T P or T Q, as, for instance, at P'Q f . Then one of the normals coincides with P Q itself, and the other has become zero, and it will be seen at once that
or if we denote by and by ,
(b.) Centroid of the Duangle.—If now, in order to find the second centroid, we suppose the line stationary, and set the angle in motion, the points passing through P and Q must move always in the direction of the arms T P and T Q themselves. The normals cut in as before. The locus of this point is now, however, that of the vertex of a triangle having a base and a vertex angle 180°——, which is evidently the circle having a diameter TO, and circumscribed about the given triangle If we denote the radius of this circle by , we have
The centroids of our supposed pair of figures, angle and triangle, are therefore, if completely constructed, two circles, having the relative magnitude 1: 2, of which the smaller rolls in the larger. The relative paths themselves are therefore trochoidal, the hypotrochoids for the rolling of in , becoming ellipses (Fig. 96), of which the one described by any point in the circumference of has a semi-axis major equal to , and a semi-axis minor equal to zero, and therefore conicides with the diameter of . For the rolling of upon the point-paths are peri-trochoids, of which the common form is the cardioid. The common, curtate and prolate forms of these curves are shown in Figs. 96 and 97.r13 The former of these cycloid problems was first treated (although by no means completely) so far as my knowledge goes, by the celebrated mathematician Cardano, in the sixteenth century.r14 As I shall frequently have to refer again to this pair of circles I shall, for the sake of shortness, call them Cardanic circles. In the figures actually
