Page:The Kinematics of Machinery.djvu/141

 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

$$T Q' = T O = \frac{P' Q'}{\sin a} = \frac{PQ}{\sin a}$$ or if we denote $$O T$$ by $$R$$ and $$PQ$$ by $$a$$, $$R=\frac{a}{\sin a}$$ (b.) Centroid of the Duangle.—If now, in order to find the second centroid, we suppose the line $$P Q$$ stationary, and set the angle $$P T Q$$ 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 $$P Q$$ and a vertex angle 180°—$$a$$—, which is evidently the circle $$Q O P T S$$ having a diameter TO, and circumscribed about the given triangle $$P Q S.$$ If we denote the radius of this circle by $$r$$, we have

$$r=\frac{T O}{2} = \frac {a}{2\sin a} =\frac{R}{2}.$$

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 $$r$$ in $$R$$, becoming ellipses (Fig. 96), of which the one described by any point in the circumference of $$r$$ has a semi-axis major equal to $$R$$, and a semi-axis minor equal to zero, and therefore conicides with the diameter of $$R$$. For the rolling of $$R$$ upon $$r$$ 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