point when its radius is diminished to zero; this form we may therefore call the concentral form of the common point-path.
Further, those roulettes also are noticeable which pass through the middle point of the whole series of curves, as No. 6 in Plate I., 2. These roulettes we shall call homocentral. In our example they take a series of forms of which some are shown in the second figures of Plates I. to IV. It must be noticed that homocentral point-paths can only be described by those points which, as the moving centroid revolves, pass through the centre of the stationary centroid; or con- versely by those points through which the centre of the stationary centroid might pass if the pair were inverted. Such points are, how- ever, only those of the concentral point-paths. In other words the points of concentral point-paths describe homocentral point-paths if the pair be inverted. Thus the homocentral curves 6, Plate I., 2, and 5, Plate II., 2, are described by points in the concentral curve M 7 of Plates III., 2 and IV., 2. The points of the concentral curves which have been used can easily be seen from the figures.
This way of looking at the curves may also be extended to the examination of trochoids, which, indeed, we should actually obtain if the centroids of the pair of elements were circles. Here the concentral roulettes would be the circles described by the centres of the rolling circles, and the homocentral paths those star-shaped figures which are described by points in the circumference of a circle concentric with the rolling circle, and having a radius equal to the difference between the radii of the two centroids. 18
Figures of Constant Breadth.
The conclusions of 21 lead us synthetically to a series of other pairs of elements, of which we may examine a few here. If upon any plane figure two parallel tangents be laid, as AB and CD, Tig. 99, the distance between them, c, measures the extension of the figure in the direction of the normals of restraint. This extension may be called the breadth of the figure, in general, it is not con- stant for the same figure. There are figures, however, in which the breadth is constant; in which, that is, all pairs of parallel tangents