that its path is really three-fold; that is, is traversed thrice in each period by the point M v This can be recognised from the looped paths 5 and 6, the tangents to which turn three times through four right angles. The path 1 is also remarkable, for the three homologous points 1, 1', and 1" lie continually in it, so that complete restraint occurs, as in Fig. 59.
Plate IV. shows the further point-paths obtained by choosing points in the line M l Q, or (what is the same thing) in the lines M l T or M l 17. It is seen at once that the principal ,axis of the figures is now turned through 90, and also that the loops form themselves about an axis perpendicular to the original one. The curve T S (Fig. 1) and its symmetrical repetitions are characteristic, the former is the circular arc described by the centre T of the rolling curve UQ.
If describing points be taken upon radii lying between A M l and T M v paths are obtained which are not, as before, symmetrical about two axes. It has not been thought necessary to give ex- amples of these; their nature will be made sufficiently clear by the analogy of the paths 1', 2', &c. in Plate II.
We have found that the point-paths of the pairs of elements which we have considered possess extraordinary variety of form, they can? however, be somewhat systematised by the use of a method and nomenclature similar to that employed for trochoidal curves. Our curves form themselves into two series, corresponding to the fixing of one or the other element, and each series divides itself into groups according to the position of the line on which its describing points are taken. The paths of points in the centroids themselves, as, e.g., the triangle U T Q, Plate I. 1, the three-cornered paths of the point m 2 in Plate II. 2, &c. are specially characteristic. These paths may be called the common form of the roulettes concerned, as in the case of the cycloid. By the same analogy we may call all paths of points which lie beyond or within the rolling centroids, curtate or prolate point-paths respectively. Among the last, one is specially characteristic, and common to all groups of point- paths, the path of the centre point of the moving centroid, M in Plates I. and IE., M l in Plates III. and IV. This roulette is always the smallest of its series; the point-paths concentrate them- selves upon it as the path of the point relatively nearest to the cen- troidal curve itself, just as a circle concentrates itself upon its centre
