coincides with P. The figures are all three-cornered, and approach more and more nearly the triangular form, which is that actually described by the point 4. In Fig. 2 the paths of three more points 5, 6, and 7 are shown on a larger scale; the last of these is the centre M of the duangle. The path of 5 contains three loops; in the case of point 6, which is so chosen as to coincide with the centre M 1 of the triangle ABC, the loops have a common point of intersection. For any describing point between 6 and 7, the curves which intersect at M 1 in the former case open out, enclosing between them a triangular space; and lastly, point 7 gives the three loops fallen together into a continuous curve, which is the smallest curve which can be described by any point of the duangle. This curve is two-fold, in a whole period, that is, the describing point passes twice through M; this can be seen by an examination of the curve 6, the tangent to which twice turns through four right angles.
If the describing point be taken further from P than 7 we simply obtain repetitions, in reverse order, of the curves already described.
By choosing describing points upon the major axis of the duangle we get a further series of curves, of which some ex- amples are shown in Plate II. Point 1 again gives us an elliptic triangle; point 2, coinciding with the end $of the axis, gives a three-cornered figure, bounded partly by straight lines and partly by elliptic arcs; point 3 gives an elliptic triangle with concave sides, which is shown on a larger scale in Fig. 2. The point 4 coincides with the end m z of the short axis of the smaller centroid. It describes the remarkable figure No. 4 shown in Fig. 2, consisting of three circular arcs (described by m 2 as centre of the a,rc Pm^ Q), and three (twofold) rectilinear continuations of them (described by m 2 as a point in the circumference of the arc Pm 2 Q). The point 5 gives a curve with three loops, intersecting in the point MI; point 6 gives a three-looped curve with an inner open tri- angular space, and point M as the centre gives again the curve shown in Fig. 2 Plate L, and there marked 7. It is to be noted that the trochoidal triangles which form the paths of the points in the major axis are turned through an angle of 60 relatively to the point-paths of the minor axis.
The paths of all points between these two axes lie between
