these two positions, they are in general of an elliptical three-cornered
form. Four of them are shown dotted in Plate II. and in-
dicated by the numbers 1', 2', 1", and 2". These figures are no
longer symmetrical about their three axes, as the former were;
this can be specially seen from No. 2". 16

### §24.

Point-paths of the Triangle relatively to the
Duangle.

(Plates III. and IV.)

To determine the point-paths of the triangle relatively to the duangle the latter must be fixed and the former set in motion The centroid, UTQ, Plate III. 1, then rolls upon the centroid, P?^ Qm 2 . The figures described are formed of arcs of peri-trochoids. All describing points lying upon the rolling centroid here describe arcs of cardioids, as we have seen in connection with Fig. 97.

It is at once noticeable how greatly these figures differ from the former ones. This forms an illustration which exactly meets a mistake made by many former writers on this subject, that the inversion of such a pair of figures, although it produces the greatest alteration, in the manner of turning, does not alter the form of the point-paths. 17 This circumstance has given me occasion to con- struct these pairs of elements, to which, otherwise, I cannot ascribe any particular use.

The figures in Plate III., show the paths of points in the axis MA of the triangle. The point 1 describes a rounded oval, consist- ing, like all the other figures, of six peri-trochoidal arcs. Point 2, coinciding with the vertex A of the triangle, gives an oval with concave sides, as does also point 3; the path of point 4 consists of two simple cardioids joined in the points m a and m 2 of the stationary centroids. The path of 4' is repeated in Fig. 2 upon a larger scale. The paths of 5 and 6 each have two loops, which in 7 fall together into one oval curve. Point 7 itself coincides with the centre point M l of the triangle A B C, and it must be noted