and. the other four times in each period. The point-path marked
I is described by the pentagon moving relatively to the square
and is (the describing point lying beyond the centroid) a curtate
curve; the I' is also a curtate roulette, it is however described
by a point of the square moving relatively to the pentagon.

Plate X., 2. Heart-shaped figure, made up of five circular arcs, in square. PSQ is an isosceles triangle with vertex angle PS Q = 53. The arcs PQ } 8T, and SR are described from S t Q, and P with a radius equal to the side A B of the square, the arcs T P and Q R from the intersection M of the lines P R and Q T y with radii equal to half the side of the square. The figure has therefore the constant breadth A B. The centroids for figures of these pro- portions are: for the curved figure a duangle 1.2 not equilateral; for the square an eight-rayed star of curves of different radii. The arcs of the centroids which roll upon each other belong always to Cardanic circles. Two point-paths / and /' are shown. The roulettes in their common form, described by the corners 1 and 2 of the duangle (centroid) are specially characteristic. They are squares having for their corners the points V 3' 5' 7' and 2' 4' 6' 8'. 19

Plate XI. Isosceles curve-triangle in rhombus. Upon an isosceles triangle 1 8 2, having a vertex angle <C 60, the circular arcs S T and S R, having radii $2 and S 1, are drawn to their intersec- tions T and R with 1.2 produced; from the same centres, and with radii 1 T and 2 R, the arcs T P and R Q are drawn until they intersect in P and Q the sides $ 1 and 8 2 of the triangle produced; lastly, P and Q are united by a circular arc drawn from the centre S. The figure thus inclosed has the constant breadth Q S. It is here paired with a rhombus having angles of 60 and 120. The centroids are somewhat complex, but consist as before of arcs of Cardanic circles, the centroid of the triangle having four such arcs, that of the rhombus eight. Two point-paths / and /', be- longing respectively to the triangle and the rhombus, are shown.

Plate XII., t shows another curve-triangle in a rhombus. The former is equilateral as in Fig. 1, but the radii of its sides (which are as before arcs described from the three corners PQR) are somewhat longer than the sides of the triangle, and the corners are rounded off with radii equal to this excess of length. The motion which occurs is exactly the same as would be given by a pair consisting of the dotted curve-triangle P Q R of the normal form and the enveloping