### §28.

Paths described by Points of the Square relatively
to the Curve-triangle.

(Plates VII. and VIII.)

Let now the triangle P Q R be fixed, and the square A B C D moved upon it. In Plate VII., six paths described by points upon the line M 4 are shown. Nos. 1 and 2 give curtate roulettes built up of peri-trochoidal arcs, No. 3 is the common form of the curve, No. 4 a prolate curve. This is repeated on a double scale in Fig. 2. No. 5 is the homocentral, No. 6 the concentral curve. The last resembles a circle very closely, but consists of peri-tro- choidal arcs which so cover one another that the curve is four- fold, being traversed by its describing point four times in every period.

The curves V and 2' belong to points in a line lying between two principal axes, the first is a curtate, the second a prolate roulette.

Plate VIII. shows seven roulettes corresponding to points on the line M l .B. Nos. 1, 2, and 3 are curtate roulettes, No. 4 the common form of the series, Nos. 5 and 6 prolate roulettes, No. 7 the concentral form, the same as No. 6 in Plate VII., 2.

### §29.

Higher Pairs of Elements:—other Curved Figures
of Constant Breadth.

(Plates IX. to XIII.)

We found in 25 that every figure of constant breadth can be constrained in a circumscribed rhombus, so that a pair of elements may be made from it and the rhombus. Eight examples of such pairs are given in Plates IX. to XIII., they are chosen as specially suited for showing the extraordinary variety of constrained motions to which this proposition leads us. On account, however, of the completeness with which we have examined the pairs already considered we shall be able to dismiss these more shortly.

In Plate IX. we have the already known equilateral curve-triangle enclosed in a rhombus with angles of 60 and 120. The form of