corners of the first polygon will give to P Q the required changes
of position relative to the fixed plane, or to any stationary figure,
as A B, lying in it.

If we examine the relation of the two polygons to each other we notice the special and important peculiarity, that each has the same properties in reference to the other, that is, that they are reciprocal. Thus in any one of the positions in which two corresponding sides coincide, the polygons show not only the position of the figure supposed to be movable relatively to the fixed figure, but also conversely the position of the fixed relatively to the movable figure. (Prop. III. of 5.) We can thus determine as many relative positions of the two figures by means of the central polygons as the latter have pairs of corresponding sides.

### §7.

Centroids; Cylindric Rolling.

The method above described affords us the means of representing a succession of given separate positions of two figures. It leaves, however, the actual changes of position undetermined, or substitutes for them a series of rotations about isolated points. But if we sup- pose the assumed positions PQ, P 1 Q lt P 2 Q 2 , etc., taken nearer and nearer together until at last the intervals between them disappear, we shall have a complete representation of the whole motion. The corners of both the central polygons will at the same time have approached each other until each is removed from its neighbour by only an indefinitely small distance, and thus the two polygons become curves, of which infinitely small parts of equal length continually fall together after infinitely small rotations about their end points, which, that is to say, turn or roll upon each other during the continuous alteration in the relative positions of the two figures. The turning which takes place about each point in the curves is not now, as before, tem- porary, but in general for an instant only, and each point is therefore called an instantaneous centre. The curves into which the polygons are transformed both pass through the whole