series of instantaneous centres point by point, and on this account we may call them the centroids of the moving figures. If these be known for any given pair of figures, their relative motions for a series of positions infinitely near together are also known, their changes of position are completely determined, and can be found by rolling one of the centroids upon the other.
It will be evident from the foregoing that in general the relative motions of two plane figures to each other are not alike, for none of the conditions of the problem necessitate the similarity of the centroids; whenever the centroids are similar, however, the relative motions become the same.
Example 1. The construction of troclioids illustrates the relative motion of plane figures of which the centroids are known. If a circular cylinder roll upon a plane, the normal sections of both figures move in a common plane, and therefore come within the conditions of our problem. The circle P Q and the straight line A B (the forms of these sections) are the centroids both of the two figures and of all figures or points connected with them. All points of P Q describe linear trochoids 13 relatively to A B; these being common, curtate, or prolate, according to whether the point lies upon, without, or within the circle. All points connected with A B describe involute srelatively to P Q; these again being common, curtate, or prolate, according as the describing point lies upon, beyond, or within the straight line.
Example 2. Two equal circles rolling upon one another have the same relative motions; points in both at equal distances from their centres describe equal epicycloids.
Our examination applies generally to the relative motions of plane figures in a common plane, or, as we shall in future call them shortly, con-plane figures, and the result of it may be summed up as follows:
All relative motions of con-plane figures may be considered to be rolling motions, and the motion of any points in them can be determined so soon as the centroids of the figures are known.
If solid bodies be laid through the supposed figures PQ and A B, and rigidly connected with them, then every pair of sections of such bodies which (like the pair of figures) lie parallel to
