the plane of motion have a pair of centroids identical with theirs. The series of centroids, which we may suppose in this way to be lying closely one behind the other, form together two cylinders (in general non-circular), which always touch along one line, and turn or roll upon one another. Each line in which the cylinders come in contact for an instant is for that instant the axis of rotation, and is called therefore the instantaneous axis of the motion. The motion itself in such a case is called a cylindric rolling. We may extend the law just enunciated for plane figures equally to this relative motion of solids. The characteristic of a series of reciprocal positions of a body undergoing cylindric rolling is that its sections normal to the instantaneous axis are figures which remain always con-plane during the motion. We can therefore say: Every relative motion of two con-plane bodies may be considered to be a cylindric rolling, and the motions of any points in them may be determined so soon as their cylinders of instantaneous axes are known.
8. The Determination of Centroids.
With the transition from irregular to continuous motion the perpendiculars (see Fig. 17) upon the lines joining pairs of consecutive positions of the points P and P1, Q and Q1, become normals to the curve-elements in which at the given instant the points P and Q are moving. In order therefore that the centroid for the motion of a figure P Q relatively to another A B may be known, there must be known for every position of P Q the directions in which at least two of its points are moving, that is the position of the tangents to their paths. The normals to such tangents for any number of points in the moving figure all intersect in the same point, from which it follows that only one pair of centroids is possible for any relative motion of con-plane figures.
Centroids can always be found by determining separately a sufficient number of points in them, and often by a general