Page:The Kinematics of Machinery.djvu/100

This page needs to be proofread.


supposing the consecutive positions of P Q to be infinitely near together, the spheric polar polygons become spheric centroids, the temporary axes become instantaneous, and the pyramids become cones (in general non-circular), which have a common vertex at A, and which roll or turn upon one another. The cones are cones of instantaneous axes, and the whole motion is called conic rolling. We arrive therefore at the following law, connecting together the phenomena we have been considering: All relative motions of two bodies which have during their motion a common point, may be considered as conic rolling, and the motions of any points in the bodies are known so soon as the corresponding cones of instantaneous axes are determined.

It is evident that our former examination of the methods of de- termining centroids and reducing them applies equally to conic and to cylindric rolling, so that it is not necessary to reconsider these matters here.


§12.
Most general Form of the Relative Motion of Rigid Bodies.

If the positions of three points in a body be known, the positions of all other points may be found by making them the vertices of triangular pyramids of which the magnitude and position of the base are fixed and the length of the edges known. We can there- fore determine the relative motion of any two rigid bodies by means of two fixed triangles, P Q R and A B C, in them. Let the body A B C be brought to rest, so that only P Q R moves relatively to us, and let the latter move into any position, as P l Q l R v Fig. 31. This change of position may take place in many ways. If, for example, we join P and P l by a straight line, and cause P Q R to undergo translation parallel to it until P falls upon P 1 and the figure takes the position P l Q' R', we have only further to turn P Q R about an axis S S (which may be found in every case), passing through P v in order that it may assume the required position P l Q l R r Thus in the most general case any motion of