Page:The Kinematics of Machinery.djvu/168

 rhombus 0-fl^O^. The rhombus is drawn with angles of 76 "and 104 instead of 60 and 120 as before. This difference makes a notable alteration in the centroids, which resemble those of Plate IX. with the corners removed. The nature of the changes of form in the point-paths from those obtained before can thus be readily traced. /

Plates XII., ^ and XIII. show three more pairs constructed analogously to those we have already examined. The first of these is remarkable, both on account of the regularity of its centroids and because the end points 1 and 2 of the smaller centroid again describe squares. The curved element in Plate XIIL, 1, is formed like that in Plate XI., but with a smaller vertex angle at #; that in Plate XIII., 2, is similar to the one in Plate X., 2. The difference between the centroids in Figs. 3 and 8 is very remarkable. The manner in which the one centroid rolls upon the other is indicated as distinctly as has been possible by the numerals. These examples show clearly the multitude of motions which can be obtained by means of the higher pairs, and show at the same time how wonder- fully the use of centroids simplifies the comprehension of these complicated motions.

§30. General Determination of Profiles of Elements for a given Motion.

The forms of the pairs of elements considered in the foregoing investigation were found synthetically; starting from the general solution of the problem of restraint we built up constrained pairs in accordance with its conditions, and afterwards ascertained their relative motions by the construction of their centroids. This latter part of our investigation was again analytical. It furnishes us however with the means of solving a further synthetical problem, the construction, namely, of pairs of elements for a given motion, i.e. for given centroids. For the forms which we found necessary for the continuous reciprocal restraint of the elements are reciprocally envelopes for one relative motion, and for that one only which is determined by the centroids. We have chosen hitherto forms conditioned only by the property of reciprocal restraint, and have from them determined their centroids. We may now reverse the problem, by assuming the centroids as given