Page:The Kinematics of Machinery.djvu/339

 ELLIPTIC CHUCK. 317

investigated by him is a very remarkable application of the mechanism before us. Use is here made of the fact that all points connected with the smaller centroid, that is in this case all points connected with the fixed link a, describe ellipses* relatively to the piece to which the larger Cardanic circle belongs. In the apparatus itself the cross is formed upon the back of the disc c. In one of its two slots it encloses the full prism 3, which is attached to the lathe-spindle. The headstock a forms the cylinder pair 2 along with the spindle 1. The cylinder of the pair 1 which belongs to a is attached to the headstock by screws ; it is made annular, so that the spindle I passes through it, in other words, it is expanded sufficiently to allow 2 to lie within 1. The piece d is made as a ring : its inner surface forms the hollow cylinder paired with a, while it carries outside the full prism of the pair 4 (divided into two) which works in the second slot of the cross c. The describing point or tool P forms a part of the fixed link or frame a. The ellipses which are described relatively to the disc by the point of P, have if P lie beyond a a difference between their semi-axes equal to the length a ; if P lie between 1 and 2, a is equal to the sum of the semi-axes. The enlargement of the pin 1 allows the magnitude of a (that is the distance 1, 2) to be varied within certain limits, and this, together with alterations in the position of P, allows very great variety in the ellipses produced by this apparatus. The link

& being the driving link, the complete formula is ((7' 2 ' P^-)r. The mechanism might also be so arranged that d, which is kinematically equal to 6, became the driving link. (Compare 76). It must be remembered that the point-paths of the disc c are those determined by the larger Cardanic circle, and are therefore peri-trochoids, in- cluding the particular case of cardioids. The path of the centre M of the disc c is the smaller centroid, through which it passes twice for each revolution of b or d.

The mechanism (C" 2 'P^-) C . We have now left only the train ob- tained by placing the chain on c. This may be called the swinging double slider-crank, or shortly swinging double slider; it is the mechanism familiar to us as the " trammel " used by draughts-

scribed in this apparatus are not ellipses, but he is mistaken. I shall afterwards ( 76) come to the form of the mechanism given by him, which dillers somewhat from the one represented above. R.
 * Laboulaye (Cintmatique, 1861, p. 863) attempts to show that the curves de-