THE KINEMATICS OF MACHINERY. 99 



the " dead points " when a and c are both standing in the 

 direction of the axis of d or b. If no means be taken to 

 prevent it, it is then possible to move the crank either in 

 one direction or the other, and the two cranks may go on 

 revolving in the same direction, or may revolve in opposite 

 directions according to circumstances. 



Such an indeterminateness is, of course, inadmissible in 

 machinery, where we therefore adopt the well-known method 

 of combining two mechanisms of the same kind, and placing 

 them with their cranks at right angles, so that they do not 

 cross the dead-points at the same time. The motions are 

 thus made determinate and the cranks revolve in similar 

 directions. We might, however, wish them to revolve in 

 opposite directions, as in the mechanism shown in Fig. 9. 

 It may be worth our while to look for a moment at the means 

 which may be used in this case to secure the determinateness 

 of the motions in the mechanism. To distinguish between 

 the two cases we may call the former " parallel cranks " and 

 represent it by the formula (Cl \\ C'), and the latter "anti- 

 parallel cranks/' (C" Z ~Z.C"^). This chain, with the link d 

 fixed, is shown in Fig. 13. 



In the case of the parallel cranks all points of the 

 centroids of b and d are at infinity, for they are at the inter- 

 sections of the parallel links a and c. We have already 

 seen, however, that in the mechanism Fig. 9 the centroids 

 are quite different, those of b and d being hyperbolae. If, 

 therefore, when the mechanism is brought into either dead 

 point, where the cranks might change from the anti-parallel 

 to the parallel position, we can only make certain that 

 the right centroids roll upon, each other, we shall get the 



FIG. 13. 



motion that we want., Fig. 13 shows an arrangement by 

 which, just in that position of possible change, a tooth made 



E 2 



