KIN EM A TIC MODELS. 27 1 



from what appears to be the fundamental principle of every machinal 

 combination or arrangement ; that each particular part of the combi- 

 nation must have at each instant only one definite motion relatively to 

 every other part. If, in any machine, there be a piece which at any 

 one instant can move in two directions, there is obviously some defect 

 in the machine. Engineers are familiar with the many devices that 

 have to be employed in connexion with certain mechanisms to carry 

 them across " dead-points," c. This condition, that it shall be 

 impossible for any point in a machine to move at any instant in any 

 direction other than that intended, is a universal one. The way in 

 which it is satisfied is by giving to certain portions of those pieces 

 which form the machine suitable geometric forms. These forms are 

 arranged in pairs, in such a way as to be reciprocally envelopes one of 

 the other. The one piece, then, so envelopes the other, on account of 

 the forms given to both, that each can move only one way relatively to 

 the other at any instant. The motion of such pieces is called 

 constrained motion. Let me take a very simple case a screw and 

 nut. These two pieces are formed in such a way that the nut can 

 move only in one way at any instant relatively to the screw, and the 

 screw in one way relatively to the 'nut. If the pitch of the screw be 

 made zero, we have simply a pair of solids of revolution, or "re volutes," 1 

 having such profiles at the end as to prevent any axial motion ; and 

 again we have a pair of mutually enveloping forms, whose relative 

 motion is absolutely constrained. If, on the other hand, the pitch be 

 made infinite, we have a pair of prisms, in which the only possible 

 relative motion is axial. In the first case, the constrained relative 

 motion is twisting ; in the second, it is turning (twisting without any 

 translation) ; and in the last case, where the pitch is infinite, it is 

 simply sliding (twisting without any turning). You recognise, in these 

 three pairs of bodies, the geometrical forms which are used in ninety- 

 nine cases out of a hundred to constrain the motions of machinery. 

 Such bodies as these Reuleaux calls pairs of kinematic elements, and 

 when they have the peculiarity that one entirely encloses the other, 

 lower pairs or closed pairs. 



It is not essential, however, that one should enclose the other. 

 There are before you a number of examples of higher pairs, the bodies 

 of which mutually constrain each other without complete enclosure. 



