THE KINEMATICS OF MACHINERY. 105 



in which, there are three infinite links. The formula 

 of the universal joint is (G'jJ- C 1 -)* j ^ corresponds exactly to 

 (6" 2 'PJ-) a and is analogous to (' 4 ') a and (C' 3 '^-L)% Figs. 

 11 and 16 respectively. I have already mentioned in passing 

 that the mechanism (C'^P^) 9 " is that of Oldham's coupling. 

 We see then clearly the close relationship in which this 

 coupling (for parallel shafts) stands to the Hooke's joint 

 or coupling for inclined shafts. It is another illustration 

 of the important results which follow naturally from Eeuleaux's 

 simple method of analysis, and which hardly seem attainable, 

 certainly not with equal directness, "by any other method. 



FIG. 19. 



We may notice one other way in which the chain (C ff 4 ), 

 which we have seen in so many forms, may be modified. In 

 some cases we may not wish to utilize the motion of all the 

 four links, but (say) only three of them ; in that case we may 

 omit the fourth link, if we carry out a proper pairing between 

 the two links thus left unconnected. If, for instance, we 

 omit the link 6 we must make a proper pairing between a 

 and d. This pairing will always be higher ; the chain 

 becomes a reduced chain. Such a chain, reduced by 6, is 

 shown in Fig. 20. The higher pairing is carried out by 

 placing upon a a suitable element, here a circular pin at 2, 

 and giving to c the form of the envelope (see p. 5) of the 

 motion of the pin relatively to it. This envelope is the 

 curved slot shown in the figure. It was not necessary to take 

 the new element at 2, but if it had been in any other 

 place, as 2 1 , the form of the slot would have been more 



