92 LECTURES TO SCIENCE TEACHERS. 



but in some cases are quite as simple. In Fig. 7, for example, 

 is shown a mechanism familiar to engineers, in which a crank 

 a drives a reciprocating bar c by means of a block b working 

 in a slot. The centroids denning the relative motions of the 

 links a and c are the two circles shown in full lines, one 

 double the diameter of the other. These two circles both 

 move as the mechanism works (supposing the link d to be 

 fixed), but always so that they roll continuously one on the 

 other. If instead of fixing d the crank a were made the 

 fixed link, the same centroids would still express the relative 

 motions of a and c. The smaller circle, the centroid of a, 

 would be stationary along with the link to which it belongs, 

 and the other would roll on it, the instantaneous centre for 

 the motion of the link c being always at their points of 

 contact. This mechanism (a being fixed) is used in Oldham's 

 coupling, in elliptic chucks, &c. Knowing these centroids 

 we know all about the motions of the two corresponding 

 links in the mechanism, not only about the motions of some 

 particular points in these links. 



The centroids of kinematic chains can in general be very 

 easily determined. Once found they make us independent to 

 a great extent of trigonometric or algebraic formulae, and 

 enable us to determine all we wish to know by purely 

 geometric graphic constructions. For technical purposes at 

 least this is frequently an immense advantage. There are very 

 few cases in which it is not more convenient for the engineer 

 to employ a construction than a formula, if both give him 

 the same result. 



Before looking at the centroids of other mechanisms it is 

 necessary to examine one particular case which often occurs. 

 Suppose that the lines P Pi and Q Q 1 in Fig. 4, or the 

 tangents to the curves at P and Q in Fig. 5, had been parallel. 

 It is obvious that the normal bisectors in the one case and the 

 normals to the curve in the other then become also parallel, or, 

 as it is for some reasons more convenient to express it, would 

 meet at an infinite distance. The temporary centre in the 

 one case and the instantaneous centre in the other are at 

 infinity. A centroid may therefore contain one or mere points 

 at an infinite distance, may have, that is, one or more infinite 

 branches. This constantly occurs in mechanism, and in some 

 cases every point in the centroid is at an infinite distance. 

 This is however a special case ; its treatment does not offer 



