THE KINEMATICS OF MACHINERY. 93 



any practical difficulty, but I cannot do more than mention 

 its existence here. 



The centroids of the connecting rod and frame of the 

 ordinary steam-engine driving mechanism (the links b and d 



FIG. 8. 



of Fig. 8) may serve as an illustration of this. When the 

 crank a is at right angles to d, the normals to the paths of the 

 two points 2 and "3 are parallel. The instantaneous centre of 

 b relatively to d is therefore at an infinite distance. Each 

 centroid has, therefore, a pair of infinite branches. 



We may look, in conclusion, at one other case which 

 possesses some special interest on account of the form taken 

 by the centroids. It is shown in Fig. 9. The chain con- 

 tains four links and four parallel cylinder pairs. The alter- 

 nate links are equal, and the two longer links are crossed so 

 that the chain forms an " anti-parallelogram " in every position, 

 the angle at 2 being always equal to that at 4, and the angle 

 at 1 to that at 3. If the link d be fixed, the links a and 

 c become two cranks which revolve in opposite directions 

 with a varying velocity-ratio. The centroids of b and d are 

 a pair of hyperbola having their foci at 2 3 and 1 4 respec- 

 tively. The one rolls upon the other as b moves, the instan- 

 taneous centre in the position shown being at the point of 

 contact 0, which is the point of intersection of 1 2 . and 

 3 4. The centroids of the two shorter links are the two 

 ellipses which are shown in dotted lines. They are confocal 

 with the hyperbolae, and their point of contact is always at 

 the intersection of 1 4 and 2 3. Their form shows at once 

 that the rotation of the axes 1 and 4 is precisely the same as 

 that which would be communicated by a pair of elliptic spur- 

 wheels having the centroids for their pitch ellipses. 



In this mechanism, as in some of the others illustrated, the 

 centroids of two adjacent links, as a and d, or b and c, are simply 



