90 LECTURES TO SCIENCE TE AGREES. 



some special use of them, more, I think, than has been made 

 by former writers. 



While the polygons only represent a series of isolated 

 positions of a body, the centroids, rolling on each other, repre- 

 sent the whole motion continuously. Like the central poly- 

 gons their properties are reciprocal. If then the centroids of 

 two figures be known, their relative motions for a series 

 of changes of position, each infinitely small, are also known, 

 i.e. their motions are completely determined. 



If A and its centroid be fixed, and the centroid of P Q 

 rolled upon it (Fig. 5), we have now the means of determin- 

 ing the path of motion of every point in the Fig. P Q 

 relative to A B, whatever may be the form of P Q. It is 



sometimes of great convenience to be able to find the motions 

 of all points in a body in such a very simple way. Eecipro- 

 cally we can determine the point paths of A B relatively to 

 P Q, which, in general, differ entirely from those of P Q 

 relatively to A B. 



If both figures be moving, as frequently happens in practice, 

 both centroids are also in motion ; their motion relative to 

 each other, however, remains unaltered. They still roll on 

 one another, and their point of contact is still the instan- 

 taneous centre of the motion of each relatively to the other. 

 Each figure moves, relatively to the other, about this point, 

 which, being common to the two centroids, is common to the 

 two figures. They might, therefore, for the instant, be 



