88 LECTURES TO SCIENCE TEACHERS. 



The figure thus found is another polygon, which we may call 

 a second central polygon. 



These polygons have important properties, the principal of 

 which can be very easily recognized. The first polygon does 

 not alter its position during the motion of the body ; it is 

 therefore fixed, so that it may be considered as a part of 

 any figure such as AB which is fixed or stationary in the plane 

 of motion. The second polygon moves with PQ and forms 

 (by construction) part of the same figure with PQ. This 

 second polygon then, by the consecutive turnings of its 

 corners upon the corresponding corners of the first (and 

 equal-sided) polygon, will give to PQ the required changes 

 of position relatively to the fixed plane or to the figure 

 AB lying in it. 



If therefore we know the central polygons for the given 

 motion, we know not only the changes of position of the 

 points P and Q, but those of every other point connected with 

 the moving figure, whatever form it may have. For at any 

 one instant every point in the figure is moving about the 

 same centre. In studying the relative motions of the figures 

 we may, therefore, quite leave out of sight their form if ws 

 only know the. central polygons for the motion. These teil 

 us, so far, all about the motion which is taking place. 



We may go further, however. We have recognised the 

 fact that the relative motion of two figures or bodies may take 

 place equally whether one or the other of them, be fixed, or 

 both moving. In the case before us we have supposed AB fixed 

 and PQ moving relatively to it. The second polygon then 

 moves on the first, and expresses the relative motion taking 

 place. If, however, we suppose PQ fixed and AB moving, 

 then the polygons still express, the relative motion ; but the 

 second is now fixed and the first rolls upon it. This follows 

 directly from the constitution of the polygons. The properties 

 of the polygons as expressing the relative motions of the bodies 

 to which they belong are therefore reciprocal. 



You will have noticed, no doubt, that the polygons do not 

 express continuous motion. They define only a series of 

 changes of position in their beginning and end, not telling us 

 of the intermediate stages. 



We may, however, take the consecutive position of the 

 figures as close together as we like. The closer together they 

 are taken the shorter become the sides of the polygons. If 



