IO 



GEOMETRY OF MOTION. 



[21. 



are taken each very near the preceding one, the angles of rota- 

 tion will be very small, and the successive centres C v C 2 , ... 

 C n will follow each other very closely. In the limit, i.e. when for 

 the series of finite displacements we substitute a continuous 

 motion of the figure, the centres C will form a continuous curve 

 (c) and the angles become the infinitely small angles between 

 the successive normals to the paths described by the points of 

 the figure. The point C about which the figure rotates in any 

 one of its positions during the motion is now called the instan- 

 taneous centre; the locus of the centres, that is the curve (c), is 

 called the centrode, or path of the centre. It is apparent that 

 in any position of the moving figure the normals to the paths of 

 all its points must pass through the instantaneous centre, and the 

 direction of motion of any such point is therefore at right angles 

 to the line joining it to the centre. 



21. The centres C are points of the fixed plane in which the 

 motion of the figure F takes place. But in any position F 1 of 



this figure some point 

 C\ of F will coincide 

 with the point C of the 

 fixed plane. Thus, in 

 the case of finite dis- 

 placements (Fig. 5), let 

 the figure F begin its 

 motion with a rotation 

 of angle O l about a point 

 C-L of the fixed plane ; 

 let C\ be the point of 

 the moving figure that 

 coincides during this 



Fig. 5. 



rotation with C v 



The next rotation, of angle 2 , takes place about a point C 2 

 of the fixed plane. The point of the moving figure that now 

 coincides with C 2 was brought into the position C 2 by the pre- 



