23-] PLANE MOTION. U 



ceding rotation. Its original position is therefore obtained by 

 turning 6\(7 2 back by an angle 6 l into the position C^C <%. 

 The rotation of angle 2 about (7 2 brings a new point (7 3 of 

 the moving figure to coincidence with the fixed centre C 3 ; and 

 the original position C' 3 of this point can be determined by 

 first turning C 2 C B back about C 2 by an angle 2 into the 

 position C 2 D, and then turning the broken line C^C^D by a 

 rotation of angle : about 6\ back into the position C\C\C\. 

 Continuing this process we obtain, besides the broken line 

 C 1 C 2 C B ... formed by joining the successive centres of rotation 

 in the fixed plane, a broken line C\CyC\... in the moving 

 figure formed by joining those points of this figure which in the 

 course of the motion come to coincide with the fixed centres. 

 The whole motion may be regarded as a kind of rolling of the 

 broken line C\C\C\ . . . over the broken line <<% 



22. In the case of continuous motion each of the broken lines 

 becomes a curve, and we have actual rolling of the curve (c f ), or 

 body centrode, over the curve (c), or space centroder The con- 

 tinuous motion of an invariable plane figure in its plane may 

 therefore always be produced by the rolling (without sliding) of 

 the body centrode over the space centrode. The point of contact 

 of the two curves is of course the instantaneous centre. 



23. It appears from the preceding articles that the continuous 

 motion of a plane figure in its plane is fully determined if we 

 know the centre of rotation for every position of the figure. 

 This centre can be found as the intersection of the normals of 

 the paths of any two points of the figure, so that the motion 

 of the figure will be known if the paths of any two of its points 

 are given. This, however, is only one out of many ways of 

 determining plane motion by two conditions. 



Thus the motion may be determined by the condition that a 

 curve of the moving figure should remain in contact with two 

 fixed curves. In this case the instantaneous centre is found as 

 the intersection of the common normals at the points of contact. 



