20.] 



PLANE MOTION. 



its centre on the perpendicular bisector of A^A^\ similarly for 

 B. Hence the intersection C of the perpendicular bisectors of 

 A Q A 1 and B^B^ is the only point by rotation about which both 

 A and B can be brought from their initial to their final posi- 

 tions. That they actually are so brought follows at once from 

 the equality of the angles A^CB Q and A 1 C^ 1 (and hence of the 

 angles A^CA l and B^CB^) which are homologous angles in the 

 equal triangles A^CB Q and A^CB V 



We thus have the proposition : Any displacement of an inva- 

 riable plane figure in its plane can be brought about by a single 

 rotation about a certain point which we may call the centre of 

 the displacement. 



19. The construction of the centre C given in the preceding 

 article becomes impossible when the bisectors coincide (Fig., 3) 

 and when they are parallel (Fig. 4). 

 In the former case, C is readily 

 found as the intersection of A^B 

 and A^B V In the latter, i.e. when- 

 ever A^A^B^B^ the centre lies at 

 infinity, and the rotation becomes 

 a translation. 



Any translation may therefore be regarded as a rotation about 

 a centre at infinity. 



20. Let the figure F pass through a series of displacements 

 F Q F lt F-)Fy ... F n _F n . Each displacement has its angle and 

 its centre. If the successive positions F Q , F v ... F n of the figure 



