XXiv INTRODUCTION. 



Resolve the translation into two components one 

 parallel to the axis of rotation, and the other perpendi- 

 cular thereto. The component perpendicular to the axis 

 of rotation will have merely the effect of transferring the 

 rotation into a parallel position. Thus the canonical 

 form of the displacement of a rigid body consists of a rota- 

 tion about an axis combined with a translation parallel to 

 that axis* 



The Canonical Form is Unique. It is easily seen that 

 there is only one axis by rotation about which, and 

 translation parallel to which, the rigid body can be 

 brought from one given position to another given posi- 

 tion ; for suppose there were two axes P and Q, which 

 possessed this property, then by the movement about P, 

 all the points of the body originally on the line P con- 

 tinue thereon ; but it cannot be true for any other line 

 that all the points of the body originally on that line 

 continue thereon after the displacement. Yet this would 

 have to be true for Q, if by rotation around Q and transla- 

 tion parallel thereto, the desired change could be effected. 

 We thus see that the displacement of a rigid body can 

 be made to assume an extremely simple form, in which 

 no arbitrary element is involved. 



*For another proof of Chasles' theorem by Professor Croft on, F. R. S., 

 see Proceedings of the London Math, Soc., Vol. v,, p. 25. 



