INTRODUCTION. \S? xxiii 



any two adjacent positions of the body, we shall prove 

 that it is possible for the body to be moved from one of 

 these positions to the other by simple rotation round 

 one axis. Describe a sphere round O as centre, and let 

 P, Q be the positions of two points on the body of the 

 sphere in the first position, and P' 9 Q' the positions of 

 the same points (still on the sphere of course) in the second 

 position ; a plane can be drawn, which shall bisect the 

 angle POP", and also be perpendicular to the line PP. 

 By suitable rotation around any axis lying in this plane 

 and passing through O, P can be made to coincide with 

 P. The next step is to rotate the body around the axis 

 OP in its new position until Q is brought to (7, which is 

 always possible, since by hypothesis PQ = PQ[ ; thus 

 by two rotations the desired change has been accom- 

 plished. But the two rotations can be compounded 

 into one, and therefore the entire change may be pro- 

 duced by one rotation. 



This proposition is also true, whatever be the magni- 

 tude of the displacements ; but the proof we have given 

 only applies to the small displacements with which we 

 are concerned. 



Reduction of any Displacement of a Rigid Body to a Rota- 

 tion and a Translation. Let P, Q y R be three points of 

 the body in the first position, and P, (7, R the three 

 positions assumed by these points after the body has been 

 displaced. By a translation the body may be moved 

 so that P coincides with P y and then by a rotation the 

 points Q and R may be brought to coincide with Q and 

 R'. Thus by the combination of a rotation, and a 

 translation, the desired change can be effected. 



The Canonical Form. In general the direction of the 

 translation will be inclined to the axis of the rotation ; 

 but an equivalent rotation and translation can be always 

 found in which this is not the case. 



