INTRODUCTION. 



XIX 



II.- ON THE REDUCTION OF THE DISPLACEMENT OF A RIGID 

 BODY TO ITS CANONICAL FORM. 



Problem. Two positions of a rigid body being given, 

 there are an infinite variety of movements by which the 

 body can be transferred from one of these positions to 

 the other. It has been discovered by Chasles* that 

 among these movements there is one of unparalleled 

 simplicity. The demonstration of this theorem is the 

 object of the present section. 



The Composition of Rotations about Intersecting Axes. 

 Suppose a body receive a small rotation through an 

 angle a about a certain axis, and another small rota- 

 tion through an angle j3 around a second axis inter- 

 secting the former one; then the position ultimately 

 attained could have been reached by a single rotation 

 from the initial position about an axis appropriately 

 chosen. 



Let OA and OB (Fig. i) represent the directions of 

 the given axes, while their lengths are proportional to 

 the angles a and |3, 

 the directions of the 

 rotations being such 

 that if an ordinary 

 screw were placed 

 with its head at O, 

 and its axis along 

 OA, then the direc- 

 tion of the rotation 

 which would make 

 the screw advance 

 from is the direc- 

 tion of the rotation 



See Appendix I. 



