ROTATIONS IN SPACE OF EVEN DIMENSIONS. 167 



dimensions 2n or 2n + 1, any proper rotation leaving the origin fixed 

 can he determined by a dyadic of the form (17). 

 Conversely any dyadic of the form 



A. — J-M 



<p - e 



represents a rotation. To show this, it is sufficient to express M as a 

 linear function of completely perpendicular planes Idj. Then 



which is equivalent to (9) with all upper signs used. 

 The dyadic 



1 • k ij = k j k i k i k i 



determines a rotation through 90° in the plane kij. This is analogous 

 toi = V — 1 which produces a rotation of 90° in the complex domain. 

 Similarlv 



represents a rotation through the angle q in the plane ki, just as e**" pro- 

 duces a rotation through the angle (p in the complex domain. We can 

 then consider qka as the vector angle of rotation. For this reason 

 M might be called the vector angle of rotation of the dyadic 



(p — e 



5. Uniqueness of exponent. The angles qa may be of any 

 size. However, if they are restricted to the region 



— 7r<g,y<7r 



we shall show that /or each rotation there is a unique value of M, that is, 

 if 



then 



il/i = il/o. 



To show this we first consider the dyadic 



I-M = i:qij(kjki-kikj). 



