160 PHILLIPS AND MOORE. 



I. Rotations in 2n Dimensions. 



1. Characteristic equation. In a space of 2n dimensions a ro- 

 tation about the origin, considered as an operation on a one-vector, is 

 represented by a matrix or dyadic of order 2?i. If r and r' are 

 corresponding vectors the transformation is represented by 



/ = (ji-r 



If this is a rotation r and r' will have the same length and therefore 



r'-/ = (0-r)- ((p-r) = r-4>c(j>-r = r-r. 

 Hence 



4>cCl^= I. (1) 



(Here (pc is used to represent the conjugate of (f), that is (j>-r = r-4>c 

 and / represents the identical transformation). 



The dyadic cj) satisfies a characteristic or Hamilton-Cayley equation 



The conjugate of (2) is 



Multiplying by 0^" and reducing by use of (1) 



1 + ai + 02 <^2 + . . . . -I- cion-l <^-"-\ + fl2n <^'" = 0. (3) 



Since this must be identical with (2) 



a2n = ± 1. (4) 



It is well known that 



(— l)-"02n = «2n 



is the determinant of the transformation. If this is positive the rota- 

 tion is called a proper rotation, if negative an improper one. In case 

 of a proper motion the characteristic equation is then reciprocal. If 



1 . 



X IS a root - is also a root. In case of an improper motion if X is a root 



A 

 1 . 



— - is also. 

 X 



