166 PHILLIPS AND MOORE. 



In case of a proper rotation 



Cl2n + 1 = — 1- 



Since the equation is reciprocal, it has a root + 1. Hence there is a 

 vector i such that 



(f>-i = i. 



The space perpendicular to i will be left iuAariant and in that space (/> 

 "will detennine a rotation with dyadic ^. Now 



(f) = ii -\-^. 



For { z + "^ converts i into i and transforms the perpendicular space 

 in the same way as ^. Also 



where 



/ = A"i /I'l -|- /i'2 /i'2 + .... + kon kin 



is the identical transformation in the space perpendicular to i. If 

 however we let 



/ = A'l A'l + A'2 A-o + .... + A-2„ A'2n + i i. 



be the identical dyadic in the complete space of 2n + 1 dimensions, 

 it is easily seen that 



For M is a linear combination of the invariant planes which He in the 

 2n — dimensional space perpendicular to / and 7-^/ will be the same 

 whether I is the 2n or the 2ti + 1 — dimensional form of the identical 

 dyadic. Then 



e^ • ^ = z i + 7 + / • 3/ + ^^^ + . . . . 



where 7 on the right signifies the 2n — dimensional form. The above 

 formula then follows at once. Hence, whether the space rotated is of 



