ROTATIONS IX SPACE OF EVEN DIMENSIONS. 171 



Instead of ai-\- a2+ . . . . +«« we could have used the function 



q:i ± ao ± .... ± a„, 



the positive and negative signs being choosen independently pro\'iding 



we used the same signs for corresponding values of jS,-. 



Hence 



Ml- 7r{ai± ao± . , . . ± a„) = .l/o- 7r(/3i± /3o± . . , ± i(3„) 



and so 



a2= 02, q;3= iSg, ... a,i= jS„. 



Since cti is the plane perpendicular to ao, as «„ and forming with 

 them a right hand system, and /3i has the same relation to /So, 183, . . ./3n, 

 we have finally ai= ^Si and so • 



.¥1= 3/2. 



If two angles are equal to tt the theorem may not be true. Thus 

 in four dimensions 



^tt/- (ai + a>) ^ ^.tt/- (/3i + ft) ^ _ J 



whether o;i+a:2= 0i-\- ^1 or not. In higher dimensions the part of 

 3/1 and J/2 in the four dimensional space of two planes rotated 

 through 180° would be subject to the same indetermination. 



6. Vector of the dyadic. If M is expressed as a linear function 



M = 2 qij kij 



of completely perpendicular planes kij, it is clear from the expression 

 of 4> in the form (9) that these planes are all left invariant and that 

 the motion in each of them is proper. 



The planes kij are invariant in the sense that if kt and kj go by the 

 rotation into ki and kj, since these ncAV vectors are perpendicular 

 and lie in the plane ^"iy, 



A" i X n i = k ; X I'' j 

 For the same reason M is invariant. Also an^' linear function of the 



