ROTATIONS IN SPACE OF EVEN DIMENSIONS. 177 



10. Geometrical significance of Mi * M2 = 0. Let Mi and M2 

 be complex two- vectors. They can be expressed in the form 



3/1 = Xi TTi + Ml (t^iJc 

 M2 = X2 7r2 -(- jU2 (7r2)c- 



Expanding and using (35) and (36) we have 



Ml * Mi = (Xi X2 + Ml M2) TTi* TTo + (Xi M2 + X2 Ml) TTi* (7r2)c, (37) 



Now TTi * 7r2 is a Hnear function of two completely perpendicular planes 

 perpendicular to and cutting both tti and 7r2. Since 



TTi* (7r2)c = (7ri*7r2)c 



it is a linear function of the same planes. Therefore if Mi and M^ 

 are resolved in any way into linear junctions of pairs of completely per- 

 pendicidar planes Mi * Mi is a linear function of two completely 

 perpendicular planes, perpendicular to and cutting the planes of each 

 pair. 



Suppose now 



Mi*il/2=0. 



The complement of this is 



(J/i),* 3/2 = 0. 



Let 



Ml = a ki2 + /S ksi] M^ = 2 Xj/ kij. 



Then {Mi)c = a A-34 + ^ kn and 



= [« 3/1 - /3 (3/1) c] * Mo = (a2 _ ^2) ^.^2 * 3/2. 



If 3/1 is not self-complementary or anti-self-complementary 



a2 _ /32 p£ 0, A-12 * 3/2 = 0. 



From (34) we then have 



Xi3 = Xi4 = X23 = X24 = 0. 



Hence 



3/2= X12 ki2 -\- X34 A*34. 



