ROTATIONS IN SPACE OF EVEN DIMENSIONS. 187 



We may call Qi the tangent vector of 0i, Q2 the tangent vector of Q2 

 etc. Then (55) has the same general form as Gibbs' relation between 

 the vector semi-tangents except for the | and that the star product 

 is used instead of the cross product. Similar results are obtained if Mi 

 and Mo are anti-self-complementary. 



If Ml is self-complementary and Mo anti-self-complementary 



Ml* Mo = 0. 

 Hence if 



01 and 02 are commutative. That is equiangular motions of opposite 

 types are ahvays commutative. If, however, Mi and M2 are of the same 

 type equation (54) shows that 



01 02 = 02 01 



when and only when 



sin qi sin qo Mi * il/2 = 0, 



If sin gi = 0, gi = or tt and 



01 = ± 7. 



Hence if neither dyadic is a multiple of the idemfactor, two equiangular 

 rotations of the same type are commutative only when 



Ml * Mo = 0. 



In this case Mi and il/2 are expressible as multiples of the same 

 completely perpendicular planes and so Mi is a multiple of M2. 



15. Expression of a rotation as a product of equiangular 

 rotations. 



Any rotation in four dimensions can be expressed as the product of two 

 equiangular rotations of opposite type. For, w^e have 



Now the dyadics 



I-i{M + Mo), I-\{M -M,) 



