ROTATIONS IN HYPERSPACE. 665 



can be resolved into the sum of mutually perpendicular planes. We 

 will now show that this resolution is always possible. 

 If the transformation 



r' = r-M = r-{X + K) 



where X is a simple plane and r any vector in X, always gives a vector 

 r' in A' then r-/v = for e\-ery r in A'. That is X is completely 

 perpendicular to K. Then the above resolution depends upon 

 whether we can find a plane left invariant by the transformation 

 r' = r- M where M is any complex 2-vector. Let 



2m 



(19) M = 2 a^ikij 



1 



2m 



and r = 2 biki. 



1 



Then if r is an invariant vector we have 



r'3I = fir 



which is equivalent to the set of equations 



2m 



^ihy = Z (lijbj 

 1 



2m 



fjLbi = 2 aojbi 

 (21) ' 



2m 



M^2m = 2 (hmjbj 

 1 



where the Vs are to be determined. The coefficients a,/ are seen to 

 satisfy the relations 



an = 0, (lij = —ciji, i ^ j. 



Equations (20) being homogeneous will have a solution provided the 

 determinant of the system vanishes. This determinant is seen to be a 

 skew determinant mth each term in the principal diagonal equal to n. 

 From the theory of such determinants ^^ it is known that it can be 

 expanded in powers of the diagonal terms. The coefficients of the 



11 Hanus, Elements of determinants. Ginn & Co., page 152. 



