ROTATIONS IN HYPERSPACE. QOZ 



in a space of 2p + 2 dimensions can be resolved into the sum of jJ + 1 

 independent simple plane vectors one of which passes through an 

 arbitrary 1 -vector. For let the complex be expressed in terms of the 

 unit coordinate planes 



2p+2 2p+2 



M = — « (iijh'ij -\- -I ciijkij. 



1 2 



The first sum is a simple 2-vector passing through A'l which can be 

 chosen arbitrarily. The second sum is a complex 2-vector in a space 

 of 2p -|- 1 dimensions and consequently can be expressed as the sum 

 of p independent simple plane vectors. Hence the whole complex 

 can be expressed as the sum of p -\- \ independent planes. We have 

 seen that a complex 2-vector in 4-space can be resolved into the sum 

 of two independent simple plane vectors one of which can be chosen 

 arbitrarily and therefore by induction we have: a complex ^-vector 

 in 2p or 2p -\- 1 dimensions can always be resolved into the sum of p 

 independent simple plane vectors. 



The condition that a 2-vector in a space of 2^ dimensions be simple^ 

 is 



MxM = 0. 



For if this condition is satisfied then the following relations are satis- 

 fied owing to the associative character of the multiplication when the 

 order of the whole product is equal to or less than 2p 



3/x3/x3/ = 

 3/xJ/xJ/xJ/ = 



MxMxMx . . .xp factors = 0. 



The last one shows that the complex must lie in a space of lower 

 dimensions and therefore can be expressed as the sum of p — 1 simple 

 plane vectors. By the same argument it can be expressed as the sum 

 of p — 2 simple plane vectors and so on until finally it can be expressed 

 as a single simple plane vector. If M is a simple plane, MxM — 0. 

 Hence this is both a necessary and a sufficient condition that a 2- 

 vector be simple. 



We shall next show that a complex 2-vector in a space of 2p dimen- 

 sions can be resolved into the sum of p mutually perpendicular simple 

 planes. Let 



M = Z miMi 

 1 



