662 MOORE. 



M = Ml ± M2 where 3/i and Mo are unit planes, then the trans- 

 formation repeated four times will be the identical transformation 

 for vectors in M^M. 



4. Complex 2-vectors in space of 2p dimensions. We shall 

 first show that if a complex 2-vector in a space of 2p dimensions can 

 be resolved into the sum of i) independent simple plane vectors one of 

 which passes through an arbitrary 1-vector then a complex 2-vector 

 in a space of 2p -\- 1 dimensions can also be resolved into the sum of p 

 inflependent simple plane vectors but in this case no one of the simple 

 2-vectors can be made to pass through an arbitrary 1-vector. To show 

 this it is only necessary to write the complex in 2p + 1 dimensions- 

 in terms of unit coordinate planes 



2p+l 2p+l 



M =■ _( flij^'i/ -p — « dijkij. 

 1 2 



The first sum represents a simple plane vector since each term contains 

 the vector ^'i. The second sum represents a complex 2-vector lying 

 in the space of 2p dimensions determined by the vectors A'o, kz, . . . I'lp+i 

 and therefore by the above assumption can be expressed as the sum 

 of p simple plane vectors one of which, A saj-, passes through the 



2p+l 



vector in which the plane 5 = S aijkij cuts the space in which 



1 



2p+l 



the complex 2 (lijkij lies. Then since A and B have a vector in 

 2 



common, their sum can be expressed as a simple plane vector. The 

 complex M is then expressed as the sum of p independent planes. 

 But p independent simple plane vectors determine a space of 2p 

 dimensions and this space of 2p dimensions is the same no matter 

 how M is expressed. For if 



Then 



MxMxM . . . 7J factors = p ! J/jX J/gX . . . xMj,; 



and since the first member is independent of how the complex is 

 expressed as the sum of p independent planes, the second must be. 



Again, if a complex 2-vector in a space of 2p dimensions can be 

 resolved into the sura of p independent simple plane vectors one of 

 which passes through an arbitrary 1-vector, then a complex 2-vector 



I 



