ROTATIONS IN HYPERSPACE. (101 



if we express the complex in terms of the unit coordinate planes we have 



5 5 4 



'../=i 1 /.y= 1 



The sum Za-oj A'sy represents a simple plane vector since each term 



4 



in it contains the vector k;,.. The sum i (lijkij represents a complex 



2-vector lying in the 4-space determined by A'l, Av, ks, A'4 and hence 

 can be expressed as the sum of two plane vectors in this 4-space, one 

 of which is arbitrary. The plane A = SasA's,- will cut the 4-space 



4 



ki^A'2^A'3^A'4 in a 1 -vector. Now resolve S (libkn into the sum of two 



simple plane, B -\- C and choose B so that it will contain the vector in 

 which A cuts the 4-space. Since A and B have a vector in common, 

 A -{- B will be a simple plane vector D and we ha^'e 31 = C -{- D 

 where C and D are simple planes. Neither of these planes can be 

 chosen arbitrarily as was the case in 4-space. It follows from the 

 fact that a complex vector is always expressible as the sum of two 

 plane vectors that it must necessarily lie in a 4-space. This 4-space 

 is the same no matter how the complex 31 is expressed as the sum of 

 two simple planes. For, if 



31 = 3Iy + 31. 

 where il/i and 31^ are simple planes, then 



31x31 = 2il/ixM2 



But 31x31 is the same however 31 is expressed hence the 4-space 

 3I]x3l2 must be the same however 31 is expressed as the sum of two 

 plane vectors. Since a plane vector in 4-space can be resolved into 

 the sum of two completely perpendicular 2-vectors the same holds 

 true for complex 2-vectors in 5-space. Just as in 4-space if 



31 = X(il/i + 3/2) or 31 = X(J/i - il/o) 



where 3Ii and 3I2 are completely perpendicular unit planes, the 

 resolution into the sum of perpendicular planes can be effected in 

 00 ~ ways. 



Besides leaving the four imaginary vectors found in 4-space invari- 

 ant, the transformation r-3I in 5-space annihilates the real vector 

 perpendicular to 31x31. The products of this transformation then 

 with itself can never be equal to the identical transformation. If 



