26o 



UNIVERSITY OF COLORADO STUDIES 



There is at least one determinant of order two in the matrix 



0.21 *^22 ^23 ^24 



which is not equal to zero. For if 



then considering a^^ =t, and a2i=r^ the expansion r^a^^+r^a^^ =0 gives 

 by means of postulate V a2i=a^2=o. But a^^ may be assumed as any 

 arbitrary non-zero element of D. 



With Cj, 62 consider €^=[aj^,a^2, (^33, 0-34] and assume a^j^^o. There 

 is at least one non-zero determinant of order three in 



^11 ^12 ^13 ^14 



(ill ^22 ^23 ^24 



For 



^31 ^32 ^33 ^34 

 ^11 ^12 ^13 



^21 ^^22 ^23 



leads to the relation a^jA^^ + aj^A^2-^a3iA3i=o, send by postulate V 

 (since A^^^^o) ^33 =0. 



Similarly it is seen that | a,-,- 1 4= o. 



§ 3. Identification with Ordinary Quaternions 



The quaternion system as thus defined is holoedrically isomorphic 

 with the quaternions of Hamilton, the coefficients belonging to the same 

 domain D. 



The quaternions e^=[i, o, o, o], e^^lo, i, o, o], e^=[o, o, i, o], 

 ^^ = [0, o, o, i] form a four dimensional system since 



1000 

 0100 

 0010 

 0001 



+ 



By postulate IV and the definition of multiplication these quater- 

 nions e have the multiplication table 



