262 UNIVERSITY OF COLORADO STUDIES 



Postulates I, IV., V, hold for the same reasons as in (II). 

 Postulate II holds since o has been added to the domain. * 

 Postulate III does not hold, since subtraction is not always per- 

 missible. 



(IV) D is the domain of complex numbers, the a^ being pure ima- 

 ginaries. 



Postulates I, II, III hold, since addition and subtraction is always 

 possible. 



Postulate V need not be considered, as no mention is made of the t's. 



Postulate IV does not hold, since the product of any two elements 

 is real. 



(V) flj arbitrary; other a's zero. 



Postulates I, II, III are true, since addition and subtraction are 

 always possible. 



Postulate IV is true, since multiplication is always possible. 



Postulate V is not true, since, if aiTj+ 02X3 + ^3X3 + 5^x^=0 where 

 02 = 53 = ^4 = 0, '^2'^^^^ may have any values. 



April, 1906. 



