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UNIVERSITY OF COLORADO STUDIES 



In order not to interrupt the argument which will follow, we point 

 out here that the unit of the domain R{i) has the multiplication table, 





e^ having the ordinary properties of unity. The units of the domain 

 R(i,i) have the multiplication table 



62 C2 "l 



where e^ has the properties of unity, e.^ the properties of i = V^ — i . 



The general number of the latter algebra is a,e, +0,^2, where a^ 

 and a^ are reals. 



Since all the numbers of the domain R(i) are represented graphically 

 by the totality of points on a straight line and all the numbers a^+ia, 

 of the domain R(i,i) are represented by the totality of points of a plane, 

 one might infer that the totality of points in space is the geometric 

 representation of some algebra in three units. Proceeding on this 

 thought Hamilton attempted to set up a three-unit system e^, e^, e^ 

 which would have this property. In the introduction to his Quater- 

 nions he proved that no such system exists when the coefficients a^, a^, 

 a^, of the general number A ^a^e^+a^e^+a^e^ belong to the domain of 

 reals i?(i). His proof consists in showing that any algebra in three 

 units, the coefficients belonging to R(i), contains numbers x=x^e^ 

 +X2e2+x^e^ and )'=3'jgi+)'je2+)'3e3,bothdififerent from zero, for which 

 the product :x:y =0. 



As Benjamin Peirce points out in his memoir on "Linear Associative 

 Algebra,"' Hamilton's investigation should be made in the field of com- 

 plex numbers instead of in the field of real numbers. In other words, 

 the coefficients a,, a^, a^, of the number A =a^e^+a2e2+a^e^ should be 

 considered as belonging to the domain R(i, i) rather than to the domain 

 R{i). 



The problem of the present paper consists in carrying out Peirce's 

 thought for one interesting algebra in three units e^, 62, e^ The algebra 



' American Journal of Mathematics, Vol. IV (1882), p. 9. footnote. 



