ON AN ALGEBRA IN THREE UNITS 



131 



in question is assumed to have the property that when e^ is deleted the 

 remaining units e^, e^ will form the ordinary complex algebra. That is 

 to say, it has, in part, the multiplication table: 



2 2 ■ ^i 



It is assumed that e^ is the unity or modulus for our algebra, i. e., e^e^ = 



e,e, =e. 



The multiplication table may, therefore, be written: 



61 6-1 62 C3 



\\) 62 €2 Ci p23l^iip232^2iP233^3 



^3 ^3 rszi^i I ^322^2 + /'323^3 /'ssi^i ~'~r332^2 +^'333^3 • 



We assume, moreover, that this algebra has in common with ordinary 

 algebra the property that all its numbers are commutative. In par- 

 ticular, then, 626^=6^62 and therefore: 



P32J "^'231 



(2) p232^^p322 



p233~p323 ■ 



It will be shown that the desired algebra has the multiplication table: 



e, 



a, b, c, being any quantities in the domain R{i, i) ; and also that this alge- 

 bra is reducible. 



* 



§ 2. The Multiplication Table of the Algebra 



I. Besides modifying the multiplication table (i) in accordance with 

 the commutative law [i. e., introducing the equalities (2)] account must 

 be taken of the associative law 



^i^j * e^^=e\ ' Sj^k • 

 The associative condition, 

 v3) ^2^2 ' ^3^^^2 ■ ^2^3 



