132 



UNIVERSITY OF COLORADO STUDIES 



gives the identity, 



(4) — e3 = /'23i^2 — /'232^i+/'233(/'23i^i+/'232«2+/'233^3) • 



Equating coefficients of e,, e^, e^, there result: 



(5) -/'232 + /'233/'23i=0> 



(6) P2il+p233p232=0, 



(7) P233p233 = -^- 



From (7), -■ __ 



(8) /'233 = "^^-I=^ 



and it is easily seen that (5) and (6) are both equivalent to the single 

 relation 



(9) p232 = ip23i- 

 The associative condition, 



(10) ^2^3 ' ^3^^^2 ' ^3^3 ) 



and the relations of commutativity, give the identity 

 p23ie3+ip23i{p23i^i+ip23i^2+iej)+i(p3^jei+p332e2 + p333^3) 



= p33t^2-p332ei+p333(p23i^i-^iP'3l^»+^h) ' 



Equating coefficients of e,, e^, e^, 



(11) ipl3i-^ip33i=-p332-^p333p23iy 



(12) - P23l+^P332 = P33i+^P333P231 > 



each of which gives, 



(13) P332 = P333P23X- Phi-ip33l ■ 



Let: • " 



p23i=(iy 



(14) P33^=^^ 



P333=<^ y 



a, b, c, being any numbers in the domain R{i, i). 



In view of (2), (8), (9), (13), (14), the multiplication table of the algebra 



takes the form: 



(15) 



