106 
ON ASSOCIATIVE IMAGINARIES. 
[822 
In order that this may be associative, we must still have the relations 
bg = cd, 
c I 2 + dg —ag — cli = 0, 
d 2 +bc — ad — bh = 0, 
which are all three of them satisfied by g = , h = ——■—— , viz. we thus have 
the associative and commutative system 
x 2 = ax + by, 
xy = yx — cx + dy, 
„ cd d 2 + be — ad 
y‘ = j*+ 1 y - 
I did not perceive how to identify this system with any of the double algebras 
of B. Peirce’s Linear Associative Algebra, see pp. 120—122 of the Reprint, Americajn 
Journal of Mathematics, t. iv. (1881); but it has been pointed out to me by Mr C. S. Peirce 
that my system, in the general case ad — be not = 0, is expressible as a mixture of two 
algebras of the form («0, see p. 120 (l.c.): whereas if ad—bc = 0, it is reducible to the 
form (c 2 ), see p. 122 (l.c.). The object of the present Note is to exhibit in the simple 
case of two imaginaries the whole system of relations which must subsist between the 
coefficients in order that the imaginaries may be associative; that is, the system of 
equations which are solved implicitly by the establishment of the several multiplication 
tables of the memoir just referred to.