70 
ON DOUBLE ALGEBRA. 
[814 
15. First, if the form be A, B, or C; there will be the two idem-or-nil symbols 
x and y, that is, we may assume 6 = 0, g = 0; and the associative conditions then 
become cd — 0, c(c — h) = 0, d(d — a) = 0, viz. for the forms A, B, C, 
A. x 2 = x, y 2 = y, 
B. x 2 — x, y 2 = 0, 
C. x 2 = 0, y 2 = 0, 
these are cd = 0, c(c —1) = 0, d(d— 1) = 0; c = 0 or 1, d = 0 or 1, 
„ cd = 0, c 2 = 0, d(d— 1) = 0; c = 0, d = 0 or 1, 
„ cd = 0, c 2 = 0, d 2 = 0 ; c = 0, d = 0. 
But for the form A, if c = 0, d = l, that is, xy = yx = y, then, writing z=x — y, we have 
z 2 — z, yz = zy = 0, y 2 = y. And similarly, if c = 1, d — 0, that is, xy = yx = x, then, writing 
z — — x + y, we have z 2 = z, zx — xz = 0, x 2 = x. That is, each of these is reduced to the 
first case c = 0, d= 0; that is, x 2 = x, xy = yx = 0, y 2 = y. 
For the form B, if c = 0, d = l, then the system is x 2 = x, xy = yx = y, y 2 = 0; and 
this cannot be reduced to the first case x 2 — x, xy — yx = 0, y 2 = 0. 
For the form C, there is only one case, as above. 
For the form E, we have a = 0, 6 = 0, (c = e, d = 0, in order that the system may 
be commutative), h — 2c, viz. the equations must be x 2 = 0, xy = yx = cx, y 2 = gx+2cy. 
The associative conditions then give c = 0; or, the system is x 2 =0, xy = yx — 0, y 2 = gx. 
Writing - instead of x, and for convenience interchanging x and y, the equations are 
x 2 = y, xy = yx = 0, y 2 = 0. 
16. The commutative associative system is thus seen to be reducible as follows :— 
A. system is x 2 — x, xy = yx = 0, y 2 = y, first mixed system, see No. 2. 
B. 
or else 
B. 
c. 
E. 
„ x 2 = x, xy = yx = y, y 2 = 0, Peirce’s system (a 2 ), 
„ x 2 = x, xy = yx = 0, y 2 = 0, second mixed system. 
„ x 2 = 0, xy = yx = 0, y 2 = 0, third mixed system. 
„ x 2 = y, xy = yx = 0, y 2 = 0, Peirce’s system (c 2 ). 
I said, at the end of my Note before referred to, that it had been pointed out 
to me “that my system [the commutative associative system], in the general case 
ad — be not = 0, is expressible as a mixture of two algebras of the form (cq), see 
American Journal of Mathematics, vol. iv., p. 120; whereas, if ad — be = 0, it is reducible 
to the form (c 2 ), see p. 122 (Z.c.).” The accurate conclusion is as above, that the 
commutative associative system is either a mixed system of one of the three forms, 
or else a system (a 2 ), or (c 2 ). 
17. Considering next the non-commutative associative systems, we have here, ante, 
No. 14, 6 = 0, g = 0 ; and the relations which remain to be satisfied then are 
cd = 0, ef— 0, c (c — h) = 0, d (d — a) = 0, e (e — h) = 0, / (/— a) = 0, 
a(c — e) — cf+ de = 0, h (/— d) — cf+ de = 0.