MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
61 
representing the two units, but not in both ; e.g., the product of (abed) and (ede) will 
be (abe). 
343. If we call the residual ( ab ) the product of a and b, and add to S a system of 
one unit 1 such that 
(aa) = 1 
(a\) = a 
whatever unit of S a may be (cf. sec. 308), we have an algebra subject to the asso¬ 
ciative law, viz., we have 
(( ab).c) = (a.(bc)) 
Some Isolated-Triad Systems.—Family Xo. 2. 
344. Another distribution of isolated triads which may exist is this —If a, b, c, 
be an isolated triad, and if o, a, l; o, b, m; o, c, n are also isolated triads, then l, m, n 
is an isolated triad. Thus (( oa)(ob)) = (oc) = (o(ab )). 
345. Here, if a, b, c, d . . . . be any satisfied component of the system, and a unit o 
be added, the residuals (oa), (ob), ( oc) .... will also compose a satisfied collection, the 
triad (oa), (ob), (oc), being isolated or not according as a, b, c is an isolated 
triad, or not. 
346. The various residuals (a(oa)) or o, (a(ob)), (a(oc)) .... are all distinct units, no 
two of these symbols denoting the same unit ; and they constitute a satisfied collection, 
the triad ( a(ob )), ( a(oc )), ( a(od)) being isolated or not according as (ob), (oc), (od) is 
isolated or not, i. e. , according as b, c, d, is isolated or not ; and similarly in the case of 
the residuals (b(oa)), (b(ob)) . 
347. We have (a(ob)) = ((ao)(ab)) (sec. 344), and thus the united collections 
a, b, c . 
(oa), (ob), (oc) .... 
(a(oa)), (b(oa)), (c(oa)) . 
constitute a satisfied collection. If the number of units in the collection a, b,c, ... . 
is n, the total number in the satisfied collection arrived at by adding o is 3 n. 
348. Thus the total number of units in any satisfied collection of the species now 
under consideration is 3 m . 
349. If we call the residual (ab) the product of a and b, and write it for sim¬ 
plicity ab, we see that ab.c * a.bc, so that the associative law does not hold, as in the 
preceding systems. But we have a.bc = ab.ac, so that the algebra is self-distribu¬ 
tive. We can have, as in the case of the preceding family of systems, 
aa = 1 
a l — 1. 
