MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
5T 
where the letters on the right-hand side of the equations depend on the particular 
unit a selected. 
291. Now we may make c„ = c, and similarly in the case of every other unit of 
S and S„; also we may represent the units X and /x by two positions , and write aj> fi 
thus aft. 2 , where the order of a x and b. 2 is immaterial, or merely ah where the order 
is material; the equations then become 
(a<m) — . 
( aab) = c 
(aac) = . 
&c. 
(aba) — . 
(abb) = . &c. 
(a be) = . 
&c. &c. 
If then we have two equations such as 
we may write 
(aab) = c and (acd) — e, 
e = (a(aab)d), 
and similarly in other cases ; so that we obtain complex expressions representing the 
form relations which the units of S hold to each other and to the units X, p, a ; which 
expressions, however, admit of considerable-simplification in certain cases. 
292. An equation such as (aab) — c I shall term a primitive equation. The whole 
collection of triads which are concerned in arriving at a will be termed a primitive 
algebra, a being termed a unified primitive algebra .* An equation, such as e — 
(a(aab)d), where only one primitive algebra is involved, may be termed a complex 
primitive equation. 
293. Those components of E which are represented when unified by the units 
of A furnish us, by the application of the preceding methods, with every possible 
primitive algebra, associative or non-associative, commutative or non-com mutative, &c., 
and we can discuss the number of the forms, and the relations of the various algebras, 
by discussing E in the case in which S is a single heap system (sec. 285). 
294. We may have any number of unified primitive algebras a, /3, y, 8, . . . which 
may be undistinguished from each other or not. If the unified primitive algebras 
are all selected from one system A^ (sec. 289), they may be said to be algebras of 
the same operation it, if from different systems A n , A„, they may be said to be 
algebras of different operations, e.g. , multiplication, addition, &c. 
295. We may deal with each of these algebras as with a and we may also deal 
with them in combination and obtain equations such as e =(a(/3a,b)d), giving compli¬ 
cated expressions for the units of S, which will represent the form relations they hold 
to each other and to the units X, p, a, /3, &c. 
* We might regard the primitive equations as units, and apply the term “ unified primitive algebra,” 
not to a, but to the unit arrived at by regarding as unified the collection of unified primitive equations 
derived from the triads concerned in arriving' at a.. 
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