52 
MR, A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
296. Where two or more primitive algebras are considered, we may speak of the 
unit arrived at by regarding them as unified as a unified compound algebra. 
297. The discussion of form is very generally carried on by the help of these 
auxiliary algebras. In some cases the units a, b, c, d, . . . of the base system under 
consideration are regarded as unified pairs Xof, X?/, Xc', .... where X is an algebra 
primitive or compound, and a, b', are units constituting a single heap system, 
which have the correspondences characteristic of the system a, b, c, . . . as long as 
X remains identically correspondent. But various devices exist which must be 
examined in each case. 
298. Where only one unified primitive algebra is considered, we may regard the 
unit a as expressed by the brackets ( ), and write an equation (a ab)=c simply as 
(ab) = c, or w r e may omit the brackets and write the equation thus ab=c. 
299. If in an algebra every equation (ab) = c is accompanied by an equation (ba)—c, 
the algebra is said to be commutative. 
300. If in an algebra any three equations such as 
(ab)=x 
(bc)=y 
( xc) = z 
are accompanied by the equation 
(ay)=z 
the algebra is said to be associative. Since ((ab) c) = (a(bc)) either expression may 
be written ( abc ), and no ambiguity arises. 
301. If 7T and p two algebras are such that the equations 
(jrab)=x ( 7 rbci) = l 
( 770 C )=y (7 tco) = m 
( Trad)—z (jrda) = n 
( j pbc) = d 
are accompanied by the equations 
(pxy) = z (plm)=n 
i.e., if p ( nab . Trac) = n (a . pbc) 
and p (7 rba . Trca) = TT (pbc . a) 
then the algebra n is said to be distributive with respect to the algebra p. 
We may take 7 Tab to be aX b, and pah to be a-fib ; we then have 
a xb-fiaX.c=a(b-fic) 
b X a-fic X a = (/> + c) X ct 
