56 
MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
The pairs (1), (5), (10), and (12) will be termed joined pairs; each is such that the 
column in which the first unit lies and the row in which the second unit lies intersect 
at an asterisk. In the literal mode of representation they are such as 
(1) ( aa)(ab), (5) ( ba)(aa ), (10) (ab)(ba), (12) ( ab)(bc), 
i.e., such that the last letter in the first bracket is of the same sort as the first letter 
in the last bracket. We may add to these the pairs such as ( aa)(aa ) which are really 
connecting pairs of two systems P A and P M . The pairs (2), (3), (4), (6), (7), (8), (11), 
may be called unjoined pairs. 
318. Each joined pair is accompanied by a unit unique with respect to it, repre¬ 
sented by the dot or asterisk in the same row as the dot or asterisk representing the 
first unit of the pah”, and in the same column as the dot or asterisk representing the 
second unit of the pair. This unit may be taken as the product of the two units 
composing the pair. In the literal mode of representation we shall have 
(ab)(bc)=(ac) 
where a, b, c, or any two of them may be letters of the same sort. 
319. The product of two unjoined pairs is taken to be a unit Z or 0, and if ( ah ) 
be any unit of P, we take 
(ab) Z = Z 
Z(ab) = Z 
ZZ —Z (sec. 308). 
320. The units (cib), (ac ), (be), . . . , when thus dealt with as multipliers, multipli¬ 
cands, and products, the products being those specified in the preceding sections, 
may be called quadrates* 
321. We may consider products of component collections of P or P n (sec. 311); and 
we may regard these as unified, and thus arrive at various primitive algebras. 
322. These quadrate algebras are all associative. 
323. It will be convenient in many cases to write a quadrate (ab) thus assimi¬ 
lating it to the ordinary algebraic fraction, and a collection of quadrates (sec. 321) 
thus, l + \ + \- 
324. Quadrates have been arrived at by considering unified pairs in the case of single¬ 
heap systems. We might consider unified pairs in the case of systems of any other 
form, but in such case P would not have the characteristic form which we have been 
considering, and we should not get products. Thus if we considered discrete heaps, 
all the unified pairs would be distinguished from each other. We, of course, can and 
do consider, unified pairs of any system S, and can take their products as if they were 
* See Pierce on “ Linear Associative Algebras,” in the ‘ Araei’ican Journal of Mathematics,’ vol. iv., p. 215. 
