54 
MR, A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
i, z, u, c, a, b, 
lie respectively on the six lines 
PQ, PS, PR, QS, PS, QR, 
passing through four coplanar points P, Q, R, S : i.e., the unit c is unique with respect 
to the collection i, z, u, a, b. Now let i, z, u, be three constants in the foregoing sense. 
Then we may call c the product of a and b. The primitive algebra which thus 
arises may readily be shown to be associative and commutative. 
Again, there is one point cl, and one only, which is such that the points 
i, i, z , d, a, b, 
lie respectively on the six lines 
LM, NO, LN, MO, LO, MN, 
passing through four coplanar points L, M, N, 0 : i.e., the unit d is unique with 
respect to the collection i, z, a, b. Here, if i and z be constants, d may be regarded as 
a second species of product of a and b, and may be termed the sum of a and b. The 
primitive algebra thus arising is also associative and commutative. 
The first of these two primitive algebras is distributive as regards the second. In 
fact the compound algebra composed of the two primitive algebras is of the same kind 
as the ordinary algebra of quantity, the units i, z, u, corresponding respectively to the 
oo, 0, and 1 of such algebra.* 
310. We may speak of the collection consisting of the products of the pairs 
connecting two component collections of a system S as the jmoduct of those collections. 
If the number of units in the multiplier collection be n, and the number of those in 
the multiplicand be m, the number in the product will not necessarily be ran, for the 
product x of a connecting pair ap may be the same unit as the product of another 
connecting pah’ bq. We may speak of the collection consisting of the products of all 
the pairs of a collection a, b, c, d . . . . together with the products (act), (bb), (cc) . . . 
as the square of the collection a, b, c, d . . . . In the same way we may have cubes, 
&c., of a collection. 
311. If we substitute for S the system S H of sec. 191, we may regard products (ap) 
and (bq) as two different units (sec. 192), and if this be done in the case of each product, 
we shall have mn units in the product. 
312. We may regard the components of S or S H as units, and then the products will 
be units, the products of pairs of units. 
Quadrates. 
313. Let S l3 S 2 , be two single heap systems, replicas of each other. Let a x , b v c x . . . 
be units of S x ; a 2 , b 2 , c 3 . . . . be corresponding units of S 2 . Considering the pairs 
connecting S T and S 3 ; these may be written 
* See a paper by the writer “ On an Extension of Ordinary Algebra ” in the Messenger of Mathematics. 
Vol. XY. New Series, p. 188. 
