865] 
ON MULTIPLE ALGEBRA. 
461 
where P, Q, X, Y are given functions of x, y, x', y. For greater simplicity the law 
of addition is taken to be 
(x, y) + (x, y') = {x+x', y+y’), 
so that as regards addition the multiple symbols are commutative and associative. But 
this is or is not the case for multiplication, according to the form of the given 
functions X, Y\ for instance, if 
(x, y) {x', y) = {xx - yy', xy' + yx), 
then in regard to multiplication the symbols will be commutative and associative. 
But if 
(x, y, z) {x, y', z') = {yz' -y'z, zx'-z'x, xy' - xy), 
then the symbols will be associative, but not commutative. 
4. I remark here that we are in general concerned with symbols of a given 
multiplicity, double symbols (x, y), triple symbols (x, y, z), ?2-tuple symbols (x 1} x 2 ,x n ), 
as the case may be, and that as well the product as the sum is a symbol ejusdem 
generis, and consequently of the same multiplicity, with the component symbols ; this 
is to be assumed throughout in the absence of an express statement to the contrary. 
It is, moreover, proper to narrow the notion of multiplication by restricting it to the 
case where the terms (X, Y, ...) of the product are linear functions of the terms 
{x, y,...) and {x', y', ...) of the component symbols respectively; any other form 
(X, Y,...) is better designated not as a product, but as a combination (or by some 
other name) of the component symbols (x, y, ...) and (x, y, ...). 
5. I assume, moreover, that, if m be any ordinary analytical magnitude, this may 
be multiplied into a multiple symbol (x, y,...), according to the law 
m {x, y, ...) = (mx, my,...). 
6. As a consequence of this last assumption and of the assumed law of addition, 
we have for instance 
(x, y, z) = (x, 0, 0) + (0, y, 0) + (0, 0, z) 
= x{l, 0, 0) + y(0, 1, 0) + z{0, 0, 1); 
that is, using single letters i, j, k for the multiple symbols (1, 0, 0), (0, 1, 0), (0, 0, 1) 
respectively, we have 
(x, y, z) = xi + yj + zk, 
where the letters i, j, k, thus standing for determinate multiple symbols, may be 
termed “extraordinaries.” Each extraordinary may be multiplied into any ordinary 
symbol x, and is commutative therewith, xi = ix\ moreover, each extraordinary may be 
multiplied into itself, or into another extraordinary, according to laws which are, in 
fact, determined by means of the assumed law of multiplication of the original multiple 
symbols ; and, conversely, the law of multiplication of the extraordinaries determines 
that of the original multiple symbols ; thus, if 
(x, y) (x, y') = {xx - yy', xy' + yx'), 
then 
(ix +jy) (ix +jy) = i {xx' - yy') +j {xy’ + yx'), 
= i~xx + ijxy' +jiyx' +j 2 yy', 
and also