50 
Proceedings of Royal Society of Edinburgh. [sess. 
y, £ pq x p xx q y for every yanda? belonging to B, therefore the matrix 
pq 
represented by £ vo {yx v ( )x q - x p { )x q y) is identically zero. 
p, a 
Hence f r (x) is formally commutative with every element of B, and 
f r (x)x=xe = x where e is the modulus. If now x is any element 
of A, f r (x) is still commutative with every element of B and 
x— f r (x)x r , since A and B have the same modulus. Hence as 
before A can be expressed as the product of B and the algebra 
composed of those elements of A which are commutative with every 
element of B. If c is the order of this algebra, which as before 
will be denoted by C, the order of A is equal to be : for suppose 
if possible that y = y x x x + y 2 x 2 + . . . + y b x b Bo, y x , y 2 , . . . being 
b 
elements of C ; then f r {y) = 0, but f r (y)= ^ f r {y r x r) = y r fr(x r ) = y r > 
r = 1 
therefore y r = 0. 
For instance, let B be the quadrate algebra (e pq ) described above, 
n n 
then f pq(x) &tpXCqt ? @rsfpq(x) @rpXC qs = f P q{x)e rs , and ^^f P q(x)e pq 
t = i P, 9=1 
n 
= e pp xe qq x. 
p,q=l 
It is interesting to note in this connection that if A is a quadrate 
algebra of the above type whose order n 2 = l 2 m 2 , it is expressible as 
the direct product of two quadrate algebras of orders Z 2 and m 1 
m 
respectively. This may be shown by setting e m — )m+t 
*=i 
i - 1 
(P,q= 1, 2, . . . I ) for the algebra B and rj pq = V^e tm+p>tm+q 
<= o 
(p,q= 1, 2 , . . . m) for C. e pq and rj pq evidently have the proper 
laws of combination, and e pm ^_ r> qm + s rjrs^p+i, q+i ~ ^-p+i,q-\-i*lrs • Hi 
general, if n=n l n 2 . . . n k , A can be expressed as the direct 
product of h quadrate algebras A 1 , A 2 , . . . A k of orders 
nf, n 2 2 , . . . n k 2 respectively. This theorem is the counterpart 
of one given by Clifford.* 
*■: American Journal , vol. i., pp. 350-58. 
( Issued separately February 9 , 1906 .) 
