1905 - 6 .] On a Theorem in Hypercomplex Numbers. 
49 
e 2 where e 1 n = e 2 n = e and e 1 e 2 = ee 2 e 1 i e being the modulus of B 
and e a primitive nth. root of unity. If x he any element of 
an algebra A which contains B and has also e as its modulus, 
y t = ^ x~ x x t xx r (t = 1, 2, . . . ri 1 ) are commutative with every 
r= 1 
element of B, if x Y , x 2 , . . . represent epe^ (a, 1, 2 , . . . n); for 
n n 
e{e 2 s Vt = e i e 2 ^j € ~ ape i ~ ae 2 ~ ^xepe/ = 2, € " a P+ a3 e 1 r ~ a e 2 s ~ Px t xepe/ 
a, /3 = 1 a, /3 = 1 
= 2 e~ a P~ ^ r e 1 - a e 2 ~ Px&ep+*ep+& = y t e{e 2 s . hf ow if x r ~ l — e{e 2 , 
a, 0 = 1 
n 2 n n 
then 2 e 2 "^i r “ ae 2 V e 2 ^ = un l ess 
<=1 a, 0 = 1 a, 0 = 1 
r and s are multiples of n, i.e. unless x r = e ; hence n 2 x 
= y, ^ xf l x r ~ 1 x t xx r = y,x t y t . It follows therefore as before 
r t t 
that x is contained in the algebra which is formed by taking the 
direct product of B and the algebra composed of those elements 
of A which are commutative with every element of B. 
It may be remarked here that the algebra B is, in the complex 
field, equivalent to the quadrate algebra e pq (p, <2 = 1, 2, . . . n ) 
where e pq e qr = e pr , e pq e rs = 0, if q 4= r and e = e u + e 22 + . . . +e nn ; in 
fact, if we set e^ — + ee 22 + . . . + e w e nn , e 2 = #j 2 + e 2 g -!-...+ , 
then e p = e 2 n = e and e Y e 2 = ee 2 e 1 . 
The second point of view is perhaps simpler. Let A be any 
algebra containing a subalgebra B which has the same modulus as 
A ; if A is expressible as the direct product of B and another 
algebra C, every element x of A can be expressed in the form 
x = ^,f r ( x ) x r where x 1 , x 2 , . . . x b is a basis of B and f r (x) 
r 
(r= 1,2, . . . b) are elements of C and are therefore commuta- 
tive with x x , x 2 . . . x b . Suppose B is such an algebra that, if 
x = ^ £ r x r is any element of B, $ r (r= 1,2, . . . b) being scalar 
co-ordinates, it is possible to express these co-ordinates as rational 
functions of x and the elements of the basis, say £ r = f r (x) : f r (x) is 
then necessarily linear in x and we may write f r (x) = ^ £ pq x p xx q . 
p,q 
If y is any element of B, then yf r ( x )=f r ( x )y, or ]?£ pq yx p xx q = 
P,q 
4 
PROC. ROY. SOC. EDIN. — YOL. XXVI. 
