48 
Proceedings of Royal Society of Edinburgh. [sess. 
On a Theorem in Hypercomplex Numbers. By J. H. 
Maclagan-Wedderburn, Carnegie Research Fellow. 
(Read January 8, 1906.) 
Scheffers in the Mathematische Annalen, vol. xxxix., pp. 364-74, 
enunciates the following theorem : — If A is an algebra containing 
the quaternion algebra B as a subalgebra, and if A and B have 
the same modulus, A can he expressed in the form B C = A = C B, 
where C is a subalgebra of A every element of which is com- 
mutative with every element of B : in other words, if i 1 , i 2 , z 3 , 
is a basis of B, it is possible to find an algebra C with the basis 
e-j , e 2 , . . . e c , such that each of its elements is commutative 
with every element of B, and such that the elements e r i s (r= 1, 
2 , ... c, s = 1, . . . 4) form a basis of A; and if a is the 
order of A, then a — 4c. 
The following is a short proof of this theorem. Let the basis 
of B be as usual 1, i,j, k where the laws of combination are the 
usual laws of quaternions. If x is an element of A, then x = 
x — ixi —jxj — kxk, x = ix + xi + kxj -jxk, x" —jx - kxi + xj + ixk , 
and xP = kx +jxi - ixj + xk are commutative with every element 
of B : further, x can be expressed in terms of x, ix", jx", and kx iv , 
in fact 4x = x - ix!' —jx" - kxP ; hence if C is the algebra of all 
elements of A which are commutative with every element of 
B, B and C satisfy the conditions required by the theorem. 
If V, Vi > 2/ 2 5 Vs are an y elements of C, x = y 4-y-f + yJ + yfc 
can only vanish if y = y Y = y 2 = = 0, for if x = 0, then 
4y = x - ixi - jxj - kxk = 0, and similarly y 1 = y 2 — y 3 = 0 ; hence 
the order of A is four times the order of C. 
In addition to being much shorter than Scheffers’ proof, which 
occupies about ten pages, this proof has the advantage of being 
rational. 
The method used in the above proof may be regarded from two 
points of view. The first of these will be best understood from 
the following extension of it. 
Let B be the algebra generated by the two elements e 1 and 
