Voh. 7, 1921 
MATHEMATICS: L. E. DICKSON 
113 
if 
n n 
Xk = 2 ( 2 7 ^ *») & = 1, . . n). (14) 
i = i * = i 
The matrix M of this substitution has the element 2jCi7*}-fc*» in the fcth 
row and ;th column. If this substitution is applied to a quadratic form 
in Xi, . . . ,X n of matrix Q, it is a standard theorem that we obtain a quadratic 
form in £i, . . . , £ M , whose matrix is M'QM, where M' is the transposed 
of M, being obtained from M by the interchange of its rows and columns. 
In our problem, Q is the identity matrix I whose elements are all zero 
except the diagonal elements which are 1. Hence, by (13), 
M'M = {x\ + . . . + x n )I. (15) 
When a homogeneous polynomial a(xi, . . . , x n ) of any degree has the 
property (4) of possessing a theorem of multiplication, the writer 8 has 
proved that we may apply a linear transformation on xi, . . . , x n which 
leave a(x) unaltered and one on £i, . . . , £ M which leaves a(£) unaltered 
such that the new algebra has the principal unit e u so that y\ jk and 7,1* 
are both 0 if j =t= k, and both unity if / = k. 
Hence M = X\M\ + • • • + x n M n , where y ijk is the element in the 
kth row and jth column of M if whence M { = 7. Thus (15) gives 
Ml = -M if M-Mi = 1, M'iMj + MjMi = 0 (t> 1, j> 1, / 4= 1). (16) 
In view of the values of yj ik , and M\ — —M i} we have, when n — 4, 
0 -1 0 0 0 0 -1 0 
0 
-1 
4 
0 - 
7323 
7323 
0 
0 
7324 
7334 
0 0 
0 —7423 
— 7424 \ 
7423 0 
~ 7434 / ' 
7424 7434 
0 
(1 0 — 7223— 7224 \ M = I 0 "~T323 "7324^ 
0 7223 0 — 7234/ 3 V 1 7323 0 —y^Uj 
0 7224 7234 0 0 7324 7334 0 
0 
M 2 
By M^Mi = I, we have M 2 * = —I, which gives 
7223 = 7224 = 7323 = 7334 = 7424 = 7434 = 0, 7* = 5 2 = € 2 = 1, 
where 7 = 7234, 5 = 7324, € = 7423- Hence 
0-100 0 0-10 000-1 
/l 0 0 o\ /o 0 0 -A i/r /0 0-e o\ 
Mo 0 0-7/ M3= (l 0 0 O/ M4== Ue 0 0/ (17) 
X 0 070 0500 1000 
The final condition (16) states that MiMj is skew-symmetric. The 
products M 2 M 3 , M 2 M 4 , M3M4 of matrices (17) are seen at once to be 
