Vol. 7, 1921 MA THEM A TICS: L. E. DICKSON 111 
n 
.2 
%' = 2%i — x = %i — ^> xfi, a = x\ — Ci% 2 i 
i = 2 i = 2 
We shall call %' the conjugate to x and <r = a(x) the norm of x. Hence 
the product of any number and its conjugate in either order equals its 
norm. We assume that the norm of a product equals the product of the 
norms of the factors : 
<r(*)<r({) = a(X), if *f = X, (4) 
and shall investigate the resulting types of linear algebras. We assume 
also that each c t =j= 0 in (3). 
5. By (2) the coordinates of X = %£ are X k = *Lx&jy ijk . Since el — c { , 
we have jm = c i} y iik — 0 (i>l, k>\). Hence 
n n 
Xi = + 2 % & c i + 2 X & 7i i 1 ' 
i = 2 i.j = 2 
^ = xiik + + 2 x &j y W' ^ 
U = 2 
i =f= i (fe> 1) 
Since this transformation is the identity X = x if £ = 1, we obtain an 
infinitesimal transformation by taking £i = 1, ^ ? = 5£, & = 0(*=fcl, /): 
t = 2 i = 2 
* =t= i * 4= 5 
71 
= { 2 t ^ } 
i = 2 
i=f=i 
For these £'s, <x(£) is unity to within an infinitesimal of the second order. 
Hence the increment to u(x) must vanish identically, so that 
Yyi = 7ijj = Tiii = 0 (1, i, j distinct), (7) 
WW + cam = 0 (l,i,j,k distinct). (8) 
By (7), (5) simplifies to 
n 
Xi = flfifi + 2 *i&i> X k = x &k + %ktl + 2 ( k>l )> ( 9 ) 
t = 2 
where, in the final sum, * and / range over distinct values from 2, . . . , n, 
excluding k. This final sum is, therefore, absent if 4 w = 3; whence <r(X) 
