u t 
71 
X/jWj (/ i, . . . , j n), 
where the \ ti are numbers of F. Elimination of the 
u'i between (18) and (19) gives 
(20) (bi=b n). 
By means of (19), any element x = 'Z% l u t of A can 
be expressed in terms of the new units u\ as follows: 
(21) ^ 2 = 2 ^' iU>i ’ ^ = 2 • 
i, i = l i = i £=i 
By (18) and (1), 
n n 
UiUj — T{ r Tj s U r U s — ' r irTj s 'Yrsh'M'h • 
r,i = i r, s, h= 1 
Replacing by its expression from (19), we get 
w « 
(22) UiUj — 7ijk'M'k ) liijk = TirTjs*YrshFhk > 
k = i 
which gives the multiplication table of the new units. 
62. Characteristic determinants are invariants. Let 
R' x and S'x be the first and second matrices of x with respect 
to the new units u' x , . . . . , u' n defined by (18). We 
seek the sum analogous to (6), but written in the accented 
letters 7' defined by (21) and (22): 
n 
Pkj~ %iyijk = 2X/jTi r Tjslirsh\ik^t )