1905 - 6 .] Dr Muir on the Theory of Alternants. 
387 
Now a glance at the final determinant suffices to show that it is 
not the eliminant sought, there being in it three types of rows, 
whereas the two equations giving rise to the said eliminant being 
both of the (n — 1 ) th degree, the coefficients of the one must occur 
in as many rows of the eliminant as the coefficients of the other. 
Further, since the coefficients of the equation f'(x) = 0 are seen 
to occur in their full number of rows, and those of the other 
equation in the last row only, it is therefore the first n - 2 rows 
that need to be changed. The set of operations requisite to effect 
this is 
n • row x - row 2n _ 3 , 
n • row 2 - row 2n _ 4 , 
n • row n _ 2 - row n . 
Brioschi (1857, Oct.). 
[Sullo svillippo di un determinante. Annali di Mat., i. pp. 9-11.] 
Brioschi enunciates without proof the proposition that the 
eveii-ordered determinant 
1 
1 
l 
1 
1 
1 
®1 
- 
Oi 
- 
«i) 2 
x 1 
- 
a 2 
(*1 
- 
a 2 f • ' 
'*i 
- 
a n 
{x x 
- «n) 2 
1 
1 
l 
1 
1 
1 
x 2 
— 
aq 
( 
- 
x 2 
- 
— 
a 2 ) 2 
x 2 
— 
a n 
(x 2 
-<) 2 
1 
1 
l 
1 
1 
1 
x 2n 
- 
( X 2n 
- 
- 
6^2 
(x 2n 
a 2 f 
x 2n 
a n 
( X 2n 
- a n ) 2 
which is seen to be a function of 2 n %’s and n a’s, is equal to 
/ n 4 (a x , a 2 , . . . , a n ) • n(aq , % 2 , ... , x 2n ) 
[ > <£>,) • tf(a 2 ) . . . <p(a n ) 
where cf>(x) = (x - x^)(x — x 2 ) . . . (x — x 2n ). He then obtains 
similar expressions for the principal minors, namely, (1) for the 
cofactor of any element in an odd-numbered column, and (2) for 
the cofactor of any element in an even-numbered column, his 
procedure being to express the minor in question in terms of 
determinants like, the original but of the order 2n—2 and then 
to make the substitutions which are thus rendered possible. 
