426 
into 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
2a 2 
3 a 3 
4 a 4 
~oa b 
/A \ 2 
1 
a 1 
a 2 
a 3 
a 4 
M x ) 
- (a i ) 
n 
(n- 1 )a 1 
(n - 2 )a 2 
(n - 3 )a, 
n 
(n - l)a x 
(n - 2 )a 2 
(n - 3 )a s 
(n - 4 )o 4 
• 
/i(») 
Zi 
z 2 
z 3 
z, = 
(■ 
'C + Q 
n ./(*), 
z 2 = 
(: X 2 + 
ape- 
+ a 2lfl( x )- 
\nx + (n 
— 1 )^] ]./(*) , 
Z 3 = 
(x 2 + ape 2 + 
ape- 
+■ a s)fi( x ) - 
\nx 2 + (n- 
l)a 1 a , + ( n 
- 2 H]/(*) • 
where 
The worthlessness of this in itself is apparent as soon as we note the 
presence of f x {x) and f(x ) : when, however, the determinant is partitioned 
into two, one having f x (x) for a factor and the other f(x), and the result 
compared with — — we obtain for N 3 an expression similar to 
that for D 0 . 
Brioschi, F. (1854, August). 
[Intorno ad alcune questioni d’ algebra superiore. Annali di sci. mat. e 
Jis., v. pp. 301-312 : or French translation of Brioschi’s Teorica dei 
Determinanti, pp. 151-170: or Opere Mat., i. pp. 127-142.] 
The questions referred to are much the same as those of his paper on 
Sturm’s functions (1854, February), the first function 
ape n + ape n ~ x + . . . +a n or a 0 (x - x 1 )(x - x 2 ) . . . (x - x n ) or f(x) 
being, however, no longer connected with the second 
bpe m + b 1 x m ~ 1 + . . . + b m or (f>(x ) where m<n. 
Putting 
Q r™ v ^(« 2 ) a. JL <V>K) 
7K) 7 W ' ' ' TkT 
he first proves the interesting theorem 
+ ^S r _ 1 + . . • + = b r+m _ n+ i(v 71 s ) . 
Then temporarily denoting 
{d>(x r )+f(Xr)y by A r 
he squares in two ways the determinant 
A: 
A 2 
• • A. 
A 1*1 
A 2^2 
. . A n x n 
A ,*? -1 
A ^.m-1 
. . aJ 1 
