284 Proceedings of the Royal Society of Edinburgh. [Sess. 
and P any other determinant of ike same order, then the sum of the m -line 
coaxial minors of QPQ is the same as the sum of the m -line coaxial minors 
of P. 
The other paragraph requiring notice concerns the determinant arising 
from Cayley’s of 1846 by subtracting 1 from each diagonal element. The 
value of this is shown (p. 65) to be 0 when n is odd, and 2 n A 0 /A when n 
is even, A being the basic determinant, and A 0 what A becomes on making 
all its diagonal elements zero. The result is easily reached on multiplying 
the given determinant by A and showing that the product is ( — l) n 2 ,l A 0 . 
Brioschi, Fr. (1854, August). 
[Note sur un theoreme relatif aux determinants gauches. Journ. (de 
Liouville) de Math., xix. pp. 253-256 : or in French translation of 
his Teorica dei Determinant, pp. 144-147 : or Opere Mat., v. 
pp. 161-164.] 
Brioschi’s subject is really the equation 
e 
i 
a 
<*>12 
<*>1 n 
W 2 1 
£ 
• to 
1 
* a 
w 2 n 
O 
II 
<*>»! 
°>n2 
• • . 
0i nn ~ X 
in which the left-hand member is the determinant of Cayley’s orthogonal 
substitution with — x affixed to each diagonal element. He notes at once, 
of course, that if the basic determinant be | a n a n . . . a nn \ , or A say, the 
equation may be changed into 
A n -y 
Aia . . 
A ln 
A 2 1 
A 22 -y • • 
A 2n 
A^i 
Ajjjj ■ ■ 
1 • A m i ~ V 
where y is put for |(1 -\rx)A . A further transformation is then effected by 
multiplying both sides by A and putting z for 1 — A/y, the result being 
Z 
a 21 • • 
■ • a nl 
a l2 
Z . . 
. a n2 
n 
■<h n • ■ 
. . z 
Using Cayley’s expansion (1847) for the determinant on the left, it is seen 
that when n is odd the equation resolves itself into z = 0 and an equation 
