1905 - 6 .] Dr Muir on the Theory of Alternants. 
(aj + ^j) -1 («i + & 2 ) -1 .... (a 1 + 6 n ) _1 
(a 2 + &i) _1 (a 2 + & 2 ) -1 .... (a 2 + 6 n ) _1 
(«» + &i) 1 (««+& 2 ) 1 ‘ • K + ^v) -1 , 
is evaluated. The process consists at the outset in subtracting 
the first column from each column after the first, removing the 
TO pf AV 
(61 -&*)(&! -6,). • • (61-&,,) 
(«i + 6j)(a 2 + ^1) • • • (#n + ^1) * 
and writing the cofactor in the form 
1 
(a 1 -H b 2 ) 1 . . 
. . (a, - K)- 1 
1 
(a-2 + h)- 1 . . 
■ ■ ( a 2 + ^«) _1 
1 
(a n + b 2 )~ 1 . . 
• • (a„ + b n )~' 
The latter determinant is then transformed by subtracting the 
first row from each row after the first, when it is found that the 
factor 
(<h ~ a z)( a i ~ a s) • • - ( a i ~«n) 
(®1 *b ^3) • * • (®1 *b ^n) 
can be removed, and that the cofactor is a determinant similar to 
the original but of the (?z-l) th order, namely, the determinant 
which is the cofactor of the element in the place (1,1) of the 
original. The final result thus obtained agrees with Cauchy’s 
save in having no sign-factor, the latter being only necessary when 
the b’ s are all made negative. 
373 
or A n say, 
Brioschi (1854, Oct.). 
[Intorno ad alcune formole per la risoluzione delle equazioni 
algebriche. Annali di sci. ejis., v. pp. 416-421.] 
All that occurs in this paper in connection with our subject is 
the statement 
s o 
*1 
. . . 
• S n _ 2 
1 
s i 
S 2 
• S n - 1 
X 
S 2 
S 3 
. . . 
• S n 
X 2 
S n - 1 
• ^2n— 4 
x n ~ 2 
1 
X 
. . . 
1 
