427 
1909-10.] The Theory of Persymmetric Determinants, 
thus obtaining 
o 
So 
Si • • 
■ • s_, 
s o 
• ■ 
• s n _i 
s x 
S 2 
• • S. 
- W • ■ 
A 2 
1 • 
S 2 ’ ■ 
• 
S„-i 
. . 
' • S 2n _2 1 
«»- 1 
• • S 2ra-2 
from which, on putting ( — . . . f'(x n ) for the deter- 
minant on the right * and substituting for the As, he deduces 
S 0 
S x 
Sj • • . S n _ x 
s 2 . . s n 
— ( — iy ntn . ^>(^ 2 ) • • • ‘M^n) • 
. . . s 2 
This result, be it noted, is not given in the original paper, but appears first 
in Combescure’s translation (1856), which contains six pages (pp. 153-159) 
more than the original. Brioschi does not point out its significance in 
connection with Euler’s first form of the resultant of f(x) = 0 , <p(x) = 0 . 
The remainder of the paper is of little interest in the present connection. 
Brioschi, F. (1855, January). 
[Sur les questions 241 et 141. Nouv. Annales de Math., xiv. pp. 20- 
24: or Opere Mat., v. pp. 107-111.] 
If for all positive integral values of r and s we have 
A r+S = « 1 A r+s _ 1 + a 2 A r+s _ 2 + . . . + a s A r , 
— in other words, if this last be a “ recurrence-formula,” — it is readily 
seen that the last column of the persymmetric determinant 
A r A r+1 
A r+1 A r _|_2 
Afs-i 
A r+S 
or A r> say 
Ar+s—i 1 A r+S 
A, 
+2s— 2 
may be legitimately changed into 
ci s A r _i , ol s A r , , ct s A r + S _2 
so that there is deducible 
and thence 
A r>s 
( 1) ^sA r _i iS , 
(-1)^^A m , 
* Brioschi unfortunately neglects the sign-factor. See Proc. Roy. Soc. Edinburgh , 
xxiii. p. 132, where the footnote might have made mention of the fact that the identity had 
already appeared in one of Cauchy's own memoirs of the year 1813. (See Journ. de Vec. 
polyt ., x. cah. 17, p. 485.) 
