The theory of determinants in the historical order of development, by Sir Thomas Muir.

350 HISTORY OF THE THEORY OF DETERMINANTS
determinant on the right * and substituting for the A's, he
deduces
So Si... Sn_
S1 S2... Sfn = (-X1) ""-1)'(xl). /(x2). )
Sn-1 Sn * * *. 2-2
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, + )(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 MJath., xiv.
pp. 20-24; or Opere mat., v. pp. 107-111.]
If for all positive integral values of r' and s we have
A,.+S = ac1A,+,_1 + c2A,.+s-2 +... + ctA,.,
-in other words, if this last be a "recurrence-formula,"-it is
readily seen that the last column of the persymmetric determinant
A,. A,+1... A,.+s-_
A,.+1 A,2... A,.+8 r y,
Or A. say,,
A,+s-1 Ar+s... Ar+2s-,
may be legitimately changed into
asA,-,_, a(A,.,. A.+s_2
so that there is deducible
A,=, = (-1)S-laA._,,
* Brioschi unfortunately neglects the sign-factor. See History, i. p. 345,
where the footnote might have made mention of the fact that the identity
there spoken of as used by Jacobi had already appeared in one of Cauchy's own
memoirs of the year 1813. (See Journ. de l'ec. polyt., x. cah. 17, p. 485.)