1910-11.] The Theory of Recurrent Determinants. 
305 
Brioschi, Fr. (1854, 1855, February). 
[Sur deux formules relatives a la theorie de la decomposition des fractions 
rationnelles. Grelle’s Journ., 1. pp. 239-242 : or Opere Math., vol. v. 
pp. 267-276. See also Nouv. Annales de Math., xiii. p. 352.] 
In a review of the second edition of Serret’s Gouts d’Algebre Superieure, 
Terquem, the editor of the Nouvelles Annales, takes occasion to enunciate 
the theorem that if s r denote the sum of the r th powers of the roots of the 
equation 
x n + apt? 1 - 1 + a 2 x n ~ 2 + .... + a n = 0 , 
then 
<q 
1 
< 2a 2 
a i 
a 2 
ra r 
«r-l 
attributing it to Brioschi, and indicating that it had been arrived at by the 
solution of a “ suite indefinie d’equations periodiques du premier degrd” * 
The origin of the determinant is thus exactly similar to that of the first 
determinant of like kind, namely, that occurring in the statement of 
Wronski’s “ loi general e des series.” 
A few months later we find on p. 240 of vol. 1. of Grelles Journal 
Brioschi himself enunciating and proving with some trouble the theorem 
that if 
<f)(x) = c 0 x n + cpf- 1 + ... + C n , 
and 
f(x) = a^x n + a x x n J + ... +a n , 
= cLq(x - x 1 )(x — x 2 ) ... (x-x n ), 
then 
X \<h( X \) + X 2 r <K X 2) + | *n<KVh) 
/'(x,) f\x 2 ) ^ ‘ + f(x n ) 
c o 
a o 
C 1 
cq 
« 0 
a r 
ct r _ 
. . . . 
«o 
C r+l 
a r+ 1 
a r 
a i 
The subject, however, is not pursued further. 
* By this, of course, is meant the set of identities known as “Newton’s formulae,” — 
q + eq 11 0 
s 2 + «i s 1 + 2a 2 = 0 
(See Newton, Arith. Univ., Tom. ii., cap. iii., § 8.) 
VOL. XXXI. 
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