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

RECURRENTS (BRIOSCHI, 1858)

209

where e1, e2,..., e,, being neither negative nor fractional, satisfy
the equation
e.1+2e2+3e3+.. +en = n,
and where le stands for e, +ee,A... + en.
BRUNO, F. FAA DI (1859).
[THE'ORIE G-E'NF',RALE DE, L'ELIMINATION... x+-224 pp. Paris.]
Of the seven carefully formulated propositions regarding symmetric functions which are the subject of Bruno's opening chapter
(pp. 1-22), the first and last involve a recurrent. The first, being
already known (Hist., ii. pp. 211-212), is only worth mentioning
because of the fact that in a footnote (p. 2) its quasi-converse is
deduced from the same set of equations, namely,
1
a I () st 2
S3 S2 s, 3...The other is taken from Brioschie by whom it was used for the
purpose of expressing in terms of the a's any symmetric function s
of the roots of the equation
+a yc~xn13-... -an (x-ai)(x-a2).. (x-a,) 0.
It is
Dr
(v -i) a, 1
s~~~D
1 ar- aQ-2...1
rar (r - 1)a...-..
*BRTOSCHI, F. Sulle funzioni simmetriche delle radici di una equazione. Annali
di sci. mat. e fis., v. (1854), pp. 422-428.
M. D. III. 0