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

142 HISTORY OF THE THEORY OF DETERMINANTS
the proof being dependent on the fact that, if any one of the a's
be put equal to 0, the determinant vanishes. The second is
1+ca, 1... 1
1 l+ct2 1... 1
1 1 l+a3... 1
1 1 S..... 1
I I I +...+ a),
aa... a,(I+ ~ -~.,
which is made to rest mainly on the fact that if any one of the
a's be put equal to 0 the determinant takes the form of the
preceding determinant. Note is taken that Sylvester's theorem
on p. 55 of the same volume is a special case of this second
result.
BRUNO, F. FAA DI (1855, Dec.).
[Addizione alla nota inserita nel fascicolo di ottobre ultimo.
Annali di Sci. mat. e fis., vi. pp. 476-479; or ~ vi. of his
THEORIE GEiNERALE DE L'ILIMINATION, x+224 pp., Paris,
1859.]
The note referred to in the title professed to be "Sulle
funzioni simmetriche delle radici di un' equazione," and contained,
besides other things, the final expansions of the resultants of two
quadrics, two cubics, and two quartics. The "addizione," on the
other hand, draws attention to the axisymmetric determinants
which represent those resultants, the author being apparently
unaware that Jacobi had already done this in 1835 and Cauchy
in 1840. His rule of formation is the same as Sylvester's of
1853 (June).*
In his " Theorie QGenrale de l'Elimination" the matter is gone
into in greater detail, the rule of formation occupying a full
page (pp. 55-56). He there also devotes a section (~ix. pp. 66,
67) to an account of Jacobi's relations between the elements of
Bezout's condensed eliminant.
* See our chapter on Persymmetric Determinants.