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

354 HISTORY OF THE THEORY OF DETERMINANTS

LEMONNIER, HI. (1875, 1878).
[The'ore'mes concernant les equations qui ont des racines communes.
Comptes Rendus.... Acad. des Sci. (Paris), lxxx. pp. 111 -112, 252-255.]
[Me'moire sur Fl'limination. Annales de l'Ecole Norm. Sup. (2), vii.
pp. 77-96, 151-214.]
Lemonnier's condition for the equations
a cemx~n. *+a.m=O, boXn +.. bn=
having k common roots is different from Trudi's, but fortunately
for comparison' is very easily expressed in Trudi's notation. It is*
that the first k determinants of
(a 0, am.-.,
(b0,... I n m-,
shall vanish, and the first determinant of
(b0, I b., k
shall not vanish. The former part of the condition recalls Zeipel's.
of 1859: the latter is an important necessary adjunct. When,,
however, the equation of the common roots
(a0 a I x~ '1...,x0) = 0
(b0,... I n, -
happens to be given along with the condition, it is less necessary to
mention the latter part, as the determinant involved is the coefficient.
of x'~ in the said equation.
The whole memoir is valuable, viewed either as anr exposition or as.
a repository.
DARBOUX, G. (1876, 1877).
[Sur la the'orie 1el~imination entre deux equations 'a une inconnue..
Bull. des Sci. Math. x. (1), pp. 56-64.]
[Sur e'~limination entre deux equations alge'briques 'a une inconnue..
Bull. des Sci. Math. (2), i. (1), pp. 54-64.]
Darboux uses Bezout's eliminant in the extended form given it.
by Cauchy for the case of unequal degrees: and he establishes the
* This is in accordance with the statement in ~ 13 of the complete memoir, and
is somewhat different from that first published.