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

LINEAR EQUATIONS (ROUCHE, 1875, 1880)

91

ROUCHE, E. (1875, 1880): MERAY, C. (1875).
[Sur la discussion des equations du premier degre. Comptes
Redus... Acad. des Sci. (Paris), lxxxi. pp. 1050-1052.]
[Note sur les equations lineaires. Journ. de l'Ec. Polyt., cah. xlviii.
pp. 221-228.]
[Sur la discussion d'un systeme d'equations lineaires simultanees.
Comptes Rendus... Acad. des Sci. (Paris), lxxxi. pp. 1203 -1204.]
Rouche is still more ambitious than Fontene, his aim being to
condense into one proposition the means of completely investigating
a set of linear equations. Simplicity in enunciation is sought to
be attained by introducing the term 'determinant principal' for
the non-zero minor of highest order in the unaugmented array, and
the term ' determinants caracteristiques ' for minors of the augmented
array but not of the unaugmented array which have the 'determinant
principal' for a primary minor. Unfortunately, the terms, like
Dodgson's, are not happily chosen. After explaining them at considerable length (pp. 221-223) Rouche says: We can now formulate
" la proposition qui renferme toute la theorie des equations lineaires."
In English the proposition is:
In order that n linear equations containing m unknowns may be
consistent it is necessary and sufficient that all the 'characteristic
determinants ' of the set shall vanish; and, if this condition be satisfied,
the set has only one solution or is indeterminate according as the order
of the said determinants is greater or less than m.
The logical foundation for the proposition is well set forth (pp.
223-227), but will readily be guessed by any one familiar with earlier
work on the subject. In the actual application of the test to a given
set of equations the first and fundamental requirement is of course
the finding of a ' dterminant principal,' and unfortunately Rouche
gives no indication of the procedure which he considered best for
attaining this end. Nor are any examples like those of Dodgson's
first appendix worked out. It is thus impossible to say whether in
the actual practice of solution the condensation and simplification
apparent in the statement of the proposition be realities of any.appreciable value. The concept, however, of a 'determinant