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

90 HISTORY OF THE THEORY OF DETERMINANTS

is examined, the test for consistency being now in regard to satisfying values that are not all zero.
VERSLUYS, J. (1870): VALERIANO, V. (1871):
DONNINI, P. (1875).
[Discussion complete d'un systeme d'equations lineaires. Archiv
d. Math. u. Phys., lii. pp. 257-277.]
[Sistema generale di n equazioni lineari fra n incognite. Giornale
di Mat., ix. pp. 371-376.]
[Un sistema particolare d'equazioni lineari. pp. (?). Livorno.]
The theorem in determinants which is made the basis of Versluys'
discussion is identical with Dodgson's of 1867 (chap. v. p. 67);
and the body of the paper may be compared in subject with
Dodgson's chapters iii. and iv. Versluys distinguishes eight different
eases in the solution of
a.x+b.y+cz = d r=3
J r==1l
and twenty-five in the solution of
a,.+by+-c.z+drw = el'=
J r'=1.
Valeriano confines himself to cases where the number of equations
and the number of unknowns are the same, beginning with equations that are homogeneous.
FONTENE, G. (1875).
[Theoreme pour la discussion d'un systeme de n equations du
premier degre a n inconnues. Nouv. Annales de Math., (2)
xiv. pp. 481-487.]
In his subject Fontene restricts himself as Valeriano did (1871);
but his aim is quite different, namely, to embody all his results in
a single proposition.' His success is more apparent than real, the
enunciation of the proposition occupying almost a full page and
not being a very compact whole. The purely determinant theorem
which he uses (pp. 485-486) is in essence identical with Kronecker's
of 1864.