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

84 HISTORY OF THE THEORY OF DETERMINANTS

It will be noticed that two useful terms are introduced incidentally
in the statement, 'augmented array' and 'unaugmented array'
representing important entities connected with the study of simultaneous linear equations.
TRUDI, N. (1862).
[TEORIA DE' DETERMINANTI,... xii+268 pp. Napoli.]
The section devoted to the Solution of Linear Equations (pp.
113-121) is fuller than in any previous exposition, and a great
improvement on Baltzer's of 1857. The fifth paragraph of it deals
with the case of
arxl + ac,22 +... - a = U, -
Jfr=
when the common denominator of the unknowns vanishes and also
one of the numerators. From this the first deduction of course is
that the other numerators must vanish also; and the proof rests on
the fact that the vanishing of the numerator of x1 gives us
ulAll + u.A21 +. * * + nA,, = 0,
and the vanishing of the common denominator gives us
A11 _ A Ai
Al, A2s ' As
The second deduction is that one of the given equations, say the
first, is derivable from the others; and the proof consists in obtaining
from these others by multiplication and addition the equality
(a21Az + a3,A3, +.. + a,,,A,,,)
+ a,,lAs + a3,,A. +.. *+ annA,,A) xn
=- U A2,+us a3s+ s. +ALs,
and thence the equality
-aAlsxl -a,2Alsx-. -.,-alAsx= -ulA,,
from which the first equation of the set at once follows if A,, is not
zero. The appearance of this extra condition at the close deserves
note, as also the fact that it is unmentioned in connection with the