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

88 HISTORY OF THE THEORY OF DETERMINANTS

minor of the unaugmented array of these n-k equations be selected,
the k+r unknowns whose coefficients are not contained in it may have
arbitrary values assigned to them, and for each such set of arbitrary
values there is one set of satisfying values for the other unknowns:
and (3) the remaining equations are dependent on the said n-k
equations. (Prop. X., Cor.)
Next we have propositions which, like the foregoing, do not
specially concern homogeneous equations, but which bear on 'redundant ' systems:
(Ibl). If there be n linear equations containing n-r unknowns,
and if the augmented array be not evanescent, the equations are inconsistent: and, conversely, if the equations be consistent, the augmented
array is evanescent. (Prop. III., XIII.)
(Ib2). If there be n linear equations containing n-r unknowns:
and if there be n-r of the equations whose unaugmented array is not
evanescent: and if when these n-r equations are taken along with
each of the remaining equations in succession, each so formed set of
n-r+1 equations has its augmented array evanescent; then (1) the
n equations are consistent: (2) there is only one set of satisfying values
for the unknowns: and (3) the remaining equations are dependent on
the said n-r equations. (Prop. VIII.)
In the third place we have propositions which refer specially to
homogeneous equations, the first being Jacobi's of 1841 (Hist., iio
p. 324):
(IIa.). If there be n linear homogeneous equations containing n +1
unknowns, then in every set of satisfying values for the unknowns the
values bear to one another one and the same set of ratios. (Prop. II.,
Cor.)
(IIa2). In every set of n linear homogeneous equations containing
n+r unknowns, there is for the unknowns a set of satisfying values
of which two at least are not zero. (Prop. XII.)
(I1a3). If n linear homogeneous equations containing n+r unknowns have for the unknowns a set of satisfying values of which one
is not zero, and if on deleting the coefficients of the unknown which takes
this non-zero value the remaining array is evanescent, then the whole
array is evanescent. (Prop. XV., Cor.)
Lastly, we have propositions which still refer specially to homogeneous equations, but now concern 'redundant' systems mainly: