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

LINEAR EQUATIONS (DODGSON, 1867)

87

contain extensions and improvements of previous incidental work
on the subject. The results are helpfully collected in tabular
form (pp. 134-138); and by way of illustration four different sets
of equations with arithmetical coefficients are investigated in full
detail (pp. 111-116). To a certain extent tabulation is facilitated
and conciseness furthered by using the term 'V-block' to stand
for the array composed of the coefficients of the unknowns, and
' B-block' for the augmented array, thus making possible such short
sentences as
V = 0, or + 0 B = 0, or + 0
V = 0, or + 0; B = 0, or 0;
llVt =0, or 70; iBlI =, = or 0.
This has its advantages, but there is also much against it, and it
has certainly not commended itself to subsequent writers. As
hitherto, therefore, we shall state the propositions at full length,
merely agreeing with Dodgson to speak of an array as being
evanescent' when all its primary minors vanish.
Taking chapter iii. (pp. 32-59) and noting first those which are
not specially concerned with equations that are homogeneous, we
have:
(Ial). If there be n linear equations containing n+r unknowns,
and if a primary minor of the unaugmented array be not zero, the
equations are consistent; further, any values being arbitrarily
assigned to the r unknowns unconnected with the said minor, one
satisfying value, and only one, can be found for each of the other unknowns. (Prop. II.)
(1a2). If there be n linear equations containing n +r unknowns
(where r may be 0), and if the unaugmented array be evanescent but
the augmented array be not, the equations are inconsistent: and, on
the other hand, if they be consistent, and have their unaugmented array
evanescent, their augmented array is also evanescent. (Prop. V., XV.)
(Ia3). If there be n linear equations containing n +r unknowns
(where r may be 0): and if there be n-k of the equations having their
unaugmented array not evanescent: and if when these n-k equations
are taken along with each of the remaining equations in succession, each
so formed set of n -k+l1 equations has its augmented array evanescent;
then (1) the equations are consistent: (2) if any non-evanescent primary