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

348 HISTORY OF THE THEORY OF DETERMINANTS

Further, he points out that the individual members of this series
can be lowered in grade by the use of his condensation-theorem,
thus providing a variant of the series. He also notes that by means
of the theorem which we have extended above into another condensation-theorem they can be transformed into
1, ao2 so 1, CCo4 so 1 1..,
1 X S1 S2 X
S~ ~2 X
82 83 X2
and so he arrives by a different route at Joachimsthal's series of
1854 (Hist., ii. p. 171).
Lastly, it may be noted that in introducing (p. 234) the discriminant of a quantic as the resultant of the first differentialcoefficients of the quantic,* and in showing like Baltzer that for a
binary n-thic, O(x, y), the discriminant must be the resultant
(bigradient or other) of
e (x, y) = 0, n (x, y)-x (x, y) =0,
he puts 1 for y and asserts that the discriminant of +(x, y) must be
identical with the resultant got by eliminating x from
(0(x, 1) = 0, n(x, l) — x- (x, 1) = 0,
ax ox
and therefore must equal the resultant of
(x,1) = 0, (, 1) = 0;
and consequently must be equivalent to the expression which with
unfortunate ambiguity had for longt been known as the 'deter* So used from 1851 by Sylvester (Hist., ii. pp. 63, 388) and so defined in the
glossary at the end of his memoir on Syzygetic Relations (1853).
t In 1801 Gauss spoke of the determinant of a function of two or more variables,
b2 - 4ac being the determinant of ax2 + 2bxy + cy2 (Hist., i. pp. 64, 65); in 1815
he spoke of the determinant of a function of one variable, - 12 + 41" being the
determinant of x2 + 21'x 1" (Werke, iii. pp. 33-56, ~ 6). By 1852 we find in use
'the determinant of the equation ax2 + 2bxy + cy2 = 0' (Salmon's Higher Plane
Curves, pp. 296-7); and by 1857 the determinant of the equation f(x) = 0 (Hist.,
ii. p. 184).
All through this there was nothing implied in regard to the precise form of the
expression so named, which might be and indeed was, even with Gauss, a product
of squares of differences. When, however, ' determinant' as thus understood
came to be expressed in a particular form which also bore the name 'determinant,'
it was clearly imperative to make a change.