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

162 HISTORY OF THE THEORY OF DETERMINANTS
where R(...;...) indicates the product of the differences got by
subtracting each quantity on the left of the semicolon from each
quantity on the right, and where the Z betokens a sum of Cq+m,.
terms of which the first is given and the others differ from the first
merely in having another set of n of the a's instead of the first set.
For example, the resultant of
ax+b = 0 and cx2+dx+e = 0
is
(cau b + ) b)( (Cy2 +dy + (e) + (a + b)(ay + b) (cP32 + d/ + e)
(y-a)(y -3) (/3-a)(/3-y)
+(a/ + b)(ay +b) ' (a2 + da +e)
(._-,3) (a -Y)
As a mere help towards acceptance it may be pointed out that
the expression is manifestly of the 0tl' degree in each of the a's,
of the nth degree in the coefficients off(x), and of the mt' degree in
the coefficients of g(x).
There is herefrom suggested to Borchardt the finding of expressions of like kind for the other 'simplified remainders' whose
vanishing is the necessary and sufficient condition for the existence
of a common factor off(x), g(x); and, this being accomplished, he
uses his paper of 1860 to show how older expressions for these
'remainders,' such as Cayley's and Sylvester's, may be reproduced.
He then passes on to the case (~ 2) where f(x), g(x) are of the same
degree. Here he not only obtains his result in the same manner
as before, but, denoting
f(x) g(y) -f(y) g() by F(x, y),
y -- x
and recalling Bezout's 'abridged method,' he arrives at a second
and quite unlike expression. Equatement being thus possible, there
is evolved the very noteworthy identity in alternants
F(a1, a+t)... (a,,,.2,) I.(a. * ~ )a,,), ( ' ' * ' a2)
R(a,..., a,;,, ~ 1), ' 2~)
The remaining half of the paper is concerned with the related
question in continued fractions.