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

344 H1STORY OF THE THEORY OF DETERMINANTS

or, in their usual form,
aq x2 + ar x + as = 0
aqx3 + (ar-+bq)x2 + (as+ br)x + bs = O
arx3 -- (as +-br)x2 + (bs +cr-dq)x + (cs-eq) = O
asx3 + bsx2 + (cs-eq)x + (ds-er) =.
We then have for the simplified remainders of the 1lt and 0th degrees
the determinants
aq ar xx + as
aq ar + bq (as- +br). x + bs
ar as+br (bs+-cr —dq) x + (cs-eq),
aq ar as
aq ar +-bq as+-br bs
ar as + br bs -+cr-dq cs-eq
as bs cs-eq ds-er
being only careful to note that both of these contain the irrelevant
factor a2. Trudi, however, does not point out that this factor
would not have troubled us if we had noted at the outset that for
the first two derived equations we might have substituted
qx2 -rx + s = 0,
qx3 + rx2 + sx = 0,
thus using Sylvester's method of derivation for the first m- n
equations and Bezout's for the remaining n, as Cauchy had shown
in 1840.
The case where B is the derivate of A receives special attention
(pp. 152-160), the object of course being to show that the quantities
D,., U,, V,. are then expressible in terms of sums of like powers of
the roots of the equation A = 0. The reason for the possibility of
this transformation lies in the fact that the coefficients
bo, bl, b2,...., bnare then equal to
aoso, a0s1 + aso, as2 + als+ + a a2s,...., asn-1 +... a-nS,