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

BIGRADIENTS (TRUDI, 1862)

335

as multiplicand, we at once find the product to be
Co C1 C2 C3
C C1 C2 C3
C C1 C2 C3... &bo b1 b2 b3 b4. bo b1 b2 b3 b4. bo b2 b6 3 b4
b0 b1 b2 b3 b4 *
which is equal to
CO C1 C2 C3
CO C1 C2 C3.
_-b6o2, c0 cC c2 C3 O= - o2
bo b0 b b3 b4
' b0o b1 ba b3 b4
as was to be proved.
The use to which this second theorem is put (pp. 132-137) is in
connection with the division-process for finding the highest-commondivisor of two integral functions, and, in particular, with the
modification of the said process employed by Sturm in obtaining
his so-called " remainders." From the general theorem* connecting
dividend, divisor, quotient, and remainder we know that the coefficients of the first remainder in such a process are proportional to
the successive determinants of a bigradient array composed of the
coefficients of the dividend and divisor. We thus also know that,
this remainder having been made the divisor and the previous
divisor the dividend, the new remainder must be expressible in
like fashion. In the second bigradient array thus arising, however,
one of the two sets of elements is complicated, being in fact the
successive determinants of the previous array: and what Trudi's
"dilatation "-theorem enables us to do is to supplant it by another
array whose elements are simply the coefficients of the original
functions. In this way the theorem finally reached is: The coefficients of the rth remainder Rr arising in the course of the performance
of Sturm's division-process on
aox~' +... + a, c box" +.. + b,
* See above under Recurrents, Trudi (1862).