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

BIGRADIENTS (TRUDI, 1862)

343

and, as by definition,
YI, a aB-bbAI
2= (aox+a1)B-(box+b,)A,
3= (aox2~alx+a2)B-(b OX2+1lx-Fb2) A,
it follows that the said remainders have still another form,
namely,
a0B - bOA,

c11 ao B - c11 bo A,
1 02, aox +a, c2, C +b,cP+b
CI "~2 ao 'B - c11 C12 bo
021 022 a 1x +a1 c21 c21 bC22 +
031 032 aOx22a1x 4-a2 C31 032 b0X22b1~i-b2

A,

-a result easily shown, by the use of the condensation-theorem, to
be in agreement with a previous one in which the determinant
coefficients of A and B are bigradients. He is also careful to note
that although here, as usual, n is taken equal to qn, no real restriction
is thereby made, the case where m > n being viewable as a case in
which the coefficients of vfl+l, l+2,..., xm in B are equal to 0.
For example, if the given equations be
ax4 4-bX3 +x2+dx+e = 0)
qX2+rx+s = of,

Bezout's derived equations (although not in Be
notation) are
a bX3 4-ox2 X2dx4-e'
qx24- rX[s - 0,
ax-F-b ox2 A-dx-e~ -0,
qX2 + r X + S
ax2 4-bx 4-c dx4- _
q1 rxs- s
aX3-bX24-cx4d e -
qx +r st

zout's nor Trudi's