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

358 HISTORY OF THE THEORY OF DETERMINANTS
o= b,1xm - alb,xml-' bn
bn-2XM-a2b xm-2 b?1_1 b.
bn-3Xm a3bnXm_3 bn2- b,,-..am.x am-2 amx3-3b3am
am amix aM2x2 -b2am
am am.,x — bla,
where, after the first column of binomial elements, there follow m -2
columns having b, for their first non-zero element, and in front of
the last column of binomial elements there are n-2 columns having
am for their last non-zero element. By mere change of b,, bn,into 1/b,, b/b,,./.. there is readily obtained a companion equation
whose roots are quotients instead of products of the roots of the two
given equations: and then, on putting x in this equal to 1 there
emerges the eliminant of the said pair of equations. For example,
the given equations being
X3-a lx2+a X-a3 = 0,1
X3-blX2+b2X-b3 = 0,
the equation whose roots are the roots of these mated by multiplication is
b2x3 -a~b3x2 b3 x3
bJx3 —a2b3x b2 ax2 X3- b3a3
x3- a3b3 b, a2x alx2- b2a3
I a3 a~x-bia3
the corresponding equation whose roots are each mated by division is
b1x3 —a x2 1 X3
b2X3-a2x b, ajx2 b3x3 -a3
b3X3 -a3 b2 a2x aibb3X2 _a3b
b3 a3 a2b3x-a3b2
and the eliminant of the given equations is
b, - a, 1 1
b2 -a2 b1 a, b3-a3 =
b3 -a. b2 a2 a,b2-a2b,
b3 a3 a2b, -a3b2