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

336 HISTORY OF THE THEORY OF DETERMINANTS

The analogous problem for even-degreed functions is next
taken up, a beginning being made with the transformation of
the quartic (a, b,., e ~ x, y)4 into the form
(p~X + q~y)4 + (P2X +q2Y)4 + 6e (plx + q1l)2 (P2X +q2Y)2.
On putting
=1 P11 P2 A=- q2X2, 2pp = i l +ki+X2 = — Si, M>~2 =2
there is obtained by equatement of like powers of x and y
b = pX4 + P2X4 + 6,xs
c Pi 4X12 + Ap4X2 2+ JUxs12 + 2ju82
d p14X14 + P2 4X/24+ 31M~s2
d Pi p4X14 + P2 4X24 + 6x1AS22
and from these by operations which lead to the elimination of
pi4' P2 4 from every consecutive triad of equations
as92 -bsl ~c -,A(8s2-2s312) =
bs2 - Cs, + d- A (492 - 12).91 =0 -cs2 -ds + e- x(8s2-2s912)82 = 0.
or, if we put vfor -/A (8s2 -2512),
as2 -bs + (C+ V) = 0
bs2 - (c - -) 81 + d = 0
(c + V)s2- ds1+ e = oJ
From the resulting cubic equation
a b C +v
v can be determined, and thence in backward order 81' 82;
In passing, note is taks-en of the fact that the said cubic when
arranged according to powers of is
V3 -(ae -4bl+ 3C2) V+ 2 b c d = 0,
cd e