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

ALTERNANTS (GORDAN, 1873)

151

If a,,3, y be common roots of the two quartic equations
ax4 -+ aX3 + a3x2 + abx + a5 = O }
blx+ b2x +bx2 - +bX -x+b5 = 0J
it at once follows that we have two arrays
a4 a3 (12 a 1 ( a(2 3 CL4 a,4 33 32 p 3 b1 2 b3 b)4 b5
74 73 y2 1,
satisfying the conditions of the said theorem, and that therefore the
ten alternants formable from the array on the left are proportional to
la4b51, -lab5,I...., ab2
Similarly taking four roots all belonging to one quartic equation
we should obtain the old result that the five alternants of the array
l a a2 a3 a4
1 /3 32 /33 /34
T 2 '3 V4
1 y y2 y3 yV4
1 8 82 o3 o4
are proportional to
a, -a2, a3, -a4, a5,
and therefore to
1, la, a3`, Ear3y, a3-yS.
In the next place, by taking the same quartic and using it thrice, the
swo arrays would be
a6 a5 a4 a a a2 a 1 1 a 2 a3 a4 a5
Y6 75 Y4 y3 y2 y 1 al a2 a3 44 a.
76 7y5 74 73 72 1 a1 a2 a3 a4 a5.
86 5 84 83 82 1,
and we should have, for example,
a, a. a4 a1
aiy2 i 1 - a2 a3 a a1 a,
a a4. a2 a3
a2 a4
al al 1 -a
a= a, 3 a5 = a _a(3 afyd
a1 a.1 1 aaf -a l y
a3 a4
a, a,