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

204 HISTORY OF THE THEORY OF DETERMINANTS
respectively. His procedure is similar to that followed by Scholtz*
in dealing with the corresponding theorem regarding I XI2Z5 14,that is to say, he multiplies the ten-line determinant by
I Y2Z3W4 I Y37P~l Y4I YI y1zw0 i
reduces the product to the form
Y1Y2 Z1Z2 W1W2 Y1Z.2 +Y2Z1 Y1i02 +YWW1 Z1W2 +Z2W1
Y1Y3 Z1Z3 W1W3 YZI3 +YJ3Z1 Zf1W3 +Y3Wi Z1W3 +Z I U
II234 15YIY4
Y2Y4
Y32Y4...
the six-line determinant of which he says is equal to
- I Y2z3w4 W I Y3z4q1 I I Y4zlw02 I ' I Y1Z26VW0
He thus obtains
S=1 12341I5
as desired.
Taking then the determinant S derived from S by substituting
for each element its cofactor in 11234 f, Hunyady multiplies D by
it, and obtains after reduction
DS = 11234 I ~E,
which, since S is equal to 1 1234 J3X, gives the relation betweeu
D and E to be
E = 12: 34 1I7.D.
Similarly, with the help of other lemmas, it is shown that
F = { 112451I113461I123561 - I 12561-113451L123461 i D,
G; =- 123451L134511'145121L151231L112341 -D.
The fifth and sixth forms of eliminant are also satisfactorily accountec
for, being shown, like the others, to be multiples of D, namely, the
one equal to
— 178901 F,
and the other equal to
-178901 18906V1 90671 106781 167891 -G.
" See above under General Determinants.