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

482 HISTORY OF THE THEORY OF DETERMINANTS

The replacement is justified by the fact that the first array when
multiplied columnwise by I xy2 13 in the form
xi2 Xly1 Y/2
2xx2 1y2 +x2y1 ' 2Y1Y2
X22 X2y2 'y22
gives a multiple of the second array by m(m-1)2. A consequence
of this, however, is that there is necessitated the use of properties
of the s-operator, for example,
(uv) = U v + v Su,

8x1y21 = 0,
( uv22W3 =; 51 SU 2 6S3 + %1 ~ 3 +
VU1 2 V3 6V1 6V2 6v3
WI W2 W3 W1 W2 W3
and the use of another transforming identity, namely,
f1l11 f"21 f3111 fi4 f2 f3 f4
fi112 f112 f3112 f4112 fi f sf3
f J2 /S /4.|/y,., 16 e fl Jf2 e3 6
fi122 f2122.3122 f4122 I X12 fi - f2 2f3 2f4
fi222 8 4^f2222 f3225 f2 3fl 82f2 1s23 32f4
f222222222 f 222 f 222 /1 8f f 83f4

Ul,
V1
6w1

U2
V2
6w2

U3
V3
6W3 ~

m2 (m- 1)2(m — 2)3

or, say,
R(fl, f2f3, f4). - i 22 16 = (m (- 1)2 ( — 2)3,
which is established by taking the multiplier I x1x2 6 in the form
x x13 Xlyl2 Y13
3x2x2 x12y2 + 2x1x2y y12x2 + 22yly2xl 3y12y2
3x1x22 x22y1 + 2x1x2y2 y22x + 2Y1Y2X2 3Y1Y2 2
X23 x22y2 X2J/22 Y23
as was shown by Schlafii (1851) to be permissible (Hist., ii. pp. 52-53).
With this preliminary explanation we shall employ Rosanes'
method to find the ratio of,x to f.
The i's being derived from the O's as the O's from thef's, and the
degree of the s's being 3(m -2), we have by the initial transformation
I 1xY2 13i1 = -3(m-2)(3m-7)2. 02 03 04
6902 603 804
8202 8203 8201)4.