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

122 HISTORY OF THE THEORY OF DETERMINANTS

X2 Y/2Z2 1 2 2 )2 2 1
Xa Y Z5 1 -3 1
X4 Y4 -4 e / 74 ~ 4
X$1 el Xl& 'X13 X1_e4 1
lx21 "XX2,* YX 25 Y-X2e4 1
Y XSel YXX3$2 Y2X3 3 YX3$4 1
lXX1 1X 6 1XYX3 Y1XX 1
where lxe, is put for x4e, + y,;7 ~ z Then performing on the
last determinant the operations which we may denote by
row - I Ex2. row., row i y 2. row5..
1 2 1 2
col - 2 * col5, colC - le2 C015,
he obtains
Y(X1- 1)2 ~(X1- d-2)2 ( e (X1- 4)21
'(X2- 1)2 l(X2-e2)2 (- 3)2 ( (X2 -)2 1
-~ ~(x3 —1)2 ~(X3-$)2 ~(X3-~5)2 ~(x3-~ ~)( 13
~(X4- ) (X4 ( _)2 l(X4 ) X(X4 4)2 1
1 1 1 1
so, that, if X,, y,, z. and, im, ~. be rectangular co-ordinates of
points in space, the result reached gives an expression for thirtysix times the product of the volumes of two tetrahedrons in
terms of the distances of the angular points of the one from the
angular points of the other. By proceeding to the case where
the two tetrahedrons are coincident, and thence to the case
where the four remaining points are situated in the same plane,
we reach Cayley's relation connecting the mutual distances of
four such points.
It is thus seen that whereas Cayley's vanishing axisymmetric
determinant was originally got as a multiple of a peculiarly
obtained square of the determinant
EX12 X1 yi 0 1
I22X2 Y2 0 1
Yx3 2 x3 J3 0 1
EX x2 g4 0 1
1 0 0 0 0'.