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

110 HISTORY OF THE THEORY OF DETERMINANTS

(Xy)12 (XY)13 (XY)14 1
(xy)21 (XY)23 (Xy)24 1
(XY)31 (XY)32 (XY)34 1
(XY)41 (XY)42 (XY)43
(XYZ)12 (XXjYZ3 (XYz)14 (XYZ)15 1
(xyz)21 (xyz)3 (xyZ)24 (xyZ)25 1
(xyz))1 (XYZ)32. (XYz)34 (Xyz)35 1
(x~Yz)41 (XYzXZ (ceyz)43 (XYz)4 1
(XYZ)51 (xyz)52 (XYV)53 (XYZ)54. 1
1 1 1 1 1
if we put
(x),.8 for (X,. x. 2
(xy),S for (x,. -x,)2 + y(y,- ys)2,
(xyz),.s for (X, - X8)2 ~ (y,. ys)2 + (Z -)2,
The fact that they are identically equal to zero is established by
showing that each one is resolvable into two factors, of which
one or both vanish; for example, that the first is equal to
x2 -2x1 1 1 1
x2 _-2x2 X1 2
X2 - x
2 2'
x32 - 2x3 1 Xr x3 x
1...... 1,
where, for the moment, we use x.' to indicate row-by-row
multiplication.
It should be noted that not more than three of the series are
contemplated by Cayley, as he viewed the identities mainly on
their geometrical side, namely, as.giving the relation between the
distances of three points in a straight line, four points in a plane,
and five points in three-dimensional space.*
* To one taking this point of view Sylvester's paper " On Standt's theorems...."
will be of interest. See Philos. Mllagazine, iv. (1852), pp. 335-345; or Collected
Math. Papers, i. pp. 382-391.