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

134 HISTORY OF THE THEORY OF DETERMINANTS
gives rise to the same result, it is easily seen how he reaches
the identity
1 Z2 y2 _
2I 2 (x+y+Z)(x+y-z-)(x-,y+Z)(-x+Y +Z).
1 Y2 x2
An alternative form of A~ is introduced by the words " Observons
*qu'on a e6videmment*
= 1- I X z =.x 'y z
x2y~z2 x. z2Xy y2XZ. z y
y Z2xy. yz y z.;Z y1XZ X2'yz. z y x
Similarly from the equation x + y + z + w=0 or any one of its,seven relatives by using the multipliers
1,zw, 'ywt, xw, ZY, zx. yx, rXyzw,and eliminating,
ctyz, ceyw, xzw, yzw, ce, y, z, w
there is obtained the result
I.. 1.z y2.
1. 1. z2
2 ~2 2 ~2* *
=(x+'y+Z+w) *(-xy+z++w)(e-'Y+Z+wV)(x+Y-Z+w)(c+y+z-wt)
It would seem preferable to multiply the columuns of A in order by xyz, x, y, z,
the quantities just eliminated.; and then divide the rows by the multipliers 1,
yz, zx, xy used in obtaining the set of equations. The advantage of this method
would be, still greater in the next ease.