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

170 HISTORY OF THE THEORY OF DETERMINANTS
seven determinants with monomial elements, he shows next that
all of the twenty-seven except six vanish; that the six contain
the common factor
(X - xi) (X - X2) (X - X3B) - (X3 - XI) (X3 - X2) (X2 Xx,)
that the aggregate of the cofactors is
2 2 + dX2X _X2X2+X1X 2
3 2 3 1 2 1 23 13 12
or
(X3 - XI) (X3 - X2) (X2 - XI);;
and that therefore finally the given determinant is equal to
{(x3- x1) )(x-2) (X2- X)} ( - i) (X - X- X3).
This is followed by the assertion that if sq were made to stand
for Sq + sq + +s" the determinant could be partitioned into n3
determinants, of which itn(n - 1)(n - 2) would be non-evanescent;
and that these could be grouped into sets of six and condensed,
the ultimate result being
sS 8 8,
1 2 83 X. 1
91 ~2 S 2 f X0 - X2) (X3 ( X,)'(x2 Xi) - x)X X)( (x - X32)
82 83 84 X2 (jjX3XXX
83 84 85 X3
A large generalisation is then made, the exact words being
"Genan eben so beweist man folgenden allgemeinen Satz:
Bezeichnet man die Potenzsumme x+x +...~ +s' durch si;
ferner das Quadrat des Productes, welches aus den li(i -1)
Differeuzen der i Grdssen x1, x2,..., x gebildet ist, durch
8 (x, x,..., x), so ist
80 SI 82.... a-1
Si 82 83..3. Sa X
82 83 84.... 8a-1 2 2(XI, X0,,Xa> (X-XX)...(X Xa)
Sa 8a+1 8a+2....~ 82a-2 I;a