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

ORTHOGONANTS (BORCHARDT, 1845)

299

generally that
(in) (i) (i)
gmx, = a21, X+ al2 x+ +.. a,,.
(min) (in) (mn)
X, =a,, )1x1 a,2 x2 +. +.. a+
where
Sl=n S2=n Sm-1=11
a(m) am> = aislas, a asm-1a8
ik k 2S9
S1=l 82=1 sm-1=1
In the next place, Y1, y2,..., gn being the roots of the resultant
of the initial set of equations, it is readily seen, from the
expression for the said resultant when arranged according to
descending powers of g, that
91 + 92 + + gn = all + a22 + *. ann,
Similarly, by considering the resultant of the second set of
equations we learn that
g + +2 +... + 1 - +a 22.+
and generally that
gm~gm+$ +gnm = a(m)+a(m)+... +a(m).
91 2 n 11 22 ) M
Consequently, if we use s8 to stand for the sum of the mth
powers of the g's we have
$ =n.a(m)
in / ii
i=1
In the third place the difference-product of the g's being
Borchardt has only to use the multiplication-theorem of determinants to obtain as an equivalent for the product of the
squared differences the determinant of the system
So 81 2 S
81 82 83 *.
82 83 84.. n+
Sn-1 8n1 8n~1 * 32? - 2