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

SKEW DETERMINANTS (CREMONA, 1864)

291

and since Ar is got from Asr by altering the signs of all the
(n_-1)2 elements and then changing -z into z, there results
when n is even,
As + Ars = 201z + 233 +....;
in other words, Asr+ As is then divisible by 2z.
Two " observations" are added, the first in regard to the case
where z= 0, and the second in regard to an alternative proof of
the first part of the foregoing. The latter is interesting in that
the expression for (As,+As,) A is not found at once as a whole,
but is viewed as consisting of two parts corresponding to AsrA
and ArsA, the reason being the known existence* of a general
theorem of determinants to the effect that if the product of
al....Inn I and b... bbn obtained in row-by-row fashion,
be Ill... Cnn\, then
As.* bl.... bnn = b1,sCr,+... +bCrn.
This is seen to be immediately applicable on making a,,... ann I
identical with A above and the b's identical with the a's; and it,
of course, implies that if the product obtained in column-by
column fashion be | c... c', then
Asr b|l.... bn I bsClr +.. bsnC'nr.
Making the said necessary specialisations and noting that the
two differently formed axisymmetric products are then identical
(in other words, that Cs = Csr, = C = C'), Torelli obtains
i=n i=n i==
ArsA = ajs1j;A,.?A1+. +ass iAsi+ + +a nsjA A,
i=l i==l i=l
AcrA = asi ArAi+! * * 4-szArAi+ E.. +c(JsAriAni;
i=1 i=1 i=1
whence by addition
i=n
(A,,8+As)A = 2assyAxAs.
i=l
* Rubini's Elementi d'Algebra, p. 277, is referred to.