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

ORTHOGONANTS (MANSION, 1877)

299

and Mansion's observation is that they are connected by the
relation
4x = I aCb2c3
This we have already seen suggested in connection with a result of
Spottiswoode's (Hist., ii. p. 308).
VOSS, A. (1877).
[Zur Theorie der orthogonalen Substitutionen. Math. Annalen,
xiii. pp. 320-374.]
It is only the first two sections (pp. 322-333) of this long paper
which strictly concern orthogonants.
A partial revise is therein given of their properties, the definition
being widened by starting with the condition that the substitution
bY=,1 Xl +a,.I-. + + *+ ArtX}is to be such that
y 2+Y22+.. + y2 S(X12+22+... +X 2).
In this way the square of each of the rows of | a1n is equal to s
instead of 1: the square of each of the columns also is s: the square
of the orthogonant as thus defined is sn: and the words 'proper'
and 'improper' as applied to a substitution correspond to the
values + J/S, -/Sn of the orthogonant.
This widening, it may be remarked, is not essentially different
from that proposed by Sylvester (see above, p. 288). For it can
readily be shown that any element of the adjugate of Voss' | an, I
is equal to the corresponding element of a,\ I itself multiplied
by s52-~, and that therefore the ratio of the two elements is
constant,-which is Sylvester's criterion.
Lagrange's determinantal equation is reached through the somewhat loose statement that "the elements x,., which by the substitution pass over into themselves, are determined by the equations
pXr = arlXl + aX2+.... +c airXn },
that is to say, by the roots of the equation
all- P (X12.... = 0.7
a21 a22- P.
~.........