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

ORTHOG.ONANTS (SPOTTISWOODE, 1853)

315

records, and which he says can be made from Cayley's result.
These may be enunciated in more modern form as follows
If Ia1 a22... a,,1I or A be a unit-axica skew determinant,
IA, A22... A,,I its adjugate, and I W022....n,, CGayley's
orthogonant formed therefrom, then
Al, + Ar+ A, Arr.. All=, } ()
AlrAis + AsrA2s +... + A,,rAs = '(A s+A().
and
CtirCCir + a2re)2r +.. a ~ rn a(rr
airwi. + a2ri)2s + + a1nrwns = ars J
The former, (a), belongs strictly to tie theory of skew determinants, as has already been mentioned in the proper place.
CAYLEY, A. (1853, November).
[On the homographic transformation of a surface of the second
order into itself. Philos. Magazine, vi. pp. 326-333; or
Collected Math. Papers, ii. pp. 105-112.]
Here Cayley recalculates the general orthogonant of the 4th
order, taking note in passing of the related identity
aCe - b~y - eZ +w)2
~ (x + vy -,uz + aw)2 + ( - VX + y + XZ + bw)2 + (fUX - NY + Z + eW)2
X2+ Y2+Z2+W2+ (ax e+by +cZ)2
+ (yy - uz + aw)2 + ( _ VX XZ + zbw)2 + (QX - Xy + Cw)2.
We may add that if the last eight squares be subtracted from
both sides of this there remains on the left-hand side a quadric
having a zero-axial discriminant.
BRIOSCHI, F. (1854, March).
[LA. TRORICA DEi DETERMINANTI, E LE SUE PRINCIPALJ APPLICAZIONI. viii +116 pp., Pavia.]
In Brioschi's text-book, the paragraphs dealing with' a
"sostituzione ortogonale" are somewhat scattered, most of them
appearing among the applications (pp. 24-26, 47-51, 62-69).