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

ORTIIOGONANTS (CAUCHY, 1855)

285

'11 - X '12.... ln
021 0'22 - X.... T2n =0
0Cll. fl2.... o-nn X
will have: all its roots real if aro. be any integral function of $ whose
form depends on r and s, and which is not altered by the interchange of r and s. One of the cases in which the -'s are linear
is taken for special comment; it should be distinguished, however,
from Sylvester's extension of 1853 (Hist., ii. p. 314).
HESSE, 0. (1861, 1869).
[VORLESUNGEN UBER ANALYTISCHE GEOMETRIE DES RAUMES,
insbesondere fiber Oberflachen zweiter Ordnung. xv+368 pp.
(2te Aufi., xvi+456 pp.) Leipzig.]
As has already been implied, there is more of Algebra in these
lectures than is usual in a book on Solid Geometry, and naturally
the algebra of orthogonal transformation in the case of three
variables receives considerable attention. A series of excerpts from
five or six of the chapters would provide an excellent exposition
of the subject.
CLEBSCH, A. (1861).
[Ueber eine Classe von Gleichungen welche nur reellen Wurzeln
besitzen. Crelle's Journ., Ixii. pp. 232-245.]
Taking the theorem (Hist., ii. pp. 449-451) regarding the reality
of the roots of the equation
al +X act1+ b19/-1 a13+ b13N-1..
(a2 -b12 - (a22 + a23+ b23./-1 0
a13 - b3 J- 1 a23- b23- 1 a~ 3................................... n
where the a's and b's are real, Clebsch makes the a's vanish, and
thence draws the conclusion that the roots of the equation. b12 b3....
-- 12 2..3...
-bl3 — b23 /.* *......................... n