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

LESS COMMON SPECIAL FORMS (HERMITE, 1854)

449

(/) DETERMINANTS WITH COMPLEX ELEMENTS.
HERMITE, C. (1854).
[Extrait d'une lettre.... sur le nombre des racines d'une equation
algdbrique comprises entre des limites donnees. Crelle's
Jowurn., lii. pp. 39-51; or (ECivres, i. pp. 397-414.]
On p. 40 it is pointed out that any determinant whose conjugate
elements are of the form a,8+b,.s/-1, (is- brssl-1, and whose
diagonal elements are therefore of the form a,., must be real, for
the reason that it is not altered in value by changing /-1
into -J-1.
HERMITE, C. (1855, August).
[Remarque sur un theoreme de M. Cauchy. Comptes rendus....
Acad. des Sci. (Paris), xli. pp. 181-183; or (Euvres, i.
pp. 479-481.]
The remark concerns the determinant just referred to, and is
to the effect that the equation
a1 -x a12 + b12i.... a + b,,i
c21 +b 2 a22-x.... cn +b2,,i
nl + bi aC2 + b7n2i.... Ca,, - x
where a,. = aC,. b.s= - b,., i= 1- 1, has all its roots real if the a's
and b's be real,-a result which degenerates into one previously
known (Lagrange, 1773; Cauchy, 1829) when all the b's vanish.
No proof is given, but it is stated that one is obtainable by
transforming "le determinant en un autre a elements reels, d'un
nombre double de colonnes et symetrique par rapport a la
diagonale." A rule is formulated for determining the number
of roots of the equation which lie between two limits. Lastly,
it is remarked that the equation arises in connection with the
study of forms of the type
x +x'i y+y'i
all a12 2- 12i - x'
a,2 + f12i a22 y - y'i,
M.D. II. 2F