314 Proceedings of the Royal Society of Edinburgh. [Sess. 
08 ) DETERMINANTS WITH COMPLEX ELEMENTS. 
Hermite, Ch. (1854). 
[Extrait dune lettre .... sur le nombre des racines dune equation 
algebrique comprises entre des limites donnees. Grelles Journ., 
lii. pp. 39-51 : or CEuvres, T. i. pp. 397-414.] 
On p. 40 it is pointed out that any determinant whose conjugate 
elements are of the form a rs + b rs J^ 1, a rs — b rs J — 1, and whose diagonal 
elements are therefore of the form a rr , must be real, for the reason that it 
is not altered in value by changing J — l into — J — 1. 
Hermite, Ch. (1855, August). 
[Remarque sur un theoreme de M. Cauchy. Gomptes rendus . . . . 
Acad, des Sci, (Paris), xli. pp. 181-183: or CEuvres , T. i. pp. 479-481.] 
The remark concerns the determinant just referred to, and is to the 
effect that the equation 
a n ~ x 
“t b x d 
• • . a in + b ln i 
a 2\ 
a 22 -x . 
, . . . a 2n + b 2n i 
a nX + b nl i 
<^n2 b n d 
. . . . a nn -x 
where a rs = a sr , b rs = — b sr , i= J — l, 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 deter- 
minant en un autre a elements reels, dun 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 + xi y + y'i 
Otjj ®12 X X 
a i2 -t a 22 y ~ y C 
a n aj 2 + 2 a 12 xy + a 22 y 2 ) 
+ a n «' 2 + 2 a 12 xy' + a 22 y 2 ) 12 
xy' . 
that is to say, 
