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

HESSIANS (SYLVESTER, 1878)

371

(r = 1) of Brioschi's result of 1854 (Hist., ii. pp. 394-396), which
is the subject of' question 6756.'
Should u have p roots each equal to a, we have only to substitute
(x- a)Pv for u, when it is at once found that the Hessian must have
2(p —) roots equal to a, as Sharp states (6077).
SYLVESTER, J. J. (1879).
[Question 6154. Educ. Times, xxxii. p. 341; or Math. from Educ.
Times, xxxiv. pp. 108-109.]
When correctly stated the theorem here given is that if the roots
of a binary quantic be all real and different, the roots of its Hessian
will be all imaginary, and the proofs rest of course on the fact that the
Hessian being always negative never changes its sign for any real
value of x/y. What is more worth noting, however, is the fact on
which one of the proofs is based, namely, that if al, a2,.., a,
be the roots of a binary quantic, its Hessian is an arithmetical
multiple of
-E(ax-a2)a2(X- C a3 )2.. a (X-a,) 2.
From this also, or from the result 6077 above, there comes the
generalisation that if m of the roots of the quantic be identical and
all the rest different, the number of imaginary roots in the Hessian
is 2(n-m-1).