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

BIGRADIENTS (TRUDI, 1862)

349

minant' of +(x, 1). Nevertheless, he does not note that in this
there is ample justification for extending the use of the term 'discriminant' to the case of the rational integral function +(x, 1), and
even in connection with the equation +(x, 1) = 0.
Trudi's work on bigradients, extending to 94 pages if both Teoria
and Applicazioni be included, has suffered undeserved neglect.
Why this should have been the case it is a little difficult to understand, its only demerits being an occasional wordiness, a not very
acceptable notation, and a paucity of concrete examples. In his
preface (p.vii) he tells us that it was first communicated in a
number of papers to the Naples Academy of Sciences in the year
1857. This being so, it was two years in advance of Zeipel's memoir
on the same subject (Hist., ii. pp. 370-372) and Bruno's text-book,
a fact which it is important for the reader to recall if any small point
of similarity between two modes of treatment should attract
attention.
SALMON, G. (1866).
[LESSONS INTRODUCTORY TO THE MODERN HIGHER ALGEBRA.
2nd ed. viii+296. Dublin.]
In a table of resultants (pp. 283-285) the final expansion of R,,
is given, and the discriminant of
ax4 + bx y + cx2y2 + dxy3 + ey4.
SARDI, C. (1866): RAJOLA, L. (1866): TORELLI, G. (1866).
[Questione 47. Giornale di Mat., iv. pp. 239-240: solution by
L. Rajola, iv. p. 297.]
[Teorema sui determinanti a due scale, e soluzione della questione 47.
Giornale di Mat., iv. pp. 294-296.]
We have already seen how, from equating two forms of the
resultant of a pair of rational integral equations, interesting identities
may be obtained (Hist., i. p. 487 at bottom: ii. pp. 369-370, 374 -375). Another instance is here reached, the forms of eliminant
used being Sylvester's bigradient and the eliminant which arises
from successively substituting the roots of one of the equations in