46 
JAMES JOSEPH SYLVESTER. 
[900 
many of these, “ beautiful exceedingly,” which, for their own sakes, one is tempted 
to refer to—or one can give titles, which, to those familiar with the memoirs 
themselves, will recall the rich stores of investigation and theory contained therein. 
A considerable number of papers, including some of the earliest ones, relate to 
the question of the reality of the roots of a numerical equation: in the several 
connexions thereof with Sturm’s theorem, Newton’s rule for the number of imaginary 
roots, and the theory of invariants. Sylvester obtained for the Sturmian functions, 
divested of square factors, or say for the reduced Sturmian functions, singularly elegant 
expressions in terms of the roots, viz. these were 
/2 (x) = '2(a — b) 2 (x — c)(x — d)..., f 3 (sc) = 2 (a — b) 2 (a — c) 2 (b — c) 2 (x — d) ..., &c.; 
but not only this: applying the Sturmian process of the greatest common measure 
(not to f (x), f(x), but instead) to two independent functions f (x), cjs (x), he obtained 
for the several resulting functions expressions involving products of differences between 
the roots of the one and the other equation, f (x) = 0, </> (x) = 0; the question then 
arose, what is the meaning of these functions ? The answer is given by his theory 
of intercalations: they are signaletic functions, indicating in what manner (when the 
real roots of the two equations are arranged in order of magnitude) the roots of 
the one equation are intercalated among those of the other. The investigations in 
regard to Newton’s rule (not previously demonstrated) are very important and 
valuable: the principle of Sturm’s demonstration is applied to this wholly different 
question: viz. x is made to vary continuously, and the consequent gain or loss of 
changes of sign is inquired into. The third question is that of the determination of 
the character of the roots of a quintic equation by means of invariants. In con 
nexion with it, we have the noteworthy idea of facultative points; viz. treating as 
the coordinates of a point in w-dimensional space those functions of the coefficients 
which serve as criteria for the reality of the roots, a point is facultative or non- 
facultative according as there is, or is not, corresponding thereto any equation with 
real coefficients: the determination of the characters of the roots depends (and, it 
would seem, depends only) on the bounding surface or surfaces of the facultative 
regions, and on a surface depending on the discriminant. Relating to these theories 
there are two elaborate memoirs, “ On the Syzygetic Relations &c.” and “ Algebraical 
Researches &c.” in the Philosophical Transactions for the years 1853 and 1864 
respectively; but as regards Newton’s rule later papers must also be consulted. 
In the years 1851—54, we have various papers on homogeneous functions, the 
calculus of forms, &c. (Camb. and Bub. Math. Journal, vols. vi. to ix.), and the 
separate work “On Canonical Forms” (London, 1851). These contain crowds of ideas, 
embodied in the new words, cogredient, contragredient, concomitant, covariant, contra- 
variant, invariant, emanant, combinant, commutant, canonical form, plexus, &c., ranging 
over and vastly extending the then so-called theories of linear transformations and 
hyperdeterminants. In particular, we have the introduction into the theory of the 
very important idea of continuous or infinitesimal variation: say that a function, 
which (whatever are the values of the parameters on which it depends) is invariant 
for an infinitesimal change of the parameters, is absolutely invariant.