412 
MR. SYLVESTER ON A THEORY OF THE CONJUGATE 
quadratic functions which are made use of in the sequel. Art. (28.) contains the proof 
of a law which, although of extreme simplicity, I do not remember to have seen, and with 
which I have not found that analysts are familiar : I mean the law of the constancy of 
signs (as regards the number of positive and negative signs) in any sum of positive and 
negative squares into which a given quadratic function admits of being transformed 
by substituting for the variables linear functions of the variables with real coeffi- 
cients. This constant number of positive signs which attaches to a quadratic func- 
tion under all its transformations, and which is a transcendental function of the 
coefficients invariable for real substitutions, may be termed conveniently inertia, 
until a better word be found. This inertia it is shown in art. (26.), by aid of a 
theorem identical with one formerly given by M. Cauchy, is measured by the 
number of combinations of sign in the series of determinants of which the first 
is the complete determinant of the function ; the second, the determinant when 
one variable is made zero ; the next, the determinant when another variable as 
well as the first is made zero, and so on, until all the variables are exhausted, 
and the determinant becomes positive unity. In art. (46.) I give some curious and 
interesting expressions for the residues and syzygetic multipliers, under the form 
of determinants communicated to me by M. Hermite ; and in art. (47-) I show how, 
by aid of the generating function which M. Hermite employs, and of the law of 
' inertia stated at the opening of the section, an instantaneous demonstration may be 
given of the applicability of my formulae for M. Sturm’s functions for discovering 
the number of real roots of fx, without any reference to the rule of common measure ; 
and moreover, that these formulae may be indefinitely varied, and give the generating 
function, out of which they may be evolved in its most general form. Had the law 
of inertia been familiar to mathematicians, this constructive and instantaneous method 
of finding formulae for determining the number of real roots within prescribed limits 
would, in all probability, have been discovered long ago, as an obvious consequence 
of such law. I then proceed in arts. (48.) and (49.), to inquire as to the nature of the 
indications afforded by the successive simplified residues to two general functions 
f and <p ; and I find that the succession of signs of these residues serves to determine 
the number of roots of/ or (p, comprised between given limits after all pairs of roots of 
either function, contained within the given limits, not separated by roots of the other 
function, have been removed, and the operation, if necessary, repeated toties quoties 
until no two roots of either function are left unseparated by roots of the other ; or in 
other words, until every root finally retained in one function is followed by a root of the 
other, or else by one of the assigned limits. The system of roots comprised between 
given limits thus reduced I call the effective scale of intercalations ; such a scale 
may begin with a root of the numerator or of the denominator of ^ ; and upon this 
and the relative magnitudes of the greatest root of (px and fx it will depend whether 
in the series of residues (among which fx and <px are for this purpose to be counted) 
