598 
PROFESSOR SYLVESTER ON THE REAL 
effect of the variable point whose coordinates are e 5 +?? 5 and lying within the area A, 
in that portion of it for which s, jj became imaginary ; viz. it is that in such case the 
equation (g, tj), which then becomes of a conjugate form, will have three simple and two 
twin roots ; and thus the unity of the interpretation is restored if we choose, as we very 
well may, to extend the use of these terms to the real roots and the paired imaginary 
roots of ordinary equations. We may neglect the curve of reality R altogether, and 
affirm that all over the area A, g, jj will have such values as will give rise to three simple 
and two coupled roots. 
(13) That part of the theorem of Newton which had received a demonstration from 
Maclaurin and Campbell in the generalized form in which I have enunciated it in this 
paper, may be easily extended to the case of conjugate equations. It will, as applied 
to them, read thus: If the (n — 1) quadratic derivatives of a conjugate form of the nth 
degree, all whose roots are simple, be multiplied respectively by the coefficients of any 
other conjugate form, all whose roots are also simple , of the degree (n— 2), and the sum 
of these products be taken as a new quadratic form, the discriminant of this latter must 
be positive, or, which is the same thing, its determinant must be negative. 
(14) So much for the case of n=5. If we were to proceed to the consideration of equa- 
tions of the 6th degree, two cases of resistance would present themselves in the demon- 
stration of Newton’s rule, viz. one in which the signs of the criteria are f- + H — , 
the other — | 1 — . In the latter it would only be necessary to show that the 
discriminant is necessarily negative, since we know from the derivatives that the equa- 
tion must have four imaginary roots, and the choice would lie between the alternatives 
of there being four or six. In the former case the derivatives only indicate the neces- 
sary existence of two real roots, and it would become requisite to prove that there must 
be four or six — an alternative which depends not on the sign of one function of the 
coefficients, but on the nature of the signs of two such functions given by Sturm’s or 
any equivalent theorem. It would thus become requisite to prove that two functions 
of the coefficients, say L, M, could not both be negative ; and this might be shown by 
demonstrating the existence of two quantities, L', M', other functions of the coefficients 
incapable of assuming any but the positive sign such that L'L-j-M'M would be necessarily 
positive. 
Past II.— ON THE LIMIT TO THE NUMBER OE REAL ROOTS IN EQUATIONS 
OE THE FORM 2(ax+b)". 
(15) I shall now proceed to the consideration of a theorem relating to a particular 
class of ordinary equations, which occurred to me in the course of and in connexion 
with the preceding investigations. The theorem itself, but unaccompanied by proof, has 
appeared in the ‘ Comptes Rendus’ of the Academy for the month of March 1864. 
Both as regards its nature and the processes involved in the proof, it stands in close 
relation to Newton’s rule, my study of which in fact led me to its discovery. It will 
therefore take its place most appropriately in this paper. 
