AND IMAGINARY ROOTS OF EQUATIONS. 
581 
have investigated and obtained a demonstration of its truth as far as the fifth degree 
inclusive, which, although presenting only a small instalment of the desired result, I am 
induced to offer for insertion in the Transactions in the hope of exciting renewed atten- 
tion to a subject so intimately bound up with the fundamental principles of algebra. 
Before commencing the inquiry I ought to state that, in addition to the rule for 
detecting the existence of a certain number of imaginary roots, Newton has given a 
remarkable subsidiary method for dividing this number into two parts, representing 
respectively how many of the positive and how many of the negative roots indicated by 
Descartes’s rule are, so to say, absorbed, and thereby obtains two distinct limits to the 
number of positive and the number of negative roots separately : of the grounds of this 
method, as far as I am aware, no one has even attempted an explanation, nor do I pro- 
pose here to enter upon it ; the rule, as I treat it, may be stated, not in Newton’s own 
words, but most simply as follows : — 
If the literal parts of the coefficients of an equation affected with the usual binomial 
coefficients be a, b, c, d, e . . . h, k, 1, and if we form the successive criteria b 2 — ac ; c 2 — bd ; 
d 2 — ce ; . . . ; k 2 — hi, or, which is the same thing differently expressed , if we write down 
the determinants (f) of all the successive quadratic derivatives of the given equation , then 
as many sequences as there are of negative signs in the arithmetical values of these criteria , 
so many pairs of imaginary roots at least there will be in the given equation. If we 
choose to consider a 2 and Z 2 also as criteria, appearing at the beginning and end of the 
series, then we may vary the expression of the rule by saying that there will be at least 
as many imaginary roots as there are variations of sign in the complete series so formed. 
It will, however, be found more convenient for our present purpose to confine the 
designation of criteria to the determinants above alluded to. 
(3) I shall deal with the homogeneous equation f(x, y) — 0 so that the question of the 
reality of the roots is that of the reality of the ratios - or It is obvious, from known 
y x 
principles, that f cannot have fewer imaginary roots than exist in j^f or ^/( 6 ), or, more 
generally, than in (J^ J r^'^'jf> from which it immediately follows ( 7 ) that if f have all its 
roots real, and the quadratic derivatives off be called Q, , Q 2 , Q„_ 1? and the coeffi- 
( 5 ) To avoid the possibility of misapprehension, I state here once for all, that in the discriminant of a form of 
any degree I suppose the sign to he so taken as to render positive the term which is a power of the product of 
the first and last coefficients ; and it may be well to remember that with this definition the number of real roots 
in any equation ^0 or 1 to modulus 4 when the discriminant is positive, and =2 or 3 when the discriminant 
is negative ; whereas the Determinant of a Quadratic form is to be taken in the same sense as that in which 
it is used by Gauss, and is the same for such form as the Discriminant with the sign changed. 
( 6 ) This rule I find merges in the following more general and symmetrical one. Let /, (p be any two quan- 
tics in x, y ; call the Jacobian of/, p J; then the difference between the number of real roots in /and the like 
number in <p, taken positively and augmented by unity, cannot exceed the number of real roots in J. "When <p 
is made equal to y, this theorem recurs to the familiar one alluded to in the text. 
( 7 ) By operating upon / successively with any (n— 2) distinct factors each of the form 
( d L d_\ 
Xjdoe^^'dy) 
4 i 2 
