AND IMAGINARY EOOTS OF EQUATIONS. 
645 
When A is negative, 25 A 3 — 3.2 10 J 3 negative and J 3 positive, no roots are imaginary. 
A is negative, 25 A 3 — 3. 2 10 J 3 positive, 25 A 3 — 2 n J 3 negative, no roots are imaginary. 
A is positive, four roots are imaginary. 
A is negative, 25 A 3 — 3.2 10 J 3 positive, 25 A 3 — 2 n J 3 positive, four roots are imagi- 
nary( 59 ). 
(71) What is the effect of the condition “ A positive or negative ,” as the case may 
be 1 ? or rather, how does this condition arise 1 ? The ground of it is simply this, that A=0 
represents a cylindrical surface passing through the curve OA (see dial figure), which 
curve is the edge of separation between two regions of opposite characters above the 
plane of D ; the cylinder in question cuts the facultative position of space below the 
plane of D, but above this plane (except along the vertical line J=0, L=0, i. e. the 
axis of D) it passes exclusively through non-facultative space, never again cutting or 
meeting the surface G (the facultative boundary). Now it is clear that any surface 
whatever which passes through OA and never meets the surface G above the plane 
D=0, except along the axis of D (i. e. the line J = 0, L=0), may be substituted for 
A( 60 ) and will serve equally well with A to distinguish between the masculine and femi- 
nine regions of space. A— g>JD will fulfil the condition of passing through the line OA, 
(69) (a) l as t four conditions ought totally (and he in effect coextensive) with the two given by me for 
the case of D positive. The third of them, viz. the case of D positive A positive, I have already noticed, as 
inferences from the dial figure ; for M. Hermite’s A, if not identical with my J, is at all events a positive mul- 
tiple of it. I do not see how the case of A negative, 25 A 3 — 3.2 10 J 3 negative with D positive, is met by this 
system of criteria, since J 3 , as well as A, may he negative consistently with the second condition. I have not 
been able to ascertain whether in the memoir such a combination is shown to be impossible. M. Hermite 
admits, and indeed has been always aware of, the existence of a lacuna in the conditions above stated, which, I 
understand from him, it is his intention at some future time to fill up, and thus to complete his original solution. 
In the meanwhile he has been led to study the question from a different point of view, and has succeeded in 
obtaining a new set of criteria adequate to a complete solution of the question without calling in the aid of the 
principle of continuity. In this new system my A criterion is replaced by an invariant of the twenty-fourth 
degree, which is of course an objection as far as it goes, but in no wise diminishes the extraordinary interest 
that attaches to this altered mode of approaching the question, which bears to his original method and my own 
the same relation as the proof of Sturm’s theorem by the law of inertia for quadratic forms bears to that given 
by Sturm himself. 
( b ) It is apparentfrom the fact that when D=0, G (M. Hermite’s I 2 ) becomes (25A 3 — 3.2 10 J 3 )(25A 3 — 2 U J 3 ) 2 
(Camb. and Dub. Journal, vol. ix. p. 206), that the factors of this product are respectively of the form aA'-f&JD, 
cA+eJD, a, b, c, e being certain numerical quantities. This gives rise to a singular reflection, to wit, that my 
own criteria for the case of D positive may be varied by the addition of a term ADJ to A (X being a numerical 
coefficient), provided A lies within certain limits, the form of the criteria in all other respects remaining un- 
changed. This proposition, fraught with the most important consequences, and not unlikely to lead to an 
entire revolution in the mode of attacking the general problem of criteria, I proceed to establish in the text. 
( 60 ) The surface to be employed will be A — f JD, which call M. A and M (or at least their upper portions above 
the plane of D) may then be regarded as the two sides of a sack, of infinite dimensions, open at the top, and 
seamed together at the bottom, along the curved line D=0, A=0, and in the vertical direction along the 
straight line J=0, L=0. The surface A serving as a screen of separation between the two upper regions, it is 
"clear that M will serve equally well as such screen, provided no superior facultative points lie in the interior 
of the sack. 
4 R 2 
