666 
ON THE EEAL AND IMAGINARY EOOTS OE EQUATIONS. 
CONTENTS. 
1. Proof (up to fifth degree inclusive) of Newton’s Eule for obtaining an inferior limit to the Arts. 
number of real roots in an equation 1 — 9 
2. Theory of the equation (l, s, s 2 , of, oj, l^.r, y) 5 and conjugate equations 10 — 14 
3. Theory of per-rotatory and trans-rotatory circulation 15 — 16 
4. On an inferior limit to the number of real roots in superlinear equations 17 — 20 
5. On the probable value of the above limit 21 — 30 
6. On the reduction of the general equation of the fifth degree to its canonical form 31 — 44 
7. Geometrical representation of the mutual limitations of the basic invariants of Quintic forms, 
and of the cause of the absence of the same for Quartic forms 45 — 54 
8. On the invariantive criteria for determining the nature of the roots of such equation 55 — 74 
9. On an endoscopic representation of the above criteria 75 — 83 
10. Geometrical determination of the arbitrary constant (limited) of the third criterion by means of 
one of the principal sections of the limiting surface of invariants 84 — 88 
11. On the forms of the other principal sections of the same 89 to end. 
Supplemental References. 
Proposed new reduced forms for binary quartics and ternary cubics (note n ). 
Theorem on the imaginary roots of odd-degreed equations (note 26 ). 
Concordance between Hermite’s invariants and those of the memoir (note 34 ). 
Identification of the latter with the corresponding numbered Tables of Professor Cayley (note 39 ( h ) and (*)). 
Proof that every invariant of a quintic is a rational integral function of the four basic invariants (note 3S ). 
Invariantive conditions for certain special forms of quintics (note 3 '). 
Conditions necessary in order that an infinitesimal variation of the coefficients of an equation may be accom- 
panied with a change of character in the roots (note 43 ). 
Schxapli’s theorem (proof and extension of) (note 52 ). 
On a number of cases capable of arising under Sturm’s theorem, and on certain questions of probability (note 61 ). 
All the invariants of a binary form vanish when more than half the roots are equal to one another, art. 48. 
Identification of section of limiting surface of invariants as a variety of the sixteenth species in Plucker’s enume- 
ration of quartic curves with two multiple points, art. 92. 
