328 THIRD REPORT — 1833. 



a collection of conditions of the reality of the roots of an equa- 

 tion of n dimensions which are ^ in number *. 



These speculations of De Gua were well calculated to show 

 the importance of examining the succession of signs of these 

 derivative equations, with a view to the discovery of their con- 

 nexion with the nature of the roots of the primitive equation. 

 The changes in the succession of signs of the coefficients of the 

 equations which resulted from the substitution of .r + « and 

 X + b, gave no certain indications of the nature and number of 

 the roots included between a and b, unless it could be shown 

 that all the roots of the primitive equation were real, a case of 

 comparatively rare occurrence, and which left the general pro- 

 blem of the separation of the roots, as preparatory to their 

 actual calculation, nearly untouched. It was the conviction that 

 all attempts to effect the solution of this problem by the aid of 

 Des Cartes's theorem would necessarily fail, which led Fourier, 

 one of the most profound and philosophical writers on analysis 

 and physical science in modern times, to the examination of the 



* Resolution des Equations Numiriques, Note viii. The equation of the 

 squares of the difFereaces of the roots of an equation will indicate the reality 



of all the roots, if its coefficients have ^ changes of sign, or be alter- 

 nately positive and negative. The succession of signs of the coefficients very 

 readily furnishes the indications of the number of impossible roots in all equa- 

 tions as far as five dimensions, as has been shown by Waring and Lagrange. 



The number of conditions of the reality of the roots of an equation of five 

 dimensions would appear from the formula in the text to be 10 ; but some of 

 these conditions, as liagrange has intimated, may, and indeed are, included 

 in the system of the others, so as to reduce them to a smaller number. La- 

 grange has assigned two conditions (not three) of the reality of the roots of 

 a cubic equation ; but the first of these is necessarily included in that of the 

 second, so as to reduce the essential conditions to one. Similar consequences 

 are found to present themselves in the examination of these conditions for 

 an equation of the fourth degree, which are three in number, and not six, as 

 the formula would appear to indicate. 



Cauchy, in the 17th cahier of the Journal de I'Ecole Poly technique, has suc- 

 ceeded, by a combined examination of the geometrical properties of the curve 

 whose equation is y = X (where X is a rational function of x of the form 

 .i;» -|- jB, a'»-i -|- . . . . pri)y and of their corresponding analytical charac- 

 ters, in the discovery of general methods, not merely for the determination of 

 the number of real roots, but likewise of the number of positive and negative 

 roots, as distinguished from each other. These methods are equally appli- 

 cable to literal and numerical equations. He has applied his method to ge- 

 neral equations of the first five degrees, and the results are in every respect, 

 as far at least as they have been examined in common, equivalent to those 

 which are derived from the equation of the squares of the differences. It is 

 impossible, however, in the space which is allowed to me in this Report, to 

 give any intelligible account of this most elaborate and able memoir, and I 

 must content myself, therefore, with this general reference to it. 



