326 THIRD REPORT — 1833. 



greater than n(n-i) :, if n denote the dimensions of the 



{2k)~ 



equation ; and in as much as H is necessarily, when the coeffi- 

 cients are whole numbers, either equal to or greater than 1, it 



1 

 ■will follow that nc-i) , will furnish a proper value of .J, 



where k has been determined by the methods described above, 

 or in any other manner. The chief objection to the use of a 

 value of A thus determined arises from its being generally much 

 too small, and from the consequent necessity of making a much 

 greater number of trials for the discovery of the limits of the 

 roots than would otherwise be necessary. 



Lagrange has proposed different methods of determining the 

 value of J, which, though much less laborious, at least for 

 equations of high orders, than the equation of the squares of 

 the differences, are still liable to great objections, in conse- 

 quence of their being indirect, difficult of application, and likely 

 to give values of J so small and so uncertain as greatly to mul- 

 tiply the number of trials which are necessai'y to be made *. It 

 is for this and other reasons that such methods have never been 

 reduced to such a form as to be competent to furnish the re- 

 quired limits by means of processes which are expressible in 

 the form of arithmetical rules, like those which are given for the 

 extraction of the square and cube root in numbers. In this re- 

 spect, therefore, they have failed altogether in satisfying the 

 great object proposed to be attained by their author, who con- 

 sidered the resolution of numerical equations as properly consti- 

 tuting a department of common arithmetic, the demonstration 

 of whose rules of operation must be subsequently sought for in 

 the general theory of algebraical equations +. 



The basis of all methods of solution of numerical equations 

 must be found in the previous separation of the roots ; and the 

 efforts of algebraists for the last two centuries and a half have 

 been directed to the discovery of rules for this purpose. The 

 methods, however, which have been proposed have been chiefly 

 directed to the separation of the roots into classes, as positive 

 and negative, real and imaginary, and not to the determination 

 of the successive limits between which they are severally placed. 

 The celebrated theorem of Des Cartes J gave a limit to the 

 number of positive and negative roots, but failed in deter- 



* Resolution des Equations Numeriques, Note iv. 



t Ibid., Introduction. 



X The proper enunciation of this theorem is the following : " Every equa- 

 tion has at least as many changes of sign from -f- to — and from — to -|- 

 as it has real and positive roots, and at least as many continuations of sign 



