for the Imaginary Roots of Numerical Equations. 451 



proposed in the present paper will, I think, prove an accept- 

 able addition to those previously given: they are not only of 

 greater simplicity, but will often succeed in detecting the pre- 

 sence of imaginary roots in cases where Newton's formulae — 

 the basis of those adverted to above — would prove inefficient. 

 These new forms are deduced as follows: — 

 Let the equation 



A„*"+A„_ 1 *"- 1 + A„_ 2 *- 2 + A„_ 3 .*»- 3 + ...A,* + A =0 



be multiplied by x — «, that is, let a new undetermined real 

 root a be introduced into the equation : whatever imaginary 

 roots are indicated in the new equation, the same of course 

 enter into the original. This new equation is 



A w ^ +1 + (A w _ 1 -aAJ," + (A ra _ 2 -«A w _ 1 )^- 1 



+ (A M _ 3 -flA M _ 2 )/- 2 +...(A o -flA 1 )a? + aA = O, 



in which a is entirely arbitrary, and may therefore be made 

 to satisfy any condition we please. It may for instance be de- 

 termined so as to render any one of these compound coeffi- 

 cients zero; and if, in conjunction with this determination of 

 a, the original coefficients be so related as to cause the com- 

 pound coefficients on each side of this zero to be of like signs, 

 we shall at once recognise the presence of imaginary roots in 

 the proposed equation. 



Equating then the several coefficients to zero, one after the 

 other, commencing with the second, and determining the ad- 

 jacent pair of coefficients in each case conformably to this 

 condition, we shall have the following sets of conditions, the 

 existence of any one of which will imply a pair of imaginary 

 roots : — 



A n =+, A^-aA^O, 



A2 „-l- A „ A w -2 = +> A n-2- aA n-l = °> 

 A2 «-2- A „-l A „-2=+> A n-3-« A M -2 = °> 



&c. &c. 



And therefore, a being always assumed so as to satisfy one 

 of the middle equations, those on each side will furnish the 

 following series of criteria of imaginary roots, viz. — 



A„=+, A,A,. ! >A',. 1 ...[1.] 



*\-2>K-i A «-V A „- 2 A„_ 4 ^A\_ 3 ... [3.] 



&c. &c 



2 G2 



