Imaginary Hoots of Equations. S3 



admit of extension : giving rise to other criteria that may 

 sometimes apply where known tests fail, and which, as well 

 on this account as for the purpose of giving completeness to 

 the former set, it may be worth while to record. 

 The general numerical equation being 



A n x n +A„_ 1< r M - 1 + A n _ 2 ^- 2 + A„_3^- 3 + ... A 1 * + A =0, 



the criteria already investigated are 



A n A re _2 >■ A w _!, 

 A ra _i A w _ 3 > A w _2» 

 A w _2 A n _4 >■ A„_ 3 , 



A 2 A > Aj , 

 which are to be applied for the detection of imaginary roots 

 exactly as the criteria of Newton are applied, and as explained 

 in the volume above referred to. To give the proposed ex- 

 tension to these forms, we have only to multiply the several 

 coefficients of the equation — in the manner so often employed 

 by Newton and Maclaurin — by the terms of an arithmetical 

 progression, as 1,2, 3, &c, and to replace the original coeffi- 

 cients by the results: we shall thus have the following set of 

 criteria to be applied in the same way as those above : — 



3A„ A n _2 > 4A w _i, 

 8A„_i A w _3 > 9A W _2> 

 15A n _2 A ra _4 > 16A W _3, 



(ra 2 -l) A 2 A >n 2 A^. 



But these are only particular cases of the following more 

 comprehensive forms, obtained by means of the general 

 arithmetical progression k, £+1, k + 2, &c. : — 



k (k + 2) A n A„_ 2 > {k + 1 ) 2 A*_ 1} 

 (k +l)(k+ 3) A B _, A n _ :j > {k + 2) 2 A*_ 2 , 

 (k + 2) {k + 4) A n _ 2 A M _ 4 > {k + 3) 2 A*_ 3 , 



{k + n - 2) (k + n) A 2 A >*(A -f n - l) 2 A^. 



and in which we may, if we please, write A , A v A 2 , &c. in 

 the place of A n , A n _i, A M _ 2) &c, and vice versa. 



It will be observed that these criteria are really distinct 

 from the former set, and are not comprehended in that set ; 

 for although whenever any one of these holds the correspond- 



Phil. Mag. S. 3. Vol. 29. No. 191 . July 1846. D 



