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XXXVII. — On the Ima ginary Hoots of Numerical Equations, with 

 an Investigation and Proof of Newton's Rule. By J. E. Young, 

 Esq., formerly Professor of Mathematics in Belfast College. 



[Read November 9, 1868.] 



( 1 ) Let the general equation of the nth degree, with numerical coef- 

 ficients, be represented by 



A n x n + A n .,x n - 1 + A n . 2 x n ~ 2 +....+ A 2 x 2 + A x x + A 0 = 0 . . [7], 



and let it be transformed into another by substituting x + r for x. 

 Then if r be determined by the condition that the second coefficient in 

 the transformed equation shall be zero, the third coefficient will be 

 found to be 



A n -i n{n-l) ( A n _ x \* 

 A n 2 [nAJ ; 



consequently, if the sign of this third coefficient be the same as that of 

 the first, the first three terms of the transformed equation, — the middle 

 term being zero, will satisfy the condition of De Gua,* and will there- 

 fore imply the existence of at least one pair of imaginary roots in the 

 transformed, and therefore of one pair also in the original equation. 

 Hence, multiplying this third coefficient by the positive quantity 

 2nA n 2 , a pair of imaginary roots will be indicated, provided the first 

 three coefficients of the proposed equation satisfy the condition 



2nA n . 2 A n >{n-\)A\. l .... [1]. 



(2) If the order of the coefficients in [7] be reversed, we shall 

 have an equation the roots of which will be the reciprocals of the 

 roots of [/] : the existence of a pair of imaginary roots in either equa- 

 tion implies therefore the existence of a pair, the reciprocals of those 

 roots, in the other. Consequently the criterion [1] may be applied, as 

 a test, as well to the last three coefficients of [/] as to the first three ; 

 so that a pair of imaginary roots in [7] will equally be indicated, pro- 

 vided that the condition 



2nA Q A 2 >(n- l)Ai 2 .... [2] 



be satisfied. 



| (3) Now, it is well known that if a limiting or derived equation 

 have imaginary roots, the primitive equation must also have imaginary 

 roots, — as many at least : taking, therefore, the several limiting equa- 

 tions, derived one after another, each from the immediately preceding, 

 in the usual way, [/] being the primitive, and applying the criterion 



* Conformably to usage, I have called this " the condition of De Gua ;" but it is 

 implied in the Rule of Newton, published many years before the researches of De Gua 

 appeared. 



R. I. A. PROC. VOL. X. 3A. 



