with some Supplementary Remarks on Newton's Rule. 123 



5(#— 1) 

 the result being — r— ; hence the value of the proposed frac- 

 tion for x=\ is 5 divided by these factors when 1 is put for x 



5 

 in them also ; that is, the value is • 



In my demonstration of Newton's rule, the case in which the 

 sign of inequality in any of the criteria is replaced by the sign of 

 equality is not adverted to ; and Newton makes no mention of 

 it. I propose here briefly to examine what inference may be 

 drawn from this circumstance. 



1. And first, let it be the leading triad of terms that furnishes 

 this equality. Then if the roots of the equation be diminished or 

 increased by that quantity which will cause the second coefficient 

 in the transformed equation to be zero, it will follow, from the 

 theorem established at the end of my paper in the May Number 

 of this Journal, that the third coefficient in that transformed 

 equation must be zero also. For, as there shown, the condition 



2nA n A n _ 2 -{n-l)Al_ l = 



necessitates the condition 



2nA' n A f n _ 2 -(n-l)A'* n _ l = 0, 



which, if A' n _ 1 = 0, can have place only when also A'„_ 2 =0. 

 Now if all the coefficients, after this third, in the transformed 

 equation turn out to be zero, the roots of the primitive equation 

 must all be equal, their common value being the nth part of the 

 second coefficient (when divided by the first) taken with contrary 

 sign. But if all the coefficients do not vanish, then, as is well 

 known, the equation must have at least one pair of imaginary 

 roots. 



The inference, therefore, is that the equation must have either 

 one pair of imaginary roots at least, or else that the roots must 

 be all real and equal. 



The same conclusion follows when it is the final triad that 

 supplies the equality, since we may reverse all the coefficients. 



2. But let it be an intermediate triad which, furnishes the 

 equality. Then by taking the derived equations in succession, 

 we shall at length arrive at one of these in which the interme- 

 diate coefficients enter the final triad; and consequently, as just 

 seen, this derived equation, if all its roots be not equal, must 

 have at least two which are imaginary. In the latter case, two 

 at least must occur in the primitive equation. In the former 

 case the derived polynomial (after division by the first coefficient, 

 if this be other than unit) will be a complete power ; its three 

 leading coefficients must therefore fulfil the condition of equality, 



