determining the number of Imaginary Roots in an Equation. 117 



and so on. And if from these inequalities the factors common 

 to each side be expunged, the conditions will be the same as those 

 already given previously in the margin. 



It follows from what has now been shown — (1) That com- 

 mencing with the second coefficient of an equation, as the mid- 

 dle one of the leading three, and applying to those three the 

 proper test given in the margin above, then, with the third 

 coefficient taken as middle term, applying in like manner the 

 suitable test, and so on till we arrive at a case of failure, we 

 may conclude that, up to this point, the existence of one pair of 

 imaginary roots, but not of more than one pair, has been detected. 

 (2) That the case of failure thus arrived at is entirely independent 

 of the preceding conclusion ; that it is wholly uninfluenced by, 

 and distinct from, every antecedent condition; the indications 

 of the imaginary pair, previously detected, have ceased to be 

 transmitted ; and that therefore if, after passing this stage, an- 

 other indication present itself, it must imply another pair of 

 imaginary roots. [I w r ould here refer to the remarks interpo- 

 lated above, and direct attention to the fact that the failure 

 here adverted to would, in the limiting cubics, necessarily origi- 

 nate in the final terms, and not in the leading terms of a cubic. 

 By altering the last term of a cubic we could make the condi- 

 tion of imaginary roots to either hold or fail, in reference to the 

 final set of three, as we please, without at all disturbing ante- 

 cedent conditions — a clear proof that the condition implied in 

 that final three is quite independent of the antecedent condi- 

 tions ; but we could not alter as we please any term of a leading 

 set without disturbing (that is, without absolutely reversing) the 

 immediately preceding condition.] 



Such, then, is my demonstration of Newton's Rule for detect- 

 ing Imaginary Roots in an equation with numerical coefficients. 

 Save the two Notes included within brackets, it is the same in 

 every essential particular as that I first published in 1843. It 

 was also printed in my ' Course of Mathematics ' (Allen and Co.) 

 in 1861. I have said that I do not regard Newton's Rule as of 

 such importance as Professor Sylvester appears to do, because 

 I think that there are some things in the work just referred to 

 that enable algebraists to be more independent of it than they 

 were formerly. 



July 5, 1865. 



