363 



Consequently there are at least six imaginary roots, two in the positive 

 region, and four in the negative region. 



4. 3a 7 + 7a 6 + 21a 5 + 35a 4 + 35a; 3 - 8a 2 + 12a - 5 = 0 

 f - + - + ' - + 



Hence there are six imaginary roots in the equation. 



As in the first and fourth of these examples the sign immediately 

 preceding the last is minus, we might, in each of these, have stopped 

 at that sign : the determination of the last sign was unnecessary, as no 

 additional imaginary roots could be indicated, whether the last sign 

 proved to be + or - ; nor could another pair have been indicated, 

 though the degree of the equation had been even instead of odd. 



(25) "We here terminate these practical details respecting Newton's 

 Eule, which rule is substantially the same as that given at page 355 

 of this paper. "We have not attempted any extension of it, but have 

 been content with ascertaining what is the utmost amount of informa- 

 tion, respecting the number of imaginary roots in a numerical equation, 

 that can be educed from it. The rule itself does not appear to the pre- 

 sent writer as capable of any extension — if by that term be meant its 

 being so enlarged as to be available for detecting the presence of ima- 

 ginary roots other than those which, in the foregoing investigations, 

 have been called primary pairs* Before, however, passing to other 

 matters, it may be well to give a practical illustration of the way in 

 which we may always ascertain whether or not any proposed polyno- 

 mial is the development of the binomial form A n (x + a) n : it was ad- 

 verted to at p. 346. 



Suppose, for instance, we wished to know whether or not the poly- 

 nomial following is a binomial development. (See p. 356.) 



64 



27a 6 - 108a 5 + 180a* - 160a 3 + 80a 2 - 2U + — 



3 27 



Mult, for the squares, 5 8 9 8 5 



„ „ products, 12 15 16 15 12 



As we find that all the triads satisfy the conditions of equality, and 

 that here 



A n _, . 108 2 



a > or -^ ls ^T27' or 3' 



we infer that the invelopment of the proposed polynomial is 27i a - - j * 

 Of course if the last term of any polynomial, when divided by A n , the 



* It will be hereafter shown, however, that non-primary pairs in an equation are 

 always convertible into primary pairs by diminishing each of the roots by a determinable 

 number. 



