290 Prof. Young on Newton's Rule for Imaginary Roots. 



second test for the final three. Since, however, the value of n 

 is 3 for every cubic equation, it follows that in the conditions 

 referred to, 2n is the same as S(n — 1), and (n— 1) the same as 

 2(n— 2) ; so that, A^ +1 being put for the middle one Of either 

 of the three consecutive coefficients in any cubic, only one con- 

 dition need be appealed to in reference to the character of the 

 roots of that cubic, namely, the condition 



3A',A' 4+2 >A' 2 t+1 (1) 



Now in prosecuting the search after primary pairs, we proceed 

 (mentally at least) as follows : — Taking the three leading coeffi- 

 cients of the proposed equation, we form from them the first 

 three terms of the commencing cubic at p. 115. From this in- 

 complete cubic we determine, by an appeal to the condition (1), 

 whether or not a primary pair of imaginaries enters the proposed 

 equation solely as a consequence of the relative values of the first 

 three coefficients of that equation. 



If a pair be seen to enter, a pair would still enter, alter the 

 subsequent coefficients as we may ; but if the terms of the in- 

 complete cubic do not satisfy the condition (1), then a primary 

 pair cannot enter the primitive equation solely as a consequence 

 of the relative values of these three coefficients. 



We now pass on to the fourth of the given coefficients, by the 

 introduction of which we complete the cubic, and then examine 

 its last three terms. If these satisfy (1), we infer one primary 

 pair, and one pair only, whether the condition for the first three 

 terms hold or fail. In the former case there is a double indica- 

 tion of the same thing; in the latter case but a single indication. 



Whatever be the character of the final three terms in reference 

 to the condition (1), I have proved that that character will of 

 necessity be transmitted to the leading three of the cubic next 

 following : these therefore need not be submitted to the test (1) ; 

 and the second cubic being completed by aid of the fifth coeffi- 

 cient of the proposed equation, we then examine > as before, the 

 final three terms. And in this way we proceed, from cubic to 

 cubic, till the last of the given coefficients is brought down. 



I have demonstrated that in this series of cubics consecutive 

 fulfilments of the condition (1) merely repeat the information 

 conveyed by the first fulfilment : they concur in testifying to 

 one and the same thing, namely, the necessary entrance of but 

 a single pair of imaginaries into the proposed equation. For 

 aught we know to the contrary, other pairs may enter, and enter, 

 too, like the primary pair, quite independently of subsequent 

 coefficients ; but we can pronounce with certainty only as to the 

 number of primary pairs that enter. 



If after a fulfilment of the condition (1), or after a series of 

 fulfilments, there occur a failure, that failure must take place 



