350 



terion of imaginary roots, n being equal to 3, be the wanting numbers 

 whatever they may, we should obtain the very same conditions [3] 

 which the original coefficients A 0 , A 1} A 2 , &c, supply; each condition 

 here being the same as that condition there, into which the same triad 

 of original coefficients enters. Mere, however, we see that the last triad 

 of terms in any cubic always involves the same coefficients of [i] as 

 the first triad in the cubic next following ; so that, if the criterion of 

 imaginary roots be satisfied by the last three terms of one cubic, it must 

 be satisfied by the first three of the next, and vice versa. As a cubic 

 equation cannot have more than one pair of imaginary roots, it follows 

 that the fulfilment of the condition by any two consecutive sets of 

 three terms of the primitive equation implies, of necessity, but one pair 

 of imaginary roots in that equation. 



When, however, each of the two triads of a cubic equation indicates 

 imaginary roots, the concurring indications imply a distinct peculiarity 

 in the pair of roots thus indicated. The peculiarity is this, namely, 

 that not only does their entrance cause the second term of the cubic 

 equation to vanish between like signs, but the entrance of their reci- 

 procals, in the reciprocal equation, causes also the second term of that 

 equation to vanish between like signs ; that is to say, the second term 

 vanishes between like signs, whether the coefficients be taken in the 

 order proposed or in the reverse order. 



"When the first triad of terms satisfies the condition of imaginary 

 roots, and the second triad fails, the reciprocal pair is not thus indicated ; 

 when the second triad satisfies the condition, and the first fails, it is in 

 the reciprocal equation alone that the second term vanishes between 

 like signs. And it is, moreover, only when such evanescence takes 

 place in the reciprocal equation that imaginarity is necessarily conveyed 

 from the one cubic to the other ; and not only in the contrary case, that 

 is, when the final triad fails to satisfy the condition, is imaginarity not 

 necessarily conveyed to the next cubic, but it cannot possibly be con- 

 veyed under any circumstances whatever. In no single instance is a 

 pair of roots in a cubic imaginary, either primary or non-primary, 

 merely in consequence of the final triad in the preceding cubic being 

 what it is, unless that preceding triad itself satisfies the condition of 

 imaginarity. This may be proved as follows : — 



(14) Let the cubic equation be 



A' z x z + A' 2 x" + A\x + A' 0 = 0. 

 in which the leading triad of terms, supplied by the final triad of the 

 immediately antecedent cubic in [Z?], fails to satisfy the condition at 

 [3] ; and let it be transformed into 



x z +px + q = 0. 



by the removal of the first coefficient, and the second term ; p will then 

 be necessarily negative, by the hypothesis. Now it is known that 



1. If ~ 3 ) > ("f - ^ ' ^ e roo ^ s w ^ a ^ ^ e real and unequal. 



