354 



of the equation next following, we can readily see whether or not ima- 

 ginarity is conveyed from the former to the latter. After the first triad 

 of [7] (or the first when the coefficients are reversed, it makes no diffe- 

 rence) each successive triad up to the last is thus repeated in [17] : it 

 is this triad — common to two consecutive cubics, which forms the con- 

 necting link mentioned, and which causes the same fulfilment or failure 

 of the condition [3], in the leading triad of the second of these cubics, 

 as in the final triad of the first. And this is the only connecting link 

 between the two : in other respects they are independent, so that when 

 the final triad of a cubic (and consequently the leading triad of the next 

 succeeding cubic) fails to satisfy the condition [3], and the final triad 

 of this succeeding cubic does satisfy that condition, an imaginary pair, 

 distinct from whatever other pair or pairs have been inferred from 

 earlier cubics in the series, must enter the primitive equation : and this 

 is the same as saying that when a fulfilment of a condition [3] by a 

 triad of the coefficients of [7] is preceded by a failure, a pair of primary 

 roots, distinct from and independent of whatever pairs may previously 

 have been detected, is indicated in the equation. 



(18) But before the passage from a fulfilment to a failure, or from 

 a failure to a fulfilment, there may have been a continuous succession 

 of such fulfilments or failures in passing from cubic to cubic ; or, which 

 is the same thing, in proceeding from term to term of the primitive 

 equation. Prom these uninterrupted concurrences we cannot infer 

 anything, as to additional imaginary pairs : such additional pairs may 

 enter the primitive, or they may not ; as is sufficiently exemplified in 

 article (10) : but we shall always be on the safe side — that is, we shall 

 never be in danger of inferring more pairs than really enter — if we 

 always regard these concurrences as merely repeated indications of one 

 and the same thing, namely, the succession of fulfilments as only so 

 many concurring proofs of the existence of but one pair of imagi- 

 nary roots, and the succession of failures as indications that no addi- 

 tional imaginary pair is to be inferred so long as the failures remain un- 

 interrupted by a fulfilment. 



We may remark, however, that when there is a continuation of ful- 

 filments, a peculiar character is impressed upon the several cubic equa- 

 tions [77], as already adverted to at (13) : the triad supplied to a cubic 

 by the antecedent cubic, imports primary imaginarity simply ; whilst 

 the triad which the new fourth term completes, so modifies the roots 

 that, whether we take the direct or the reciprocal equation, the second 

 term in either case vanishes between like signs; and imaginarity cannot 

 be expelled, whether we change the final term or the leading term ; it 

 is what it is in virtue of both triads satisfying the condition indepen- 

 dently ; and is as much a consequence of one triad as of the other. In- 

 dependent imaginarity in any one of the cubics after the first cubic 

 can be inferred only when the imaginarity is due exclusively to the 

 final triad, and may therefore be expelled from the equation by merely 

 modifying the final term, that is, the absolute number which completes 

 the cubic. Of course it will be understood throughout these remarks 



