352 



transmission of imaginarity, from the one to the other, if the triad 

 satisfy the condition [_a] ; a truth which is indeed sufficiently obvious 

 from the first of the formulae at ( 1 4) ; for removing the middle term of the 

 transferred triad, the resulting p is Decessarily negative; and therefore, 

 by merely altering the value of q (that is of A' 0 ), if it be found to need 

 altering for the purpose, without meddling with the transferred triad, all 

 the roots may always be made real ; which could not be done if the trans- 

 ference of the triad ever involved conveyance of imaginarity. The 

 conclusion, therefore, is irresistible, that if each of these two cubics has 

 imaginary roots — the second, in virtue of its final triad alone, satisfying 

 the condition of a primary pair — the two pairs must be entirely inde- 

 pendent of one another : the entrance of the second pair cannot possibly 

 be a consequence of the entrance of the first pair. 



(16) Let now the first of these two cubics, taken from be re- 



presented by (7=0, and the second by C x = 0 ; the imaginary pair in the 

 former by and the imaginary pair in the latter by i", ; then, as just 

 seen, the entrance of I x into the cubic C x - 0 is not a consequence of the 

 entrance of /into C- 0, but is entirely independent of the entrance of 

 I. Calling the two reciprocal equations from which these cubics have 

 been respectively deduced R = 0, and R x - 0, we may, by reversing the 

 process by which (7 has been derived from R, derive this latter from the 

 former, adding in at each reverse step that particular constant (or final 

 term) which in the direct step was made to disappear. In the equa- 

 tion of the fourth degree, the result of the first step from (7, there enters 

 an imaginary pair — a primary pair, necessarily and exclusively depen- 

 dent for its character as such upon the pair in G- 0. 



But the pair in C x = 0 is not dependent for its imaginarity on this pair 

 in the biquadratic ; for if it were, it would be dependent on the pair in 

 O=0, which it is not. In like manner, the equation of the fifth 

 degree, in the next reverse step, has an imaginary pair dependent on 

 the before-mentioned pair in the preceding result, and therefore on the 

 pair in (7 = 0: the pair in C\ = 0 is therefore equally independent of 

 this pair ; and so on throughout all the reverse steps up to R - 0, that 

 is, there is a pair of imaginary roots in R = 0, of which the imaginary 

 pair in C x = 0, derived from R x = 0, is independent. 



But imaginary roots can enter R x = 0 only as a consequence of ima- 

 ginary roots entering R = 0 ; and imaginary roots can enter O x = 0 only 

 as a consequence of imaginary roots entering R x = 0 ; and therefore only 

 as a consequence of imaginary roots entering R = 0. 



But it was shown that the pair in C x = 0 does not enter as a conse- 

 quence of that particular pair in R = 0, of which the pair in C = 0 (in 

 the reverse process of derivation) is the source : hence the pair in C x = 0 

 must be the consequence of some other pair in R = 0 ; which equation 

 has therefore at least two pairs of imaginary roots. Consequently the 

 primitive equation has at least two pairs of imaginary roots. 



The particular imaginary pair in the equation R = 0, here adverted 

 to, is that pair the entrance of which is indicated by the evanescence of 

 the second term of the equation R = 0 between like signs : in other 



