351 



2. If 



two of the roots will be imaginary. 



3. If - — ) = ( ~ , the roots will all be real, and two of 



them equal. 



The second of these conditions shows that, in the proposed hypo- 

 thesis, the three coefficients, A' 3f A' 2 , A' u can never alone suffice to 

 introduce imaginary roots into the complete cubic, since that condition 

 implies that a suitable value of g, and consequently of A' 0i is indispen- 

 sably necessary to such introduction. Hence the final triad of one of the 

 cubics [77] can never introduce imaginarity into the cubicnext follow- 

 ing, unless that triad itself satisfies the condition of imaginarity. On the 

 other hand, whenever the triad does satisfy that condition, no value of 

 the absolute term (or q) can ever prevent imaginary roots from enter- 

 ing the cubic to which that triad is transferred ; that is, under this con- 

 dition imaginarity is of necessity introduced by the former cubic into 

 the latter, let q, or A f 0 , in this latter cubic, be whatever it may : it may 

 indeed be anything or nothing, p being necessarily positive ; and on ac- 

 count of this sign of p it is, and on this account solely, that two of the 

 roots are of necessity imaginary ; the value or sign of q (and therefore 

 of A' 0 ) having nothing at all to do with the matter. 



(15) It thus follows that when in any cubic equation 



has place, it is impossible that imaginary roots can enter independently 

 of the value of A f 0 ; so that imaginarity can never be introduced 

 into the cubie (if the absolute term A' Q be arbitrary), whatever values 

 we may give to the three leading coefficients, provided only the condi- 

 tion [_a] is preserved. But when, on the contrary, the condition is 



then it is impossible that imaginary roots can be excluded, let A' 0 

 take whatever value it may; that is, under the condition [6], imaginary 

 roots must enter the equation independently of the value of A r 0 . The 

 first of these two conclusions is that to which attention is here more 

 especially invited. It is indispensably necessary to the inference which 

 it is our main object here to deduce, that it should be clearly seen that 

 the transference of the final triad of any one of the cubics [77] to the 

 position of leading triad of the cubic next following can never be a 



* The general condition 2nA n -2<(n- 1) A 2 n -\, obviously becomes, in the .case of 

 the cubic, the particular condition in the text ; and it is further obvious, from the group 

 of conditions [5], that whenever the exponent n is odd, the multipliers are each of them 

 even, and therefore divisible by 2. 



E. I. A. PEOC VOL. X. 3 B 



A' z x s + A' 2 x* + A\ x + A' 0 = 0, 



the condition 



3 A't A' 3 < AV 



M 



3 A', A' s > AV . . . . [6], 



