346 



the corrseponding triad in the preceding equation, and conversely ; and 

 from this it further follows that in the development of (x + all the 

 triads must satisfy the conditions of equality. For they are all satisfied 

 in the case 



(x + aY - x 3 + Sax 2 + Sa 2 x + a 3 , 



and consequently, from the above inference, all (except the last triad, 

 which has no correspondent here) must satisfy the conditions of equa- 

 lity in 



(x + a) 4 - = x i + 4ax z + 6a 2 x 2 + 4a 3 x + a*, 



of which the former multiplied by 4 is the derived equation ; and thence 

 the conditions of equality are satisfied in {x + «) 5 , (x + a) 6 , &c, by 

 whatever factors these be multiplied. And since in each case the last 

 triad always satisfies the same condition as the first, it follows that all 

 the triads in the development of A n (x + a) n satisfy the conditions of 

 equality ; and this development being fixed in form, these tests of equa- 

 lity may be employed to ascertain whether a polynomial is really the 

 development of a binomial, or of a binomial multiplied by a factor, or 

 not. 



(8) From the theorem just established, we see that whatever ima- 

 ginary roots the conditions [3] may enable us to detect the existence of 

 in the equation [/], these roots always have a peculiar character ; they 

 are distinguished from other imaginary roots in this, — namely, that 

 on account of their entrance into the primitive equation, imaginarity* is 

 necessarily transmitted to the first derived equation, thence to the se- 

 cond derived equation, and so on, till one of the three coefiicients of the 

 primitive, which supply the condition [3], disappears, as it at length 

 must do in the process of successive derivation. As long as the three 

 coefiicients fulfilling one of the conditions [3] are all preserved in the 

 subsequent equations, so long will each of those equations have a pair 

 of imaginary roots. With regard, therefore, to this particular class of 

 imaginary roots, it is true not only that a pair in any derived equation 

 implies, of necessity, the entrance of a pair in the primitive, bat, con- 

 versely, that a pair in the primitive necessitates a pair in every derived 

 equation down to that one from which the third coefficient in the primi- 

 tive triad has disappeared. 



"With other imaginary pairs, that is, with those pairs the existence 

 of which is not indicated by any of the conditions [3], the case is dif- 

 ferent: we know that the equation may have imaginary roots, and 

 yet not transmit imaginarity to any of the equations derived from 

 it — the roots of these may all be real ; if a pair of them be ima- 

 ginary, then, indeed, a pair also imaginary must necessarily enter 



* This term, "imaginarity," is not employed by English algebraists : its equivalent, 

 imagwaritt, is, however, of frequent occurrence in French works, and deserves to be im- 

 ported into our language. 



