361 



(c being positive or negative according as 



A Q = A n r n + c, or A 0 = 



A n r n - c. 



In the former case A Q is in excess, in the latter case, in defect) 



which shows that when n is even, and, moreover, c positive, all the n 

 roots will be imaginary ; but, if in this case of n even, o be negative, 

 only n - 2 of the roots will be imaginary. And that when n is odd, then 

 whether c be positive or negative, x will have one real value, and one 

 only. From this it appears that — 



When n is even, and it be found necessary to subtract a positive 

 number (c) from A 0 to make the triad one of equality, the equation 

 will have all its n roots imaginary ; but if it be necessary to add a posi- 

 tive number (c) to A 0 for this purpose, then the equation will have n - 2 

 imaginary roots, and no greater number. 



When n is odd, the equation will have n - 1 imaginary roots, what- 

 ever be the value of A 0 , or, which is the same thing, whether c be sub- 

 tractive or additive. 



(24) The conclusions, then, are quite analogous to those deduced 

 above from the case in which the n - 1 terms between A n x n , and A Q , 

 are zeros, instead of significant terms having the peculiar relations to 

 one another here supposed. If n is even, and A 0 in excess, the sign 

 to be placed under the last of these intervening terms is to be - ; and 

 if A Q is in defect, the sign is to be + ; but if n is odd, then, in the case 

 we are considering, as in the case of the zeros, it is matter of indiffe- 

 rence whether the sign under the last of the intervening terms ba + or - ,• 

 the number of imaginary roots indicated being the same, whichever sign 

 be chosen. For the immediately preceding sign in the row will always be 

 -, inasmuch as the sign under the first of the odd number of interven- 

 ing terms is itself - ; and the underwritten signs are alternately - and 

 +. But, having in view the case to be considered in the article next 

 following, and in order to preserve uniformity in both cases, it will 

 always be better to write + or according as A 0 is in defect or in ex- 

 cess, just as in the case of n being even. 



The foregoing conclusions may be arrived at in another way. Put- 

 ting the proposed equation in the form given to it above, namely, 

 A n (x + r) w + o = 0, and supplying the n - 1 zero-terms, it becomes 



which, x + r being here in the place of x, is identical with the form 

 [1], at page 357, and whatever values, real or imaginary, x has in the 

 former equation, so many must x + r have in this, and vice versd, since 

 the value of r is always real. 



Suppose, now, that triads of equality occur anywhere among the 



A H {x + r) n + 0 + 0 + 0 + . .. + 0 + 0 = 0, 



