362 



terms of an equation; then, by taking the successive limiting equations, 

 as in the case of consecutive zeros, we can reach an equation in which 

 all the triads are triads of equality, except the last triad,* and can thus 

 return to the case just considered. And it is plain, from what has 

 already been shown, that the signs to be written under the terms which 

 intervene between the first term and the last term of this derived equa- 

 tion will be alternate signs, like as if these intervening terms were so 

 many zeros ; and, as in the case of the zeros, these are the signs to be 

 written under those terms of the primitive equation from which the 

 terms here spoken of, in the derived equation, have been deduced. But 

 one thing must be attended to here, which, in the case of intervening 

 zeros, requires no special notice. The signs to be written under the 

 first term of the leading triad, and under the last term of the series of 

 triads we are here considering, are not necessarily +, as in the case of 

 the zeros, but may be either both -, or one + and the other - : which 

 sign is to be underwritten, the general rule at (19) will, of course, 

 enable us to readily ascertain. If the former of these two signs is seen 

 to be -, the sign immediately next following must be + ; and so on, 

 alternately, till the latter of the two is, in like manner, determined by the 

 rule, and underwritten. In the case of the zeros, the first of these two signs 

 was invariably + (and so was the last) ; and the immediately next sign, 

 - ; but it may be otherwise here, as the general rule must be obeyed. 

 It may be observed, however, that an underwritten - always implies a 

 pair of imaginary roots, as a + must have preceded it ; for it is with a + 

 that the row of signs under the terms of the proposed equation com- 

 mences ; so that no fewer imaginary pairs can ever be indicated by the 

 signs under the terms of the primitive than would be indicated by the 

 signs under the terms of the final derived equation. 



We shall now give an example or two by way of practical illus- 

 tration : — 



1. a 5 + 10a 4 + 40a 3 f 80a 2 + 80a + 36 = 0 



+ - + - - [Here A 0 - 36, is in excess.] 



Hence there are four imaginary roots. 



2. 16a 4 - 96a 3 + 216a 2 + 216a - 80 = 0 



Z + - + + [Here A 0 = - 80, is in defect.] 



So that the equation has but two imaginary roots. 



3. a 8 + 5a* - 3a 8 + 56a 6 + 70a* * 56a 8 + 28a 3 + 6a + 4 = 0 



[Here the 4 is in excess.] 



* If the last triad satisfy the condition of equality, and not all the triads, then, just 

 as in the case of the first triad being a triad of equality, an imaginary pair will be indi- 

 cated by that triad alone (Art. 6). 



