355 



th at transference, or conveyance, of primary imaginarity always implies 

 transference of a triad of terms. 



(19) From what has now been established, we deduce the follow- 

 ing rule for determining (at least approximately) the number of imagi- 

 nary roots in a numerical equation, from the mere examination of the 

 coefficients. 



Rule 1. Under the leading term of the equation, write the sign 

 plus, as the first of a row of signs. 



2. Then, taking the second coefficient of the equation as the middle 

 one of the first three coefficients, apply to those three the proper test 

 [3] or [5]. If the condition be satisfied, write minus under the second 

 term ; if it be not satisfied, write plus. In other words, plus or minus 

 is to be written under the second coefficient according as its square, 

 multiplied by the proper factor, is greater or less than the product of the 

 adjacent coefficients multiplied by the proper factor. 



3. Passing to the third coefficient; take that as the middle of the 

 two adjacent coefficients ; and apply, in like manner, the next follow- 

 ing test ; and as before, annex to the former signs minus, or plus, ac- 

 cording as the condition holds or fails. And in this way proceed till all 

 the coefficients have been employed. Then as many changes as there 

 are in this completed row of signs, from plus to minus (not from minus 

 to plus), so many pairs of imaginary roots must enter the equation : it 

 may have more pairs, but it cannot have fewer. 



Note. — The last triad of coefficients need not be tested whenever 

 the row of signs already written down terminates in a minus sign ; and 

 it is well to remember that the test for the last triad is always the same 

 as that for the first ; for the last but one, the same as that for the 

 second ; and so on. 



It may further be observed that it is impossible for fulfilments in 

 the positive region of the roots to be succeeded by fulfilments in the 

 negative region, or for fulfilments in the negative region to be suc- 

 ceeded by fulfilments in the positive region, without a separating 

 failure ; for whether permanencies of sign in the equation, are suc- 

 ceeded by variations, or variations by permanencies, the sign which is 

 the termination of the one set, and the commencement of the other set, 

 must evidently always have the sign adjacent to it on the one side of 

 opposite character to that adjacent to it on the other side ; so that the 

 sign written under the middle one of the three must always be +. 

 The region, therefore, in which a pair of imaginary roots lies, or in 

 which the pair is indicated, is sufficiently marked by the collocation of 

 signs in the equation. 



(20) The following are applications of the foregoing rule: — 



1. x 5 - 4x* + 4a? - 2x 2 - 5x - 4 = 0 



+ + - + + 



The equation has one pair of imaginary roots, at least. By the rule o 

 Descartes, there are but three roots in the positive region; two of these 

 are those here found to be imaginary. 



