359 



underwritten signs in conformity with, the rule at page 355, till we come 

 to that term which immediately precedes the zeros ; under this (what- 

 ever be its sign) to write +; under the first zero, and so on alter- 

 nately, till we reach the last zero, under which is to be written +, if the 

 signs of the two terms which bound the zeros are unlike, and - if they 

 are like, and then to proceed according to the rule. The sign under the 

 term which immediately follows the zeros, as well as that under the 

 term which immediately precedes them, will, of course, always be +. 



Note. — It was directed above that when the first of the boundary- 

 terms is minus, the sign of the other boundary-term is that which is 

 always to be written under the last zero. If it should be for the 

 moment thought that what has just been said is inconsistent with that 

 direction, the reader has only to reflect that, in the case of A' n , negative, 

 if A' 0 be +, the signs of the two terms will be unlike, and that if A' 0 be 

 that the signs will be like ; so that in the former case, + is to be 

 written, and in the latter case -, the sign being always (in the case of 

 A' n , negative), the same as the sign of A' 0 . 



We shall now give some examples of the application of these pre- 

 cepts. 



1 . x 5 + ax* + 0 + 0 + 0 + e = 0 ; 

 + + - + - 



2. x" + ax* + 0 + 0 + 0 - e = 0 

 + + - + + 



The first of these equations has four imaginary roots ; the second two, 

 at least ; of the other three, one is positive, and the remaining two 

 belong to the negative region. [It may be observed here that if each 

 of the roots of an equation be diminished by 8, the number 8 may ob- 

 viously be taken so small, that in carrying on the transforming process, 

 by Horner's method, each addend may be made as small as we please ; 

 so small, therefore, that the signs of the significant terms of the original 

 shall all be preserved unaltered in the transformed equation ; in which 

 case, what was a zero in the original, will, in the transformed equation, 

 be a significant term, with the same sign as the significant term imme- 

 diately preceding the zero. There will thus be a permanence of sign ; 

 and in this way permanencies will replace the arbitrary signs of all the 

 zeros.] 



3. x 1 - 2x Q + 3x> - 2x* + x 3 + 0 f 0 - 3 «= 0 

 + + - + - + 



Therefore six of the roots are imaginary. 



4. x 1 - 2x 6 + Sx 5 - 2x* - x 3 + 0 + 0 - 3 = 0 

 + + + + - - 



Here the equation has four imaginary roots, at least ; one real root is 

 positive ; the other two roots are doubtful, and belong to the negative 

 region. 



E. I. A. PKOC. VOL. X. 3 C 



