384 I'HILOSOPHICAL TRANSACTIOXS. [aNNO 1778. 



between the least of the affirmative and the greatest of the negative roots of the 

 equation (a). 



^ 5. It is possible even, that the equation (b) may have imaginary roots, at 

 the same time that all the roots of the equation a are real, which is contrary to 

 what all algebraical writers have thought. For instance, the roots of the equation 

 X* — 6 X — 7 = are 7 and — 1, and if the terms of this equation be mul- 

 tiplied by 1, — 1, — 3 (an arithmetical series where the common difference of 

 the terms is equal to 2) the resulting equation will be x' -\- 6x -\- 23, the roots 

 of which are evidently impossible. 



^ 6. However, the equation (b) can never have more than 2 imaginary roots, 

 when the roots of the equation (a) are real. For suppose these last roots to be 

 -|- a, + A, + c, + (/, — e, — f, he. in their order from the greatest to the least, 

 and since the results whicli arise from the successive substitution of these quan- 

 tities are always alternately negative and positive, that case only excepted where 

 dand — e are substituted, it is manifest, that we shall always have n — 2 of the 

 roots of the equation (b) which will be limits of the equation (a). 



^ 7- It is remarkable, that whenever the equation a has all its terms complete, 

 its roots real, and some of them positive, and others negative, if / + w??i be 

 assumed equal to O, the equation b will always have one of its roots either 

 greater than the greatest affirmative root, or less than the least negative root of 

 the equation (a). Thus, in the quadratic x^ -^ 6x — 7 = 0, assume any 

 arithmetical progression 0, 1,2, the first term of which is equal to nothing, 

 then the equation b in this case is 6^^ — 14 := O, and x = ■§-, which is greater 

 than 1, the greatest affirmative root of the assumed equation. 



(^ 8. The roots of the equation (a) being still supposed a, b, c, d, — e, — y, 

 &c. let wi be taken equal to unity, and / any positive integer whatever; then in 

 that case, 2 of the roots of the equation b will lie between the roots d and — e, 

 one of which will be positive, and the other negative. 



For example, the quadratic equation .r' -\- 6x — 7 = O has its roots 1 and 

 — 7 ; and if the terms of this equation be multiplied into 3,2, 1 ; 4, 3, 2 ; or 

 5, 4, 3, successively, the resulting quadratic in every case will have its 2 roots 

 between the roots of the given equation, and one of them will be positive, and 

 the other negative. 



§ 9. The equation b, which in the last article was deduced from the equation 

 A by taking ?« equal to 1, and / any positive integer, may itself be treated in 

 the same way, and the resulting equation will, a fortiori, have 2 of its roots 

 between the roots (/ and — e of the original equation, one of which will be 

 positive, and the other negative. 



■^ 10. Let a' — px -\- (/ = o represent any quadratic equation, tlie real rootvS 



