366 



polynomial were the power [5], as is obvious ; and hence the truth of 

 the theorem announced above. 



(27) An immediate deduction from this theorem is, that if the con- 

 dition of imaginarity hold or fail for the first three coefficients of any 

 equation, it will, in like manner, hold or fail for the first three coeffi- 

 cients of every transformed equation which can result from increasing 

 or diminishing the roots by any quantity (r) whatever. 



For the proposed equation being 



A n x n + A n _ x x n ~ x + A n .,x n ' 2 + . . . + A 0 = 0, 



we know that the first three coefficients of the transformed equation 

 / (a? + r) = 0 will be the last three of the second development [1] ; and, 

 therefore, writing the final coefficient first, that they will be of the 

 forms 



A n , ar + A n _ l} hr 2 + cr + A n _ 2 . 



But, by the foregoing general principle, if the condition hold or fail for 

 the three original coefficients, A m A n _^, A n _ 2 , it must equally hold or 

 fail for these, inasmuch as that 2n times the product of the first and 

 third of them differs from n — 1 times the square of the second by 

 exactly the same amount that 2nA n A n _ 2 differs from (n -l)A 2 n _ y ; 

 for the two terms involving r disappear from the difference, as just 

 proved. 



Hence, for every transformation, these two functions of the coeffi- 

 cients have the same constant difference. 



Suppose we had developed one of the roots of a cubic equation by 

 Horner's method, and that we wished to ascertain whether the roots of 

 the quadratic equation 



A 2 x 2 + A' 2 x + A\ = 0, 



to which the process would conduct us, were real or imaginary ; that 

 is, whether 4A\A 3 be less or greater than A' 2 2 . Now we know, from 

 the foregoing principle, that 



3A, A 3 - A, 2 = ?>A\A Z - A'i [6] 



and, therefore, having calculated the first of these expressions from the 

 original coefficients, we see that we have only to ascertain whether 

 A\ A s added to it will make the result positive or negative. If positive, 

 the other two roots of the cubic equation are imaginary ; if negative, 

 they are real. For example : In the writer's treatise on " The Analysis 

 and Solution of Cubic Equations" (p. 172), the development of a root 

 of the equation 



a s + 8x 2 + 6x- 75-9 == 0 

 conducts to the quadratic 



x* + 15'2771.r + 62-46326147 



