368 



X 0 = 4ar 1 -10a; 3 + 9x 2 -3x + l = 0 

 then X 1 = 16a; 3 - 30a; 2 + 18a; - 3 

 „ X 2 = 24a; 2 - 30a; + 9 

 ,, X 3 = 16a? - 10 



When a? = 0, these give the original coefficients, namely, 



X± X 3 X% X x X 0 

 4-10 + 9-3 + i 



and for underwritten signs, + + + + , showing no imaginary roots. 

 But when a? = 1, the results are — 



X± X 3 X 2 X± X 0 

 4+6+3+1 +J 



and the underwritten signs, + + - , showing two imaginary 



roots. 



The other two roots are found to be real ; one lies between *1 and -2, 

 and the other is -5. 



The coefficients last written are those of that transformed equation, 

 which we should get by diminishing each root of the original equation 

 by 1 — an operation much more readily performed than that above. And 

 we see, by this example, that non-primary pairs of imaginary roots in 

 an equation may become primary pairs in the transformed equation that 

 would result from diminishing or increasing each root by some number ; 

 but what this number is we can discover only by trial, or by a previous 

 analysis of the equation. Yet, by developing a real root by Horner's 

 method of approximation, whenever we find that two variations of sign 

 are lost or gained, in passing from one step of the operation to the next, we 

 may always ascertain, as here, whether the two roots passed over are cer- 

 tainly imaginary, or possibly real ; and in thus testing the several triads 

 in any of the transformed equations, the first triad need never be examined, 

 since, by the foregoing theorem, the results of such examination is 

 already known from the three leading coefficients of the primitive equa- 

 tion. And, as regards other triads, any nice calculation of the squares 

 and products of the coefficients will very seldom be necessary ; a mere 

 glance at those coefficients will often suffice to assure us whether the 

 criterion of imaginarity is satisfied or not. We can generally see, by 

 inspection, whether the square of the middle coefficient of any triad is 

 less (or not greater) than the product of the other two coefficients ; and, 

 since the numerical multiplier of the square is always less than the 

 numerical multiplier of the product, we may thus, in most instances, 

 have sufficient indication of the presence of a pair of imaginary roots 

 without any actual numerical work. 



(29) We have stated above that in developing a real root by 

 Horner's method, it would be well, so soon as the two variations are 



