369 



lost or gained in a transformed equation, to apply the test of imagi- 

 narity anew ; but we need not wait till such changes occur : the test 

 will be as likely to discover the presence of an imaginary pair 

 before any change of variations takes place, as after; so that when 

 in the case of doubtful roots in the assigned interval, we carry on the 

 work of approximation in uncertainty as to whether these doubtful 

 roots may eventually turn out to be real or not, reference to the criteria 

 should always be made at each completed step. Thus, take the ex- 

 ample at page 308 of the " Theory of Equations," namely, 



12a 3 f 24z 2 - 5Sx + 25 = 0, 



in which it is doubtful whether the two roots indicated between *7 and *8 

 are real or imaginary. Proceeding on the supposition that they are real, 

 the first figure of each root must be # 7. Diminishing by this number, 

 the transformed equation is 



12z 3 + 49-2^ - 6*76a? + '276 = 0. 



The next figure, still presuming the roots to be real, is '06 ; and the 

 next transformed equation is 



12x 3 + 51-36^ - -7264x + -050112 = 0. 



Taking the first of these three equations, we see at a glance that 58 s is 

 greater than 24 x 25, and greater even than four times that product, or 

 24 x 100 ; so that here there is no indication of imaginary roots. Tak- 

 ing the second equation, the conclusion is similar : it is seen at once 

 that 6*76 2 must exceed not only 4*92 x 2 # 76, but also three times this 

 product. But as respects the third equation, the conclusion is different ; 

 •7264 2 is obviously less than 5*136 x -50112 ; and therefore we may be 

 certain that the two roots, hitherto in doubt, are imaginary. 



Suppose, however, instead of the inferior limit *7, between which 

 and *8 the roots are indicated, we had taken the superior limit, and had 

 diminished each root by -8 : the transformed equation would have been 



12x 3 + 52-8x 2 + 3'Ux + -104 = 0, 



in which the condition of imaginarity is satisfied ; for three times 

 52-8 x -104 is greater than 3*44 2 , since it is pretty obvious that 3*44 2 can- 

 not be so great as even 15. 



(30) Of all the known methods for determining the numerical 

 values of such of the roots of an equation as may be real — after by 

 trial, or by a previous analysis, the intervals in which they lie are as- 

 certained, Horner's method of continuous development is to be preferred. 

 And if this method be employed in combination with the criteria of 

 imaginarity in the case of doubtful roots, we shall always be led to 

 satisfactory conclusions respecting all the roots of a numerical equation. 



But even without applying these tests, the true character of the 

 doubtful roots may always be discovered by proceeding onwards with 



