98 Mr. J. R. Christie on the Extension of Budan's Criterion 



two more reciprocal transformations will give |8iv v' — l as 

 the imaginary part of the corresponding pair of roots, (3i V de- 

 pending in value on fi u as /3 n does upon /3; we get therefore, 

 from the equation j3 n = "134, the value 



/3 =-033 

 as that which exceeds all values of (3 which can make /3iv less 

 than *5. 



It appears therefore that, in the case of a greater than *5, 

 a small odd number of reciprocal transformations can hardly 

 fail to detect the imaginary roots, supposing «„ always greater 

 than *5. 



Taking now the case a not greater than *5, we shall obtain 

 the least value which fi x can in this case hold, by making in 

 it a. = '5; it becomes then > 



which is precisely the same as the inferior limit to the value 

 of /3 n in the preceding case: it follows therefore from what 

 has been there shown, that if a be not greater than -5 the se- 

 cond reciprocal equation must detect the imaginary roots, un- 

 less /3 be less than *134. 



On a similar hypothesis the third reciprocal equation can- 

 not fail unless jS, be less than *134-, which involves the con- 

 dition |3 less than *033 : and so on. 



In thus developing the changes which this limit of |3 suc- 

 cessively undergoes, it has been assumed that «„, the real part 

 of the imaginary roots in the nth reciprocal equation, retains 

 the character assigned to it in each particular case ; but it is 

 manifest that if it does not retain its character, the change 

 will only have the effect of altering the hypothesis from u^ '5 

 to a < *5, or vice versa. 



Independently of the additional value which these consider- 

 ations give to the criterion of Budan, there is yet another most 

 difficult case which the same operations tend to elucidate, viz. 

 that in which two or more roots are nearly equal to one another. 

 In fact, let a and b be two roots very nearly equal, both of them 

 positive and less than unity, a condition always attainable ; in 

 the first reciprocal equation these roots will appear under the 



11 7 



form — and -7-, and their difference becomes r~, greater 



a a ° 



than before, since a b is a proper fraction. Now to whatever 

 extent the roots of this equation are diminished, their differ- 

 ence is unaltered; if therefore this difference should still be 

 less than unity, another reciprocal transformation will again 

 increase it; so that each transformation must of necessity 



