for the Imaginary Roots of an Equation, fyc. 97 



Supposing that the variations (which correspond to those of 

 the original equation whose indications of roots, real or ima- 

 ginary, we are attempting to discover by aid of the criterion) 



disappear from the equation in (p +.1), the roots in the 



immediately preceding equation will be of the form 



a 2 + /3 * P- a « + j 8 2 



or «-p(a 2 +/3 2 ) + /3j/-l 



a 2 4- |S 2 ' 



and in the second reciprocal equation 



(a* + ff 2 ) . {q - p (a 2 + (?) + g S^l] 



which finally reduces to 



(1 -p a) 2 + f /3 2 -{l-paf+p /3 2 ' 



Now p evidently represents the integer next less than — - — 



a 2 + /3' 2 , 



to which, if we assume a. greater than *5, the superior limit is 

 2; consequently, in this case, p = 1, and the above expression 

 becomes 



q-(a 2 + /3 2 ) +/3 4/:=! 

 (1 - a) 2 + /3 2 



or a n + n */^-\, 



making « u = ( ,_ a)2 + /32 and ft, - (1 _ a)2 + /3 *. 



It is easily seen that /3 n decreases with the value of a, and 

 the lower limit of its value will therefore, in this case, be 



<S 

 •25 + /3 2 ' 



which decreases with the decrease of /3; solving therefore 

 the equation 



£ _. 5 

 .25 + /3 2_ * 

 we obtain (3 = 1 ± *866 ; and since /3 must be less than '5, 

 we have /3 = • 134 as the value which /3 cannot exceed if /3 U 

 is less than *5. 



Should /3 U be less than '5 (^ and u n being greater than *5) 

 Pfo7. Mag. S. 3. Vol. 21. No. 136. Aug. 1842. H 



