REPORT ON CERTAIN BRANCHES OF ANALYSIS. 335 



the corresponding scheme will be as follows : 



X'' V'v Viii Vii Yi Y 



96 42 

 (-1) + - + _ + - 



(-i) 



36 9 



+ - + - + 



12 2 2 



(0) + + + - + 



24 6 4 2 



12 



a) 



I 



+ + + + + - 



36 10 



(1) + + + + + + 



If we take the interval from (— 1) to 0, we find two roots in- 



42 6 . 

 eluded within it ; but since qp + hi is less than the interval, 



no certain conclusion can be drawn with respect to the nature 

 of the corresponding roots. If we now consider the interval 



from — -^ to 0, which includes the same roots, we shall find 



9 6 1 

 — + — = —, a quantity equal to the whole interval, and we 



are consequently authorized in concluding that the correspond- 

 ing roots are imaginary. In a similar manner, we find the in- 

 dication of the existence of two roots between and -^ ; and 



2 1 



in as much as -j- = -5- = the whole interval, we at once con- 

 clude that the two roots in question are imaginary*. 



It thus appears that we are enabled, by the processes just 

 described, to separate all the real roots of an equation and to 



* When we speak of the existence of imaginary roots between two limits, 

 we do not mean that such limits comprehend the moduli of these roots, but 

 merely that the real roots which would be found between those limits, if cer- 

 tain conditions were satisfied, are wanting, and that there are as many ima- 

 ginary roots of the equation which may be said to correspond to them which 

 are sufficient to complete the required number of changes of siga which are 

 lost. The theory of Fourier as given in his work, determines nothing con- 

 cerning the values or limits of the moduli, or of the peculiar nature of the 

 signs of affection, of such imaginary roots. 



