174 



Mr. L. R. Manlove on Fourier's Th 



teorem 



The process may be shortly described as follows: — 



(a) Ascertain by Fourier's original theorem the pairs of 

 consecutive integers between which the roots must lie. 



(b) Where the number of roots in any interval is doubtful 

 proceed as if approximating by Lagrange's method to the 

 roots in that interval. The real roots will be separated ; the 

 imaginary roots will lead to derivative equations which can be 

 seen to have unity for the superior limit of their positive roots. 



(c) Apply the test for equal roots only if and when in the 

 course of the investigation the existence of equal roots appears 

 probable. 



It is submitted that the process is as systematic as that of 

 Sturm's theorem without involving the labour frequently 

 required by Sturm's method. 



Equations of the 17th and 18th degrees have been treated 

 without difficulty by Fourier's theorem with the modification 

 described above. 



Example. 

 * 6 -6* 5 -f* 4 + 50* 3 -62* 2 - 104* + 170 = 0. 

 (Positive roots.) 

 f(x) = x 6 - 6* 5 + * 4 + 50* 3 - 62x 2 _ 1 04* + 1 70, 

 f(x) = 6* 5 -30* 4 + 4* 3 + 150* 2 - 124*- 104, 

 f"(x) = 30a* - 120* 3 + 12* 2 + 300* - 124, 

 /'"(*) = 120* 3 --360* 2 + 24* + 300, 

 /**(*) = 360* 2 — 720* + 24, 

 f%x) =720*- 720, 

 / VI (*) = 720. 



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