Fourier* 's Theorem and Roots of Equations. 173 



oscillations occur on a sphere with no charge, throws some 

 light on the point here involved. 



The subject is, however, rather irrelevant to the main 

 objects o£ this paper and need not be pursued. As before 

 stated, the oscillations which do occur are very soon damped 

 out, and we have then only to deal with the quite determinate 

 particular solution of the equations involved. 



The problem of the sphere accelerated from any initia 

 velocity can easily be deduced from that given above if 

 attention is paid to the few remarks offered by Walker in 

 explanation of the single difficulty involved. 



In a third and concluding paper I intend to give solutions 

 on the above lines of the various types of oscillatory motion 

 discussed by Walker. A short account of the radiation 

 taking place from the various motions under review will also 

 be added. 



Sheffield, 



1911. 



— 



XII. Proposed Method for the better practical Application of 

 Fourier's Theorem concerning the Roots of an Algebraical 

 Equation. By L. R. MANLOVE *. 



HOW to determine the situation and number of the real 

 roots of any equation is a problem which has engaged 

 the. attention of many mathematicians. 



" Sturm's Theorem " is a complete solution of the problem, 

 but its application to equations of a high degree is very 

 laborious. 



" Fourier's Theorem " (it has been said) " has the ad- 

 vantage that the auxiliary functions employed in it can be 

 formed by inspection, so that the method can be applied 

 nearly with equal ease to an equation of any degree. The 

 objection to this method is that by its immediate application 

 we only find a limit which the number of real roots in a given 

 interval cannot exceed, and not the actual number ; and 

 that the subsidiary propositions by which this defect is 

 supplied are not of the same simple character as the original 

 theorem," 



It is suggested that the following method provides a simple 

 practical substitute for the " subsidiary propositions " above 

 referred to, thus completing Fourier's Theorem and, perhaps, 

 making it fit to replace Sturm's Theorem in the textbooks. 



* Communicated bv the Author. 



