C(58 Practical application of Fourier's 7 heorem. 



The ratio of the energy transmitted per unit of time per degree 

 of freedom to the energy stored in that freedom takes here 

 the place of the element of energy, and there is no limitation 

 upon its finitude. There is necessarily equipartition of energy 

 amongst a ]l freedoms for which that ratio has one and the 

 same value. 



LXV. Supplemental Note on a Proposed Method for the 

 better practical application of Fourier s Theorem. By 

 L. R. Manlove * . 



rT^HE method suggested by the writer in the Phil. Mag. for 

 JL July last involves this assumption : — 



i{ When, being uncertain whether there are any real roots 

 of an equation between two given consecutive integers, we 

 proceed as if approximating by Lagrange's method to the 

 roots in that interval, then in case no such roots actually 

 exist we shall ultimately obtain a derivative equation which 

 can be seen to have unity for the superior limit of it's 

 positive roots. " 



No formal proof is attempted, but the following consider- 

 ations show that the assumption is well founded. 



Ess hypothesi we have a series of derivative equations none 

 of which has a positive root greater than unity, and for the 

 pre ent purpose we may treat these as independent equations. 



Given that an equation has in fact no real root greater 

 than unity, and that nothing further is known, what is the 

 probability that it can be seen to have unity as a superior 

 limit of the roots ? 



Suppose that this probability is p for each equation ; then 

 the probability of the first equation failing is l—p\ and the 

 probabi ity of the first n equations all failing is (1 —p) n which 

 may be made as small a quantity as we please by taking n 

 large enough. 



In testing the method with some scores of examples the 

 writer has only in one case found it necessary to go so far as 

 the 3rd derivative equation. Of forty equations taken at 

 random only five required more than one derivative to clear 

 up an interval. 



* Communicated by the Author. 



