652 Mr. Gr. H. Livens on tlie 



3. Equi-partition of energy amongst the oscillations. 



It has always appeared to me very difficult to see how the 

 simple application of a direct physical principle, such as 

 the theorem of the equi-partition of energy employed by 

 Eayleigh and Jeans, should lead to a result which is entirely 

 at variance with the fundamental assumptions on which the 

 theorem is based, and in a search for the origin of the dis- 

 crepancy I find that an oversight occurs in the usual proofs 

 advanced by Rayleigh and Jeans, a consideration of which 

 will largely remove the difficulties inherent in their method*. 

 This oversight, as is clear from the results obtained, is 

 mainly concerned with the convergence of the Fourier series 

 expansions usually employed, and may be sufficiently ex- 

 plained by a reference to a simple illustrative problem 

 examined by Rayleigh, viz. that of the partition of energy 

 among the vibrations of a string of length I stretched between 

 two points. The form of the string at any instant may be 

 regarded as defined by the coefficients in the Fourier ex- 

 pansion of the displacement y at a point distance x from one 

 end, viz. 



y= 2 <£„sm — -, 



n=l An 



21 

 where X„= — : the kinetic and potential energies are then 



of the form 



M=l «=1 



a, b being definite constants, the forms of which are not 

 however required in the present connexion. Applying then 

 the general theorem to determine the partition of energy 

 among the various oscillations in a steady chaotic motion of 

 the string in equilibrium with a system at temperature T, 

 Rayleigh is led to the result that on the average the values 



of a<f> n 2 , bn 2 <f> n 2 (n = l, 2, ) are all equal to &T, k being 



the usual absolute constant in gas theory. This result is, 

 however, incompatible with finite values for T and V. 



In obtaining such a result, however, no notice has been 

 taken of the fundamentally necessary restrictions placed on 



the values of the coefficients <£ l5 $ 2 > </>*> by the 



assumption of the existence of the Fourier expansion for the 



* It would appear that a similar difficulty has presented itself to 

 Prof. Love, but he does not appear to offer any considerations in solution 

 of it (Phys. Zeit. xiv. 1913, pp. 1300-1301). 



