and the Partition of Energy in Continuous Media. 241 



arrangement selected at random shall have a u cc 2 , ... a n as the 

 numbers of molecules in the successive n cells, is * 



N! n~* 



a x \ a 2 . . . . a n ! 



a-i a 9 



Hence the probabilitv that =^, ^ . . . shall lie within limits 



<§>■<©•••> ' 



Let us now transform variables from a h a. 2 . . . a n to 

 ?i> ?2? • • • ?«* where f l9 f 2 ? • • • f» are given by the equations 



a £ = ^+"S"f sin2!5,&e. ( s =l, 2,...n). . (17) 

 0) ?<ft) 2=) 7 n 



In the limit, when n is made infinite, this equation becomes 



q-n x 



v— v =2£ sm T qTT, 



q = l y l 



where vis the molecular-density at a distance x along the tube 

 (supposed of the total length /), and v Q is the average value of 

 v (cf. equation (14)). Thus f x , £ 2 , . . . f„ ultimately l.)ecome 

 proportional to the amplitudes of waves of wave-lengths 

 21, i(20, i(il), &c. 



Let expression (16), transformed to variables £ u £ 2 j ■ • • ?»> 

 be supposed to become 



/(fi,6»... *.)<*& <*&,...<*&; 



so that this expression will measure the probability that 

 fi! ?s» • • • £n shall lie within limits rff l5 </f 2 , . . . rff». From 

 equation (17.) we have 



so that 



^/ai a, \ 



where A is the determinant whose (s, q) term is sin (sqir/n), 

 a pure number and a constant. 



* ' Dynamical Theory of Gases/ p. 39. 



