Theoretical Evaluation ofy. 641 



factor either F— /(E), or else some power of this factor, or 

 else some other function of this factor which vauishes when 

 the factor is put equal to zero. In order that a simply 

 periodic variation in F and E may be possible [i. e. in 

 order that sound may be propagated in the gas) the first of 

 these alternatives must be the true one. In this case, assuming 

 the values of E and F to vary only slightly from their equi- 

 librium values E 0; F , equation (1) may be written in the 

 form 



§ = -Xl"E«)[F-/(E)], .... (5 ) 



where x(-^o) is some function of E which is positive if the 

 gas tends to return to the steady state in which E = E , F = F . 

 Assuming as a general solution 



E = E + E 1 ^ f , 



F = F + F 1 «v<, 



we find from (5) the equation 



*pF 1 =- x (B )[F 1 -/(E„)E 1 ]; 



or, what is the same thing, 



Fi=/3E„ (6) 



where 



*=W( 1+ *fc) c7) 



§ 4. Let w, p be the pressure and density of the element 

 of gas under consideration, and let us assume as a solution 

 for isr, p the values 



•sw. + w^ (8) 



p =p +p l e i » t (9) 



The quantities nr, p, and E are connected by the relation 



=*% (10) 



uJ: 



where X is a constant. Substituting the assumed values of 

 -or. p, and E, we find at once the relation 



^ = f + <^ (ii) 



We further suppose that no heat enters or leaves the 

 element in question, so that the total energy of the element 



