184 Theory of a Non-Conservative Gas [OH. vm 



By differentiation, the volume for which E = E V while 3> lies between <J\ 

 and 4> t + d<&, will be 



2 1 2M ~ 3 



fL-JiC (<&.-<&,) d. 



Hence the aggregate mass of fluid representing systems for which E=E l and 

 <I> lies between <I> and <&!, is 



2n-l 



r*o 



1 



The integrand vanishes at <E> = <1> . After this, as <f> decreases, the integrand 

 increases until it reaches its only maximum, which occurs when 



*,-*=i-i, I 



and then decreases. In the limit when n = oo , this maximum is given by 



3> = 4>o-l (451), 



and if this value lies between <f> and ^> 1 , the whole value of the integral (450) 

 is given by contributions from the immediate neighbourhood of this maximum, 

 and is equal to 



.(452). 



This result, however, only holds subject to the condition that we may neglect 

 terms of order higher than the second in equation (443). 



217. Assuming for the moment that this condition is satisfied, our result 

 shews that all the fluid for which E ' = E lt and which is in the neighbourhood 

 of the point of maximum density <& = <J> may be supposed to lie within the 

 boundary defined by equation (445). In other words the fluid is seen to flow 

 into small clusters of stream lines, a cluster surrounding each maximum value 

 of <E>. All except an infinitesimally small proportion of the fluid for which 

 E = E will lie in one or other of these clusters. 



The amount of fluid surrounding the maximum <I> = 5> is given by 

 expression (452). In comparing the amounts surrounding two different 

 maxima <E> and ^o', the ratio of the two factors e n *, e n<s> ' will of course be 

 infinite or zero according as 3> <J> ' is positive or negative, but it is unfortu- 

 nately impossible to determine the ratio of the remaining factors, for the ratio 

 of the two (7's (equation (449)) may be infinite or zero. We therefore see 

 that in general the ratio of two expressions of the form of (452) will be either 

 infinite or zero, but it is impossible to determine which. In other words, we 

 can see that all except an infinitesimally small amount of the fluid will be in 

 a single cluster, but it is impossible to identify the particular cluster. 



