216-220] General Dynamical Theory 185 



218. An exception to the statement that all the fluid will be in a single 

 cluster occurs when there are a number of clusters which are similar in every 

 respect, so that expression (452) is identical for all. This will, for instance, 

 occur in the case of a gas, owing to the possibility of interchanging the roles 

 of the different molecules. 



219. We must now return to equations (443) and (445), and examine 

 whether or not the condition is satisfied that we may neglect terms of degree 

 higher than the second in (p^ PI), etc. Upon the assumption that we may 

 do so, we have found the value of < <J> at the boundary of a cluster to be 

 equal to 1 (equation (451)). Let us suppose that the corresponding value of 

 (pi PI) is of the order of magnitude Sp, then the first term which has been 



neglected is (piPi) 3 * 3 or (8p) 3 ? 3 . Failing definite knowledge it is 

 opi op\ 



natural to suppose that ^-5- is of the order of magnitude of ,^ . 3 , and 



hence that the neglected term is of the same order of magnitude as 1. In 

 other words, the neglect of the remaining terms in equation (445) cannot be 

 justified. 



But this only affects the detail of the result we obtain : the main result 

 remains. The value 4> being still a maximum for <I>, the locus <I> = 4>j will 

 surround the point <1> , although it will only be ellipsoidal when <E> ^ is 

 very small. The integral in expression (450) must be replaced by an integral 

 of the form 



/' 



J < 



(453), 



in which f(x) is an unknown, but always positive, function of x, which 



becomes identical with x 2 in the neighbourhood of x = 0. The integrand 

 in this integral will, as before, have a maximum, and the integral will derive 

 its whole value from the contributions supplied by values of < in the neigh- 

 bourhood of this maximum. Also, as before, the value of the integral 

 involves <I> merely through the factor e n *. Again, then, it appears that the 

 fluid will flow into clusters surrounding maximum values of <E>, and the 

 strength of such a cluster will be proportional jointly to e n * and to another 

 factor. All except an infinitesimal fraction of the fluid will be in a single 

 cluster, or, in cases such as that contemplated in 218, family of similar 

 clusters, but as already explained, it is impossible to identify this cluster 

 or family of clusters by general arguments of the kind which have so far 

 been used. 



220. Under the special assumption that we may neglect terms of degree 

 higher than the second in equation (443), we have at any point inside a 

 cluster (equation (445)), 



h[^~*?:|s+*^ 



by equation (451). 



