Statistical mechanics 



55 



42. Integration regarding b. 



The expression (111) for Vyfc ^aay be written in tlie form 



(116) Vp= \i \j \k j{Siji]fJJ>^'^à.,AB,db, 



if Sijk is iutroduced instead of Ryu . 



We consider especially the coefficient V200 which (together with V020 ^"*^ 

 V002) i^ particular interest in statistical meclianics. 

 We have 



and 



['^200] = ^ — [1] = 

 so that, according to (113), 



[Ä,oJ = ^ sin 2& cos 2^ (— 2«2 + y2 + 10^) . 



In the integration regarding h the values of ii^, v^, w^, u^, v^, tv^ are to be 

 kept constant, whereas is dependent on h. Hence we liave to consider the integral 



(117) 7 = / sin ■^O' cos 2^ 6(0 



(the quantity w is here brought within the sign of integration though independent 

 of h). Having evaluated this integral we get 



QC-j _ Zl . 1 



200 „2 



(118) 



V200 ==' I j(- + v' + w') If J, A^, Zfs, 



It may be observed that in all developments hitherto made the law of Newton 

 is taken into account only at one point, namely for deducing the relation oi = oi' . 

 This relation, however, holds good as soon as the potential of the forces vanishes 

 at infinite distance. It is valid for elastic bodies as well as for the repulsion law 

 of Maxwell used in the kinetic theory of the gases. We hence conclude that 

 all developments hitherto made hold true unaltered for all such laws of attraction or 

 reptdsion. It is at the integration regarding h that a difference first occurs for 

 different laws. Here this difference takes place at the evaluation of the integral I. 



It happens for that of Maxwell adopted law (1 : r^) that 1 is reduced to a 

 numerical constant. It is hence especially independet of co. Through that circum- 

 stance the whole problem is considerably simplified as is shown at the integration 

 of (118). There is no doubt that this simplification has been a contributive reason 

 why Maxwell has supposed that the molecules of a gas repel each other inversely 

 as the 5:th power of the distance. Another advantage of this law is that the limits 

 of the integral I may be assigned as (0 and) 00 , which, as we shall find, is not 

 allowable for the law of Newton. 



