82 PROFESSOR TAIT ON THE 



22. The investigation of the separate values of the parts of this expression 

 is a little more troublesome, as the numerators now involve second partial 

 differential coefficients of I x ; but it is easy to see that we have 



- = _i \dh" til-) Tl " 2 ( 3 dh ~ dk) l * /3 + Is/5 ii + 2k 



16 I 3 /3 2h(h + k) 



- 1 a-0 I .-<a^.)v3-3i,/5 t 



uv = - T , 



16 y3 2(h + k) 



and, from these, the above result again follows. 



[It is clear, from the investigation just given, that the expression for the 

 value of u 2 — uv would be the same (to a numerical factor iwes) whatever law we 

 assumed for the probability of the line of centres having a definite position, 

 and thus that Maxwell's Theorem would be true, provided only that the law 

 were a function of y alone, and not of <f> (i.e., that the possible positions of 

 the line of centres were symmetrically distributed round the direction of 

 relative motion of the impinging particles). In my first non-approximate 

 investigation (read to the Society on Jan. 18, and of which an Abstract 

 appeared in Nature, Jan. 21, 1886) I had inadvertently assumed that the possible 

 positions of C were equally distributed over the surface of the hemisphere of 

 which V is the pole, instead of over the surface of its diametral plane. The 

 forms, however, of u 2 and of uv separately, suffer more profound modifications 

 when such assumptions are made.] 



V. Bate of Equalisation of Average Energy per particle in two 



Mixed Systems. 



23. To obtain an idea of the rate at which a mixture of two systems 

 approaches the Maxwell final condition, suppose the mixture to be complete, 

 and the systems each in the special state, but the average energy per particle 

 to be different in the two. As an exact solution is not sought, it will be 

 sufficient to adopt, throughout, roughly approximate expressions for the various 

 quantities involved. We shall choose such as lend themselves most readily to 

 calculation. 



It is easy to see, by making the requisite slight modifications in the formula 

 of § 12, that, if m be the number of Ps and n that of Qs in unit volume, the 

 number of collisions per second between a P and a Q is 



2mn W Jk J > 

 where s now stands for the sum of the radii of a P and of a Q. For if, in the 



