6Q REPORT — 1894. 



7. He next assumes that the momenta (now denoted by a,, ... a„) 

 may be so chosen as to reduce the kinetic energy to a sum of squares, or 



T = i2Mrar^ (4) 



With this assumption, he integrates (3) with respect to the momenta, and 

 finds by Dirichlet's method 



p 'c?a2 . . . da„_^ 2-iT(ln) (E- Y-l;n„a„2)i(u-3) 



where j,=/xia[. Hence he infers that if ^„=^/i,„«„2 is the part of the 

 kinetic energy arising out of the momentum o„, then the number of systems 

 in a given configuration, in which k,, lies within limits differing by dk„, is 



r(in) (E-v-^J^'^. _,^j. 



mnun-1)] (E-Y)'<" 



(5) 



and that since this expression only involves k,„ therefore the law of distri- 

 bution of the kinetic energy is the same for all the momenta. Multiplying 

 the above expression by k„, and integrating from A;„^0 to A;„^T=E — V, 

 we find that the mean value of k„ is 



K=^(E-V) = 1t . . . (6) 



71 n 



the maximum value being of course equal to T, because the portions of 

 the kinetic energy due to the other momenta cannot be negative. Hence 

 Maxwell infers ' that the average kinetic energy corresponding to any one 

 of the variables is the same for every one of the variables of the system.' 

 This result is commonly called Clerk MnxiielVs Theorem. 



8. In Part II. of the paper Maxwell deals with a free system, con- 

 sisting of n particles not acted on by external forces. For such a system 

 not only tlie energy but also the velocity- components of the centre of mass 

 and the components of angular momentum round this point in any three 

 fixed directions will be constant throughout the motion. Maxwell therefore 

 assumes them the same for every system. Under these circumstances the ^n 

 momentum-components of the system are not all independent, but seven 

 of them can be expressed in terms of the rest by means of the seven 

 equations of condition, and the law of permanent distribution is expres- 

 sible in terms of the multiple difierential of the Sw co-ordinates and 3w — 7 

 of the velocity components of the particles. The algebra is very long 

 and laborious, and need not be examined in detail here. The objections 

 to Maxwell's investigations can be much more easily discussed and 

 criticised with reference to the simpler case considered in Part I., and 

 the law of distribution in a free system can be treated more simply by 

 alternative methods {vide §§ 16-18, § 45, and appendices A, B below). 



The Assumption that the System passes through every Phase consistent 

 iviih the Equation of Energy. 



9. This assumption probably presents greater difiiculties than any 

 other part of the Kinetic Theory, and it is therefore advisable to com- 

 mence by stating under what circumstances it requires to be made 



