70 REPORT— 1894. 



Now, supposing that t,qy, . . . q„ are kept constant during differen- 

 tiation, we have 



9Q, drji 9Q, 9j2 9Q2 



-^'90; "*" ^'dcr, '" 



^.^ 9(P|,P2, ■ ■ ■ P„ ) ^ 9(gl, g 2 • • • g») ^ 9(gl» ? 2 • • • ?») 



^{lh,P2, • • • ;j«) 9(Q„(52 • . .0,,) 9(Q„Q2 . . . Q„> 

 .-. 9( Pi,P2, • ■ • P.,) X 9 (Qi,Q2, ■ . . Q„ )_^_ Q_j,j) 



9bl.P2, • • . Pn) 9(9i, ?2, • • . ?«) 



T/^e Most General Law of Permanent Distribution for 

 Non-colliding Systems. 



15. The possible laws of permanent distribution of the co-ordinates 

 and momenta among a large number of such systems may now be esta- 

 blished thus : — 



We know that for any one system the total energy is independent of 

 the time or 



E= constant, 



and the determinantal relation shows that the multiple differential 



dpi dp2 . . . dq„ 



is also independent of the time. 



Therefore, if the law of distribution be such that the number of 

 systems included within the multiple differential at any instant of time 

 (0 is 



f{^)dp, . . .dq,, . . . . (9) 



where /denotes any function whatever, the same law will hold good at 

 any subsequent instant of time {t'). 



16. The above proof depends only on the fact tliat E=constant is an 

 integral of the equations of motion of the system and not on any other 

 property peculiar to E. But the equations of motion may have other 

 integrals as well. In such cases (3) does not represent the most general 

 law of permanent distribution. 



For if the integrals in question be 



7ij=const. /t.j — const., »tc. 



then any distribution expressed by the formula 



/(E, A,, 7^2, . . . .) djj^ . . . . dq„ . . . (10> 



will also be independent of the time, and therefore permanent. 



This is the most general form of the law of permanent distribution in 

 a system of non-colliding bodies. It is applicable in particular to the free 

 systems considered in Part II. of Maxwell's paper, where /»,, /tg • . • may 

 denote the velocity components of the centre of gravity, and the component 

 angular momenta about the centre of gravity. 



17. Again, take the case of a number of particles distributed unifornily 

 throughout infinite space and moving uniformly in straight lines under no 



