10 i REPORT — 1894. 



=He~''' at time 0, where g is the total energy of this molecule ; H and h 

 are constants ; therefore at time the number of vessels in which a free 

 molecule is in the state (1) is 



Tj,c?\V=mHe~'''c?a dh . . . dp dq . . , . . . (3) 



Similarly the number of vessels in which a free molecule appears with 

 co-ordinates and momenta between 



a' and a' + da', b' and b' + db' . . . p' and jj' + dj/, q' and q' + dq' . . . (4) 



(condition 4) is 



ndW'=nB.e-"«'da'db' . . . dp'dq' . . . 



where g' is the total energy of this second free molecule. Finally, the 

 number of vessels in which one free molecule is in condition (1) and a 

 second one in condition (4) is 



n dW c?W'=«H2e-"<'+"'' dadb . . . dp dq . . . da'db' . . . dp'dq' ... (5) 



Let a", b", . . . a'", V", ... be values of the co-ordinates such that a 

 molecule with the former co-ordinates acts on or encounters a molecule 

 with the latter co-ordinates. And let us assume that at the time the 

 number of vessels which contain a pair of molecules whose co-ordinates 

 and momenta respectively lie between 



«"and a" + da". b" and J" +db" p" andp" + dp", Q"andq" + dq" . . . \ ^r\ 



a'" and a'" + da'", b'" and b'" + db'", . . .p'" andj/" + dp'", q"'&ndq"' + dq'". . . J \^) 



is 



'nB.h-''fda"db" . . . dp"dq" . . . da'"db"' . . . dp"'dq"' ... . (7) 



where/ is the total energy of the two molecules. We proceed to prove 

 that a stationary state is defined by these formuUe. Consider a duration 

 of time t long enough to permit of encounters between a finite number of 

 molecules, but not so long as to permit of many molecules colliding more 

 than once. We must demonstrate that after this time t, the number of 

 vessels in which the state of a molecule lies between certain limits is 

 exactly the same as before this time. We distinguish between four kinds 

 of molecules : — 



(i) Molecules which are free at the beginning and at the end, and during 

 the whole time t. For any of these molecules let the co-ordinates and 

 momenta lie at the time between 



AandA + c?A, BandB + cm, . . . P and P + (ZP, Q and Q -f- dQ ... (8) 



and at the time t between 



aanda-j-c?a, 6andi-|-c?i, . . .^j and^? + (f^9, 5'and5' + (/5' . . . . (9) 



Let G and g represent the energy of such a molecule at the times 

 and t respectively ; G, g being equal. According to equation (3) the number 

 of vessels in which at the time the co-ordinates and momenta of a 

 molecule lie between the limits (8) is «He~''^c?A o?B, . . . dP cZQ. . . . But, 

 by hypothesis, the co-ordinates and momenta of these same molecules lie 

 between the limits (9) at the time t ; hence the above expression gives 

 also the number l^ of vessels in which, at the time t, co-ordinates and 

 momenta of a molecule lie between the limits (9). But we have Gr^g, 



