ON OUK KNOWLEDGE OF THERMODYNAMICS. 105 



and by a well-known theorem (cf. Watson, 'Kinetic Theory of Gases,' 

 2nd edition, p. 22) 



da db . . . dp dq . . .= dAdB . . . dF dQ 



Therefore the number ^ is equal to 



nKe-'-sdadb . . .dpdq... 



But this last expression gives at the time the number of vessels for 

 which the co-ordinates and momenta of a molecule lie between the limits 

 (9) (according to formula 3). We see that this number remains con- 

 stant during the time t ; and since the same is true for all values of a, 

 b, . . . p, q, . . . the theorem holds good for all molecules of the first kind, 

 (ii) We call all those molecules ' molecules of the second kind' which are 

 free at the time 0, but which are in process of encounter at the time t. 

 For a pair of such molecules let the co-ordinates and momenta lie at the 

 time between the limits 



A, andA,-l-<^A„B, andBi-t-^B, . . . P, and P, +<^P|, Q, and Q, -l-^Q, . . . "I ,,p^, 

 and Aj and A^ + dA„, B., and B, + dB., . . . P^ and P^ + dP„, Q, and Q, + f/Q^ . . . J \^ ^/ 



respectively, and at the time t between the limits 



a, and «, + da^, J, and b, + dl, . . . p, and jo, + dp^, q^ and q^ + dq^ . . .\ ,-,-,^ 



and a„ and a., + da„, b.^ and k, + db^ .. . p., and p„ + dp.,, q„ and q., + dq.,...\ ' v ^ ^ / 



respectively. Because these molecules were free at the time 0, the 

 number of vessels in which at time a pair of molecules fulfils the 

 condition (10) is, according to formula (5), 



nHV^'«'-«"'c^AirfB, . . . d^.dq, . . . dA^dBo . . . dP.,dQ., . . . 



Gi and G.2 are the energies of the molecules at the time 0. But the 

 above-mentioned vessels are identical with the vessels for which at the 

 time t a. pair of molecules fulfil the conditions (11). The number of the 

 last kind of vessels is therefore also given by the above expression. It is 

 easily seen that this expression is equal to 



nlPe~''^da^dbi . . . dp^dqi . . . da.^db^ . . . dp^dq.^ . . . 



where / is the whole energy of the two molecules at time t. Compari- 

 son with formula (7) shows that the last formula gives also the number of 

 vessels in which at time a pair of molecules fulfilled the condition 

 (11). Therefore the theorem also holds good for the molecules of the 

 second kind. 



(iii) Molecules which are in process of encounter at time 0, but are 

 free at time t ; 



(iv) Molecules which are free at times and t, but which have been 

 encountered by another molecule between these two instants of time. 



It is easily seen that our theorem can be proved in the same way as 

 before for every pair of molecules of the third or fourth kind. 



To calculate the mean vis viva T of a molecule we put 



Xi=kiip+ki2q+ . . ., x.2=k2iP + ko2q • . ., &c. 



The coefficients k may be chosen to be functions of the co-ordinates such 

 that T acquires the form h {)n^x^^ + 1n^x/+ . . . m^x/), where/ is the 

 number of degrees of freedom of a molecule. The probability that for a 



