304 Dr. L. Natanson on the Dynamical 



systems require closer examination. Suppose Q to represent 

 a quantity characteristic for a system in a given state ; i. e. a 

 function, first of the initial conditions of encounter, and 

 secondly of the time t, which in the system under considera- 

 tion has elapsed from the beginning moment of encounter. 

 If we multiply the above number (1) by 



T$f(v)dvdt, (3) 



we shall have the number of encounters which occurred at 

 moments lying infinitely near one another and under initial 

 conditions differing but infinitely little. In all these encoun- 

 ters the simultaneous values of Q are practically equal, hence 



TSQf(v)dvdt, (4) 



multiplied by (1), is the instantaneous Q-sum for the 

 v, 10, ^, ^-subclass considered; and the sum for the v, io, -^r-class 

 is found by integration from to r. Now let Q denote the 

 mean value the variable quantity Q assumes in the course of 

 the encounter ; since, then, 



f 



Q#=Qt, (5) 



the Q-sum for the total mass of gas is seen to be 



7 L^L^Qrwf(v)¥(v,w)smf cos f dv dw df. . (6) 



3. The kinetic energy of two molecules suffering mutual 

 interaction we shall divide in two parts, which will correspond 

 respectively to the mutual relative motion of the molecules 

 and to the motion of their centre of inertia. Call L the 

 mutual relative kinetic energy of both molecules and S the 

 virial of the forces between them ; then, with the above nota- 

 tion for averages, and with m to denote the masses of the points 

 (supposed to be equal), we have 



Ij — o= 2t ' ^ ' 



by a theorem which is closely connected, but cannot be said 

 to be identical with the ordinary Yirial-theorem. (See Wiede- 

 mann'' s Annalen, vol. xxxiii. p. 698.) 



4, The virial-equation is now easily constructed. If we 

 use E to denote the mean kinetic energy of a free molecule, 

 E c the mean kinetic energy of the centre of inertia of a 

 bimolecular system, and p the external (normal and uniform") 

 pressure, we find 



