ARISING FROM INEQUALITIES OF TEMPERATURE. 689 



by the number of encounters of each molecule per second in which p. lies 

 between p, and p. + dp., and integrating for all values of p. from 1 to + 1 . 



This operation requires, in general, a knowledge of the law of force be- 

 tween the molecules, and also a knowledge of the distribution of velocity 

 among the molecules. 



When, as in the present investigation, we suppose both the molecules to 

 be of the same kind, and take both molecules into account in the final 

 summation, the spherical harmonics of odd orders will disappear, so that if we 

 restrict our calculations to functions of not more than three dimensions, the 

 effect of the encounters will depend on harmonics of the second order only, in 

 which case I*(p.)- I =f (/t 2 - 1) = -f sin 2 20. Note added May, 1879.] 



(2) Number of Encounters in Unit of Time. 



We now abandon the dynamical method and adopt the statistical method. 

 Instead of tracing the path of a single molecule and determining the effects 

 of each encounter on its velocity-components and their combinations, we fix 

 our attention on a particular element of volume, and trace the changes in 

 the average values of such combinations of components for all the molecules 

 which at a given instant happen to be within it. The problem which now 

 presents itself may be stated thus : to determine the distribution of velocities 

 among the molecules of any element of the medium, the current-velocity and 

 the temperature of the medium being given in terms of the coordinates and 

 the time. The only case in which this problem has been actually solved is 

 that in which the medium has attained to its ultimate state, in which the 

 temperature is uniform and there are no currents. 



Denoting by 



dN =/, ( r), x, y, z, t) ddi)dt,dxdydz 



the number of molecules of the kind M l which at a given instant are within 

 the element of volume dxdydz, and whose velocity-components lie between the 

 limits g$d, "n : kd'n> ^> Boltzmann has shewn that the function / must 

 satisfy the equation 



............ (8) 



VOL. II. 87 



