s/\ 



and its Application to the Molecular Motions of Gases. 115 



velocity-component referring to the ^-direction lies between a 

 value x s and an infinitesimally different value x 1 ' + dx' is repre- 

 sented by the formula 



1 _^ 2 

 — -p=. e « 2 dot?, 



a\/7r 



wherein e is the base of the natural logarithms, and a a constant 

 which depends on the vivacity of the motion, while x' may have 

 any value from — x to + go . An expression of the same form 

 then holds also for the relative velocities of the different combi- 

 nations of two molecules each. If we apply this law to the 

 above-introduced quantity z, it reads : — The probability that the 

 quantity denoted by z lies between the values z and z-\-dz is 

 represented by the formula 



I e-Wdz, 



IT 



wherein z may have any value from to go . Accordingly, ad- 

 mitting this law, we obtain for the determination of the function 

 / the equation 



M = \/-o-i" (85) 



V 7T 



We will now, for a single one of the moving points whose mo- 

 tions are independent of one another, form the equation express- 

 ing the theorem of the mean ergal ; and we will keep to the form 

 (IIIc), which, if U provisionally signify the ergal of a single 

 point, is applicable to this case, and reads : — 



A dc 



Herein, for our present purpose, the first sum on the right-hand 

 side refers to the three coordinate-directions, and the second to 

 the six boundary-planes of the parallelepiped. 



But now, in the present case, not only are the motions of the 

 different points independent of one another, but also, for each 

 point, the components of the motion along the different coordi- 

 nate-directions can be independently determined. Accordingly, 

 with respect to one of the coordinate-directions (which we will 

 again name the ^-direction) we can form one of the preceding 

 equations corresponding to it. Therein we must put 



1 — 7 m fdxV 

 and the quantity u has the signification 



(86) 



12 



\dt) 



