26 Ladislaus Natanson on the Kinetic Theory 



mean value V 2 is therefore for all molecules, 



f f f . . rY^e-^^x^, y,...)dYdxdy... 



V 2 =-J^ ; . (3) 



| J J • - • rV**-*»l*x(m,y, . . .)dV dxdy . . . 



the limits of integration with respect to %, y, . . . being x , a? 2 ; 

 y^y x \ ... according to the hypothesis (/3), and f , £ l3 t; , ^ . . . 

 on the (a) hypothesis. According to the (ft) hypothesis, t is 

 independent of V ; but on the other assumption (a), t depends 

 upon the time which elapses between the collisions, and there- 

 fore on V. If we accept the (ft) hypothesis as the true one, 

 we obtain 



V 2 ={« 2 ; (4) 



whence 



E s =mY 2 = im** = E 1 (5) 



If, on the contrary, we take the (a) hypothesis, we must 

 write iT for t, where T denotes the time between two succes- 

 sive collisions of a molecule whose velocity is V ; and ^ + 1 is 

 the number of collisions which the molecule experiences before 

 breaking up, reckoning as first collision that in which the 

 molecule is formed. We can treat t as dependent on V only 

 as T depends on V, and then i will be determined by the 

 quantities a?, y, . . . . We have then 



**=%* (6) 



\ TW 2V2 / a W 

 Jo 



Now since the relation 



(" > TV i e- 2y2 l« 2 dY= |« 2 CTY 2 e-^/«* d y 



+ j f " V3 • |^- 2V2/a2 ^v- ~ Prvv 2 ^/« 2 1" 



holds good for all values of T, and T must be finite at V=0 

 and equal to zero at Y = oo (as follows from the definition of 

 T), we must have 



V^=|« 2 (l+|) and E^E^l+l), . . (7) 



