28 Ladislaus Natanson on the Kinetic Theory 



If we obtain the expression for the differential coefficient 



j r v/a 



* r Y^e-^^F (V) } and apply it to the cases F (V) = I e ~ x2 dx 

 d V r v/$ Jo 



and F(Y}= l e~ x2 dx, we shall be able to calculate J x and J 2 



easily. Their values are 



j.= 



IT 



8v/3 



77. 



NiEi,V ; 



1^12 



J 2 =-"N 2 R 2 2 



n/« 2 + 2/3 2 



(8) 

 (9) 



and substituting the values of Bj and B 2 from (3) and (4) in 

 the expression (5) for e, 



N 



6 = 



^ 2 Vrfc 



N,E, 



V 3(2 + ^NjRu 2 + V 6^N 2 E* S 



(10) 



It follows from the definition of fi (§4) that e = 3(l— fi), and 

 therefore equation (10) gives us a means of calculating the 

 value of ft. As the general expression for //, is complicated 

 we shall content ourselves with taking the two extreme cases. 

 It can be shown that the minimum value of \x corresponds to 

 Nj = ? the maximum value to ]ST 2 = 0. In the first case the 

 equation for jjl is 



18/<1 + ,*)(W) 2 =1; 



and in the second case, 



27(2+m)(1-/0 1 =1; 



from which the value of & must always lie between 0*805 and 

 0-886. 



We can check this calculation in the following manner. 

 The probability of a molecule having a velocity between V 

 and VH-cZV is (from § 4) equal to 



c 



TY 2 e- 2y2 ^ 2 dY 



Since (B 1 + B S )T=1, we have 

 W JUr _ =z: where J 



"I "bt+b; 



dv. 



(12) 



