Constant in the Kinetic Theory of Gases 451 



Now cos x = vlv<' = Vl^ =" ^ (say)- This substitution gives 



AQ" (7) = - ^(j'^ttV \ ipt^ (1 - t^)^ (2«2- i)dt. 

 Jo 



It is now necessary to specialize the law of force. 



§ 6. The Inverse Power Law. We take 



f(r) = ^/r«, 0) (r) = k/{n- l)'r«-i, 



where s,. is the coefficient of I*" in the expansion of (1 — |) 2^. Thus 



^ 72 w + 2r a^+2'- F2a"-ion+2r ' 

 AQ" (7) = - |!^/^^| .-3_A_(1 _ ,.)i (2,._ 1) ,,, 



which readily evaluates to 



"^ r + 1 



where A„ is the sum of the series 2 -r^ — t^^tto — , rrx /o , q\/o — ; — \"- 

 " (2'-+7)(2r+5)(2r+3)(2r+n) 



The next step is the evaluation of 



^R" ^ r ^a"{V)V^e-i^''^^"dv 



• 



cr'^r^ J 



" ct'*-^ [hmj 

 Now O (ct) =- hjin- l)a"-^ Substituting, 



where /.,' = 24 (n - 1) S (2, + 7) (2r + 5) (2r + 3) (2r + ri) * 

 This will be found to lead'to 



Now h = 1/2RT, O (a) = 2</. {a)/m. Thus 

 ^ ^ (g) 1— hn 

 R 3 ' 



