FOUNDATIONS OF THE KINETIC THEORY OF GASES. 1035 



The first integral we have already had as part of the encounter. To simplify the 

 second, let s/a = cos a, and it becomes 



/C cot a 

 udue ~ w2/2 (u? + c 2 —u 2 sec 2 a)' i > 



which, with c 2 — u 2 tan 2 <x = z 2 , gives 



+5(z 2 -C 2 )cot 2 a 



cot 2 a / z*dze 



or 



<V-COt a 



(|) tan 3 a |-*»«**-P / aAfajS* 1 



The whole is now 



R = 



/<V 7 , . *,? Z* ^ cot « I 



\ /9\l -?£-COt2a / 2 I 



= -^^{ e e2 y|+ V2e - V2(/ o *"*% + v '2«tan*a- ^""'Wa^ ."&>} 



= -^cos 2 a{g ea v /^+V2e 8 ec 2 a- N /2gy s^dy- J2r* e0t \s,n*aj e x2 dxl , 

 which, when e = and cos a = 1 , becomes 



~ P V 21 

 as in § 30. 



It would at first sight appear that the value of the impact is finite (=— Pe./V) 



when there is no nucleus (i.e. a=«A But, in such a case, we must remember that the 



second part of the first expression for E above has no existence. In fact the value of 

 the second of the two integrals is ^2 tan 3 a . e cot a, when e cot a is small ; and this 

 destroys the apparently non-vanishing term. 



XVII. Effect of Encounters on the Free Path. 



§ 61. If two particles of equal diameters impinge on one another, the relative path 

 must obviously be shortened on the average by 



f ' 2 2tt sin 6 cos 2 6d0 



2s 

 3 ' 



7 2tt sin 6 cos l 

 o 



But if v, Vi be their speeds, and v their relative speed, the paths are shortened respect- 

 ively by the fractions v/v and v^Vq of this. The average values must be equal, so that 

 we need calculate one only. 



