FOUNDATIONS OF THE KINETIC THEORY OF GASES. 1037 



Hence the average space passed over in that interval is 



72 s 



f» fiv. v jvldv.l\*=J- 

 ZuJ vv 1 1 3 



If we put a for s in this expression we have the amount to be subtracted from the 

 average path between two encounters in consequence of the finite size of the region of 

 encounter. 



XVIII. Average Duration of Entanglement, and consequent Average Kinetic Energy. 



§ 63. We have next to find the average duration of entanglement of two particles : — 

 i.e., the interval during which their centres are at a distance less than a. 

 The whole relative path between the entering and leaving encounters is 



2(a cos 0— s cos -^r) , 

 or 



la cos , 



according as there is, or not, an impact. 



Hence the whole time of entanglement is the quotient, when one or other is divided 

 by io. And the average value, for relative speed u, is 



t = — 2 / (a Jw 2 — u 2 sin 2 6 - Jw*s 2 - a 2 u 2 sin 2 6 ) cos 6 sin $d0, 



=sM 5 (w3 - c3) - »( wh * - (^ 2 -^ 2 ) f ) } 



when ivs>au', 



and 



= ~ 2 \ I ajw 2 —u 2 sin 2 6 cos dsinOdO—/ Jw 2 s 2 — a 2 u 2 sin 2 6 cos 6 sin 6d6 c , 



when ws <aw . 



These must be multiplied by the chance of relative speed u, as in § 58, and the result is 



cs 





'0 



or, with the notation of S 60 



2ah 2 ft« 2 /2 



/"* co r* c cosec o 



{J c ^(^ (l - COs3a) - e3 y^ l2+ J c 1Z (c 2 -w 2 sm 2 afe- hwi/2 } 

 \ / 3 / c 2 — * 



= 2ah? ekC 2 l2 JdAuf 3( 1 _ cos 3 a )_ c 3V-^ 2 /2 + 2a^ 2 g -2 :c0t20 / g 4 cfe e +fa»c«ec*,/2 

 VOL. XXXV. PART IV. (NO. 22). 7 g 



