1194 



These are, accordingly, the Iwo time integrals of the square of 

 velocity ii' before and during the collision. 



Hence we shall have for the mean square of velocity li" ^= - = — - — 



t ^ + ^, 



u» = k ".*- — — -7= -r^ '(6) 



beine- with (p = — I X — the required expression tor tr, expres- 



sed in u^\ and which will be valid for small volumes « Vk) for 

 all temperatures. 



§ 5. Two Important Limiting Cases. 

 a. High temperatures. 



For ^^, = od (r/^ = 0) we now get : 



M* = è «0 7= ' 



as — log { — y -|-^^^ +'/'') then approaches to — /ö(/ (1 — f/) = ^Z, and 

 likewise /0// (>/? -|- ^ ^1-f 7') and (py'l-\-<f\ For y near the first 

 terms will be cancelled by the second, and m' will, therefore, 

 approach to 



(r = oo) Ü' = i u,\ (7) 



so that the time average of the square of velocity for small volumes 

 and high temperatures amounts to only half the square of velocity 

 in the neutral point. In consequence of the disappearance of the 

 terms with / by the side of those with b the time average is namely 

 chiefly formed by the diminution of velocity (hiring the collision, 

 and not by the increase before the impact in consequence of the 

 attraction. This hitter increase lasts so short that it may be neglected 

 with respect to the subsequent important diminution of velocity 

 (down to 0). 



Now for a linear system /Vw w' is not =3R7\ hut simply = RT, 

 and in the general relation 



'— * y m / 



for the vis viva at the beginning of the collision, i.e. for the sum 



