306 Dynamical Illustration of the Isothermal Formula. 



For passive molecules t=2R cos yjr/w and b = ; for hard 

 elastic spheres t = and b = ^co, because, as will imme- 

 diately be shown, e=-J. This is in accordance with Yan der 

 Waals's result, since his molecular volume corresponds to 



5. Let us consider the number of encounters which happen 

 per unit time between molecules of class i\ and class t* 2 re- 

 spectively with the further restriction of the angle between 

 the Vi- and v 2 -directions being >A, and <\+t/X. If, for 

 reasons" presently to be seen, we substitute v 2 cf>(v) for f(y), 

 as written in the above, we may put that number 



= iirWR 2 Vi 2 v 2 husm\(f){v 1 )(l>{v 2 )dv 1 clv 2 d\. . . (16) 



Now, in (16), we introduce the relative velocity w of the 

 two molecules, and the velocity V of their centre of inertia, 

 by means of 



Vl 2 + v 2 2 = iw 2 + 2Y 2 (17) 



and v 2 -\-v 2 — 2^2 cos \=w 2 ; .... (18) 



it follows that the number of ^^V-encounters per unit time 



is irWWvyV^v^tv^dv^wdV. . . . (19) 



We have next to integrate this with respect to v x 



from v 1= \/(iw 2 + Y 2 ^ivY) to r 1 = \' (\io 2 + Y 2 + w V), 



and this may be done directly, the result being simply wY, 

 because the product <£(tfi) . 0(i? 2 ), by a method of proof 

 originally given by Clerk Maxwell and since frequently used 

 in the Kinetic Theory, may be shown to be of the form 

 yfr (zy)^(V) . Hence the number of w?,V-encounters per unit 

 time is ^ 7rWBPw s Y 2 '4r(w)^(Y)dw dV, . . . (20) 



and the law of distribution for the V-velocities is expressed 

 by Y 2X P(Y)dY with a proper constant prefixed. Now the 

 only relation between i?„ v 2 , w, and V is clearly that given by 

 equation (17). From this equation, then, and from 



log^ 1 )+log^ 2 )=logf^) + log^(V), . .(21) 

 the form, of the functions may be inferred, and the con- 

 clusion may be drawn that the mean value of V 2 is equal to 

 half that of v 2 or v 2 . Hence 



E = E c and e=^ (22) 



This result shows, besides, that we are justified in taking 

 NE = fBi say, where t is the absolute thermodynamical 

 temperature. 



