164 Prof. Ludwig Boltzmann on the 



begins are then 



m + m' m m + m! m 



m! m! m 2 m 



m + m' m fc * 



w x = i—r ,w. . (b) 



For the number of impacts which take place in unit volume 

 during time St, under the conditions specified above, we find 

 without difficulty the value 



dn = D 2 . ¥(x, y, z, u", v", iv n , u, v, w)/(u v v ly u\) 



x Yedx . . . diu" du dv dw du Y dv 1 dw 1 d\ St. 



In this expression V is the relative velocity of the two 

 atoms at the moment of collision, and e the cosine of the acute 

 angle between the line of motion and the line of centres. 

 If we introduce instead of u ± , v x , u\ the variables p, q, r, from 

 equation (6) we find 



dn = D 2 F/i ( ffl + ™) Ve dx... div" da do dw dp da dr dX St, 



in which the suffix 1 denotes that the values of the variables 

 as given in equations (6) are to be substituted in the function. 



In each of the impacts just specified a shell loses velocity- 

 components u, v, w, and hence in all the dn impacts 2 log F 

 will be diminished by dn log F. 



After each of the impacts considered let the velocity- 

 components of the shell be between u' and u ! + du ! , v' and v + du', 

 id and id + did. In order that shells may be formed with 

 these new velocities % log F must be increased by dnlogF', 

 where the affix denotes that the variables x . . . w", u 1 , v', id 

 are to be substituted in the function F. "We assume that the 

 collisions are instantaneous, in which case the variables x . . . %d\ 

 are unaltered by the impacts. The total increase of S log F 

 by reason of the impacts considered is, therefore, 



dnf}og¥'— logF). 



In like manner we find that Slog/ receives an increase 



dn (log// -log/0 



during time St from the same impacts, the affix and suffix 

 meaning that the following values of the velocity-components 

 of the impinging single atom after impact are to be substi- 

 tuted in the function f. 



