the Molecular Velocities in Gases. 61 



(6), (8), (9), (12) will again give the total number m of col- 

 lisions between the molecules of different species, and the 

 number m! will be that of the collisions at which one of the 

 new velocities is comprised between x and x + dx. 



Fifth Disposition. 



Let us now pass to the real case, in which the velocities are 

 infinitely variable. Let N be the total number of the mole- 

 cules of a gas. On making the velocity vary from to infinity 

 by equal small increments, the molecules will be divided into 

 a like number of groups; and we will designate by Scf)(v)dx 

 the numbers of those whose velocity at the commencement of 

 the time t is comprised between x and x + dx. The collisions 

 against the sides do not alter the velocity] but if the whole of 

 the velocities of all the molecules together be considered, every 

 collision between these has the effect of suppressing two velo- 

 cities and creating out of them two others. The number of 

 of velocities comprised between x and x + dx which are sup- 

 pressed during the time t is that of the collisions which have 

 happened between the single group numbering Scfi(x)dx, on 

 the one hand, and all the groups, on the other. It will there- 

 fore be the sum of the values of m. on replacing therein n, n' 

 by S6(,v)dx. S(j)(y)dv, u by x, and making v to vary from 

 to infinity. According to formulas (6) and (8) this sum will 

 be S 2 atTJr(x)dx. putting 



f(x)=<j>(x)^f(x,v)cf>(v)dv. 



(13) 



As to the velocities comprised between x and x + dx which 

 are created during the time t. they may proceed from colli- 

 sions happening between any two groups whatever, corre- 

 sponding to the velocities u and u + du, v and v + dv. The 

 number sought will therefore be obtained by replacing in ex- 

 pression (12) n and n' by T$<f>(u)du } ~S<f>(v)dv, then adding 

 those values of the result which correspond to all those of u 

 and v. In this way, however, every possible association of 

 two groups will be found to have been reckoned twice; so 

 that half of the sum must be taken. According to formulas 

 (6) and (12) the number sought will be 

 TPatyfr f (x)dx, 



P lltth3 S r „ r - 



>lr'(x) = 2x\ \ F(u,v)<f>(u)<f>(v)dudv. . (U) 



«- i- 



The number of the velocities comprised between x and .i' + dx 

 will constantlv increase or decrease until the instant when an 



