the Molecular Velocities in Gases. 57 



For brevity let us agree to designate by 8, for any collision 

 whatever, the numerical difference of the squares of the new 

 velocities ; in the present case we have 



S=yY'cos.0', whence 8 = or < V V. 

 Nextj let 2 be a given positive number. If the number of 

 collisions for ^vhich 8<z is required, the answer will be 8= ft 

 if :>YY'. since the condition is satisfied by all the colli- 

 sions; but if :<YY', on the acute angle a being chosen so 

 that we have V V cos u=z, the relation 8 < z gives cos & < cos « 

 or 6' > a; and, as we have seen, the number of collisions sought 



will be fju cos «, or vfv^- On replacing z by z + dz, those num- 

 bers in the two cases will be /j, or ^~ ~ -- } taking the differ- 

 ence, we shall have the number yJ' of collisions for which 8 

 is comprised between z and z + dz: consequently 



p"=Oifs>W; n"=^tfz<YY f . . (5) 



Thied Disposition. 



The molemles are again of two hinds, in number n, n'; for 

 each kind the velocities are all equal, viz. u for the first, v for 

 the second. For the first hind all are parallel and in the same 

 direction; for the second they are in all directions indifferently. 

 We neglect the collisions occurring between the molecules of 

 the second kind. 



Let us suppose a typical sphere described having for its 

 centre a fixed point 0, and for its radius unity; let us divide 

 it into elements co, and take each of them for the typical region 

 of a group of velocities — that is to say, of the group of mole- 

 cules of the second kind to which those velocities belong. 

 The velocities being in all directions indifferently, the number 



of molecules of the group will be -j — , 47r being the surface of 



the sphere. The number of collisions effected between this 

 group and the molecules of the first kind returns into the 

 second disposition, and will be deduced from fj, by replacing 



n' by n' Putting 7 ,. , R \ 



J 4-Tj- ° h = annt, (6) 



formula (1) will give /j,= ; the number of collisions for 



which 8 is comprised between z and z + dz will be, according 

 to equations (5), /x // = or =Tfv=7, f° r the sni gl e group. Let us 



