*»i PROFESSOR TAIT ON THE 



27. Partly as a matter of curiosity, but also because we shall require a 

 case of it, it may be well to mention here that similar processes (in which it is 

 no longer necessary to break the y integration into two parts) lead to the 

 companion formula 



hn _ /g-A*2 xdx / g -t y 2 ydJ x + J" -x-y n 

 2/i J o J o 



n-l 



= tt_ 1 .3.5. .. (2n-l) (h+k) 

 ~ 4 2" 2n + 1 



(Kk) 2 



And we see, by Wallis' Theorem, that (when n is increased without limit) 

 I 2n is ultimately the geometric mean between !,„_! and I 2 „ +1 . 



VII. Mean Path in a Mixture of two Systems. 



28. If we refer to § 10, we see that, instead of what was there written as — ehx, 

 we must now write —(e + e^Sx; where e v which is due to stoppage of a 

 particle of the first system by particles of the second, differs from e in three 

 respects only. Instead of the factor 4s 2 , which appears in e, we must now 

 write (s + Sif; where s-^ is the diameter of a particle of the second system. 

 Instead of h and n we must write h 1 and n x respectively. 



Hence the mean free path of a particle of the first system is 







which, when the values of e and e x are introduced, and a simplification 

 analogous to those in §§ 9, 11, is applied, becomes 



in which 



Thus the values tabulated at the end of the paper for the case of a single 

 system enable us to calculate the value of this expression also. 



VIII. Pressure in a System of Colliding Particles. 



29. There are many ways in which we may obtain, by very elementary 

 processes, the pressure in a system of colliding particles. 



