436 MATHEMATICAL APPENDICES 40* 



be traversed, that is, if ut>r. From this we obtain the required 

 pressure p by multiplying by 2ww cos s, dividing by df, and 

 integrating. If, again, we introduce polar coordinates, we have 



f " 



o 



and this quintuple integration gives, as before, 

 p = 



f e'^^du (*6-*B1r l dt [ ni dr [**ds sin s cos 2 s f 

 Jo Jo Jo Jo J 



41*. Number and Mean Collision-impulse of the 

 Colliding Particles 



The number of particles, by the collision of which against the 

 walls of the vessel this pressure is set up, is easily calculated by 

 the same methods. We obtain it from the foregoing formulae 

 by suppressing the factor 2wo> cos s in them. The result of the 

 integration is that the unit area is struck by NQ particles 

 in a unit of time ; and since these give rise to the pressure 

 p = ^TrNmft 2 , the mean value of the impulse of a single particle 

 is 



These considerations show that it is not right under all circum- 

 stances to calculate mean values in the mode invented (see 10) 

 by Joule and Kronig. 



42*. Another Calculation of the Mean Free Path 



Our formulae may be employed with proportionate ease to 

 calculate in another way the value of the mean free path. 



The number of all the particles issuing in unit time from unit 

 volume is 



Be ~ *"*** 



o o 



where B is again put for /3w, and the sum of the paths traversed 

 by them till they next collide is 



47T- *N(lcm)* f Be- **"Vtf w fV*^ dr. 



J o J o 



The values of these integrals respectively are 



2N/(27r)N 2 s 2 /&w and 

 and their ratio 



is the mean free path, as has already been found. 



