348 On the foundations of the Kinetic Theory of Gases. 



These values are deduced from particular cases of the 

 curious expression 



J e-^x dx( \ *e-«V */ (^HH/ 2W+ ' ~^/ 2n+ ' ) 



•J^ e-^ydy(y+^ n+1 -y~x 2n+1 )j 



2n— I 



2 



which, in its turn, may be made to depend upon the following 

 pair of fundamental theorems : — 



f 6-P x2 xdx\ *e-w 2 dy= X 



Jo Jo 4/? V? 



f 



*/o 



±p S/p + q 



e-P x 'dx\ €-™ydy = 



-av 2 ». ,7„._ V 77 " 



o J* 4^ Vp + q 



6. Another question of importance in the theory regards 

 the proper definition of the Mean Free Path. From the point 

 of view adopted by Maxwell, and since taken by Meyer, 

 Watson, and others, the mean free path in a system of equal 

 spheres is 



average speed of a sphere 

 average number of collisions per sphere per second 



It seems to be more in accordance with the usual sense of the 

 word " mean " to define the mean path as the sum of the pro- 

 ducts of the mean path for each speed into the chance of that 

 speed. And those who adopt the above deviation from the 

 ordinary usage must, I think, face the question, — Why not 

 deviate in the opposite direction, and define the mean path as 

 (average time of describing a free path) x (average speed) ? 



The numerical values deduced from these three definitions 

 bear to one another, in the above order, the ratios 



0-707 : 0-677 : 0-647 ; 



unit being the length of the mean path of a particle (whatever 

 its speed) if all the others were reduced to rest, and evenly 

 distributed throughout the space which they occupied while 

 in motion. 



