458 MATHEMATICAL APPENDICES 52* 



52*. Signification and Value of the Integral 



The integral that occurs in the last formula has a simple 

 meaning, in virtue of which we are enabled to solve the problem 

 in 95, viz. to find the mean perpendicular distance, represented 

 by A, by which the original layers of the molecules that diffuse 

 through any section are distant from that section. 



In the preceding paragraph we first found the number of the 

 particles meeting the section considered ; to this was added the 

 factor rcoss, or the perpendicular distance of one of the original 

 layers, by the development of the function N. Consequently the 

 somewhat altered expression 



I""" 



would be nothing else than the sum of the perpendicular distances 

 for all the particles that meet the unit area in unit time. If we 

 divide this sum by the number of particles that meet the area, 

 which by 41* is ^N^l, we obtain the mean value we are seeking, 



A = | 



Employing as before the symbol M to denote the mean value of 

 a magnitude within brackets, as calculated on the assumption of 

 Maxwell's law, we may write for this 



where I is the mean free path of the molecules moving with the 

 speed w. 



Since the integration cannot be carried out, we are obliged to 

 estimate its value by approximations. It seems sufficient to put 

 for B its mean value, which was represented by F in 29*. We 

 then get 



and hence, since by 19* the mean value of w 2 is 



M(^) = G 2 

 we obtain 



A = 



where is the mean free path of the molecules. 



