672 BELL SYSTEM TECHNICAL JOURNAL 



the value which must be assigned to the cross-section of these spheres, 

 in order to make the calculated values of "missing current" agree with 

 the observed ones. 



An elastic-sphere model is also used in the kinetic theory of gases: 

 one visualizes a gas as a flock of spheres, and for their cross-section one 

 chooses the particular numerical value which, when inserted into the 

 kinetic-theory formula for the viscosity of that gas, gives a figure agree- 

 ing with the measured viscosity. This is the so-called "gas-kinetic 

 cross-section," which I will denote by o-q. One should not expect it 

 to be identical with the quantity a which has just been defined, nor be 

 surprised at finding differences — even differences in order-of-magnitude 

 - — between the two. The elastic-sphere model is good for many pur- 

 poses; but it has its limitations. 



The ratio of <j to o-q is occasionally used instead of o- as a measure of 

 the likelihood of interception. Much more frequent of usage ^ is the 

 product of this a by N\, the number of molecules in a cc. of a gas at 0° 

 Centigrade and one mm. Hg (A'^i = 3.56- 10^®). So also is the reciprocal 

 of A^icr, the so-called "mean free path " of the electrons under the stated 

 conditions, zero Centigrade and one millimetre pressure. It should be 

 called "mean free path for interception," but the qualifying word is 

 usually left out. The reciprocals of Na^ and of iVicro are also frequently 

 used as standards of comparison, sometimes with the name "gas- 

 kinetic mean free path." Since the concept of mean free path is used 

 quite often in stating the results of these experiments, I will give the 

 reasoning whence follow at once its definition, and its relation to the 

 quantity c. 



Returning to equation (1), rewrite it thus: 



dQ = - aNQdx. (2) 



Here the quantity {Q — R), the number of electrons lost per unit time 

 from the beam between the planes x and x -\- dx, is written as — dQ, 

 the negative of the difference between the numbers which in unit time 

 cross the planes x -\- dx and x respectively. Integrating, we get: 



Q,/Q2 = exp [ - Na{xi - X2)] (3) 



for the ratio between the numbers crossing any two planes separated 

 by the distance (xi — X2). 



For simplicity, suppose that it is at the plane x = that the corpus- 

 cles enter the gas, and denote by Qo the number entering per unit time. 

 Then the number Q{x) reaching any plane x is this: 



Q{x) = (2oexp(- Nax), (4) 



^ Especially by Rcimsauer, by Briiche, and other Germans generally. 



