FAILURE OF OHM's LAW AT HIGH CURRENT DENSITIES. 169 



nal force. The elements that enter our analysis are the acceleration of 



the electron (a), its mean free path (/), and its normal velocity in the 



absence of the external impressed force (v). The departure from 



Ohm's law is to be expressed in terms of these. As we have defined 



it, the departure from Ohm's law, D, is a ratio of similar quantities, 



and is hence dimensionless. The only dimensionless combination of a, 



al ^^ 

 Z, and z) is --;. Hence we have 



D=f 



Considerations of symmetry show that the unknown function must be 

 an even function of its argument, since reversing the direction of the 

 acceleration leaves the resistance unaltered. Also the acceleration 

 of the electron is determined by its charge, mass, and the applied force 

 by the equation 



Ee 



a = — • 



7n 



Hence finally we have the result 



/Eel\ 

 D = even function I — : I . 



\mry 



MaxW'ClI assumed, as was perfectly natural, that the function was 

 algebraic, and hence that the first term was proportional to E-, it 

 being an experimental fact that the constant term vanishes. In the 

 light of the experiments above, however, it is exceedingly question- 

 able whether this assumption is justified. It seems to me that the 

 curve hugs the axis at the origin much more closely than an algebraic 

 curve, and in fact may have contact of an infinitely high order, like 

 an exponential curve. I have not succeeded in satisfactorily repro- 

 ducing the course of the curve with two or three terms, although this 

 may be done with a fair degree of approximation. I found this signi- 

 ficant thing in trying to fit an algebraic curve to the experimental 

 results. If the first term, that is the square term, corresponds to 

 anything real physically, then its magnitude in a three constant 

 formula should not be very different from its value in a two constant 

 formula, the terms of higher order constituting merely correction 

 terms. This is not the case. The square term in a three constant 

 formula fitting the results for 8 X 10-*^ gold was only about one fifth 

 as large as in a two constant formula. It seems to be likely, therefore, 

 both from the appearance of the curve and this evidence from com- 



