the Electrical Conductivity of Metals. 445 



To obtain the current density across the plane PM due to 

 the electrons coming from the left of the plane, we must 

 multiply 8a by e, and after replacing x by R cosijr, integrate 

 first with respect to R from to co , and then from yjr = 

 to yjr = 7r/2. Doing this we obtain 



nve r 1 2Xe\ \ ,^ ^ 



T\2 + "WJ l j 



The corresponding quantity due to the flow of electrons 

 from right to left is obtained by replacing X by —X in the 

 above, and the difference of these two current densities 

 represents the resultant current density. Taking this differ- 

 ence and dividing by X we obtain for the conductivity <r 



2 ne 2 \v ne 2 \v * 



<7 = 



3 rnv* 3a# 



It will perhaps now be well to indicate the exact reason for 

 the discrepancy between this result and the result indicated 

 by equation (1) and found by Drude. In the deduction of 

 (1) the argument usually given is equivalent to the following: 

 " The electrons may be looked upon as all travelling the 

 distance A between two collisions, the time taken to travel 

 this distance being Xjv. The velocity created by the field in 



"XT" -v 



this time is , and the average velocity created by the 



mv ' » J J 



field in all the electrons to be found at an instant in any 



place is ~— . The current density is consequently ne .. , 



... 2mv „e*Xv„ W 



which gives a= ■. ~ . 



The portions in inverted commas must not be looked upon 

 as the arguments of the present author. They simply re- 

 present the arguments from which (1) may be considered 

 to have been deduced. It will be noticed that the assumption 



X^A 



that the average velocity is half is equivalent to the 



assumption that the electrons to be found at any point have 

 on the average travelled a distance A/2, which appears at 



* We are not altogether justified in writing mv 2 =2u3 y since the 

 existence of ?. steady temperature involves that all the electrons shall not 

 have the same velocity. The same assumption is, however, involved 

 in (1). It is not this assumption with which I am here particularly 

 concerned, though in Part II. of this paper I have worked out the problem 

 taking account of the velocity distribution. 



Phil. Mag. S. 6. Vol. 27. No. 159. March 1914. 2 H 



