(T 



of the Optical Properties of Metals. 839 



The amount o£ heat produced per c.c. per sec. will be equal 

 to the mean value of Neu a cos pt and to JaV where <r 

 denotes the conductivity. Hence we get using (2) and 

 taking the mean value 



When p = this gives 



//7 \l/2 .2 7 



u \7r / m 



Hence 



f y 6 - ? V2^y 



: -^°J 1 + ymV^VW' • • W 



If we take all the velocities to be equal we get instead 

 (T =Ne 2 L/mY ; and 



__ °j) (A\ 



ff "l+/mW/W W 



which is the expression found by Jeans. Drude expressed 

 uas a sum of terms like (4), one term for each class of 

 electrons. If we regard each group dN as constituting a 

 class, then Drnde's expression becomes identical with (3) 

 allowing for the change of notation. Drude got it by 

 assuming the motion of the electrons in each class to be 

 opposed by a viscous resistance proportional to u. 



Equation (4) has been used by Schuster and Jeans to get 

 estimates of N from the values of cr deduced from optical 

 observations. The integral in (3) unfortunately cannot be 

 expressed in finite terms, but it can be found of course by 

 graphical methods. I find that (3) gives values of N about 

 double those given by (4) in most cases. 



The conductivity for any frequency can also be obtained 

 in another way by calculating the heat produced directly. 

 Suppose as before that the metal contains N free electrons 

 per c.c, and let I denote the length of a free path and V the 

 velocity of an electron. The kinetic energy of an electron 

 will be altered during a free path by the action of the electric 

 force. If we calculate the total gain of kinetic energy for 

 all the free paths traversed by all the N electrons in one 

 second, the result will be the amount of heat energy produced. 



* II. A. Lorentz, in ' Theory of Electrons/ gets <r less by the 

 factor 2/3. 



