Electron Theory of Metallic Conduction. 119 



substitution of the values of K, u, A, and q previously found 

 or known, as a function of the time. The solution of this 

 last equation is easily obtained in the form 



Jt t-t x 



The suffix indicates that all the quantities affected are to be 

 interpreted as functions of t, which is taken as the argument 

 of the integration. 



This is the most generally suitable form of the function <f> 

 which is consistent with the above differential equation. 

 We must, however, choose the constants properly in order 

 to make it represent the more particular function which is 

 to be the characteristic function of our specified group of 

 electrons. To do this we must choose the constants in the 

 integrals of the equations of motion so that the position and 

 velocity of each electron is such that it occurs in the group 

 specified by (dV, dv) at the instant t. Then, again, in order 

 to remove entirely the effect of the initial conditions of the 

 system under investigation, we must choose the lower limit 

 of the integral for <f> as ( — go ). This gives us the complete 

 physical solution of our problem in the most general possible 

 form. A more convenient form for the function <f>, however, 

 expresses it as a double integral in the form 



/»« -— dr C* 



JO T '" Jt-T 



This is easily verified to be equivalent to the form given 

 above, and has an important physical significance which has 

 been discussed at length in a previous communication. It 

 is equivalent to the statement that the distribution of 

 velocities at the initial instants of beginning the free-path 

 motions being pursued at any instant t is precisely that 

 specified by Maxwell's law. This fact can be used forja 

 direct evaluation of the function <f>. 



The general form of the law of distribution of velocities is 

 thus obtained, and according to it the number of electrons 

 per unit volume of the metal at any point in it with their 

 velocity-points in the small volume dv of the diagram is 



r 



fdv = [Ae-'"'+ f" — -" dr f 'xi^'^^h'l 



L Jo T >» Jt-T J 



As stated above, the function </> is the expression for the 

 change which an external force or difference of temperature 

 produces in the state of motion of the system of electrons. 



