Electrons concerned in Metallic Conduction. 217 



o£ the quantities concerned are such, in nearly all cases, as 

 to reduce the integrals to first terms in their series repre- 

 sentations. 



The results thus obtained represent, in the opinion of the 

 writer, the only possible consequences of a rigorous treatment 

 of the problem, and are in a form which admits of ready 

 comparison with the results of other theories. This com- 

 parison is the object of the latter part of this paper, in which 

 it is shown, moreover, that these formulae alone give a good 

 representation of the experimental phenomena. The paper 

 does not propose to deal with the values of other magnitudes, 

 such as the mean free path of the electrons within the solid, 

 or the emissivity of a plate, its scope being strictly limited 

 to an examination of the number of electrons which are 

 effective in conveying the current. This will account for 

 the absence of certain important references, which bear upon 

 this special problem in a more indirect way. 



Wilson's mode of treatment of the problem of conduction 

 is essentially that of Jeans, with the added hypothesis that 

 collisions do not sensibly alter the velocities of the electrons. 

 Collisions with atoms do not, because of the much greater 

 mass of an atom, and collisions with atoms are the more 

 numerous. . Thus if N is the number of electrons in a unit 

 of volume, and e?]N the number with a resultant velocity 

 between V and V + dY, then the group tZN has, in a sense, 

 a permanent existence, and can take the place of a " class " 

 of the ions in Drude's theory, although individuals ma}' be 

 entering or leaving the group at any time. If u is the mean 

 velocity in the group, along the direction x of the electric 

 force X, the equation of motion of the group becomes 



d/dt(mud^)=Xed^-umYd^/l m , . . (1) 

 as derived from a consideration of momentum gained and 

 lost by the group. The mass of the electron, of whatever 

 nature, is m, a magnitude to be regarded as effectively con- 

 stant. The charge on any electron is e, and l m denotes 

 (irnR 2 )' 1 , n being the number of atoms in the unit of volume, 

 and R the sum of radii of electron and atom. 



For motion under a periodic force X = a cos pt, of frequency 

 pl'2iT, the solution of this equation is 



udN = eadNcos(2>t-S)/>»(p 2 + V 2 / l m 2 ) h , • • (2) 

 where tan B=pl m /Y. 



The mean velocity along x of the electrons in the whole 

 volume is u , where 



Nao== jtfdST,. '....... (3) 



where the integration extends over all the groups. If the 



