Theory of Metallic Conduction, 553 



Now the result obtained above in the more restricted case 

 and the general form o£ the function on the left-hand side of 

 the differential equation suggest that we may take 



where (f> 2 is]a linear function of (£, 77, f) and </>! a function 

 of the resultant velocity. If we do this we find at once that 



("•27T r*> 



whence by comparison with the left-hand side of the 

 differential equation, which is to be equalled to (b — a), the 

 general form of <p l and (j> 2 are easily deduced in the form 

 which verifies the statement tentatively made above. 



It follows then that 111 the most general possible case if 

 t ot is properly interpreted so as to include an account of the 

 factor (1 — e) expressing the persistence of the velocities, we 

 may use 



b a = fo-f 



Tin 



so that the above differential equation may be written in 

 the form 



of d?? d? oa oy o~ ot t« 



This is the fundamental differential equation of the theory 

 in a form suitable for application in the most general case 

 where it is possible to calculate t«. A good deal of infor- 

 mation crtn of course be gained in certain general types of 

 problem by a mere consideration of dimensions, but as a 

 general rule a full specification of the dynamical character 

 of the collision between an electron and an atom is necessary 

 to determine this quantity. We can in many cases make a 

 good deal of progress without any special knowledge of r m . 



The main restrictions tacitly assumed in the above argument 

 are firstly that the metal is isotropic in constitution but not 

 necessarily in condition, and, secondly, that the atoms are 

 of such comparatively large mass that the collisions of the 

 electrons with them are not effective in altering either their 

 velocities or the direction of their motion ; this latter condi- 

 tion underlies the assumption made above, that the energy of 

 an electron is unaltered by collision. Mutual collisions between 



