120 Mr. G. H. Livens on the 



Now, as a matter of actual experience, this change is known 

 to be extremely small in all real cases ; so that various 

 methods of approximation may be adopted in evaluating the 

 integrals in the expression for /. General rules can hardly 

 be given for such methods, as they will essentially depend on 

 the conditions of the particular problem to be dealt with ; 

 but the few particular cases worked out in detail below will 

 illustrate the points quite clearly. It may, however, be 

 worth noticing two conditions which the physical circum- 

 stances of the case necessarily imply for the function </>. 

 We know that whatever be the new state of the motion of 

 the swarm of electrons, we shall always have a definite 

 number N of electrons per unit volume, and a definite value 

 of the mean square of their velocities. These conditions 

 imply that 



§<j>dv = 0, §<j)u 2 dv = 0, 



the integrations being over the whole of the space in the 

 velocity diagram. This implies, generally speaking, that 

 <f> is an odd function of (f, rj, f); as a matter of actual 

 experience, a linear function is usually found sufficient. 

 The implied non-existence of square terms in the expression 

 shows that squares and products of direct accelerating forces 

 are negligible, as are also the products of these forces by the 

 gradient of the conditions in the metal. 



The formula; thus far are perfectly general and involve 

 probably none but very easily justifiable assumptions. A 

 few applications to special cases are worth noticing. 



1. If the acceleration is due to a steady uniform electric 

 field of intensity E, and the conditions of the metal at each 

 point are also constant in time, then the integrations for <j> 

 are immediately effected. In this case 



<>E 

 R=— , 

 m 



and thus 



X=^( M E)-i(«V)A+«>V)?. 



Thus x depends on the time only through the terms involving 

 f , 7], f, and u. As, however, these quantities only vary from 

 their undisturbed values by quantities of the order of E, we 

 may neglect their variable parts altogether. With this 

 approximation % is independent of the time, and we thus 

 have 



f=Ae-^[l + r, xX ], 



