RECENT STATISTICAL THEORIES 733 



angle l3 = arc cos (^/v) between the initial path of the electron and 

 the axis of x, as follows: 



cos CO = cos /3 cos \p + sin /3 sin \p cos 0. (105) 



We now have everything necessary to do the integration in equation 

 (103), and we find: 



b - a = (v^ll)g(v). • (106) 



Substituting this into equation (93), the condition that the distribution- 

 function (/o + ^g) shall be stable by virtue of perfect balance between 

 the rates at which electrons are shifted from compartment to com- 

 partment by the impacts and by the accelerating field, we get: 



(eE/m) ^ (/o + ^g) = (v^/l)g. (107) 



If the term ^g{v) is truly small by comparison with the term fo(v), 

 we may neglect the second term on the left; and since (dfo/dv) = 

 (dfo!dv)(dvjd^) = (^/v) (dfo/dv), the culmination of all the argument is 

 in the formula: 



{g(„) = i 'A f (108) 



V- m dv ^ ^ 



for the alteration which the applied electric field imposes on the 

 distribution. Notice that g involves ^ and r? and f only in the combi- 

 nation v; this justifies the procedure of Lorentz. 



Now each electron which during unit time crosses any surface 

 imagined in the metal contributes an amount e to the current through 

 that surface; but the contributions made by electrons crossing in 

 opposite senses are opposite in sign — what we perceive as current is 

 net current, the excess of the flow of charge one way over the flow the 

 other. Conceive a plane surface-element of area da, normal to the 

 field, therefore normal to the axis of x. We must classify the electrons 

 which traverse it according to their values of ^. Let H{^)d^da 

 represent the number passing through in unit time, and having at the 

 moment of passage A;-components of velocity in the range di, at ^. 

 This is equal to the number which at any instant have their x-compo- 

 nents of velocity in this range, and are situated in the right prism 

 having da for its base and extending a distance ^ down the direction 

 of X : -^ 



HiOd^da = ^dad^ I dv I di'fi^, v, f). (109) 



23 This would be immediately obvious, if all the electrons were moving parallel 

 to the X-axis and made no impacts. Electrons ha\ing y and z components of velocity 

 in addition to the .r-component will tlrift obliquely out of the prism, and electrons 

 making impacts will be thrown out of the range d^; but each electron thus lost will 

 be balanced by another coming in from outside. 



