490 Dr. W. F. G. Swann on the Electrical 



enough will strike the plane on the area represented by DE. 

 In the notation of fig. 7 this area amounts to 



cos y T 



We shall, in the present problem, make the assumption 

 that when the field is on, the number of electrons starting off 

 from an element with any given velocity is the same as if 

 the field were absent, and that the initial directions of 

 ejection are symmetrically arranged. This assumption is 

 analogous to the assumption made in Drude's calculation of 

 the conductivity of the metal in bulk. We shall also assume 

 that the thickness of the metal slabs which the gaps separate 

 is sufficiently large compared with the mean free path to 

 enable us, in treating the flow across any gap, to assume that 

 the metal extends on each side of the gap in a continuous con- 

 dition to infinity. This assumption is very approximately 

 true unless \jl is as small as one or two, in virtue of the very 

 small number of electrons which travel distances appreciably 

 greater than the mean free path. Lastly, we shall make the 

 assumption that the mean free path for any given velocity is 

 independent of the velocity. It is not absolutely necessary 

 to make this assumption ; I have considered the problem when 

 X is a function of the velocity c, and as the general conclu- 

 sions are not materially modified thereby, it is perhaps as well 

 to avoid introducing this complication. 



The number of electrons which pass through one square 

 centimetre per second due to the group referred to above is 



A cSn sin 6 . cos -dr dO ,,,. ~ ~ ,.,_. 



&n=— -^ = — — f e-^Sco*. . 12 



47rXK 2 sm -^rdty 



Now if t is the time taken for an electron to go from to E r 

 and if ES = */ we have 



*=<*cos0+^* i , (13) 



zm 



y=ctsm0, (14) 



also 



C0S *=(^P- ■ ( 15 ) 



Substituting in (15) from (13) and (14), and neglecting in 



* In the expression e - ,A , R should strictly speaking be replaced by 

 the length of the dotted path, but it is easy to see tbat this only differs 

 from the length R by a quantity of the second order of small quantities. 



