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



The current, density so obtained (see Appendix, Problem 2) is 



2{kmy L 2 6 L 



+ |(A,Y) 2 {e-«-g]}, . . . (,) 



where a * = hnic + 2/^, 1= f° e"^, 



and A, is the mean free path, and where Xi is the same thing 

 asX- T . 



In the above equation quantities smaller than (JieY) z 

 compared with unity are neglected. The maximum value of 

 V is XZ, so that these quantities are at most only of the 

 order (helX)*. For a field of 1 volt per cm. we have 

 heX = U, since 7i=l'4x 10 13 at 0° C. and <> = 10- 20 . I is in 

 general probably a quantity of the order 10 ~ 5 cm., so that 

 these terms are quite negligible. We have not neglected 

 terms of the order (JieY)' 2 because we wish at a later stage 

 of the paper to discuss the question of the deviations from 

 Ohm's law. 



When there are no gaps, a = 0, V = 0*, and consequently 



Ae 2 h\ ' . /~2 ve 2 \v 



represents the conductivity for the metal in bulk, the dif- 

 ference between the numerical factor and that usually given 

 being due to a circumstance already discussed by the author 

 in a previous paper f. Thus in the metal between the gaps, 



the current densitv will be f * * ■ . Equating- this to i as 



'6(hmY x 



given bj' (4) , we obtain after a little reduction 



x -j= x >=^)t 1+w/(a) }' • • • (5) 



* In so far as I/a is infinite when a is zero, it may be thought that 

 the term in (4) involving this quantity becomes infinite under these 

 conditions. It turns out, however, that when a is infinitesimal, V 2 

 becomes an infinitesimal of a sufficiently low order of magnitude to insure 

 the vanishing of Y 2 I/a. It has not been thought necessary to give the 

 formal analytical proof of this, especially as we shall only have occasion 

 to use the expression for cases where a is of the order unity. It is of 

 ceurse obvious, without analytical proof, that Y'~I/a must be zero when 

 a is zero, from consideration of the fact that in this case our results 

 must reduce to those for the ca.se of the metal without gaps. 



t Phil. Mag. March 1914, p. 441. 



