2ttC 3 N 

 n k = 



P 



502 Z. P. Wheeler — Dispersion of Metals. 



and substitute the value of a in equation (1), we obtain on 

 separating the real and imaginary parts 



n\= 3 a S 1? (5) andrc 3 (l-#c a ) = K 5 S 2 , (6) 



p m <r p m ' 



in which, in the first terms of each series the value of a is 

 quoted from equation (3). Equation (6) differs from that given 

 by Nicholson (loc. cit.) in that the second term on the right 

 hand side is ir times as great.* 1 am unable, however, to find 

 any error in the formula as given above. 



The hypothesis of equal velocities for the electrons yields on 

 similar treatment 



C"NV/ TrX- 1 •. 2 . _ 47rC 2 Ne 2 / ttX" 1 



^— = — H ) and^ 2 (l— k 2 )=K ; H ) 



s mV \ 4a / V ' p 2 m \ 4a / 



Thus neglecting the correction terms the values of n*K are, 

 on this hypothesis, 7r/4 times as great as those given by the 

 assumption of the Maxwellian distribution ; while to the same 

 approximation the values of n 3 (l — tc*) are the same on both 

 hypotheses. Now Nicholson, owing to the absence of the 

 factor 7r in equation (6), of course finds that the value of 

 K-\-n 2 (fc* — 1) on the equal velocity hypothesis is it times as 

 great as on the other. He then uses the values of K yielded 

 by the two formulae to discriminate between the two hypoth- 

 eses. This argument, if the formulae as given above are 

 correct, is illusory. Further, since the value of N is not in- 

 dependently known, it is impossible to discriminate between 

 the two hypotheses by means of the difference between the two 

 results for u 2 k. The equations resulting from the assumption 

 of the Maxwellian distribution have been adopted for the com- 

 putations of this paper, because that assumption seems on the 

 whole to the present author to have more inherent probability. 



In using equations (3), (5), and (6) for numerical evaluations 

 it is convenient to use, following Schuster, the ratio of the 

 number of free electrons to the number of molecules per unit 

 volume in place of N. Calling this ratio r, we have N=^/Mt, 

 where M is the mass of the hydrogen atom and r the relative 

 atomic volume. If, to simplify further, we gather together 

 those constants which are the same for all metals, writing 

 <2=:6 4 /7r 3 m 2 M 2 C 2 , and use the wave length in vacuo (X) in place 

 of p, then equations (3), (5) and (6) become 



, (7) n\= r S l5 (8) K + rc 2 (K 2 -l)= S 2 , (9) 



ar 2 y 



* In addition there are minor differences in the values of the numerical 

 coefficients in Si and S 2 . 



