42 
Proceedings of the Royal Society of Edinburgh. [Sess. 
the same result as (11). Neglecting the factors that vary slowly and 
putting j/=l, this becomes 
rXNe 2 f sin ±(p - g)r\ 2 
K = ^n ( J !/' -!')* ) (13) 
The expression in the bracket is of the same form as the solution of a well- 
known diffraction problem. Its maximum value is 1. Hence 
_ rA m Ne 2 
4cm 
(14) 
k has half its maximum value for the values of g given by 
(P - gy = V 2 sin - g)r. 
The value of (p — g)r satisfying this equation is found to be 2232. 
Consequently 
* m(K-Kn) _T* Ne 2 2-232 = -28Ne 2 
A m 3 4cm 27 tctA 7 rm 
approximately, if e be changed to electromagnetic units. But according 
to equation (7) 
K m(^i ~~ ■bn) __ Ne 2 
A m 3 4-7rm ’ 
i.e. we have now -28 instead of -25. So that the new treatment leads to 
substantially the same result for the number of electrons per atom as the 
former treatment, which was based on the ordinary friction term. 
It is obvious from (13) that the greater T is, the narrower is the band. 
Also, if (13) gave k accurately, there should be small maxima on each side 
of the principal one. These are, however, not to be expected on account 
of the neglected terms in (10), and also on account of variations in the 
value of r. 
It is obviously not necessary in the above discussion that the intervals 
t should follow consecutively. There may be — indeed the numerical 
values of the quantities involved demand that there should be — intervals of 
irregular motion, i.e. of no absorption, in between the intervals t. For 
example, in the case of the FeK band r is about 2*5 times the free period 
of the k electron, but the intervals r constitute only about four-fifths of 
the total time. During the rest of the time the electron is giving back as 
much energy to the wave as it is taking from it. And the value of r 
above is an average one ; an exceptionally large value of r would permit 
of a correspondingly longer period of no absorption. This consideration 
may help to remove the difficulty alluded to in § 3. 
( Issued separately February 23, 1920.) 
