THE COEFFICIENT OF OPACITY 
235 
(161*1) Q„ is proportional to (ZV) x ( Z/V 3 ) = Z 2 /V 2 in agreement with 
(155-42). Kramers’ result is thus confirmed in every detail. 
It remains to test the absolute value of Q. This is more difficult since 
it depends on absolute instead of differential experiments. According to 
a direct comparison the experiments give about twice as much radiation 
as the theory ; but as Kramers points out they are not strictly comparable 
and the actual agreement is probably closer. In any case the experimental 
confirmation of this part of Kramers’ theory is so close as to constitute 
a remarkable triumph for the theory. 
162. One doubtful point remains which may conceivably have astro 
nomical importance. In addition to the radiation here described Kulen- 
kampff found a considerable emission at or very near to the limiting 
frequency v 0 ; that is to say, J v does not rise uniformly from zero value 
at v 0 but starts almost abruptly at a finite value. This radiation, which he 
calls spectrum B, must be emitted by electrons which just lose their whole 
energy. Presumably they may be considered as captured in high quantum 
orbits; if so, the atoms capturing them become negatively charged*. 
But there is no provision for the corresponding radiation in either spectrum 
a or spectrum /3 of the classical theoryj*. It may be that capture in ordinary 
orbits being blocked, the spectrum ¡3 heaps itself up at the limit v 0 ; but 
this is not in accordance with Kramers’ ideas. The slight reference to 
spectrum B in Kramers’ paper ( loc . cit. p. 870) does not seem to elucidate 
the phenomenon. We cannot foresee what will happen to this radiation 
when we are dealing with ions instead of complete atoms, so it is impossible 
to say what part (if any) it will play in stellar opacity. 
163. The spectrum between v 0 and v x is not emitted under laboratory 
conditions ; and the theoretical predictions cannot be tested directly. But 
since coefficients of emission and absorption are connected by Einstein’s 
relation (38-4) we may make equivalent tests on the corresponding ab 
sorption spectrum. Consider the emission and absorption of a line w T hich 
replaces a stretch of continuous spectrum of extent fip x in energy units 
or fifjjh in frequency units (cf. (159-3)). Here ifj x = - Xx is the negative 
energy of a 1-quantum orbit. To calculate the emission for this line dv 
must be replaced by fifjjh in (155-42). Consider a cubic centimetre of 
material containing s fully-ionised atoms and n free electrons of velocity 
V to V + dV. In unit time n' V electrons will traverse the cubic centimetre 
so that the emission per cu. cm. per second is by (155-42) 
m'Vfi/t! 32tt 2 W 
h Zy/Zc 2 m 2 V 2 ' 
* It is known from positive ray experiments that atoms can become negatively 
charged. 
f The part of spectrum /3 assignable to the high quantum orbits is extremely small. 
f