THE COEFFICIENT OF OPACITY 
231 
and so on. We have no great confidence that the proper limits are precisely 
at the half-way mark—more especially as regards the first limit for the 
K line; but fortunately this uncertainty does not matter much in astro 
nomical applications. 
If this view is right the classical radiation will cease altogether at a 
frequency v x given by 
hv x = imV 2 + 0i/(!) 2 , 
since beyond this it is not heaped up into any line. If, however, the two 
K orbits are already occupied captures at the K level are impossible, and 
the K line cannot be emitted. The guillotine then falls at 
hv x = >F 2 + 0i /(f) 2 , 
or rather (since the simple theory of the hydrogen atom no longer applies 
strictly) at some point about half-way between the K and L lines which 
cannot be very definitely specified. If the eight L orbits are occupied the 
limit v x is between the L and M lines. Presumably if some of the L orbits 
are occupied the stretch of spectrum corresponding to L is emitted but 
wfith proportionately reduced intensity. 
Having duly placed the guillotine-frequency v x according to the 
ionisation, Kramers’ theory asserts that the total emission is equal to the 
classical radiation up to frequency v x , but between v 0 and v x it is emitted 
in line spectrum instead of continuous spectrum. 
When we are dealing with electrons having a Maxwellian distribution 
of velocities the varying value of the initial energy |wF 2 spreads the lines 
into bands. Thus the Maxwellian spread of the initial energies to a large 
extent undoes the quantum concentration of the final energies. When, 
moreover, we have to do with a mixture of elements having their spectral 
lines in different places there can be very little trace left of concentration 
to particular values of v. It appears then that in the end the classical 
continuous spectrum is re-established practically unchanged; all that 
remains of Kramers’ modifications is the “guillotine” cutting off the 
radiation beyond a frequency v x determined by the state of ionisation of 
the atoms—or determined by a half-quantum orbit if they are completely 
ionised. 
Even the guillotine will not concern us in astronomy if it is placed 
beyond the range of frequencies contributing sensibly to the opacity. If it 
is not placed so high it will reduce the opacity and consequently increase 
the discordance between theory and observation reached in § 158. It will 
be found that for Capella is so high that there is very little correction 
required; but there are other stars (including the sun) which should suffer 
a considerable reduction of opacity. 
Apart from the guillotine effect the astronomical results obtained from 
the classical theory in § 158 equally represent Kramers’ theory. 
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