RADIATIVE EQUILIBRIUM 
111 
frequencies, which might be difficult and uncertain. Of course we cannot 
ignore the regions beyond jq and v 2 entirely, since anything like a trans 
parent gap, wherever it occurred, might make k 2 very much less than k'. 
But it is easy to convince ourselves that all such windows are blocked up 
by the study of a single process of absorption without exhaustive treat 
ment. 
The contrast between an arithmetic mean k ± and a harmonic mean k 2 
is that, roughly speaking, in an arithmetic mean we have to fear infinite 
values and in a harmonic mean infinitesimal values. The latter fear is 
much more easily allayed than the former. 
As an application of this, it has been pointed out by Rosseland that 
fine line absorption has no important effect on the opacity. For taking 
together the spectral lines (of all the elements) in the range zq to v 2 — chiefly 
K and L lines of the X ray spectrum which are not unduly numerous— 
they will if of the usual narrowness cover up only a small part of the 
range. However opaque they may be they can only increase the mean 
opacity proportionately to the area, or more strictly the weight, which 
they block out. This leads us to expect that continuous absorption due 
to ionisation, and not line absorption due to excitation, will be the main 
cause of stellar opacity. But it is doubtful if we ought to dismiss line 
absorption quite so summarily. It is well known that in certain conditions 
spectral lines may become very broad; and although the theory of the 
broadening is not at present well understood we may suspect that in the 
stellar interior the L lines of the elements are broad. In that case the 
lines of the various elements may together cover the whole frequency 
range. 
79. 
so that 
By Planck’s^Law 
I(v) 
Cv 3 
e hv/RT _ 1 » 
dl (v) Gh v i e hv l RT 
~W~ ~ RT 2 (e hv i RT - l) 2 ' 
Writing hv/RT = x, the weighted any range dx to be used in forming the 
mean value k 2 is proportional to 
x^dx 
(e x — l) 2 
(79-1). 
From this Table 8 has been calculated giving in the second column the 
relative weight for each value of x and in the third column the weight of 
the range from 0 to x. 
We see from the third column that 69 per cent, of the weight is con 
tributed by frequencies between 
2 -5RT/h and IRTjh. 
/