THE COEFFICIENT OF OPACITY 
233 
' ■ 
abruptly and shaded off according to this law on the high-frequency side. 
It cannot be pretended that the continuous representation now proposed 
gives a very accurate summation of these effects for individual lines. But 
it serves the main purpose of avoiding a failure of the representation at 
very high frequencies which would have led to our integrals diverging. 
Introducing this modification into the investigation of § 157 we have 
in place of (157-15) 
k 0 
[ Vi T . .dl(v)j 
l 0 7 ^~dT~ dv + 
° e H V - Vl )/RT / ( v ) d JJ¡y) dv 
di{v) 
dv 
: /» dT 
...(160-2), 
where v x is the frequency corresponding to i p. From this it is deduced that 
the opacity is decreased in the ratio 
[X 1 
.'0 
e~ 2x (1 — e~ x )~ 3 dx + e~ Xi [ x 7 e~ x (1 — e~ x )~ 3 dx : [ x 7 e~ 2x (1 — e~ x ) 3 dx 
J Xi • 0 
...(160-3), 
where x 1 = ip/RT. 
The reducing factors calculated from (160-3) are— 
Table 29. 
Guillotine-Factors. 
ip/ET Factor 
8 1-025 
6 1-33 
4 4-51 
2 30-8 
When Ip < 6 RT the guillotine begins to have a serious effect on the opacity 
and the luminosity of the star should be multiplied by the factor here 
given. 
An alternative (but very crude) way of allowing for the guillotine is to 
suppose that the opacity is cut down so as to be proportional to the weight 
of the region of the spectrum which survives, i.e. a spectrum occupying 
a domain of only half the total weight is considered equivalent to a 
spectrum of half the intensity occupying the whole domain. By Table 8 
the factors for the above 4 values of ifi/RT are then 1T0, T36, 2-42, 11-9. 
This procedure involves an incongruous mixture of harmonic and arith 
metic means; but it confirms the rapidly increasing factor found by the 
other method which is possibly more accurate. 
It may be stated at once that the observational evidence does not 
support these factors (§179). Stars for which ip/RT is small appear to 
agree with the uncorrected law k <x p¡T\ There is perhaps a small reduction 
of opacity, but it is not at all comparable with the large reductions 
here predicted. This is not altogether surprising. Whilst Kramers’ theory