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SOLUTION OF THE EQUATIONS 
121 
in k 0 for the series of stars from M to A, and therefore to an equal range 
of L. This is equivalent to a range of 1-4 bolometric magnitudes. No idea 
was formed as to which direction of variation was the more likely. 
Having found that k 0 is tolerably constant throughout a 12 -fold range 
of temperature it would seem natural to conclude that at stellar tempera 
tures it tends to a constant value independent of physical conditions 
altogether. The opacity would then be the same for all stars whatever 
their masses. This conclusion, which was accepted in the writer’s earlier 
papers, turns out to be fallacious. We are now convinced that k does 
indeed vary rapidly with the temperature, but it also varies with the 
density. The major part of the variation is proportional to pjT z which is 
a constant for stars of the same mass (and also within any one star). 
Thus the main variation conceals itself in our first study of the observational 
data, and it springs a surprise on us when we begin to compare stars of 
different masses. 
88 . The approximate constancy of k 0 from star to star in the giant 
series which we have found must be distinguished from the approximate 
constancy of k (and more strictly of rjk) within a single star which we have 
assumed. A little consideration will show that there has been no vicious 
circle. If rjk is variable within a star we may still expect that some kind 
of average value of it will be determined by (83-4); then comparison with 
observation gives us the unexpected result that nearly the same average 
value holds in other stars in spite of great differences of physical condition. 
But having now found that the absorption changes very little from 
star to star notwithstanding great differences of temperature, it seems so 
much the more likely that we were right in assuming that its variations 
within a single star could be neglected. A tenfold range of temperature 
covers all the interior of a star with which it is necessary for us to deal, 
and we have verified the approximate constancy over a tenfold range of 
temperature from star to star. We have verified it for stars presumed to 
be of the same mass and therefore subject to the limitation p/T s = con 
stant; but that is just the condition satisfied by different parts of the same 
star. Thus we have already evidence that our first wild shot is likely to 
be a tolerable approximation. We now turn to some results which suggest 
that it may be an especially good approximation. 
The Approximation rjk = constant. 
89. Anticipating the results of the theoretical investigation of absorp 
tion in Chapter ix we accept as the law of absorption 
kccp/jiT* (89-1),