SOLUTION OF THE EQUATIONS 
127 
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from the same star if the source obeys the law e « T, this being the law 
for which our ordinary method is most accurate. The result is 
logio KL' = 63-0467, 
so that log (L'/L) = 0-388, 
corresponding to a magnitude difference 0 m -97. 
The point-source gives the lowest possible brightness and we may take 
the law e oc fas giving the greatest possible brightness, since the concen 
tration of the source to the centre can scarcely be less than this. We 
could split the difference by adopting a = 2-75 in (90-1). The extreme 
uncertainty (not the probable uncertainty) due to ignorance of the law 
of subatomic energy would then be ± 0 m -5. 
The foregoing numerical solution applies only to stars of mass 5-02 x O, 
but it can be adapted to any assigned radius of the star since it is easily 
proved that an alteration of radius leaves the solution homologous. The 
magnitude difference 0 m -97 between L and L ' does not depend on the 
radius. It will no doubt change with the mass of the star, presumably 
being less for the less massive stars. This could only be tested by repeating 
the whole work with other constants. But our main purpose is achieved 
by calculation for a single mass since the discrepancy referred to at the 
beginning of § 91 is shown by all stars. In particular it was desired to 
investigate the problem for a mass near that of Capella for which the best 
observational data are available*. 
Attention may be called to a few points of interest in Table 11. The 
ratio of p R to p G in the outer parts settles down to a value near 0-37; for 
the usual solution for the same star the ratio (1 — yS)/yS is 0-52 throughout. 
In the main part of the mass the point-source gives slightly more uniform 
temperature and considerably more uniform density than the usual 
solution. 
93. The numerical results obtained in the trial solutions bring home 
to us very vividly the fact that the general internal conditions determine 
the surface conditions and not vice versa. We find that two solutions which 
are scarcely distinguishable to four significant figures through ^ of the 
mass of the star will diverge rapidly in opposite directions before the 
boundary is reached. Practically any kind of distribution in the outer 
layers can be tacked on to the same solution for the main part of the mass 
(i.e. the same to the number of significant figures commonly employed). 
At first sight it might be thought that the solution must depend on 
the surface conditions because we actually employed a boundary condition, 
viz. that p R and p G vanish together, to decide the value of T corresponding 
to r = 0-9, p = 0-01; other trial values of T were rejected because they 
* It was also thought best to choose a star with fairly large radiation pressure, 
since this brings in a complication which makes it less easy to prophesy the result.