134 
SOLUTION OF THE EQUATIONS 
deviations from the gas laws were much smaller. The reader who cares 
to examine the development of the ideas may perhaps be interested to trace 
how by progress of the theory and improvement of the observational data 
p 0 was raised from 4 to 13 to 83 and then practically to infinity*; otherwise 
there is not much profit in going over the old ground. Our concern here 
has been to show that if the dwarf stars were affected by deviations from 
the perfect gas law as large as those affecting terrestrial gases the conse 
quent effect on their luminosities should be easily detectable by observation. 
Principal Results. 
98. We now resume the main discussion from the point reached in 
§ 88, adopting the approximations pk — const., p, = const., and the perfect 
gas condition. Accepting the law of absorption suggested by physical 
investigations (Chapter ix) 
k = CpffxTi (98-1), 
where C is a constant, we have by (87-1) 
Ca ¡3 1 
so that by (90-1) 
k 39U - 0 T* ' 
4:ttcG 391 iff (1 - /3) 2 
.(98-2), 
Ti (98-3), 
a Ca ¡3 
where a may be taken to be about 2-5f. 
By (84-6) the factor iff (1 — ¡3) 2 /[3 is a function of iff and /x only. 
The radius of the star need only be known roughly, the main dependence 
of L being on iff. The radius is involved because it settles the internal 
temperature which appears directly in the factor T^\ also a general 
knowledge of internal temperature and density is needed as a guide to the 
ionisation to be expected, and is a basis for estimating the best value of 
fx to adopt, but we can scarcely go far wrong over this. As regards the 
factor T in order to change L by as much as 1 magnitude it would be 
necessary to change T c from, say, 25 million to 4 million degrees; hence 
it is not likely that our calculations of internal temperature can be so 
much in error as to affect L seriously^. Granting this, our calculation of 
L for a star of accurately known mass should be trustworthy to well 
* Monthly Notices, 84, p. 308. 
f Whatever the actual central temperature may be, the value of T c to be used 
in (98-3) should be calculated on the assumption that the distribution is that of the 
polytrope n — 3. Different models of internal structure are then represented by 
slightly modifying the factor a. For example, in the point-source model treated in 
§91 the actual central temperature is infinite; but using the fictitious T c , equation 
(98-3) applies if we put a = 4-2. 
$ In this connection the calculations of minimal temperature in §§ 65, 66 are of 
interest.