SOLUTION OF THE EQUATIONS 
119 
Alteration of the linear dimensions in the ratio Z will alter p in the ratio 
l~ 3 and g in the ratio Z~ 2 . To preserve homology p Q and p R at corresponding 
points must keep the same ratio to one another. This means that pT must 
change in the same ratio as T i . Hence T changes in the ratio Z _1 , and 
p R and p G in the ratio Z -4 . Inserting these changes in the equations of 
equilibrium (81*1) and (81-7), the first continues to be satisfied and the 
second is satisfied if 7 r 
T)kL 
is unaltered at every point. The assumptions stated above secure that 
7] is unaltered and that k is altered in the same ratio at every point. Hence 
the sufficient condition is that kL is unaltered, or that L is inversely 
proportional to k. 
In putting forward the giant and dwarf theory of evolution Russell 
and Hertzsprung laid stress on the fact that observational statistics show 
a series of bright (giant) stars which have about the same luminosity from 
type M to type A, as well as a series of rapidly diminishing brightness 
from A to M (dwarfs). The giants all have low density so that our results 
for a perfect gas are unquestionably applicable to them. Assuming that 
there is no important change of average mass along the giant series*, 
the approximate constancy of L shows that k 0 must also be approximately 
constant along the series. 
This evidence is admittedly rough. In Russell’s type-luminosity dia 
gram the giant series lies along a fairly horizontal line indicating constant 
absolute visual magnitude. This must be corrected to reduce to bolometric 
magnitude and also for effects of selection of the data. Then the weak 
point arises that we have very little direct evidence as to how constant is 
the average mass along the line. Perhaps also allowance should be made 
for a slightly smaller molecular weight at the hotter end of the series. 
But rough as it is this indication is remarkably suggestive, because 
between type M and type A there is a great change in the internal con 
dition of the star and, as we shall presently see, the internal temperature 
rises tenfold. It would be something to the good to be able to say definitely 
that the change of k 0 is not more than in the ratio 20 : 1. In physical 
experiments X ray absorption is a rapidly varying function of the wave 
length of the radiation and therefore of the temperature; a range of 
1000 : 1 is by no means excluded. 
87. To proceed further we must be able to compare the internal 
temperatures of stars of different spectral types. The fundamental formulae 
so far obtained in this chapter are 
* We have purposely stopped short at type A since beyond this (in types B and 
O) the average mass is known to be considerably greater. 
1 — j8 = -00309 (Jf/O) 2 /tyS 4 , 
L = 4:7tcGM (1 — ¡3 )/k 0 .