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8 SURVEY OF THE PROBLEM 
Much additional confirmation is obtained. The required bifurcation of 
density has been verified by the researches of Russell and Shapley on 
eclipsing variable stars. The sun and a number of other dwarf stars of 
type G have densities near that of water; but at least three eclipsing 
variables of type G are found to have densities less than that of air. There 
is evidence that this is not due to continuous range of density but is a 
definite bifurcation; intermediate densities belong to the higher types 
F, A, B which are traversed between the two stages of G. As already 
mentioned, the startling bulk ascribed by this theory to the giant stars 
has been verified by interferometer measurements. 
The giants and dwarfs can now be distinguished by special differences 
in their spectra of a kind not considered in the Draper classification into 
types. This is a particular application of the spectroscopic method of 
determining absolute magnitude. 
We shall find later that it is difficult to accept the giant and dwarf 
theory in its entirety. The ascending series presents no difficulty; but the 
descending series does not seem to be explicable in the manner that Lockyer, 
Russell and Hertzsprung supposed, because we now have evidence that 
the sun and other stars assigned to this branch behave as though con 
stituted of perfect gas, notwithstanding that their densities are greater 
than water. In fact, the conditions in the stellar interior are such that the 
gas laws should continue to hold at much higher densities than under 
terrestrial conditions. The theory of stellar evolution is now in a very 
confused state, and the difficulties will be considered in due course. 
8 . The broad principles used by Lane in calculating the internal dis 
tribution of temperature have been followed in all later researches. We 
consider the case of a star composed of perfect gas. Then any one of the 
three variables, pressure (P), density ( p ), temperature (T), can be calcu 
lated from the other two by the law 
P = 9ipP//x (8-1), 
where 91 is the universal gas constant 8-26 . 10 7 and /x the molecular 
weight in terms of the hydrogen atom. Thus effectively there are only two 
independent variables determining the state of the material. The differential 
equations satisfied by them are obtained by expressing two conditions: 
(1) the mechanical equilibrium of the star, which requires that the pressure 
at any internal point is just sufficient to support the weight of the layers 
above, and (2) the thermal equilibrium of the star, which requires that the 
temperature distribution is capable of maintaining itself automatically 
notwithstanding the continual transfer of heat from one part of the star 
to another. It is necessary to formulate and integrate the two equations 
expressing these conditions; and they suffice to determine the two in 
dependent variables specifying the condition of the material at any point.