170 
THE MASS-LUMINOSITY RELATION 
average separation have a mutual potential energy nearly equal to the 
average kinetic energy of 4 free molecules. At the average separation 
this mutual energy merely cancels that due to the free electrons in the 
neighbourhood since the average potential is zero. Now let the two ions 
approach to a distance 0-42.10 -8 cm. Their mutual energy being pro 
portional to r -1 increases by 50 per cent.; and there is no corresponding 
increase in the cancelling term, since the negative charge being divided 
between 23 free electrons has a comparatively non-fluctuating distribution. 
The increase is thus equal to the average kinetic energy of 2 free molecules, 
and the 2 ions can just make the approach at the expense of all their 
kinetic energy if they were originally endowed with the average amount. 
Hence on the average two ions cannot approach nearer than § of their 
mean distance, which means that effectively an ion is barred out from 
^ of the whole volume. 
When a molecule is barred from a third of the volume by finite size 
of other molecules the constant b in the gas equation is %v, so that the 
pressure equation becomes %pv — RT . Accordingly, the pressure is in 
creased 20 per cent. But in the present case only ^ of the gas pressure 
comes from the ions; the rest is from the free electrons which are not 
barred from any appreciable volume. Thus the increase in the pressure 
would be less than 1 per cent. 
We repeat, however, that this is not a calculation of the true electro 
static effect. It deals with a particular objection which arises in most 
minds, viz. that ions will act as though they had large volumes. A rather 
difficult mathematical investigation will show that the objection is a 
phantasm (§ 184); meanwhile, we take the easier course of showing that, 
phantasm or not, it is at any rate not of large order of magnitude. The 
barring out of ions from close approach to one another has actually an 
effect which would scarcely have been anticipated. It means that in very 
small stars the ions are constrained to keep at the greatest possible distance 
from one another, whilst the electrons can wander as they like. The 
repulsive forces are thus kept down to a minimum, whilst the attractive 
forces have a good chance of exceeding the minimum. The result is that 
attractive forces predominate and assist the compression of the material. 
White Dwarfs. 
117. If stellar matter at the density of platinum has still the com 
pressibility of a perfect gas, the limiting density must be much higher. 
It is therefore possible that matter in the stars may attain densities un 
paralleled in terrestrial experience. Conversely, if we can discover in the 
universe matter of transcendently high density, it will be the strongest 
possible confirmation of our conclusion that in the ordinary dwarf stars