284 
IONISATION, DIFFUSION, ROTATION 
P and p are constant over a level surface and, since in material of homo 
geneous composition two variables suffice to define the state, all other 
scalar properties T, ¡ 1 , k, p R , ... are constant over a level surface. Vector 
properties are formed by introducing the factor dn representing normal 
distance to a neighbouring level surface; this cannot be constant over the 
surface in a rotating star. Hence we have vectors such as H and g whose 
ratio is constant on a level surface although they themselves are not. The 
next step is to show that their ratio is constant not only over the level 
surface but from one surface to the next. The proof depends on the fact 
that the divergences of H and g are scalar quantities pe and — 477 Op + 2co, 
which are constant over a level surface. After this step it follows that 
H/g and />€/(— 47 rOp + 2co) are constant everywhere. The theorem would 
remain true even if the flow of heat were due to conduction. 
For slow rotation (198-1) approximates to the law e = constant, except 
in a thin film near the surface where very low density is reached. For the 
sun, with rotation period about 25| days, we obtain 
eoc (1 - -0000195/p), 
so that e is constant to within 10 per cent, in all parts where the density is 
above -0002. It is a mathematical curiosity that if we imagine a star with 
strictly zero rotation the argument breaks down and no limitation is 
imposed on e. We take this to signify that as to becomes smaller the 
condition becomes more and more nearly e = const., but at the same time 
the consequences of violating the condition become less serious and de 
terrent; so that at to = 0, when the condition becomes exact, the star is 
able to violate it with impunity. 
199. We can scarcely believe that von Zeipel’s condition is fulfilled in 
actual stars. For example, it requires that e shall be negative in the outer 
parts of a rotating star, that is to say, subatomic energy is absorbed 
instead of being liberated. It requires that in a slowly rotating star the 
liberation of energy shall be nearly constant through a wide range of 
temperature and density; and if the unknown laws of subatomic energy 
are obliging enough to fulfil this condition, how can they modify them 
selves so as to provide the right distribution in fast rotating stars ? Thus 
the question is raised, Will anything very awful happen to a star which 
does not satisfy von Zeipel’s condition? At present all we know is that 
it cannot remain rotating as a rigid body in statical equilibrium. 
The angular velocity of the sun’s surface varies with the latitude, and 
no doubt this variation extends into the interior; thus the sun has not 
a constant co. It is possible that for the actual distribution of co in the sun 
the condition corresponding to (198-1) might be satisfied*. The sun might 
* The condition will involve dco/dx, etc. so that it is not obtained by merely 
inserting the varying co in (198-1).