282 
IONISATION, DIFFUSION, ROTATION 
Rotating Stars. 
198. We prove first a very beautiful theorem due to H. von Zeipel*. 
If a star, rotating as a rigid body with angular velocity co, is in static 
equilibrium, the rate of liberation of energy e at points in the interior is given by 
It is assumed that the physical characteristics of the material (opacity, 
molecular weight, liberation of subatomic energy, etc.) depend on T and 
p only; this would be true of a star of strictly homogeneous composition. 
We take axes rotating with the star and include centrifugal along with 
gravitational force so that the combined potential is 
where </> 0 is the pure gravitational potential. Poisson’s equation is then 
so that dP = 0, when d<f> = 0. Hence P is constant over a level surface; 
that is to say, P is a function of <f> only. 
so that p is a function of </> only. 
Since P and p are functions of </> only, T must be a function of f> only; 
and all other physical characteristics which depend only on the two 
variables T and p defining the statistical state of the material will be 
functions of f> only and constant over a level surface. The gradients of any 
of these quantities will be normal to the level surface. 
The flow of radiation H being along the normal, we have by (7IT) 
where dn is along the outward normal to the level surface. We can write 
(198-1). 
f> = <f >0 + \w 2 (x 2 + y 2 ), 
V 2 </> = V 2 </> 0 + 2 a; 2 
= — 4:TrGp + 2co 2 
The surfaces over which <j> is constant are called level surfaces. 
From the usual hydrostatic equations 
(198-2). 
dP ty dPdj> dP df 
dx P dx ’ dy dz ^ dz ’ 
df> 
we have 
dP = pdf 
(198-3), 
c dp R _ c dp R df> 
kp dn kp df> dn ’ 
so that 
(198-42). 
(198-41), 
* Festschrift für H. v. Seeliger, p. 144 (1924).