It is to be noticed that deft/dn will not be a function of </> only unless the 
distance from one level surface to the next is the same for every point 
on it. This could not happen in a rotating star. 
It is convenient to resolve H into rectangular components 
It will be seen that H x represents the net flow across a unit area normal 
to the ir-axis; for the lines of flow cross such an area obliquely at an angle 
, . . dd> idcb 
whose cosine is / v- . 
ox! dn 
If no additional radiation were being generated the equation of con 
tinuity of flow would be 
dH x dH y a H z 
- ^ - H = 0 
3a; ■ dz ’ 
but since the rate of generation is pe per unit volume the condition becomes 
+ + (198-5). 
ox oy oz 
Now by (198-43) 
Hence (198-5) becomes 
=/w^5 + 
/(«&+/' №(!)• 
/(«VV-/'W(^/ = p E (198-6), 
since ( dcfi/dn ) 2 is the square of the resultant force and therefore equal to 
the sum of the squares of its components dcf)/dx, etc. Then by (198-2) 
4:7rGp + 2a> 2 ) -/'(</>) ( ^ ~ 
pe (198-7). 
We have seen that in a rotating star dcfrldn is not constant over a level 
surface. But the other quantities in (198-7) are constant over the level 
surface. Hence (198-7) can only be satisfied if 
r (</>) = 0 , 
so that, integrating / (</>) = const (198-8). 
Accordingly (198-7) becomes 
pe = const, x (4:7rGp — 2o> 2 ), 
or € = C (l - 
which proves the theorem. 
The following summary of von Zeipel’s analysis will serve to show its 
extreme generality. The condition of mechanical equilibrium shows that