SOLUTION OF THE EQUATIONS 129 
by (55-6) and (55-42). The suffix 0 here indicates central values. Hence 
{GMy 
VM'J 
('n + l) 3 
4 t tG 
a 
:i -1) 
39? (1 - /3)) w + 1 f 9? T 0 1+s ) 3 
a/x xi S 
= (n + I) 3 3 ( 1 - /3) / jR y „ 
4:7tG a V/SyU-x/ 0 
= (^± l) 3 3 (1 - /3) / 9? \ 4 
4:7tG a 
This gives 
where 
a 
.(94-5), 
Wo/ 
1 - ^ = (7 n (ilf/O) Vo 4 /3 4 (94-4), 
4TrG%0 2 
39? 4 (n + l) 3 
and M n is the value of M' for the law P oc p 1+1 / w . 
We have thus a quartic equation (94-4) for ¡3 analogous to (84-4), the 
coefficient C n depending on the law adopted for the change of molecular 
weight by (94-3). Constant molecular weight corresponds to n = 3. But 
(94-4) contains the central (minimum) molecular weight ¿x 0 . It would 
make a fairer comparison with the results for constant molecular weight 
if we used an average value of /x, instead of the minimum value. 
The effect of varying molecular weight can best be appreciated if we 
use in all caseL the standard formula 
1 - j8 = -00309 (Jf/0)V 4 /3 4 (94-6), 
and indicate how to choose the average value ¡x so that this may be exact. 
The condition reconciling (94-4) and (94-6) is 
C> 0 4 = C 3/ x 4 (94-71), 
since -00309 = C 3 . The required value of /x occurs at a point in the star 
where the temperature T n is such that 
so that 
In Table 12 results are given for various laws /x oc T~ s . The columns 
M n and p c /p m are taken from Emden’s tables, except for s = | which is 
a rough interpolation. We have included a range of laws much wider 
than is likely to be required, including laws in which /x decreases out 
wards. The fourth column gives the coefficient to be used in place of 
•00309 in (94-4). Alternatively if we use (94-6) /x must be calculated for 
a point in the star where the temperature T n is as given in the sixth 
column. It will be seen that if we use the value of /x appropriate to a 
region of the star where the temperature is f of the central temperature, 
the result will serve for any law between 5 = 0 and s = ^. 
The value s = | would give a reasonable representation of the expected 
effect. With this law a molecular weight 2-1 (iron retaining 1 or 2 electrons) 
GP s 
1 o — 
(94-72), 
T n 
% ~ 
(cl) 
(94-8). 
}