SOLUTION OF THE EQUATIONS 123 
/ 
of e in the two spheres. (Central values are denoted by the suffix c.) By 
(62-1) T m /T c = 0-584. Hence 
Vc = 1-7. 
If we proceed from the centre to a distance including 95 per cent, of 
the mass the temperature drops in the ratio 1 : 0-21 (Table 6). Hence by 
(89-2) k increases in the ratio 1 : 2-2. Meanwhile 77 undergoes practically 
its full decrease in the ratio 1-7 : 1. Evidently r]k keeps remarkably steady 
in the main part of the mass of the star. A more detailed calculation for 
intermediate points is given in Table 10. 
The law e cc T represents a moderate concentration of the source of 
energy to the centre*. If the source is subatomic we can well imagine that 
a much stronger concentration occurs. It is desirable therefore to consider 
e oc T 2 and e oc T 4 . Results obtained by rough calculation of 77 are given for 
four different laws in Table 10. The star is divided into 10 shells of equal 
mass, and the values of T , 77 and 77 T~- are given for the limit of each shell, 
the central temperature being taken as unity|. 
Table 10. 
Test of constancy of yT K 
M r /M 
T 
e = const. 
e <x T 
€ OC 
rp2 
€ OC 
J '4 
vT~- 
V 
V 
7] T~b 
V 
7] T~* 
0-0 
1-00 
1-00 
1-70 
1-70 
2-57 
2-57 
4-71 
4-71 
0-1 
0-88 
1-07 
1-57 
1-68 
2-18 
2-32 
3-38 
3-60 
0-2 
0-80 
1-12 
1-49 
1-67 
1-97 
2-20 
2-81 
3-14 
0-3 
0-73 
1-17 
1-41 
1-65 
1-81 
2-12 
2-40 
2-81 
0-4 
0-66 
1-23 
1-36 
1-67 
1-67 
2-06 
2-07 
2-55 
0-5 
0-60 
1-29 
1-31 
1-69 
1-54 
1-99 
1-80 
2-33 
0-6 
0-53 
1-37 
1-25 
1-72 
1-41 
1-94 
1-58 
2-17 
0-7 
0-46 
1-47 
1-19 
1-76 
1-31 
1-93 
1-40 
2-06 
0-8 
0-38 
1-62 
1-13 
1-84 
1-20 
1-95 
1-24 
2-02 
0-9 
0-28 
1-89 
1-07 
2-02 
M0 
2-08 
111 
2-10 
Mean 
1-32 
1-74 
2-12 
2-75 
The constancy of 77 T - is best for e oc T, but it is also reasonably close 
for the other laws. In Table 10 T c has been taken as unit, so that 
T-* = */*,. 
* The law e oc T also corresponds to the contraction theory of stellar energy if 
the star passes through a series of homologous states. 
f The solution based on the approximation -qk — const, is used to calculate these 
values; so that, for example, the values of are only to be trusted if they turn 
out to be reasonably constant. The table tests the first approximation; it would be 
a dubious procedure to use it as a starting-point for a second approximation.