132 
SOLUTION OF THE EQUATIONS 
Thus, having found any pair of corresponding values of M and k we 
can find k for any other star of different mass but of the same mean 
density. To pass to another mean density a new quadrature is necessary. 
From k we find 1 — j8 by (84-3). 
Results of a number of such calculations are given in Table 13*. 
Table 13. 
Solution for Dense Stars ( k = 2-22.1(U 8 ). 
Pc/Po 
R 
M 
Pm^Po 
© 
II 
V 
o 
¿0 
O, 
0-9 
2-294 
29-70 
0-587 
13-5564 
0-8 
1-800 
8-325 
0-340 
13-9247 
0-7 
1-717 
4-112 
0-194 
14-1289 
0-6 
1-790 
2-592 
0-1078 
14-2625 
0-5 
1-958 
1-884 
0-0600 
14-3549 
0-4 
2-223 
1-483 
0-0322 
14-4241 
0-3 
2-616 
1-245 
0-0166 
14-4748 
0-0 
— 
0-884 
— 
14-5739 
The values used in the calculations for this table were k = \G and 
p 0 = 1. These give “stars” of mass only a few grams, but the results can 
afterwards be transformed to the stellar scale. Columns 2 and 3 give the 
radius in centimetres and the mass in grams resulting from the quadratures. 
From these we find at once the mean density tabulated in column 4. We 
can now find k for any other mass by the rule k cc M%, and the last column 
contains the values of k corresponding to the sun’s mass, e.g. in the first 
line, raising the mass from 29-70 gm. to 1-98.10 33 gm. involves raising k 
from 2-22.HU 8 to the value shown. 
If p 0 and all densities| are altered in the same ratio by altering lengths, 
leaving the masses unchanged, (95-3) continues to be satisfied with the 
same value of k. Hence the table is applicable for any value of p 0 , except 
that the auxiliary column B should be altered in the appropriate ratio. 
In practical applications of the table we are concerned only with the last 
two columns. 
96. As an example, let us calculate what deviation from a perfect 
gas would be necessary to decrease the brightness of the sun 3 magnitudes 
below that of a perfectly gaseous star of the same mass. (This is about 
the difference assumed on the giant and dwarf theory.) Assume first that 
k is the same for both. The reduction in L corresponding to 3 magnitudes 
is in the ratio 1:16, and hence by (83-4) we must suppose 1 — /3 decreased 
* Zeits. ficr Physik, 7, p. 379. The later values of the constants (Appendix i) 
would increase all the entries in the last column of Table 13 by -0096. 
| The previous statement that alteration of mean density involves a new 
quadrature referred to problems in which p 0 is unaltered.