248 
, and so tabulated. In the same way, or p/pg might be 
tabulated. Indeed ¢’ ‘=6. Again p or px*.dx may be 
0 
tabulated. Mr. Lane has done this work for several 
values of y. Solution by means of series of powers of x 
can be relied upon only till x=1. After that one must 
work indirectly. Lord Kelvin, in a paper published in the 
Philosophical Magazine, 1887, vol. xxiii. p. 287, gives 
numbers calculated by his assistant, Mr. Magnus Maclean, 
from which, with the help of Mr. J. Lister’s or Mr. Homer 
Lane’s values at x=1, I could give a table like the 
following for the case of y=1'4. There are outside 
limits for x and p, which Mr. Lane calls 2 and wp’. 
Knowing the value of @ for x=1, I find that Lord Kelvin’s 
numbers give x’ as 5°24, and the corresponding p’ as 
2°165, whereas Mr. Lane gives x’ as 5°35, and p' as 2'188. 
Mr. Lane does not publish the other values, and his 
curves are drawn to too small a scale for us to be able to 
make out tables of the values of 6 or ¢. Lord Kelvin 
from x=o, and Mr. Lane for values beyond +=1, ob- 
tained their results by methods such that errors may 
have increased as the work proceeded. 
On the whole, I am disposed to take Lord Kelvin’s 
numbers with an 2’, which is the mean of those just 
given, or 5°30, and p’ as 2°177. 
TABLE I.—For Gaseous Stuff whose Specific Heat Ratio ts 1"4. 
x 6 iy BB 
10) eee 1000 1*000 vee oO 
795 “904 777 “136 
"883 "884 "734 "184 
993 857 ‘679 *252 
I'lq. - ‘819 “607 355 
1°33 "763 “508 “512 
1°59 “681 "385 ‘758 
1°99 “562 237 1°133 
2°65 "384 0916 1'666 
3'97 on “141 ee "0074 2°117 
5°30 Rite {e) ae fo) 25577; 
_ We know now that for any star whose stuff behaves 
like a perfect gas 
a(2 e Bs,3%p,-2/2 
7= Al — ane — Deel 
a : OPO Ei acy eS) 
7=1)8, p=poh 
Where 
N= a , and B=4rA3, 
4na(y—1) 
we see that A and B, x’ and p’ depend merely on the 
nature of the gas. We have 
if R is the outside radius and M is the whole mass. 
We may choose values of 7) and py, and calculate R 
and M, or it is easy to see that if we know R and M, we 
may calculate the internal density and temperature by 
x M 
0 TTA No) 
4nmAty’ OR 
SNE, (isan tees (11) 
PO amu’ RS 
It will be noticed that, « being proportional to the 
molecular volume (being sixteen times as great in 
hydrogen as in oxygen), py is independent of o, whereas 
7) is inversely proportional to o. If we consider our 
own sun to be made of hydrogen, and if the laws of 
perfect gases could be applied as we have applied them, 
% = 3°25 x 10’ degrees centigrade, p, = 33, that is, 50 
per cent. greater than the density of platinum (see how I 
blush). Whereas if it were made of oxygen, py is the 
NO. 1550, VOL. 60] 
NATURE 
[Juty 13, 1899 
same as before, but 4% is 2°03 x 10° degrees. It is 
sometimes good to employ, instead of (8) 
ne 
a Mx’ Mx oh epapet ag (22) 
~ gmA2Ry!’ PS peRSa! 
The above tables and these formulz enable us to find 
the temperature and density at any point in any gaseous 
star of any mass, size and material (if y is 1°4). The 
curve connecting @ and x is the 4,7 curve for any star ; 
the curve connecting @ and x is the p, 7 curve for any 
star ; the scales of measurement are given in (12). 
The intrinsic energy (not including any gravitational 
energy) of the whole mass being %, since the intrinsic 
energy of unit mass at temperature ¢/ is #7, if & is the 
specific heat (in ergs) at constant volume, and ¢ is 
%y Por Y pY~*, 
rR 
h=4rnktopy "| rp .dr 
0 
or 
h=4nkA*t)?p)X 
if X stands for 
Bs 
| Oy Dax 
0 
a known number depending only on the value of y. 
Hence 
Ee Ete 
TA? | 
If W is the work done by gravitation in bringing all the 
stuff into its present position from an infinite distance, 
M? 
Reel 
» (13) 
= es 
We=age | pmr. dr=aV i . + (14) 
0 
where 
oa! 
yi— i xp. dx 
0 
a known number depending only on the value of y. 
We can now speculate on these results. If the pieces 
of stuff which come together to form the nebula are not 
mere molecules, but of the size of meteors such as reach 
our earth, W will not be much less than what is here 
stated. Indeed, we may say that even when a star 
ceases to be gaseous, and throughout its whole history 
the value of W is so nearly what is given in (14), that 
(14) may be used generally in such speculations as these. 
A gaseous star doubles all its temperatures and its 
intrinsic heat energy when its radius is halved. We see 
that if all stars are of the same gaseous stuff, the ratio 
of % to W is constant for all stars at all times. Let us 
_. M2 ee 
put W =a R? Ae RR 
As W =’+H if H is the total energy lost by the 
star by radiation, then 
os. ene a oo ol@G)) 
As part of this heat was lost by the stuff before it 
became a spherical gaseous star, we may take as the 
heat lost from time T = 0 when the radius was Ry to the 
present time T, when the radius is R 
Deed 
a-B M( = 2 x) 
( ) RR 
In the mass M there are surfaces whose areas are 
proportional to R*, and whose temperatures are pro- 
portional to + I shall assume as quite reasonable, 
that 
Total radiation per 
year from siete } ccareas x (temperatures)” . (17) 
where 7 is some constant. 
