297 
1907-8.] The Problem of a Spherical Gaseous Nebula. 
©'1.5(g) (2*707) is less than it ought to be by about 1 in 600; the value of 
g(*18676) in Table II. is too high, the error being less than 1 in 3000, while 
©'2.5 (q) (2T88) is probably too high, the true value lying close to 2T88. 
In Tables III. and IV. q and ©'(g) may be taken as correct to about 1 in 200. 
In Table V., the final values of the Boylean Function are believed to be 
correct to one per cent. 
Greater accuracy has been aimed at in the case /c = 2*5 than in any of 
the other cases, owing to its theoretical importance, being applicable to stars 
composed of gases such as our terrestrial atmosphere. This was the case 
dealt with by Lord Kelvin in his 1887 paper referred to in § 2 above. The 
numerical results given in that paper were obtained by an approximate pro- 
cess using radii of curvature calculated for successive small arcs of the curve ; 
and they are in satisfactory agreement with the results given in Table II. 
§ 16. One important numerical result which can be derived from the 
values of q K and ®' K (q) is the value of the ratio cen ^ ra ^ density a ne p u | a 
mean density 
composed of any gas for which k is 1*5, 2*5, 3, or 4. This ratio is given by 
the expression 
P 
central density 
1 
3.©'* (■ q ).q i 
mean density 
For k — 1*5 (monatomic gases) 
P = 3.2*707. -2737 3 = 6 ‘ 006, 
For k — 2*5 (diatomic gases) 
P = 3.2-188.-1S676 3 = 23 ' 39, 
or, accurate to nearest figure, 
3.2*188. T8673 3 
= 23*40. 
(For k = 2*44 or h = 1*41, Ritter gives the value of p as 23.) 
For k: = 3, and k = 4, the values of the ratio are 54*2 and 625 respectively. 
For /c = 5, the ratio is infinite. 
§ 17. The following table of results is given for comparison : — 
?1.5 
®'i -s(g) 
P 
%5 
® 2 - 5 ( 0 ) 
P 
Homer Lane, 
*2735 
2*741 
5*943 
*18674 
2*188 
23*40 
See, 
*27368 
2*7097 
6*0014 
Ritter, 
*274 
2*70 
6 
* 
