MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
23 
§ 22. Let us, in the first place, consider the “series” of nebulae such that u has a 
different constant value for each. This series includes a single free nebula, for it 
appears from Darwin’s paper that there is a nebula such that u — 1 at every point. 
This nebula, it is true, has infinite density at the centre, but this objection disappears 
when the innermost shells of gas are replaced by a solid core, the mean density of the 
core being equal to three times the density of the gas at its surface, and therefore 
finite. Let us, in the first instance, simplify the problem by supposing that the core 
is held at rest in space. The boundary equations (35 and 53) which have to be 
satisfied at r — B 0 now take the forms 
(A),, k =0 ..(80), 
= 0 . 
__ ?*—R 0 
( 81 ). 
independently of the value of n. The value of u in equation (55) being now 
independent of r, we may write the solution (56) in the form 
.(82), 
£ = Ep’' 1 + E 2 W. 
in which p, p' are the roots of the quadratic, 
t (t — 1) = n (n + 1) — 2u . 
We accordingly have 
p -f p — 1 j p — p 2 \/(yi -f- -g-)~ — 2 u j pp — n {ii -j- I) -(- 2 u 
Equation (79) now takes the form 
Ej (p + n) Bf 1 *”- 1 + E 2 (p' + n) R 1 '‘ ,+n-1 = 0 ... 
while equation (81) becomes 
E t (p - n - 1) Bo' 1 -’ 1 " 2 + E 2 (p ; - n - 1) Bp— 2 = 0 . . 
The elimination of Ej and E 3 from these equations gives 
jqy-M' ^ (/ +n )(f l - n - 1) 
R 0 / — (p + n) (p' - n — 1) 
(83) . 
(84) . 
(85) , 
(80). 
(87). 
The fraction on the right hand can be simplified by the help of equations (84); it 
is equal to 
2 (u — (n + D 3 ) + (p — &') ( n + i) 
2 (u — (n + i) 2 ) — (p - p') {n + D 
Now the left-hand member of (87) may be replaced by 
cosh {i(p - a') log (Ph/Rq)} + sinh (p - pQ log (RfiRp)} 
cosh D(p-p') log (RfiRg)} — sinh (p — p') log 
