MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
34 
and 
so that 
dr 
X ~~ J >• xl> 2 ’ 
_1 cl 
dr XT r 2 do; 
then the above equation may be written 
d?y 
-j- 
dor 
„ d ( 0 dV'\ 
tt€ j ) a — "7- ( r 2 -7 = 0 . 
dr \ dr ] 
( 108 ). 
At infinity we are supposing XT to have a definite and finite limit, so that the 
L(r^\ 
dr \ dr ! 
limiting value of x is l/ATr. Let us further suppose that ~(r z ~) has a definite 
limit given by 
d [ „ dV' 
r - 
dr\ dr 
-AL _ y" 
I 
109 ), 
and that squares of V" may be neglected. Then the limiting form of ( 108 ) at 
infinity is 
Write 
then 
and 
d 2 y AiTred 
dx* (XLW 
V = 7 ) + log 
V" 
- 3 - 0 
X 4 TU 2 
2tt 
d*y dhj _2jP , . 
dx* ~ dx* 0 ? 4 ^ ° 8 ’ ( 
dx* 
4-71 _ 2e r ’ 
(\Tx) i = h? 
Equation (110) is now transformed into 
g+ ->-!) +4 J>g(XT)-^ = 0 
d 2 
In the special case in which —- log (XT) vanishes, this may be written 
d ~ V 4 - — ( n 4 - W 3 4 - W 3 4 - — — ' ^ = 0 
r 2 1 7 ,2 \ y 1 ~y 1 w • • • ‘/> x 2 t 2 * ^ ■ 
dx 2 
and at infinity (be., for very small values of x), the solution is 
’ = 2^+ V i eoa (id 7 Vg | 
(no). 
(111). 
( 112 ) 
where A, B are the two constants of integration. 
