44 
MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
This equation may be transformed in the same way as equation (39). If we 
write 
«' r i x Pn<f>l\ 
y = ZV r '-*~*)' 
we find that the above equation is equivalent to (cf equations (47), (48), (49), 
(54), (56)) 
dry Gy 
~d? ~~7~~ 
477our 
together with the two equations 
'1 *a\' 
r~ dr ^ ^ 
r 6 J r (;yr s ) 
4 ? {y - 1 } •'' 
■ • (125), 
= 0. 
0 
. . (126), 
— 0 . 
. . (127). 
,■ = 0 
Writ 
tmg 
equation (119) becomes 
y _ fW' 
d 
dr 
■y y 
h (6 - 2 u) = 
_ ,? ’Pn 0 r 
(128). 
Referring to the table of values for u, which will be found on p. 15 of 
Professor Darwin’s paper, it appears that u increases from a zero value at the origin 
up to a maximum value of about 1'66 ; it then decreases to a minimum of about "8, 
and after this increases to 1, its value at infinity. Thus the factor 6 — 2 u has a range 
of values from 6 to about 2§. 
Now the solution of 
&y 
dr~ 
4 n(n + 1) = 
is easily found to be 
477/ 
y — 
- 4 til J _L [' Pah: r 'u +a c j r f + r n f 
2n + 1 [r ,i+1 Jo 3/d ^ 
OKT7 
3 /e 2 
( 129 ) 
^dr' | + Cp-"" + C 3 r ’' +1 . 
( 130 ), 
in which Cj and 0 3 are constants of integration, which may at once be put equal to 
zero, if n is positive, and if y is to satisfy conditions (126) and (127). 
Comparing (128) with (129), we see that if u had a constant value v 0 at every 
point of the nebula, the value of y would be given by equation (130), in which C 1? Ch 
would lie put equal to zero, and n would be the positive root of 
n (n -f- 1) = 6 — 2w 0 , 
provided only that 6 — 2w 0 were positive. 
