14 
MR. ,J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
Divide (44) by r 2> ‘ and differentiate with respect to r, then 
47rJ 
or, writing £ for Lr, and simplifying 
n (n + 1) 
— 47T?'J 
(«)> 
(V). 
and this same equation could have been deduced from (45) instead of (44). 
Equation (47) is more general than (43) since the two constants K 0 , K, have 
disappeared. In fact equation (47), being a differential equation of the second order, 
will contain two arbitrary constants in its solution, and these correspond to the two 
missing constants K 0 and K 1 . We can, however, determine K 0 , K, in terms of 
these two arbitrary constants, and if these constants are chosen so as to give the 
right values for K 0 , K L , the solution of (47) will be equivalent to the original 
equation (43). 
To determine K n , K,, put r = R 1 in (44) and we obtain 
and similarly from (45) 
inK, 
~l d 
dr 
(48), 
47tKa = — 
di 
:(&- ( " +,) ) 
i'=B n 
(«)• 
Hence we see that equation (43) is exactly equivalent to the three equations 
(47), (48), and (49). 
§ 16. Comparing (42) with (43), it appears that (42) is exactly equivalent to the 
following equations :— 
+ Bp ) 
/ 
T dm j dp 
p dr / dr 
(50). “ 
n (n + 1) 
, o 
0 
- 4771' (A + Bp i 
(51). 
4ttK, = 
Cyll\ 
~A(p — a j) 
i— 
J 
’•= R 1 
r n ~ 1 
47tK 0 — — 
,,3 m + 3 
d_ 
dr 
:{€ r {n+1) ) = A (p — «x 0 ) i 
J>-=Ro 
a %n + 2 
,=n u 
(52) . 
(53) . 
The right-hand member of (51) is equal to 
