ME. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
15 
so that if we introduce a new quantity u, defined by 
o o dp / dm 
u = 2rrpr-~/- 
equation (51) may be written in the form 
{n (n + 1) — 2 u) $ 
( 54 ). 
(55). 
The solution of this will be of the form 
f = K.'/j (>') + J'-.'/'. (''■) 
(56), 
in which E l5 Eo are constants of integration. We have, from the definition of 
A f + B ' , = w = sW‘< r > + E *W> .... (57), 
and the elimination of B from this equation and (39) gives 
fiv C = !< XT >{ A ! - 
(58). 
If we imagine this value for C substituted in equation (36), we shall have a 
differential equation of the second order for A. The solution of this will be of the 
form 
A 
— + Eo J 2 (r) + E 3 / 3 (r) + Ej./ 4 (r) 
(59), 
in which E 3 and E^ are the new constants of integration. From this value of A we 
can deduce the values of B and C (equations (57) and (58)) without introducing any 
further constants of integration. 
Turning to the boundary conditions, we now find that there are six boundary- 
equations to be satisfied (equations (32), (33), (34), (35), (52), (53)) and only three 
arbitrary constants at our disposal, namely, the ratios of the four E’s. If we 
eliminate these E’s we shall be left with three equations to determine the configura¬ 
tion of the nebula at which instability sets in, and these equations will iii general be 
inconsistent. 
§ 17. In order to put the right interpretation upon this result, it will be necessary 
to return to the general equations of free vibrations found in § 12. 
Il we eliminate F from equations (29) and (30), we obtain 
