279 
1907-8.] The Problem of a Spherical Gaseous Nebula. 
§ 49. With these values of t and dm substituted in the third member 
of equation (42), the expression for the internal energy, i, of the gas within 
a sphere of radius r becomes 
/ 
i = K,, / dm t 
_K„E( K +l)S(rC- i<, ‘ 
1 T 
I dz%'\z)%(z) 
• (53). 
By putting ®"(z) — — [®(z)] K /z 4 in this, and then integrating by parts as in 
§ 40, equation (43), we may write i in the form — 
K v E(k+1)S<tC-^~ 5) 
-[©(*)] 
3z 3 
K + l 
(54), 
Similarly, from the third member of (47), with the values of m and dm 
given in (51) and (52) above, we obtain the following expression for the 
gravitational work, w, done in collecting the gas within a sphere of radius r 
from infinite space — 
E(«+1)2SVC [©( 2 )] 
dz 
©'(2) 
. (55). 
It is easy to verify from these equations for % and w that with S = K,, — K„, 
as in § 42, 
l_K v E(K + l)3o-C- i(K - 5) [©(.)p+ 1 
e 2 3z s 
. (56). 
§ 50. For the complete mass of gas, M, which can he in convective 
equilibrium under the influence of its own gravitation only, with central 
temperature C, we have the following results : — 
M _ (k + l)S<rC - ^ K ~ 3 ! 
E e 2 
• (57); 
e _ o-C~ k ' , ' 1) 
I = K,E( K+ l) 2 S<rC-«- ^^ z | 
3 e 2 J z 3 
e( K+ i) 2 svc-^- 5 > f dz mi &(z) 
e 2 J z 3 v ’ 
with 
2 _5-6.e( K + l)A K 
(58); 
. (59): 
• (60); 
[(22) repeated]. 
