a Spherical Gaseous Nebula. 709 
centre, in terms of the notation of §§ 23 ... 25 above. 
Thus, taking as our solution with central temperature C 
(equation 26), 
t = CS(z) (50), 
where 
and where <r is given in terms o£ the Adiabatic Constant, A, 
by (22) ; we have from equations (25) and (50) 
m = (»-H)S^C-*'-'-" &(z) _ _ _ (51)) 
±j e" 
and bj differentiating this we obtain — 
dm («+l)So-C- i(,c - 3) 
E 
W(z)dz . . . (52). 
§ 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 
Jo e J z . 
By putting &'(z) = — [<d(z)~] K /z i in this, and then integrating 
by parts as in §40, equation (43), w r e may write i in the 
form — 
.j^o^^^rm^ + ^%m^ &i ^ (54 ). 
e L oz o j z z -* 
Similarly, from the third member of (47), wdth 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 — 
ic=— J - — 9 — - 1 dz± — 4^®(~) . 0>o)- 
It is easy to verify from these equations for i and w that 
withS = Kp-K„ as in §42, 
* = ? ^_ + ^«, . ( 6); 
Phil. Mag. S. 6. Vol. 15. No. 90. June 1908. 3 B 
