710 The Problem of a Spherical Gaseous Nebula. 
§ 50. For the complete mass of gas, M, which can be 
in convective equilibrium under the influence of its own 
gravitation only, with central temperature C, we have the 
following results : — 
M_ («+l)B < rC-«'-^ , . 
R=^Y~ (58); 
with 
w= E(.- t -i)^vc-^-^p &I e( | )r @ , (2)< (60) . 
g , = 5-8^(« + l)A' .... [(22) repeated]. 
The two equations (59) and (60) give as before 
i=l < 61 >- 
§ 51. The equations of §§ 48... 50, with equation (19), give 
the solution of Homer Lane's problem for all values of k for 
which the function ® K (z) and its derivative ©^(s) have been 
completely determined, namely for /c=l and /c = 5, referred 
to in § § 28, 29 above, and for the values 1*5, 2*5, 3, 4, for 
which the Homer Lane functions and their derivatives are 
given in the Appendix to the present paper (Tables I IV.). 
It is important to remark that these equations indicate clearly 
the critical case # = 3, and that they also reveal some inter- 
esting peculiarities of the case k=5 ; which we have found 
to be the smallest value of k, for which a finite mass of gas 
is unable to arrange itself in equilibrium within a finite 
boundary (see §§ 27, 29). 
Equation (57) shows that in spherical nebulas for whose 
gaseous- stuff k = 3 the total mass of any gas which can exist 
in the equilibrium condition corresponding to a definite 
central temperature, when so distributed throughout its 
whole volume that the temperature and density at every 
point are related to each other in accordance with a chosen 
value of the adiabatic constant A, can also be brought into 
the equilibrium condition corresponding to any smaller 
central temperature, through gradual loss of energy, without 
