280 Proceedings of the Royal Society of Edinburgh. [Sess. 
The two equations (59) and (60) give as before 
IP K 
W 3 
• ( 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 ©**( 0 ) have been completely determined, namely for k — 1 
and k — 5, referred to in §§ 28, 29 above, and for the values 1*5, 25, 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 k = 3, and that they 
also reveal some interesting 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 disturbing the relation of temperature and density at 
any point of the mass. 
Equations (59) and (60) show that in spherical gaseous nebulas, for 
whose gaseous stuff k = 5 , the total internal energy, and the gravitational 
work, corresponding to each equilibrium distribution of gas, has the same 
value, whatever be the central temperature or total mass, provided tem- 
perature and density at each point within the mass are related to each other 
in accordance with the same value of the adiabatic constant in each case. 
§ 52. We may now apply the above equations to obtain the complete 
solution of our problem of § 21 : — to determine for any spherical gaseous 
nebula of given mass, initially in convective equilibrium, exactly what its 
radius was, what its central temperature and density were, and what were 
the temperature and density at any distance from the centre, at the time 
when a stated quantity of heat has been radiated into space. Looking to 
equation (57), we see that throughout all approximate equilibrium conditions 
of a constant total mass the relation 
o-C 3) = Jl (a constant) .... (62) 
holds: and, with this condition, equation (51) shows that, during the 
gradual ]oss of heat from the nebula, the value of 0 for each stated mass m, 
