a Spherical Gaseous Nebula. 701 
§ 29. For the case k = 5, Schuster found a solution in finite 
terms, which with our present notation may be written as 
follows : — 
§^ = 5W = - 7 J/| I) . . . (27). 
This makes £ = 1 at the centre (<r/r = # = oo ). At very great 
distances from the centre (x '■= 0) it makes 
^•8-^, and , = g ) 5 =(¥)V = (¥)? W 
Using (27) in (25), we find 
m_ (x:-fl)S<x ^/3 / 9 qv 
E~ <?* (3<* 2 + l) 3 / 2 * ' ' * ^ ^' 
and i£ in this we put #=0, we find 
where M denotes the whole mass of the fluid. Thus w r e see 
that while the temperature and density both diminish to zero 
at infinite distance from the centre, the whole mass o£ the 
fluid is finite. 
§ 30. It is both mathematically and physically very inter- 
esting to pursue our solutions beyond tc = 5, to larger and 
larger values of k up to /c = co : though we shall see in § 43 
below, that, for all values of k greater than 3 (or &<:1J), 
insufficiency of gravitational energy causes us to lose the 
practical possibility of a natural realization of the convective 
equilibrium on which we have been founding. But notwith- 
standing this large failure of the convective approximate 
equilibrium, we have a dynamical problem of true fluid 
equilibrium, continuous through the whole range of k from 
— 1 to -co, and from +oo to ; that is to say, for all 
values of k from to cc . In fact, looking back to the 
hydrostatic equation (10), and the physical equations (1), or 
(7), and (16), we have the whole foundations of equations 
(17) to {26), in which we may regard t merely as a con- 
venient mathematical symbol defined by (6') in § 4. Any 
positive value of h is clearly admissible in (1), if we concern 
ourselves merely with a conceivable fluid having any law of 
relation between pressure and density which we please to 
give it, subject only to the condition that pressure is increased 
