124 
SOLUTION OF THE EQUATIONS 
Hence i]T ^ = 7]k/k c = k 0 /k c , 
where k 0 is the constant introduced in § 83. 
If rjT~- is not quite constant the closest approximation to k 0 /k c will 
be given by using the mean value of rjT ~* given at the foot of the table. 
Hence (83-4) may be written 
(90-1), 
where a = 1-32 1-74 2-12 2-75 
for e oc const. T T 2 T 4 
respectively. The brightness diminishes as the concentration of the source 
to the centre increases, as already foreseen. When numerical results are 
required we shall generally adopt a = 2-5 representing a fairly strong 
concentration to the centre. 
Point-source of Energy. 
91. It will appear in later Chapters that the observed brightness of 
the stars is less than that predicted by present physical theories and that 
it is very difficult to find a plausible explanation of the discordance. The 
predicted brightness is decreased by concentrating the source of energy 
towards the centre and the discordance thereby lessened. We inquire 
what is the maximum possible change that can be made in this way on 
the most extreme suppositions. We therefore consider now the limit when 
the concentration is complete and the source of stellar energy is a point- 
source at the centre*. 
This problem can only be solved by very laborious numerical calcula 
tions; but it seems worth while to carry out an accurate calculation for 
this limiting case in order that we may know the extreme margin of error 
entailed by our present ignorance of the laws of liberation of subatomic 
energy. 
The differential equations of the theory are 
dpu = — kpHdr/c (91T), 
dP = — gpdr (91*2). 
For a point-source at the centre emitting the whole of the energy L 
ultimately radiated by the star 
H = L/4:Trr 2 } 
and the adopted law of absorption is 
k = k lP /T\ 
Monthly Notices, 85, p. 408.