RADIATIVE EQUILIBRIUM 
103 
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The point to remember is that when we 'prove a result without under 
standing it—when it drops unforeseen out of a maze of mathematical 
formulae—we have no ground for hoping that it will apply except when 
the mathematical premises are rigorously fulfilled—that is to say, never, 
unless we happen to be dealing with something like aether to which 
‘‘perfection” can reasonably be attributed. But when we obtain by 
mathematical analysis an understanding of a result—when we discern 
which of the conditions are essentially contributing to it and which are 
relatively unimportant—we have obtained knowledge adapted to the fluid 
premises of a natural physical problem. 
I think the idea that the purpose of study is to arrive at a string of 
proofs of propositions is a little overdone even in pure mathematics. Our 
purpose in studying the physical world includes much that is not com 
prised in so narrow an ideal. We might indeed say that, whereas for 
the mathematician insight is one of the tools and proof the finished 
product, for the physicist proof is one of the tools and insight the finished 
product. The tool must not usurp the place of the product, even though 
we fully recognise that disastrous results may occur when the tool is 
badly handled. 
73. We now give an alternative derivation of the fundamental equation 
(71T) which enables some points of detail to be discussed. 
In isotropic radiation of density E the density of that part travelling 
in directions included within a solid angle da> is 
Since the flow of energy in a star is not perfectly isotropic but depends 
on the angle 6 between dœ and the direction of the radius we shall denote 
the energy-density of radiation within the solid angle dœ by 
Consider a small cylinder of length ds and cross-section dS with its 
length in the direction 6. The infinitesimal dœ is considered to be small 
compared with dS/ds 2 , so that the divergence of the beam in the cylinder is 
negligible and the radiation within dœ travels along the cylinder. 
The amount entering the cylinder per second through the base is 
gh in- 
th the 
The amount leaving at the top is 
(73-2).