42 
THERMODYNAMICS OF RADIATION 
Expanding by Taylor’s theorem, and neglecting squares of the infinite 
simals s and dT 
J (v\ T) + dT^J (F, T) = J (F, T) I g (s) ds 
+ jJ (*', T) + F^J (/, T)| | sg (s) ds. 
Omitting the accents as no longer necessary, this reduces by (33-5) to 
dJ 
dT 
dT=s 0 (j+v%), 
or 
0 (vJ) _ vs 0 0 ( vJ) .ft\ 
dT dT dv ( ’’ 
Now s 0 /dT is independent of v but we have no reason to suppose it 
independent of T and we must therefore take it to be an unknown function 
of T. Then (33-6) can be written 
0 ( vJ) vd (vJ) 0 ( vJ ) 
0(lo g f(T)) dv ~ d (log v)- 
The solution of this partial differential equation is 
v J= F (v/f (T)) (33-71), 
where F is another unknown function. 
The energy is obtained by multiplying the number of quanta by v 
(according to our definition above). Hence 
vJ (v, T) = vl (v, T) (33-72), 
where v is the volume of the enclosure and I (v, T) dv the energy-density 
of radiation between v and v + dv. Integrating for all values of v and 
setting E for the whole energy-density 
Ev - f°° vJdv = i "*F (v/f (T)) dv 
Jo J o 
= Cf(T) (33-73), 
G = [ F (x) dx. 
' n 
where 
o 
Since the change is adiabatic, we have by setting dQ = 0 in (30-2) 
p%v = const., 
or since E = 3p E x v = const (33-74). 
Hence by (33-73) and (33-74) Eft is proportional to f (T). But by Stefan’s 
Law E* is proportional to T. Hence / (T) is a constant multiple of T, 
and without loss of generality we can set in (33-71) 
f(T)~T.