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RADIATIVE EQUILIBRIUM 109 
radiation is of slightly lower average frequency than the outward-flowing, 
because it is made up of emissions from matter at slightly lower tempera 
ture. Since k varies according to the character of the radiation that is 
being absorbed the coefficient to be used for each stream should be that 
appropriate to its composition. 
It has been pointed out by S. Rosseland* that this has important 
consequences. Consider the equations (75-5) 
j = ckE Ì 
„ c dE\ (77-1). 
H 3 kp dr) 
If the quantities j, H, E refer to the amount of radiation between narrow 
limits of frequency v to v + dv these equations will be correct since k is 
then a constant k v . They are therefore correct in the differential form 
dj — ck v I ( v) dv 1 
¿H. « "W* < 77 ' 15 >- 
3 k v p dr J 
where as usual we write dE = 1 (v) dv for the energy-density between 
v and v + dv. Hence integrating 
f 00 \ 
j = c k„ I (v) dv 
<77 ' 2) ' 
Putting (77-2) in the form of (77*1) we write 
3 = chB (77-31), 
H = < 77 ' 32 >’ 
where /,'1 — j lc.l W) dv ~ j I (v) dv ' 
1 fia/M, f 0/ (v) , i (77 ' 4 >’ 
the latter being obtained since 
dl (y) _ a I (v) dT 
dr ~ dT dr' 
The result in brief is that the two equations (77-1) require that k shall 
be averaged in a different manner and if we use them simultaneously our 
results will be in error by an averaging factor. To distinguish the two 
mean values we call 
k x the mean coefficient of absorption, 
k 2 the coefficient of opacity. 
* Monthly Notices, 84, p. 525.