54 
QUANTUM THEORY 
where, since c is the velocity of propagation, 
v 2 = c 2 (?q 2 + n 2 2 + n 2 )/l 2 ...(40-4). 
Since the waves contain two independent components polarised in per 
pendicular planes, we have (allowing for the double signs) 16 independent 
waves for each set of positive integral values of n x ,n 2 ,n z . 
The number of combinations of integers satisfying 
n \ + n 2 2 + n z 2 < n 2 
approximates when n is large to the volume of an octant of a sphere of 
radius n, viz. 
& 
Hence by (40-4) the number of independent waves of frequency less than 
v becomes 
16 X ^7T ( vl/c ) 3 , 
and the number between frequencies v and v + dv is thus 
8tt1 3 
v 2 av. 
c 3 
According to the classical law of equipartition of energy each of these 
independent vibrations of the aether will receive on the average the energy 
RT*. Hence the energy in the enclosure of frequency v to v + dv is 
8TrRTl 3 v 2 dvlc 3 . 
By the Correspondence Principle this must agree with the limit of Planck’s 
formula which by (40-2) gives 
PCRTv 2 dvlh. 
Hence C = Snh/c 3 (40-5). 
Having evaluated G we can now give more explicitly the relation 
between the coefficients of absorption and emission resulting from Einstein’s 
equation. By (38-4) 
A- n. a . S-n-^.v.. 3 
,(40-61). 
^21 __ p 3 _ 
. Kio 
a 12 q 2 q 2 c - 5 
By (36-21) a-^n^I (v 12 ) is the number of quanta absorbed per unit time, and 
therefore a 12 %/ (v 12 ) . hv 12 is the energy absorbed per unit time—a time 
during which a quantity of monochromatic radiant energy cl (v 12 ) dv has 
passed through a square centimetre. Hence if the % atoms (in state 1 ) 
form an absorbing screen of area 1 sq. cm. the fraction absorbed is 
ct 12 hv 12 /cdv * 
The atomic absorption coefficient, or absorption coefficient per atom per 
sq. cm. for the monochromatic radiation, is thus 
a = a 12 hv 12 /cdv (40-62). 
* I.e. \RT kinetic + \RT potential. A free particle receives \RT kinetic for 
each of its three degrees of freedom (39-4).