QUANTUM THEORY 
49 
/ 
X2> Xl)- 
T)J 
.(37-1), 
.(37-2). 
■1) in the 
y assume 
increases 
he second 
taking T 
..(37*3). 
)• 
..(37-4), 
...(37-5), 
1 
>/(” tal T ) 
...(37*6). 
mperatures 
i the three 
arguments 
^dependent 
n it depend 
Meeting C 23 - 
,e a definite 
nsferred from 
rent only to v- 
it it increases 
natural constant involved in the unknown function f (v/T). We have 
accordingly 
C 12 = C 23 = ^13 = @ (37*7). 
Hence (37-4) becomes 
( 1+ /F)X 1+ /§)M 1+ /lSTw) (37 ' 8) ’ 
where a = v 12 /T, ¡3 = v^jT. 
It is well known that the only solution of this equation is the ex 
ponential function 
1 + 
where & is a constant. Hence 
C 
/(«) 
= e K 
f(a) = CI(e»*- 1). 
Wien’s Law thus reduces to 
1 ( v > T ) = ¿S/S^Tl (37-9). 
The radiation law is thus fully determined except for the two constants 
G and k which must later be identified. The form (37-9) is Planck’s Law. 
38 . We can now find the relative 
at temperature T. By (36-4) 
Ti-\ dtyx f , G \ 
proportions of 
= ^21 e ku i2 /T _ ®21 
®12 ®12 
atoms in states 1 and 2 
g(X2— Xl)/RT, 
where E = h/k (38-1). 
And generally 
^ = ( ^el»-xr)lRT ( 38 . 2 ). 
7l s &rs 
Let q 1 ,q 2 ,...q r be the proportions of atoms in states 1, 2 ... r at 
infinite temperature. Then by (38-2) 
a sr la rs = q r lq s (38-25), 
so that, reverting to finite temperature, 
n 1 :n 2 : ... : n r = q x e~^ T : q 2 e~x*l RT : ... : q r e~xrl R T ...(38-3). 
The factors q 1} q 2 , ... are called the weights of the respective states. The 
theory of these weighting factors will be considered later. They are deter 
mined when the constitution at any given temperature is known; and 
(38-3) then shows how the constitution changes with temperature. The 
result (38-3) is called Boltzmann’s formula. 
In Einstein’s original paper* Boltzmann’s formula (38-3) was quoted 
as a result established in statistical mechanics and the derivation of 
* Phys. Zeits. 18, p. 122 (1917). Einstein was following the converse procedure 
so as to deduce the quantum law (36-1) from his equation.