SECTIONAL TRANSACTIONS.—A. 429 
where the symbol of differentiation 5 refers to two neighbouring solutions of the 
equations (2). By this relation the quantities J are fixed apart from arbitrary additive 
constants. These constants, however, are fixed by the further condition 
Do,.Jp= 29: . . . . . ° (6) 
k t 
where the member of the right side represents the mean value of the expression under 
the horizontal line taken over a time interval long compared with the fundamental 
periods of the motion. 
According to the quantum relations (1) and (4) we get now for the frequency of the 
radiation emitted by a transition between two stationary states characterized by the 
quantum numbers n’,;. . .”, andn). . . n”,, respectively 
Lepr et [ ' my. 
v=; (E/E i Yoth= Sm eather? soophaa Mnees) 
where the mean value of the last expression is to be taken over such solutions of the 
equations (2) which in the r-dimensional J-space are represented by a straight line 
connecting the points (J’, . . . J’,) and (J’; . . . J”,) indicating the two 
stationary states involved in the process. 
In the limit where the values of the quantum numbers are large compared with 
their differences »’,—n”;, we may consider the frequencies w, as constants in the 
mean value in equation (7), and get the asymptotical relation 
Vm Xo»;,(n",—n”;,). . . . . ° . (8) 
k 
In this limit the frequency of the radiation will accordingly coincide asymptotically 
with the frequency of that harmonic component in the motion represented by (3) 
for which the relations 
é 
T=1N',—Nn"), (cle ret) : é ‘ sial@) 
are fulfilled. 
This result opens a possibility in the limit of large quantum numbers to obtain 
a connection between the statistical results of the quantum theory and the classical 
theory of radiation. It must be emphasised, however, that here we have by no means 
to do with a gradual disappearance in this limit of the fundamental difference between 
the quantum theory and the classical theory. In fact, according to the latter theory 
the radiation from the atom will take place continuously and consists of the simul- 
taneous emission of a multitude of wave systems with different frequencies, each 
corresponding to one of the harmonic components in the motion, while on the quantum 
theory each train of waves is emitted by an independent process of transition between 
stationary states, the relative occurrence of the different processes being governed by 
laws of probability. Just this circumstance leads us to consider the connection between 
the harmonic components of the motion and the various processes of transition traced 
in the region of large quantum numbers, as evidence of a general law holding for all 
quantum numbers. According to this law, the so-called ‘‘ correspondence principle,” 
every transition process between two stationary states given by (4) can be co-ordinated 
with a corresponding harmonic component in the motion defined by (9). This co- 
ordination involves that the probability of occurrence of the transition depends on 
the amplitude of the corresponding harmonic component in a way analogous to that 
in which, according to the classical theory, the intensity of the radiation emitted from a 
particle performing a harmonic oscillation would depend on its amplitude. At the same 
time, the state of polarisation of the radiation emitted during the transition is assumed 
to depend on the shape and orientation of the corresponding harmonic oscillation in a 
way analogous to that in which, on the classical theory, the polarisation of the radiation 
would depend on the orbit of tne emitting particle. 
In this discourse it was shown by examples from the investigation of the spectra 
of the elements and of the effects of electric and magnetic fields on spectral lines, how 
this correspondence principle has been supported to an extent that seems to justify 
us in using it as a guide also in more complicated cases, which we meet in the theory 
of atomic constitution, and where it has not yet been possible to fix the stationary 
states in an unambiguous way by use of symbols borrowed from classical mechanics. 
