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% The Project Gutenberg EBook of On the Quantum Theory of Line-Spectra, Part 1 and 2, by Niels Henrik David Bohr
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% Title: On the Quantum Theory of Line-Spectra, Part 1 and 2              %
%                                                                         %
% Author: Niels Henrik David Bohr                                         %
%                                                                         %
% Release Date: November 22, 2014 [EBook #47167]                          %
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% Language: EN                                                            %
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% *** START OF THIS PROJECT GUTENBERG EBOOK QUANTUM THEORY OF LINE-SPECTRA ***
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\begin{PGtext}
The Project Gutenberg EBook of On the Quantum Theory of Line-Spectra, Part 1 and 2, by Niels Henrik David Bohr

This eBook is for the use of anyone anywhere in the United States and most
other parts of the world at no cost and with almost no restrictions
whatsoever.  You may copy it, give it away or re-use it under the terms of
the Project Gutenberg License included with this eBook or online at
www.gutenberg.org.  If you are not located in the United States, you'll have
to check the laws of the country where you are located before using this ebook.



Title: On the Quantum Theory of Line-Spectra, Part 1 and 2

Author: Niels Henrik David Bohr

Release Date: November 22, 2014 [EBook #47167]

Language: EN

Character set encoding: ISO-8859-1

*** START OF THIS PROJECT GUTENBERG EBOOK QUANTUM THEORY OF LINE-SPECTRA ***
\end{PGtext}
\end{minipage}
\end{center}
\newpage
%% Credits and transcriber's note %%
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\begin{minipage}{\textwidth}
\begin{PGtext}
Produced by Andrew D. Hwang
\end{PGtext}
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\vfill
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%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%
\PageSep{i}
\FrontMatter
\iffalse % [** TN: Publisher's titlepage with series header and price]
\begin{center}
Mémoires de l'Académie Royale des Sciences et des Lettres de Danemark. Copenhague.

Section des Sciences, 3\ieme~série, t.~IV, \numero~1, fasc.~1.

\Large ON \\
\LARGE THE QUANTUM THEORY \\
OF LINE-SPECTRA \\
\footnotesize
BY \\
\large
N. BOHR \\
\vfill

\textsf{PART I} \\
\vfill

\footnotesize
\textsc{D. Kgl.\ Danske Vidensk.\ Selsk.\ Skripter, naturvidensk.\ og mathem.\ Afd.,~8. Række,~IV.}
\vfill\vfill

\large
KØBENHAVN \\
\textsc{hovedkommissionær}: ANDR. FRED. HØST \&amp;~SON, \textsc{kgl., hof-boghandel} \\
BIANCO LUNOS BOGTRYKKERI \\
1918

Pris: 2~Kr.\ 25~Ore.
\end{center}
\cleardoublepage
\fi % [** TN: End of publisher's titlepage]
\PageSep{ii}
%[Blank page]
\PageSep{iii}
\begin{center}
\large ON
\vfill
\Huge THE QUANTUM THEORY \\
OF LINE-SPECTRA
\vfill

\footnotesize
BY
\vfill
\large
N. BOHR \\
\vfill\vfill\vfill

\scriptsize
\textsc{D. Kgl.\ Danske Vidensk.\ Selsk.\ Skripter, naturvidensk.\ og mathem.\ Afd.,~8. Række,~IV.}
\vfill\vfill\vfill

\normalsize
KØBENHAVN \\
\scriptsize
\textsc{hovedkommissionær}: ANDR. FRED. HØST \&amp;~SON, \textsc{kgl., hof-boghandel} \\
\textsc{bianco lunos bogtrykkeri} \\
1918
\end{center}
\newpage
\PageSep{iv}
\null
\vfill
\begin{center}
\itshape
\footnotesize
DEDICATED TO THE MEMORY OF~MY~VENERATED~TEACHER \\[2\baselineskip]
%[** TN: Small caps in the original]
\normalsize
Professor C. CHRISTIANSEN \\[\baselineskip]
\footnotesize
OCTOBER~9, 1843\qquad \dag\ NOVEMBER~28, 1917
\upshape\vfill

UDGIVET PAA CARLSBERGFONDETS BEKOSTNING
\end{center}
\PageSep{v}
\MainMatter


\Introduction

\First{In} an attempt to develop certain outlines of a theory of line-spectra based on
a suitable application of the fundamental ideas introduced by Planck in his theory
of temperature-radiation to the theory of the nucleus atom of Sir \Name{Ernest Rutherford},
the writer has shown that it is possible in this way to obtain a simple interpretation
of some of the main laws governing the line-spectra of the elements, and
especially to obtain a deduction of the well known Balmer formula for the hydrogen
spectrum\footnote
  {\Name{N. Bohr}, Phil.\ Mag., XXVI, pp.\ 1, 476, 857~(1913), XXVII, p.~506 (1914), XXIX. p.~332 (1915), XXX.
  p.~394 (1915).}
The theory in the form given allowed of a detailed discussion
only in the case of periodic systems, and obviously was not able to account in
detail for the characteristic difference between the hydrogen spectrum and the
spectra of other elements, or for the characteristic effects on the hydrogen spectrum
of external electric and magnetic fields. Recently, however, a way out of this difficulty
has been opened by \Name{Sommerfeld}\footnote
  {\Name{A.~Sommerfeld}, Ber.\ Akad.\ München, 1915, pp.~425, 459, 1916, p.~131. 1917. p.~83. Ann.\ de Phys.,
  LI. p.~1 (1916).}
who, by introducing a suitable generalisation
of the theory to a simple type of non-periodic motions and by taking the
small variation of the mass of the electron with its velocity into account, obtained
an explanation of the fine-structure of the hydrogen lines which was found to be
in brilliant conformity with the measurements. Already in his first paper on this
subject, \Name{Sommerfeld} pointed out that his theory evidently offered a clue to the
interpretation of the more intricate structure of the spectra of other elements. Briefly
afterwards \Name{Epstein}\footnote
  {\Name{P.~Epstein}, Phys.\ Zeitschr.\ XVII, p.~148 (1916), Ann.\ d.\ Phys.~L, p.~489. LI. p.~168 (1916).}
 and \Name{Schwarzschild},\footnote
  {\Name{K.~\Typo{Schwarzchild}{Schwarzschild}}, Ber.\ Akad.\ Berlin, 1916, p.~548.}
independent of each other, by adapting
\Name{Sommerfeld}'s ideas to the treatment of a more extended class of non-periodic
systems obtained a detailed explanation of the characteristic effect of an electric
field on the hydrogen spectrum discovered by \Name{Stark}. Subsequently \Name{Sommerfeld}\footnote
  {\Name{A.~Sommerfeld}, Phys.\ Zeitschr.\ XVII, p.~491 (1916).}
himself and \Name{Debye}\footnote
  {\Name{P.~Debye}, Nachr.\ K. Ges.\ d.~Wiss.\ Göttingen, 1916, Phys.\ Zeitschr.\ XVII, p.~507 (1916).}
have on the same lines indicated an interpretation of the
\PageSep{4}
effect of a magnetic field on the hydrogen spectrum which, although no complete
explanation of the observations was obtained, undoubtedly represents an important
step towards a detailed understanding of this phenomenon.

In spite of the great progress involved in these investigations many difficulties
of fundamental nature remained unsolved, not only as regards the limited applicability
of the methods used in calculating the frequencies of the spectrum of
a given system, but especially as regards the question of the polarisation and intensity
of the emitted spectral lines. These difficulties are intimately connected with
the radical departure from the ordinary ideas of mechanics and electrodynamics
involved in the main principles of the quantum theory, and with the fact that it
has not been possible hitherto to replace these ideas by others forming an equally
consistent and developed structure. Also in this respect, however, great progress
has recently been obtained by the work of \Name{Einstein}\footnote
  {\Name{A.~Einstein}, Verh.\ d.\ D. phys.\ Ges.\ XVIII, p.~318 (1916), Phys.\ Zeitschr.\ XVIII, p.~121 (1917).}
and \Name{Ehrenfest}.\footnote
  {\Name{P.~Ehrenfest}, Proc.\ Acad.\ Amsterdam, XVI. p.~591 (1914), Phys.\ Zeitschr.\ XV. p.~657 (1914), Ann.\
  d.\ Phys.\ LI. p.~327 (1916)\Add{,} Phil.\ Mag.\ XXXIII. p.~500 (1917).}
On this
state of the theory it might therefore be of interest to make an attempt to discuss
the different applications from a uniform point of view, and especially to consider
the underlying assumptions in their relations to ordinary mechanics and electrodynamics.
Such an attempt has been made in the present paper, and it will be shown
that it seems possible to throw some light on the outstanding difficulties by trying
to trace the analogy between the quantum theory and the ordinary theory of
radiation as closely as possible.

The paper is divided into four parts.
\begin{description}
\item{Part I} contains a brief discussion of the general principles of the theory and
deals with the application of the general theory to periodic systems of one
degree of freedom and to the class of non-periodic systems referred to
above.

\item{Part II} contains a detailed discussion of the theory of the hydrogen spectrum in
order to illustrate the general considerations.

\item{Part III} contains a discussion of the questions arising in connection with the explanation
of the spectra of other elements.

\item{Part IV} contains a general discussion of the theory of the constitution of atoms and
molecules based on the application of the quantum theory to the nucleus
atom.
\end{description}

{\footnotesize Copenhagen, November 1917.}
\PageSep{5}

\Part{I.}{On the general theory.}

\Section{1.}{General principles.}

The quantum theory of line-spectra rests upon the following fundamental
assumptions:

\begin{Hypothesis}{I.}
That an atomic system can, and can only, exist permanently
in a certain series of states corresponding to a discontinuous series
of values for its energy, and that consequently any change of the
energy of the system, including emission and absorption of electromagnetic
radiation, must take place by a complete transition
between two such states. These states will be denoted as the
``stationary states'' of the system.
\end{Hypothesis}

\begin{Hypothesis}{II.}
That the radiation absorbed or emitted during a transition
between two stationary states is ``unifrequentic'' and possesses a
frequency~$\nu$, given by the relation
\[
E' - E'' = h\nu,
\Tag{(1)}
\]
where $h$~is \Name{Planck}'s constant and where $E'$~and $E''$ are the values of
the energy in the two states under consideration.
\end{Hypothesis}

As pointed out by the writer in the papers referred to in the introduction,
these assumptions offer an immediate interpretation of the fundamental \Emph{principle
of combination of spectral lines} deduced from the measurements of the
frequencies of the series spectra of the elements. According to the laws discovered
by \Name{Balmer}, \Name{Rydberg} and \Name{Ritz}, the frequencies of the lines of the series spectrum
of an element can be expressed by a formula of the type:
\[
\nu = f_{\tau''}(n'') - f_{\tau'}(n'),
\Tag{(2)}
\]
where $n'$ and $n''$ are whole numbers and $f_{\tau}(n)$~is one among a set of functions
of~$n$, characteristic for the element under consideration. On the above assumptions
this formula may obviously be \Typo{interpretated}{interpreted} by assuming that the stationary states
of an atom of an element form a set of series, and that the energy in the \Ord{$n$}{th}~state
of the \Ord{$\tau$}{th}~series, omitting an arbitrary constant, is given by
\[
E_{\tau}(n) = -hf_{\tau}(n).
\Tag{(3)}
\]

We thus see that the values for the energy in the stationary states of an atom
may be obtained directly from the measurements of the spectrum by means of
relation~\Eq{(1)}. In order, however, to obtain a theoretical connection between these
\PageSep{6}
values and the experimental evidence about the constitution of the atom obtained from
other sources, it is necessary to introduce further assumptions about the laws which
govern the stationary states of a given atomic system and the transitions between
these states.

Now\Pagelabel{6} on the basis of a vast amount of experimental evidence, we are forced
to assume that an atom or molecule consists of a number of electrified particles in
motion, and, since the above fundamental assumptions imply that no emission of
radiation takes place in the stationary states, we must consequently assume that
\Emph{the ordinary laws of electrodynamics cannot be applied} to these states
without radical alterations. In many cases, however, the effect of that part of the
electrodynamical forces which is connected with the emission of radiation will
at any moment be very small in comparison with the effect of the simple electrostatic
attractions or repulsions of the charged particles corresponding to \Name{Coulomb}'s
law. Even if the theory of radiation must be completely altered, it is therefore a
natural assumption that it is possible in such cases to obtain a close approximation
in the description of the motion in the stationary states, by retaining only the
latter forces. In the following we shall therefore, as in all the papers mentioned
in the introduction, for the present \Emph{calculate the motions of the particles
in the stationary states as the motions of mass-points according to
ordinary mechanics} including the modifications claimed by the theory of relativity,
and we shall later in the discussion of the special applications come back to
the question of the degree of approximation which may be obtained in this way.

If next we consider a transition between two stationary states, it is obvious
at once from the essential discontinuity, involved in the assumptions I~and~II, that
in general it is impossible even approximately to describe this phenomenon by
means of ordinary mechanics or to calculate the frequency of the radiation absorbed
or emitted by such a process by means of ordinary electrodynamics. On the other
hand, from the fact that it has been possible by means of ordinary mechanics and
electrodynamics to account for the phenomenon of temperature-radiation in the
limiting region of slow vibrations, we may expect that any theory capable of describing
this phenomenon in accordance with observations will form some sort of natural
generalisation of the ordinary theory of radiation. Now the theory of temperature-radiation
in the form originally given by \Name{Planck} confessedly lacked internal
consistency, since, in the deduction of his radiation formula, assumptions of
similar character as I~and~II were used in connection with assumptions which were
in obvious contrast to them. Quite recently, however, \Name{Einstein}\footnote
  {\Name{A.~Einstein}, loc.\ cit.}
has succeeded,
on the basis of the assumptions I~and~II, to give a consistent and instructive deduction
of \Name{Planck}'s formula by introducing certain supplementary assumptions about
the \Emph{probability of transition of a system between two stationary states}
and about the manner in which this probability depends on the density of radiation
\PageSep{7}
of the corresponding frequency in the surrounding space, suggested from
analogy with the ordinary theory of radiation. \Name{Einstein}\Pagelabel{7} compares the emission or
absorption of radiation of frequency~$\nu$ corresponding to a transition between two
stationary states with the emission or absorption to be expected on ordinary
electrodynamics for a system consisting of a particle executing harmonic vibrations
of this frequency. In analogy with the fact that on the latter theory such a system
will without external excitation emit a radiation of frequency~$\nu$, \Name{Einstein} assumes
in the first place that on the quantum theory there will be a certain probability~$A_{n''}^{n'}\, dt$
that the system in the stationary state of greater energy, characterised by
the letter~$n'$, in the time interval~$dt$ will start \Emph{spontaneously} to pass to the stationary
state of smaller energy, characterised by the letter~$n''$. Moreover, on ordinary
electrodynamics the harmonic vibrator will, in addition to the above mentioned
independent emission, in the presence of a radiation of frequency~$\nu$ in the surrounding
space, and dependent on the accidental phase-difference between this
radiation and the vibrator, emit or absorb radiation-energy. In analogy with this,
\Name{Einstein} assumes secondly that in the presence of a radiation in the surrounding
space, the system will on the quantum theory, in addition to the above mentioned
probability of spontaneous transition from the state~$n'$ to the state~$n''$, possess
a certain probability, depending on this radiation, of passing in the time~$dt$
from the state~$n'$ to the state~$n''$, as well as from the state~$n''$ to the state~$n'$. These
latter probabilities are assumed to be proportional to the intensity of the surrounding
radiation and are denoted by $\rho_{\nu} B_{n''}^{n'}\, dt$ and $\rho_{\nu} B_{n'}^{n''}\, dt$ respectively, where $\rho_{\nu}\, d\nu$~denotes
the amount of radiation in unit volume of the surrounding space distributed on
frequencies between $\nu$~and $\nu + d\nu$, while $B_{n''}^{n'}$~and $B_{n'}^{n''}$ are constants which, like~$A_{n''}^{n'}$,
depend only on the stationary states under consideration. \Name{Einstein} does not introduce
any detailed assumption as to the values of these constants, no more than to the
conditions by which the different stationary states of a given system are determined
or to the ``a-priori probability'' of these states on which their relative occurrence
in a distribution of statistical equilibrium depends. He shows, however,
how it is possible from the above general assumptions, by means of \Name{Boltzmann}'s
principle on the relation between entropy and probability and \Name{Wien}'s well known
displacement-law, to deduce a formula for the temperature radiation which apart
from an undetermined constant factor coincides with \Name{Planck}'s, if we only assume
that the frequency corresponding to the transition between the two states is determined
by~\Eq{(1)}. It will therefore be seen that by reversing the line of argument,
\Name{Einstein}'s theory may be considered as a very direct support of the latter relation.

In the following discussion of the application of the quantum theory to determine
the line-spectrum of a given system, it will, just as in the theory of temperature-radiation,
not be necessary to introduce detailed assumptions as to the mechanism
of transition between two stationary states. We shall show, however, that the
conditions which will be used to determine the values of the energy in the stationary
states are of such a type that the frequencies calculated by~\Eq{(1)}, in the limit
\PageSep{8}
where the motions in successive stationary states comparatively differ very little
from each other, will tend to coincide with the frequencies to be expected on
the ordinary theory of radiation from the motion of the system in the stationary
states. In order to obtain the necessary relation to the ordinary theory of radiation
in the limit of slow vibrations, we are therefore led directly to certain conclusions
about the probability of transition between two stationary states in this limit. This
leads again to certain general considerations about the connection between the
probability of a transition between any two stationary states and the motion of the
system in these states, which will be shown to throw light on the question of the
polarisation and intensity of the different lines of the spectrum of a given system.

In the above considerations we have by an atomic system \Typo{tacidly}{tacitly} understood
a number of electrified particles which move in a field of force which, with the
approximation mentioned, possesses a potential depending only on the position of
the particles. This may more accurately be denoted as a system under constant
external conditions, and the question next arises about the variation in the stationary
states which may be expected to take place during a variation of the external
conditions, \eg\ when exposing the atomic system to some variable external field
of force. Now, in general, we must obviously assume that this variation cannot
be calculated by ordinary mechanics, no more than the transition between two
different stationary states corresponding to constant external conditions. If, however,
the variation of the external conditions is very slow, we may from the necessary
stability of the stationary states expect that the motion of the system at any
given moment during the variation will differ only very little from the motion in
a stationary state corresponding to the instantaneous external conditions. If now,
moreover, the variation is performed at a constant or very slowly changing rate,
the forces to which the particles of the system will be exposed will not differ at
any moment from those to which they would be exposed if we imagine that the
external forces arise from a number of slowly moving additional particles which
together with the original system form a system in a stationary state. From this
point of view it seems therefore natural to assume that, with the approximation
mentioned, the motion of an atomic system in the stationary states can be calculated
by direct application of ordinary mechanics, not only under constant external
conditions, but in general also during a slow and uniform variation of these
conditions. This assumption, which may be denoted as the principle of the
``\Emph{mechanical transformability}'' of the stationary states, has been introduced
in the quantum theory by \Name{Ehrenfest}\footnote
  {\Name{P.~Ehrenfest}, loc.\ cit. In these papers the principle in question is called the ``adiabatic hypothesis''
  in accordance with the line of argumentation followed by \Name{Ehrenfest} in which considerations
  of thermodynamical problems play an important part. From the point of view taken in the present
  paper, however, the above notation might in a more direct way indicate the content of the principle
  and the limits of its applicability.}
and is, as it will be seen in the following
sections, of great importance in the discussion of the conditions to be used to fix
\PageSep{9}
the stationary states of an atomic system among the continuous multitude of
mechanically possible motions. In this connection it may be pointed out that
the principle of the mechanical transformability of the stationary states allows
us to overcome a fundamental difficulty which at first sight would seem to
be involved in the definition of the energy difference between two stationary states
which enters in relation~\Eq{(1)}. In fact we have assumed that the direct transition
between two such states cannot be described by ordinary mechanics, while on
the other hand we possess no means of defining an energy difference between two
states if there exists no possibility for a continuous mechanical connection between
them. It is clear, however, that such a connection is just afforded by \Name{Ehrenfest}'s
principle which allows us to transform mechanically the stationary states of a
given system into those of another, because for the latter system we may take one
in which the forces which act on the particles are very small and where we may
assume that the values of the energy in all the stationary states will tend to coincide.

As regards the problem of the statistical distribution of the different stationary
states between a great number of atomic systems of the same kind in temperature
equilibrium, the number of systems present in the different states may be deduced
in the well known way from \Name{Boltzmann}'s fundamental relation between entropy
and probability, if we know the values of the energy in these states and the
\Emph{a-priori probability} to be ascribed to each state in the calculation of the probability
of the whole distribution. In contrast to considerations of ordinary statistical
mechanics we possess on the quantum theory no direct means of determining these
a-priori probabilities, because we have no detailed information about the mechanism
of transition between the different stationary states. If the a-priori probabilities
are known for the states of a given atomic system, however, they may be deduced
for any other system which can be formed from this by a continuous transformation
without passing through one of the singular systems referred to below. In fact,
in examining the necessary conditions for the explanation of the second law of thermodynamics
\Name{Ehrenfest}\Pagelabel{9}\footnote
  {\Name{P.~Ehrenfest}, Phys.\ Zeitschr.\ XV p.~660 (1914). The above interpretation of this relation is
  not stated explicitly by \Name{Ehrenfest}, but it presents itself directly if the quantum theory is taken in
  the form corresponding to the fundamental assumption~I.}
has deduced a certain general condition as regards the
variation of the a-priori probability corresponding to a small change of the external
conditions from which it follows, that the a-priori probability of a given stationary
state of an atomic system must remain unaltered during a continuous transformation,
except in special cases in which the values of the energy in some of the stationary
states will tend to coincide during the transformation. In this result we possess,
as we shall see, a rational basis for the determination of the a-priori probability
of the different stationary states of a given atomic system.
%[** TN: Printer's mark]
% D.K.D. Vidensk. Selsk. Skr., naturvidensk. og mathem. Ard., 8. Række, IV. 1.
\PageSep{10}


\Section{2.}{Systems of one degree of freedom.}

As the simplest illustration of the principles discussed in the former section
we shall begin by considering systems of a single degree of freedom, in which case
it has been possible to establish a general theory of stationary states. This is
due to the fact\Chg{,}{} that \Emph{the motion will be simply periodic}, provided the
distance between the parts of the system will not increase infinitely with the time,
a case which for obvious reasons cannot represent a stationary state in the sense
defined above. On account of this, the discussion of the mechanical transformability
of the stationary states can, as pointed out by \Name{Ehrenfest},\footnote
  {\Name{P.~Ehrenfest}, loc.\ cit.\ Proc.\ Acad.\ Amsterdam, XVI, p.~591 (1914).}
for systems of
one degree of freedom be based on a mechanical theorem about periodic systems due
to \Name{Boltzmann} and originally applied by this author in a discussion of the bearing of
mechanics on the explanation of the laws of thermodynamics. For the sake of the
considerations in the following sections it will be convenient here to give the proof
in a form which differs slightly from that given by \Name{Ehrenfest}, and which takes
also regard to the modifications in the ordinary laws of mechanics claimed by the
theory of relativity.

Consider for the sake of generality a conservative mechanical system of $s$~degrees
of freedom, the motion of which is governed by \Name{Hamilton}'s equations:
\[
\frac{dp_{k}}{dt} = -\frac{\dd E}{\dd q_{k}},\quad
\frac{dq_{k}}{dt} =  \frac{\dd E}{\dd p_{k}},\qquad
(k = 1, \dots, s)
\Tag{(4)}
\]
where $E$~is the total energy considered as a function of the generalised positional
coordinates $q_{1}$,~\MyDots $q_{s}$ and the corresponding canonically conjugated momenta $p_{1}$,~\MyDots $p_{s}$.\Pagelabel{10}
If the velocities are so small that the variation in the mass of the particles due to
their velocities can be neglected, the $p$'s~are defined in the usual way by
\[
p_{k} = \frac{\dd T}{\dd q_{k}},\qquad
(k = 1, \dots, s)
\]
where $T$~is the kinetic energy of the system considered as a function of the
generalised velocities $\dot{q}_{1}$,~\MyDots $\dot{q}_{s}$ $\left(\dot{q}_{k} = \dfrac{dq_{k}}{dt}\right)$ and of $q_{1}$,~\MyDots $q_{s}$. If the relativity modifications
are taken into account the $p$'s~are defined by a similar set of expressions
in which the kinetic energy is replaced by
% [** TN: Set in-line in the original]
\[
T' = \sum m_{0} c^{2}\bigl(1 - \sqrt{1 - v^{2}/c^{2}}\bigr),
\]
where
the summation is to be extended over all the particles of the system, and $v$~is the
velocity of one of the particles and $m_{0}$~its mass for zero velocity, while $c$~is the
velocity of light.

Let us now assume that the system performs a periodic motion with the
period~$\sigma$, and let us form the expression
\[
I = \int_{0}^{\sigma} \sum_{1}^{s} p_{k} \dot{q}_{k}\, dt,
\Tag{(5)}
\]
\PageSep{11}
which is easily seen to be independent of the special choice of coordinates $q_{1}$,~\MyDots $q_{s}$
used to \Typo{decribe}{describe} the motion of the system. In fact, if the variation of the mass
with the velocity is neglected we get
\[
I = 2 \int_{0}^{\sigma} T\, dt,
\]
and if the relativity modifications are included, we get a quite analogous expression
in which the kinetic energy is replaced by $T'' = \sum \frac{1}{2} m_{0} v^{2} \sqrt{1 - v^{2}/c^{2}}$.

Consider next some new periodic motion of the system formed by a small
variation of the first motion, but which may need the presence of external forces
in order to be a mechanically possible motion. For the variation in~$I$ we get then
\[
\delta I = \int_{0}^{\sigma} \sum_{1}^{s} (\dot{q}_{k}\, \delta p_{k} + p_{k}\, \delta \dot{q}_{k})\, dt
  + \sum_{1}^{s} p_{k}\, \dot{q}_{k}\, \delta t\Big|_{0}^{\sigma},
\]
where the last term refers to the variation of the limit of the integral due to the
variation in the period~$\sigma$. By partial integration of the second term in the bracket
under the integral we get next
\[
\delta I = \int_{0}^{\sigma} \sum_{1}^{s} (\dot{q}_{k}\, \delta p_{k} - \dot{p}_{k}\, \delta q_{k})\, dt
  + \sum_{1}^{s} p_{k}(\dot{q}_{k}\, \delta t + \delta q_{k})\Big|_{0}^{\sigma},
\]
where the last term is seen to be zero, because the term in the bracket as well as~$p_{k}$
will be the same in both limits, since the varied motion as well as the original
motion is assumed to be periodic. By means of equations~\Eq{(4)} we get therefore
\[
\delta I = \int_{0}^{\sigma} \sum_{1}^{s}
  \left(\frac{\dd E}{\dd p_{k}}\, \delta p_{k}
      + \frac{\dd E}{\dd q_{k}}\, \delta q_{k}\right) dt
  = \int_{0}^{\sigma} \delta E\, dt.
\Tag{(6)}
\]

Let\Pagelabel{11} us now assume that the small variation of the motion is produced by a
small external field established at a uniform rate during a time interval~$\theta$, long
compared with~$\sigma$, so that the comparative increase during a period is very small.
In this case $\delta E$~is at any moment equal to the total work done by the external
forces on the particles of the system since the beginning of the establishment of
the field. Let this moment be $t = -\theta$ and let the potential of the external field
at $t \geq 0$ be given by~$\Omega$, expressed as a function of the~$q$'s. At any given moment
$t &gt; 0$ we have then
\[
% [** TN: Correction to integrand numerator retrieved October 17, 2014 from
%     web.ihep.su/dbserv/compas/src/bohr18/eng.pdf]
\delta E
  = -\int_{-\theta}^{0} \frac{\theta \Add{ + t}}{\theta} \sum_{1}^{s} \frac{\dd \Omega}{\dd q_{k}}\, \dot{q}_{k}\, dt
  - \int_{0}^{t} \sum_{1}^{s} \frac{\dd \Omega}{\dd q_{k}}\, \dot{q}_{k}\, dt,
\]
which gives by partial integration
\PageSep{12}
\[
\delta E = \frac{1}{\theta} \int_{-\theta}^{0} \Omega\, dt - \Omega_{t},
\]
where the values for the $q$'s to be introduced in~$\Omega$ in the first term are those
corresponding to the motion under the influence of the increasing external field, and
the values to be introduced in the second term are those corresponding to the
configuration at the time~$t$. Neglecting small quantities of the same order as the
square of the external force, however, we may in this expression for~$\delta E$ instead of
the values for the $q$'s corresponding to the perturbed motion take those corresponding
to the original motion of the system. With this approximation the first term
is equal to the mean value of the second taken over a period~$\sigma$, and we have
consequently
\[
\int_{0}^{\sigma} \delta E\, dt = 0.
\Tag{(7)}
\]

From \Eq{(6)} and \Eq{(7)} it follows that $I$~will remain constant during the slow
establishment of the small external field, if the motion corresponding to a constant
value of the field is periodic. If next the external field corresponding to~$\Omega$ is considered
as an inherent part of the system, it will be seen in the same way that $I$~will
remain unaltered during the establishment of a new small external field, and
so on. Consequently \Emph{$I$~will be invariant for any finite transformation
of the system which is sufficiently slowly performed}, provided the
motion at any moment during the process is periodic and the effect of the variation
is calculated on ordinary mechanics.

Before we proceed to the applications of this result we shall mention a simple
consequence of~\Eq{(6)} for systems for which every orbit is periodic independent of the
initial conditions. In that case we may for the varied motion take an undisturbed
motion of the system corresponding to slightly different initial conditions. This
gives $\delta E$~constant, and from~\Eq{(6)} we get therefore
\[
\delta E = \omega\, \delta I,
\Tag{(8)}
\]
where $\omega = \dfrac{1}{\sigma}$ is the frequency of the motion. This equation forms a simple relation
between the variations in $E$ and~$I$ for periodic systems, which will be often
used in the following.

Returning now to systems of one degree of freedom, we shall take our starting
point from \Name{Planck}'s original theory of a \Emph{linear harmonic vibrator}. According to
this theory the stationary states of a system, consisting of a particle executing linear
harmonic vibrations with a constant frequency~$\omega_{0}$ independent of the energy, are
given by the well known relation
\[
E = nh\omega_{0},
\Tag{(9)}
\]
where $n$~is a positive entire number, $h$~\Name{Planck}'s constant, and $E$~the total energy
which is supposed to be zero if the particle is at rest.
\PageSep{13}

From \Eq{(8)} it follows at once\Chg{,}{} that \Eq{(9)}~is equivalent to
\[
I = \int_{0}^{\sigma} p\dot{q}\, dt
  = \int p\, dq
  = nh,
\Tag{(10)}
\]
where the latter integral is to be taken over a complete oscillation of~$q$ between
its limits. On the principle of the mechanical transformability of the stationary
states we shall therefore assume, following \Name{Ehrenfest}, that \Eq{(10)}~holds not only for a
\Name{Planck}'s vibrator but for \emph{any periodic system of one degree of freedom} which
can be formed in a continuous manner from a linear harmonic vibrator by a gradual
variation of the field of force in which the particle moves. This condition is immediately
seen to be fulfilled by all such systems in which the motion is of oscillating
type \ie\ where the moving particle during a period passes twice through
any point of its orbit once in each direction. If, however, we confine ourselves to
systems of one degree of freedom, it will be seen that systems in which the
motion is of rotating type, \ie\ where the particle during a period passes only once
through every point of its orbit, cannot be formed in a continuous manner from
a linear harmonic vibrator without passing through singular states in which the
period becomes infinite\Note{[sic]} long and the result becomes ambiguous. We shall not here
enter more closely on this difficulty which has been pointed out by \Name{Ehrenfest}, because
it disappears when we consider systems of several degrees of freedom, where we
shall see that a simple generalisation of~\Eq{(10)} holds for any system for which
every motion is periodic.

As regards the application of~\Eq{(9)} to statistical problems it was assumed in
\Name{Planck}'s theory that the different states of the vibrator corresponding to different
values of~$n$ are \Emph{a-priori equally probable}, and this assumption was strongly
supported by the agreement obtained on this basis with the measurements of the
specific heat of solids at low temperatures. Now it follows from the considerations
of \Name{Ehrenfest}, mentioned in the former section, that the a-priori probability of a
given stationary state is not changed by a continuous transformation, and we shall
therefore expect that for any system of one degree of freedom the different states
corresponding to different entire values of~$n$ in~\Eq{(10)} are a-priori equally probable.

As pointed out by \Name{Planck} in connection with the application of~\Eq{(9)}, it is
simply seen that statistical considerations, based on the assumption of equal probability
for the different states given by~\Eq{(10)}, will show the necessary relation to
considerations of ordinary statistical mechanics in the limit where the latter theory
has been found to give results in agreement with experiments. Let the configuration
and motion of a mechanical system be characterised by $s$~independent variables
$q_{1}$,~\MyDots $q_{s}$ and corresponding momenta $p_{1}$,~\MyDots $p_{s}$ and let the state of the system be
represented in a $2s$-dimensional phase-space by a point with coordinates $q_{1}$,~\MyDots $q_{s}$,
$p_{1}$,~\MyDots $p_{s}$. Then, according to ordinary statistical mechanics, the probability for
this point to lie within a small element in the phase-space is independent of the
\PageSep{14}
position and shape of this element and simply proportional to its volume, defined
in the usual way by
\[
\delta W = \int dq_{1} \dots dq_{s}\, dp_{1} \dots dp_{s}.
\Tag{(11)}
\]
In the quantum theory, however, these considerations cannot be directly applied,
since the point representing the state of a system cannot be displaced continuously
in the $2s$-dimensional phase-space, but can lie only on certain surfaces of lower
dimensions in this space. For systems of one degree of freedom the phase-space is
a two-dimensional surface, and the points representing the states of some system
given by~\Eq{(10)} will be situated on closed curves on this surface. Now, in general, the
motion will differ considerably for any two states corresponding to successive entire
values of~$n$ in~\Eq{(10)}, and a simple general connection between the quantum theory
and ordinary statistical mechanics is therefore out of question. In the limit, however,
where $n$~is large, the motions in successive states will only differ very little
from each other, and it would therefore make little difference whether the points
representing the systems are distributed continuously on the phase-surface or situated
only on the curves corresponding to~\Eq{(10)}, provided the number of systems which
in the first case are situated between two such curves is equal to the number which
in the second case lies on one of these curves. But it will be seen that this condition
is just fulfilled in consequence of the above hypothesis of equal a-priori probability
of the different stationary states, because the element of phase-surface
limited by two successive curves corresponding to~\Eq{(10)} is equal to
\[
\BreakMath{%
\delta W = \int dp\, dq
  = \left[\int p\, dq\right]_{n} - \left[\int p\, dq\right]_{n-1}
  = I_{n} - I_{n-1}
  = h,
}{%
\begin{aligned}
\delta W = \int dp\, dq
  &amp;= \left[\int p\, dq\right]_{n} - \left[\int p\, dq\right]_{n-1} \\
  &amp;= I_{n} - I_{n-1}
   = h,
\end{aligned}
}
\Tag{(12)}
\]
so that on ordinary statistical mechanics the probabilities for the point to lie within
any two such elements is the same. We see consequently that the hypothesis of
equal probability of the different states given by~\Eq{(10)} gives the same result as
ordinary statistical mechanics in all such applications in which the states of the
great majority of the systems correspond to large values of~$n$. Considerations of
this kind have led \Name{Debye}\footnote
  {\Name{P.~Debye}, Wolfskehl-Vortrag. Göttingen 1913.}
to point out that condition~\Eq{(10)} might have a general
validity for systems of one degree of freedom, already before \Name{Ehrenfest}, on the
basis of his theory of the mechanical transformability of the stationary states,
had shown that this condition forms the only rational generalisation of \Name{Planck}'s
condition~\Eq{(9)}.

We shall now discuss the relation between the theory of \Emph{spectra of atomic
systems of one degree of freedom}, based on \Eq{(1)}~and \Eq{(10)}, and the ordinary
theory of radiation, and we shall see that this relation in several respects shows a
close analogy to the relation, just considered, between the statistical applications of~\Eq{(10)}
and considerations based on ordinary statistical mechanics. Since the values
\PageSep{15}
for the frequency~$\omega$ in two states corresponding to different values of~$n$ in~\Eq{(10)} in
general are different, we see at once that we cannot expect a simple connection
between the frequency calculated by~\Eq{(1)} of the radiation corresponding to a
transition between two stationary states and the motions of the system in these
states, except in the limit where $n$~is very large, and where the ratio between the
frequencies of the motion in successive stationary states differs very little from unity.
Consider now a transition between the state corresponding to $n = n'$ and the state
corresponding to $n = n''$, and let us assume that $n'$~and $n''$ are large numbers and
that $n' - n''$ is small compared with $n'$~and~$n''$. In that case we may in~\Eq{(8)} for
$\delta E$ put $E' - E''$ and for $\delta I$ put $I' - I''$, and we get therefore from \Eq{(1)}~and \Eq{(10)} for
the frequency of the radiation emitted or absorbed during the transition between
the two states
\[
\nu = \frac{1}{h} (E' - E'')
  = \frac{\omega}{h} (I' - I'')
  = (n' - n'') \omega.
\Tag{(13)}
\]

Now in a stationary state of a periodic system the displacement of the particles
in any given direction may always be expressed by means of a \Name{Fourier}-series
as a sum of harmonic vibrations:
\[
\xi = \sum C_{\tau} \cos 2\pi(\tau\omega t + c_{\tau}),
\Tag{(14)}
\]
where the $C$'s and $c$'s are constants and the summation is to be extended over all
positive entire values of~$\tau$. On the ordinary theory of radiation we should therefore
expect the system to emit a spectrum consisting of a series of lines of frequencies
equal to~$\tau\omega$, but, as it is seen, this is just equal to the series of frequencies
which we obtain from~\Eq{(13)} by introducing different values for $n' - n''$. As far as
the frequencies are concerned we see therefore that in the limit where $n$~is large
there exists a close relation between the ordinary theory of radiation and the theory
of spectra based on \Eq{(1)} and~\Eq{(10)}. It may be noticed, however, that, while on the
first theory radiations of the different frequencies~$\tau\omega$ corresponding to different
values of~$\tau$ are emitted or absorbed at the same time, these frequencies will on
the present theory, based on the fundamental assumption I~and~II, be connected
with entirely different processes of emission or absorption, corresponding to the
transition of the system from a given state to different neighbouring stationary states.

In order to obtain the necessary connection, mentioned in the former section,
to the ordinary theory of radiation in the limit of slow vibrations, we must further
claim that a relation, as that just proved for the frequencies, will, in the limit of
large~$n$, hold also for the intensities of the different lines in the spectrum. Since
now on ordinary electrodynamics the intensities of the radiations corresponding
to different values of~$\tau$ are directly determined from the coefficients~$C_{|tau}$ in~\Eq{(14)}, we
must therefore expect that for large values of~$n$ these coefficients will on the quantum
theory determine the \Emph{probability of \Typo{spontanuous}{spontaneous} transition} from a given
stationary state for which $n = n'$ to a neighbouring state for which $n = n'' = n' - \tau$.
Now this connection between the amplitudes of the different harmonic vibrations into
\PageSep{16}
which the motion can be resolved, characterised by different values of~$\tau$, and the
probabilities of transition from a given stationary state to the different neighbouring
stationary states, characterised by different values of $n' - n''$, may clearly be expected
to be of a general nature. Although, of course, we cannot without a detailed
theory of the mechanism of transition obtain an exact calculation of the latter probabilities,
unless $n$~is large, we may expect that also for small values of~$n$ the amplitude
of the harmonic vibrations corresponding to a given value of~$\tau$ will in some
way give a measure for the probability of a transition between two states for which
$n' - n''$ is equal to~$\tau$. Thus in general there will be a certain probability of an
atomic system in a stationary state to pass spontaneously to any other state of
smaller energy, but if for all motions of a given system the coefficients~$C$ in~\Eq{(14)}
are zero for certain values of~$\tau$, we are led to expect that no transition will be
possible, for which $n' - n''$ is equal to one of these values.

A simple illustration of these considerations is offered by the linear harmonic
vibrator mentioned above in connection with \Name{Planck}'s theory. Since in this case
$C_{\tau}$~is equal to zero for any $\tau$ different from~$1$, we shall expect that for this system
only such transitions are possible in which $n$~alters by one unit. From \Eq{(1)}~and
\Eq{(9)} we obtain therefore the simple result that the frequency of any radiation emitted
or absorbed by a linear harmonic vibrator is equal to the constant frequency~$\omega_{0}$.
This result seems to be supported by observations on the absorption-spectra of
diatomic gases, showing that certain strong absorption-lines, which according to
general evidence may be ascribed to vibrations of the two atoms in the molecule
relative to each other, are not accompanied by lines of the same order of intensity
and corresponding to entire multipla of the frequency, such as it should be expected
from~\Eq{(1)} if the system had any considerable tendency to pass between non-successive
states. In this connection it may be noted that the fact, that in the absorption
spectra of some diatomic gases faint lines occur corresponding to the double frequency
of the main lines,\footnote
  {See \Name{E.~C. Kemble}, Phys.\ Rev., VIII, p.~701, 1916.}
obtains a natural explanation by assuming that for
finite amplitudes the vibrations are not exactly harmonic and that therefore the
molecules possess a small probability of passing also between non-successive states.


\Section{3.}{Conditionally periodic systems.}

If we consider systems of several degrees of freedom the motion will be periodic
only in singular cases and the general conditions which determine the stationary states
cannot therefore be derived by means of the same simple kind of considerations
as in the former section. As mentioned in the introduction, however, \Name{Sommerfeld}
and others have recently succeeded, by means of a suitable generalisation of~\Eq{(10)}, to
obtain conditions for an important class of systems of several degrees of freedom,
\PageSep{17}
which, in connection with~\Eq{(1)}, have been found to give results in convincing agreement
with experimental results about line-spectra. Subsequently these conditions
have been proved by \Name{Ehrenfest} and especially by \Name{Burgers}\footnote
  {\Name{J.~M. Burgers}, Versl.\ Akad.\ Amsterdam, XXV, pp.~849, 918,~1055. (1917), Ann.\ d.\ Phys.\ LII. p.~195
  (1917), Phil.\ Mag.\ XXXIII, p.~514 (1917).}
to be invariant for
slow mechanical transformations.

To the generalisation under consideration we are naturally led, if we first consider
such systems for which the motions corresponding to the different degrees of freedom
are dynamically independent of each other. This occurs if the expression for the
total energy~$E$ in \Name{Hamilton}\Add{'}s equations~\Eq{(4)} for a system of $s$~degrees of freedom
can be written as a sum $E_{1} + \dots + E_{s}$, where $E_{k}$~contains $q_{k}$~and $p_{k}$ only. An
illustration of a system of this kind is presented by a particle moving in a field
of force in which the force-components normal to three mutually perpendicular
fixed planes are functions of the distances from these planes respectively. Since in
such a case the motion corresponding to each degree of freedom in general will
be periodic, just as for a system of one degree of freedom, we may obviously expect
that the condition~\Eq{(10)} is here replaced by a set of $s$~conditions:
\[
I_{k} = \int p_{k}\, dq_{k} = n_{k} h,\qquad
(k = 1, \MyDots s)
\Tag{(15)}
\]
where the integrals are taken over a complete period of the different $q$'s respectively,
and where $n_{1}$,~\MyDots $n_{s}$ are entire numbers. It will be seen at once that these conditions
are invariant for any slow transformation of the system for which the independency
of the motions corresponding to the different coordinates is maintained.

A more general class of systems for which a similar analogy with systems of
a single degree of freedom exists and where conditions of the same type as~\Eq{(15)}
present themselves is obtained in the case where, although the motions corresponding
to the different degrees of freedom are not independent of each other, it is possible
nevertheless by a suitable choice of coordinates to express each of the momenta~$p_{k}$
as a function of $q_{k}$~only. A simple system of this kind consists of a particle moving
in a plane orbit in a central field of force. Taking the length of the radius-vector
from the centre of the field to the particle as~$q_{1}$, and the angular distance of this
radius-vector from a fixed line in the plane of the orbit as~$q_{2}$, we get at once
from~\Eq{(4)}, since $E$~does not contain~$q_{2}$, the well known result that during the motion
the angular momentum~$p_{2}$ is constant and that the radial motion, given by the
variations of $p_{1}$~and $q_{1}$ with the time, will be exactly the same as for a system
of one degree of freedom. In his fundamental application of the quantum theory
to the spectrum of a \Emph{non-periodic system} \Name{Sommerfeld} assumed therefore that
the stationary states of the above system are given by two conditions of the form:
\[
I_{1} = \int p_{1}\, dq_{1} = n_{1} h,\quad
I_{2} = \int p_{2}\, dq_{2} = n_{2} h.
\Tag{(16)}
\]
%[** TN: Printer's mark]
%D. K. D. Vidensk. Selsk. Skr., naturvidensk. og mathem. Afd., 8. Række. IV. 1.
\PageSep{18}
While the first integral obviously must be taken over a period of the radial motion,
there might at first sight seem to be a difficulty in fixing the limits of integration
of~$q_{2}$. This disappears, however, if we notice that an integral of the type under
consideration will not be altered by a change of coordinates in which $q$~is replaced
by some function of this variable. In fact, if instead of the angular distance of the
radius-vector we take for~$q_{2}$ some continuous periodic function of this angle with
period~$2\pi$, every point in the plane of the orbit will correspond to one set of
coordinates only and the relation between $p$~and~$q$ will be exactly of the same
type as for a periodic system of one degree of freedom for which the motion is of
oscillating type. It follows therefore that the integration in the second of the
conditions~\Eq{(16)} has to be taken over a complete revolution of the radius-vector,
and that consequently this condition is equivalent with the simple condition that
the angular momentum of the particle round the centre of the field is equal to an
entire \Chg{multiplum}{multiple} of~$\dfrac{h}{2\pi}$. As pointed out by \Name{Ehrenfest}, the conditions~\Eq{(16)} are
invariant for such special transformations of the system for which the central
symmetry is maintained. This follows immediately from the fact that the angular
momentum in transformations of this type remains invariant, and that the equations
of motion for the radial coordinate as long as $p_{2}$~remains constant are the
same as for a system of one degree of freedom. On\Pagelabel{18} the basis of~\Eq{(16)}, \Name{Sommerfeld}
has, as mentioned in the introduction, obtained a brilliant explanation of the fine
structure of the lines in the hydrogen spectrum, due to the change of the mass
of the electron with its velocity.\footnote
  {In this connection it may be remarked that conditions of the same type as~\Eq{(16)} were proposed
  independently by \Name{W.~Wilson} (Phil.\ Mag.\ XXIX p.~795 (1915) and XXXI p.~156 (1916)), but by him applied
  only to the simple Keplerian motion described by the electron in the hydrogen atom if the relativity
  modifications are neglected. Due to the singular position of periodic systems in the quantum theory
  of systems of several degrees of freedom this application, however, involves, as it will appear from the
  following discussion, an ambiguity which deprives the result of an immediate physical interpretation.
  Conditions analogous to~\Eq{(16)} have also been established by \Name{Planck} in his interesting theory of the
  ``physical structure of the phase space'' of systems of several degrees of freedom (Verh.\ d.\ D.\ Phys.\
  Ges.\ XVII p.~407 and p.~438 (1915), Ann.\ d.\ Phys.\ L p.~385, (1916)\Add{)}. This theory, which has no direct
  relation to the problem of line-spectra discussed in the present paper, rests upon a profound analysis
  of the geometrical problem of dividing the multiple-dimensional phase space corresponding to a system
  of several degrees of freedom into ``cells'' in a way analogous to the division of the phase surface of
  a system of one degree of freedom by the curves given by~\Eq{(10)}.}
To this theory we shall come back in Part~II\@.

As pointed out by \Name{Epstein}\footnote
  {\Name{P.~Epstein}, loc.\ cit.}
and \Name{Schwarzschild}\footnote
  {\Name{K.~Schwarzschild}, loc.\ cit.}
the central systems considered
by \Name{Sommerfeld} form a special case of a more general class of systems for
which conditions of the same type as~\Eq{(15)} may be applied. These are the so\Typo{}{ }called
\Emph{conditionally periodic systems}, to which we are led if the equations of motion
are discussed by means of the \Name{Hamilton}-\Name{Jacobi} partial differential equation.\footnote
  {See f.\ inst.\ \Name{C.~V.~L. Charlier}, Die Mechanik des Himmels, Bd.~I, Abt.~2.}
In
the expression for the total energy~$E$ as a function of the~$q$'s and the~$p$'s, let the
latter quantities be replaced by the partial differential coefficients of some function~$S$
\PageSep{19}
with respect to the corresponding $q$'s respectively, and consider the partial
differential equation:\Pagelabel{19}
\[
E \left(q_{1}, \MyDots q_{s}, \frac{\dd S}{\dd q_{1}}, \MyDots \frac{\dd S}{\dd q_{s}}\right) = \alpha_{1},
\Tag{(17)}
\]
obtained by putting this expression equal to an arbitrary constant~$\alpha_{1}$. If then
\[
S = F (q_{1}, \MyDots q_{s}, \alpha_{1}, \MyDots \alpha_{s}) + C,
\]
where $\alpha_{2}$,~\MyDots $\alpha_{s}$, and~$C$ are arbitrary constants like~$\alpha_{1}$, is a total integral of~\Eq{(17)}, we
get, as shown by \Name{Hamilton} and \Name{Jacobi}, the general solution of the equations of
motion~\Eq{(4)} by putting
\[
\frac{\dd S}{\dd \alpha_{1}} = t + \beta_{1},\quad
\frac{\dd S}{\dd \alpha_{k}} = \beta_{k},\qquad (k=2, \MyDots s)
\Tag{(18)}
\]
and
\[
\frac{\dd S}{\dd q_{k}} = p_{k},\qquad (k = 1, \MyDots s)
\Tag{(19)}
\]
where $t$~is the time and $\beta_{1}$,~\MyDots $\beta_{s}$ a new set of arbitrary constants. By means of~\Eq{(18)}
the $q$'s are given as functions of the time~$t$ and the $2s$~constants $\alpha_{1}$,~\MyDots $\alpha_{s}$,
$\beta_{1}$,~\MyDots $\beta_{s}$ which may be determined for instance from the values of the $q$'s and
$\ddot{q}$'s at a given moment.

Now the class of systems, referred to, is that for which, for a suitable choice
of orthogonal coordinates, it is possible to find a total integral of~\Eq{(17)} of the form
\[
S = \sum_{1}^{s} S_{k}(q_{k}, \alpha_{1}, \MyDots \alpha_{s}),
\Tag{(20)}
\]
where $S_{k}$~is a function of the $s$~constants $\alpha_{1}$,~\MyDots $\alpha_{s}$ and of $q_{k}$~only. In this case, in
which the equation~\Eq{(17)} allows of what is called ``separation of variables'', we get
from~\Eq{(19)} that every~$p$ is a function of the~$\alpha$'s and of the corresponding $q$ only. If
during the motion the coordinates do not become infinite in the course of time
or converge to fixed limits, every~$q$ will, just as for systems of one degree of freedom,
oscillate between two fixed values, different for the different~$q$'s and depending on
the~$\alpha$'s. Like in the case of a system of one degree of freedom, $p_{k}$~will become
zero and change its sign whenever $q_{k}$~passes through one of these limits. Apart
from special cases, the system will during the motion never pass twice through
a configuration corresponding to the same set of values for the $q$'s and~$p$'s, but it
will in the course of time pass within any given, however small, distance from any
configuration corresponding to a given set of values $q_{1}$,~\MyDots $q_{s}$, representing a point
within a certain closed $s$-dimensional extension limited by $s$~pairs of $(s - 1)$-dimensional
surfaces corresponding to constant values of the~$q$'s equal to the above mentioned
limits of oscillation. A motion of this kind is called ``conditionally periodic''. It
will be seen that the character of the motion will depend only on the~$\alpha$'s and not
on the~$\beta$'s, which latter constants serve only to fix the exact configuration of the
\PageSep{20}
system at a given moment, when the~$\alpha$'s are known. For special systems it may
occur that the orbit will not cover the above mentioned $s$-dimensional extension
everywhere dense, but will, for all values of the~$\alpha$'s, be confined to an extension
of less dimensions. Such a case we will refer to in the following as a case of
``degeneration''.

Since for a conditionally periodic system which allows of separation in the
variables $q_{1}$,~\MyDots $q_{s}$ the $p$'s are functions of the corresponding $q$'s only, we may,
just as in the case of independent degrees of freedom or in the case of quasi-periodic
motion in a central field, form a set of expressions of the type
\[
I_{k} = \int p_{k}(q_{k}, \alpha_{1}, \MyDots \alpha_{s})\, dq_{k},\qquad
(k = 1, \MyDots s)
\Tag{(21)}
\]
where the integration is taken over a complete oscillation of~$q_{k}$. As, in general, the
orbit will cover everywhere dense an $s$-dimensional extension limited in the characteristic
way mentioned above, it follows that, except in cases of degeneration, a
separation of variables will not be possible for two different sets of coordinates
$q_{1}$,~\MyDots $q_{s}$ and $q'_{1}$,~\MyDots $q'_{s}$, unless $q_{1} = f_{1}(q'_{1})$,~\MyDots $q_{s} = f_{s}(q'_{s})$, and since a change of
coordinates of this type will not affect the values of the expressions~\Eq{(21)}, it will be
seen that the values of the~$I$'s are completely determined for a given motion of the
system. By putting
\[
I_{k} = n_{k} h,\qquad
(k = l, \MyDots s)
\Tag{(22)}
\]
where $n_{1}$,~\MyDots $n_{s}$ are positive entire numbers, we obtain therefore a \Emph{set of conditions
which form a natural generalisation of condition~\Eq{(10)}} holding
for a system of one degree of freedom.

Since the~$I$'s, as given by~\Eq{(21)}, depend on the constants $\alpha_{1}$,~\MyDots $\alpha_{s}$ only and
not on the~$\beta$'s, the $\alpha$'s may, in general, inversely be determined from the values
of the~$I$'s. The character of the motion will therefore, in general, be completely
determined by the conditions~\Eq{(22)}, and especially the value for the total energy, which
according to~\Eq{(17)} is equal to~$\alpha_{1}$, will be fixed by them. In the cases of degeneration
referred to above, however, the conditions~\Eq{(22)} involve an ambiguity, since in
general for such systems there will exist an infinite number of different sets of
coordinates which allow of a separation of variables, and which will lead to different
motions in the stationary states, when these conditions are applied. As we shall
see below, this ambiguity will not influence the fixation of the total energy in the
stationary states, which is the essential factor in the theory of spectra based on~\Eq{(1)}
and in the applications of the quantum theory to statistical problems.

A well known characteristic example of a conditionally periodic system is
afforded by a particle moving under the influence of the attractions from two
fixed centres varying as the inverse squares of the distances apart, if the relativity
modifications are neglected. As shown by \Name{Jacobi} this problem can be solved by a
separation of variables if so called elliptical coordinates\Pagelabel{21} are used, \ie\ if for $q_{l}$~and~$q_{2}$
we take two parameters characterising respectively an ellipsoid and a hyperboloid
of revolution with the centres as foci and passing through the instantaneous position
\PageSep{21}
of the moving particle, and for~$q_{3}$ we take the angle between the plane through
the particle and the centres and a fixed plane through the latter points, or, in
closer conformity with the above general description, some continuous periodic function
of this angle with period~$2\pi$. A limiting case of this problem is afforded by
an electron rotating round a positive nucleus and subject to the effect of an additional
homogeneous electric field, because this field may be considered as arising
from a second nucleus at infinite distance apart from the first. The motion in this
case will therefore be conditionally periodic and allow a separation of variables in
parabolic coordinates, if the nucleus is taken as focus for both sets of paraboloids
of revolution, and their axes are taken parallel to the direction of the electric force.
By applying the conditions~\Eq{(22)} to this motion \Name{Epstein} and \Name{Schwarzschild} have,
as mentioned in the introduction, independent of each other, obtained an explanation
of the effect of an external electric field on the lines of the hydrogen spectrum,
which was found to be in convincing agreement with \Name{Stark}'s measurements. To
the results of these calculations we shall return in Part~II\@.

In the above way of representing the general theory we have followed the
same procedure as used by \Name{Epstein}. By introducing the so called ``angle-variables''
well known from the astronomical theory of perturbations, \Name{Schwarzschild} has given
the theory a very elegant form in which the analogy with systems of one degree
of freedom presents itself in a somewhat different manner. The connection between
this treatment and that given above has been discussed in detail by \Name{Epstein}.\footnote
  {\Name{P.~Epstein}, Ann.\ d.\ Phys.\ LI, p.~168 (1916). See also Note on \PageRef{29} of the present paper.}

As mentioned above the conditions~\Eq{(22)}, first established from analogy with
systems of one degree of freedom, have subsequently been proved generally to be
\Emph{mechanically invariant for any slow transformation for which the
system remains conditionally periodic}. The proof of this in variance has
been given quite recently by \Name{Burgers}\footnote
  {\Name{J.~M. Burgers}, loc.\ cit.\ Versl.\ Akad.\ Amsterdam, XXV, p.~1055 (1917).}
by means of an interesting application of the
theory of contact-transformations based on \Name{Schwarzschild}'s introduction of angle
variables. We shall not enter here on these calculations but shall only consider some
points in connection with the problem of the mechanical transformability of the
stationary states which are of importance for the logical consistency of the general
theory and for the later applications. In~\SecRef{2} we saw that in the proof of the mechanical
invariance of relation~\Eq{(10)} for a periodic system of one degree of freedom, it was
essential that the comparative variation of the external conditions during the time of one
period could be made small. This may be regarded as an immediate consequence of the
nature of the fixation of the stationary states in the quantum theory. In fact the answer
to the question, whether a given state of a system is stationary, will not depend
only on the motion of the particles at a given moment or on the field of force in
the immediate neighbourhood of their instantaneous positions, but cannot be given
before the particles have passed through a complete cycle of states, and so to speak
\PageSep{22}
have got to know the entire field of force of influence on the motion. If thus, in
the case of a periodic system of one degree of freedom, the field of force is varied
by a given amount, and if its comparative variation within the time of a single
period was not small, the particle would obviously have no means to get to know
the nature of the variation of the field and to adjust its stationary motion to it,
before the new field was already established. For exactly the same reasons it is a
necessary condition for the mechanical invariance of the stationary states of a
conditionally periodic system, that the alteration of the external conditions during
an interval in which the system has passed approximately through all possible
configurations within the above mentioned $s$-dimensional extension in the coordinate-space
can be made as small as we like. This condition forms therefore also
an essential point in \Name{Burgers}' proof of the invariance of the conditions~\Eq{(22)} for
mechanical transformations. Due to this we meet with a characteristic difficulty
when during the transformation of the system we pass one of the cases of
degeneration mentioned above, where, for every set of values for the~$\alpha$'s, the orbit
will not cover the $s$-dimensional extension everywhere dense, but will be confined
to an extension of less dimensions. It is clear that, when by a slow transformation
of a conditionally periodic system we approach a degenerate system of this
kind, the time-interval which the orbit takes to pass close to any possible configuration
will tend to be very long and will become infinite when the degenerate
system is reached. As a consequence of this \Emph{the conditions~\Eq{(22)} will generally
not remain mechanically invariant when we pass a degenerate
system}, what has intimate connection with the above mentioned ambiguity in the
determination of the stationary states of such systems by means of~\Eq{(22)}.

A typical case of a degenerate system, which may serve as an illustration of
this point, is formed by a system of several degrees of freedom for which every
motion is \Emph{simply periodic}, independent of the initial conditions. In this case, which
is of great importance in the physical applications, we have from \Eq{(5)}~and \Eq{(21)}, for
any set of coordinates in which a separation of variables is possible,
\[
I = \int_{0}^{\sigma} (p_{1} \dot{q}_{1} + \dots + p_{s} \dot{q}_{s})\, dt
  = \kappa_{1} I_{1} + \dots + \kappa_{s} I_{s},
\Tag{(23)}
\]
where the integration is extended over one period of the motion, and where
$\kappa_{1}$,~\MyDots $\kappa_{s}$ are a set of positive entire numbers without a common divisor. Now we
shall expect that every motion, for which it is possible to find a set of coordinates in
which it satisfies~\Eq{(22)}, will be stationary. For any such motion we get from~\Eq{(23)}\Pagelabel{22}
\[
I = (\kappa_{1} n_{1} + \dots + \kappa_{s} n_{s}) h = nh,
\Tag{(24)}
\]
where $n$~is a whole number which may take all positive values if, as in the applications
mentioned below, at least one of the~$\kappa$'s is equal to one. Inversely, if
the system under consideration allows of separation of variables in an infinite continuous
multitude of sets of coordinates, we must conclude that generally every
\PageSep{23}
motion which satisfies~\Eq{(24)} will be stationary, because in general it will be possible
for any such motion to find a set of coordinates in which it satisfies also~\Eq{(22)}.
It will thus be seen that, for a periodic system of several degrees of freedom,
condition~\Eq{(24)} forms a simple generalisation of condition~\Eq{(10)}. From relation~\Eq{(8)},
which holds for two neighbouring motions of any periodic system, it follows
further that the energy of the system will be completely determined by the value
of~$I$, just as for systems of one degree of freedom.

Consider\Pagelabel{23} now a periodic system in some stationary state satisfying~\Eq{(24)}, and
let us assume that an external field is slowly established at a continuous rate and
that the motion at any moment during this process allows of a separation of variables
in a certain set of coordinates. If we would assume that the effect of the
field on the motion of the system at any moment could be calculated directly by
means of ordinary mechanics, we would find that the values of the~$I$'s with respect to
the latter coordinates would remain constant during the process, but this would
involve that the values of the~$n$'s in~\Eq{(22)} would in general not be entire numbers,
but would depend entirely on the accidental motion, satisfying~\Eq{(24)}, originally possessed
by the system. That mechanics, however, cannot generally be applied directly
to determine the motion of a periodic system under influence of an increasing external
field, is just what we should expect according to the singular position of
degenerate systems as regards mechanical transformations. In fact, in the presence
of a small external field, the motion of a periodic system will undergo slow variations
as regards the shape and position of the orbit, and if the perturbed motion is
conditionally periodic these variations will be of a periodic nature. Formally, we
may therefore compare a periodic system exposed to an external field with a simple
mechanical system of one degree of freedom in which the particle performs a slow
oscillating motion. Now the frequency of a slow variation of the orbit will be seen
to be proportional to the intensity of the external field, and it is therefore obviously
impossible to establish the external field at a rate so slow that the comparative
change of its intensity during a period of this variation is small. The process which
takes place during the increase of the field will thus be analogous to that which
takes place if an oscillating particle is subject to the effect of external forces which
change considerably during a period. Just as the latter process generally will give
rise to emission or absorption of radiation and cannot be described by means of
ordinary mechanics, we must expect that the motion of a periodic system of several
degrees of freedom under the establishment of the external field cannot be determined
by ordinary mechanics, but that the field will give rise to effects of the same kind
as those which occur during a transition between two stationary states accompanied
by emission or \Typo{absorbtion}{absorption} of radiation. Consequently we shall expect that, during
the establishment of the field, \Emph{the system will in general adjust itself
in some unmechanical way} until a stationary state is reached in which the
frequency (or frequencies) of the above mentioned slow variation of the orbit has a
relation to the additional energy of the system due to the presence of the external
\PageSep{24}
field, which is of the same kind as the relation, expressed by \Eq{(8)}~and~\Eq{(10)}, between
the energy and frequency of a periodic system of one degree of freedom. As it will
be shown in Part~II in connection with the physical applications, this condition
is just secured if the stationary states in the presence of the field are determined
by the conditions~\Eq{(22)}, and it will be seen that these considerations offer a means
of fixing the stationary states of a perturbed periodic system also in cases where
no separation of variables can be obtained.

In consequence of the singular position of the degenerate systems in the
general theory of stationary states of conditionally periodic systems, we obtain
a means of \Emph{connecting mechanically two different stationary states
of a given system} through a continuous series of stationary states without
passing through systems in which the forces are very small and the energies in all
the stationary states tend to coincide (comp.\ \PageRef{9}). In fact, if we consider a given
conditionally periodic system which can be transformed in a continuous way into
a system for which every orbit is periodic and for which every state satisfying~\Eq{(24)}
will also satisfy~\Eq{(22)} for a suitable choice of coordinates, it is clear in the first
place that it is possible to pass in a mechanical way through a continuous series
of stationary states from a state corresponding to a given set of values of the~$n$'s
in~\Eq{(22)} to any other such state for which $\kappa_{1} n_{1} + \dots + \kappa_{s} n_{s}$ possesses the same value. If,
moreover, there exists a second periodic system of the same character to which the
first periodic system can be transformed continuously, but for which the set of~$\kappa$'s
is different, it will be possible in general by a suitable cyclic transformation to
pass in a mechanical way between any two stationary states of the given conditionally
periodic system satisfying~\Eq{(22)}.

\begin{Remark}
To obtain an example of such a cyclic transformation let us take the system consisting
of an electron which moves round a fixed positive nucleus exerting an attraction varying as
the inverse square of the distance. If we neglect the small relativity corrections, every orbit
will be periodic independent of the initial conditions and the system will allow of separation
of variables in polar coordinates as well as in any set of elliptical coordinates, of the
kind mentioned on \PageRef{21}, if the nucleus is taken as one of the foci. It is simply seen that
any orbit which satisfies~\Eq{(24)} for a value of $n &gt; 1$, will also satisfy~\Eq{(22)} for a suitable choice
of elliptical coordinates. By imagining another nucleus of infinite small charge placed at the
other focus, the orbit may further be transformed into another which satisfies~\Eq{(24)} for the
same value of~$n$, but which may have any given value for the eccentricity. Consider now
a state of the system satisfying~\Eq{(21)}, and let us assume that by the above means the orbit
is originally so adjusted that in plane polar coordinates it will correspond to $n_{1} = m$ and
$n_{2} = n - m$ in~\Eq{(16)}. Let then the system undergo a slow continuous transformation during which
the field of force acting on the electron remains central, but by which the law of attraction
is slowly varied until the force is directly proportional to the distance apart. In the final
state, as well as in the original state, the orbit of the electron will be closed, but during the
transformation the orbit will not be closed, and the ratio between the mean period of
revolution and the period of the radial motion, which in the original motion was equal to
one, will during the transformation increase continuously until in the final state it is equal to
two. This means that, using polar coordinates, the values of $\kappa_{1}$~and~$\kappa_{2}$ in~\Eq{(22)} which for the
first state are equal to $\kappa_{1} = \kappa_{2} = 1$, will be for the second state $\kappa_{1} = 2$ and $\kappa_{2} = 1$. Since
during the transformation $n_{1}$~and~$n_{2}$ will keep their values, we get therefore in the final state
$I = h\bigl(2m + (n - m)\bigr) = h(n + m)$. Now in the latter state, the system allows a separation of
\PageSep{25}
variables not only in polar coordinates but also in any system of rectangular Cartesian
coordinates, and by suitable choice of the direction of the axes, we can obtain that any orbit,
satisfying~\Eq{(24)} for a value of $n &gt; l$, will also satisfy~\Eq{(22)}. By an infinite small change of the
force components in the directions of the axes, in such a way that the motions in these
directions remain independent of each other but possess slightly different periods, it will
further be possible to transform the elliptical orbit mechanically into one corresponding to
any given ratio between the axes. Let us now assume that in this way the orbit of the
electron is transformed into a circular one, so that, returning to plane polar coordinates, we
have $n_{1} = 0$ and $n_{2} = n + m$, and let then by a slow transformation the law of attraction
be varied until again it is that of the inverse square. It will be seen that when this state
is reached the motion will again satisfy~\Eq{(24)}, but this time we will have $I = h (n + m)$ instead
of $I = nh$ as in the original state. By repeating a cyclic process of this kind we may pass
from any stationary state of the system in question which satisfies~\Eq{(24)} for a value of $n &gt; 1$
to any other such state without leaving at any moment the region of stationary states.
\end{Remark}

The theory of the mechanical transformability of the stationary states gives us a
means to discuss the question of the \Emph{a-priori probability} of the different states
of a conditionally periodic system, characterised by different sets of values for the~$n$'s
in~\Eq{(22)}. In fact from the considerations, mentioned in~\SecRef{1}, it follows that, if
the a-priori probability of the stationary states of a given system is known, it
is possible at once to deduce the probabilities for the stationary states of any
other system to which the first system can be transformed continuously without
passing through a system of degeneration. Now from the analogy with systems of
one degree of freedom it seems necessary to assume that, for a system of several
degrees of freedom for which the motions corresponding to the different coordinates
are dynamically independent of each\Typo{}{ }other, the a-priori probability is the same for all
the states corresponding to different sets of~$n$'s in~\Eq{(15)}. According to the above
we shall therefore assume that the a-priori probability is the same for all states,
given by~\Eq{(22)}, of any system which can be formed in a continuous way from a system
of this kind without passing through systems of degeneration. It will be observed
that on this assumption we obtain exactly the same relation to the ordinary theory
of statistical mechanics in the limit of large~$n$'s as obtained in the case of systems
of one degree of freedom. Thus, for a conditionally periodic system, the volume
given by~\Eq{(11)} of the element of phase-space, including all points $q_{1}$,~\MyDots $q_{s}$, $p_{1}$,~\MyDots $p_{s}$
which represent states for which the value of~$I_{k}$ given by~\Eq{(21)} lies between $I_{k}$
and~$I_{k} + \delta I_{k}$, is seen at once to be equal to\footnote
  {Comp.\ \Name{A.~Sommerfeld}, Ber.\ Akad.\ München, 1917, p.~83.}
\[
\delta W = \delta I_{1}\, \delta I_{2} \dots \delta I_{s},
\Tag{(25)}
\]
if the coordinates are so chosen that the motion corresponding to every degree of
freedom is of oscillating type. The volume of the phase-space limited by $s$~pairs
of surfaces, corresponding to successive values for the~$n$'s in the conditions~\Eq{(22)},
will therefore be equal to~$h^{s}$ and consequently be the same for every combination
of the~$n$'s. In the limit where the~$n$'s are large numbers and the stationary states
%[** TN: Printer's mark]
%D. K. D. Vidensk. Selsk. Skr., naturvidensk. og mathem. Afd 8. Række, IV. 1.
\PageSep{26}
corresponding to successive values for the~$n$'s differ only very little from each
other, we thus obtain the same result on the assumption of equal a-priori probability
of all the stationary states, corresponding to different sets of values of
$n_{1}$,~\MyDots $n_{s}$ in~\Eq{(22)}, as would be obtained by application of ordinary statistical
mechanics.

The\Pagelabel{26} fact that the last considerations hold for every non-degenerate conditionally
periodic system suggests the assumption that in general \Emph{the a-priori
probability will be the same for all the states determined by~\Eq{(22)}},
even if it should not be possible to transform the given system into a system of
independent degrees of freedom without passing through degenerate systems. This
assumption will be shown to be supported by the consideration of the intensities
of the different components of the \Name{Stark}-effect of the hydrogen lines, mentioned
in the next Part. When we consider a degenerate system, however, we cannot
assume that the different stationary states are a-priori equally probable. In such a
case the stationary states will be characterised by a number of conditions less
than the number of degrees of freedom, and the probability of a given state must
be determined from the number of different stationary states of some non-degenerate
system which will coincide in the given state, if the latter system is continuously
transformed into the degenerate system under consideration.

In order to illustrate this, let us take the simple case of a degenerate system
formed by an electrified particle moving in a plane orbit in a central field, the
stationary states of which are given by the two conditions~\Eq{(16)}. In this case the
plane of the orbit is undetermined, and it follows already from a comparison with
ordinary statistical mechanics, that the a-priori probability of the states \Chg{characterized}{characterised}
by different combinations of $n_{1}$ and $n_{2}$ in~\Eq{(16)} cannot be the same. Thus the
volume of the phase-space, corresponding to states for which $I_{1}$~lies between $I_{1}$~and
$I_{1} + \delta I_{1}$ and for which $I_{2}$~lies between $I_{2}$~and $I_{2} + \delta I_{2}$, is found by a
simple calculation\footnote
  {See \Name{A.~Sommerfeld}, loc.\ cit.}
to be equal to $\delta W = 2I_{1}\, \delta I_{1}\, \delta I_{2}$, if the motion is described
by ordinary polar coordinates. For large values of $n_{1}$~and~$n_{2}$, we must therefore
expect that the a-priori probability of a stationary state corresponding to a
given combination $(n_{1}, n_{2})$ is proportional to~$n_{2}$. The question of the a-priori probability
of states corresponding to small values of the~$n$'s has been discussed by
\Name{Sommerfeld} in connection with the problem of the intensities of the different components
in the fine structure of the hydrogen lines (see Part~II). From considerations
about the volume of the extensions in the phase-space, which might be
considered as associated with the states characterised by different combinations
$(n_{1}, n_{2})$, \Name{Sommerfeld} proposes several different expressions for the a-priori probability
of such states. Due to the necessary arbitrariness involved in the
choice of these extensions, however, we cannot in this way obtain a rational
determination of the a-priori probability of states corresponding to small values of
\PageSep{27}
$n_{1}$~and~$n_{2}$. On the other hand, this probability may be deduced by regarding the
motion of the system under consideration as the degeneration of a motion characterised
by three numbers $n_{1}$,~$n_{2}$ and~$n_{3}$, as in the general applications of the conditions~\Eq{(22)}
to a system of three degrees of freedom. Such a motion may be obtained
for instance by imagining the system placed in a small homogeneous magnetic
field. In certain respects this case falls outside the general theory of conditionally
periodic systems discussed in this section, but, as we shall see in Part~II, it can
be simply shown that the presence of the magnetic field imposes the further condition
on the motion in the stationary states that the angular momentum round the
axis of the field is equal to~$n'\, \dfrac{h}{2\pi}$, where $n'$~is a positive entire number equal to or
less than~$n_{2}$, and which for the system considered in the spectral problems must
be assumed to be different from zero. When regard is taken to the two opposite
directions in which the particle may rotate round the axis of the field, we see
therefore that for this system a state corresponding to a given combination of
$n_{1}$~and~$n_{2}$ in the presence of the field can be established in $2n_{2}$~different ways.
The a-priori probability of the different states of the system may consequently for
all combinations of $n_{1}$~and~$n_{2}$ be assumed to be proportional to~$n_{2}$.

The\Pagelabel{27} assumption just mentioned that the angular momentum round the axis
of the field cannot be equal to zero is deduced from considerations of systems for
which the motion corresponding to special combinations of the~$n$'s in~\Eq{(22)} would
become physically impossible due to some singularity in its character. In such cases
we must assume that no stationary states exist corresponding to the combinations
$(n_{1}, n_{2}, \MyDots n_{s})$ under consideration, and on the above principle of the invariance
of the a-priori probability for continuous transformations we shall accordingly
expect that the a-priori probability of any other state, which can be transformed
continuously into one of these states without passing through cases of degeneration,
will also be equal to zero.

Let us now proceed to consider \Emph{the spectrum of a conditionally periodic
system}, calculated from the values of the energy in the stationary states by means of
relation~\Eq{(1)}. If $E(n_{1}, \MyDots n_{s})$ is the total energy of a stationary state determined by~\Eq{(22)}
and if $\nu$~is the frequency of the line corresponding to the transition between
two stationary states characterised by $n_{k} = n'_{k}$ and $n_{k} = n''_{k}$ respectively, we have
\[
\nu = \frac{1}{h}\bigl[E(n'_{1}, \MyDots n'_{s}) - E(n''_{1}, \MyDots n''_{s})\bigr].
\Tag{(26)}
\]
In general, this spectrum will be entirely different from the spectrum to be expected
on the ordinary theory of electrodynamics from the motion of the system.
Just as for a system of one degree of freedom we shall see, however, that in
the limit where the motions in neighbouring stationary states differ very little from
each other, there exists a close relation between the spectrum calculated on the
quantum theory and that to be expected on ordinary electrodynamics. As in~\SecRef{2}
\PageSep{28}
we shall further see, that this connection leads to certain general considerations
about the probability of transition between any two stationary states and about
the nature of the accompanying radiation, which are found to be supported by
observations. In order to discuss this question we shall first deduce a general expression
for the energy difference between two neighbouring states of a conditionally
periodic system, which can be simply obtained by a calculation analogous to that
used in~\SecRef{2} in the deduction of the relation~\Eq{(8)}.

Consider some motion of a conditionally periodic system which allows of
separation of variables in a certain set of coordinates $q_{1}$,~\MyDots $q_{s}$, and let us assume
that at the time $t = \theta$ the configuration of the system will to a close approximation
be the same as at the time $t = 0$. By taking $\theta$~large enough we can make
this approximation as close as we like. If next we consider some conditionally
periodic motion, obtained by a small variation of the first motion, and which allows
of separation of variables in a set of coordinates $q'_{1}$,~\MyDots $q'_{s}$ which may differ
slightly from the set $q_{1}$,~\MyDots $q_{s}$, we get by means of \Name{Hamilton}'s equations~\Eq{(4)}, using
the coordinates $q'_{1}$,~\MyDots $q'_{s}$,
\[
\BreakMath{%
\int_{0}^{\theta} \delta E\, dt
  = \int_{0}^{\theta} \sum_{1}^{s} \left(
    \frac{\dd E}{\dd p'_{k}}\, \delta p'_{k} + \frac{\dd E}{\dd q'_{k}}\, \delta q'_{k}
  \right) dt
  = \int_{0}^{\theta} \sum_{1}^{s} (\dot{q}'_{k}\, \delta p'_{k} - \dot{p}'_{k}\, \delta q'_{k})\, dt.
}{%
\begin{aligned}
\int_{0}^{\theta} \delta E\, dt
  &amp;= \int_{0}^{\theta} \sum_{1}^{s} \left(
    \frac{\dd E}{\dd p'_{k}}\, \delta p'_{k} + \frac{\dd E}{\dd q'_{k}}\, \delta q'_{k}
  \right) dt \\
  &amp;= \int_{0}^{\theta} \sum_{1}^{s} (\dot{q}'_{k}\, \delta p'_{k} - \dot{p}'_{k}\, \delta q'_{k})\, dt.
\end{aligned}
}
\]
By partial integration of the second term in the bracket this gives:
\[
\int_{0}^{\theta} \delta E\, dt
  = \int_{0}^{\theta} \sum_{1}^{s} \delta(p'_{k}\dot{q}'_{k})\, dt
  - \left\lvert\sum_{1}^{s} p'_{k}\, \delta q'_{k}\right\rvert_{t=0}^{t=\theta}.
\Tag{(27)}
\]
Now we have for the unvaried motion
\[
\int_{0}^{\theta} \sum_{1}^{s} p'_{k} \dot{q}'_{k}\, dt
  = \int_{0}^{\theta} \sum_{1}^{s} p_{k} \dot{q}_{k}\, dt
  = \sum_{1}^{s} N_{k} I_{k},
\]
where $I_{k}$~is defined by~\Eq{(21)} and where $N_{k}$~is the number of oscillations performed
by~$q_{k}$ in the time interval~$\theta$. For the varied motion we have on the other hand:
\[
\int_{0}^{\theta} \sum_{1}^{s} p'_{k} \dot{q}'_{k}\, dt
  = \int_{t=0}^{t=\theta} \sum_{1}^{s} p'_{k} dq'_{k}
  = \sum_{1}^{s} N_{k} I'_{k}
  + \left\lvert \sum_{1}^{s} p'_{k}\, \delta q'_{k}\right\rvert_{t=0}^{t=\theta}\!\!,
\]
where the~$I$'s correspond to the conditionally periodic motion in the coordinates
$q'_{1}$,~\MyDots $q'_{s}$, and the~$\delta q$'s which enter in the last term are the same as those in~\Eq{(27)}.
Writing $I'_{k} - I_{k} = \delta I_{k}$, we get therefore from the latter equation
\[
\int_{0}^{\theta} \delta E\, dt
  = \sum_{1}^{s} N_{k}\, \delta I_{k}.
\Tag{(28)}
\]
\PageSep{29}

In the special case where the varied motion is an undisturbed motion belonging
to the same system as the unvaried motion we get, since $\delta E$~will be constant,
\[
\delta E = \sum_{1}^{s} \omega_{k}\, \delta I_{k},
\Tag{(29)}
\]
where $\omega_{k} = \dfrac{N_{k}}{\theta}$ is the mean frequency of oscillation of~$q_{k}$ between its limits, taken
over a long time interval of the same order of magnitude as~$\theta$. This equation
forms a simple generalisation of~\Eq{(8)}, and in the general case in which a separation
of variables will be possible only for one system of coordinates leading to a complete
definition of the~$I$'s it might have been deduced directly from the analytical
theory of the periodicity properties of the motion of a conditionally periodic
system, based on the introduction of angle-variables.\Pagelabel{29}\footnote
  {See \Name{Charlier}, Die Mechanik des Himmels, Bd.~I Abt.~2, and especially \Name{P.~Epstein}, Ann.\ d.\
  Phys.\ LI p.~178 (1916). By means of the well known theorem of \Name{Jacobi} about the change of variables in
  the canonical equations of \Name{Hamilton}, the connection between the notion of angle-variables and the
  quantities~$I$, discussed by \Name{Epstein} in the latter paper, may be briefly exposed in the following elegant
  manner which has been kindly pointed out to me by Mr.\ \Name{H.~A. Kramers}. Consider the function
  $S(q_{1}, \MyDots q_{s}, I_{1}, \MyDots I_{s})$ obtained from~\Eq{(20)} by introducing for the~$\alpha$'s their expressions in terms of the~$I$'s
  given by the equations~\Eq{(21)}. This function will be a many valued function of the~$q$'s which increases
  by~$I_{k}$ if $q_{k}$~describes one oscillation between its limits and comes back to its original value
  while the other~$q$'s remain constant. If we therefore introduce a new set of variables $w_{1}, \MyDots w_{s}$
  defined by
  \[
  w_{k} = \frac{\dd S}{\dd I_{k}},\qquad
  (k = 1, \MyDots s)
  \Tag[I.]{(1^{*})}
  \]
  it will be seen that $w_{k}$~increases by one unit while the other~$w$'s will come back to their original
  values if $q_{k}$~describes one oscillation between its limits and the other~$q$'s remain constant. Inversely
  it will therefore be seen that the~$q$'s, and also the $p$'s which were given by
  \[
  p_{k} = \frac{\dd S}{\dd q_{k}},\qquad
  (k = i, \MyDots s)\Chg{,}{}
  \Tag[I.]{(2^{*})}
  \]
  when considered as functions of the $I$'s and~$w$'s will be periodic functions of every of the~$w$'s with
  period~$1$. According to \Typo{Fourier's}{\Name{Fourier}'s} theorem any of the~$q$'s may therefore be represented by an $s$-double
  trigonometric series of the form
  \[
  q = \sum A_{\tau_{1}, \MyDots \tau_{s}} \cos 2\pi(\tau_{1} w_{1} + \dots \Add{+} \tau_{s} w_{s} + \alpha_{\tau_{1}, \MyDots \tau_{s}}),
  \Tag[I.]{(3^{*})}
  \]
  where the $A$'s and~$\alpha$'s are constants depending on the~$I$'s and where the summation is to be extended
  over all entire values of $\tau_{1}$,~\MyDots $\tau_{s}$. On account of this property of the~$w$'s, the quantities $2\pi w_{1}$,~\MyDots
  $2\pi w_{s}$ are denoted as ``angle variables''. Now from \Eq[I.]{(1^{*})}~and \Eq[I.]{(2^{*})} it follows according to the above mentioned
  theorem of \Name{Jacobi} (see for instance \Name{Jacobi}, Vorlesungen über Dynamik §\;37) that the variations
  with the time of the $I$'s and~$w$'s will be given by
  \[
  \frac{dI_{k}}{dt} = -\frac{\dd E}{\dd w_{k}},\quad
  \frac{dw_{k}}{dt} =  \frac{\dd E}{\dd I_{k}},\qquad
  (k = 1, \MyDots s)
  \Tag[I.]{(4^{*})}
  \]
  where the energy~$E$ is considered as a function of the $I$'s and~$w$'s. Since~$E$, however, is determined
  by the $I$'s~only we get from~\Eq[I.]{(4^{*})}, besides the evident result that the~$I$'s are constant during the motion,
  that the~$w$'s will vary linearly with the time and can be represented by
  \[
  w_{k} = \omega_{k} t + \delta_{k},\quad
  \omega_{k} = \frac{\dd E}{\dd I_{k}},\qquad
  (k = 1, \MyDots s)
  \Tag[I.]{(5^{*})}
  \]
  where $\delta_{k}$~is a constant, and where $\omega_{k}$~is easily seen to be equal to the mean frequency of oscillation
  of~$q_{k}$. From~\Eq[I.]{(5^{*})} equation~\Eq{(28)} follows at once, and it will further be seen that by introducing~\Eq[I.]{(5^{*})} in~\Eq[I.]{(3^{*})}
  we get the result that every of the~$q$'s, and consequently also any one-valued function of the~$q$'s,
  can be represented by an expression of the type~\Eq{(31)}.

  In this connection it may be mentioned that the method of \Name{Schwarzschild} of fixing the stationary
  states of a conditionally periodic system, mentioned on \PageRef{21}, consists in seeking for a given system
  a set of canonically conjugated variables $Q_{1}$,~\MyDots $Q_{s}$, $P_{1}$,~\MyDots $P_{s}$ in such a way that the positional
  coordinates of the system $q_{1}$,~\MyDots $q_{s}$, and their conjugated momenta $p_{1}$,~\MyDots $p_{s}$, when considered as functions
  of the $Q$'s~and~$P$'s, are periodic in every of the~$Q$'s with period~$2\pi$, while the energy of the
  system depends only on the~$P$'s. In analogy with the condition which fixes the angular momentum in
  \Name{Sommerfeld}'s theory of central systems \Name{Schwarzschild} next puts every of the~$P$'s equal to an entire
  \Chg{multiplum}{multiple} of~$\dfrac{h}{2\pi}$. In contrast to the theory of stationary states of conditionally periodic systems based
  on the possibility of separation of variables and the fixation of the~$I$'s by~\Eq{(22)}, this method does not
  lead to an absolute fixation of the stationary states, because, as pointed out by \Name{Schwarzschild} himself,
  the above definition of the~$P$'s leaves an arbitrary constant undetermined in every of these quantities.
  In many cases, however, these constants may be simply determined from considerations of mechanical
  transformability of the stationary states, and as pointed out by \Name{Burgers} (loc.\ cit.\ Versl.\ Akad.\ Amsterdam
  XXV p.~1055 (1917)\Typo{}{).} \Name{Schwarzschild}'s method possesses on the other hand the essential advantage of
  being applicable to certain classes of systems in which the displacements of the particles may be
  represented by trigonometric series of the type~\Eq{(31)}, but for which the equations of motion cannot be
  solved by separation of variables in any fixed set of coordinates. An interesting application of this to
  the spectrum of rotating molecules, given by \Name{Burgers}, will be mentioned in Part~IV\@.}
From~\Eq{(29)} it follows moreover
\PageSep{30}
that, if the system allows of a separation of variables in an infinite continuous
multitude of sets of coordinates, the total energy will be the same for all motions
corresponding to the same values of the~$I$'s, independent of the special set of
coordinates used to calculate these quantities. As mentioned above and as we have
already shown in the case of purely periodic systems by means of~\Eq{(8)}, the total
energy is therefore also in cases of degeneration completely determined by the
conditions~\Eq{(22)}.

Consider now a transition between two stationary states determined by~\Eq{(22)}
by putting $n_{k} = n'_{k}$ and $n_{k} = n''_{k}$ respectively, and let us assume that $n'_{1}$,~\MyDots $n'_{s}$,
$n''_{1}$,~\MyDots $n''_{s}$ are large numbers, and that the differences $n'_{k} - n''_{k}$ are small compared
with these numbers. Since the motions of the system in these states will differ
relatively very little from each other we may calculate the difference of the energy
by means of~\Eq{(29)}, and we get therefore, by means of~\Eq{(1)}, for the frequency of the
radiation corresponding to the transition between the two states
\[
\nu = \frac{1}{h} (E' - E'')
  = \frac{1}{h} \sum_{1}^{s} \omega_{k} (I'_{k} - I''_{k})
  = \sum_{1}^{s} \omega_{k} (n'_{k} - n''_{k}),
\Tag{(30)}
\]
which is seen to be a direct generalisation of the expression~\Eq{(13)} in~\SecRef{2}.

Now, in complete analogy to what is the case for periodic systems of one degree
of freedom, it is proved in the analytical theory of the motion of conditionally
periodic systems mentioned above that for the latter systems the coordinates
$q_{1}$,~\MyDots $q_{s}$, and consequently also the displacements of the particles in any given
direction, may be expressed as a function of the time by an $s$-double infinite
\Name{Fourier} series of the form:
\PageSep{31}
\[
\xi = \sum C_{\tau_{1}, \MyDots \tau_{s}}
\cos 2\pi \bigl\{(\tau_{1} \omega_{1} + \dots \Add{+} \tau_{s} \omega_{s})t + c_{\tau_{1}, \MyDots \tau_{s}}\bigr\},
\Tag{(31)}
\]
where the summation is to be extended over all positive and negative entire values
of the~$\tau$'s, and where the~$\omega$'s are the above mentioned mean frequencies of oscillation
for the different~$q$'s. The constants $C_{\tau_{1}, \MyDots \tau_{s}}$ depend only on the~$\alpha$'s in the equations~\Eq{(18)}
or, what is the same, on the~$I$'s, while the constants $c_{\tau_{1}, \MyDots \tau_{s}}$ depend on
the~$\alpha$'s as well as on the~$\beta$'s. In general the quantities $\tau_{1} \omega_{1} + \dots \Add{+} \tau_{s} \omega_{s}$ will be
different for any two different sets of values for the~$\tau$'s, and in the course of time
the orbit will cover everywhere dense a certain $s$-dimensional extension. In a
case of degeneration, however, where the orbit will be confined to an extension of
less dimensions, there will exist for all values of the~$\alpha$'s one or more relations of the
type $m_{1} \omega_{1} + \dots \Add{+} m_{s} \omega_{s} = 0$ where the~$m$'s are entire numbers and by the introduction
of which the expression~\Eq{(31)} can be reduced to a \Name{Fourier} series which is
less than $s$-double infinite. Thus in the special case of a system of which every
orbit is periodic we have $\dfrac{\omega_{1}}{\kappa_{1}} = \dots = \dfrac{\omega_{s}}{\kappa_{s}} = \omega$, where the~$\kappa$'s are the numbers
which enter in equation~\Eq{(23)}, and the \Name{Fourier} series for the displacements in
the different directions will in this case consist only of terms of the simple form
$C_{\tau} \cos 2\pi \{\tau \omega t + c_{\tau}\}$, just as for a system of one degree of freedom.

On the ordinary theory of radiation, we should expect from~\Eq{(31)} that the
spectrum emitted by the system in a given state would consist of an $s$-double infinite
series of lines of frequencies equal to $\tau_{1} \omega_{1} + \dots + \tau_{s} \omega_{s}$. In general, this
spectrum would be completely different from that given by~\Eq{(26)}. This follows already
from the fact that the~$\omega$'s will depend on the values for the constants $\alpha_{1}$,~\MyDots $\alpha_{s}$
and will vary in a continuous way for the continuous multitude of mechanically
possible states corresponding to different sets of values for these constants. Thus
in general the~$\omega$'s will be quite different for two different stationary states
corresponding to different sets of~$n$'s in~\Eq{(22)}, and we cannot expect any close relation
between the spectrum calculated on the quantum theory and that to be expected
on the ordinary theory of mechanics and electrodynamics. In the limit,
however, where the~$n$'s in~\Eq{(22)} are large numbers, the ratio between the~$\omega$'s for
two stationary states, corresponding to $n_{k} = n'_{k}$ and $n_{k} = n''_{k}$ respectively, will tend
to unity if the differences $n'_{k} - n''_{k}$ are small compared with the~$n$'s, and as seen
from~\Eq{(30)} the spectrum calculated by \Eq{(1)}~and \Eq{(22)} will in this limit just tend to
coincide with that to be expected on the ordinary theory of radiation from the
motion of the system.

As far as the frequencies are concerned, we thus see that for conditionally
periodic systems there exists a connection between the quantum theory and the
ordinary theory of radiation of exactly the same character as that shown in~\SecRef{2}
to exist in the simple case of periodic systems of one degree of freedom. Now on
ordinary electrodynamics the coefficients $C_{\tau_{1}\Add{,} \dots, \tau_{s}}$ in the expression~\Eq{(31)} for the
displacements of the particles in the different directions would in the well known
way determine the intensity and polarisation of the emitted radiation of the
corresponding frequency $\tau_{1} \omega_{1} + \dots \Add{+} \tau_{s} \omega_{s}$. As for systems of one degree of freedom
\PageSep{32}
we must therefore conclude that, in the limit of large values for the~$n$'s, the probability
of spontaneous transition between two stationary states of a conditionally
periodic system, as well as the polarisation of the accompanying radiation, can be
determined directly from the values of the coefficient $C_{\tau_{1}, \MyDots \tau_{s}}$ in~\Eq{(31)} corresponding
to a set of~$\tau$'s given by $\tau_{k} = n'_{k} - n''_{k}$, if $n'_{1}$,~\MyDots $n'_{s}$ and $n''_{1}$,~\MyDots $n''_{s}$ are the numbers which
characterise the two stationary states.

Without\Pagelabel{32} a detailed theory of the mechanism of transition between the stationary
states we cannot, of course, in general obtain an exact determination of the \Emph{probability
of spontaneous transition} between two such states, unless the~$n$'s are
large numbers. Just as in the case of systems of one degree of freedom, however,
we are naturally led from the above considerations to assume that, also for
values of the~$n$'s which are not large, there must exist an intimate connection
between the probability of a given transition and the values of the corresponding
\Name{Fourier} coefficient in the expressions for the displacements of the particles in the
two stationary states. This allows us at once to draw certain important conclusions.
Thus, from the fact that in general negative as well as positive values for the~$\tau$'s
appear in~\Eq{(31)}, it follows that we must expect that in general not only such
transitions will be possible in which all the~$n$'s decrease, but that also transitions
will be possible for which some of the~$n$'s increase while others decrease. This
conclusion, which is supported by observations on the fine structure of the hydrogen
lines as well as on the \Name{Stark} effect, is contrary to the suggestion, put forward
by \Name{Sommerfeld} with reference to the essential positive character of the~$I$'s, that
every of the~$n$'s must remain constant or decrease under a transition. Another
direct consequence of the above considerations is obtained if we consider a system
for which, for all values of the constants $\alpha_{1}$,~\MyDots $\alpha_{s}$, the coefficient $C_{\tau_{1}, \MyDots \tau_{s}}$ corresponding
to a certain set $\tau_{1}^{0}$,~\MyDots $\tau_{s}^{0}$ of values for the~$\tau$'s is equal to zero in the expressions
for the displacements of the particles in every direction. In this case we
shall naturally expect that no transition will be possible for which the relation
$n'_{k} - n''_{k} = \tau_{k}^{0}$ is satisfied for every~$k$. In the case where $C_{\tau_{1}^{0}, \MyDots \tau_{s}^{0}}$ is equal to zero
in the expressions for the displacement in a certain direction only, we shall expect
that all transitions, for which $n'_{k} - n''_{k} = \tau_{k}^{0}$ for every~$k$, will be accompanied by
a radiation which is polarised in a plane perpendicular to this direction.

A simple illustration of the last considerations is afforded by the system mentioned
in the beginning of this section, and which consists of a particle executing
motions in three perpendicular directions which are independent of each other. In
that case all the \Name{Fourier} coefficients in the expressions for the displacements
in any direction will disappear if more than one of the~$\tau$'s are different from
zero. Consequently we must assume that only such transitions are possible for
which only one of the~$n$'s varies at the same time, and that the radiation corresponding
to such a transition will be linearly polarised in the direction of the displacement
of the corresponding coordinate. In the special case where the motions
in the three directions are simply harmonic, we shall moreover conclude that none
\PageSep{33}
of the~$n$'s can vary by more than a single unit, in analogy with the considerations
in the former section about a linear harmonic vibrator.

Another\Pagelabel{33} example which has more direct physical importance, since it includes
all the special applications of the quantum theory to spectral problems mentioned
in the introduction, is formed by a conditionally periodic system possessing an axis
of symmetry. In all these applications a separation of variables is obtained in a
set of three coordinates $q_{1}$,~$q_{2}$ and~$q_{3}$, of which the first two serve to fix the position
of the particle in a plane through the axis of the system, while the last is
equal to the angular distance between this plane and a fixed plane through the
same axis. Due to the symmetry, the expression for the total energy in \Name{Hamilton}'s
equations will not contain the angular distance~$q_{3}$ but only the angular momentum~$p_{3}$
round the axis. The latter quantity will consequently remain constant during
the motion, and the variations of $q_{1}$~and~$q_{2}$ will be exactly the same as in a conditionally
periodic system of two degrees of freedom only. If the position of the
particle is described in a set of cylindrical coordinates $z$,~$\rho$,~$\theta$, where $z$~is the displacement
in the direction of the axis, $\rho$~the distance of the particle from this axis
and $\theta$~is equal to the angular distance~$q_{3}$, we have therefore
\[
\begin{aligned}
z &amp;= \sum C_{\tau_{1}, \tau_{2}} \cos 2\pi \bigl\{(\tau_{1} \omega_{1} + \tau_{2} \omega_{2}) t + c_{\tau_{1}, \tau_{2}}\bigr\} \\
\llap{\text{and}\quad}
\rho &amp;= \sum C'_{\tau_{1}, \tau_{2}} \cos 2\pi \bigl\{(\tau_{1} \omega_{1} + \tau_{2} \omega_{2}) t + c'_{\tau_{1}, \tau_{2}}\bigr\},
\end{aligned}
\Tag{(32)}
\]
where the summation is to be extended over all positive and negative entire values
of $\tau_{1}$~and~$\tau_{2}$, and where $\omega_{1}$~and~$\omega_{2}$ are the mean frequencies of oscillation of the
coordinates $q_{1}$~and~$q_{2}$. For the rate of variation of~$\theta$ with the time we have further
\[
\BreakMath{%
\frac{d\theta}{dt} = \dot{q}_{3}
  = \frac{\dd E}{\dd p_{3}}
  = f(q_{1}, q_{2}, p_{1}, p_{2}, p_{3})
  = ± \sum C''_{\tau_{1}, \tau_{2}} \cos 2\pi \bigl\{(\tau_{1} \omega_{1} + \tau_{2} \omega_{2}) t + c''_{\tau_{1}, \tau_{2}}\bigr\},
}{%
\begin{aligned}
\frac{d\theta}{dt} = \dot{q}_{3}
  &amp;= \frac{\dd E}{\dd p_{3}}
   = f(q_{1}, q_{2}, p_{1}, p_{2}, p_{3}) \\
  &amp;= ± \sum C''_{\tau_{1}, \tau_{2}} \cos 2\pi \bigl\{(\tau_{1} \omega_{1} + \tau_{2} \omega_{2}) t + c''_{\tau_{1}, \tau_{2}}\bigr\},
\end{aligned}
}
\]
where the two signs correspond to a rotation of the particle in the direction of increasing
and decreasing~$q_{3}$ respectively, and are introduced to separate the two
types of symmetrical motions corresponding to these directions. This gives
\[
± \theta = 2\pi \omega_{3} t + \sum C'''_{\tau_{1}, \tau_{2}} \cos 2\pi \bigl\{(\tau_{1} \omega_{1} + \tau_{2} \omega_{2}) t + c'''_{\tau_{1}, \tau_{2}}\bigr\},
\Tag{(33)}
\]
where the positive constant $\omega_{3} = \dfrac{1}{2\pi} C''_{0,0}$ is the mean frequency of rotation round the
axis of symmetry of the system. Considering now the displacement of the particle
in rectangular coordinates $x$,~$y$ and~$z$, and taking as above the axis of symmetry
as $z$-axis, we get from \Eq{(32)}~and~\Eq{(33)} after a simple contraction of terms
\[
\BreakMath{%
\begin{alignedat}{3}
x &amp;= \rho \cos \theta
  &amp;&amp;=  &amp;&amp;\sum D_{\tau_{1}, \tau_{2}} \cos 2\pi \bigl\{(\tau_{1} \omega_{1} + \tau_{2} \omega_{2} + \omega_{3}) t + d_{\tau_{1}, \tau_{2}}\bigr\} \\
\llap{\text{and}}\
y &amp;= \rho \sin \theta
  &amp;&amp;= ± &amp;&amp;\sum D_{\tau_{1}, \tau_{2}} \sin 2\pi \bigl\{(\tau_{1} \omega_{1} + \tau_{2} \omega_{2} + \omega_{3}) t + d_{\tau_{1}, \tau_{2}}\bigr\},
\end{alignedat}
}{%
\begin{aligned}
x &amp;= \rho \cos \theta \\
  &amp;= \sum D_{\tau_{1}, \tau_{2}} \cos 2\pi \bigl\{(\tau_{1} \omega_{1} + \tau_{2} \omega_{2} + \omega_{3}) t + d_{\tau_{1}, \tau_{2}}\bigr\} \\
\llap{\text{and}}\
y &amp;= \rho \sin \theta \\
  &amp;= ± \sum D_{\tau_{1}, \tau_{2}} \sin 2\pi \bigl\{(\tau_{1} \omega_{1} + \tau_{2} \omega_{2} + \omega_{3}) t \rlap{$+ d_{\tau_{1}, \tau_{2}}\bigr\},$}
\end{aligned}
}
\Tag{(34)}
\]
where the $D$'s and~$d$'s are new constants, and the summation is again to be extended
over all positive and negative values of $\tau_{1}$~and~$\tau_{2}$.
%[** TN: Printer's mark]
%D. K. D. Vidensk. Selsk. Skr., naturvidensk. og mathem. Afd., 8. Række, IV. 1. 5
\PageSep{34}

From \Eq{(32)} and~\Eq{(34)} we see that the motion in the present case may be considered
as composed of a number of linear harmonic vibrations parallel to the axis
of symmetry and of frequencies equal to the absolute values of $(\tau_{1} \omega_{1} + \tau_{2} \omega_{2})$,
together with a number of circular harmonic motions round this axis of frequencies
equal to the absolute values of $(\tau_{1} \omega_{1} + \tau_{2} \omega_{2} + \omega_{3})$, and possessing the same direction
of rotation as that of the moving particle or the opposite if the latter expression
is positive or negative respectively. According to ordinary electrodynamics
the radiation from the system would therefore consist of a number of components
of frequency $\tau_{1} \omega_{1} + \tau_{2} \omega_{2}$ polarised parallel to the axis of symmetry, and a number
of components of frequencies $\tau_{1} \omega_{1} + \tau_{2} \omega_{2} + \omega_{3}$ and of circular polarisation round
this axis (when viewed in the direction of the axis). On the present theory we
shall consequently expect that in this case only two kinds of transitions between
the stationary states given by~\Eq{(22)} will be possible. In both of these $n_{1}$~and~$n_{2}$
may vary by an arbitrary number of units, but in the first kind of transition, which
will give rise to a radiation polarised parallel to the axis of the system, $n_{3}$~will
remain unchanged, while in the second kind of transition $n_{3}$~will decrease or increase
by one unit and the emitted radiation will be circularly polarised round
the axis in the same direction as or the opposite of that of the rotation of the particle
respectively.

In the next Part we shall see that these conclusions are supported in an
instructive manner by the experiments on the effects of electric and magnetic fields
on the hydrogen spectrum. In connection with the discussion of the general theory,
however, it may be of interest to show that the formal analogy between the ordinary
theory of radiation and the theory based on \Eq{(1)}~and \Eq{(22)}, in case of systems possessing
an axis of symmetry, can be traced not only with respect to frequency
relations but also by considerations of \Emph{conservation of angular momentum}.\Pagelabel{34}
For a conditionally periodic system possessing an axis of symmetry the angular
momentum round this axis is, with the above choice of coordinates, according to~\Eq{(22)}
equal to $\dfrac{I_{3}}{2\pi} = n_{3}\, \dfrac{h}{2\pi}$. If therefore, as assumed above for a transition corresponding
to an emission of linearly polarised light, $n_{3}$~is unaltered, it means that
the angular momentum of the system remains unchanged, while if $n_{3}$~alters by
one unit, as assumed for a transition corresponding to an emission of circularly
polarised light, the angular momentum will be altered by~$\dfrac{h}{2\pi}$. Now it is easily
seen that the ratio between this amount of angular momentum and the amount of
energy~$h\nu$ emitted during the transition is just equal to the ratio between the amount
of angular momentum and energy possessed by the radiation which according to
ordinary electrodynamics would be emitted by an electron rotating in a circular
orbit in a central field of force. In fact, if $a$~is the radius of the orbit, $\nu$~the frequency
of revolution and $F$~the force of reaction due to the electromagnetic field
of the radiation, the amount of energy and of angular momentum round an
axis through the centre of the field perpendicular to the plane of the orbit, lost
\PageSep{35}
by the electron in unit of time as a consequence of the radiation, would be equal to
$2\pi\nu aF$ and $aF$~respectively. Due to the principles of conservation of energy and of
angular momentum holding in ordinary electrodynamics, we should therefore expect
that the ratio between the energy and the angular momentum of the emitted radiation
would be~$2\pi\nu$,\footnote
  {Comp.\ \Name{K.~Schaposchnikow}, Phys.\ Zeitschr.\ XV, p.~454 (1914).}
but this is seen to be equal to the ratio between the energy~$h\nu$ and
the angular momentum~$\dfrac{h}{2\pi}$ lost by the system considered above during a transition
for which we have assumed that the radiation is circularly polarised. This agreement
would seem not only to support the validity of the above considerations but
also to offer a direct support, independent of the equations~\Eq{(22)}, of the assumption
that, \Emph{for an atomic system possessing an axis of symmetry, the total
angular momentum round this axis is equal to an entire multiple of~$\dfrac{h}{2\pi}$}.

A further illustration of the above considerations of the relation between
the quantum theory and the ordinary theory of radiation is obtained if we consider
a conditionally periodic system subject to the \Emph{influence of a small perturbing
field of force}. Let us assume that the original system allows of separation of
variables in a certain set of coordinates $q_{1}$,~\MyDots $q_{s}$, so that the stationary states are
determined by~\Eq{(22)}. From the necessary stability of the stationary states we must
conclude that the perturbed system will possess a set of stationary states which
only differ slightly from those of the original system. In general, however, it will
not be possible for the perturbed system to obtain a separation of variables in any
set of coordinates, but if the perturbing force is sufficiently small the perturbed
motion will again be of conditionally periodic type and may be regarded as a superposition
of a number of harmonic vibrations just as the original motion. The displacements
of the particles in the stationary states of the perturbed system will
therefore be given by an expression of the same type as~\Eq{(31)} where the fundamental
frequencies~$\omega_{k}$ and the amplitudes $C_{\tau_{1}, \MyDots \tau_{s}}$ may differ from those corresponding to
the stationary states of the original system by small quantities proportional to the
intensity of the perturbing forces. If now for the original motion the coefficients
$C_{\tau_{1}, \MyDots \tau_{s}}$ corresponding to certain combinations of the~$\tau$'s are equal to zero for all
values of the constants $\alpha_{1}$,~\MyDots $\alpha_{s}$, these coefficients will therefore for the perturbed
motion, in general, possess small values proportional to the perturbing forces. From
the above considerations we shall therefore expect that, in addition to the main
probabilities of such transitions between stationary states which are possible for the
original system, there will for the perturbed system exist small probabilities of new
transitions corresponding to the above mentioned combinations of the~$\tau$'s. Consequently
we shall expect that the effect of the perturbing field on the spectrum
of the system will consist partly in a small displacement of the original lines\Add{,}
partly in the appearance of new lines of small intensity.

A simple example of this is afforded by a system consisting of a particle moving
in a plane and executing harmonic vibrations in two perpendicular directions with frequencies
\PageSep{36}
$\omega_{1}$~and~$\omega_{2}$. If the system is undisturbed all coefficients $C_{\tau_{1}, \tau_{2}}$ will be zero,
except $C_{1,0}$ and~$C_{0, 1}$. When, however, the system is perturbed, for instance by an
arbitrary small central force, there will in the \Name{Fourier} expressions for the displacements
of the particle, in addition to the main terms corresponding to the
fundamental frequencies $\omega_{1}$~and~$\omega_{2}$, appear a number of small terms corresponding
to frequencies given by $\tau_{1} \omega_{1} + \tau_{2} \omega_{2}$ where $\tau_{1}$~and~$\tau_{2}$ are entire numbers which may
be positive as well as negative. On the present theory we shall therefore expect
that in the presence of the perturbing force there will appear small probabilities
for new transitions which will give rise to radiations analogous to the so\Typo{}{ }called
harmonics and combination tones in acoustics, just as it should be expected on the
ordinary theory of radiation where a direct connection between the emitted radiation
and the motion of the system is assumed. Another example of more direct
physical application is afforded by the effect of an external homogeneous electric
field in producing new spectral lines. In this case the potential of the perturbing
force is a linear function of the coordinates of the particles and, whatever is the
nature of the original system, it follows directly from the general theory of perturbations
that the frequency of any additional term in the expression for the perturbed
motion, which is of the same order of magnitude as the external force, must correspond
to the sum or difference of two frequencies of the harmonic vibrations into
which the original motion can be resolved. With applications of these considerations
we will meet in Part~II in connection with the discussion of \Name{Sommerfeld}'s theory
of the fine structure of the hydrogen lines and in Part~III in connection with the
problem of the appearance of new series in the spectra of other elements under the
influence of intense external electric fields.

As mentioned we cannot without a more detailed theory of the mechanism of transition
between stationary states obtain quantitative information as regards the general
question of the intensities of the different lines of the spectrum of a conditionally
periodic system given by~\Eq{(26)}, except in the limit where the~$n$'s are large numbers,
or in such special cases where for all values of the constants $\alpha_{1}$,~\MyDots $\alpha_{s}$ certain
coefficients $C_{\tau_{1}, \MyDots \tau_{s}}$ in~\Eq{(31)} are equal to zero. From considerations of analogy, however,
we must expect that it will be possible also in the general case to obtain an
\Emph{estimate of the intensities} of the different lines in the spectrum by comparing
the intensity of a given line, corresponding to a transition between two stationary
states characterised by the numbers $n'_{1}$,~\MyDots $n'_{s}$ and $n''_{1}$,~\MyDots $n''_{s}$ respectively, with the
intensities of the radiations of frequencies $\omega_{1} (n'_{1} - n''_{1}) + \dots + \omega_{s} (n'_{s} - n''_{s})$ to be expected
on ordinary electrodynamics from the motions in these states; although of
course this estimate becomes more uncertain the smaller the values for the~$n$'s are.
As it will be seen from the applications mentioned in the following Parts this is
supported in a general way by comparison with the observations.
\vfill
\begin{center}
  \footnotesize
  Færdig fra Trykkeriet d.~27, April 1918.
\end{center}

% [** TN: Publisher's titlepage with series header and price]
\iffalse%****
\begin{center}
Mémoires de l'Académie Royale des Sciences et des Lettres de Danemark. Copenhague. \\
Section des Sciences, 8\ieme~série, t.~IV, \numero~1, fasc.~2.

\Large ON \\
\LARGE THE QUANTUM THEORY \\
OF LINE-SPECTRA \\
\footnotesize
BY \\
\large
N. BOHR \\
\vfill

\textsf{PART II} \\
\vfill

\footnotesize
\textsc{D. Kgl.\ Danske Vidensk.\ Selsk.\ Skripter, naturvidensk.\ og mathem.\ Afd.,~8. Række,~IV. 1, 2.}
\vfill\vfill

\large
KØBENHAVN \\
\textsc{hovedkommissionær}: ANDR. FRED. HØST \&amp;~SON, \textsc{kgl., hof-boghandel} \\
BIANCO LUNOS BOGTRYKKERI \\
1918

Pris: 4~Kr.
\end{center}
%\cleardoublepage
\fi % [** TN: End of publisher's titlepage]

\Part{II.}{On the hydrogen spectrum.}

\Section{1.}{The simple theory of the series~spectrum~of~hydrogen.}

As well known, the frequencies of the lines of the series spectrum of hydrogen
may, if we look apart from the fine structure of the single lines revealed by instruments
of high dispersive power, be represented by the formula
\[
\nu = K\left(\frac{1}{n''^{2}} - \frac{1}{n'^{2}}\right),
\Tag{(35)}
\]
where $K$~is a constant, and $n'$~and $n''$ a set of two entire numbers, different for the
different lines of the spectrum. According to the general principles of the quantum
theory of line spectra discussed in the first section of Part~I, we shall therefore
expect that this spectrum is emitted by a system which possesses a series of stationary
states in which the numerical value of the energy in the \Ord{$n$}{th}~state, omitting
an arbitrary constant, with a high degree of approximation is given by
\[
E_{n} = \frac{Kh}{n^{2}},
\Tag{(36)}
\]
where $h$~is \Name{Planck}'s constant which enters in the fundamental relation~\EqI{(1)}.

Now according to \Name{Rutherford}'s theory of atomic structure, a neutral hydrogen
atom must be expected to consist of an electron and a positive nucleus of a mass
very large compared with that of the electron, which move under the influence of
a mutual attraction inversely proportional to the square of the distance apart. Assuming
that the motion in the stationary states may be determined by ordinary
mechanics, and neglecting for the moment the small modifications claimed by the
theory of relativity, we find that each of the particles will describe an elliptical
orbit with their common centre of gravity at one of the foci, and from the well
known laws for a Keplerian motion we have that the frequency of revolution~$\omega$
and the major axis~$2\alpha$ of the relative orbit of the particles, quite independent of
the degree of eccentricity of this orbit, are given by
\[
\omega = \sqrt{\frac{2W^{3} (M + m)}{\pi^{2} N^{2} e^{4} Mm}},\quad
2\alpha = \frac{Ne^{2}}{W},
\Tag{(37)}
\]
where $W$~is the work necessary to remove the electron to infinite distance from the
nucleus, while $Ne$~and $M$ are the charge and the mass of the nucleus, and $-e$~and
$m$ the charge and the mass of the electron.
% [** TN: Printer's mark]
% D. K. D. Vidensk. Selsk. Skr., naturvidensk. og mathem. Afd., 8. Raekke, IV. 1. 6
\PageSep{38}

As explained in Part~I, there will in general be no simple connection between
the motion of a system in the stationary states and the spectrum emitted during
transitions between these states; such a connection, however, must be expected to
exist in the limit where the motions in successive stationary states differ comparatively
little from each other. In the present case this connection claims in the first
place that the frequency of revolution tends to zero for increasing~$n$. According to
\Eq{(36)}~and \Eq{(37)} we may therefore put the value of~$W$ in the \Ord{$n$}{th}~stationary state equal to
\[
W_{n} = \frac{Kh}{n^{2}}.
\Tag{(38)}
\]
Moreover, since \Eq{(35)}~can be written in the form
\[
\nu = (n' - n'') K\, \frac{n' + n''}{n'^{2} n''^{2}},
\]
it is seen to be a necessary condition that the frequency of revolution for large
values of~$n$ is asymptotically given by
\[
\omega_{n} \sim \frac{2K}{n^{3}},
\Tag{(39)}
\]
if we wish that the frequency of the radiation emitted during a transition between
two stationary states, for which the numbers $n'$~and $n''$ are large compared with
their difference $n' - n''$, shall tend to coincide with one of the frequencies of the
spectrum which on ordinary electrodynamics would be emitted from the system in
these states. But from \Eq{(37)}~and \Eq{(38)} it will be seen that \Eq{(39)}~claims the fulfilment
of the relation
\[
K = \frac{2\pi^{2} N^{2} e^{4} Mm}{h^{3} (M + m)}
  = \frac{2\pi^{2} N^{2} e^{4} m}{h^{3} (1 + m/M)}.
\Tag{(40)}
\]

As shown in previous papers, this relation is actually found to be fulfilled
within the limit of experimental errors if we put $N = 1$ and for $e$,~$m$, and~$h$ introduce
the values deduced from measurements on other phenomena; a result
which may be considered as affording a strong support for the validity of the
general principles discussed in Part~I, as well as for the reality of the atomic model
under consideration. Further it was found that, if in formula~\Eq{(35)} for the hydrogen
spectrum the constant~$K$ is replaced by a constant which is four times larger, this
formula represents to a high degree of approximation the frequencies of the lines of a
spectrum emitted by helium, when this gas is subject to a condensed discharge. This
was to be expected on \Name{Rutherford}'s theory, according to which a neutral helium
atom contains two electrons and a nucleus of a charge twice that of the nucleus
of the hydrogen atom. A helium atom from which one electron is removed will
thus form a dynamical system perfectly similar to a neutral hydrogen atom, and
may therefore be expected to emit a spectrum represented by~\Eq{(35)} if in~\Eq{(40)} we put
$N = 2$. Moreover a closer comparison of the helium spectrum under consideration
\PageSep{39}
with the hydrogen spectrum has shown that the value of the constant~$K$ in the
former spectrum was not exactly four times as large as that in the latter, but that
the ratio between these constants within the limit of experimental errors agreed
with the value to be expected from~\Eq{(40)}, when regard is taken to the different masses
of the nuclei of the atoms of hydrogen and helium corresponding to the different
atomic weights of these elements.\footnote
  {For the \Typo{litterature}{literature} on this subject the reader is referred  to the papers cited in the introduction.}

Introducing the expression for~$K$ given by~\Eq{(40)} in the formulæ \Eq{(37)} and \Eq{(38)},
we find for the values of $W$,~$\omega$ and~$2\alpha$ in the stationary states
\[
\BreakMath{%
W_{n} = \frac{1}{n^{2}}\, \frac{2\pi^{2} N^{2} e^{4} Mm}{h^{2} (M + m)},\quad
\omega_{n} = \frac{1}{n^{3}}\, \frac{4\pi^{2} N^{2} e^{4} Mm}{h^{3} (M + m)},\quad
2\alpha_{n} = n^{2}\, \frac{h^{2} (M + m)}{2\pi^{2} N e^{2} Mm}.
}{%
\begin{gathered}
W_{n} = \frac{1}{n^{2}}\, \frac{2\pi^{2} N^{2} e^{4} Mm}{h^{2} (M + m)}, \\
\omega_{n} = \frac{1}{n^{3}}\, \frac{4\pi^{2} N^{2} e^{4} Mm}{h^{3} (M + m)},\quad
2\alpha_{n} = n^{2}\, \frac{h^{2} (M + m)}{2\pi^{2} N e^{2} Mm}.
\end{gathered}
}
\Tag{(41)}
\]
Now for a mechanical system as that under consideration, for which every motion
is periodic independent of the initial conditions, we have that the value of the total
energy will be completely determined by the value of the quantity~$I$, defined
by equation~\EqI{(5)} in Part~I\@. As mentioned this follows directly from relation~\EqI{(8)},
which shows at the same time that for a system for which every motion is periodic
the frequency will be completely determined by~$I$ or by the energy only. For the
value of~$I$ in the stationary states of the hydrogen atom we get by means of~\EqI{(8)}
from \Eq{(37)}~and \Eq{(41)}, since in this case $I$~will obviously become zero when $W$~becomes
infinite,
\[
\BreakMath{%
I = \int_{W_{n}}^{\infty} \frac{dW}{\omega}
  = \sqrt{\frac{\pi^{2} N^{2} e^{4} Mm}{2(M + m)}} \int_{W_{n}}^{\infty} W^{-3/2}\, dW
  = \sqrt{\frac{2\pi^{2} N^{2} e^{4} Mm}{W_{n}(M + m)}}
  = nh.
}{%
\begin{aligned}
I = \int_{W_{n}}^{\infty} \frac{dW}{\omega}
  &amp;= \sqrt{\frac{\pi^{2} N^{2} e^{4} Mm}{2(M + m)}} \int_{W_{n}}^{\infty} W^{-3/2}\, dW \\
  &amp;= \sqrt{\frac{2\pi^{2} N^{2} e^{4} Mm}{W_{n}(M + m)}}
   = nh.
\end{aligned}
}
\]
This result will be seen to be consistent with condition~\EqI{(24)} which, as mentioned
in Part~I, presents itself as a direct generalisation to periodic systems of several
degrees of freedom of condition~\EqI{(10)} which determines the stationary states of a
system of one degree of freedom, and which again on \Name{Ehrenfest}'s principle of
the mechanical transformability of the stationary states forms a rational generalisation
of \Name{Planck}'s fundamental formula~\EqI{(9)} for the possible values of the energy of a
linear harmonic vibrator.

In this connection it will be observed, that the relation discussed above between
the hydrogen spectrum and the motion of the atom in the limit of small frequencies
is completely analogous to the general relation, discussed in \SecRef[I]{2} in Part~I,
between the spectrum which on the quantum theory would be emitted by a system
of one degree of freedom, the stationary states of which are determined by~\EqI{(10)}, and
the motion of the system in these states. It will at the same time be noted that, in
case of hydrogen, this relation implies that the motion of the particles in the
stationary states of the atom will not in general be simply harmonic, or in other
words that the orbit of the electron will not in general be circular. In fact if the
motion of the particles were simply harmonic, as the motion of a \Name{Planck}'s vibrator,
\PageSep{40}
we should expect on the considerations in Part~I that no transition between two
stationary states of the atom would be possible for which $n'$~and~$n''$ differ by more
than one unit; but this would obviously be inconsistent with the observations,
since for instance the lines of the ordinary Balmer series, according to the theory,
correspond to transitions for which $n'' = 2$ while $n'$~takes the values $3$,~$4$, $5$,~\dots\Add{.}
In connection with this consideration it may be remarked that, adopting a terminology
well known from acoustics, we may from the point of view of the quantum
theory regard the higher members of the Balmer series ($n' = 4$, $5$,~\dots) as the
``harmonics'' of the first member ($n' = 3$), although of course the frequencies of
the former lines are by no means entire multipla of the frequency of the latter line.

While in the above way it was possible to obtain a simple interpretation of
certain main features of the hydrogen spectrum, it was not found possible in this way
to account in detail for such phenomena in which the deviation of the motion of
the particles from a simple Keplerian motion plays an essential part. This is the
case in the problem of the fine structure of the hydrogen lines, which is due to
the effect of the small variation of the mass of the electron with its velocity, as
well as in the problems of the characteristic effects of external electric and magnetic
fields on the hydrogen lines. As mentioned in the introduction, a progress of
fundamental importance in the treatment of such problems was made by \Name{Sommerfeld},
who obtained a convincing explanation of the fine structure of the hydrogen
lines by means of his theory of the stationary states of central systems, in which
the single condition $I = nh$ was replaced by the two conditions~\EqI{(16)}; and the
theory was further developed by \Name{Epstein} and \Name{Schwarzschild}, who on this line
established the general theory, based on the conditions~\EqI{(22)}, of the stationary states
of a conditionally periodic system for which the equations of motion may be
solved by means of separation of variables in the \Typo{Hamilton-Jacobi}{\Name{Hamilton}-\Name{Jacobi}} partial differential
equation. If the hydrogen atom is exposed to a homogeneous electric or to a
homogeneous magnetic field, the atom forms a system of this class, and, as shown
by \Name{Epstein} and \Name{Schwarzschild} as regards the \Name{Stark} effect and by \Name{Sommerfeld} and
\Name{Debye} as regards the \Name{Zeeman} effect, the theory under consideration leads to values
for the total energy of the atom in the stationary states, which together with relation~\EqI{(1)}
lead again to values for the frequencies of the radiations emitted during
the transitions between these states, which are in agreement with the measured frequencies
of the components into which the hydrogen lines are split up in the presence
of the fields. As pointed out in Part~I, it is possible moreover to throw light
on the question of the intensities and polarisations of these components on the
basis of the necessary formal relation between the quantum theory of line spectra
and the ordinary theory of radiation in the limit where the motions in successive
stationary states differ very little from each other. In the following sections the
mentioned problems will be discussed in detail. As regards the fixation of the
stationary states we shall not, however, follow the same procedure as used by the
authors just mentioned, which rests upon the immediate application of the conditions~\EqI{(22)},
\PageSep{41}
but it will be shown how the conditions which fix the stationary states of
the perturbed atom may be obtained by a direct examination of the small deviations
of the motion of the electron from a simple Keplerian motion. In this way it
seems possible to obtain a more direct illustration of the principles discussed in
Part~I; and we shall see moreover that the treatment in question may be used also
in cases where the method of separation of variables cannot be applied.

In Part~III the problem of the series spectra of other elements will be treated
from a similar point of view. As pointed out by the writer in an earlier paper, a
simple explanation of the pronounced analogy between these spectra and the hydrogen
spectrum is offered by the fact, that the atomic systems, involved in the
emission of the spectra under consideration, in a certain sense may be regarded as
a perturbed hydrogen atom. On the other hand, a clue to the interpretation of the
characteristic difference between the hydrogen spectrum and the spectra of other
elements was first obtained by \Name{Sommerfeld}'s theory of the stationary states of central
systems referred to above. As shown by \Name{Sommerfeld}, it is possible on this
theory to account in general outlines for the well known laws governing the frequencies
of the series spectra of the elements; and, as it will be shown in Part~III, it
is also possible, on the basis of the formal relation between the quantum theory and
the ordinary theory of radiation, in this way to obtain a simple interpretation of the
laws governing the remarkable differences in the intensities with which the various
series of lines appear, which on the combination principle would constitute the
complete spectra under consideration. As regards the detailed discussion of these
spectra, however, it is necessary to bear in mind that the part played by the inner
electrons in the atoms of the elements in question forms a far more intricate problem
than the perturbing effect of a fixed external field on the hydrogen atom.
For the treatment of this problem the theory of conditionally periodic systems
based on the conditions~\EqI{(22)} does not seem to suffice, while, as it will be shown
in Part~III, it appears that the method of perturbations exposed in the following
lends itself naturally also to this case.


\Section{2.}{The stationary states of a perturbed~periodic~system.}

In Part~I it was shown that the problem of the fixation of the stationary
states of a periodic system of several degrees of freedom, which is subject to the
perturbing influence of a small external field, cannot be treated directly on the basis
of the general principle of the mechanical transformability of the stationary states
by considering the influence, which on ordinary mechanics a slow establishment
of the external field would exert on the motion of some arbitrarily chosen stationary
state of the undisturbed system (see Part~I, \PageRefI{23}). This is an immediate
consequence of the fact, mentioned in the former section, that the stationary states
\PageSep{42}
of the perturbed system are characterised by a greater number of extra-mechanical
conditions than the stationary states of the undisturbed system. On the other hand,
we were led to assume from the general formal relation between the quantum theory
of line spectra and the ordinary theory of radiation, that it is possible to obtain
information about the stationary states of the perturbed system from a direct
consideration of the slow variations which the periodic orbit undergoes as a
consequence of the mechanical effect of the external field on the motion. Thus, if
these variations are of periodic or conditionally periodic type, we may expect that,
in the presence of the external field, the values for the additional energy of the
system in the stationary states are related to the small frequency or frequencies of
the perturbations, in a manner analogous to the relation between energy and frequency
in the stationary states of an ordinary periodic or conditionally periodic system.

If\Pagelabel{42} the equations of motion for the perturbed system can be solved by means
of separation of variables, it is easily seen that the relation in question is fulfilled
if the stationary states are determined by the conditions~\EqI{(22)}. Consider thus a system
for which every orbit is periodic, and let us assume that in the presence of a given
small external field a separation of variables is possible in a certain set of coordinates
$q_{1}$,~\MyDots $q_{s}$. For the undisturbed system we have then, according to equation~\EqI{(23)},
that the quantity~$I$, defined by~\EqI{(5)}, is equal to $\kappa_{1} I_{1} + \dots + \kappa_{s} I_{s}$, where
$I_{1}$,~\MyDots $I_{s}$ are defined by~\EqI{(21)} and calculated with respect to the set of coordinates
just mentioned, and where the~$\kappa$'s are a set of entire positive numbers without a
common divisor. For simplicity let us assume that at least one of the~$\kappa$'s, say~$\kappa_{s}$,
is equal to one, and that consequently, as mentioned on \PageRefI[page]{22}, the number~$n$ in~\EqI{(24)},
which characterises the stationary states of the undisturbed system, may take
all positive values. This condition will be fulfilled in case of all the applications to
spectral problems discussed below; it will be seen, however, that the extension to
problems where this condition is not fulfilled will only necessitate small modifications
in the following considerations. By use of~\EqI{(29)} we get now for the difference
in the total energy of two slightly different states of the perturbed system
\[
\delta E = \sum_{1}^{s} \omega_{k}\, \delta I_{k}
  = \omega_{s} \sum_{1}^{s} \kappa_{k}\, \delta I_{k}
  + \sum_{1}^{s-1} (\omega_{k} - \kappa_{k} \omega_{s})\, \delta I_{k}.
\Tag{(42)}
\]
Since for the undisturbed system $\omega_{k} = \kappa_{k} \omega_{s}$, the differences $\omega_{k} - \kappa_{k} \omega_{s}$ appearing
in the last term will, for the perturbed system, be small quantities which will just
represent the frequencies of the slow variations which the orbit undergoes in the
presence of the external field. These quantities will in the following be denoted by~$\frakv_{k}$.
Consider now the multitude of states of the perturbed system for which $\sum_{1}^{s} \kappa_{k} I_{k}$
is equal to~$nh$, where $n$~is a given entire positive number. This multitude will be
seen to include all possible stationary states of the perturbed system, which satisfy~\EqI{(22)},
and the motion of which differs at any moment only slightly from some
stationary motion of the undisturbed system, satisfying~\EqI{(24)} for the given value of~$n$.
\PageSep{43}
Denoting the value of the energy of the undisturbed system in such a state by~$E_{n}$,
and the value of the energy of the perturbed system in a state belonging to
the multitude under consideration by $E_{n} + \frakE$, we get from~\Eq{(42)}
\[
\delta \frakE = \sum_{1}^{s-1} \frakv_{k}\, \delta I_{k}
\Tag{(43)}
\]
for the energy difference between two neighbouring states of this multitude. Since
this relation has the same form as~\EqI{(29)}, we see consequently that by putting $I_{1}$,~\MyDots $I_{s-1}$
equal to entire multipla of~$h$, as claimed by the conditions~\EqI{(22)}, we obtain exactly
the same relation between the additional energy~$\frakE$ and the small frequencies~$\frakv_{k}$,
impressed on the system by the external field, as that which holds between the
total energy and the fundamental frequencies in the stationary states of a conditionally
periodic system of $s - 1$~degrees of freedom.

\begin{Remark}
As\Pagelabel{43} a simple illustration of these calculations let us consider the system consisting
of a particle moving in a plane and subject to an attraction from a fixed point, which varies
proportional to the distance apart. If undisturbed, the motion of this system will be periodic
independent of the initial conditions, and the particle will describe an elliptical orbit with its
centre at the fixed point. Moreover the equations of motion of the undisturbed system may
be solved by means of separation of variables in polar coordinates, as well as in any set of
rectangular coordinates. In the first case we have, taking for~$q_{1}$ the length of the radius vector
from the fixed point to the particle and for~$q_{2}$ the angular distance of this radius vector
from a fixed direction, $\kappa_{1} = 2$ and $\kappa_{2} = 1$, while in the second case we have $\kappa_{1} = \kappa_{2} = 1$. In
the presence of an external field the orbit will in general not remain periodic, but will in the
course of time cover a continuous extension of the plane. If the external field is sufficiently
small, however, the orbit will at any moment only differ little from a closed elliptical orbit,
but in the course of time the lengths and directions of the principal axes of this ellipse will
undergo slow variations. In general the perturbed system will not allow of separation of
variables, but two cases obviously present themselves in which such a separation is still
possible; in the first case the external field is central with the fixed point as centre, and a
separation is possible in polar coordinates; in the second case the external field of force is
perpendicular to a given line and varies as some function of the distance from this line, and
separation is possible in a set of rectangular coordinates with the axes parallel and perpendicular
to the given line. In the first case the perturbations will not affect the lengths of
the principal axes of the elliptical orbit and will only produce a slow uniform rotation of
the directions of these axes, while in the second case the lengths of the principal axes as well
as their directions will perform slow oscillations. It will consequently be seen that, by fixing
the stationary states of the perturbed system by means of the conditions~\EqI{(22)}, the cycles of
shapes and positions which the orbit of the particle will pass through in the stationary states
will be entirely different in the two cases. In both cases, however, it will be seen that the
frequency $\frakv = \omega_{1} - \kappa_{1} \omega_{2}$ will be equal to the frequency with which the orbit at regular
intervals re-assumes its shape and position. By fixing the stationary states by~\EqI{(22)} we obtain
therefore, as seen from~\Eq{(43)}, in both cases that the relation between this frequency and the
additional energy of the system due to the presence of the field will be the same as the
relation between energy and frequency in the stationary states of a system of one degree of
freedom; and it will be seen that the above considerations offer a dynamical interpretation
of the characteristic discontinuity involved in the application of the method of separation of
%[** TN: Footnote mark before punctuation in the original]
variables to the fixation of the stationary states of perturbed periodic systems.\footnote
  {In this connection it may be of interest to note that the possibility of a rational interpretation of
  the discontinuity in question would seem to be essentially connected with the form of the principles
  of the quantum theory adopted in this paper. If for instance the quantum theory is taken in the
  form proposed by \Name{Planck} in his second theory of temperature radiation, the consequent development
  to periodic systems of several degrees of freedom would seem to involve a serious difficulty as regards
  the question of the necessary stability of the temperature equilibrium among a great number of systems
  for small variations of the external conditions. In fact, in connection with the development of his theory
  of the ``physical structure of the phase space'', mentioned in Part~I on \PageRefI[page]{18}, in which conditions
  of the same type as~\EqI{(22)} are established, \Name{Planck} has deduced expressions for the total energy of a
  great number of systems in temperature equilibrium, which, if applied to systems of the same kind
  as those considered in the above example, show a dependency of this energy on the temperature which
  is different, according to whether polar coordinates or rectangular coordinates are used as basis for the
  structure of the phase space.}
\end{Remark}
\PageSep{44}

In general it will not be possible to solve the equations of motion of the perturbed
system by means of separation of variables in a fixed set of positional coordinates,
but we shall see that the problem of the fixation of the stationary states
of the perturbed system may be attacked by a direct examination of the additional
energy of the system and its relation to the slow variations of the orbit, on the
basis of the usual theory of perturbations well known from celestial mechanics.
Consider a system for which every orbit, if undisturbed, is periodic independent of
the initial conditions, and let us assume that the equations of motion for some set
of coordinates $q_{1}$,~$q_{2}$,~\MyDots $q_{s}$ are solved by means of the \Typo{Hamilton-Jacobi}{\Name{Hamilton}-\Name{Jacobi}} partial
differential equation, given by formula~\EqI{(17)} in Part~I\@. The motion of the system is
then determined by the equations~\EqI{(18)}, and the orbit is characterised by means of
the constants $\alpha_{1}$,~\MyDots $\alpha_{s}$, $\beta_{1}$,~\MyDots $\beta_{s}$. If now the system is subject to some small
external field of force, the motion will no more be periodic, but, defining in the
usual way the osculating orbit at a given moment as the periodic orbit which
would result if the external forces vanished suddenly at this moment, we find that
the constants $\alpha_{1}$,~\MyDots $\alpha_{s}$, $\beta_{1}$,~\MyDots $\beta_{s}$, characterising the osculating orbit, will vary
slowly with the time. Assuming for the present that the external forces possess a
constant potential~$\Omega$ given as a function of the~$q$'s, we have according to the theory
of perturbations that the rates of variation of the orbital constants of the osculating
orbit will be given by\footnote
  {See f.\ inst.\ \Name{C.~V.~L. Charlier}, Die Mechanik des Himmels, Bd.~I, Abt.~1, §~10.}
\[
\frac{d\alpha_{k}}{dt} = -\frac{d\Omega}{d\beta_{k}},\quad
\frac{d\beta_{k}}{dt}  =  \frac{d\Omega}{d\alpha_{k}},\qquad
(k = 1, \MyDots s)
\Tag{(44)}
\]
where $\Omega$~is considered as a function of $\alpha_{1}$,~\MyDots $\alpha_{s}$, $\beta_{1}$,~\MyDots $\beta_{s}$ and~$t$, obtained by
introducing for the~$q$'s their expressions as functions of these quantities obtained
by solving~\EqI{(18)}. The equations~\Eq{(44)} allow to follow completely the perturbing
effect of the external field on the motion of the system. For the problem under
consideration, however, a detailed examination of the perturbations is not necessary.
In fact, we shall not be concerned with the small deformation of the orbit
characterised by the small oscillations of the orbital constants within a time interval
of the same order of magnitude as the period of the osculating orbit, but only
\PageSep{45}
with the so\Typo{}{ }called ``secular perturbations'' of the orbit, characterised by the total
variation of these constants taken over a time interval long compared with the
period of the osculating orbit. As we shall see below, these variations may, with
an approximation sufficient for our purpose, be obtained directly by taking mean
values on both sides of the equations~\Eq{(44)}. Before entering on these calculations,
however, it may be observed that the part played by the constants $\alpha_{1}$ and $\beta_{1}$ differs
essentially from that played by the other orbital constants $\alpha_{2}$,~\MyDots $\alpha_{s}$, $\beta_{2}$,~\MyDots $\beta_{s}$.
Thus from the formulæ \EqI{(17)} and \EqI{(18)} on \PageRefI[page]{19}, it follows that $\alpha_{1}$~is the total
energy corresponding to the osculating orbit, while $\beta_{1}$~will represent the moment
in which the system would pass some distinguished point in this orbit. If for instance
we consider the perturbations of a Keplerian motion, we may for~$\beta_{1}$ take
the so called time of perihelium\Note{[sic]} passage. When discussing the secular perturbations
of the shape and position of the orbit, we see therefore in the first place that
the variations of~$\beta_{1}$ may be left out of consideration. Further, it follows from the
principle of conservation of energy, that $\alpha_{1} + \Omega$ will remain constant during the
motion, and that consequently during the perturbations $\alpha_{1}$~will change only by
small quantities of the same order as~$\lambda\alpha_{1}$, where $\lambda$~denotes a small constant of the
same order of magnitude as the ratio between the external forces and the internal
forces of the system. Moreover, since the period~$\sigma$ of the undisturbed motion depends
on $\alpha_{1}$~only, it follows that the period of the osculating orbit will remain
constant during the perturbations, with neglect of small quantities of the same
order as~$\lambda\sigma$. On the other hand it follows from~\Eq{(44)} that, in a time interval of the
same order as~$\sigma/\lambda$, the constants $\alpha_{2}$,~\MyDots $\alpha_{s}$, $\beta_{2}$,~\MyDots $\beta_{s}$ will in general undergo variations
of the same order of magnitude as the values of these constants themselves.

As mentioned above, the total variations of the constants $\alpha_{2}$,~\MyDots $\alpha_{s}$, $\beta_{2}$,~\MyDots $\beta_{s}$,
which characterise the \Emph{secular perturbations of the shape and position of
the orbit}, may be obtained by taking mean values on both sides of the equations~\Eq{(44)}.
Introducing a function~$\psi$ of the~$\alpha$'s and~$\beta$'s, equal to the mean value of the
potential~$\Omega$ taken over a period~$\sigma$ of the motion of the undisturbed system and
defined by the formula
\[
\psi = \frac{1}{\sigma} \int_{t}^{t+\sigma} \Omega\, dt,
\Tag{(45)}
\]
it is easily seen, since $\sigma$~depends only on~$\alpha_{1}$, that the mean values of the partial
differential coefficients of~$\Omega$ with respect to $\alpha_{2}$,~\MyDots $\alpha_{s}$, $\beta_{2}$,~\MyDots $\beta_{s}$, taken over an approximate
period of the perturbed motion, may, if we look apart from small
quantities proportional to~$\lambda^{2}$, be replaced by the values of the corresponding partial
differential coefficients of~$\psi$ at some moment within this period. With the approximation
mentioned we get therefore
\[
\frac{D\alpha_{k}}{Dt} = -\frac{\dd \psi}{\dd \beta_{k}},\quad
\frac{D\beta_{k}}{Dt}  =  \frac{\dd \psi}{\dd \alpha_{k}},\qquad
(k = 2, \MyDots s)
\Tag{(46)}
\]
% [** TN: Printer's mark]
% D. K. D. Vidensk. Selsk. Skr., naturvidensk. og mathem. Afd., 8.~Række, IV.~1. 7
\PageSep{46}
where the differential symbols on the left sides are written to indicate mean values
of the rates of variation of the orbital constants during an approximate period of
the perturbed motion. From the definition of~$\psi$ it follows that this quantity in
general will depend on~$\alpha_{1}$ as well as on $\alpha_{2}$,~\MyDots $\alpha_{s}$, $\beta_{2}$,~\MyDots $\beta_{s}$, but that it will not
depend upon~$\beta_{1}$. From the above considerations it follows further that, with the
approximation in question, $\alpha_{1}$~may be considered as a constant in the expressions
on the right sides of~\Eq{(46)}, while for $\alpha_{2}$,~\MyDots $\alpha_{s}$, $\beta_{2}$,~\MyDots $\beta_{s}$ we may take a set of values
corresponding to some moment within the period to which the mean values on the
left sides refer.

It will be seen that the equations~\Eq{(46)} allow to follow the secular perturbations
during a time interval sufficiently long for the external forces to produce a considerable
change in the shape and position of the original orbit, if in the total
variations of the orbital constants $\alpha_{2}$,~\MyDots $\alpha_{s}$, $\beta_{2}$,~\MyDots $\beta_{s}$ we look apart from small
quantities of the same order as the small oscillations of these constants within a
single period. As a consequence of the secular variations, the orbit will pass through
a cycle of shapes and positions, which will depend on its original shape and position
and on the character of the perturbing field, but not on the intensity of this
field. In fact, as seen from~\Eq{(46)}, the variations in the shape and position of the orbit
will remain the same if $\psi$~is multiplied by a constant factor, which will only influence
the rate at which these variations are performed. It will further be observed
that the problem of determining the secular perturbations by means of~\Eq{(46)} consists
in solving a set of equations of the same type as the Hamiltonian equations
of motion for a system of $s - 1$~degrees of freedom. In these equations the quantity~$\psi$
plays formally the same part as the total energy in the usual mechanical problem,
and in analogy with the principle of conservation of energy it follows directly from~\Eq{(46)}
that, with neglect of small quantities proportional to~$\lambda^{2}$, \Emph{the value of~$\psi$ will
remain constant during the perturbations}, even if the external forces act
through a time interval of the same order as~$\sigma/\lambda$. In fact, with neglect of small
quantities proportional to~$\lambda^{2}$, we have
\[
\BreakMath{%
\frac{D\psi}{Dt}
  = \sum_{2}^{s} \left(\frac{\dd \psi}{\dd \alpha_{k}}\, \frac{D \alpha_{k}}{Dt}
                    + \frac{\dd \psi}{\dd \beta_{k}}\, \frac{D \beta_{k}}{Dt}\right)
  = \sum_{2}^{s} \left(-\frac{\dd \psi}{\dd \alpha_{k}}\, \frac{\dd \psi}{\dd \beta_{k}}
                     + \frac{\dd \psi}{\dd \beta_{k}}\, \frac{\dd \psi}{\dd \alpha_{k}}\right)
  = 0.
}{%
\begin{aligned}
\frac{D\psi}{Dt}
  &amp;= \sum_{2}^{s} \left(\frac{\dd \psi}{\dd \alpha_{k}}\, \frac{D \alpha_{k}}{Dt}
                     + \frac{\dd \psi}{\dd \beta_{k}}\, \frac{D \beta_{k}}{Dt}\right) \\
  &amp;= \sum_{2}^{s} \left(-\frac{\dd \psi}{\dd \alpha_{k}}\, \frac{\dd \psi}{\dd \beta_{k}}
                      + \frac{\dd \psi}{\dd \beta_{k}}\, \frac{\dd \psi}{\dd \alpha_{k}}\right)
  = 0.
\end{aligned}
}
\]

Since at any moment $\psi$~will differ only by small quantities proportional to~$\lambda^{2}$
from the mean value of the potential of the external forces taken over an approximate
period of the perturbed motion, it follows from the above that, with neglect
of small quantities of this order, also the mean value of the inner energy~$\alpha_{1}$ of the
perturbed system, taken over an approximate period, will remain constant during
the perturbations, even if the perturbing forces act through a time interval long
enough to produce a considerable change in the shape and position of the orbit.
In the special case, where the perturbed system allows of separation of variables,
this last result may be shown to follow directly from formula~\EqI{(28)} in Part~I\@.
Taking for the time interval~$\theta$ in this formula the period~$\sigma$ of the undisturbed
\PageSep{47}
motion, we get $N_{k} = \kappa_{k}$, where $\kappa_{1}$,~\MyDots $\kappa_{s}$ are the numbers entering in formula~\EqI{(23)}.
Comparing a given perturbed motion of the system with some undisturbed
motion of which it may be regarded as a small variation, we get therefore from~\EqI{(28)},
with neglect of small quantities proportional to the square of the intensity
of the external forces,
\[
\int_{0}^{\sigma} \delta E\, dt
  = \sum_{1}^{s} \kappa_{k}\, \delta I_{k},
\Tag{(47)}
\]
where the $I$'s are calculated with respect to a set of coordinates in which a separation
can be obtained for the perturbed motion, and where $\delta E$~is the difference
between the total energy of the undisturbed motion and the energy which the
system would possess in its perturbed state, if the external forces vanished suddenly
at the moment under consideration, and which in the above calculations was
denoted by~$\alpha_{1}$. Now the energy~$E$ of the undisturbed motion is determined completely
by the value of $I = \sum \kappa_{k} I_{k}$. If therefore the perturbed motion is all the
time compared with a neighbouring undisturbed motion of given constant energy,
it follows directly from~\Eq{(47)}, that, with neglect of small quantities of the same order
as the square of the external forces, the integral on the left side, taken over an
approximate period of the perturbed motion, will remain unaltered during the perturbations
through any time interval, however long.

Before\Pagelabel{47} proceeding with the applications of the equations~\Eq{(46)} which apply to
the case of a constant perturbing field, it will be necessary to consider \Emph{the effect
of a slow and uniform establishment of the external field}. Let us
assume that, within the interval $0 &lt; t &lt; \theta$ where $\theta$~denotes a quantity of the same
order as~$\sigma/\lambda$, the intensity of the external field increases uniformly from zero to the
value corresponding to the potential~$\Omega$. Since the variation in the perturbing field
during a single period will only be a small quantity of the same order as~$\lambda^{2}$, we
see in the first place that the secular variations of the constants $\alpha_{2}$,~\MyDots $\alpha_{s}$, $\beta_{2}$,~\MyDots $\beta_{s}$,
with the same approximation as for a constant field, will be given by a set of equations
of the same form as~\Eq{(46)}, with the only difference that $\psi$~is replaced by~$\dfrac{t}{\theta} \psi$.
Moreover it may be shown that in these equations the quantity~$\alpha_{1}$ may be considered
as constant, just as in the equations which hold for a constant perturbing
field. In fact the total variation in~$\alpha_{1}$ at any moment~$t$ will be equal to the total
work performed by the external forces since the beginning of the establishment
of the perturbing field, and will therefore be given by
\[
\Delta_{t} \alpha_{1}
  = -\int_{0}^{t} \frac{t}{\theta} \sum_{1}^{s} \frac{\dd \Omega}{\dd q_{k}}\, \dot{q}_{k}\, dt
  = \frac{1}{\theta} \int_{0}^{t} \Omega\, dt - \frac{t}{\theta} \Omega_{t},
\Tag{(48)}
\]
where the expression on the right side is obtained by partial integration; but, since both
terms in this expression are of the same order of magnitude as~$\lambda \alpha_{1}$, we see that the
\PageSep{48}
total variation in~$\alpha_{1}$ within the interval in question will, just as in case of a constant
perturbing field, be only a small quantity of this order. We get therefore
the result, that, for the same shape and position of the original orbit, the cycle of
shapes and positions passed through by the orbit during the increase of the external
field will be the same as that which would appear for a constant perturbing field,
and that, with neglect of small quantities proportional to~$\lambda^{2}$, the value of the
function~$\psi$ will consequently remain constant during the establishment of the field.
With this approximation we get therefore from~\Eq{(48)}, putting $t = \theta$,
\[
\Delta_{\theta} \alpha_{1} + \Omega_{\theta}
  = \frac{1}{\theta} \int_{0}^{\theta} \Omega\, dt
  = \psi,
\]
which shows that the change in the total energy of the system, due to the slow and
uniform establishment of the external field, is just equal to the value of the function~$\psi$,
and consequently equal to the mean value of the potential of the external
forces taken over an approximate period of the perturbed motion. This result may
also be expressed by stating, that, with neglect of small quantities proportional to the
square of the external forces, the mean value of the inner energy taken over an
approximate period of the perturbed motion will be equal to the energy possessed
by the system before the establishment of the perturbing field.

Returning now to the problem of the fixation of the stationary states of a
periodic system subject to the influence of a small external field of constant potential,
we shall base our considerations on the fundamental assumption that these states
are distinguished between the continuous multitude of mechanically possible states
by a relation between the additional energy of the system due to the presence of
the external field and the frequencies of the slow variations of the orbit produced
by this field, which is analogous to the relation discussed on \PageRef[page]{42} in the special case
in which the perturbed system allows of separation of variables in a fixed set of
coordinates. On this assumption we shall expect in the first place that, apart from
small quantities proportional to~$\lambda$, the cycles of shapes and positions of the orbit
belonging to the stationary states of the perturbed system will depend only on the
character of the external field, but not on its intensity. Since now, as shown above,
such a cycle will remain unaltered during a slow and uniform increase of the intensity
of the external field if the effect of the external forces is calculated by means
of ordinary mechanics, we are therefore, with reference to the principle of the
mechanical transformability of the stationary states, led to the conclusion that it is
possible by direct application of ordinary mechanics, not only to follow the secular
perturbations of the orbit in the stationary states corresponding to a constant external
field, but also to calculate the variation in the energy of the system in the
stationary states which results from a slow and uniform change in the intensity of
this field. If we denote the energy in the stationary states of the perturbed system
by $E_{n} + \frakE$, where $E_{n}$~is the value of the energy in the stationary state of the undisturbed
\PageSep{49}
system characterised by a given entire value of~$n$ in the condition $I = nh$,
we may therefore conclude from the above that \Emph{the additional energy~$\frakE$ in
the stationary states of the perturbed system will be equal to the
value in these states of the function~$\psi$ defined by~\Eq{(45)}, if we look
apart from small quantities proportional to the square of the intensity
of the external forces}. It will be seen that this result is equivalent
to the statement, that the mean value of the inner energy taken over an approximate
period of the perturbed motion will be equal to the value~$E_{n}$ of the energy in the
corresponding stationary state of the undisturbed system. In case of the perturbed
system allowing of separation of variables in a fixed set of coordinates, this result
may be simply shown to be a direct consequence of the fixation of the stationary
states by means of the conditions~\EqI{(22)}. In fact, if we assume that the undisturbed
motion, considered in~\Eq{(47)}, corresponds to some stationary state, satisfying~\EqI{(24)}
for a given value of~$n$, and that the perturbed motion is also stationary and satisfies~\EqI{(22)},
we see that the right side of~\Eq{(47)} will be zero, and we get the result that
the mean value of the inner energy in the stationary states of the system, with the
approximation mentioned, will not be altered in the presence of the external field.

Due to the above result that the additional energy~$\frakE$ in the stationary states
of the perturbed system, with neglect of small quantities proportional to~$\lambda^{2}$, may
be taken equal to the value in these states of the function~$\psi$ entering in the
equations~\Eq{(46)} which determine the secular perturbations of the orbits, we are
now able to draw further conclusions from the fact, mentioned above, that these
equations are of the same type as the Hamiltonian equations of motion for a
mechanical system of $s - 1$~degrees of freedom. In fact, we see that \Emph{the fixation
of the stationary states of the perturbed system is reduced to a problem
which is formally analogous to the fixation of these states for
a mechanical system of less degrees of freedom}. As it will appear from
the following applications this problem may, quite independent of the possibility of
separation of variables for the perturbed system, be treated directly on the basis of
the fundamental relation between energy and frequency in the stationary states of
periodic or conditionally periodic systems, discussed in Part~I, if only the solution
of the equations~\Eq{(46)} is of a periodic or conditionally periodic character. In this connection
it may once more be emphasised that these equations, according to the
manner in which they were deduced, allow to follow the secular perturbations only
through a time interval of the same order of magnitude as that sufficient for the
external forces to produce a finite alteration in the shape and position of the orbit.
With reference to the necessary stability of the stationary states of an atomic system,
it seems justified, however, to conclude that any possible small discrepancy between
the motion to be expected from a rigorous application of ordinary mechanics and that
determined by a calculation of the secular perturbations, based on the equations~\Eq{(46)}, cannot cause a material change in the character of the stationary states as
fixed by a consideration of the periodicity properties of these perturbations. On the
\PageSep{50}
other hand, from the point of view of the general formal relation between the
quantum theory and the ordinary theory of radiation, we must be prepared to find
that the motion and the energy in the stationary states of a perturbed periodic
system, for which we only know that the secular perturbations as determined by~\Eq{(46)}
are of conditionally periodic type, will not be as sharply defined as the motion
and the energy in the stationary states of a conditionally periodic system for which
the equations of motion allow of a rigorous solution by means of the method of
separation of variables. Thus, if we consider a large number of similar atomic
systems of the type in question, we may be prepared to find that the values of the
additional energy in a given stationary state will for the different systems deviate
from each other by small quantities; but it must be expected that the values of
the additional energy for the large majority of systems will differ from the value
of~$\psi$, as determined by the method indicated above, only by small quantities proportional
to~$\lambda^{2}$, and that only for a small fraction (at most of the same order as~$\lambda^{2}$)
of the systems the values of the additional energy will show deviations from
this value of~$\psi$, which are of the same order as~$\lambda$.

\medskip
As\Pagelabel{50} to the application of the preceding considerations to special problems, it will
be seen in the first place that in case of a \Emph{perturbed periodic system possessing
two degrees of freedom}, as for instance that considered in the example
on \PageRef[page]{43}, the problem of the fixation of the stationary states of the perturbed
system in the presence of a small external field allows of a general solution on
the basis of the method developed above, because in this case the secular perturbations
will in general be \Emph{simply periodic}. In fact, in this case the shape and
position of the orbit are characterised by two constants $\alpha_{2}$ and~$\beta_{2}$, and from the
equations~\Eq{(46)}, which will be analogous to the equations of motion of a system of
one degree of freedom, it follows directly that during the perturbations $\alpha_{2}$~will be
a function of~$\beta_{2}$ and that in general these quantities will be periodic functions of
the time with a period~$\fraks$ which, besides on~$\alpha_{1}$, will depend on the value of $\psi$~only.
Considering two slightly different states of the perturbed system for which the
corresponding states of the undisturbed system (\ie\ the states which would appear
if the external forces vanished at a slow and uniform rate) possess the same energy
and consequently the same value for the quantity~$I$ defined by~\EqI{(5)}, we get therefore
by a calculation completely analogous to that leading to relation~\EqI{(8)} in Part~I, which
was deduced directly from the Hamiltonian equations, for the difference in the
values of the function~$\psi$ for these two states
\[
\delta \psi = \frakv\, \delta \frakI,
\Tag{(49)}
\]
where $\frakv = \dfrac{1}{\fraks}$ is the frequency of the secular perturbations, and where the quantity~$\frakI$
is defined by
\[
\frakI = \int_{0}^{\fraks} \alpha_{2}\, \frac{D\beta_{2}}{Dt}\, dt
  = \int \alpha_{2}\, D\beta_{2},
\Tag{(50)}
\]
\PageSep{51}
where the latter integral is taken over a complete oscillation of~$\beta_{2}$. In order to fix
the stationary states, it will now be seen in the first place that, among the multitude
of states of the perturbed system for which the value of~$I$ in the corresponding
states of the undisturbed system is equal to~$nh$ where $n$~is a given positive integer,
the state for which $\frakI = 0$ must beforehand be expected to be a stationary state. In
fact, for this value of~$\frakI$, the shape and position of the orbit will not undergo secular perturbations
but will remain unaltered for a constant external field as well as during a slow
and uniform establishment of this field. In contrast to what in general will take place
during a slow establishment of the external field, we may therefore expect that, for
this special shape and position of the orbit, a direct application of ordinary mechanics
will be legitimate in calculating the effect of the establishment of the field, since
there will in this case obviously be nothing to cause the coming into play of some
non-mechanical process, connected with the mechanism of a transition between
two stationary states accompanied by the emission or absorption of a radiation of
small frequency. With reference to relation~\Eq{(49)} we see therefore that, by fixing the
stationary states of the perturbed system by means of the condition
\[
\frakI = \frakn h,
\Tag{(51)}
\]
where $\frakn$~is an entire number, we obtain a relation between the additional energy
$\frakE = \psi$ of the system in the presence of the field and the frequency~$\frakv$ of the secular
perturbations, which is exactly of the same type as that which holds between the energy
and frequency in the stationary states of a system of one degree of freedom, and
which is expressed by \EqI{(8)}~and~\EqI{(10)}. By means of~\Eq{(51)} it is possible, with neglect of
small quantities proportional to the square of the perturbing forces, directly to
determine the value of the additional energy in the stationary states of a periodic
system of two degrees of freedom subject to an arbitrarily given small external field
of force, and consequently with this approximation, by use of the fundamental relation~\EqI{(1)},
to determine the effect of this field on the frequencies of the spectrum of the
undisturbed periodic system. In general this effect will consist in a splitting up of
each of the spectral lines into a number of components which are displaced from
the original position of the line by small quantities proportional to the intensity of
the external forces.

When we pass to \Emph{perturbed periodic systems of more than two degrees
of freedom}, the general problem is more complex. For a given external field,
however, it may be possible to choose a set of orbital constants $\alpha_{2}$,~\MyDots $\alpha_{s}$,
$\beta_{2}$,~\MyDots $\beta_{s}$ in such a way, that during the motion every of the~$\beta$'s will depend on
the corresponding $\beta$ only, while every of the~$\beta$'s will oscillate between two fixed
limits. From analogy with the theory of ordinary conditionally periodic systems
which allow of separation of variables, the perturbations may in such a case be
said to be \Emph{conditionally periodic}, and, from a calculation quite analogous to
that leading to equation~\EqI{(29)} in Part~I which is based entirely on the use of the
Hamiltonian equations, we get for the difference in~$\psi$ for two slightly different states
\PageSep{52}
of the perturbed system, for which the value of~$I$ in the corresponding states of the
undisturbed system is the same,
\[
\delta \psi = \sum_{1}^{s-1} \frakv_{k}\, \delta \frakI_{k},
\Tag{(52)}
\]
where $\frakv_{k}$~is the mean frequency of oscillation of~$\beta_{k+1}$ between its limits, and where
the quantities~$\frakI_{k}$ are defined by\Pagelabel{52}
\[
\frakI_{k} = \int \alpha_{k+1}\, D\beta_{k+1},\qquad
(k = 1, \MyDots s-1)
\Tag{(53)}
\]
where the integral is taken over a complete oscillation of~$\beta_{k+1}$. In analogy with
the expression~\EqI{(31)} for the displacements of the particles of an ordinary conditionally
periodic system which allows of separation of variables, we get further in the
present case that every of the $\alpha$'s and~$\beta$'s may be expressed as a function of the
time by a sum of harmonic vibrations of small frequencies
\[
\BreakMath{%
\left.
\begin{gathered}
\alpha \\
\beta
\end{gathered}
\right\}
  = \sum \frakC_{\frakt_{1}, \MyDots \frakt_{s-1}} \cos 2\pi \bigl\{(\frakt_{1} \frakv_{1} + \dots \Add{+} \frakt_{s-1} \frakv_{s-1})t + \frakc_{\frakt_{1}, \MyDots \frakt_{s-1}}\bigr\},
}{%
\begin{split}
\MathBack
\left.
\begin{gathered}
\alpha \\
\beta
\end{gathered}
\right\}
  = \sum \frakC_{\frakt_{1}, \MyDots \frakt_{s-1}} \cos 2\pi \bigl\{(\frakt_{1} \frakv_{1} + \dots \Add{+} \frakt_{s-1} \frakv_{s-1})t \\
  + \frakc_{\frakt_{1}, \MyDots \frakt_{s-1}}\bigr\},
\end{split}
}
\Tag{(54)}
\]
where the $\frakC$'s~and $\frakc$'s are constants, the former of which, besides on~$I$, depend on
the $\frakI$'s~only, and where the summation is to be extended over all positive and
negative entire values of the~$\frakt$'s. If therefore the secular perturbations are conditionally
periodic, we may conclude that the stationary states of the perturbed
system, corresponding to a given stationary state of the undisturbed system, will be
characterised by the $s - 1$~conditions
\[
\frakI_{k} = \frakn_{k} h,\qquad
(k = 1, \MyDots s - l)
\Tag{(55)}
\]
where $\frakn_{1}$,~\MyDots $\frakn_{s-1}$ form a set of entire numbers. In fact, as seen from~\Eq{(52)}, we obtain
in this way a relation between the additional energy and the frequencies of the
secular perturbations of exactly the same type as that holding for the energy and
frequencies of ordinary conditionally periodic systems and expressed by \EqI{(22)}~and
\EqI{(29)}; moreover we may conclude beforehand that the state in which every of the
quantities~$\frakI_{k}$, defined by~\Eq{(53)}, is equal to zero must belong to the stationary states of the
perturbed system, because in this case the orbit will not undergo secular perturbations
for a constant external field, nor during a slow and uniform establishment of this
field. Since the conditions~\Eq{(55)}, with neglect of small quantities proportional to the
square of the intensities of the external forces, allow to determine the additional
energy of the system due to the presence of the external field, we see therefore that
the effect of this field on the spectrum of the undisturbed system, if the secular
perturbations are conditionally periodic, will consist in a splitting up of each spectral
line in a number of components, in analogy with the effect of a perturbing
field on the spectrum of a periodic system of two degrees of freedom. In general,
however, the perturbations, which a periodic system of more than two degrees of
\PageSep{53}
freedom undergoes in the presence of a given external field, cannot be expected
to be conditionally periodic and to exhibit periodicity properties of the type expressed
by formula~\Eq{(54)}. In such cases it seems impossible to define stationary
states in a way which leads to a complete fixation of the total energy in these
states, and we are therefore led to the conclusion, that the effect of the external
field on the spectrum will not consist in the splitting up of the spectral lines of the
original system into a number of sharp components, but in a diffusion of these
lines over spectral intervals of a width proportional to the intensity of the external
forces.

In special cases in which the secular perturbations of a perturbed periodic
system of more than two degrees of freedom are of conditionally periodic type, it
may occur that these perturbations are characterised by a number of fundamental
frequencies, which is less than~$s - 1$. In such cases, in which the perturbed periodic
system from analogy with the terminology used in Part~I may be said to be
\Emph{degenerate}, the necessary relation between the additional energy and the frequencies
of the secular perturbations is secured by a number of conditions less than that
given by~\Eq{(55)}, and the stationary states are consequently characterised by a number
of conditions less than~$s$. With a typical example of such systems we meet if, for
a perturbed periodic system of more than two degrees of freedom, the secular perturbations
are \Emph{simply periodic} independent of the initial shape and position of
the orbit. In direct analogy to what holds for perturbed periodic systems of two
degrees of freedom, the difference between the values of~$\psi$ in two slightly different
states of the perturbed system, corresponding to the same value of~$I$, will in the
present case be given by
\[
\delta \psi = \frakv\, \delta \frakI,
\Tag{(56)}
\]
where $\frakv$~is the frequency of the secular perturbations, and where $\frakI$~is defined by
\[
\frakI = \int_{0}^{\frakv} \sum_{2}^{s} \alpha_{k} \frac{D\beta_{k}}{Dt}\, dt,
\Tag{(57)}
\]
where $\fraks = 1/\frakv$ is the period of the perturbations. We may therefore conclude that
the stationary states of the perturbed system, corresponding to a given stationary
state of the undisturbed system, will be characterised by the single condition
\[
\frakI = \frakn h,
\Tag{(58)}
\]
in which $\frakn$~is an entire number, and which will be seen to be completely analogous
to the condition which fixes the stationary states of ordinary periodic systems of
several degrees of freedom.

\medskip
In the following sections we shall apply the preceding considerations to the
problem of the fixation of \Emph{the stationary states of the hydrogen atom},
when the relativity modifications are taken into account, and when the atom is exposed
to small external fields. In this discussion we shall for the sake of simplicity
% [** TN: Printer's mark]
% D. K. D. Vidensk. Selsk. Skr., naturvidensk. og mathem. Afd., 8. Række, IV. 1. 8
\PageSep{54}
consider the mass of the nucleus as infinite in the calculations of the perturbations
of the orbit of the electron.\Pagelabel{54} This involves, in the expression for the additional
energy of the system, the neglect of small terms of the same order as the product
of the intensity of the external forces with the ratio between the mass of the electron
and the mass of the nucleus, but due to the smallness of the latter ratio
the error introduced by this simplification will be of no importance in the comparison
of the results with the measurements. Since in the case under consideration
the system possesses three degrees of freedom, the equations which determine the
secular perturbations of the orbit of the electron will correspond to the equations
of motion of a system of two degrees of freedom, and it will therefore not be possible
to give a general treatment of the problem of the stationary states. Thus, for
any given external field, we meet with the question whether the perturbations are
conditionally periodic and, if so, in what set of orbital constants this periodicity
may be conveniently expressed. Now, in many spectral problems, the external field
possesses \Emph{axial symmetry round an axis through the nucleus}, and in this
case it is easily shown that the problem of the fixation of the stationary states
allows of a general solution. A choice of orbital constants which is suitable for the
discussion of this problem, and which is well known from the astronomical theory
of planetary perturbations, is obtained by choosing for~$\alpha_{2}$ the total angular momentum
of the electron round the nucleus and for~$\alpha_{3}$ the component of this angular momentum
round the axis of the field. For the set of~$\beta$'s, corresponding to this set of~$\alpha$'s,
we may take $\beta_{2}$~equal to the angle, which the major axis makes with the line
in which the plane of the orbit cuts the plane through the nucleus perpendicular
to the axis of the field, and $\beta_{3}$~equal to the angle between this line and a fixed
direction in the latter plane. For the problem under consideration it will be seen
that, with this choice of constants, the mean value~$\psi$ of the potential of the perturbing
field will, besides on~$\alpha_{1}$, generally depend on $\alpha_{2}$~and~$\beta_{2}$ as well as on~$\alpha_{3}$,
but due to the symmetry round the axis it will obviously not depend on~$\beta_{3}$. In
consequence of this, the equations~\Eq{(46)}, which determine the secular perturbations,
will possess the same form as the Hamiltonian equations of motion for a particle
moving in a plane and subject to a central field of force. Thus corresponding to the
conservation of angular momentum for central systems, we get in the first place
from~\Eq{(46)} that $\alpha_{3}$~will remain unaltered during the perturbations. Next corresponding
to the simple periodicity of the radial motion in central systems, we see from~\Eq{(46)},
if $\alpha_{3}$ as well as~$\alpha_{1}$ is considered as a constant, that during the perturbations $\alpha_{2}$~will
be a function of~$\beta_{2}$ and vary in a simple periodic way with the time. The perturbations
of the orbit of the electron produced by an external field which possesses
axial symmetry will therefore always be of \Emph{conditionally periodic} type,
quite independent of the possibility of separation of variables for the perturbed
system. As regards the form of the conditions which fix the stationary states, it may
be noted, however, that with the choice of orbital constants under consideration
the~$\beta$'s will not, as it was assumed for the sake of simplicity in the general discussion
\PageSep{55}
on \PageRef[page]{52}, oscillate between fixed limits, but it will be seen that $\beta_{2}$~during
the perturbations may either oscillate between two such limits or increase (or decrease)
continuously, while $\beta_{3}$~will always vary in the latter manner. This constitutes,
however, only a formal difficulty of the same kind as that mentioned in
Part~I in connection with the discussion of the conditions~\EqI{(16)}, which fix the
stationary states of a system consisting of a particle moving in a central field of
force. Thus from a simple consideration it will be seen that, in complete analogy
to the relations \Eq{(52)} and~\Eq{(53)}, we get in the present case for the difference between
the energy of two slightly different states of the perturbed system, which correspond
to the same value of~$I$,
\[
\delta \psi = \frakv_{1}\, \delta \frakI_{1} + \frakv_{2}\, \delta I_{2},
\Tag{(59)}
\]
where $\frakv_{1}$~is the frequency with which the shape of the orbit and its position relative
to the axis of the field repeats itself at regular intervals and which is characterised
by the variation of $\alpha_{2}$~and~$\beta_{2}$, while $\frakv_{2}$~is the mean frequency of rotation of
the plane of the orbit round this axis characterised by the variation of~$\beta_{3}$, and
where $\frakI_{1}$~and~$\frakI_{2}$ are defined by the equations
\[
\frakI_{1} = \int \alpha_{2}\, D\beta_{2},\quad
\frakI_{2} = \int_{0}^{2\pi} \alpha_{3}\, D\beta_{3} = 2\pi \alpha_{3}.
\Tag{(60)}
\]
In case $\beta_{2}$~varies in an oscillating manner with the time, the first integral must be
taken over a complete oscillation of this orbital constant, while, if $\beta_{2}$~during the
perturbations increases or decreases continuously, the integral in the expression for~$\frakI_{1}$
must be taken over an interval of~$2\pi$, just as the integral in the expression
for~$\frakI_{2}$. By fixing the stationary states of the perturbed system by means of the two\Pagelabel{55}
conditions\footnote
  {Quite apart from the problem of perturbed periodic systems, the second of these conditions
  would also follow directly from certain interesting considerations of \Name{Epstein} (Ber.\ d.\ D. Phys.\ Ges.~XIX,
  p.~116 (1917)) about the stationary states of systems which allow of what may be called ``partial separation
  of variables''. In this case it is possible to choose a set of positional coordinates $q_{1}$,~\MyDots $q_{s}$ in such
  a way that, for some of the coordinates, the conjugated momenta may be considered as functions of
  the corresponding $q$'s~only, so that, for these coordinates, quantities~$I$ may be defined by~\EqI{(21)} in the
  same way as for systems for which a complete separation of variables can be obtained. From analogy
  with the theory of the stationary states of the latter systems, \Name{Epstein} proposes therefore the assumption,
  that some of the conditions to be fulfilled in the stationary states of the systems in question may
  be obtained by putting the $I$'s~thus defined equal to entire multipla of~$h$. It will be seen that, in case
  of systems possessing an axis of symmetry, this leads to the second of the conditions~\Eq{(61)}, which expresses
  the condition that in the stationary states the total angular momentum round the axis must
  be equal to an entire multiple of~$h/2\pi$. As pointed out in Part~I on \PageRefI[page]{34}, this condition would also
  seem to obtain an independent support from considerations of conservation of angular momentum
  during a transition between two stationary states.}
\[
\frakI_{1} = \frakn_{1} h,\quad
\frakI_{2} = \frakn_{2} h,
\Tag{(61)}
\]
where $\frakn_{1}$~and~$\frakn_{2}$ are entire numbers, it will therefore be seen that we obtain the right
relation between the additional energy $\frakE = \psi$ of the perturbed atom and the frequencies
\PageSep{56}
of the secular perturbations of the orbit of the electron. It will moreover be
seen that a state in which the electron moves in a circular orbit perpendicular to
the axis of the field, and which beforehand must be expected to belong to the
stationary states of the perturbed atom since this orbit will not undergo secular
perturbations during a uniform establishment of the external field, will be included
among the states determined by~\Eq{(61)}. In fact, if $n$~is the number which characterises
the corresponding stationary state of the undisturbed system, this state of the
perturbed system will correspond to $\frakn_{1} = 0$, $\frakn_{2} = n$ or to $\frakn_{1} = n$, $\frakn_{2} = n$, according
to whether $\beta_{2}$~during the perturbations oscillates between fixed limits, or increases
(or decreases) continuously. As regards the application of the conditions~\Eq{(61)} it is
of importance to point out that, from considerations of the invariance of the a-priori
probability of the stationary states of an atomic system during continuous transformations
of the external conditions (see Part~I, \PageRefI[page]{9} and~\PageRefI[]{27}), it seems necessary
to conclude that no stationary state exists corresponding to $\frakn_{2} = 0$. For this value
of~$\frakn_{2}$ the motion of the electron would take place in a plane through the axis,
but for certain external fields such motions cannot be regarded as physically
realisable stationary states of the atom, since in the course of the perturbations the
electron would collide with the nucleus (compare \PageRef[page]{68}).

A special case of an external field possessing axial symmetry, in which the
secular perturbations are very simple, presents itself if the external forces form
\Emph{a central field with the nucleus at the centre}. In this case the solution
of the problem of the fixation of the stationary states is given by \Name{Sommerfeld}'s
general theory of central systems, discussed \Typo{i}{in} Part~I, which rests upon the fact that
these systems allow of separation of variables in polar coordinates. In connection
with the above considerations it may be of interest, however, to consider the problem
in question directly from the point of view of perturbed periodic systems, because
it presents a characteristic example of a \Emph{degenerate} perturbed system. In the present
case $\psi$~will, besides on~$\alpha_{1}$, depend on $\alpha_{2}$~only, and from the equations~\Eq{(46)} we get
therefore the well known result, that the angular momentum of the electron and the
plane of its orbit will not vary during the perturbations, and that the only secular
effect of the perturbing field will consist in a slow uniform rotation of the direction
of the major axis. For the frequency of this rotation we get from~\Eq{(46)}
\[
\frakv = \frac{1}{2\pi}\, \frac{D\beta_{2}}{Dt}
  = \frac{1}{2\pi}\, \frac{\delta\psi}{\delta\alpha_{2}},
\Tag{(62)}
\]
from which we get directly for the difference between the values of~$\psi$ for two neighbouring
states of the perturbed system, for which the corresponding value of~$I$ is the
same,
\[
\delta\psi = 2\pi \frakv\, \delta\alpha_{2}.
\Tag{(63)}
\]
This relation, which corresponds to~\Eq{(56)}, is seen to coincide with~\Eq{(59)}, since in the
present case $\frakv_{2} = 0$ and $\frakI_{1} = 2\pi\alpha_{2}$. From~\Eq{(63)} it follows that the necessary relation
between the additional energy of the atom and the frequency of the perturbations
\PageSep{57}
is secured if the stationary states in the presence of a small external central
field are characterised by the condition\Pagelabel{57}
\[
\frakI = 2\pi \alpha_{2} = \frakn h,
\Tag{(64)}
\]
where $\frakn$~is an entire number. This condition, which is equivalent with the second
of \Name{Sommerfeld}'s conditions~\EqI{(16)}, corresponds to~\Eq{(58)} and is seen to coincide
with the first of the conditions~\Eq{(61)}, while the second of the latter conditions in the
special case under consideration \Typo{looses}{loses} its validity corresponding to the fact that
the orientation of the plane of the orbit in space is obviously arbitrary. Since, for
a Keplerian motion, the major axis of the orbit depends on the total energy only
while the minor axis is proportional to the angular momentum, it will be seen from~\Eq{(64)}
that the presence of a small external field imposes the restriction on the motion
of the atom in the stationary states, that the minor axis of the orbit of the electron
must be equal to an entire multiple of the \Ord{$n$}{th}~part of the major axis, which was
given by~$2\alpha_{n}$ in~\Eq{(41)}. This result has been pointed out by \Name{Sommerfeld} as a consequence
of the application of the conditions~\EqI{(16)}.

\medskip
In the preceding it has been shown how it is possible to attack the problem
of the stationary states of a perturbed periodic system by an examination of the
secular perturbations of the shape and position of the orbit, and to fix these states
if the perturbations are of periodic or conditionally periodic type. While these considerations
allow to determine the possible values for the total energy of the perturbed
system and thereby the \Emph{frequencies} of the components into which the
lines of the spectrum of the undisturbed system are split up in the presence of the
external field, it is necessary, however, for the discussion of the \Emph{intensities} and
\Emph{polarisations} of these components to consider more closely the motion of the
particles in the perturbed system and the relation of the total energy of this system
to the fundamental frequencies which characterise the motion. In the first place it
will be seen that, if the secular perturbations as determined by the equations~\Eq{(46)}
are of conditionally periodic type, the displacements of the particles of the system
in any given direction may, with neglect of small quantities proportional to the
intensity of the external forces, be represented, within a time interval sufficiently
large for these forces to produce a considerable change in the shape and position
of the orbit, as a sum of harmonic vibrations by expressions of the type:
\[
\BreakMath{%
\xi = \sum C_{\tau, \frakt_{1}, \MyDots \frakt_{s-1}} \cos 2\pi \bigl\{(\tau\omega_{P} + \frakt_{1} \frakv_{1} + \dots \Add{+} \frakt_{s-1} \frakv_{s-1})t
  + c_{\tau, \frakt_{1}, \MyDots \frakt_{s-1}}\bigr\},
}{%
\MathBack
\begin{split}
\xi = \sum C_{\tau, \frakt_{1}, \MyDots \frakt_{s-1}} \cos 2\pi \bigl\{(\tau\omega_{P} + \frakt_{1} \frakv_{1} + \cdots \\
  {} \Add{+} \frakt_{s-1} \frakv_{s-1})t
  + c_{\tau, \frakt_{1}, \MyDots \frakt_{s-1}}\bigr\},
\end{split}
}
\Tag{(65)}
\]
where the summation is to be extended over all positive and negative entire values
of $\tau$,~$\frakt_{1}$,~\MyDots $\frakt_{s-1}$, and where the $C$'s~and~$c$'s are two sets of constants, the former of
which depend only on the values of the quantities $\frakI_{1}$,~\MyDots $\frakI_{s-1}$ defined by~\Eq{(53)} and
on the value of the quantity~$I$, which characterises the corresponding state of the
undisturbed system which would appear if the external field vanished at a slow and
uniform rate. While the quantities $\frakv_{1}$,~\MyDots $\frakv_{s-1}$ are the same as those which appear
\PageSep{58}
in the formula~\Eq{(54)}, and represent the small frequencies of the secular perturbations
of the shape and position of the orbit, the quantity~$\omega_{P}$ may be considered as
representing the mean frequency of revolution of the particles in their approximately
periodic orbit. As regards the total energy of the perturbed system, it may next be
proved that, looking apart from small quantities proportional to the square of the
intensity of the external forces, the difference in the total energy in two slightly
different states of the perturbed system, for which the values of $I$, $\frakI_{1}$,~\MyDots $I_{s-1}$ differ
by $\delta I$, $\delta \frakI_{1}$,~\MyDots $\delta \frakI_{s-1}$ respectively, is given by the relation\Typo{.}{}\Pagelabel{58}\footnotemark
\[
\delta E = \omega_{P}\, \delta I
  + \sum_{1}^{s-1} \frakv_{k}\, \delta \frakI_{k},
\Tag{(66)}
\]
\footnotetext{From a comparison with formula~\EqI{(8)}, holding for the energy difference between two neighbouring
  states of the undisturbed system, and with formula~\Eq{(52)}, it will be seen that \Eq{(66)}~implies the condition
  $\omega_{P} = \omega + \dd\psi/\dd I$, where $\omega$~is the frequency of revolution in the corresponding state of the undisturbed
  system characterised by the given value of~$I$, and where, in the partial differential coefficient, $\psi$~is considered
  as a function of~$I$ and $\frakI_{1}$,~\MyDots $\frakI_{s-1}$. This relation can be verified by means of a consideration
  based on the perturbation equations~\Eq{(44)}, which takes into account the simple relation between $\alpha_{1}$~and~$I$
  for the undisturbed system, as well as the relation between the mean rate of variation of~$\beta_{1}$ with
  the time and the difference between $\omega_{P}$~and~$\omega$. We shall not enter, however, on the details of the
  rather intricate calculations involved in such a consideration, since the problems in question allow of
  a more elegant treatment by means of another analytical method. Thus it will be shown by Mr.~\Name{H.~A.
  Kramers}, in the paper mentioned in the end of~\SecRef{4}, that, quite independent of the possibility of separation
  of variables for the perturbed system in a fixed set of positional coordinates, the theory of secular
  perturbations exposed in this section offers---if these perturbations as determined by~\Eq{(46)} are of conditionally
  periodic type---a means of disclosing a set of \Emph{angle variables}, which may be used to
  describe the motion of the perturbed system with the same degree of approximation as that involved
  in the preceding calculations. According to the definition of angle variables, mentioned in the Note on
  \PageRefI[page]{29} in Part~I, this means that it is possible, in stead of the positional coordinates $q_{1}$,~\MyDots $q_{s}$ of the
  perturbed system and their conjugated momenta $p_{1}$,~\MyDots $p_{s}$ to introduce a new set of $s$~variables in
  such a way, that the $q$'s~and~$p$'s are periodic in every of the new variables with period~$1$, when they are
  considered as functions of these variables and of their canonically conjugated momenta. These momenta
  will just coincide with the quantities denoted above by $I$, $\frakI_{1}$,~\MyDots $\frakI_{s-1}$, and the corresponding angle
  variables may conveniently be denoted by $w$, $\frakw_{1}$,~\MyDots $\frakw_{s-1}$ respectively. Introducing the new variables,
  the total energy of the perturbed system will be a function of~$I$, $\frakI_{1}$,~\MyDots $\frakI_{s-1}$ only, if we look apart
  from small quantities proportional to~$\lambda^{2}$. With this approximation we get consequently by a calculation,
  analogous to that given in the Note referred to, that the angle variables $w$, $\frakw_{1}$,~\MyDots $\frakw_{s-1}$ may be represented
  as linear functions of the time within an interval of the same order as~$\sigma/\lambda$. Denoting the rates
  of variation of $w$, $\frakw_{1}$,~\MyDots $\frakw_{s-1}$ by $\omega$, $\frakv_{1}$,~\MyDots $\frakv_{s-1}$ respectively, the \Chg{formulae}{formulæ} \Eq{(65)} and \Eq{(66)} are therefore
  directly obtained, just as the corresponding \Chg{formulae}{formulæ} \EqI{(31)} and \EqI{(29)} in Part~I\@. In this connection it
  will be observed that, due to the possibility of introduction of angle variables, the conditions~\Eq{(67)} appear
  in the same form as that in which the conditions, which fix the stationary states of ordinary conditionally
  periodic systems which allow of separation of variables, have been formulated by \Name{Schwarzschild},
  and which, as mentioned in the Note in Part~I, has already been applied by \Name{Burgers} to certain systems
  for which such a separation cannot be obtained.}
which coincides with~\Eq{(52)} if $\delta \frakI = 0$, and which will be seen to be completely
analogous with formula~\EqI{(29)} in Part~I, holding for an ordinary conditionally periodic
system which allows of separation of variables in a fixed set of positional coordinates;
just as \Eq{(65)}~is analogous to formula~\EqI{(31)} representing the displacements of the particles
\PageSep{59}
for such a system. Since moreover, in complete analogy to the conditions~\EqI{(22)},
the stationary states of the perturbed system are characterised by\Pagelabel{59}
\[
I = nh,\quad
\frakI_{k} = \frakn_{k} h,\qquad
(k = 1, \MyDots s - 1)
\Tag{(67)}
\]
we see consequently that, for sufficiently small intensity of the external forces, we
obtain in the region of large values of~$n$ and of the~$\frakn$'s a connection between the
frequencies of the components of the spectral lines, determined on the quantum theory
by means of relation~\EqI{(1)}, and those to be expected on ordinary electrodynamics,
which is of exactly the same type as the analogous connection, discussed in Part~I,
in case of ordinary conditionally periodic systems which allow of separation of
variables. In perfect analogy with the general considerations in Part~I, we are
therefore led directly to certain simple conclusions as regards the intensities and
polarisations of the components into which the lines of the spectrum of the undisturbed
periodic system are split up in the presence of the external field. Thus we
shall expect that there will exist an intimate connection between the probability of
spontaneous transition between two stationary states of the perturbed system, for
which $n = n'$, $\frakn_{k} = \frakn_{k}'$ and $n = n''$, $\frakn_{k} = \frakn_{k}''$ respectively, and the values in these
states of the coefficient $C_{\tau, \frakt_{1}, \MyDots \frakt_{s-1}}$ in the expressions for the displacements of the
particles, for which $\tau = n' - n''$ and $\frakt_{k} = \frakn_{k}' - \frakn_{k}''$. If for instance, for a certain set
of values of~$\tau$ and $\frakt_{1}$,~\MyDots $\frakt_{s-1}$, the coefficient $C_{\tau, \frakt_{1}, \MyDots \frakt_{s-1}}$ in the expressions for the
displacements in every direction will be equal to zero for all motions of the perturbed
system, we shall expect that the corresponding transitions between two
stationary states will be impossible in the presence of the given external field; and
if this coefficient is zero for the displacements of the particles in a certain direction
only, we shall expect that the corresponding transitions will give rise to the
emission of a radiation which is polarised in a plane perpendicular to this direction.

With a characteristic example of these considerations we meet in the case of
the spectrum of a hydrogen atom exposed to an external field of force which possesses
axial symmetry round an axis through the nucleus. In analogy with the
resolution of the motion of an ordinary conditionally periodic system which possesses
an axis of symmetry in its constituent harmonic vibrations, discussed in
Part~I on \PageRefI[page]{33}, it follows from the discussion of the general character of the
secular perturbations on \PageRef[page]{54} that the motion of the electron in the perturbed atom
in this case can be resolved in a number of linear harmonic vibrations parallel to
the axis with frequencies $\tau\omega_{P} + \frakt_{1} \frakv_{1}$ and in a number of circular harmonic rotations
perpendicular to the axis with frequencies $\tau\omega_{P} + \frakt_{1}\frakv_{1} + \frakv_{2}$. In complete analogy with
the considerations in Part~I, we are therefore led to conclude that in the present
case only two types of transitions between the stationary states of the perturbed
atom are possible. In the transitions of the first type $\frakn_{2}$~will remain unaltered and
the emitted radiation will give rise to components of the hydrogen lines which will
show linear polarisation parallel to the axis. In the transitions of the second type
$\frakn_{2}$~will change by one unit and the emitted radiation will show circular polarisation
when viewed in the direction of the axis. Remembering that, according to
\PageSep{60}
the conditions~\Eq{(61)}, the angular momentum of the system round the axis in the
stationary states is equal to~$\frakn_{2}\, \dfrac{h}{2\pi}$, it will be seen moreover that, also in the present
case, these conclusions obtain an independent support from a consideration of conservation
%[** TN: Footnote mark before punctuation in the original.]
of angular momentum during the transitions (Compare Part~I \PageRefI[page]{34}).\footnote
  {\Emph{Note added during the proof.} In an interesting paper by \Name{A.~Rubinowicz} (Phys.\ Zeitschr.\
  XIX, p.~441 and p.~465 (1918)) which has just been published, a similar consideration of conservation
  of angular momentum has been used to draw conclusions, as regards the possibility of transitions
  between the stationary states of a conditionally periodic system possessing an axis of symmetry, and
  as regards the character of the polarisation of the radiation accompanying these transitions. In this way
  \Name{Rubinowicz} has arrived at several of the results discussed in the present paper; in this connection,
  however, it may be remarked that, from a consideration of conservation of angular momentum, it is
  not possible, even for systems possessing axial symmetry, to obtain as complete information, as regards
  the number and polarisation of the possible components, as from a consideration based on the resolution
  of the motion of the electron in harmonic vibrations.}
In the following we will meet with applications of these considerations when discussing
the effect of electric and magnetic fields on the hydrogen lines. In the latter
case, however, the preceding considerations need some modifications due to the
fact, that the external forces acting on the electron cannot be derived from a
potential expressed as a function of its positional coordinates; to this point we shall
come back in~\SecRef{5}.

Before\Pagelabel{60} leaving the general theory of perturbed periodic systems we shall still
consider the problem of the effect on the spectrum of a periodic system, undergoing
secular perturbations of conditionally periodic type under the influence of a
given small external field, if this system is further subject to \Emph{the influence of
a second external field which is small compared with the first field},
but the perturbing effect of which is yet large compared with the small effects
on the motion, proportional to the square of the intensity of the first perturbing
field, which were neglected in the preceding calculations. This problem is
closely analogous to the problem, briefly discussed in Part~I, of the effect of a
small perturbing field on the spectrum of an ordinary conditionally periodic
system which allows of separation of variables. As mentioned on \PageRefI[page]{34}, we have
in this case, quite independent of the possibility of separation of variables for the
perturbed system, that in general the motion under the influence of the external
field may still be represented as a sum of harmonic vibrations by a formula of
the type~\EqI{(31)}, if we look apart from small terms proportional to the square of the
perturbing forces. Corresponding to this we have in the case under consideration
that, independent of the nature of the second external field, the resultant secular
perturbations may in general be expressed as a sum of harmonic vibrations of small
frequencies of the type~\Eq{(54)}, if we look apart from small terms of the same order
as the product of the secular perturbations produced by the first external field with
the square of the ratio between the intensities of the forces due to the first and
those due to the second external field. Let us denote this ratio by~$\mu$ and let, as
above, $\lambda$~represent a small constant of the same order as the ratio between the external
\PageSep{61}
forces due to the first field and the internal forces of the system. On the
basis of the general relation between energy and frequency in the stationary states,
we may then expect that it is possible to fix the motion in these states for the perturbed
periodic system in the presence of both external fields with neglect of small
terms of the same order as the largest of the quantities $\mu^{2}$~and~$\lambda$, and to fix the
corresponding values for the energy with neglect of small terms of the same order
as the largest of the quantities $\lambda\mu^{2}$ and~$\lambda^{2}$.\Pagelabel{61}\footnote
  {In analogy with the considerations on \PageRef[page]{50} it may be expected, however, that these limits
  for the definition of the energy in the stationary states will hold only for the great majority among
  a large number of atomic systems. Thus in the present case we must be prepared to find that for a small
  fraction of the systems of the same order as~$\mu^{2}$ (if $\mu^{2} &gt; \lambda$) the energy will differ from that fixed by
  the method under consideration by small quantities of the same order as~$\mu\lambda$.}
In general, however, the effect on the
spectrum of the perturbed system, produced by the second external field, may be
calculated without considering the perturbing effect of this field in detail. In fact, it
is in general possible, by means of the principle of the mechanical transformability
of the stationary states, with the approximation mentioned to determine the alteration
of the energy of the system, due to the presence of the second external
field, directly from the character of the secular perturbations produced by the first
external field only. Thus let us assume that the second field is slowly established
at a uniform rate within a time interval of the same order of magnitude as
that in which the system will pass approximately through any state belonging
to the cycle of shapes and positions, which the orbit passes through in the
stationary states in the presence of the first external field only. Denoting a time
interval of this order by~$\theta$ and the potential of the first perturbing field by~$\Omega$ and
that of the second by~$\Delta\Omega$, we get then, by a calculation quite analogous to that
given in Part~I on \PageRefI[page]{11} for the alteration in the mean value of the energy of a
periodic system during a slow establishment of a small external field, that the
alteration in the mean value of $\alpha_{1} + \Omega$ taken over a time interval of the same
order as~$\theta$, due to the establishment of the second external field, will be a
small quantity of the same order of magnitude as~$\theta(\Delta\Omega)^{2}$; but with the notation
used above this means, in general, a small quantity of the same order as~$\lambda\mu^{2}$. It
follows consequently that, with this approximation, the alteration in the energy in
a given stationary state, due to the presence of the second perturbing field, is equal
to the mean value of the potential of this field taken over the cycle of shapes and
positions, which the orbit would pass through in the corresponding stationary state
of the perturbed system under the influence of the first external field only. In
general, the effect on the spectrum will therefore consist in a small displacement of
the original components proportional to the intensity of the forces due to the second
perturbing field; and as regards the degree of approximation with which these displacements
are defined, it will be seen from the above that, if $\mu$~is smaller than~$\sqrt{\lambda}$,
the fixation of the energy in the stationary states in the presence of the second external
field, and therefore also the determination of the frequencies of the spectral
% [** TN: Printer's mark]
% D. K. D. Vidensk. Selsk. Skr., naturvidensk. og mathem, Afd. 8. Række, IV. 1. 9
\PageSep{62}
lines by means of~\EqI{(1)}, allow of the same degree of approximation as the fixation
of the energy in the stationary states of the original perturbed periodic system.
If $\mu$~is larger than~$\sqrt{\lambda}$, however, the stationary states will in general not be as well
defined as for the original system, and from relation~\EqI{(1)} we may therefore expect
that the components will be diffuse, although, as long as $\mu$~remains small compared
with unity, the width of the components will remain small compared with the
displacements from their positions in the presence of the first external field alone.
Only when $\mu$~becomes of the same order as unity, the simultaneous effect of both
perturbing fields may be expected to consist in a diffusion of the lines of the undisturbed
periodic system;\Pagelabel{62} unless of course the secular perturbations due to the
simultaneous presence of both fields are still of conditionally periodic type, as it
may happen in special problems. In certain cases the second external field will not
only give rise to small displacements of the original components but also to the
appearance of new components of small intensities proportional to~$\mu^{2}$. This occurs
if for the original perturbed periodic system, due to some \Typo{pecularity}{peculiarity} of the motion,
some of the coefficients $C_{\tau,\frakt_{1}, \MyDots \frakt_{s-1}}$ in the expressions~\Eq{(65)} for the displacements
of the particles as a sum of harmonic vibrations, corresponding to certain combinations
of the numbers $\tau$, $\frakt_{1}$,~\MyDots $\frakt_{s-1}$, are equal to zero, while in the presence
of the second external field these coefficients are small quantities proportional to~$\mu$
(\Typo{Compare}{compare} Part~I, \PageRefI[page]{34}).\footnote
  {As regards the degree of definition with which the positions of the new components will be
  determined, we must be prepared to find that the frequencies of these components are only defined
  with neglect of small quantities proportional to~$\lambda\mu$. Compare the detailed discussion of the example
  in~\SecRef{5} on \PageRef[page]{97}.}
In the preceding considerations it has been assumed
that the perturbed system in the presence of the first external field is non-degenerate.
In case, however, this system is \Emph{degenerate}, it is obviously impossible,
by a direct application of the principle of the mechanical transformability of the
stationary states, to determine the alteration in the energy in the stationary states
of the system, which will be due to the presence of a second external field small
compared with the first field; because, as mentioned, the stationary states of the
system, in the presence of this field only, will be determined by a number of conditions
which is less than the number~$s$ of degrees of freedom, and that consequently
the cycles of shapes and positions, which the orbit will pass through in these states,
will not be completely determined. For the calculation of the energy in the \Typo{stationay}{stationary}
states it will therefore be necessary to consider the secular perturbing effect of the
second external field on these cycles. In the special case where the secular perturbations
due to the first field are simply periodic, it wall in this way be seen that the
problem of the fixation of the stationary states in the presence of the second external
field, by means of the method exposed in this section, may be reduced to
the problem of the fixation of the stationary states of a system of $s - 2$~degrees of
freedom. If, as in the applications considered below, $s$~is equal to~$3$, this problem
allows of a general solution, and we must therefore expect that in this case the
\PageSep{63}
effect on the spectrum of the perturbed system produced by an arbitrary second
external field, which is small compared with the first, will consist in the splitting
up of every component into a number of separate components, just as the effect of
an arbitrary small external field on the lines of the spectrum of a simple periodic
system of two degrees of freedom. We will meet with applications of the above
considerations when considering the effect on the hydrogen spectrum of the combined
action of different external fields and when considering the effect of an external
field on the spectra of other elements, which latter problem will be discussed
in Part~III\@.


\Section{3.}{The fine structure of the~hydrogen~lines.}

An instructive application of the calculations in the last section may be made
in connection with the fine structure of the hydrogen lines, which, according to
\Name{Sommerfeld}'s theory mentioned in Part~I on \PageRefI[page]{18}, may be explained by taking into
account the small variation of the mass of the electron with its velocity, claimed by
the theory of relativity. In this connection it must first of all be remarked that all
the general considerations in the preceding sections, as regards relations between
energy and frequency and as regards the mechanical transformability of the stationary
states, hold unaltered if the relativity modifications are taken into account.
This follows from the fact that the Hamiltonian equations~\EqI{(4)}, which are taken
as a basis for all the previous calculations, may be used to describe the motion
also in this case. If, when the relativity modifications are taken into account, the
motion of the system is simply periodic independent of the initial conditions, we
shall consequently expect that the stationary states are characterised by the condition
$I = nh$ only, and that the energy and frequency are the same for all states
corresponding to a given value of~$n$ in this equation. Further the stationary states
will also in the relativity case be fixed by~\EqI{(22)}, if the system is conditionally
periodic and allows of separation of variables; while the stationary states of a
perturbed periodic system, also in the relativity case, will be characterised by the
conditions~\Eq{(67)}, if the secular perturbations are of conditionally periodic type.

Now, when the relativity modifications are taken into account, the motion of
the particles in the hydrogen atom will not, as assumed in~\SecRef{1}, be exactly periodic, but
the orbit of the electron will be of the same type as that, which would appear on
ordinary Newtonian mechanics, if the law of attraction between the particles differed
slightly from that of the inverse square. If, for the moment, we consider the mass
of the nucleus as infinite, the system will allow of a separation of variables in polar
coordinates, and the stationary states may consequently be fixed by the conditions~\EqI{(16)}.
In this way \Name{Sommerfeld} obtained an expression for the total energy in the
stationary states, which, with neglect of small quantities of higher order than the
\PageSep{64}
square of the ratio of the velocity of the electron and the velocity of light~$c$, is
given by\footnote
  {\Name{A.~Sommerfeld}, Ann.\ d.\ Phys.\ LI, p.~53 (1916). Compare also \Name{P. Debye}, Phys.\ Zeitschr., XVII, p.~512
  (1916). In the special case of circular orbits ($n_{1} = 0$), this expression coincides with an expression
  previously deduced by the writer (Phil.\ Mag.\ XXIX p.~332 (1915)), by a direct application of the condition
  $I = nh$ to these periodic motions.}
\[
\llap{$E$} = -\frac{2\pi^{2} N^{2} e^{4}m}{h^{2} (n_{1} + n_{2})^{2}}
  \left[1 + \frac{\pi^{2} N^{2} e^{4}}{c^{2} h^{2} (n_{1} + n_{2})^{2}}
  \left(1 + 4\frac{n_{1}}{n_{2}}\right)\right]\!,
\Tag{(68)}
\]
where, as in the calculations in~\SecRef{1}, the charge and the mass of the electron are
denoted by $-e$ and~$m$, and for sake of generality the charge of the nucleus by~$Ne$.
Further $n_{1}$~and~$n_{2}$ are the integers appearing on the right side of the conditions~\EqI{(16)}
as factors to \Name{Planck}'s constant. While $n_{1}$~may take the values $0$,~$1$, $2$,~\dots, it
will be seen that $n_{2}$~can only take the values $1$,~$2$,~\dots, because in the present case
there will obviously not correspond any stationary state to $n_{2} = 0$, since in such
a state the electron would collide with the nucleus. Introducing the experimental
values for $e$,~$h$ and~$c$, it is found that $e^{2}/hc$ is a small quantity of the same order as~$10^{-3}$;
and, unless $N$~is large number, the second term within the bracket on the
right side of~\Eq{(68)} will consequently be very small compared with unity. Putting
$n_{1} + n_{2} = n$, it will further be seen that the factor outside the bracket will coincide
with the expression for~$W_{n}$ given by~\Eq{(41)} in~\SecRef{1}, if we look apart from the
small correction due to the finite mass of the nucleus. Due to the presence of the
second term within the bracket, we thus see that, for any value of~$n$, formula~\Eq{(68)} gives
a set of values for~$E$ which differ slightly from each other and from~$-W_{n}$. \Name{Sommerfeld}'s
theory leads therefore to a direct explanation of the fact, that the hydrogen
lines, when observed by instruments of high dispersive power, are split up in a
number of components situated closely to each other; and, by means of formula~\Eq{(68)}
in connection with relation~\EqI{(1)}, it was actually found possible, within the limits
of experimental errors, to account for the frequencies of the components of this
\Typo{socalled}{so called} \Emph{fine structure} of the hydrogen lines. Moreover the theory was supported
in the most striking way by \Name{Paschen}'s\footnote
  {\Name{F.~Paschen}, Ann.\ d.\ Phys.\ L, p.~901 (1916). See also \Name{E.~J. Evans} and \Name{C.~Croxson}, Nature, XCVII,
  p.~56 (1916).}
recent investigation of the fine structure
of the lines of the analogous helium spectrum, the frequencies of which are
represented approximately by formula~\Eq{(35)}, if in the expression for~$K$, given by~\Eq{(40)},
we put $N = 2$. As it should be expected from~\Eq{(68)}, the components of these lines
were found to show frequency differences several times larger than those of the
hydrogen lines, and from his measurements \Name{Paschen} concluded, that it was possible
on \Name{Sommerfeld}'s theory to account completely for the frequencies of all the components
observed.

We shall not enter here on the details of the calculation leading to~\Eq{(68)}, but
shall only show how this formula may be simply interpreted from the point of
\PageSep{65}
view of perturbed periodic systems. Thus, by a simple application of relativistic
mechanics, it is found that, if the equation of a Keplerian ellipse in polar coordinates
is given by $r = f(\theta)$, the equation of the orbit of the electron in the
case under consideration will be given by $r = f(\gamma\theta)$ where $\gamma$~is a constant given
by $\gamma^{2} = 1 - \left(\dfrac{Ne^{2}}{pc}\right)^{2}$, in which expression~$p$ denotes the angular momentum of the
electron round the nucleus.\footnote
  {See f.\ inst.\ \Name{A.~Sommerfeld}, loc.\ cit.\ p.~47.}
Now in the stationary states the quantity in the
bracket, which is of the same order of magnitude as the ratio between the velocity
of the electron and the velocity of light, will be very small, unless $N$~is a large
number, and it will therefore be seen that the orbit of the electron can be described
as a periodic orbit on which a slow uniform rotation is superposed. Denoting the
frequency of revolution in the periodic orbit by~$\omega$ and the frequency of the superposed
rotation by~$\frakv_{R}$, we have, with neglect of small quantities of higher order
than the square of the ratio between the velocity of the electron and the velocity
of light,
\[
\frakv_{R} = \omega(1 - \gamma) = \frac{1}{2} \omega \left(\frac{Ne^{2}}{pc}\right)^{2}.
\Tag{(69)}
\]
Comparing this formula with equation~\Eq{(62)} and remembering that, with the approximation
in question, $p$~may be replaced by the quantity denoted in~\SecRef{2} by~$\alpha_{2}$, we
see that the frequency of the secular rotation of the orbit will be the same as
that which would appear, if the variation of the mass of the electron was neglected,
but if the atom was subject to a small external central force the mean value of
the potential of which, taken over a revolution of the electron, was equal to
\[
\psi = -\omega \frac{\pi N^{2}e^{4}}{c^{2} \alpha_{2}}.
\Tag{(70)}
\]
This is simply shown, however, to be equal to the expression for~$\psi$ corresponding
to a small attractive force varying as the inverse cube of the distance. In fact, let
the potential of such a force be given by $\Omega = C/r^{2}$, where $C$~is a constant and $r$~the
length of the radius vector from the nucleus to the electron. By means of
the relation $\alpha_{2} = mr^{2} \dot{\theta}$, where $\theta$~is the angular distance of the radius vector from
a fixed line in the plane of the orbit, we get then
\[
\psi = \frac{1}{\sigma} \int_{0}^{\sigma} \frac{C}{r^{2}}\, dt
  = \frac{\omega mC}{\alpha_{2}} \int_{0}^{2\pi} d\theta
  = \frac{2\pi \omega mC}{\alpha_{2}},
\]
which expression is seen to coincide with~\Eq{(70)}, if $C = -\dfrac{N^{2} e^{4}}{2c^{2}m}$.

If the relativity modifications are taken into account, and if for a moment
we would imagine that the nucleus, in addition to its usual attraction, exerted
\PageSep{66}
a small repulsion on the electron, proportional to the inverse cube of the distance
and equal and opposite to the attraction just mentioned, we would therefore obtain
a system for which, with neglect of small quantities of higher order than
the square of the ratio between the velocity of the electron and the velocity of
light, every orbit would be periodic independent of the initial conditions, and for
which consequently the stationary states would be fixed by the single condition
$I = nh$.\Pagelabel{66} Now the actual hydrogen atom may obviously be considered as a \Emph{perturbed
system}, formed by this periodic system, when it is exposed to a small
central field for which the value of~$\psi$ is given by~\Eq{(70)}. With the approximation
mentioned, we get therefore for the total energy in the stationary states of the atom
\[
E = E_{n}' - \frac{8\pi^{4} N^{4} e^{8}m}{h^{4} c^{2}}\, \frac{1}{n^{3} \frakn},
\Tag{(71)}
\]
where $E_{n}'$~is the energy in the stationary states of the periodic system just mentioned,
and where the last term is obtained by introducing in~\Eq{(70)} the value of~$\alpha_{2}$ given
by~\Eq{(64)} and the value of~$\omega_{n}$ given by~\Eq{(41)}, neglecting the small correction due to
the finite mass of the nucleus. Remembering that in our notation $n_{1} + n_{2} = n$
and $n_{2} = \frakn$, it will be seen that, as regards the small differences in the energy of
the different stationary states corresponding to the same value of~$n$, formula~\Eq{(71)}
gives the same result as \Name{Sommerfeld}'s formula~\Eq{(68)}. In fact, comparing \Eq{(68)}~and~\Eq{(71)},
we get
\[
E_{n}' = -\frac{2\pi^{2} N^{2} e^{4} m}{h^{2} n^{2}}
  \left(1 - \frac{3\pi^{2} N^{2} e^{4}}{c^{2} h^{2} n^{2}}\right),
\Tag{(72)}
\]
which is seen to be a function of $n$~only. This expression might also have been deduced
directly from the condition $I = nh$ by considering, for instance, a circular orbit,
in which case the calculation can be very simply performed.

In connection with the above calculations, it may be remembered that the fixation
of the stationary states, leading to the formulæ \Eq{(68)} or~\Eq{(71)}, is based on the assumption,
that the motion of the electron can be determined as that of a mass point
which moves in a conservative field of force, according to the laws of ordinary
relativistic mechanics, and that we have looked apart from all such forces which,
according to the ordinary theory of electrodynamics, would act on an accellerated\Note{[sic]}
charged particle, and which constitute the reaction from the radiation which on this
theory would accompany the motion of the electron. Some procedure of this kind,
which means a radical departure from the ordinary theory of electrodynamics, is
obviously necessary in the quantum theory in order to avoid dissipation of energy
in the stationary states. Since we are entirely ignorant as regards the mechanism
of radiation, we must be prepared, however, to find that the above treatment will
allow to determine the motion in the stationary states, only with an approximation
which looks apart from small quantities of the same order as the ratio between
the radiation forces in ordinary electrodynamics and the main forces on the electron
\PageSep{67}
due to the attraction from the nucleus.\footnote
  {Compare Part~I, \PageRefI[p.]{6}. It may in this connection be noted that the degree of approximation, involved
  in the determination of the frequencies of an atomic system by means of relation~\EqI{(1)} if in the fixation
  of the stationary states we look apart from small forces of the same order of magnitude as the radiation
  forces in ordinary electrodynamics, would appear to be intimately connected with \Emph{the limit of
  sharpness of the spectral lines}, which depends on the total number of waves contained in the
  radiation emitted during the transition between two stationary states. In fact, from a consideration
  based on the general connection between the quantum theory and the ordinary theory of radiation, it
  seems natural to assume that the rate, at which radiation is emitted during a transition between two
  stationary states, is of the same order of magnitude as the rate, at which radiation would be emitted
  from the system in these states according to ordinary electrodynamics. But this will be seen to imply
  that the total number of waves in question will just be of the same order as the ratio between the
  main forces acting on the particles of the system and the reaction from the radiation in ordinary
  electrodynamics.}
Now it is easily shown that this ratio
will be a small quantity of the same order of magnitude as $N^{2}\left(\dfrac{e^{2}}{pc}\right)^{3}$, and it would
therefore beforehand seem justified in the expression for the total energy in the
stationary states to retain small terms of the same order as the second term in~\Eq{(71)},
while at the same time it might appear highly questionable, whether, in the
complete expression for the total energy in the stationary states deduced by \Name{Sommerfeld}
and \Name{Debye} on the basis of the conditions~\EqI{(16)}, it has a physical meaning to
retain terms of higher order than those retained in formula~\Eq{(68)}; unless $N$~is a large
number, as in the theory of the Röntgenspectra to be discussed in Part~III\@.

While the preceding considerations, which deal with the determination of the
energy in the stationary states of the hydrogen atom, allow to determine \Emph{the frequency
of the radiation} which would be emitted during a transition between
two such states, they leave quite untouched the problem of the actual occurrence
of these transitions in the luminous gas, and therefore give no direct information
about the \Emph{number and relative intensities of the components} into which
the hydrogen lines may be expected to split up as a consequence of the relativity
modifications. This problem has recently been discussed by \Name{Sommerfeld},\footnote
  {\Name{A.~Sommerfeld}, Ber.\ Akad.\ München, 1917, p.~83.}
who in
this connection emphasises the importance of the different a-priori probabilities of
the stationary states, characterised by different sets of values of the~$n$'s in the conditions~\EqI{(16)}.
Thus \Name{Sommerfeld} attempts to obtain a measure for the relative intensities
of the components of the fine structure of a given line, by comparing the
intensities observed with the products of the values of the a-priori probabilities of
the two states, involved in the emission of the components under consideration; and he
tries in this connection to test different expressions for these a-priori probabilities
(See Part~I, \PageRefI[\Chg{pag.}{page}]{26}). In this way, however, it was not found possible to account
in a satisfactory manner for the observations; and the difficulty in obtaining an explanation
of the intensities on this basis was also strikingly brought out by the
fact, that the number and relative intensities of the components observed varied in
a remarkable way with the experimental conditions under which the lines were
\PageSep{68}
excited. Thus \Name{Paschen} found a greater number of components in the fine structure
of the helium lines, mentioned above, when the gas was subject to a condensed
interrupted discharge, than when a continuous voltage was applied. It would seem, however,
that all the facts observed obtain a simple interpretation on the basis of the
general considerations about the relation between the quantum theory of line spectra
and the ordinary theory of radiation discussed in Part~I\@. According to this relation,
we shall assume that the probability, for a transition between two given stationary
states to take place, will depend not only on the a-priori probability of these states,
which is determining for their occurrence in a distribution of statistical equilibrium,
but will also depend essentially on the motion of the particles in these states,
characterised by the harmonic vibrations in which this motion can be resolved.
Now, in the absence of external forces, the motion of the electron in the hydrogen
atom forms a special simple case of the motion of a conditionally periodic system
possessing an axis of symmetry, and may therefore be represented by trigonometric
series of the type deduced for such motions in Part~I\@. Taking a line through the
nucleus perpendicular to the plane of the orbit as $z$-axis, we get from the calculations
on \PageRefI[page]{32}
\[
z = \text{const.}
\]
and
\[
\BreakMath{
 x = \sum C_{\tau} \cos 2\pi \bigl\{(\tau\omega_{1} + \omega_{2})t + c_{\tau}\bigr\},\quad
±y = \sum C_{\tau} \sin 2\pi \bigl\{(\tau\omega_{1} + \omega_{2})t + c_{\tau}\bigr\},
}{
\begin{aligned}
 x &amp;= \sum C_{\tau} \cos 2\pi \bigl\{(\tau\omega_{1} + \omega_{2})t + c_{\tau}\bigr\}, \\
±y &amp;= \sum C_{\tau} \sin 2\pi \bigl\{(\tau\omega_{1} + \omega_{2})t + c_{\tau}\bigr\},
\end{aligned}
}
\Tag{(73)}
\]
where $\omega_{1}$~is the frequency of the radial motion and $\omega_{2}$~is the mean frequency of
revolution, and where the summation is to be extended over all positive and negative
entire values of~$\tau$. It will thus be seen that the motion may be considered as a
superposition of a number of circular harmonic vibrations, for which the direction
of rotation is the same as, or the opposite of, that of the revolution of the electron
round the nucleus, according as the expression $\tau \omega_{1} + \omega_{2}$ is positive or negative
respectively. From\Pagelabel{68} the relation just mentioned between the quantum theory of line
spectra and the ordinary theory of radiation, we shall therefore in the present case
expect that, if the atom is not disturbed by external forces, only such transitions
between stationary states will be possible, in which the plane of the orbit remains
unaltered, and in which the number~$n_{2}$ in the conditions~\EqI{(16)} decreases or increases
by one unit; \ie\ where the angular momentum of the electron round the nucleus
decreases or increases by~$h/2\pi$. From the relation under consideration, we shall
further expect that there will be an intimate connection between the probability of
a spontaneous transition of this type between two stationary states, for which $n_{1}$~is
equal to $n_{1}'$~and $n_{1}''$ respectively, and the intensity of the radiation of frequency
$(n_{1}' - n_{1}'') \omega_{1} ± \omega_{2}$, which on ordinary electrodynamics would be emitted by the
atom in these states, and which would depend on the value~$C_{\tau}$ of the amplitude
of the harmonic rotation, corresponding to $\tau = ±(n_{1}' - n_{1}'')$, which appears in the
motion of the electron. Without entering upon a closer examination of the numerical
values of these amplitudes, it will directly be seen that the amplitudes of the
harmonic rotations, which have the same direction as the revolution of the electron,
\PageSep{69}
in general, are considerably larger than the amplitudes of the rotations in the opposite
direction, and we shall accordingly expect that the probability of spontaneous
transition will in general be much larger for transitions, in which the angular
momentum decreases, than for transitions in which it increases. This expectation
is verified by \Name{Paschen}'s observations of the fine structure of the helium lines,
which show that, for a given line, the components corresponding to the transitions
of the former kind are by far the strongest. On \Name{Paschen}'s photographs, however,
especially in the case of the application of a condensed discharge to the
vacuum tube containing the gas, there appear, in addition to the main components
corresponding to transitions for which the angular momentum changes by~$h/2\pi$,
a number of weaker components, corresponding to transitions for which the
angular momentum remains unchanged or changes by higher multipla of~$h/2\pi$. This
fact obtains a simple interpretation on the considerations in Part~I on \PageRefI[page]{34}
about the influence of small external forces on the spectrum of a conditionally
periodic system. Thus, in the presence of small perturbing forces, the motion will
generally not remain in a plane, and in the trigonometric series representing
the displacement of the electron in space, there will occur small terms corresponding
to frequencies $(\tau_{1} \omega_{1} + \tau_{2} \omega_{2})$, where $\tau_{2}$~may be different from one. In
the presence of such forces, we shall therefore expect that, in addition to the regular
probabilities of the above mentioned main transitions, there will appear small
probabilities for other transitions.\footnote
  {\Emph{Note added during the proof.} As remarked in Part~I, this consideration obtains a striking
  confirmation by the observation of the appearance of new series of lines in the ordinary series spectra
  of helium and other elements, when the atoms are exposed to an intense external electric field. As it
  will be discussed more closely in Part~III, it is possible in this way to account in detail for the manifold
  results, regarding the appearance of such series in the helium spectrum, which have been published
  quite recently by \Name{J.~Stark} (Ann.\ d.\ Phys.\ LVI, p.~577 (1918)) and by \Name{G.~Liebert} (ibid.\ LVI, p.~589 and
  p.~610 (1918)).}
A detailed discussion of these problems will be
given in a later paper by Mr.\ \Name{H.~A. Kramers},\Pagelabel{69} who on my proposal has kindly
undertaken to examine the resolution of the motion of the electron in its constituent
harmonic vibrations more closely, and who has deduced \Typo{explicite}{explicit} expressions for
the amplitudes of these vibrations, not only for the motion of the electron in the
undisturbed atom, but also for the perturbed motion in the presence of a small
external homogeneous electric field. As it will be shown by \Name{Kramers}, these calculations
allow to account in particulars\Note{[sic]} for the observations of the relative intensities
of the components of the fine structure of the hydrogen lines and the analogous
helium lines, as well as for the characteristic way in which this phenomenon is
influenced by the variation of the experimental conditions.


\Section{4.}{The effect of an external electric~field on~the~hydrogen~lines.}

As mentioned in the introduction, a detailed theory of the characteristic effect
of an external homogeneous electric field on the hydrogen spectrum, discovered by
% [** TN: Printer's mark]
% D. K. D. Vidensk. Selsk. Skr., naturvidensk. og mathem. Afd., 8. Række, IV. 1. 10
\PageSep{70}
\Name{Stark}, has been given by \Name{Epstein} and \Name{Schwarzschild} on the basis of the general
theory of conditionally periodic systems which allow of separation of variables.
Before we enter on the discussion of the results of the calculations of these authors,
we shall first, however, show how the problem may be treated in a simple way
by means of the considerations about perturbed periodic systems, developed in~\SecRef{2}.

Consider an electron of mass~$m$ and charge~$-e$, rotating round a positive
nucleus of infinite mass and of charge~$Ne$, and subject to a homogeneous electric field
of intensity~$F$, and let us for the present neglect the small effect of the relativity
modifications. Using rectangular coordinates, and taking the nucleus as origin and
the $z$-axis parallel to the external field, we get for the potential of the system
relative to the external field, omitting an arbitrary constant,
\[
\Omega = eFz.
\]

Calculating now the mean value of~$\Omega$ over a period~$\sigma$ of the undisturbed
motion, we see at once, from considerations of symmetry, that this mean value~$\psi$
will depend only on the component of the external electric force in the direction
of the major axis of the orbit. We have therefore
\[
\psi = eF \cos\phi\, \frac{1}{\sigma} \int_{0}^{\sigma} r \cos\theta\, dt,
\]
where $\phi$~is the angle between the $z$-axis and the major axis, taken in the direction
from the nucleus to the aphelium, and where $r$~is the length of the radius-vector
from the nucleus to the electron, and $\theta$~the angle between this radius-vector and
the major axis. By means of the well known equations for a Keplerian motion
\[
r \cos\theta = \alpha(\cos u + \eps),\quad
\frac{dt}{\sigma} = (1 + \eps \cos u)\frac{du}{2\pi},
\]
where $2\alpha$~is the major axis, $\eps$~the eccentricity and $u$~the \Typo{socalled}{so called} eccentric anomaly,
this gives
\[
\BreakMath{
\psi = eF \cos\phi\, \frac{1}{2\pi} \int_{0}^{2\pi} \alpha(\cos u + \eps)(1 + \eps \cos u)\, du
  = \frac{3}{2} \eps\alpha eF \cos\phi.
}{%
\begin{aligned}
\psi &amp;= eF \cos\phi\, \frac{1}{2\pi} \int_{0}^{2\pi} \alpha(\cos u + \eps)(1 + \eps \cos u)\, du \\
  &amp;= \frac{3}{2} \eps\alpha eF \cos\phi.
\end{aligned}
}
\Tag{(74)}
\]

We see thus that $\psi$~is equal to the potential energy relative to the external
field, which the system would possess, if the electron was placed at a point, situated
on the major axis of the ellipse and dividing the distance~$2\eps a$ between the foci in
the ratio~$3:1$. This point may be denoted as the ``electrical centre'' of the orbit.
From the approximate constancy of~$\psi$ during the motion, proved in~\SecRef{2}, it follows
therefore in the first place that, with neglect of small quantities of the same order
of magnitude as the ratio between the external force and the attraction from the
nucleus, \Emph{the electrical centre will during the perturbations of the
orbit remain in a fixed plane perpendicular to the direction of the
external force}. From the considerations in~\SecRef{2} it follows further, that the total
\PageSep{71}
energy in the stationary states of the system in the presence of the field, with neglect
of small quantities proportional to~$F^{2}$, will be equal to $E_{n} + \psi$, where $E_{n}$~is the energy
of the hydrogen atom in its undisturbed stationary state. Since both $\eps$~and $\cos\phi$ are
numerically smaller than one, we obtain therefore at once from~\Eq{(74)} a lower and
an upper limit for the possible variations of the energy in the stationary states, due
to the field. Introducing from~\Eq{(41)} the values of $E_{n}$~and~$\alpha_{n}$, and neglecting, here
as well as in the following calculations in this section, the small correction due
to the finite mass of the nucleus---not only in the expression for the additional
energy but, for the sake of brevity, also in the main term---we get for these limits
\[
E = -\frac{2\pi^{2} N^{2} e^{4} m}{h^{2} n^{2}}
    ± \frac{3h^{2} n^{2}}{8\pi^{2} Nem}\, F,
\Tag{(75)}
\]
which formula coincides with the expression previously deduced by the writer by
applying the condition $I = nh$ to the two (physically not realisable) limiting cases,
corresponding to $z = 1$ and $\cos\phi = ±1$, in which the orbit remains periodic in
the presence of the field.\footnote
  {See \Name{N.~Bohr}, Phil.\ Mag.\ XXVII, p.~506 (1914) and XXX, p.~394 (1915). Compare also \Name{E.~Warburg},
  Verh.\ d.\ D.\ Phys.\ Ges.\ XV, p.~1259 (1913), where it was pointed out, for the first time, that the effect of
  an electric field on the hydrogen lines to be expected on the quantum theory was of the same order
  of magnitude as the effect observed by \Name{Stark}.}

In order to obtain further information as to the values of the energy in the
stationary states in the presence of the field, it is necessary to consider more closely
the variation of the orbit during the perturbations. Since the external forces possess
axial symmetry, the problem of the stationary states might be treated by means of
the procedure indicated in~\SecRef{2} on \PageRef[page]{55}. In the present special case, however,
the stationary states of the atom may be very simply determined, due to the fact
that the secular perturbations are simply periodic independent of the initial shape
and position of the orbit, so that we are concerned with a \Emph{degenerate} case of a
perturbed periodic system. This property of the perturbations follows already from
some calculations given by \Name{Schwarzschild}\footnote
  {\Name{K.~Schwarzschild}, Verh.\ d.\ D.\ Phys.\ Ges.\ XVI, p.~20 (1914).}
in a previous attempt to explain the
\Name{Stark} effect of the hydrogen lines, without the help of the quantum theory, by
means of a direct consideration of the harmonic vibrations into which the motion
may be resolved, according to the analytical theory of conditionally periodic
systems. Starting from the above result, that the electrical centre moves in a plane
perpendicular to the direction of the external field, the periodicity of the perturbations
may also be proved in the following way, by means of a simple consideration
of the variation of the angular momentum of the electron round the nucleus, due
to the effect of the external electric force.

\begin{Remark}
Using again rectangular coordinates with the nucleus at the origin and the $z$-axis parallel
to the direction of the electric force, and calling the coordinates of the electrical centre $\xi$,~$\eta$,~$\zeta$,
we have according to formula~\Eq{(74)}
\PageSep{72}
\[
\xi^{2} + \eta^{2} + \zeta^{2}
  = \left(\frac{3}{2} \eps\alpha\right)^{2},\qquad
\zeta = \text{const.}
\Tag[II.]{(1^{*})}
\]
Denoting the components parallel to the $x$~$y$ and $z$-axis of the angular momentum of the
electron round the nucleus, considered as a vector, by $P_{x}$,~$P_{y}$ and~$P_{z}$, we have next
\[
P_{x}^{2} + P_{y}^{2} + P_{z}^{2} = (1 - \eps^{2})(2\pi m\alpha^{2}\omega)^{2},\qquad
P_{z} = \text{const.}
\Tag[II.]{(2^{*})}
\]
Since the angular momentum is perpendicular to the plane of the orbit, we have further
\[
\xi P_{x} + \eta P_{y} + \zeta P_{z} = 0.
\Tag[II.]{(3^{*})}
\]
Now we have for the mean values of the rates of variation of $P_{x}$~and $P_{y}$ with the time
\[
\frac{DP_{x}}{Dt} = eF\eta,\quad
\frac{DP_{y}}{Dt} = -eF\xi.
\Tag[II.]{(4^{*})}
\]
From this we get, differentiating \Eq[II.]{(1^{*})}~and \Eq[II.]{(2^{*})} with respect to the time, and remembering that $\alpha$~and~$\omega$
remain constant during the perturbations,
\[
\BreakMath{%
\xi\, \frac{D\xi}{Dt} + \eta\, \frac{D\eta}{Dt}
  = -K^{2} \left(P_{x}\, \frac{DP_{x}}{Dt} + P_{y}\, \frac{DP_{y}}{Dt}\right)
  = -eFK^{2} (\eta P_{x} - \xi P_{y}),
}{%
\begin{aligned}
\xi\, \frac{D\xi}{Dt} + \eta\, \frac{D\eta}{Dt}
  &amp;= -K^{2} \left(P_{x}\, \frac{DP_{x}}{Dt} + P_{y}\, \frac{DP_{y}}{Dt}\right) \\
  &amp;= -eFK^{2} (\eta P_{x} - \xi P_{y}),
\end{aligned}
}
\Tag[II.]{(5^{*})}
\]
where
\[
K = \frac{3}{4\pi m\alpha\omega}.
\Tag[II.]{(6^{*})}
\]
On the other hand we have, differentiating~\Eq[II.]{(3^{*})} and introducing~\Eq[II.]{(4^{*})},
\[
P_{x}\, \frac{DP_{x}}{Dt} + P_{y}\, \frac{DP_{y}}{Dt} = 0,
\]
which together with~\Eq[II.]{(5^{*})} gives
\[
\frac{D\xi}{Dt} = eFK^{2} P_{y},\quad
\frac{D\eta}{Dt} = -eFK^{2} P_{x},
\]
from which we get, by means of~\Eq[II.]{(4^{*})},
\[
\frac{D^{2} \xi}{Dt^{2}} = -e^{2} F^{2} K^{2} \xi,\quad
\frac{D^{2} \eta}{Dt^{2}} = -e^{2} F^{2} K^{2} \eta,
\]
the solution of which is
\[
\xi  = \frakA \cos 2\pi(\frakv t + \fraka),\quad
\eta = \frakB \cos 2\pi(\frakv t + \frakb),
\Tag[II.]{(7^{*})}
\]
where $\frakA$,~$\fraka$,~$\frakB$ and~$\frakb$ are constants, and where, introducing~\Eq[II.]{(6^{*})}, we have
\[
\frakv = \frac{eFK}{2\pi}
  = \frac{3eF}{8\pi^{2} m\alpha\omega}.
\Tag[II.]{(8^{*})}
\]
\end{Remark}

\Emph{During the perturbations the electrical centre will thus perform
slow harmonic vibrations perpendicular to the direction of the
electric force}, with a frequency which is proportional to the intensity of the
electric field, but, for a given value of~$F$, quite independent of the initial shape of
the orbit and its position relative to the direction of the field. For the value of
this frequency in the multitude of states of the perturbed system, for which the
mean value of the inner energy is equal to the energy~$E_{n}$ in a stationary state of
the undisturbed system corresponding to a given value of~$n$, we get from the above
calculation, introducing for $\alpha$~and~$\omega$ the values of $\alpha_{n}$~and $\omega_{n}$ given by~\Eq{(41)},
\PageSep{73}
\[
\frakv_{F} = \frac{3hn}{8\pi^{2} Nem}\, F.
\Tag{(76)}
\]
Now\Pagelabel{73} from the periodic motion of the electrical centre we may conclude that, in the
presence of the field, the system will be able to emit or absorb a radiation of
frequency~$\frakv_{F}$, and that accordingly the possible values of the additional energy of
the system in the presence of the field will be given directly by \Name{Planck}'s fundamental
formula~\EqI{(9)}, holding for the possible values of the total energy of a linear
harmonic vibrator, if in this formula $\omega$~is replaced by the above frequency~$\frakv_{F}$. Since
further a circular orbit, perpendicular to the direction of the electric force, will not
undergo secular perturbations during a slow establishment of the field, and therefore
must be included among the stationary states of the perturbed system, we get
for the total energy of the atom in the presence of the field
\[
E = E_{n} + n\frakv_{F} h
  = -\frac{2\pi^{2} N^{2} e^{4} m}{n^{2} h^{2}} + \frac{3h^{2} n\frakn}{8\pi^{2} Nem}\, F,
\Tag{(77)}
\]
where $\frakn$~is an entire number which in the present case may be taken positive as
well as negative. From a comparison between \Eq{(75)}~and \Eq{(77)}, we see that the presence
of the external field imposes the restriction on the motion of the atom in the stationary
states, that the plane in which the electrical centre of the orbit moves
must have a distance from the nucleus equal to an entire multiple of the \Ord{$n$}{th}~part
of its maximum distance~$\dfrac{3}{2} \alpha_{n}$.

The result, contained in formula~\Eq{(77)}, is in agreement with the expression for
the total energy in the stationary states, deduced by \Name{Epstein} and \Name{Schwarzschild}
by means of the general theory of conditionally periodic systems based on the
conditions~\EqI{(22)}. The treatment of these authors rests upon the fact, that, as mentioned
in Part~I, the equations of motion for the electron in the present problem may be
solved by means of separation of variables in parabolic coordinates (compare \PageRefI[page]{21}).
Taking for $q_{1}$~and~$q_{2}$ the parameters of the two paraboloids of revolution, which
pass through the instantaneous position of the electron and which have their foci
at the nucleus and their axes parallel to the direction of the field, and for~$q_{3}$ the
angular distance between the plane through the electron and the axis of the system
and a fixed plane through this axis, the momenta $p_{1}$,~$p_{2}$,~$p_{3}$ will during the motion
depend on the corresponding $q$'s only, and the stationary states will be fixed by
three conditions of the type~\EqI{(22)}. With neglect of small quantities proportional to
higher powers of~$F$, the final formula for the total energy, obtained by \Name{Epstein} in
this way, is given by
\[
\BreakMath{%
E = -\frac{2\pi^{2} N^{2} e^{4} m}{h^{2} (n_{1} + n_{2} + n_{3})^{2}}
  - \frac{3h^{2} (n_{1} + n_{2} + n_{3})(n_{1} - n_{2})}{8\pi^{2} Nem}\, F,\quad\footnotemark
}{%
\begin{aligned}
E &amp;= -\frac{2\pi^{2} N^{2} e^{4} m}{h^{2} (n_{1} + n_{2} + n_{3})^{2}} \\
  &amp;\qquad\qquad- \frac{3h^{2} (n_{1} + n_{2} + n_{3})(n_{1} - n_{2})}{8\pi^{2} Nem}\, F,\quad\footnotemark
\end{aligned}
}
\Tag{(78)}
\]
\footnotetext{\Name{P.~Epstein}. Ann.\ d.\ Phys.\ L, p.~508 (1916).}
\PageSep{74}
where $n_{1}$,~$n_{2}$,~$n_{3}$ are the positive entire numbers which occur as factors to \Name{Planck}'s
constant on the right sides of the mentioned three conditions.

As regards the possible values of the total energy of the hydrogen atom in
the presence of the electric field, it will be seen that \Eq{(78)}~coincides with~\Eq{(77)} if we
put $n_{1} + n_{2} + n_{3} = n$ and $n_{1} - n_{2} = \frakn$. At the same time it will be observed,
however, that the motion in the stationary states, as fixed by the procedure followed
by \Name{Epstein}, is more restricted than was necessary in order to secure the right
relation between the additional energy and the frequency of the secular perturbations.
Thus, in addition to the condition which fixes the plane in which the electrical
centre moves, \Name{Epstein}'s\Pagelabel{74} theory involves the further condition, that the angular
momentum of the electron round the axis of the perturbed system is equal to an
entire multiple of~$h/2\pi$; which multiple is seen to be even or uneven, according as
$n + \frakn$ is an even or an uneven number respectively. This circumstance is intimately
connected with the fact that, although the perturbed system under consideration is
degenerate if we look apart from small quantities proportional to the square of the
intensity of the external force, the degenerate character of the system does not reveal
itself from the point of view of the theory of stationary states based on the conditions~\EqI{(22)},
because the system under consideration allows of separation of variables
only in one set of positional coordinates. On the other hand, this degenerate
character of the system has been emphasised by \Name{Schwarzschild}\footnote
  {\Name{K.~Schwarzschild}, Ber.\ Akad.\ Berlin, 1916, p.~548.}
on the basis
of the theory of stationary states based on the introduction of angle variables, in
which the periodicity properties of the motion play an essential part. In a later
discussion of this point \Name{Epstein}\footnote
  {\Name{P.~Epstein}, Ann.\ d.\ Phys.\ LI, p.~168 (1916).}
calls attention to the fact that, if small quantities
proportional to the square of the electric force are taken into account, the system
appears no more as degenerate; and he finds therein a justification of the fixation of
the stationary states by means of~\EqI{(22)}. From the point of view of perturbed systems,
this would mean that the motion in the stationary states of the system in question,
as fixed by~\EqI{(22)}, would certainly be stable for infinitely small disturbances, but that
we should expect finite deviations from the motion in these states, already if the
system was exposed to a second perturbing field, the intensity of which was only
of the same order as the product of the external electric force with the ratio between
this force and the attraction from the nucleus. A closer consideration, however, in
which regard is taken to the influence of the relativity modifications, learns that
the degree of stability of the motion in the stationary states, as determined by~\EqI{(22)},
actually is often much higher, the order of magnitude of the external force, necessary
to cause finite deviations from this motion, being of the same order as the product
of the attraction from the nucleus with the square of the ratio of the velocity of
the electron and the velocity of light. To this point we shall come back at the end
of this section, when considering the simultaneous perturbing influence on the
\PageSep{75}
motion of the election in the hydrogen atom, due to the relativity modifications
and an external electric field.

In the deduction of formula~\Eq{(78)} there is looked apart, not only from the effect
on the motion of the electron due to the small modifications in the laws of mechanics
claimed by the theory of relativity, but also from the effect of possible forces
which might act on the electron, corresponding to the reaction from the radiation
in ordinary electrodynamics. If, however, for the moment we exclude all stationary
states for which the angular momentum round the axis of the system would be
equal to zero ($n_{3} = 0$), the total angular momentum of the electron round the
nucleus will during the perturbations always remain larger than or equal to~$h/2\pi$,
just as in the stationary states considered in the theory of the fine structure; and,
according to the considerations on \PageRef[page]{66}, we shall therefore expect that the effect
of the neglect of possible ``radiation'' forces will be small compared with the effect
of the relativity modifications. On the other hand, if the intensity of the electric field is
of the same order of magnitude as that applied in \Name{Stark}'s experiments, the effect of
these modifications must again be expected to be very small compared with the
total effect of the electric force on the hydrogen lines, since the perturbing effect
of this force on the Keplerian motion of the electron will be very large compared
with the corresponding effects of the relativity modifications. If, on the
contrary, we would consider a state of the atom for which $n_{2}$~was equal to zero,
the orbit would be plane and would during the perturbations assume shapes, for
which the total angular momentum round the nucleus was very small, and in which
the electron during the revolution would pass within a very short distance from
the nucleus. In such a state the effect of the relativity modifications on the motion of
the electron would be considerable, but quite apart from this a rough calculation shows
that the amount of energy, which, on ordinary electrodynamics, would be emitted
during the intervals in which the angular momentum during the perturbations of
the orbit remains small, is so large that it would hardly seem justifiable to calculate
the motion and the energy in these states by neglecting all forces corresponding to
the radiation forces in ordinary electrodynamics. We need not, however, enter more
closely on these difficulties, because, on the general considerations in Part~I about
the a-priori probability of the different stationary states, we are forced to conclude
that, for any value of the external electric field, \Emph{no state which would correspond
to $n_{3} = 0$ will be physically possible}; since any such state might be
transformed continuously, and without passing through a degenerate system, into
a state which obviously cannot represent a physically realisable stationary state
(compare \PageRefI[\Chg{pag.}{page}]{27}). In fact, if we imagine that an external central field of force,
varying as the inverse cube of the distance from the nucleus, is slowly established,
it would be possible to compensate the secular effect of the relativity modifications
and to obtain orbits in which the electron would pass within any given, however
small, distance from the nucleus. As regards the other stationary states fixed
\PageSep{76}
by~\EqI{(22)}, which correspond to $n_{3} &gt; 1$, we shall according to the considerations in Part~I
expect that their a-priori probabilities are all equal.\Pagelabel{76}\footnote
  {By a simple enumeration it follows from this result, that the total number of different stationary
  states of the hydrogen atom, subject to a small homogeneous electric field, which corresponds to a stationary
  \Typo{states}{state} of the undisturbed atom, characterised by a given value of~$n$ in the condition $I = nh$,
  is equal to $n(n + 1)$. This expression is directly obtained, if we remember that $n = n_{1} + n_{2} + n_{3}$
  and if we count each state, characterised by a given combination of the positive integers $n_{1}$,~$n_{2}$,~$n_{3}$, as
  double, corresponding to the two possible opposite directions of rotation of the electron round the axis
  of the field. With reference to the necessary stability for a small variation of the external conditions
  of the statistical distribution of the values of the energy among a large number of atoms in temperature
  equilibrium (see Note on \PageRef[page]{43}), it will be seen that the expression $n(n + 1)$ may be taken as
  a measure for \Emph{the relative value of the a-priori probability of the different stationary
  states of the undisturbed hydrogen atom}, corresponding to different values of~$n$. The problem of
  the determination of this a-priori probability has been discussed by \Name{K.~Herzfeld} (Ann.\ d.\ Phys.\
  LI, p.~261 (1916)) who, by an examination of the volumes of the different extensions in the phase space
  which might be considered as belonging to the different stationary states of the hydrogen atom, has
  arrived at an expression for the a-priori probability of these states which differs from the above. From
  the point of view, as regards the principles of the quantum theory, taken in the present paper, a consideration
  of this kind, however, does not, as explained in Part~I on \PageRefI[page]{26}, afford a rational means
  of determining the a-priori probability of the stationary states of an atomic system.}

As regards the comparison between the theory and the experiments, it will be
remembered that \Name{Stark} found that every hydrogen line in the presence of an
electric field was split up in a number of polarised components, in a way different
for the different lines. When viewed parallel to the direction of the field, there appeared
a number of components polarised parallel to the field and a number of components
polarised perpendicular to the field; when viewed in the direction of the
field, only the latter components appeared, but without showing characteristic
polarisation. Apart from the marked symmetry of the resolution of every line,
the distances between successive components and their relative intensities varied
in an apparently irregular way from component to component. As pointed out by
\Name{Epstein} and \Name{Schwarzschild}, however, it is possible by means of~\Eq{(78)}, in connection
with relation~\EqI{(1)}, to account in a convincing way for \Name{Stark}'s measurements as regards
the \Emph{frequencies} of the components. Especially a closer examination of these measurements
showed that all the differences between the frequencies of the components
were equal to entire multipla of a certain quantity, which was the same for all lines
in the spectrum and, within the limits of experimental errors, equal to the theoretical
value $\dfrac{3hF}{8\pi^{2} Nem}$. On the other hand, the theories of \Name{Epstein} and \Name{Schwarzschild} gave
no direct information as regards the question of the polarisation and intensity
of the different components. Comparing formula~\Eq{(78)} with \Name{Stark}'s observations,
\Name{Epstein} pointed out, however, that the \Emph{polarisation} of the different components
observed could apparently be accounted for by the rule: that a transition between
two stationary states gives rise to a component polarised parallel to the field, if $n_{3}$~remains
unchanged or is changed by an even number of units; while a component,
corresponding to a transition in which $n_{2}$~is changed by an uneven number of units,
\PageSep{77}
is polarised perpendicular to the field. This result may be simply interpreted on
the basis of the general formal relation between the quantum theory of line
spectra and the ordinary theory of radiation. In fact, it was shown in Part~I
that, for a conditionally periodic system possessing an axis of symmetry, we
shall expect only two types of transitions to be possible. In transitions of the
first type $n_{3}$~remains unchanged, and the emitted radiation is polarised parallel to
the axis of symmetry, while the transitions of the second type, in which $n_{3}$~varies
by one unit, give rise to a radiation of circular polarisation in a plane perpendicular
to this axis (see \PageRefI[page]{34}). In order to show that this agrees with the empirical rule
of \Name{Epstein}, it may be noted in the first place that, for any component which might
be ascribed to a certain transition in which $n_{3}$~changes by a given entire number
of units, there exists always another transition which will give rise to a radiation
of the same frequency but in which $n_{3}$~remains unchanged or changes by one
unit, according to whether the given number is even or uneven. Next it will be
seen that, in case of the effect of an electric field on the hydrogen spectrum, we
cannot detect by means of direct observations the circular polarisation of the
radiation corresponding to transitions of the second type; because, for each transition
giving rise to a radiation of circular polarisation in one direction, there will
exist another transition giving rise to a radiation which possesses the same frequency
but is polarised in the opposite direction. Besides on the problem of the
polarisations of the different components into which the hydrogen lines are split up
in the presence of the electric field, the general considerations in Part~I allow also to
throw light on the question of the \Emph{relative intensities} of these components, by
considering the harmonic vibrations into which the motion of the electron in the
stationary states can be resolved. Compared with the problem of the relative intensities
of the components of the fine structure of the hydrogen lines, the present
problem is simpler in that respect, that the stationary states may be assumed to
be a-priori equally probable. Since the different components, into which a given
hydrogen line is split up in the electric field, correspond to transitions between pairs of
states which for all components have very nearly the same values for the total energy,
these states may therefore be expected to be of approximately equal occurrence in
the luminous gas. According to the considerations in Part~I, we shall consequently
assume that for a given hydrogen line the relative intensities of the different \Name{Stark}
effect components, corresponding to transitions between different pairs of stationary
states characterised by $n_{1} = n_{1}'$, $n_{2} = n_{2}'$, $n_{3} = n_{3}'$ and $n_{1} = n_{1}''$, $n_{2} = n_{2}''$, $n_{3} = n_{3}''$
respectively, will be intimately connected with the intensities of the radiations of
frequency $(n_{1}' - n_{1}'') \omega_{1} + (n_{2}' - n_{2}'') \omega_{2} + (n_{3}' - n_{3}'') \omega_{3}$, which on ordinary electrodynamics
would be emitted by the atom in the two states involved in the transition
in question; $\omega_{1}$,~$\omega_{2}$, and~$\omega_{3}$ being the fundamental frequencies entering in the expression~\EqI{(31)}
for the displacement of the electron. In order to test how far such
a connection is actually brought out by the observations, it is necessary to determine
the numerical values of the amplitudes of the harmonic vibrations into which the
% [** TN: Printer's mark]
% D. K. D. Vidensk. Selsk. Skr., naturvidensk. og mathem. Afd., 8. Række, IV. 1. 11.
\PageSep{78}
motion of the electron can be resolved. The examination of this problem has been
undertaken by Mr.\ \Name{H.~A. Kramers}, who has deduced complete expressions for these
amplitudes, by means of which it was found possible, for each of the hydrogen
lines $H_{\alpha}$,~$H_{\beta}$,~$H_{\gamma}$ and~$H_{\delta}$, to account in a convincing way for the apparently capricious
laws which govern the intensities of the components observed by \Name{Stark}.\footnote
  {\Emph{Note added during the proof.} In recent papers \Name{H.~Nyquist} (Phys.\ Rev.\ X, p.~226 (1917))
  and \Name{J.~Stark} (Ann.\ d.\ Physik, LVI, p.~569 (1918)) have published measurements on the effect of an
  electric field on certain lines of the helium spectrum which is given by~\Eq{(35)}, if in~\Eq{(40)} we put $N = 2$.
  As will be seen from~\Eq{(78)}, the differences between the frequencies of the components into which these
  lines are split up will, for the same intensity of the external electric field, be smaller than for the
  hydrogen lines. In conformity with this it was not possible, with the experimental arrangement used
  by the authors mentioned, to observe separately the numerous components to be expected on the
  theory, but only to obtain certain rough features of the resolution of the lines in question. For the
  interpretation of these observations a detailed consideration of the relative intensities to be expected
  for the different theoretical components is therefore essential; and, as it will be shown in \Name{Kramers}'
  paper, it is possible, on the basis of the calculation of the amplitudes of the harmonic vibrations into
  which the motion of the electron in the stationary states can be resolved, to account satisfactorily for
  \Name{Nyquist}'s and \Name{Stark}'s results.}
This agreement
offered at the same time a direct experimental support for the conclusions
mentioned above: that there exist no stationary states corresponding to $n_{3} = 0$, while
the stationary states corresponding to other values of~$n_{2}$ are a-priori equally probable;
and that transitions can only take place between pairs of stationary states
for which $n_{3}$~is the same or differs by one unit. A general discussion of these problems
will be given by \Name{Kramers} in the paper, mentioned on \PageRef[page]{69} in the last
section, in which also the problem of the intensity of the fine structure components
is treated in detail.

In the former section and in the present we have seen, how the problems of
the influence of the relativity modifications on the lines of the hydrogen spectrum
and of the influence of an external electric field on this spectrum can be treated,
by regarding the motion of the electron as a perturbed periodic motion, and by
fixing the stationary states on the basis of the relation between the energy and the
frequencies of the secular perturbations. As it was done originally by \Name{Sommerfeld}
and \Name{Epstein}, both these problems can also be treated by means of the theory of
the stationary states of conditionally periodic systems which allow of separation
of variables in a fixed set of positional coordinates. If, however, we consider the
problem of \Emph{the simultaneous influence on the hydrogen spectrum of
the relativity modifications and a homogeneous electric field} of any
given intensity, there does not exist a set of coordinates for which a separation of
variables can be obtained. On the other hand it is possible, also in this case, to
apply the general considerations about perturbed periodic systems developed in the
preceding. In fact, with reference to the treatment given in~\SecRef{3} of the problem of
the fine structure of the hydrogen lines, it will be seen that the deviations of the
orbit of the electron from a Keplerian ellipse in the problem under consideration
\PageSep{79}
will be the same as the secular perturbations produced on a Keplerian motion by
the simultaneous influence of an external homogeneous field of force and an external
central force proportional to the inverse cube of the distance from the nucleus.
Since these two fields together form a perturbing field possessing axial symmetry,
it follows therefore that the secular perturbations, when the relativity \Typo{modications}{modifications}
are taken into account, will be conditionally periodic and that the problem of the
stationary states may be treated by means of the method mentioned in~\SecRef{2} on
\PageRef[page]{55}. In this way we obtain in the first place the result, that, for any value of
the intensity of the external electric field, we must expect that the hydrogen lines
will be split up in a number of sharp components. Next, since for any value of
this intensity different from zero the system will be non-degenerate, it follows from
the conditions~\Eq{(61)}, that we must assume that the angular momentum round the
axis of the field is always equal to an entire multiple of~$h/2\pi$; in consistence with
the assumption of the validity of the analogous condition involved in the fixation
of the stationary states by means of the method of separation of variables, when
applied to an explanation of the \Name{Stark} effect with neglect of the relativity modifications
(compare \PageRef[page]{74}). On the basis of the conditions~\Eq{(61)} it is possible to
predict in detail, how the fine structure of the hydrogen lines will be influenced
by an increasing electric field until, for a sufficiently large intensity of this field,
the phenomenon develops gradually into the ordinary \Name{Stark} effect. The problem
of this transmutation will be treated in a later paper by Mr.\ \Name{H.~A. Kramers},\footnote
  {Besides the discussion of this problem, the paper in question will contain a general treatment
  of the theory of perturbed periodic systems from the point of view of the possibility of describing
  the motion by means of angle variables (compare Note on \PageRef[page]{58}).}
who
has kindly drawn my attention to this interesting application of the method of
perturbations, and has thereby given a valuable impetus to the detailed elaboration
of this method as regard the treatment of more complicate problems.


\Section{5.}{The effect of a magnetic~field on~the~hydrogen~spectrum.}

A theory of the \Name{Zeeman} effect of the hydrogen fines based on the quantum
theory of line spectra has, as mentioned in the introduction, been given independently
by \Name{Sommerfeld} and by \Name{Debye}. The calculations of these authors rest upon
the fact, that it is possible, also in the presence of a magnetic field, to write the
equations of motion of the electron in the canonical Hamiltonian form given by~\EqI{(4)},
if the momenta $p_{1}$,~$p_{2}$,~$p_{3}$ which are conjugated to the positional coordinates of
the electron $q_{1}$,~$q_{2}$,~$q_{3}$, are defined in a suitable way. In complete analogy to the
problem of the fixation of the stationary states of an atomic system when the relativity
modifications are taken into account, it follows therefore that, if these equations
\PageSep{80}
can be solved by the method of separation of variables, we obtain, by fixing the
stationary states by means of the conditions~\EqI{(22)}, a relation between the total energy
of the atom in the presence of a magnetic field and the fundamental frequencies
characterising the motion of the electron, which is exactly the same as that holding
between the energy and frequencies in the stationary states of an ordinary conditionally
periodic system. By a procedure analogous to that applied by \Name{Burgers}
in his proof of the mechanical invariance of the relations~\EqI{(22)} for slow changes of
the external conditions, mentioned in Part~I on \PageRefI[page]{21}, it may further be proved
that also in the presence of a magnetic field these relations are invariant, when
regard is taken to the effect of the induced electric forces which, according to the
ordinary theory of electrodynamics, will accompany a variation of the magnetic
field. In the following, however, we shall not treat the problem of the influence of
an external magnetic field on the hydrogen spectrum by means of the method of
separation of variables, but in analogy to the treatment of the problems of the fine
structure and of the \Name{Stark} effect of the hydrogen lines, given in the preceding
sections, we shall treat the problem from the point of view of the theory of perturbed
periodic systems. Before entering on the detailed discussion of the necessary
modifications to be introduced in the general considerations in~\SecRef{2}, in order that
they may be applied also to the problem of the fixation of the stationary states of
the atom in the presence of external magnetic forces, we shall for the sake of
illustration first show how it is possible in certain cases to treat the problem of
the effect of a homogeneous magnetic field on the hydrogen spectrum in a simple
way, which will be seen to present a close formal analogy with the theory originally
devised by \Name{Lorentz} on the basis of the classical theory of electrons.

In these considerations we shall make use of a well known theorem of \Name{Larmor},
which states that, if we look apart from small quantities proportional to the
square of the intensity of the magnetic field, the motion of a system of electrons
moving in a conservative field of force possessing axial symmetry round a fixed
axis will, in the presence of an external homogeneous magnetic field parallel to
this axis, differ from a mechanically possible motion of the system without field,
only by a superposed uniform rotation of the entire system round the axis, the
frequency of which is given by
\[
\frakv_{H} = \frac{e}{4\pi me}H,
\Tag{(79)}
\]
where $H$~is the intensity of the magnetic field and $c$~the velocity of light, while $-e$~and
$m$ represent the charge and the mass of an electron.\footnote
  {\Name{J.~Larmor}, Aether~\&amp; Matter, Cambridge 1900, p.~341. This theorem, which was established in
  connection with an attempt to develop a general theory of the \Name{Zeeman} effect based on the ordinary theory
  of electrodynamics, is directly proved by observing that, with the degree of approximation in question,
  the accellerations of the electrons due to the presence of the magnetic field are equal to the changes in
  the accellerations of the particles due to the superposed rotation of the system.}
If the magnetic field
\PageSep{81}
is not constant, but if its intensity increases slowly and uniformly from zero, it is
further simply shown that the electric induction forces, which will accompany the
change in the intensity of the magnetic force, will just effect that a rotation as
that described will be impressed on the original motion of the system.\footnote
  {Compare \Name{P.~Langevin}, Ann.\ de Chim\Add{.}\ et de Phys.\ V, p.~70 (1905), who has deduced this result in
  connection with his well known theory of the magnetic properties of atomic systems based on the
  classical theory of electrons.}
Moreover,
as regards the effect of the magnetic field on the total energy of the system,\footnote
  {In an earlier paper (Phil.\ Mag.\ XXVII, p.~506 (1914)) the writer had assumed that the total
  energy in the stationary states of the hydrogen atom in the presence of a magnetic field would not be
  different from the energy in the corresponding states without field, as far as small quantities proportional
  to the intensity of the magnetic force are concerned; the effect on the kinetic energy of the electron
  due to the superposed rotation being assumed to be compensated by some kind of ``potential'' energy of
  the whole atom relative to the magnetic field. This assumption seemed not only suggested by the absence
  of \Emph{paramagnetism} in many elements, the atoms and molecules of which, according to the
  theory to be discussed in Part~IV, must be expected to possess a resultant angular momentum, but it
  was especially thought to be supported by the fact, that the spectrum, emitted by hydrogen in the
  presence of a magnetic field, apparently did not form a combination spectrum of the type which should
  be expected, it the frequency of the radiation, emitted during a transition between two stationary states
  of the atom in the presence of the field, could be calculated directly from the values of the energy in
  these states by means of relation~\EqI{(1)}. As remarked by \Name{Debye} (Phys.\ Zeitschr.\ XVII, p.~511 (1916)), this
  view, however, would not be reconcilable with \Name{Einstein}'s theory of temperature radiation (see Part~I,
  \PageRefI[page]{7}) which implies the general validity of relation~\EqI{(1)}; and, moreover, as will be shown in the following,
  the \Name{Zeeman} effect of the hydrogen lines may actually be considered, not as involving a deviation
  from the combination principle, but rather as affording an instructive example of a systematic disappearance
  of certain possible combination lines, for which a simple explanation can be obtained from
  a consideration based on the general formal relation between the quantum theory of line spectra and
  the ordinary theory of radiation. Further, with reference to this relation---and remembering that on
  ordinary electrodynamics the magnetic field will not directly influence the exchange of energy during
  a process of radiation, since the forces due to this field, being always perpendicular to the direction
  of the velocity, will not perform work on the moving electron---it seems also natural to assume that
  it is possible, simply from the effect of the superposed rotation on the kinetic energy of the electron,
  to determine the effect of the magnetic field, as regards the \Emph{differences} between the values of the
  energy in the different stationary states of the atom. Now, in a discussion of the spectrum to be expected
  on the quantum theory, we are concerned only with these differences and not with the
  \Emph{absolute values} of the additional energy of the system due to the presence of the magnetic field.
  It would therefore be possible to escape from the difficulty, mentioned above, as regards the absence
  of paramagnetism, by assuming that only the energy in the \Typo{socalled}{so called} ``normal'' state of an atomic
  system (\ie\ the stationary state of the system which possesses the smallest value for the total energy;
  see Part~IV) is not altered in the presence of a magnetic field, as far as small quantities proportional
  to the intensity of the magnetic force are concerned. On this view, the absence of paramagnetism would
  thus be a special property of the normal state, connected with the impossibility of spontaneous transitions
  from this state to other stationary states of the system. To this question we shall come back in
  the following parts of this paper; for the sake of simplicity, however, we shall not, in the considerations
  of this section, enter more closely on the consequences of the mentioned hypothesis, which would
  imply small modifications in the form of the following considerations, but would not affect the results.}
it will\Pagelabel{81}
be observed that the superposed rotation under consideration will not affect the
mutual potential energy of the particles, while, with neglect of small quantities proportional
to~$H^{2}$, it will produce a change in the kinetic energy equal to~$2\pi P\frakv_{H}$,
\PageSep{82}
where $P$~represents the total angular momentum of the system round the axis,
taken in the same direction as that of the superposed rotation.

From these results it follows that the motion of the electron in any stationary
state of a hydrogen atom, which is exposed to a \Emph{homogeneous magnetic
field}, will---if we look apart from small quantities proportional to the square
of the intensity of the magnetic force and to the product of this intensity with the
ratio between the mass of the electron and that of the nucleus---differ from the
motion in some stationary state of the atom in the absence of the field, only by
a superposed uniform rotation round an axis through the nucleus parallel to the
magnetic force with a frequency given by~\Eq{(79)}. Due to the degenerate character of
the system formed by the atom in the absence of the magnetic field, it is not possible,
however, from a consideration of the mechanical effect produced on the
motion of the electron by a slow and uniform establishment of the magnetic field,
to fix the stationary states of the perturbed atom completely, but in order to fix
these states we must consider more closely the relation between the additional
energy of the system due to the presence of the magnetic field and the character
of the secular perturbations produced by this field on the orbit of the electron.
On the basis of \Name{Larmor}'s theorem the discussion of this problem is very simple.
In fact, since the frequency~$\frakv_{H}$ is independent of the shape and position of the
orbit, we may proceed in a manner which is completely analogous to that applied
in the fixation of the stationary states of the hydrogen atom in the presence of a
homogeneous electric field. Thus, looking apart from the effect of the relativity
modifications, we may conclude at once that the total energy in the stationary
states of the atom will be given by
\[
E = E_{n} + \frakn \frakv_{H}h,
\Tag{(80)}
\]
where $\frakn$~is an entire number which can be positive as well as negative, while $E_{n}$~will
be equal to the energy in the corresponding stationary state of the undisturbed
atom, which is given by $-W_{n}$~in~\Eq{(41)}. As in the case of the \Name{Stark} effect,
it will moreover be seen that this formula includes the values of the energy in
such states of the atom, in which the electron moves in a circular orbit perpendicular
to the direction of the field, and which beforehand must be expected to be
included among the stationary states of the perturbed system, since such orbits
during a slow and uniform establishment of the external field will not undergo
secular perturbations as regards shape and position (compare \PageRef[page]{73}). In fact,
since in these cases we have $P = ±nh/2\pi$, where $n$~is the entire number characterising
the stationary states of the undisturbed hydrogen atom, it follows from the
above that the total energy in the special stationary states under consideration will
just be represented by the formula~\Eq{(80)}, if we put $\frakn = ±n$. From this formula it
will be seen at the same time, that the presence of the external magnetic field imposes
the restriction on the motion in the stationary states of the hydrogen atom,
that, with neglect of small quantities proportional to~$H$, the angular momentum
\PageSep{83}
of the electron round the axis of the field will be equal to an entire multiple
of~$h/2\pi$.

As regards the expression for the total energy of the hydrogen atom in the
presence of the magnetic field, formula~\Eq{(80)} is in agreement with the formulæ obtained
by \Name{Sommerfeld} and \Name{Debye} on the basis of the conditions~\EqI{(22)}, holding for conditionally
periodic systems which allow of separation of variables. As shown by these
authors, a system, which consists of an electron moving under the influence of
the attraction from a fixed nucleus and of a homogeneous magnetic field, allows of
separation of variables in polar coordinates, if the polar axis is chosen parallel to
the magnetic field. Looking apart from the effect of the relativity modifications,
and choosing for $q_{1}$,~$q_{2}$, and~$q_{3}$ the length of the radius vector from the nucleus to
the electron, the angle between this radius vector and the axis of the system, and
the angle which the plane through the electron and this axis makes with a
fixed plane through the axis respectively, they obtain the following expression for
the total energy:\footnote
  {\Name{A.~Sommerfeld}, Phys.\ Zeitschr.\ XVII, p.~491 (1916) and \Name{P.~Debye}, Phys.\ Zeitschr.\ XVII, p.~507
  (1916). While \Name{Debye} proceeds directly by the application of the conditions~\EqI{(22)} in a fixed set of positional
  polar coordinates, \Name{Sommerfeld} determines the stationary states by applying these conditions to
  the motion of the system relative to a set of coordinates which rotates uniformly round the polar axis
  with the frequency~$\frakv_{H}$; a procedure which in the special case under consideration is simply shown to
  give the same result as the direct application of~\EqI{(22)} to fixed polar coordinates.}
\[
E = -\frac{2\pi^{2} N^{2} e^{4} m}{h^{2} (n_{1} + n_{2} + n_{3})^{2}}
  ± \frac{ehn_{3}}{4\pi mc} H,
\Tag{(81)}
\]
where $n_{1}$,~$n_{2}$, and~$n_{3}$ are the integers which appear as factors to \Name{Planck}'s constant
on the right side of the conditions~\EqI{(22)}. As mentioned this formula gives the same
result as~\Eq{(80)}; in fact, if we put $n = n_{1} + n_{2} + n_{3}$ and if we look apart from the
small correction due to the finite mass of the nucleus, the first term in~\Eq{(81)} is seen
to coincide with the expression for~$-W_{n}$ given by~\Eq{(41)}, while the last term in~\Eq{(81)}
coincides with the last term in~\Eq{(80)}, if we put $\frakn = n_{3}$. It will be observed, however, that,
while in the theories of \Name{Sommerfeld} and \Name{Debye} the stationary states are characterised
by three conditions, only two conditions were necessary on the above considerations
in order to secure the right relation between the energy and frequencies
of the system in the stationary states. Thus, besides the conditions which prescribe
the length of the major axis of the rotating orbit and the value of the angular
momentum of the system round the axis of the field, the theories of the mentioned
authors involve the further condition, that the value of the total angular momentum
of the electron round the nucleus must be equal to an entire multiple of~$h/2\pi$; and
that consequently the minor axis of the orbit has the same values as in a hydrogen
atom perturbed by a small external central field (compare \PageRef[page]{57}). This is due to
the circumstance, that the perturbed atom forms a \Emph{degenerate system} if we
look apart from the effect of the relativity modifications, because the secular perturbations
\PageSep{84}
are simply periodic. From the point of view of separation of variables,
this degenerate character of the system is in the present case, in contrast to the
analogous case of the \Name{Stark} effect, also directly revealed by the fact, that a separation
can be obtained, not only in polar coordinates,\Pagelabel{84} but in any set of axial elliptical
coordinates for which one focus is placed at the nucleus and the other at
some point on the axis of the field. Just as in the case of the \Name{Stark} effect, however,
the system is no more degenerate as soon as the relativity modifications are
taken into account, in which case a separation of variables will still be possible
but only in polar coordinates. To this point we shall come back below.

The observations on the \Name{Zeeman} effect of the hydrogen lines show that, if the
fine structure is neglected, each line is in the presence of a magnetic field split up
in a normal \Name{Lorentz} triplet; \ie\ each line is resolved in three components of
which the one is undisplaced and polarised parallel to the direction of the field,
while the two other components possess frequencies, which differ from that of the
original line by~$\frakv_{H}$, and are circularly polarised in opposite directions in a plane
perpendicular to the direction of the field. As pointed out by \Name{Sommerfeld} and by
\Name{Debye}, the frequencies of a \Name{Lorentz} triplet are included among the frequencies of the
components deduced from~\Eq{(81)} by application of relation~\EqI{(1)}. In addition to the
observed components, however, we might from \Eq{(81)}~and~\EqI{(1)} expect the appearance
of a number of components, displaced from the original positions of the lines by
higher multipla of~$\frakv_{H}$. For the non-appearance of these components the theories of
\Name{Sommerfeld} and \Name{Debye} offered no explanation, no more than for the polarisation of
the components observed; except that \Name{Sommerfeld} in this connection draws attention
to the fact, that the law governing the observed polarisations exhibits a certain
analogy to the \Typo{emperical}{empirical} rule of \Name{Epstein} concerning the observed polarisations of
the components of the \Name{Stark} effect of the hydrogen lines (see \PageRef[page]{76}). On the other
hand, just as in case of the latter effect, an explanation of the number of the components
observed and their characteristic polarisations is directly obtained on the
basis of the general formal relation between the quantum theory of line spectra and
the ordinary theory of radiation. In the first place we have at once from \Name{Larmor}'s
theorem, denoting the frequency of revolution of the electron in a stationary state of
the undisturbed hydrogen atom by~$\omega$, that the motion of the electron, in a corresponding
stationary state of the atom in the presence of the field, may be resolved
in a number of linear harmonic vibrations parallel to the direction of the magnetic
force with frequencies~$\tau\omega$, where $\tau$~is a positive integer, and in a number of circular
harmonic rotations perpendicular to this direction with frequencies $\tau\omega + \frakv_{H}$ or $\tau\omega - \frakv_{H}$,
according as the direction of rotation is the same as or the opposite of that of the
superposed rotation. Next, with neglect of small quantities proportional to~$H^{2}$ we
have for the difference in the total energy between two neighbouring states of the
perturbed system under consideration
\[
\delta E = \delta E_{0} + \delta \frakE
  = \omega\, \delta I + \frakv_{H}\, \delta \frakI,
\Tag{(82)}
\]
\PageSep{85}
where $E_{n}$~and~$\omega$ are the values of the energy and frequency and $I$~is the value of the
quantity defined by~\EqI{(5)}, all corresponding to the state of the undisturbed system
which would appear if the magnetic force vanished at a slow and uniform rate,
while $\frakE$~is the additional energy due to the presence of the magnetic field and $\frakI$~the
angular momentum of the system round the axis of the field multiplied by~$2\pi$
and taken in the same direction as that of the superposed rotation. Since \Eq{(82)}~has
exactly the same form as relation~\Eq{(66)}, and since in the stationary states we have
$I = nh$ and $\frakI = \frakn h$, we are therefore from a consideration, quite analogous to
that given in~\SecRef{2} on \PageRef[page]{59}, led to the conclusion, that, in the presence of
the magnetic field, only two types of transitions between stationary states are
possible. For both types of transitions the integer~$n$ may change by any number
of units, but in transitions of the first type the integer~$\frakn$ will remain constant and
the emitted radiation will be polarised parallel to the direction of the field, while
in transitions of the second type $\frakn$~will decrease or increase by one unit and the
emitted radiation will be circularly polarised in a plane perpendicular to the field, the
direction of the polarisation being the same as or the opposite of that of the superposed
rotation respectively. Remembering that, with neglect of small quantities proportional
to the magnetic force, the angular momentum of the system round the
axis of the field remains unaltered in transitions of the first type and changes by~$h/2\pi$
in transitions of the second type, it will be seen that this conclusion is independently
supported by a consideration of conservation of angular momentum
during the transitions, like that given in Part~I on \PageRefI[page]{34}.

With reference to formula~\Eq{(80)}, it will be seen that the above results are in
complete agreement with the experiments on the \Name{Zeeman} effect of the hydrogen
lines, as regards the \Emph{frequencies and polarisations} of the observed components.
On the other hand, the observed \Emph{intensities} are directly accounted for, independent
of any special theory about the origin of the lines. In fact, from a consideration of
the necessary ``stability'' of spectral phenomena, it follows that the total radiation of
the components, in which a spectral line, which originally is unpolarised, is split up
in the presence of a small external field, cannot show characteristic polarisation
with respect to any direction. In case of the \Name{Zeeman} effect of the hydrogen lines, it
is therefore necessary beforehand to expect that the intensity of the radiation,
summed over all directions, corresponding to each of the three components in
which every line is split up must be the same. From the point of view of the
quantum theory of line spectra, it will be seen that by means of considerations
of this kind we may inversely obtain a certain amount of direct quantitative information
as regards the probabilities of spontaneous transition between different
sets of stationary states, holding also in the region where the integers characterising
these states are not large and where consequently the estimate of the values of
these probabilities, based on the formal relation between the quantum theory and
the ordinary theory of radiation, gives results which are only of an approximative
character. This point will be discussed more closely in \Name{Kramers}' paper on the relative
% [** TN: Printer's mark]
% D. K. D. Vidensk. Selsk. Skr., naturvidensk. og mathem. Afd. 8. Række, IV. 1. 12
\PageSep{86}
intensities of the components of the fine structure and the \Name{Stark} effect of the
hydrogen lines.

A procedure quite analogous to that applied above may be used to treat the
problem of the effect of a homogeneous magnetic field on the hydrogen spectrum,
also when the relativity modifications are taken into account, and when the atoms
at the same time are exposed to a small external field of force of constant potential,
which possesses axial symmetry round an axis through the nucleus parallel to the
magnetic force; because also in this case we can obviously make direct use of
\Name{Larmor}'s theorem. We shall not, however, proceed in this way, but shall come
back to these questions when we have shown how, by a simple modification
of the general considerations of perturbed periodic systems given in~\SecRef{2}, it is possible
to represent the theory of the stationary states of the hydrogen atom in the
presence of a small magnetic field on a form, which allows to discuss the effect on
the hydrogen spectrum also if the atom is exposed to a magnetic field which is
not homogeneous, or to discuss the effect of a homogeneous magnetic field if electric
forces, which do not possess axial symmetry round an axis through the nucleus
parallel to the magnetic field, are acting on the atom at the same time.

In order to examine the general problem of the secular perturbations of the
orbit of the electron in the hydrogen atom which take place if the atom is exposed
to small external forces which, entirely or partly, are of magnetic origin,
we shall, as in the usual theory of planetary perturbations, take our starting point
in the equations of motion in their canonical form. Now the equations of motion
of an electron of charge~$-e$, which besides by an electric field of potential~$V$ is
acted upon by a magnetic field of vector potential~$\frakA$ (defined by $\div \frakA = 0$ and
$\curl \frakA = \frakH$, where $\frakH$~is the magnetic force considered as a vector), can be written
in the Hamiltonian form given by~\EqI{(4)}, if, just as in the absence of the magnetic
field, $E$~is taken equal to the sum of the kinetic energy~$T$ of the electron and its
potential energy~$-eV$ relative to the electric field, while the momenta which are
conjugated to the positional coordinates $q_{1}$,~$q_{2}$,~$q_{3}$ of the electron in space are
defined by the equations\footnote
  {See f.\ inst.\ \Name{G.~A. Schott}: Electromagnetic Radiation, App.~F (Cambridge, 1912).}
\[
p_{k}' = p_{k} - \frac{e}{c}\, \frac{\dd(\frakv \frakA)}{\dd \dot{q}_{k}},
\qquad
(k = 1, 2, 3)
\Tag{(83)}
\]
where the~$p$'s are the momenta defined in the usual way (compare \PageRefI[page]{10}), and
where $(\frakv \frakA)$~represents the scalar product of the velocity of the electron~$\frakv$ and the
vector potential~$\frakA$, considered as a function of the~$q$'s and of the generalised velocities
$\dot{q}_{1}$,~$\dot{q}_{2}$,~$\dot{q}_{3}$. If we now assume that the effect of the magnetic forces on the
motion of the electron is so small compared with the effect of the electric forces,
that in the calculations we may look apart from all terms proportional to~$\frakH^{2}$, it
is simply seen that the energy function~$E$ in~\EqI{(4)}, obtained by introducing the
momenta defined by~\Eq{(83)}, will differ from the corresponding function, holding in
\PageSep{87}
the absence of the magnetic field, only by the addition of a term which is linear
in the momenta and equal to~$\dfrac{e}{c}(\frakv \frakA)$. In fact, denoting $E$~expressed as a function
of the $q$'s~and~$p$'s by~$\phi(p, q)$, we get from~\Eq{(83)} together with~\EqI{(4)}, with the approximation
under consideration,
\[
\BreakMath{%
E - \phi(p', q)
  = -\sum_{1}^{3} \frac{\dd \phi}{\dd p_{k}'}(p_{k}' - p_{k})
  = \sum_{1}^{3} \frac{\dd E}{\dd p_{k}'}\, \frac{e}{c}\, \frac{\dd(\frakv \frakA)}{\dd \dot{q}_{k}}
  = \frac{e}{c} \sum_{1}^{3} \dot{q}_{k}\, \frac{\dd(\frakv \frakA)}{\dd \dot{q}_{k}}
  = \frac{e}{c} (\frakv \frakA).
}{%
\begin{aligned}
E - \phi(p', q)
  &amp;= -\sum_{1}^{3} \frac{\dd \phi}{\dd p_{k}'}(p_{k}' - p_{k})
  = \sum_{1}^{3} \frac{\dd E}{\dd p_{k}'}\, \frac{e}{c}\, \frac{\dd(\frakv \frakA)}{\dd \dot{q}_{k}} \\
  &amp;= \frac{e}{c} \sum_{1}^{3} \dot{q}_{k}\, \frac{\dd(\frakv \frakA)}{\dd \dot{q}_{k}}
  = \frac{e}{c} (\frakv \frakA).
\end{aligned}
}
\]

From this it follows that, with neglect of small quantities proportional to the
square of the magnetic forces, the perturbations of the orbit of the electron in a
hydrogen atom, which besides to a small external electric field of potential is
exposed to a small external magnetic field of vector potential~$\frakA$, are given by a
set of equations of the same form as~\Eq{(44)} in~\SecRef{2}, but where the $\alpha$'s~and~$\beta$'s are
replaced by a set of quantities $\alpha_{1}'$,~$\alpha_{2}'$,~$\alpha_{3}'$, $\beta_{1}'$,~$\beta_{2}'$,~$\beta_{3}'$, which are related to the $q$'s
and~$p'$'s and the time in the same way as the orbital constants $\alpha_{1}$,~$\alpha_{2}$,~$\alpha_{3}$, $\beta_{1}$,~$\beta_{2}$,~$\beta_{3}$
for the undisturbed atom are related to the $q$'s and~$p$'s and the time through the
equations~\EqI{(18)}, and where $\Omega$~is replaced by the expression $-e\Phi + \dfrac{e}{c} (\frakv \frakA)$, considered
as a function of the $\alpha'$'s~and $\beta'$'s and the time. Since now, at any moment, the
quantities $\alpha_{1}'$,~$\alpha_{2}'$,~$\alpha_{3}'$, $\beta_{1}'$,~$\beta_{2}'$,~$\beta_{3}'$ differ from the corresponding orbital constants $\alpha_{1}$,~$\alpha_{2}$,~$\alpha_{3}$, $\beta_{1}$,~$\beta_{2}$,~$\beta_{3}$ only by small terms proportional to the intensity of the magnetic
field, we see therefore that, with neglect of small quantities of the same order as
the variation in the orbital constants within a single period, the \Emph{secular perturbations
of the shape and position of the orbit of the electron will
again be given by the equations~\Eq{(46)}}, if in the present case $\psi$~is taken equal
to the sum of the mean value~$\psi_{E}$ of the potential energy~$-e\Phi$ of the electron
relative to the external electric forces and the mean value~$\psi_{M}$ of the quantity~$\dfrac{e}{c} (\frakv \frakA)$,
both taken over an osculating orbit corresponding to some moment during
the revolution and expressed as functions of $\alpha_{1}$,~$\alpha_{2}$,~$\alpha_{3}$, $\beta_{1}$,~$\beta_{2}$,~$\beta_{3}$.\footnote
  {If the relativity modifications are taken into account, the orbit of the electron in the undisturbed
  atom is not strictly periodic, but it will be seen that the secular variations of this orbit are
  still obtained from the equations~\Eq{(46)}, if only, to the expression for~$\psi$ as defined in the text, a term is
  added which is equal to the expression for~$\psi$ given by formula~\Eq{(70)} in~\SecRef{3}.}
The latter mean
value, however, is easily seen to allow of a simple interpretation. In fact, we have
\[
\psi_{M} = \frac{e}{c}\, \frac{1}{\sigma} \int_{0}^{\sigma} (\frakv \frakA)\, dt
  = -\frac{e\omega}{c} B,
\Tag{(84)}
\]
where $\omega$~is the frequency of revolution of the electron in the osculating orbit, and
where $B$~represents the total flux of magnetic force through this orbit, taken in
\PageSep{88}
the same direction as that of the magnetic force which would arise from the motion
of the electron according to ordinary electrodynamics.

From the considerations in~\SecRef{2} it follows now in the first place that, with
neglect of small quantities proportional to the square of the external forces, $\psi = \psi_{E} + \psi_{M}$
will remain \Emph{constant} during the perturbations within a time interval, sufficiently
long for the perturbing forces to produce a considerable change in the shape and
position of the orbit of the electron; \ie\ in a time interval of the same order as~$\sigma/\lambda$,
if $\lambda$, just as in~\SecRef{2}, denotes a small quantity of the same order as the ratio
between the external forces acting on the electron and the attraction from the nucleus.
From a consideration analogous to that given in~\SecRef{2}, we may further conclude that,
in the stationary states of the perturbed system, the quantity $\psi = \psi_{E} + \psi_{M}$ may
be taken equal to the \Emph{additional energy} of the system due to the presence of the
external fields. In fact, let us imagine that these fields are slowly established at a
uniform rate within a time interval from $t = 0$ to $t = \theta$, where $\theta$~is a quantity
of the same order as~$\sigma/\lambda$. For the total alteration in the inner energy of the system
during this process we get then, with neglect of small quantities proportional to~$\lambda^{2}$,
\[
\Delta_{\theta} \alpha_{1}
  = e\int_{0}^{\theta} \frac{t}{\theta} \sum_{1}^{3} \frac{\dd \Phi}{\dd q_{k}}\, \dot{q}_{k}\, dt
  - \frac{e}{c} \int_{0}^{\theta} \frac{\omega B}{\theta}\, dt,
\]
where the first term represents the work done on the system by the slowly increasing
external electric forces, while the second term represents the work performed
by the induced electric forces which accompany the variation in the intensity
of the magnetic field. By partial integration of the first term, we get from
this equation, with the approximation under consideration,
\[
\BreakMath{%
\Delta_{\theta} \alpha_{1} - e\Phi_{\theta}
  = -\frac{e}{\theta} \int_{0}^{\theta} \left(\Phi + \frac{\omega}{c}\, B\right) dt
  = \frac{1}{\theta} \int_{0}^{\theta} (\psi_{E} + \psi_{M})\, dt
  = \frac{1}{\theta} \int_{0}^{\theta} \psi\, dt.
}{%
\begin{aligned}
\Delta_{\theta} \alpha_{1} - e\Phi_{\theta}
  &amp;= -\frac{e}{\theta} \int_{0}^{\theta} \left(\Phi + \frac{\omega}{c}\, B\right) dt \\
  &amp;= \frac{1}{\theta} \int_{0}^{\theta} (\psi_{E} + \psi_{M})\, dt
  = \frac{1}{\theta} \int_{0}^{\theta} \psi\, dt.
\end{aligned}
}
\Tag{(85)}
\]
Now the expression on the left side of this equation is equal to the change in the
total energy of the system due to the establishment of the external field. Since the
expression on the right side is seen to be a small quantity of the same order as~$\lambda \alpha_{1}$,
it follows therefore from\Eq{(85)} in the first place that the secular variations of
$\alpha_{2}$,~$\alpha_{3}$, $\beta_{2}$,~$\beta_{3}$ during the increase of the fields will, just as in the case considered
in~\SecRef{2} (see \PageRef[page]{47}), be given by a set of equations of the same form as (46), where
$\psi$~is replaced by~$\dfrac{t}{\theta} \psi$, and where again $\alpha_{1}$~may be considered as a constant. Also in
the present case it follows therefore that $\psi$~will remain constant during the establishment
of the external fields, and we see consequently that the expression on the
right side of~\Eq{(85)} will be simply equal to~$\psi$, a result which, with reference to the
principle of the mechanical transformability of the stationary states, leads to the
\PageSep{89}
conclusion mentioned above, that the value of the additional energy in the stationary
states of the perturbed system is given by the value of~$\psi$ in these states.

From the above considerations it follows that the problem of the stationary
states of the hydrogen atom in the presence of external electric and magnetic forces
may be treated in a manner, which is exactly analogous to that applied in~\SecRef{2}
in case of a periodic system exposed to a small external field of constant potential.
Thus, if the secular perturbations as determined by~\Eq{(46)} are of conditionally periodic
type, we shall expect that, with neglect of small quantities proportional to~$\lambda$, the
cycles of shapes and positions which the orbit of the electron passes through
in the stationary states of the perturbed system will be characterised by the
conditions~\Eq{(55)}, and that the possible values of the additional energy of the
atom in the stationary states will be fixed by these conditions with neglect of small
quantities proportional to~$\lambda^{2}$. We shall therefore conclude that, also in the presence
of external magnetic forces, the lines of the hydrogen spectrum will, if only the secular
perturbations are of conditionally periodic type, be split up in a number of sharp
components, the frequencies of which are determined by means of the conditions~\Eq{(67)}
together with relation~\EqI{(1)}. As regards the problem of the intensities and polarisation
of these components, we may further proceed in a way quite analogous to that followed
in~\SecRef{2}. In fact, if the secular perturbations are of conditionally periodic type,
the displacement of the electron in any given direction may be represented as a
sum of harmonic vibrations by an expression of the same type as~\Eq{(65)}. Moreover it
can be proved that the difference in the total energy of two neighbouring states
of the perturbed atom will again be given by the expression~\Eq{(66)}.\footnote
  {Compare Note on \PageRef[page]{58}. Also in the presence of small magnetic forces, it will be possible
  to describe the motion of the perturbed system by means of a suitably chosen set of angle variables,
  if only the secular perturbations are of conditionally periodic type.}
The general
considerations in~\SecRef{2} will therefore apply without alterations to the problem of
the intensity and polarisation of the components into which the hydrogen lines
are split up in the presence of small external forces, also if these forces are entirely
or partly of magnetic origin. Similarly, it will be seen that the effect on the spectrum
of a perturbed hydrogen atom, which will be due to the presence of a second
external field small compared with the first, also in this case may be discussed
directly by means of the considerations at the end of~\SecRef{2}.

We meet with a direct application of the preceding considerations, if the
hydrogen atom is exposed to \Emph{the simultaneous influence of an external
electric and an external magnetic field, which possess axial symmetry
round a common axis through the nucleus}. Introducing the same
set of orbital constants as described in~\SecRef{2} on \PageRef[page]{54}, we get in this case that
$\psi_{M}$, as well as~$\psi_{E}$, and consequently the function $\psi = \psi_{E} + \psi_{M}$ which enters in the
equations~\Eq{(46)}, will, besides on~$\alpha_{1}$, depend on $\alpha_{2}$,~$\beta_{2}$ and~$\alpha_{3}$ but not on~$\beta_{3}$. The general
character of the secular perturbations of the orbit of the electron will therefore be
the same as in the case, considered in~\SecRef{2}, where the atom is exposed only to an
\PageSep{90}
electric field of axial symmetry, and the conditions which fix the stationary states of the
perturbed atom will again be expressed by the relations~\Eq{(61)}. As regards the question
of the probability of spontaneous transition between the stationary states, we get
moreover, just as in~\SecRef{2}, from a consideration of the harmonic vibrations into which
the motion of the electron can be resolved, that only two types of transitions will
be possible; in transitions of the first type $\frakn_{2}$~remains unaltered, and the accompanying
radiation is polarised parallel to the direction of the common axis of the
perturbing fields; in transitions of the second type $\frakn_{2}$~decreases or increases by one
unit, and the accompanying radiation will be circularly polarised in a plane perpendicular
to this axis. In this connection it may be remarked, however, that the
number of components, into which a given hydrogen line is split up in the presence
of a magnetic field, will in general be double as large as the number of components
which appear in the presence of an external electric field of axial symmetry. In
fact, in the latter case the motions of the electron in two stationary states of the
perturbed atom, corresponding to the same value of~$n$, will be symmetrical with
respect to a plane through the axis, and these states will possess the same values
for the additional energy, if $\frakn_{1}$~is the same while the values of~$\frakn_{2}$ are numerically
equal but have opposite signs. On the other hand, if the atom is exposed also to
a magnetic field, this will not hold, because the value of the function~$\psi_{M}$, in contrast
to the value of~$\psi_{E}$, will not possess the same sign for two orbits which have
the same shape and position relative to the axis, but for which the direction of
revolution of the electron is reversed. Considering two states of the perturbed atom
for which the values of~$\frakn_{1}$ are the same and the values of~$\frakn_{2}$ are numerically equal
but have opposite signs, we get therefore, if the atom is exposed only to a magnetic
field of axial symmetry, that the values of the additional energy will be equal
with exception of the sign; while, if the atom is exposed to a magnetic as well
as to an electric field, the additional energy in two such states will in general differ
also as regards its numerical value. In contrast to what in general will take place
if the atom is exposed to an electric field of axial symmetry, it will thus be seen
that, if the hydrogen atom is exposed only to a magnetic field possessing axial
symmetry, the ensemble of components into which a given hydrogen line is resolved
will be completely symmetrical with respect to the position of the original line, as
regards the frequencies as well as the intensities and polarisations. Moreover it follows
from the above, that if we consider a hydrogen atom exposed to an electric
field of axial symmetry and imagine that an external magnetic field, which possesses
symmetry round the same axis, is gradually established, each component which
appears in the presence of the first field only will split up into two components,
in such a way that each component polarised parallel to the axis will split up into
two components of the same polarisation, while each component polarised perpendicular
to the axis, and which originally showed no polarisation when viewed
in a direction parallel to the axis, will split up into two components showing
circular polarisations in opposite directions. If the magnetic field is small, the new
\PageSep{91}
components will be placed symmetrically with respect to the position of the original
components and their intensities will be approximately equal, but when the perturbing
influence of the magnetic forces on the motion of the electron becomes of
the same order of magnitude as that of the external electric forces, the components
in question will in general be placed unsymmetrically with respect to their original
position, and their intensities may differ considerably.

An\Pagelabel{91} especially simple example of a magnetic field which possesses axial symmetry
is afforded by the case of a \Emph{homogeneous magnetic field}, discussed in the
beginning of this section. In this case we have that the total magnetic flux of force
through the orbit of the electron is equal to the product of the intensity~$H$ of the
magnetic field and the area of the projection of the orbit on a plane perpendicular
to this field. Since this area is equal to~$\alpha_{3}/2m\omega$, we get consequently from~\Eq{(84)}
\[
\psi_{M} = \frac{e\alpha_{3}}{2cm} H.
\Tag{(86)}
\]
From the equations~\Eq{(46)} it follows therefore that the effect of a homogeneous magnetic
field, which acts upon a hydrogen atom which at the same time is exposed
to an external electric field possessing axial symmetry round an axis through the
nucleus parallel to the magnetic force, will consist in a superposition of a uniform
rotation of the orbit round the axis with a frequency equal to
\[
\frakv_{H}
  = \frac{1}{2\pi}\, \frac{\dd \psi_{M}}{\dd \alpha_{3}}
  = \frac{e}{4\pi mc} H
\]
on the secular perturbations which would take place in the absence of the magnetic
field. This result follows also directly from \Name{Larmor}'s theorem, on which the simple
considerations about the effect of a homogeneous magnetic field in the beginning
of this section were based. Since a superposed rotation as that in question will
not influence the shape of the orbit of the electron or its position relative to the
axis, it follows from~\Eq{(61)} that the value of~$\psi_{E}$ in the stationary states of the atom
will not be affected by the presence of the magnetic field, and that consequently the
effect of this field on the additional energy of the system will simply consist in the
addition of a term given by
\[
\psi_{M} = \frac{e}{2mc}\, \frac{\frakn_{2} h}{2\pi} H
  = \frakn_{2} \frakv_{H} h.
\Tag{(87)}
\]
This result was also to be expected from a simple consideration of the mechanical
effect produced on the motion by a slow and uniform establishment of the magnetic
field (compare \PageRef[page]{81}). With reference to the above considerations as regards
the probability of transition between stationary states, it will be seen to follow from~\Eq{(87)},
that the presence of the homogeneous magnetic field will leave the components
polarised parallel to the axis unaltered, but will cause every component, which in
the absence of the field was polarised perpendicular to the axis, to split up in a
\PageSep{92}
symmetrical doublet the members of which will show circular polarisation in opposite
directions, when viewed in the direction of the axis, and will be displaced
from the position of the original component by an amount corresponding to a
frequency difference equal to~$\frakv_{H}$.

A simple application of the last result is afforded by the problem of \Emph{the
simultaneous effect on the hydrogen lines of a homogeneous electric
and a homogeneous magnetic field which have the same direction}.
Thus, if the intensities of the fields are so large that we may look apart from the
small modifications claimed by the theory of relativity, we shall from the above expect
that the effect in question will differ from the ordinary \Name{Stark} effect of the hydrogen
lines, only therein that every component polarised perpendicular to the field is split up
in two symmetrical components corresponding to the outer members of a \Name{Lorentz}
triplet. This seems to agree with some observations of the effect of two such fields
on the hydrogen line~$H_{\alpha}$, published by \Name{Garbasso}.\footnote
  {\Name{A.~Garbasso}, Phys.\ Zeitschr.\ XV, p.~123 (1914).}
The problem in question might
also have been treated by means of the method of separation of variables, because,
as may be easily shown, the perturbed system---if the relativity modifications
are neglected---allows of separation of variables in parabolic coordinates, just as
in the presence of the electric field only. If, on the other hand, the relativity modifications
are taken into account, the method of separation of variables cannot be applied,
but, with reference to the considerations at the end of the last section, it will
be seen that it is possible, also in this case, to predict at once the modification in
the effect of an electric field on the fine structure of the hydrogen lines, which
would result from the simultaneous presence of a parallel magnetic field. Passing
to the limiting case where the intensity of the electric field is equal to zero, it will
thus be seen at once from the preceding, that \Emph{the effect of a homogeneous
magnetic field on the fine structure of the hydrogen lines} will consist
in the splitting up of every component in a normal \Name{Lorentz} triplet. As far as
the frequencies of the components are concerned, this result has been predicted by
\Name{Sommerfeld} and \Name{Debye}, who have treated the problem under consideration by
means of separation of variables in polar coordinates (compare \PageRef[page]{84}). In connection
with the fixation of the stationary states in this problem, it may be remarked
that we must assume that no stationary state will exist for which the angular
momentum round an axis through the nucleus parallel to the magnetic field would
be equal to zero. In fact, as seen in~\SecRef{4}, we must assume that, in case of a hydrogen
atom exposed to a homogeneous electric field, no such states will be possible; and
by imagining that the electric field decreases slowly to zero, while at the same
time a magnetic field parallel to the electric field is slowly established, it would
be possible, without passing through a degenerate system, to obtain a continuous
transformation of the stationary states of the perturbed atom during which the
angular momentum of the electron round the axis would remain unaltered. With
\PageSep{93}
reference to the invariance of the a-priori probability of the stationary states during
such a transformation (see Part~I, \PageRefI[page]{9}~and~\PageRefI[]{27}), we must therefore conclude that,
also in the case of a hydrogen atom in the presence of a magnetic field, no stationary
states exist for which the angular momentum round the axis would be equal to
zero, although these slates in mechanical respect do not exhibit singularities from
which we might anticipate that they are physically unrealisable.\footnote
  {\Emph{Note added during the proof.} In a dissertation which has just appeared, \Name{J.~M. Burgers} (Het
  Atoommodel van \Name{Rutherford}-\Name{Bohr}, Haarlem 1918) has given a very interesting general survey of the
  applications of the quantum theory to the problem of the constitution of atoms, and has in this connection
  entered upon several of the questions discussed in the present paper; for instance on the
  question of the relation between the spectrum of an atomic system, deduced by application of relation~\EqI{(1)}
  from the values of the energy in the stationary states, and the frequencies of the harmonic vibrations
  into which the motion in these states can be resolved; and on the question of the determination
  of the relative values for the a-priori probability of the different stationary states of an atomic system
  by means of \Name{Ehrenfest}'s principle of the invariance of these values during a continuous transformation
  of the system. As an illustration of the latter considerations, \Name{Burgers} has deduced an expression
  for the relative values of the a-priori probability of the different stationary states of the undisturbed
  hydrogen atom, by means of an enumeration of the states, determined by the conditions~\EqI{(22)} when
  applied in connection with a separation of variables in polar coordinates, which correspond to a stationary
  state of the undisturbed atom, characterised by a given value of~$n$ in the condition $I = nh$.
  Excluding only such states for which the total angular momentum of the electron round the nucleus
  would be equal to zero, \Name{Burgers} (loc.\ cit.\ p.~259) finds in this way for the value of the a-priori probability
  in question $(n + 1)^{2} - 1$. In connection with the analogous consideration, given in the Note on
  \PageRef[page]{76} of the present paper, which leads to a different result, it may be of interest to remark that
  the necessary conformity between the relative values for the a-priori probability of the different stationary
  states of the undisturbed hydrogen atom, deduced from an enumeration of the stationary states
  of the atom which appear in the presence of a small external electric field or in the presence of a
  small magnetic field respectively, cannot be obtained if in both cases we would exclude only such
  states in which the angular momentum of the electron round the nucleus is always equal to zero. In
  fact, while in case of a magnetic field this would give $(n + 1)^{2} - 1$ different states corresponding to
  a given value of~$n$, it would in case of an electric field give only $(n + 1)^{2} - 2$ such states. On the other
  hand, if the possible stationary states are selected in the manner explained in the text, the conformity
  in question will obviously be obtained.}

In case we consider the general problem of the effect on a hydrogen atom of
a small electric or magnetic field, which do not possess axial symmetry round an
axis through the nucleus, or of the simultaneous effect of two such fields, which
do not possess such symmetry round a common axis, we must expect that the
secular perturbations of the orbit of the electron will in general not be of conditionally
periodic type. In such a case we cannot obtain a complete fixation of the
stationary states, and we may conclude that the presence of the external forces will
not give rise to the splitting up of the hydrogen lines into a number of sharp
components but to a diffusion of these lines. With a simple example, in which the
secular perturbations of the atom seem not to be of conditionally periodic type, we
meet if we consider \Emph{the simultaneous effect on the hydrogen spectrum of
an external homogeneous electric field and a homogeneous magnetic
field, the directions of which make an angle with each other}. If
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% D. K. D. Vidensk. Selsk. Skr., naturvidensk. og mathem. Afd, 8. Række, IV. 1. 13
\PageSep{94}
the effects of the two fields on the motion of the electron are of the same order
of magnitude we may in this case expect that the hydrogen lines will not be
resolved into sharp components but will become diffuse. From the considerations
on \PageRef[page]{60} of the effect on the spectrum of a perturbed periodic system due
to a second external field, the perturbing effect of which is small compared with
that of the first, we may conclude, however, that, if the effect of one of the fields on
the motion of the electron is large compared with that of the other, the hydrogen
lines will still show a resolution in a number of components, the spectral widths
of which are small compared with the displacements which they have undergone due to
the presence of the weaker of the external fields. In the discussion of this problem
we shall for simplicity neglect the influence of the relativity modifications, assuming
that the effect on the spectrum produced by each external field separately is large
compared with the inherent fine structure of the hydrogen lines. Denoting, as in~\SecRef{2},
by~$\mu$ a small constant of the same order as the ratio between the forces on
the electron due to the weaker of the external fields and those due to the stronger
of these fields, and by~$\lambda$ a small constant of the same order as the ratio between
the latter forces and the attraction from the nucleus, we have, as shown on \PageRef[page]{61},
that, with neglect of small quantities of the same order of magnitude as~$\lambda\mu^{2}$,\footnote
  {Rigorously this result holds with neglect of small quantities of the same order of magnitude
  as the largest of the quantities $\lambda^{2}$ and~$\lambda\mu^{2}$, but for the sake of simplicity it is here and in the following
  assumed that $\mu$~is not smaller than~$\sqrt{\lambda}$ (compare \PageRef[page]{61}).}
the change in the additional energy of the atom due to the presence of the weaker
field is, in general, directly obtained by taking the mean value of the function~$\psi$,
corresponding to the weaker field, over the cycle of shapes and positions which
the orbit of the electron passes through in the stationary states of the atom in the
presence of the stronger field only. In the special case under consideration, however,
the perturbed system, formed by the atom in the presence of the stronger field
only, is degenerate, the secular perturbations of the orbit of the electron being of a
simple periodic character. The mean value in question will therefore not be completely
determined, but will be different for the different periodic cycles of shapes and
positions of the orbit, which represent the continuous multitude of stationary motions
which the electron may perform in each of the stationary states of the atom in
the presence of the stronger field only. In order to fix the stationary states in the
presence of both fields and the change in the additional energy of the atom due to
the presence of the weaker field, it will thus, as mentioned on \PageRef[page]{62}, be necessary
to examine the relation between the mean value in question and the frequency
of the slow periodic ``secular'' variations which the cycles under consideration will
undergo under the influence of the weaker of the external fields. Now, in the
special case under consideration this problem may be treated very simply, if we
imagine the weaker field as composed of two homogeneous fields of which the one
is parallel and the other perpendicular to the stronger field, and if we consider
separately the secular effect due to each of these fields. In fact, due to the symmetry
\PageSep{95}
with respect to the axis of the stronger field, exhibited by the periodic cycle of
shapes and positions which the orbit of the electron would pass through if the
atom were exposed to this field only, it is easily seen that the contribution, which
the perpendicular component of the weaker field gives to the mean value of~$\psi$
corresponding to the latter field, will vanish. From this it follows that the secular effect
of the weaker field, with neglect of small quantities proportional to~$\mu^{2}$ will be
the same as if only the parallel component of this field was acting on the atom;
and we see consequently that, in the stationary states of the atom in the presence
of both fields, the possible cycles of shapes and positions of the orbit of the electron
will be characterised in the same way as if the weaker field was parallel to the
stronger. The problem, however, of the fixation of the stationary states of a hydrogen
atom in the presence of a homogeneous electric field and a homogeneous magnetic
field, which are parallel to each other, is very simple. In fact, as it appears from the
considerations on \PageRef[page]{91}, the stationary states will in this case be fixed completely
by two conditions, of which the one, in the same way as in the simple theory of
the \Name{Stark} effect, defines the position of the plane in which the electrical centre of
\Typo{he}{the} orbit of the electron moves, while the other defines the value of the angular
momentum of the electron round the axis of the fields in the same way as in the
simple theory of the \Name{Zeeman} effect. In connection with the problem under consideration
here, it may be useful for the sake of illustration to note, that, if the
perturbing effect of the electric field is large compared with that of the magnetic,
the second of these conditions may be said to be imposed on the system by the
slow and uniform rotation, which the magnetic field produces on the periodic cycle
of shapes and positions of the orbit of the electron, which would appear if the
atom was exposed to the electric field only. If, on the other hand, the effect of the
magnetic field is large compared with that of the electric field, the first condition
may be said to be imposed on the system by the slow periodic oscillation in the
shape and position relative to the axis, which the electric field produces on the
uniformly rotating orbit which the electron would describe if the atom was exposed
to the magnetic field only.

If we consider a hydrogen atom which is exposed to the simultaneous influence
of a homogeneous electric field of intensity~$F$ and a homogeneous magnetic
field of intensity~$H$, the direction of which makes an angle~$\phi$ with the direction of
the electric field, it follows from the above that, if the perturbing influence of the
electric field is large compared with that of the magnetic field, the main effect produced
by the latter field on the spectrum may be described as the splitting up of
each \Name{Stark} effect component, polarised perpendicular to the axis of the electric field,
into two circularly polarised components, corresponding to the outer members of a
\Name{Lorentz} triplet which would be produced by a magnetic field of intensity $H\cos \phi$.
On the other hand, if the perturbing effect of the magnetic field is large compared
with that of the electric, it follows that the main effect, produced by the latter field
on the spectrum, may be described as the resolution of the middle component and
\PageSep{96}
of each of the outer components of the normal \Name{Zeeman} effect into a number of
components, corresponding to the parallel and perpendicular components respectively
of a \Name{Stark} effect produced by an electric field of intensity $F\cos \phi$.

The effects just described, however, which are the same as would take place
if only the parallel component of the weaker field was acting on the atom, will
not be the only effects of the presence of the weaker field on the spectrum. In
fact, although the perpendicular component of the weaker field, apart from small
quantities proportional to~$\mu^{2}$, will not have any secular effect on the cycle of shapes
and positions which the orbit of the electron would pass through if the atom was
exposed to the stronger field only, it will obviously produce alterations in the
motion of the electron within this cycle which are proportional to~$\mu$. Thus, if the
weaker field was parallel to the stronger, the motion of the electron in the perturbed
atom would be composed of a number of linear harmonic vibrations parallel
to the direction of the fields, the frequencies of which are of the type $|\tau \omega_{P} + \frakt_{1} \frakv_{1}|$,
and of a number of circular harmonic rotations perpendicular to this direction, the
frequencies of which are of the type $|\tau \omega_{P} + \frakt_{1} \frakv_{1} + \frakv_{2}|$ (compare \PageRef[page]{59}). In the
general case, however, where the weaker field is not parallel to the stronger, there
will, in the expression for the displacement of the electron in any given direction,
in addition appear a number of harmonic vibrations the amplitudes of which are
proportional to~$\mu$ and the frequencies of which, as a closer consideration of the
perturbations learns, are equal to the sum or difference of the frequency of one of
the harmonic vibrations, in which the motion in this direction could be resolved if
the external fields were parallel to each other, and one of the small frequencies of
type $|\frakt_{1} \frakv_{1} + \frakv_{2}|$, which appear in the expression for the secular perturbations of the
electron in this case. A part of these additional vibrations will again possess frequencies
of the types $|\tau \omega_{P} + \frakt_{1} \frakv_{1}|$ and $|\tau \omega_{P} + \frakt_{1} \frakv_{1} + \frakv_{2}|$, and will cause that the motion,
instead of consisting of vibrations which are exactly linear and exactly circular
as in the case where the external fields are parallel to each other, will be composed
of elliptical harmonic vibrations which partly are nearly linear and parallel
to the direction of the stronger field and partly nearly circular and perpendicular
to this direction. On account of this we shall expect that, due to the
presence of the perpendicular component of the weaker field, the different components
mentioned above will not be sharply polarised. Further there will, in the
motion of the perturbed atom, also appear a number of circular harmonic rotations
perpendicular to the stronger field, the amplitudes of which are small quantities
proportional to~$\mu$, and the frequencies of which are of the type $|\tau \omega_{P} + \frakt_{1} \frakv_{1} + 2\frakv_{2}|$.
From this we shall expect the appearance in the spectrum of a number of new
weak components, corresponding to a type of transition between stationary states
which would not be possible if the two external fields were parallel to \Typo{eachother}{each other}.
When considering more closely the frequencies of these new components, it must
be remembered, however, that, as mentioned above, the present method of treating
the problem of the perturbations assures us of the conditionally periodic character
\PageSep{97}
of the motion of the electron within a time interval of the same order of
magnitude as~$\sigma/\lambda$, only if we look apart from small quantities of the same order as~$\mu^{2}$;
and we must therefore be prepared to find that the frequencies of the vibrations
of small amplitudes will not be defined with the same degree of approximation
as the frequencies of the vibrations of large amplitudes. Thus, while the frequencies
of the latter vibrations are defined with neglect of small quantities proportional
to~$\lambda\mu^{2}$, the frequencies of the small vibrations under consideration are
obviously defined only with neglect of small quantities proportional to~$\lambda\mu$. In intimate
connection with the general want of definition of the energy in the stationary
states for perturbed systems of the type in question, we must accordingly be prepared
to find that, in contrast to the strong components, for which we may expect
that by far the larger part of the intensity is contained within a spectral interval
of a width proportional to~$\lambda\mu^{2}$, the new components will be diffused over spectral
intervals of a width proportional to~$\lambda\mu$.\Pagelabel{97}\footnote
  {Compare Note on \PageRef[page]{61}. With reference to the general validity of relation~\EqI{(1)}, it will be seen
  that the assumption, that the weak components possess this degree of diffusion, implies the assumption,
  that the corresponding transitions (the probability of occurrence of which is very small compared with
  the probability of the transitions responsible for the strong components) will generally take place
  between two states of the perturbed atom, which do not both belong to the well defined ensemble of
  stationary states in which at any moment the great majority among a large number of atoms will be
  present. Thus, in case the effect of the external electric field is large compared with that of the magnetic
  field, we may expect that, in both states involved in the transitions in question, the positions of
  the plane in which the electrical centre moves will coincide with positions of this plane in states
  belonging to the ensemble just mentioned, while the angular momentum of the electron round the
  axis of the electric field will generally change by an amount which will not be equal to an entire
  multiple of~$h/2\pi$. On the other hand, if the effect of the magnetic field is the larger, the angular
  momentum of the electron round the axis of this field will, in the transitions in question, change by
  two times~$h/2\pi$, while we may expect that the plane in which the electrical centre moves will generally,
  in at least one of the states involved in these transitions, differ from the positions of this plane in the
  ensemble of stationary states referred to.}
Thus, in case the effect of the external
electric field is large compared with that of the magnetic field, we might expect at
first sight that, on each side of every of the \Name{Stark} effect components polarised
parallel to the electric force, there would appear a weak component which would be
circularly polarised and be displaced from this component by an amount twice that
of the displacement of the strong components into which the perpendicularly
polarised \Name{Stark} effect components are split up as a consequence of the small
magnetic field. We must be prepared, however, to find that these weak components
will be so diffuse, that they are not separated from the weak perpendicular component
which has the same frequency as the strong parallel components on each
side of which the weak components under consideration would lie, and which
appears as a consequence of the above mentioned want of sharpness as regards the
polarisation of the strong components. On the other hand, if the effect of the magnetic
field is large compared with that of the electric field, any weak component
of the type under consideration, which corresponds to transitions in which the
\PageSep{98}
angular momentum of the electron round the axis of the magnetic field changes
by two times~$h/2\pi$, will lie at a distance from the original hydrogen line, which is
approximately twice as large as that of the outer components of the normal \Name{Zeeman}
effect, and will therefore be distinctly separated from the strong components into
which each of the components of the normal \Name{Zeeman} effect is split up in the presence
of the small electric field. We must be prepared, however, to find that the
weak components will not, as it might be expected at first sight, form two sets of
distinctly separated lines, but that they will only appear as two diffuse lines of circular
polarisation in opposite directions and of a spectral width proportional to~$\lambda\mu$.\footnote
  {No experiments, which allow to test the preceding results in detail, seem to have been recorded,
  but it would appear that the above considerations afford an explanation of the general character of the
  remarkable deviations from a normal \Name{Zeeman} effect, observed by \Name{F~Paschen} and \Name{E.~Back} (Ann.\ d.\ Phys.\
  XXXIX, p.~897 (1912)) in experiments in which the hydrogen lines were excited by passing a powerful
  condensed discharge through a capillary tube placed at right angles with the direction of the magnetic
  field. Besides the characteristic want of sharpness of the polarisation of the middle component,
  exhibited by all the spectrograms published by \Name{Paschen} and \Name{Back}, especially one of their photographs
  (Tafel VIII, Bild~4) seems to suggest the presence of a weak, perpendicularly polarised, diffuse line on
  each side of the original line and at a distance from it twice that of the outer components of the
  normal effect.}


\Section{6.}{The continuous hydrogen~spectrum.}

We shall conclude the considerations of this Part by a brief discussion of the
characteristic continuous spectrum of hydrogen in the ultra violet region, which is
intimately connected with the series spectrum given by~\Eq{(35)}. This spectrum consists
of a radiation, the frequencies of which are continuously distributed over a spectral
interval extending from the head of the Balmer series in the direction of higher
frequencies.\footnote
  {This spectrum has been observed as an emission spectrum in spectra of solar protuberances
  and planetary nebulae (See \Name{J.~Evershed}, Phil.\ Trans.\ Roy.\ Soc.\ 197~A, p.~399 (1901) and \Name{W.~H. Wright},
  Lick Observatory Bulletin, No.~291 (1917)) as well as in direct laboratory experiments on spectra excited
  by positive rays (See \Name{J.~Stark}, Ann.\ d.\ Phys.\ LII, p.~255 (1917)). Further it has been observed
  as an absorption spectrum in the spectra of several stars (see \Name{W.~Huggins}, An Atlas of Representative
  Stellar Spectra, p.~85 (1899) and \Name{J.~Hartmann}, Phys.\ Zeitschr.\ XVIII p.~429 (1917)).}
The existence of a continuous spectrum of this type is just what
should be expected from a natural generalisation of the principles underlying the
quantum theory of series spectra.\footnote
  {Compare \Name{N.~Bohr}, Phil.\ Mag.\ XXVI, p.~17 (1913); and also \Name{P.~Debye}, Phys.\ Zeitschr.\ XVIII, p.~428
  (1917).}
Thus the spectrum under consideration may be
directly explained by application of relation~\EqI{(1)}, if we assume that the complete
spectrum, emitted by a system consisting of a nucleus and of an electron, originates
not only from radiations, emitted during transitions between two states belonging
to the multitude of stationary states in which the electron describes a closed orbit,
characterised by the condition $I = nh$, but also from radiations emitted during
\PageSep{99}
transitions between two states, one (or both) of which belong to the multitude of
states in which the electron possesses sufficient energy to remove to infinite distance
from the nucleus. While the electron in the states of the type first mentioned may
be said to be ``bound'' by the nucleus to form an atom, it may in the states of the
last mentioned type be described as ``free''. In order to account for the appearance
of the continuous spectrum, it is necessary to assume that the motions
in the latter states are not restricted by extra-mechanical conditions of the type
holding for the former states, but that all motions, which are consistent with the
application of ordinary mechanics, will represent physically possible states. This
assumption would also seem to present itself naturally from the point of view on
the principles of the quantum theory, taken in the present paper.\footnote
  {A view contrary to this has been taken by \Name{Epstein}, who in a recent paper (Ann.\ d.\ Phys.\ L,
  p.~815 (1916)) has made an attempt to obtain an explanation of certain observations on the photoelectric
  effect of hydrogen occluded in metals, by applying conditions of the same type as~\EqI{(22)} to
  states of the hydrogen atom in which the electron describes a hyperbolic orbit, and has tried in a
  similar way to develop a theory of the characteristic $\beta$-ray spectra of radioactive substances.}
Thus it will in
the first place be observed that any attempt to discriminate between the different
states of the type in question, by means of considerations of the mechanical stability
of stationary states for slow transformations of the external conditions, would fail
on account of the essentially non-periodic character of the motion, which is irreconcilable
with the idea of invariance of extra-mechanical conditions for such transformations.
Next, with reference to the formal analogy between the quantum theory
and the ordinary theory of radiation, it will be seen that the fact, that the motion
of a free electron in its hyperbolic orbit cannot be resolved in a sum of harmonic
vibrations of discontinuously varying frequencies but can only be represented by
a Fourier integral extended over a continuous range of frequencies, suggests beforehand
that the free electron may pass, under emission or absorption of radiation, to
any one among a continuous multitude of other states corresponding to a continuous
multitude of values for the energy of the system. From the preceding considerations
we may infer, by application of~\EqI{(1)}, that the complete spectrum emitted by the
hydrogen atom will, besides the series spectrum and the continuous ultra-violet
spectrum mentioned above, which corresponds to transitions from a state in which
the electron is free to a stationary state characterised by $n = 2$ in~\Eq{(41)}, contain a
set of continuous spectra, corresponding to transitions from free states to other stationary
states, and each extending in the direction of larger frequencies from one
of the values of the frequency, given by~\Eq{(35)} if we put $n' = \infty$. Moreover, we may
expect the presence of a weak continuous spectrum, extending as a continuous back
ground over the whole region of frequencies, which will correspond to transitions
between two different states in both of which the electron is free. The relative intensities
of these different continuous spectra, and the laws according to which the intensity
is distributed within each of them, may be expected to vary to a large extent
according to the different conditions under which the radiation is excited. Thus,
\PageSep{100}
while the continuous spectrum of hydrogen, when observed as emission spectrum
in stars, shows a abrupt beginning at the head of the Balmer series, the continuous
spectrum, observed by \Name{Stark} in his experiments referred to above, was
not sharply limited but showed a pronounced maximum in the spectral region
which corresponds to transitions between two states, in the first of which the velocity
of the free electron relative to the nucleus, before the ``collision'' with the
latter, was of the same order of magnitude as the velocity of the positive rays by
means of which the spectrum was excited.

Besides the series spectrum and the connected continuous spectrum just considered,
there exists, as well known, another hydrogen spectrum, the \Typo{socalled}{so called} many-line
spectrum, which on account of its complex structure and its resemblance with
the band spectra, emitted by other elements and combinations of elements, is generally
ascribed to the hydrogen molecule and not to the atom. This assumption would
also seem to present itself directly from the point of view of the quantum theory,
according to which the simple structure of the series spectrum is directly connected
with the simple periodic character of the motion of the particles in the
atom, while a spectrum of a complexity of the order exhibited by the many-line
spectrum must be assumed to originate from a system the motion of which does
not show such simple periodicity properties. The problem of the constitution of the
hydrogen molecule, to be expected on the quantum theory, and the possible motions
of the particles of this system will be treated in Part~IV\@. In this connection we
shall also consider the problem of dispersion of light in hydrogen gas and the problem
of the voltage necessary to produce the lines of the series spectrum of hydrogen
by an electric discharge in this gas.
\vfill
\begin{center}
\footnotesize
Færdig fra Trykkeriet d.~30, December 1918.
\end{center}

%[** TN: Catalog text not manually inspected]
\iffalse
Det Kgl. Danske Videnskabernes Selskabs Skrifter.

Naturvidenskabelig og: mathematisk Afdeling,

8de Række.

Kr. 0re

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skab. Avec un resume en fran§ais. 1917 8. »

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1. Jergensen, S. M.: Det kemiske Syrebegrebs Udviklingshistorie indtil 1830. Efterladt Manuskript,
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3. \Name{Kramers}, H. A.: Intensities of Spectral Lines. On the application of the Quantum Theory to
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1. Christensen, Carl: A Monograph of the genus Dryopteris. Part II. 1920 8. 25.
\fi

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