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Bájatelo

On the Constitution of Atoms and Molecules

N. Bohr,
Dr. phil. Copenhagen
(Received July 1913)

Introduction

In order to explain the results of experiments on scattering of $\alpha$ rays by matter Prof. Rutherford¹has given a theory of the structure of atoms. According to this theory, the atom consist of a positively charged nucleus surrounded by a system of electrons kept together by attractive forces from the nucleus; the total negative charge of the electrons is equal to the positive charge of the nucleus. Further, the nucleus is assumed to be the seat of the essential part of the mass of the atom, and to have linear dimensions exceedingly small compared with the linear dimensions of the whole atom. The number of electrons in an atom is deduced to be approximately equal to half the atomic weight. Great interest is to be attributed to this atom-model; for, as Rutherford has shown, the assumption of the existence of nuclei, as those in question, seems to be necessary in order to account for the results of the experiments on large angle scattering of the $\alpha$ rays.²

In an attempt to explain some of the properties of matter on the basis of this atom-model we meet, however, with difficulties of a serious nature arising from the apparent instability of the system of electrons: difficulties purposely avoid in atom-models previously considered, for instance, in the one proposed by Sir. J.J. Thomson³ According to the theory of the latter the atom consist of a sphere of uniform positive electrification, inside which the electrons move in circular orbits.

The principal difference between the atom-models proposed by Thomson and Rutherford consist in the circumstance that the forces acting on the electrons in the atom-model of Thomson allow of certain configurations and motion of the electrons for which the system is in a stable equilibrium; such configurations, however, apparently do not exist for the second atom- model. The nature of the difference in question will perhaps be most clearly seen by noticing that among the quantities characterizing the fist atom a quantity appears - the radius of the positive sphere - of dimensions of a length and of the same order of magnitude as the linear extension of the atom, while such a length does not appear among the quantities characterizing the second atom, viz. the charges and masses of the electrons and the positive nucleus; nor can it do determined solely by help of the latter quantities.

The way of considering a problem of this kind has, however, undergone essential alterations in recent years owing to the development of the theory of the energy radiation, and the direct affirmation of the new assumptions introduced in this theory, found by experiments on very different phenomena such as specific heats, photoelectric effect, , & c. The result of the discussion of these questions seems to be a general acknowledgment of the inadequacy of the classical elecrtodynamics in describing the behaviour of system of atomic size.⁴ Whatever the alteration in the laws of motion of the electrons may be, it seems necessary to introduce in the laws in question a quantity foreign to the classical electrodynamics, i.e., Planck's constant, or as it often is called the elementary quantum of action. By the introduction of this quantity the question of the stable configuration of the electrons in the atoms is essentially changed, as this constant is of such dimensions and magnitude that it, together with the mass and charge of the particles, can determine a length of the order of magnitude required.

This paper is an attempt to show that the application of the above ideas to Rutherford's atom-model affords a basis for a theory of the constitution of atoms. It will further be shown that from this theory we are led to a theory of the constitution of molecules.

In the present first part of the paper the mechanism of the binding of electrons by a positive nucleus is discussed in relation to Planck's theory. It will be shown that it is possible from the point of view taken to account in a simple way for the law of the line spectrum of hydrogen. Further, reason are given for a principal hypothesis on which the considerations contained in the following parts are based.

I wish here to express my thinks to Prof. Rutherford for his kind and encouraging interest in this work.

Part I. - Binding of Electrons by Positive Nuclei.

$\S~1$. General Considerations

The inadequacy of the classical electrodynamics in accounting for the properties of atoms from an atom-model as Rutherford's, will appear very clearly if we consider a simple system consisting of a positively charged nucleus of very small dimensions and an electron describing closed orbits around it. For simplicity, let us assume that the mass of the electron is negligibly small in comparison with that of the nucleus, and further, that the velocity of the electron is small compared with that of light.

Let us at first assume that there is no energy radiation. In this case the electron will describe stationary elliptical orbits. The frequency of revolution $\omega$ and the major-axis of the orbit $2a$ will depend on the amount of energy $W$ which must be transferred to the system in order to remove the electron to an infinitely great distance apart from the nucleus. Denoting the charge of the electron and of the nucleus by - $e$ and $E$ respectively and the mass of the electron by $m$, we thus get

$$\omega = \frac{\sqrt{2}}{\pi} \cdot \frac{W^{3/2}}{eE \sqrt{m}}, ~~~~ 2a = \frac{eE}{W}.$$ (1)

Further, it can easily be shown that the mean value of the kinetic energy of the electron taken for a whole revolution is equal to $W$. We see that if the value of $W$ is not given, there will be no values of $\omega$ and $a$ characteristic for the system in question.

Let us now, however, take the effect of the energy radiation into account, calculated in the ordinary way from the acceleration of the electron. In this case the electron will no longer describe stationary orbits. $W$ will continuously increase, and the electron will approach the nucleus describing orbits of smaller and smaller dimensions, and with greater and greater frequency; the electron on the average gaining in kinetic energy at the same time as the whole system loses energy. This process will go on until the dimensions of the orbit are the same order of magnitude as the dimensions of the electron or those of the nucleus. A simple calculation shows that the energy radiated out during the process considered will be enormously great compared with that radiated out by ordinary molecular processes.

It is obvious that the behaviour of such a system will be very different from that of an atomic system occurring in nature. In the first place, the actual atoms in their permanent state to have absolutely fixed dimensions and frequencies. Further, if we consider any process, the result seems always to be that after a certain amount of energy characteristic for the systems in question is radiated out, the system will again settle down in a stable state of equilibrium, in which the distance apart of the particles are of the same order of magnitude as before the process.

Now the essential point in Planck's theory of radiation is that the energy radiation from an atomic system does not take place in the continuous way assumed in the ordinary electrodynamics, but that it, on the contrary, takes place in distinctly separated emissions, the amount of energy radiated out from an atomic vibrator of frequency $\nu$ in a single emission being equal to $\tau h \nu$, where $\tau$ is an entire number, and $h$ is a universal constant.⁵

Returning to the simple case of an electron and a positive nucleus considered above, let us assume that the electron at the beginning of the interaction with the nucleus was at a great distance apart from the nucleus, and had no sensible velocity relative to the latter. Let us further assume that the electron after interaction has taken place has settled down in a stationary orbit around the nucleus. We shall, for reasons referred to later, assume that the orbit in question is circular: this assumption will, however, make no alteration in the calculations for system containing only a single electron.

Let as now assume that, during the binding of the electron, a homogeneous radiation is emitted of a frequency $\nu$, equal to half the frequency of revolution of the electron in its final orbit; then from Planck's theory, we might expect that the amount of energy emitted by the process considered is equal to $\tau h \nu$, where $h$ is Planck's constant an entire number. If we assume that the radiation emitted is homogeneous, the second assumption concerning the frequency of the radiation suggests itself, since the frequency of revolution of the electron at the beginning of the emission is 0. The question, however, of the rigorous validity of both assumptions, and also of the application made of Planck's theory, will be more closely discussed in $\S~3$.

Putting

$$W = \tau h \frac{\omega}{2},$$ (2)

we get by help of the formula (1)

$$W = \frac{2 \pi^2 me^2E^2}{\tau^2 h^2}, ~~~ \omega = \frac{4... ...2 E^2} {\tau^3 h^3}, ~~~ 2a = \frac{\tau^2 h^2}{2 \pi^2 me E}.$$ (3)

If in these expressions we give $\tau$ different values, we get a series of values for $W$, $\omega$, and $a$ corresponding to a series of configurations of the system. According to the above considerations, we are led to assume that these configurations will correspond to states of the system in which there is no radiation of energy; states which consequently will be stationary as long as the system is not disturbed from outside. We see that the value of $W$ is greatest if $\tau$ has its smallest value 1. This case will therefore correspond to the most stable of the system, i.e., will correspond to the binding of the electron for the breaking up of which the greatest amount of energy is required.

Putting in the above expressions $\tau = 1$ and $E = e$, and introducing the experimental values

$$e = 4.7 \cdot 10^{-10}, ~~~ \frac{e}{m} = 5.31 \cdot 10^{17}, ~~~ h = 6.5 \cdot 10^{-27},$$

we get

$$2a = 1.1 \cdot 10^{-8}~ \mbox{cm}, ~~~ \omega = 6.2 \cdot 10^{15}~ \frac{1}{\mbox{sec}}, ~~~ \frac{W}{e} = 13~ \mbox{volt}.$$

We see that these values are of the same order of magnitude as the linear dimensions of the atoms, the optical frequencies, and the ionization- potentials.

The general importance of Planck's theory for the discussion of the behaviour of atomic system was originally pointed out by Einstein.⁶The considerations of Einstein have been developed and applied on a number of different phenomena, especially by Stark, Nernst, and Sommerfield. The agreement as to the order of magnitude between values observed for the frequencies and dimensions of the atoms, and values for these quantities calculated by considerations similar to those given above, has been the subject of much discussion. It was first pointed out by Haas,⁷ in ann attempt to explain the meaning and the value of Planck's constant on the basis of J.J. Thomson's atom-model, by help of the linear dimensions and frequency of an hydrogen atom. Systems of the kind considered in this paper, in which the forces between the particles vary inversely as the square of the distance, are discussed in relation to Planck's theory by J.W. Nicholson.⁸ In a series of papers this author has shown that it seems to be possible to account for lines of hitherto unknown origin in the spectra of the stellar nebulae and that of the solar corona, by assuming the presence in these bodies of certain hypothetical elements of exactly indicated constitution. The atoms of these elements are supposed to consist simply of a ring of a few electrons surrounding a positive nucleus of negligibly small dimensions. The ratios between the frequencies corresponding to the lines in question are compared with the ratios between the frequencies corresponding to different modes of vibration of the ring of electrons. Nicholson has obtained a relation to Planck's theory showing that the ratios between the wave-lenth of different sets of lines of the coronal spectrum can be accounted for with great accuracy by assuming that the ratio between the energy of the system and the frequency of rotation of the ring is equal to an entire multiple of Planck's constant. The quantity Nicholson refers to as the energy is equal to twice the quantity which we have denoted above by W. In the latest paper cited Nicholson has found it necessary to give the theory a more complicated form, still, however, representing the ratio of energy to frequency by a simple function of whole numbers.

The excellent agreement between the calculated and observed values of the ratios between the wave-length in question seems a strong argument in favour of the validity of the foundation of Nicholson's calculations. Serious objections, however, may be raised against the theory. These objections are intimately connected with the problem of the homogeneity of the radiation emitted. In Nicholson's calculations the frequency of lines in a line-spectrum is identified with the frequency of vibration of a mechanical system in a distinctly indicated state of equilibrium. As a relation from Planck's theory is used, we might expect that the radiation is sent out in quanta; but systems like those considered, in which the frequency is a function of the energy, cannot emit a finite amount of a homogeneous radiation; for, as soon as the emission of radiation is started, the energy and also the frequency of the system are altered. Further, according to the calculation of Nicholson, the systems are unstable for some modes of vibration. Apart from such objections - which may be only formal (see p. 23)?????? - it must be remarked, that the theory in the form given dies not seem to be able to account for the well-known laws of Balmer and Rydberg connecting the frequencies of the lines in the line- spectra of the ordinary elements.

It will now be attempted to show that the difficulties in question disappear if we consider the problems from the point of view taken in this paper. Before proceeding it may be useful to restate briefly the ideas characterizing the calculations on p. 5. The principal assumptions used are:

(1)

That the dynamical equilibrium of the systems in the stationary states can be discussed by help of the ordinary mechanics, while the passing of the systems between different stationary states cannot be treated on that basis.

(2)

That the latter is followed by the emission of a homogeneous radiation, for which the relation between the frequency and the amount of energy emitted is the one given by Planck's theory.

The first assumption seems to present itself; for it is known that the ordinary mechanism cannot have an absolute validity, but will only hold in calculations of certain mean values of the motion of the electrons. On the other hand, in the calculations of the dynamical equilibrium in a stationary state in which there is no relative displacement of the particles, we need not distinguish between the actual motions and their mean values. The second assumption is in obvious constant to the ordinary ideas of electrodynamics, but appears to be necessary in order to account for experimental facts.

In the calculations on page 5 we have further made use of the more special assumptions, viz., that the different stationary states correspond to the emission of a different number of Planck's energy-quanta, and that the frequency of the radiation emitted during the passing of the system from a state in which no energy is yet radiated out to one of the stationary states, is equal to half the frequency of revolution of the electron in the latter state. We can, however (see $\S~3$), also arrive at the expressions (3) for the stationary states by using assumptions of somewhat different from. We shall, therefore, postpone the discussion of the spacial assumptions, and first show how by the help of the above principal assumptions, and of the expressions (3) for the stationary states, we can account for the line-spectrum of hydrogen.

$\S~2$.Emission of Line-spectra

Spectrum of Hydrogen. - General evidence indicates that an atom of hydrogen consist simply of a single electron rotating round a positive nucleus of charge $e$.⁹ The reformation of a hydrogen atom, when the electron has been removed to great distances away from the nucleus - e.g. by the effect of electrical discharge in a vacuum tube - will accordingly correspond to the binding of an electron by a positive nucleus considered on p. 5. If in (3) we put $E = e$, we get for the total amount of energy radiated out by the formation of one of the stationary states,

$$W_r = \frac{2 \pi^2 me^4}{\tau^2 h^2}.$$

The amount of energy emitted by the passing of the system from a state corresponding to $\tau = \tau_1$ to one corresponding to $\tau = \tau_2$, is consequently

$$W_{r_2} - W_{r_1} = \frac{2 \pi^2 me^4}{h^2} \cdot \left( \frac{1}{\tau^2_2} - \frac{1}{\tau^2_1} \right).$$

If now we suppose that the radiation is question is homogeneous, and that the amount of energy emitted is equal to $h \nu$, where $\nu$ is the frequency of the radiation, we get

$$W_{r_2} - W_{r_1} = h \nu$$

and from this

$$\nu = \frac{2 \pi^2 me^4}{h^3} \cdot \left( \frac{1}{\tau^2_2} - \frac{1}{\tau^2_1} \right).$$ (4)

We see that this expression accounts for the law connecting the lines in the spectrum of hydrogen. If we put $\tau_2 = 2$ and let $\tau_1$ vary, we get the ordinary Balmer series. If we put $\tau_3 = 3$, we get the series in the ultra-red observed by Paschen¹⁰and previously suspected by Ritz. If we put $\tau_2 = 1$ and $\tau = 4, 5, \ldots,$ we get series respectively in the extreme ultraviolet and the extreme ultra-red, which are not observed, but the existence of which may be expected.

The agreement in question is quantitative as well as qualitative. Putting

$$e = 4.7 \cdot 10^{-10}, ~~~ \frac{e}{m} = 5.31 \cdot 10^{17} ~~~ \mbox{and} ~~~ h = 6.5 \cdot 10^{-27},$$

we get

$$\frac{2 \pi^2 me^4}{h^3} = 3.1 \cdot 10^{15}.$$

The observed value for the factor outside the bracket in the formula (4) is

$$3.290 \cdot 10^{15}.$$

We agreement between the theoretical and observed values is inside the uncertainty due to experimental errors in the constants entering in the expression for the theoretical value. We shall in $\S~3$ return to consider the possible importance of the agreement in question.

It may be remarked that the fact, that it has not been possibly to observe more than 12 lines of the Balmer series in experiments with vacuum tubes, while 33 lines are observed in the spectra of some celestial bodies, is just what we should expect from the above theory. According to the equation (3) the diameter of the orbit of the electron in the different stationary states is proportional to $\tau^2$. For $\tau = 12$ the diameter is equal to $1.6 \cdot 10{-6}$ cm, or equal to mean distance between the molecules in a gas at a pressure of about 7 mm mercury; for $\tau = 33$ the diameter is equal to $1.2 \cdot 10{-5}$ cm, corresponding to the mean distance of the molecules at a pressure of about 0.02 mm mercury. According to the theory the necessary condition for the appearance of a great number of lines is therefore a very small density of the gas; for simultaneously to obtain an intensity sufficient for observation the space filled with the gas must be very great. If the theory is right, we may therefore never expect to be able in experiments with vacuum tubes to observe the lines corresponding to high numbers of the Balmer series of the emission spectrum of hydrogen; it might, however, be possible to observe the lines by investigation of the absorption spectrum of this gas. (see $\S~4$).

It will be observed that we in the above way do not obtain other series of lines, generally ascribed to hydrogen; for instance, the series first observed by Pickering¹¹ in the spectrum of the star $\zeta$ Puppis, and the set of series recently found by Fowler¹² by experiments with vacuum tubes containing a mixture of hydrogen and helium. We shall, however, see that, by help of the above theory, we can account naturally for these series of lines if we ascribe them to helium.

A neutral atom of the latter element consists, according to Rutherford's theory, of a positive nucleus of charge $2e$ and two electrons. Now considering the binding of a single electron by a helium nucleus, we get putting $E = 2e$ in the expressions (3) on page 5, and proceeding in in exactly the same way as above,

$$\nu = \frac{8 \pi^2 me^4}{h^3} \cdot \left( \frac{1}{\tau^2_... ...ight)^2} - \frac{1}{\left(\frac{\tau_1}{2}\right)^2} \right).$$

If we in this formula put $\tau_1 = 1$ or $\tau_2 = 2$, we get series of lines in the extreme ultra-violet. If we put $\tau_2 = 3,$ and let $\tau_1$ vary, we get a series which includes 2 of the series observed by Folwer, and denoted by him as the first and second principal series of the hydrogen spectrum. If we put $\tau_2 = 4$, we get the series observed by Pickering in the spectrum of $\zeta$ Puppis. Every second of the lines in this series is identical with a line in the Balmer series of the hydrogen spectrum; the presence of hydrogen in the star in question may therefore account for the fact that these lines are of a greater intensity than the rest of the lines in the series. The series is also observed in the experiments of Fowler, and denoted in his paper as the Sharp series of the hydrogen spectrum. If we finally in the above formula put $\tau_2 = 5,~6, \ldots$, we get series, the strong lines of which are to be expected in the ultra-red.

The reason why the spectrum considered is not observed in ordinary helium tubes may be that in such tubes the ionization of helium is not so complete in the star considered or in the experiments of Fowler, where a strong discharge was sent through a mixture of hydrogen and helium. The condition for the appearance of the spectrum is, according to the above theory, that helium atoms are present in a state in which they have lost both their electrons. Now we must assume that the amount of energy to be used in removing the second electron from a helium atom is much greater than that to be used in removing the first. Further, it is known from experiments on positive rays, that hydrogen atoms can acquire a negative charge; therefore the presence of hydrogen in the experiments of Fowler may effect that more electrons are removed from some of the helium atoms than would be the case if only helium were present.

Spectra of other substances. -- in case of systems containing more electrons we must - in conformity with the result of experiments - expect more complicated laws for the line-spectra than those considered. I shall try to show that the point of view taken above allows, at any rate, a certain understanding of the laws observed. According to Rydberg's theory -- with the generalization given by Ritz¹³ - the frequency corresponding to the lines of the spectrum of an element can be expressed by

$$\nu = F_{\tau}(\tau_1) - F_s(\tau_2),$$

where $\tau_1$ and $\tau_2$ are entire numbers, and $F_1,~F_2,~F_3, \ldots$ are functions of $\tau$ which approximately are equal to $\frac{K}{(\tau + a_1)^2}$, $\frac{K}{(\tau + a_2)^2}, \ldots$ $K$ is a universal constant, equal to the factor outside the bracket in the formula (4) for the spectrum of hydrogen. The different series appear if we put $\tau_1$ or $\tau_2$ equal to a fixed number and let the other vary.

The circumstance that the frequency can be written as a difference between two functions of entire numbers suggests an origin of the lines in the spectra in question similar to the one we have assumed for hydrogen; i.e. that the lines correspond to a radiation emitted during the passing of the system between two different stationary states. For system containing more than one electron the detailed discussion may be very complicated, as there will be many different configurations of the electrons which can be taken into consideration as stationary states. This may account for the difference sets of series in the line spectra emitted from the substances in question. Here I shall only try to show how, by help of the theory, it can be simple explained that the constant $K$ entering in Rydberg's formula is the same for all substances. Let us assume that the spectrum in question corresponds to the radiation emitted during the binding of an electron; and let us further assume that the system including the electron considered is neutral. The force on the electron, when at a great distance apart the nucleus and the electrons previously bound, will be very nearly the same as the above case of the binding of an electron by a hydrogen nucleus. The energy corresponding to one of the stationary states will therefore for $\tau$ great be very nearly equal to that given by the expression (3) on p. 5, if we put $E = e$. For $\tau$ great we consequently get

$$\lim[\tau^2 \cdot F_1(\tau)] = \lim [\tau^2 \cdot F_2(\tau)] = \ldots = \frac{2 \pi^2 me^4}{h^3},$$

in conformity with Rydberg's theory.

$\S~3$.General Considerations Continued

We shall now return to the discussion (see p. 7) of the special assumptions used in deducing the expression (3) on p. 5 for the stationary states of a system consisting of an electron rotating round a nucleus.

For one, we have assumed that the different stationary states correspond to an emission of a different number of energy-qyanta. Considering systems in which the frequency is a function of the energy, this assumption, however, may be regarded as improbable; for as soon as one quantum in sent out the frequency is altered. We shall now see that we can leave the assumption used and still retain the equation (2) on p. 5, and thereby the formal analogy with Planck's theory.

Firstly, it will be observed that it has not been necessary, in order to account for the law of the spectra by help of the expressions (3) for the stationary states, to assume that in any case a radiation is sent out corresponding to more than a single energy-quantum, $h \nu$. Further information on the frequency of the radiation may be obtained by comparing calculations of the energy radiation in the region of slow vibrations based on the above assumptions with calculations based on the ordinary mechanics. As is known, calculations on the latter basis are in agreement with experiments on the energy radiation in the named region.

Let us assume that the ratio between the total amount of energy emitted and the frequency of revolution of the electron for the different stationary states is given by the equation $W = f(\tau) \cdot h \omega$, instead of by the equation (2). Proceeding in the same way as above, we get in this case instead of (3)

$$W = \frac{\pi^2 me E^2}{2h^2f^2(\tau)}, ~~~ \omega = \frac{\pi^2 me^2 E^2}{2h^3 f^3(\tau)}.$$

Assuming as above that the amount of energy emitted during the passing of the system from a state corresponding to $\tau = \tau_1$ to one for which $\tau = \tau_2$ is equal to $h \nu$, we get instead of (4)

$$\nu = \frac{\pi^2 me^2 E^2}{2h^3} \cdot \left( \frac{1}{f^2(\tau_2)} - \frac{1}{f^2(\tau_1)} \right).$$

We see that in order to get an expression of the same form as the Balmer series we must put $f(\tau) = c \tau$.

In order to determine $c$ let us now consider the passing of the system between two successive stationary states corresponding to $\tau = N$ and $\tau = N - 1$; introducing $f(\tau) = c \tau$, we get for the frequency of the radiation emitted

$$\nu = \frac{\pi^2 me^2 E^2}{2c^2h^3} \cdot \frac{2N - 1}{N^2(N - 1)^2}.$$

For the frequency of revolution of the electron before and after the emission we have

$$\omega_N = \frac{\pi^2 me^2 e^2}{2c^3 h^3 N^3} ~~~ \mbox{and} ~~~ \omega_{N - 1} = \frac{\pi^2 me^2 E^2}{2c^3 h^3(N - 1)^3}.$$

If $N$ is great the ratio between the frequency before and after the emission will be very near equal to 1; and according to the ordinary electrodynamics we should therefore expect that the ratio between the frequency of radiation and the frequency of revolution also very nearly equal to 1. This condition will only be satisfied if $c = 1/2$. Putting $f(\tau) = \tau/2$, we, however, again arrive at the equation (2) and consequently at the expression (3) for the stationary states.

If we consider the passing of the system between two states corresponding to $\tau = N$ and $\tau = N - n$, where $n$ is small compared with $N$, we get with the same approximation as above, putting $f(\tau) = \tau/2$,

$$\nu = n \omega.$$

The possibility of an emission of a radiation of such a frequency may also be interpreted from analogy with the ordinary electrodynamics, as an electron rotating round a nucleus in an elliptical orbit will emit a radiation which according to Fourier's theorem can be resolved into homogeneous components, the frequency of which are $n \omega$, if $\omega$ is the frequency of revolution of the electron.

We are thus led to assume that the interpretation of the equation (2) is not that the different stationary states correspond to an emission of different numbers of energy-quanta, but that the frequency of the energy emitted during the passing of the system from a state in which no energy is yet radiated out to one of the different stationary states, is equal to different multiples of $\omega/2$, where $\omega$ is the frequency of revolution of the electron in the state considered. From this assumption we get exactly the same expressions as before for the stationary states, and from these by help of the principal assumptions on p. 7 the same expression for the law of the hydrogen spectrum. Consequently we may regard our preliminary considerations on p. 5 only as a simple from of representing the results of the theory.

Before we leave the discussion of this question, we shall for a moment return to the question of the significance of the agreement between the observed and calculated values of the constant entering in the expressions (4) for the Balmer series of the hydrogen spectrum. From the above consideration it will follow that, taking the starting-point in the form of the law of the hydrogen spectrum and assuming that the different lines correspond to a homogeneous radiation emitted during the passing between different, stationary states, we shall arrive at exactly the same expression for the constant in question as that given by (4), if we only assume (1) that the radiation is sent out in quanta $h \nu$, and (2) that the frequency of the radiation emitted during the passing of the system between successive stationary states will coincide with the frequency of revolution of the electron in the region of slow vibrations.

As all the assumptions used in this latter way of representing the theory are of what we may call a qualitative character, we are justified in expecting -- if the whole way of considering is a sound one - an absolute agreement between the values calculated and observed for the constant in question, and not only an approximate agreement. The formula (40 may therefore be of value in the discussion of the results of experimental determinations of the constants $e$, $m$, and $h$.

While there obviously can be no question of a mechanical foundation of the calculations given in this paper, it is, however, possible to give a very simple interpretation of the result of the calculation on p. 5 by help of symbols taken from the ordinary mechanics. Denoting the angular momentum of the electron round the nucleus by $M$, we have immediately for a circular orbit $\pi M = T/\omega$, where $\omega$ is the frequency of revolution and $T$ the kinetic energy of the electron; for a circular orbit we further have $T = W$ (see p. 3) and from (2), p. 5, we consequently get

$$M = \tau M_0,$$

where

$$M_0 = \frac{h}{2 \pi} = 1.04 \cdot 10^{-27}.$$

If we therefore assume that the orbit of the electron in the stationary states is circular, the result of the calculation on p. 5 can be expressed by the simple condition: that the angular momentum of the electron round the nucleus in a stationary state of the system is equal to an entire multiple of a universal value, independent of the charge on the nucleus. The possible importance of the angular momentum in the discussion of atomic systems in relation to Planck's theory is emphasized by Nicholson.¹⁴

The great number of different stationary states we do not observe expect by investigation of the emission and absorption of radiation. It most of the other physical phenomena, however, we only observe the atoms of the matter in a single distinct state, i,e., the state of the atoms at low temperature. From the preceding considerations we are immediately led to the assumption that the `` permanent'' state is the one among the stationary states during the formation of which the greatest amount of energy is emitted. According to the equation (3) on p. 5, this state is the one which corresponds to $\tau = 1$.

$\S~4$. Absorption of Radiation

In order to account for Kirchhoff's law it is necessary to introduce assumptions on the mechanism of absorption of radiation which correspond to those we have used considering the emission. Thus we must assume that a system consisting of a nucleus and an electron rotating round it under certain circumstances can absorb a radiation of a frequency equal to the frequency of the homogenous radiation emitted during the passing of the system between different stationary states. Let us consider the radiation emitted during the passing of the system between two stationary states $A_1$ and $A_2$ corresponding to values for $\tau$ equal to $\tau_1$ and $\tau_2,~\tau_1 > \tau_2$. As the necessary condition of the radiation in question was the presence of systems in the state $A_1$, we must assume that the necessary condition for an absorption of the radiation is the presence of systems in the state $A_2$.

These considerations seems to be in conformity with experiments on absorption in gases. In hydrogen gas at ordinary conditions for instance there is no absorption of a radiation of a frequency corresponding to the line-spectrum of this gas; such an absorption is only observed in hydrogen gas in a luminous state. This is what we should expect according to the above. We have on p. 9 assumed that the radiation in question was emitted during the passing of the systems between stationary states corresponding to $\tau \geq 2$. The state of the atoms in hydrogen gas at ordinary conditions should, however, correspond to $\tau = 1$; furthermore, hydrogen atoms at ordinary conditions combine into molecules, i.e., into system in which the electrons have frequencies different from those in the atoms (see Part III.) From the circumstance that certain substances in a non-lumimous state, as, foe instance, sodium vapour, absorb radiation corresponding to lines in the line-spectra of the substances, we may, on the other hand, conclude that the lines in question are emitted during the passing of the system between two states, one of which is the permanent state.

How much the above considerations differ from an interpretation based on the ordinary electrodynamic of perhaps most early shown by the fact that we have been forced to assume that a system of electrons will absorb a radiation of a frequency different from the frequency of vibration of the electrons calculated in the ordinary way. It may in this connexion be of interest to mention a generalization of the considerations to which we are led by experiments on the photo-electric effect and which may be able to throw some light on the problem in question. Let us consider state of the system in which the electron is free, i.e., in which the electron possesses kinetic energy sufficient to remove to infinite distances from the nucleus. If we assume that the motion of the electron is governed by the ordinary mechanics and that there is no ( sensible) energy radiation, the total energy of the system - as in the above considered stationary states - will be constant. Further, there will be perfect continuity between the two kinds of states, as the difference between frequency and dimensions of the system in successive stationary states will diminish without limit if $\tau$ increases. In the following considerations we shall for the sake of brevity refer to the two kinds of states in question as `` mechanical'' states; by this notation only emphasizing the assumption that the motion of the electron in both cases can be assumed for by the ordinary mechanics.

Tracing the analogy between the two kinds of mechanical states, we might now expect the possibility of an absorption of radiation, not only corresponding to the passing of the system between two different stationary states, but also corresponding to the passing between one of the stationary states and a state in which the electron is free; and as above, we might expect that the frequency of this radiation was determined by the equation $E = h \nu$, where $E$ is the difference between the total energy of the system in the two states. As it will be see, such an absorption of radiation is just what is observed in experiments on ionization by ultra-violet light and by Röntgen rays. Obviously, we get in this way the same expression for the kinetic energy of an electron ejected from an atom by photo-electron effect as that deduced by Einstein¹⁵i.e., $T = h \nu - W$, where $T$ is the kinetic energy of the electron ejected, and $W$ the total amount of energy emitted during the original binding of the electron.

The above considerations may further account for the result of some experiments of R.W. Wood¹⁶ on absorption of light by sodium vapour. In these experiments, an absorption corresponding to a very great number of lines in the principal series of the sodium spectrum is observed, and in addition a continuous absorption which begins at the head of the series and extends to the extreme ultra-violet. This is exactly what we should expect according to the analogy in question, and, as we shall see, a closer consideration of the above experiments allows us to trance the analogy still further. As mentioned on p. 9 the radii of the orbits of the electrons will for stationary states, corresponding to high values for $\tau$ be very great compared with ordinary atomic dimensions. This circumstance was used as an explanation of the non-appearance in experiments with vacuum-tubes of lines corresponding to the higher numbers in the Balmer series of the hydrogen spectrum. This is also in conformity with experiments on the emission spectrum of sodium; in the principal series of the emission spectrum of this substance rather few lines are observed. Now in Wood's experiments the pressure was not very low, the states corresponding to high values for $\tau$ could therefore not appear; yet in the absorption spectrum about 50 lines were detected. In the experiments in question we consequently observe an absorption of radiation which is not accompanied by a complete transition between two different stationary states. According to the present theory we must assume that this absorption is followed by an emission of energy during which the systems pass back to the original stationary state. If there are no collisions between the different systems this energy will be emitted as a radiation of the same frequency as that absorbed, and there will be no true absorption but only a scattering of the original radiation; a true absorption will not occur unless the energy in question is transformed by collisions into kinetic energy of free particles. In analogy we may now from the above experiments conclude that a bound electron - also in cases in which three is no ionization - will have an absorbing (scattering) influence on a homogeneous radiation, as soon as the frequency of the radiation is greater than $W/h$, where $W$ is the total amount of energy emitted during the binding of the electron. This would be highly in favour of a theory of absorption as the one sketched above, as there can in such a case be no question of a coincidence of the frequency of the radiation and a characteristic frequency of vibration of the electron. If will further be seen that the assumption, that there will be an absorption (scattering) of any radiation corresponding to a transition between two different mechanical states, is in perfect analogy with the assumption generally used that a free electron will have an absorbing (scattering) influence on light of any frequency. Corresponding considerations will hold for the emission of radiation.

In analogy to the assumption used in this paper that the emission of line- spectra is due to the re-formation of atoms after one or more of the lightly bound electrons are removed, we may assume that the homogeneous Röntgen radiation is emitted during the setting down of the systems after one of the firmly bound electrons escapes, e.g. by impact of cathode particles.¹⁷ In the next part in this paper, dealing with the constitution of atoms, we shall consider the question more closely and try to show that a calculation based on this assumption is in quantitative agreement with the results of experiments: here we shall only mention briefly a problem with which we meet in such a calculation.

Experiments on the phenomena of X-rays suggest that not only the emission and absorption of radiation cannot be treated by the help of the ordinary electrodynamics, but not even the result of a collision between two electrons of which the one is bound in an atom. This is perhaps most early shown by some very instructive calculations on the energy of $\beta$-particles emitted from radioactive substances recently published by Rutherford.¹⁸ These calculations strongly suggest that an electron of great velocity in passing through an atom and colliding with the electrons bound will loose energy in distinct finite quanta. As is immediately seen, this is very different from what we might expect if the result of the collisions was governed by the usual mechanical laws. The failure of the classical mechanics in such a problem might also be expected beforehand from the absence of anything like equipartition of kinetic energy between free electrons and electrons bound in atoms. From the point of view of the ``mechanical'' states we see, however, that the following assumption - which is in accord with the above analogy - might be able to account for the result of Rutherford's calculation and for the absence of equipartition of kinetic energy; two colliding electrons, bound or free, will, after the collision as well as before, be in mechanical states. Obviously, the introduction of such an assumption would not make any alteration necessary in the classical treatment of a collision between two free particles. But, considering a collision between a free and a bound electron, it would follow that the bound electron by the collision could not acquire a less amount of energy than the difference in energy corresponding to successive stationary states, and consequently that the free electron which collides with it could not lose a less amount.

The preliminary and hypothetical character of the above considerations needs not to be emphasized. The intention, however, has been to show that the sketched generalization of the theory of the stationary states possibly may afford a simple basis of representing a number of experimental facts which cannot be explained by help of the ordinary electrodynamics, and that assumptions used do not seem to be inconsistent with experiments on phenomena for which a satisfactory explanation has been given by the classical dynamics and the wave theory of light.

$\S~5$.The permanent State of an Atomic System

We shall now return to the main object of this paper - the discussion of the ``permanent'' state of a system consisting of nuclei and bound electrons. For a system consisting of a nucleus and an electron rotating round it, this state is, according to the above, determined by the condition that the angular momentum of the electron round the nucleus is equal to $h/2 \pi$.

On the theory of this paper the only neutral atom which contains a single electron is the hydrogen atom. The permanent state of this atom should correspond to the values of $a$ and $\omega$ calculated on p. 5. Unfortunately, however, we know very little of the behaviour of hydrogen atoms on account of the small dissociation of hydrogen molecules at ordinary temperatures. In order to get a closer comparison with experiments, it is necessary to consider more complicated systems.

Considering systems in which more electrons are bound by a positive nucleus, a configuration of the electrons which presents itself as a permanent state is in which the electrons are arranged in a ring round the nucleus. In the discussion of this problem on the basis of the ordinary electrodynamics, we meet- apart from the question of the energy radiation - with new difficulties due to the question of the stability of the ring. Disregarding for a moment this latter difficulty, we shall first consider the dimensions and frequency of the systems in relation to Planck's theory of radiation.

Let us consider a ring consisting of $n$ electrons rotating round a nucleus of charge $E$, the electrons being arranged at equal angular intervals the circumference of a circle of radius $a$.

The total potential energy of the system consisting of the electrons and the nucleus is

$$P = - \frac{ne}{a} \cdot \left( E - es_n \right),$$

where

$$s_n = \frac{1}{4} \sum^{s=n-1}_{s=1} \mbox{cosec} \frac{s \pi}{n}.$$

For the radial force exerted on an electron by the nucleus and the other electrons we get

$$F = - \frac{1}{n} \cdot \frac{dP}{da} = - \frac{e}{a^2} \cdot \left( E - es_n \right).$$

Denoting the kinetic energy of an electron by $T$ and neglecting the electromagnetic forces due to the motion of the electrons (see Part II), we get, putting the centrifugal force on an electron equal to the radial force,

$$\frac{2T}{a} = \frac{e}{a^2} \cdot \left( E - es_n \right),$$

or

$$T = \frac{e}{2a} \cdot \left( E - es_n \right).$$

>From this we get for the frequency of revolution

$$\omega = \frac{1}{2\pi} \cdot \sqrt{\frac{e \left( E - es_n \right)}{ma^3}}.$$

The total amount of energy $W$ necessary transferred to the system in order to remove the electrons to infinite distances apart from the nucleus and from each other is

$$W = - P - nT = \frac{ne}{2a} \cdot \left( E - es_n \right) = nT,$$

equal to the total kinetic energy of the electrons.

We see that the only difference in the above formula and those holding for the motion of a single electron in a circular orbit round a nucleus is the exchange of $E$ for $E - es_n$. It is also immediately seen that corresponding to the motion of an electron in an elliptical orbit round a nucleus, there will be a motion of the $n$ electrons in which each rotates in an elliptical orbit with the nucleus in the focus, and the $n$ electrons at any moment are situated at equal angular intervals on a circle with the nucleus as the centre. The major axis and frequency of the orbit of the single electrons will for this motion be given by the expressions (1) on p. 3 if we replace $E$ by $E - es_n$ and $W$ by $W/n$. Let us now suppose that the system of $n$ electrons rotating in a ring round a nucleus is formed in a way analogous to the one assumed for a single electron rotating round a nucleus. It will thus be assumed that the electrons, before the binding by the nucleus, were at a great distance apart from the latter and possessed no sensible velocities, and also that during the binding a homogeneous radiation is emitted. As in the case of a single electron, we have here that the total amount of energy emitted during the formation of the system is equal to the final kinetic energy of the electrons. If we now suppose that during the formation of the system the electrons at any moment are situated at equal angular intervals on the circumference of a circle with the nucleus in the centre, from analogy with the considerations, on p. 5 we are here led to assume the existence of a series of stationary configurations in which the kinetic energy per electron is equal to $\tau h \omega/2$, where $\tau$ is an entire number, $h$ Planck's constant, and $\omega$ the frequency of revolution. The configuration in which the greatest amount of energy is emitted is, as before, the one in which $\tau = 1$. This configuration we shall assume to be the permanent state of the system if the electrons in this state are arranged in a single ring. As for the case of a single 3electron we get that the angular momentum of each of the electrons is equal to $h/2 \pi$. It may be remarked that instead of considering the single electrons we might have considered the ring as an entity. This would, however, lead to the same result, for in this case the frequency of revolution $\omega$ will be replaced by the frequency $n \omega$ of the radiation from the whole ring calculated from ordinary electrodynamics, and $T$ by the total kinetic energy $nT$.

There may be many other stationary states corresponding to other ways of forming the system. The assumption of the existence of such states seems necessary in order to account for the line-spectra of systems containing more than one electron (p. 11); it is also suggested by the theory of Nicholson mentioned on p. 6, to which we shall return in a moment. The consideration of the spectra, however, gives, as far as I can see, no indication of the existence of stationary states in which all the electrons are arranged in a ring and which correspond to greater values for the total energy emitted than the one we above have assumed to be the permanent state. Further, there may be stationary configurations of a system of $n$ electrons and a nucleus of charge $E$ in which all the electrons are not arranged in a single ring. The question, however, of the existence of such stationary configurations is not essential for our determination of the permanent state, as long as we assume that the electrons in this state of the system are arranged in a single ring. Systems corresponding to more complicated configurations will be discussed on p. 24.?????

Using the relation $T = h \omega/2$ we get, by help of the above expressions for $T$ and $\omega$, values for $a$ and $\omega$ corresponding to the permanent state of the system which only differ from those given by the equations (3) on p. 5, by exchange of $E$ for $E - es_n$.

The question of stability of a ring of electrons rotating round a positive charge is discussed in great detail by Sir. J.J. Thomson¹⁹ An adaption of Thomson's analysis for the case here considered of a ring rotating round a nucleus of negligibly small linear dimensions is given by Nicholson.²⁰ The investigation of the problem in question naturally divides in two parts: one concerning the stability for displacements of the electrons on the plane of the ring; one concerning displacements perpendicular to this plane. As Nicholson's calculations show, the answer to the question of stability differs very much in the two cases in question. While the ring for the latter displacements in general is stable if the number of electrons is not great; the ring is in no case considered by Nicholson stable for displacement of the first kind.

According, however, to the point of view taken in this paper, the question of stability for displacements of the electrons in the plane of the ring is most intimately connected with the question of the mechanism of the binding of the electrons, and like the latter cannot be treated on the basis of the ordinary dynamics. The hypothesis of which we shall make use in the following is that the stability of a ring of electrons rotating round a nucleus is secured through the above condition of the universal constancy of the angular momentum, together with the further condition that the configuration of the particles is the one by the formation of which the greatest of energy is emitted. As will be shown, this hypothesis is, concerning the question of stability for a displacement of the electrons perpendicular to the plane of the ring, equivalent to that used in ordinary mechanical calculations.

Returning to the theory of Nicholson on the origin of lines observed in the spectrum of the solar corona, we shall now see that the difficulties mentioned on p. 7 may be only formal. In the first place, from the point of view considered above the objection as to the instability of the systems for displacements of the electrons in the plane of the ring may not be valid. Further, the objection as to emission of the radiation in quanta will not have reference to the calculations in question, if we assume that in the coronal spectrum we are not dealing with a true emission but only with a scattering of radiation. This assumption seems probable if we consider the conditions in the celestial body in question: for on account comparatively few collisions to disturb the stationary states and to cause a true emission of light corresponding to the transition between different stationary states; on the other hand there will in the solar corona be intense illumination of light of all frequencies which may excite the natural vibrations of the systems in the different stationary states. If the above assumption is correct, we immediately understand the entirely different from for the laws connecting the lines discussed by Nicholson and those connecting the ordinary line-spectra considered in this paper.

Proceeding to consider systems of more complicated constitution, we shall make use of the following theorem, which can be very simply proved; - ``In every system consisting of electrons and positive nuclei, in which the nuclei are at rest and the electrons move in circular orbits with a velocity small compared with the velocity of light, the kinetic energy will be numerically equal to half the principal energy.''

By help of this theorem we get - as in the previous cases of a single electron or of a ring rotating round a nucleus - that the total amount of energy emitted, by the formation of the systems from a configuration in which the distances apart of the particles are infinitely great and in which the particles have no velocities relative to each other, is equal to the kinetic energy of the electrons in the final configuration.

In analogy with the case of a single ring we are here led to assume that corresponding to any configuration of equilibrium a series of geometrically similar, stationary configuration of the system will exist in which the kinetic energy of every electron is equal to the frequency of revolution multiplied by $\tau/2 h$ where $\tau$ is an entire number and $h$ Planck's constant. In any such series of stationary configurations the one corresponding to the greatest amount of energy emitted will be the one in which $\tau$ for every electron is equal to 1. Considering that the ratio of kinetic energy to frequency for a particle rotating in a circular orbit is equal to $\pi$ times the angular momentum round the center of the orbit, we are therefore led to the following simple generalization of the hypotheses mentioned on pp. 15 and 22. ??????

``In any molecular system consisting of positive nuclei and electrons in which the nuclei are at rest relatire to each other and the electrons more in circular orbits, the angular momentum of every electron round the centre of its orbit will in the permanent state of the system be equal to $h/2 \pi$, where $h$ is Planck's constant.''²¹

In analogy with the considerations on p. 23, we shall assume that a configuration satisfying this condition is stable if the total energy of the system is less than in any neighbouring configuration satisfying the same condition of the angular momentum of the electrons.

As mentioned in the introduction, the above hypothesis will be used in a following communication as a basis for a theory of the constitution of atoms and molecules. It will be shown that it leads to results which seem to be in conformity with experiments on a number of different phenomena.

The foundation of the hypothesis has been sought entirely in its relation with Planck's theory of radiation; by help of considerations given later it will be attempted to throw some further light on the formation of it from another point of view.

Part II. - Systems containing only a Single Nucleus²²

$\S~1$ General Assumptions

Following the theory of Rutherford, we shall assume that the atoms of the elements consist of a positively charged nucleus surrounded by a cluster of electrons. The nucleus is the seat of the essential part of the mass of the atom, and has linear dimensions exceedingly small compared with the distance apart of the electrons in the surrounding cluster.

As in the previous paper, we shall assume that the cluster of electrons is formed by the successive binding by the nucleus of electrons initially nearly at rest, energy at the same time being radiated away. This will go on until, when the total negative charge on the bound electrons is numerically equal to the positive charge on the nucleus, the system will be neutral and no longer able to exert sensible forces on electrons at distances from the nucleus great in comparison with the dimensions of the orbits of the bound electrons. We may regard the formation of helium from $\alpha$ rays as an observed example of a process of this kind, an $\alpha$ particle on this view being identical with the nucleus of a helium atom.

On account of the small dimensions of the nucleus, its internal structure will not be of sensible influence on the constitution of the cluster of electrons, and consequently will have no effect on the ordinary physical and chemical properties of the atom. The latter properties on this theory will depend entirely on the total charge and mass of the nucleus; the internal structure of the nucleus will be of influence only on the phenomena of radioactivity.

>From the result of experiments on large-angle scattering of $\alpha$-rays, Rutherford²³ found an electric charge on the nucleus corresponding per atom to a number of electrons approximately equal to half the atomic weight. This result seems to be in agreement with the number of electrons per atom calculated from experiments on scattering of Röntgen radiation.²⁴ The total experimental evidence supports the hypothesis²⁵ that the actual number of electrons in a neutral atom with a few exceptions is equal to the number which indicated the position of the corresponding element in the series of element arranged in order of increasing atomic weight. For example on this view, the atom of oxygen which is the eighth element of the series has eight electrons and a nucleus carrying eight unit charges.

We shall assume that the electrons are arranged at equal angular intervals in coaxial rings rotating round the nucleus. In order to determine the frequency and dimensions of the rings we shall use the main hypothesis of the first paper, viz.; that in the permanent state of an atom the angular momentum of every electron round the centre of its orbit is equal to the universal value $h/2 \pi$, where $h$ is Planck's constant. We shall take as a condition of stability, that the total energy of the system in the configuration in question is less than in any neighbouring configuration satisfying the same condition of the angular momentum of the electrons.

If the charge on the nucleus and the number of electrons in the different rings is known, the condition in regard to the angular momentum of the electrons will, as shown in $\S~2$, completely determine the configuration of the system. i.e., the frequency of revolution and the linear dimensions of the rings. Corresponding to different distributions of the electrons in the rings, however, there will, in general, be more than one configuration which will satisfy the condition of the angular momentum together with the condition of stability.

In $\S~3$ and $\S~4$ it will be shown that, on the general view of the formation of the atoms, we are led to indications of the arrangement of the electrons in the rings which are consistent with those suggested by the chemical properties of the corresponding element.

In $\S~5$ will be shown that it is possible from the theory to calculate the momentum velocity of cathode rays necessary to produce the characteristic Röntgen radiation from the element, and that this is in approximate agreement with the experimental values.

In $\S~6$ the phenomena of radioactivity will be briefly considered in relation of the theory.

$\S~2$ Configuration and Stability of the System

Let us consider an electron of charge $e$ and mass $m$ which moves in a circular orbit of radius $a$ with a velocity $v$ small compared with the velocity of light. Let us denote the radial force acting on the electrons by $e^2/a^2 F$; $F$ will in general be dependent on $a$. The condition of dynamical equilibrium gives

$$\frac{mv^2}{a} = \frac{e^2}{a^2}F.$$

Introducing the condition of universal constancy of the angular momentum of the electron, we have

$$mva = \frac{h}{2 \pi}.$$

>From these two conditions we now get

$$a = \frac{h^2}{4 \pi^2 e^2 m} \cdot F^{-1} ~~~ \mbox{and} ~~~ v = \frac{2 \pi e^2}{h} \cdot F;$$ (5)

and for the frequency of revolution $w$ consequently

$$\omega = \frac{4 \pi^2 e^2 m}{h^2} \cdot F^2.$$ (6)

If $F$ is known, the dimensions and frequency of the corresponding orbit are simply determined by (1) and (2). For a ring of $n$ electrons rotating round a nucleus of charge $ne$ we have (comp. Part I., p. 20)????

$$F = N - s_n, ~~~ \mbox{where} ~~~ s_n = \frac{1}{4} \cdot \sum^{s=n-1}_{s=1} \mbox{cosec} \frac{s \pi}{n}.$$

The values for $s_n$ from $n = 1$ to $n = 16$ are given in the table 1.

For systems consisting of nuclei and electrons in which the first are at rest and the latter move in circular orbits with a velocity small compared with the velocity of light, we have shown (see part I., p. 21)???? that the total kinetic energy of the electrons is equal to the total amount of energy emitted during the formation of the system from an original configuration in which all the particles are at rest and at infinite distances from each other. Denoting this amount of energy by $W$, we consequently get

$$W = \sum \frac{m}{2} v^2 = \frac{2 \pi^2 e^4m}{h^2} \sum F^2.$$ (7)

Putting in (1), (2), and (3) $e = 4.7 \cdot 10^{-10}$, $\frac{e}{m} = 5.31 \cdot 10^{-17}$, and $h = 6.5 \cdot 10^{-27}$ we get

$$\begin{array}{ll} a = 0.55 \cdot 10^{-8} F^{-1},&v = 2.1 \cd... ...& W = 2.0 \cdot 10^{-11} {\displaystyle \sum} F^2. \end{array}$$ (8)

In neglecting the magnetic forces due to the motion of the electrons we have in Part I. assumed that the velocities of the particles are small compared with the velocity of light. The above calculations show that for this to hold, $F$ must be small compared with 150. As will be seen, the latter condition will be satisfied for all the electrons in the atoms of elements of low atomic weight and for a greater part of the electrons contained in the atoms of the other elements.

If the velocity of the electrons in not small compared with the velocity of light, the constancy of the angular momentum no longer involved a constant ratio between the energy and the frequency of revolution. Without introducing new assumptions, we cannot therefore in this case determine the configuration of the systems on the basis of the consideration in Part I. Considerations given later suggest, however, that the constancy of the angular momentum is the principal condition. applying this condition for velocities not small compared with the velocity of light, we get the same expression for $v$ as that given by (1), while the quantity $m$ in the expressions for $a$ and $\omega$ is replaced by $m/\sqrt{(1 - v^2/c^2)}$, and in the expression for $W$ by

$$m \cdot 2 \frac{c^2}{v^2} \cdot \left( 1 - \sqrt{1 - \frac{v^2}{c^2}} \right).$$

As stated in Part I., a calculation based on the ordinary mechanics given the result, that a ring of electrons rotating round a positive nucleus in general is unstable for displacement of the electrons in the plane of the ring. In order to escape from this difficulty, we have assumed that the ordinary principles of mechanics cannot be used in the discussion of the problem in question, any more than in the discussion of the connected problem of the mechanism of binding of electrons. We have also assumed that the stability for such displacement is secured through the introduction of the hypothesis of the universal constancy of the angular momentum of the electrons.

As is easily shown, the latter assumption in included in the condition of stability in $\S~1$. Consider a ring of electrons rotation round a nucleus, and assume that the system is in dynamical equilibrium and that the radius of the ring is $a_0$, the $v_0$, the total kinetic energy $T_0$, and the potential energy $P_0$. As shown in Part i. (p. 21) we have $P_0 = - 2T_0$. Next consider a configuration of the system in which the electrons, under influence of extraneous forces, rotate with the same angular momentum round the nucleus in a ring of radius $a = \alpha a_0$. In this case we have $P = \frac{1}{\alpha} P_ 0$, and on account of the uniformity of the angular momentum $v = 1/\alpha \cdot v_0$ and $T = 1/\alpha^2 \cdot T_0$. Using the relation $P_0 = - 2T_0$, we get

$$P + T = \frac{1}{\alpha} \cdot P_0 + \frac{1}{\alpha^2} T_0 = P_0 + T_0 + T_0 \cdot \left( 1 - \frac{1}{\alpha} \right)^2.$$

We see that the total energy of the new configuration is greater than in the original. according to the condition of stability in $\S~1$ the system is consequently stable for the displacement considered. In this connexion, it may be remarked that in Part I. we have assumed that the frequency of radiation emitted or absorbed by the systems cannot be determined from the frequencies of vibration of the electrons in the plane of the orbits, calculated by help of the ordinary mechanics. We have, on the contrary, assumed that the frequency of the radiation is determined by the condition $h \nu = E$, where $\nu$ is the frequency, $h$ Planck's constant, and $E$ the difference in energy corresponding to two different ``stationary'' states of the system.

In considering the stability of a ring of electrons rotating round a nucleus for displacements of the electrons perpendicular to the plane of the ring, imagine a configuration of the system in which the electrons are displaced by $\delta z_1, \delta z_2, \ldots \delta z_n$ respectively, and suppose that the electrons, under influence of extraneous forces, rotate in circular orbits parallel to the original plane with the same radial and the same angular momentum round the axis of the system as before. The kinetic energy is unaltered by the displacement, and neglecting powers of the quantities $\delta z_1, \ldots \delta z_n$ higher than the second, the increase of the potential energy of the system is given by

$$\frac{1}{2} \cdot \frac{e^2}{a^3} \cdot N \sum (\delta z)^2 ... ...\pi(r - s)}{n} \mid \left( \delta z_r - \delta z_s \right)^2,$$

where $a$ is the radius of the ring, $Ne$ the charge on the nucleus, and $n$ the number of electrons. According to the condition of stability in $\S~1$ the system is stable for the displacement considered, if the above expression is positive for arbitrary values of $\delta z_1, \ldots \delta z_n$. By a simple calculation it can be shown that the latter condition is equivalent to the condition

$$N > p_{n, 0} - p_{n, m},$$ (9)

where $m$ denotes the whole number (smaller than $n$) for which

$$p_{n,k} = \frac{1}{8} \sum^{s=n-1}_{s=1} \cos 2k \cdot \frac{s \pi}{n} \mbox{cosec}^3 \frac{s \pi}{n}$$

has its smallest value. This condition is identical with the condition of stability for displacements of the electrons perpendicular to the plane of the ring, deduced by help of ordinary mechanical considerations.²⁶

A suggestive illustration is obtained by imagining that the displacements considered are produced by the effect of extraneous forces acting on the electrons in a direction parallel to the axis of the ring. If the displacements are produced infinitely slowly the motion of the electrons will at any moment be parallel to the original plane of the ring, and the angular momentum of each of the electrons round the centre of its orbit will obviously be equal to its original value; the increase in the potential energy of the system will be equal to the work done by the extraneous forces during the displacements we are led to assume that the ordinary mechanics can be used in calculating the vibrations of the electrons perpendicular to the plane of the ring - contrary to the ease of vibrations in the plane of the ring. This assumptions is supposed by the apparent agreement with observations obtained by Nicholson in his theory of the origin of lines in the spectra of the solar corona and stellar nebulae (see Part I. pp. 6 & 23).?????? In addition it will be shown later that the assumption seems to be in agreement with experiments on dispersion.

The following table gives the values of $s_n$ and $P_{n,0}$ - $P_{n, m}$ from $n = 1$ to $n = 16.$

Table 1.

n	s$_n$	$p_{n,0} - p_{n,m}$	n	$s_n$	$p_{n,0} - p_{n,m}$