N. Bohr, On the
Constitution of Atoms and Molecules,*
Munksgaard, 1963. pp. 1 – 25.
*
Communicated by Prof. E. Rutherford, F.R.S.
(Original
paper published in Phil. Mag. S. 6,
Vol. 26, No. 151, July 1913)
[収録箇所]
Introduction
(全文),
§
1, General Consideration から
pp.7 – 8,
§
4, Absorption of Radiation からpp. 16 – 17,
§
5, The Permanent State of an Atomic System から pp.24 – 25
付録 Bohrの仮定の根拠についての考察
§
1, General Consideration から pp.3 – 6,
§2.
Emission of Line-spectra から pp.8 – 9.
Introduction.
In order to explain the results of
experiments on scattering of α rays by matter Prof.
(1)
E. Rutherford, Phil. Mag., xxi. p.
669 (1911),
(2)
See also Geiger and Marsden, Phil. Mag.,
April 1913.
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 avoided in atom-models previously
considered, for instance, in the one proposed by Sir J. J. Thomson.(3)
According to the theory of the latter the atom consists 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 consists in the circumstance
that the forces acting on the electrons in the atom-model of Thomson allow of
certain configurations and motions 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
first 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 be determined solely by help of the latter quantities.
(3)
J. J. Thomson, Phil. Mag. vii, p. 237
(1904).
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, Röntgen-rays,
&c. The result of the discussion of these questions seems to be a general
acknowledgment of the inadequacy of the classical electrodynamics in describing
the behavior of systems 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.
(4)
See f. inst., ` Théorie du rayonnement et les quanta.' Rapports de la réunion à Bruxelles, Nov. 1911. Paris, 1912.
This paper is an attempt to show that
the application of the above ideas to
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, reasons are given for a principal hypothesis on which the
considerations contained in the following parts are based.
I wish here to express my thanks to
Prof. Rutherford for his kind and encouraging interest in this work.
§ 1, General Consideration pp. 7 – 8
-
- - - - - - - - - - - - -
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 process 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 mechanics 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 contrast 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 Section 3), also arrive
at the expressions (3)* for the stationary states by using
assumptions of somewhat different form. We shall, therefore, postpone the
discussion of the special 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,
-
- - - - - - - - - - - - - -
* (引用注) Expression (3): W = 2π2me2E2/τ2h2
, ω = 4π2me2E2/τ2h2, 2a = τ2h2/2π2meE (τ is an entire number).
§ 4, Absorption of Radiation pp. 16 – 17
- - - - - - - - - - -
How much the
above considerations differ from an interpretation based on the ordinary
electrodynamics is perhaps most clearly 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 connection 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 a 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 systems in successive stationary states will diminish without
limit if T 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 accounted 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 = hν, where E is the difference between the total energy of the system in the
two states. As it will be seen, 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-electric effect as that deduced by
Einstein*, i. e. T = hν – 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.
- - - - - - - - -
*
A. Einstein, Ann. d. Phys. xvii, p.
146 (1905)
§ 5, The
-
- - - -
Proceeding to consider systems of a 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
potential energy."*
*[引用注] これはビリアル定理をクーロンポテンシャルの場合に適用した形の結論である。ビリアル定理は、「ビリアルと運動エネルギーの平均値とは相等しい」ことを示す。ここで、ビリアルは力と座標の積の平均値として定義される。
By help of this theorem we gets – 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 configurations of the
system will exist in which the kinetic energy of every electron is equal to the
frequency of revolution multiplied by τh/2
where τ 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
τ 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 π
times the angular momentum round the centre 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 relative to each other and the electrons move 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/2π where h is
Planck's constant”*[*]
* In the considerations leading to this
hypothesis we have assumed that the velocity of the electrons is small compared
with the velocity of light. The limits of the validity of this assumption will
be discussed in
[*] (引用注)この結論が一般には（s-stateなどに対しては）正しくないことは後に明らかになり、量子力学により解決されたことである。
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 neighboring 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 foundation of it from another point of view.
April
5, 1913.
付録
Bohrの仮定の根拠についての考察
上に引用したのは、Bohrの長い論文の中の、理解しやすい部分である。この部分から分かるように、Bohrの原子模型の根幹は、（1）Rutherfordの原子模型に有限の大きさを与えるために何らかの尺度を導入することが必要なことを認識したことと、（2）電磁波の放射が電子の状態間の遷移によるとしたことにある。そのために彼が注目したのが、Planckが仮定した作用量子h であった。
その作用量子をどのようにして原子構造に取り入れるかが、彼の最も苦心した点ではなかったろうか。彼の論理では、安定軌道の存在、無限遠に静止した状態と安定軌道とのエネルギー差Wをnｈν（整数をnとする）としたこと(式(2))に加えて、次の仮定が本質的な役割を果たしている：
Let us now assume that, during the
binding of the electron, a homogeneous radiation is emitted of a frequency ν,
equal to half the frequent ω of revolution of the electron in its final orbit;-
- - -
（下に§1. General
Considerations の冒頭の4ページを引用したが、その中の下線部。モデルの全体を理解するのには不適当で、完成した量子論から見て根拠が曖昧なので、上のテキストには引用しなかったもの。)
この仮定の根拠について、少し考察しよう。
まず、Bohrの原子論の契機は、Rutherfordの原子模型と古典論におけるその不安定性を妥協させる仮定として、いかにPlanckの作用量子ｈを生かすかを考えた点にあることは上述した。彼が参考にすることのできたのは、(a)Planckの仮定した（導入した）「振動数νの振動子のエネルギー準位がｈνの間隔でならんでいる」ことと、(b)Einsteinの仮定した「振動数νの光のエネルギーがｈνであること」とであった。
Bohrはこの二つを原子内の電子にたいして適用するのだが、その他に余分な仮定をしている。Bohrの論理を整理すると、次のようになる。
まず、金属からの電子放射（光電効果）の説明に用いられたEinsteinの式T = hν – W（このWは仕事関数）を原子からの電子励起に適用し、hν ＝ Wを仮定する（このWは原子内電子を無限遠の静止状態に励起するエネルギー）。次に、ビリアル定理を使ってWが電子の運動エネルギーの平均値に等しいことを示す（(1)式の下の段落の説明）。３番目に、光の振動数νと電子の軌道運動の振動数 ωの間に上に引用した文章で表された関係ν = ω/2 を仮定する。4番目に、Planckの仮定を§１の式(2)の形で（整数が入っていることに注意）原子内電子に適用する。すると、式(3)の第一式が得られ、これを使ってスペクトル系列を説明することができる。
導入の仕方で、今日の立場から見て正しい仮定は、定常状態が存在することと、無限遠に静止した電子と原子内の定常状態とのエネルギー差をｈνと置いたことである[Bohrの元の仮定(2)でn = 1の場合で、仮定１とする]。
仮定１.無限遠に静止した電子と、原子内の定常状態（安定と仮定した）とのエネルギー差Wはその間に放射される振動数νの光のエネルギーｈνに等しい。
ビリアル定理でWを安定軌道の運動エネルギーと関係付けることができていることに注意する必要がある。
実証性を重んじるならば、この仮定１と放射スペクトル（例えば§2の(４)のBalmer系列など）とから、安定軌道のエネルギー固有値系列(下に引用した§2 の式で[1]と置いた)を決めるのが筋道であった。その場合、上の仮定[仮定２とする]
仮定２. 電子の回転の振動数ωと放射光の振動数νの間には次の関係がある：ν = ω/2
は、根拠のない、不必要なものとなる。
後知恵でこういうのは容易だが、当時の状況を考えると、そう簡単に批判はできないというべきだろう。
不可解な現象の説明を手探りで探求するときの錯雑した思考過程の一端を示すのが、この仮定２ではなかろうか。上述のように、当時知られていた量子論的な現象は、温度Tの物体と熱平衡状態にある電磁波のスペクトルを説明するためにPlanckが導入した振動子のエネルギー・スペクトルのnｈνと、光電効果を説明するためにEinsteinが導入した振動数νの光のエネルギーｈνとであった。前者は調和振動子であり、後者は自由電子(当時の金属電子論の常識であった)の金属外への放射であると考えると、それらは、原子の構造にくらべて格段に簡単な系であることに注意する必要がある。
したがって、Bohrが仮定１に重ねて、仮定２を用いたのは、理解できないことでもない。仮定１だけではBalmer系列などを説明できないというわけである。仮定２で、振動数比を勝手に1/２にしたのは、Balmer系列などを説明するためだったことは明らかである。
なお、Bohrモデルについての発表当時の物理学者の反応については、次のページを参照してください。
http://www.geocities.jp/hjrfq930/Science/sciencee/sciencee03.htm
余談。原子構造と常温核融合現象
常温核融合現象という、現段階では何が本質なのかの見当もつかない対象を取り扱うことを考えてみよう。常温核融合現象の起こる系に比べれば原子は格段に簡単な系である。原子構造の説明のために、この長い、分かりにくい論文が必要だったことを考えると、Bohrが原子構造モデルを作った作業に準じて、常温核融合現象を現象論的なアプローチで説明しようとしたとき、より多くの曖昧さが伴うことは明らかである。後に不要なことが分かるような仮定をしている可能性が十分にある。
原子という簡単な系で成功したBohrの原子模型の場合でもこんなに苦労したのかと、Bohrの論文の分かりにくさの意味が身にしみるのである。Bohrの同時代人の多くが、Bohrの論文に拒絶反応を示したと書かれているが、なるほどと頷ける。（2006.4）
補足引用文
§1.
General Considerations.
The inadequacy of the classical
electrodynamics in accounting for the properties of atoms from an atom-model as
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 ω 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
ω = ((2)1/2
/π)(W3/2/eE(m)1/2
), 2a = 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 value of ω 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 of 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 behavior 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 seem to
have absolutely fixed dimensions and frequencies. Further, if we consider any
molecular process, the result seems always to be that after a certain amount of
energy characteristic for the systems in question is radiated out, the systems
will again settle down in a stable state of equilibrium, in which the distances
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 ν in a
single emission being equal to τhν, where τ is an entire number, and h is a universal constant*.
*
See f. inst., M. Planck, Ann. d. Phys,
xxxi. p. 758 (1910) ; xxxvii. p. 642 (1912) ; Verh. deutsch. Phys. Ges. 1911, p. 138.
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 the 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 systems
containing only a single electron.
Let us now
assume that, during the binding of the electron, a homogeneous radiation is
emitted of a frequency ν, equal to half the frequent 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 τhν, where h is Planck's constant and τ 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
§3.
Putting
W = τhω/2, - - - - - - -
- - - - - - - - - - - - - - - - - - - - - -- - - -(2)
we
get by help of the formula (1)
W = 2π2me2E2/τ2h2,
ω = 4π2me2E2/τ2h2,
2a = τ2h2/2π2meE- - - - (3).
If in these expressions we give τ
different values, we get a series of values for W, ω, 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 τ has its smallest value 1. This case will therefore correspond to
the, most stable state 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 τ = 1
and E = e, and introducing the
experimental values
e = 4.7×10–10, e/m = 5.31×1017, h
= 6.5×10–27,
we
get
2a = 1.1×10–8 cm, ω = 6.2×1015 1/sec, W/e
= 13 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 behavior of atomic systems was originally
pointed out by Einstein*. The considerations of Einstein have been developed
and applied on a number of different phenomena, especially by Stark, Nernst,
and Sommerfeld. 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**, in an 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.
*
A. Einstein, Ann. d. Phys. xvii. p.
132 (1905); xx. p. 199 (1906) ; xxii. p. 180 (1907).
**
A. E. Haas, Jahrb. d. Rad. u. El.
vii. p. 261 (1910). See further, A. Schidlof, Ann. d. Phys. xxxv. p. 90 (1911); E.
Wertheimer, Phys. Zeitschr.
xii. p. 409 (1911), Verh. deutsch. Phys. Ges.
1912, p. 431; F. A. Lindemann, Verh. dentsch. Phys. Ges. 1911, pp.
482, 1107; F. Haber, Verh. deutsch. Plays. Ges.
1911, p. 1117.
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*. 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-length 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.
*
J. W. Nicholson, Month. Not.
-
- - - - - - - - - - - - - - - - - - - - - --
§2.
Emission of Line-spectra.
Spectrum
of Hydrogen. General evidence indicates that an atom of hydrogen consists
simply of a single electron rotating round a positive nucleus of charge e*.
*
See f. inst. N. Bohr, Phil. Mag. xxv.
p. 24 (1913). The conclusion drawn in the paper cited is strongly supported by
the fact that hydrogen, in the experiments on positive rays of Sir J. J.
Thomson, is the only element which never occurs with a positive charge
corresponding to the loss of more than one electron (comp. Phil. Mag. xxiv. p. 672 (1912)).
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,
Wr
= 2π2me4/h2τ2 [1]
The amount of energy emitted by the
passing of the system from a state corresponding to τ = τ1 to one
corresponding to τ = τ2, is consequently
Wr2 – Wr1 = 2π2me4/h2(1/τ22 – 1/τ12
). [2]
If now we suppose that the radiation in
question is homogeneous, and that the amount of energy emitted is equal to hν, where ν is
the frequency of the radiation, we get
Wr2 – Wr1 = hν,
and
from this
ν = 2π2me4/h3(1/τ22
– 1/τ12 ). - - - - - - - - - - (4)
We see that this expression accounts for
the law connecting the lines in the spectrum of hydrogen. If we put τ2
= 2 and let τ1 vary, we get the ordinary Balmer series. If we put τ2=
3, we get the series in the ultra-red observed by Paschen* and previously
suspected by Ritz. If we put τ2 =1 and τ1 = 4, 5, .., 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.
*
F. Paschen, Ann. d. Phys. xxvii, p.
565 (1908).
The agreement in question is quantitative
as well as qualitative. Putting
e = 4.7×10–10, e/m = 5.31×1017, and h
= 6.5×10–27,
we
get
2π2me4/h3 = 3.l×1015.
The
observed value for the factor outside the bracket in the formula (4) is
3.290×1015.
The 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 §3
return to consider the possible importance of the agreement in question.
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