Such a connection had previously been known to exist in the fact that, for long wave-lengths, the radiation formula of Planck reduces practically to the Rayleigh Jeans Law which can be derived from electrodynamics. This is related to the fact that when ν is small (long wave-lengths), the energy quantum hν is very small, and hence the character of the radiation emitted will approach more and more nearly to a continuous “unquantized” radiation. One might then expect that the Bohr theory also should lead in the limit of long wave-lengths and small frequencies to results resembling those of the ordinary electrodynamic theory of the radiation process. On the Bohr theory we get the long wave-lengths for transitions between two stationary states of high numbers (numbers which also differ little from each other). Thus suppose n is a very large number. Then the transition from the orbit n to the orbit n - 1 will give rise to radiation of great wave-length. For in this case Aₙ and Aₙ₋₁ differ very little, and accordingly hν is very small, as must ν be also. According to the electrodynamic theory of radiation, the revolving electron should emit radiation whose frequency is equal to the electron’s frequency of revolution. According to the Bohr theory it is impossible to fulfil this condition exactly, since radiation results from a transition between two stationary orbits in each of which the electron has a distinct revolutional frequency. But if n is a large number, the difference between the frequencies of revolution ωₙ and ωₙ₋₁ for the two orbits n and n - 1, respectively, becomes very small; for example, for n = 100, it is only 3 per cent. For a certain high value of n, then, the frequency of the emitted radiation can therefore be approximately equal to the frequency of revolution of the electron in both the two orbits, between which the transition takes place. But even if this proved correct for values of n about 100, one could not be sure beforehand whether it would work out right for still larger values of n, for example, 1000.

In order to investigate this latter point we must look into the formulæ for the revolutional frequency ω in a stationary orbit and for the radiation frequency ν. Since, according to the Bohr theory, we can apply the usual laws of mechanics to revolution in a stationary orbit, it is an easy matter to find an expression for ω. From a short mathematical calculation we can deduce that ω = R/n³, where R is the frequency of revolution for the first orbit (n = 1). We find ν, on the other hand, by substituting in the Balmer-Ritz formula the numbers n and n - 1, and a simple calculation shows that for great values of n, the expression for ν will approach in the limit the simple form ν = 2K/n&sup3. For large orbit numbers, ν accordingly varies as ω, i.e., inversely proportional to the third power of n, and by equating R and 2K, we find that the values for ν and ω tend more and more to become equal.

In this way the value of K, the Balmer constant, may be computed. It is found that

K = 2π²e⁴ m
h³

where e is the charge on the electron, m the mass of the electron, and h is Planck’s constant. Upon the substitution of the experimental values for these quantities, a value of K is determined which agrees with the experimental value (from the spectral lines investigation) of 3·29 × 10¹⁵ within the accuracy to which e, m and h are obtainable. This agreement has from the very first been a significant support for the Bohr theory.

One might now object that we have here considered radiation due to a transition between two successive stationary states, e.g., No. 100 and No. 99, or the like (a “single jump” we might call it). On the other hand, for transitions between states whose numbers differ by 2, 3, 4 or more (as in a double jump, or a triple jump) the agreement found above will wholly disappear, and doubt be cast on its value. For in such cases of high orbit numbers the frequency of revolution will remain approximately the same even for a difference of 2, 3, 4 or more in orbit number; but the radiation frequency for a double jump will be nearly twice that for a single jump, while that for a triple jump will be nearly three times, etc. Accordingly, for approximately the same revolutional frequency ω we shall have in these cases for the radiation frequency very nearly ν₁ = ω, ν₂ = 2ω, ν₃ = 3ω, etc. We must, however, remember that when the orbit in the stationary states is not a circle, but an ellipse (as must in general be assumed to be the case), the classical electrodynamics require that the electron emits besides the “fundamental” radiation of frequency ν₁ = ω, the overtones of frequencies ν₂ = 2ω, ν₃ = 3ω .... We then also here see the outward similarity between the Bohr theory and the classical electrodynamics. We may say that the radiation of frequency ν, produced by a single jump, corresponds to the fundamental harmonic component in the motion of the electron, while the radiation of frequency ν₂, emitted by a double jump, corresponds to the first overtone, etc.

The similarity is, however, only of a formal nature, since the processes of radiation, according to the Bohr theory, are of quite different nature than would be expected from the laws of electrodynamics. In order to show how fundamental is the difference, even where the similarity seems greatest, let us assume that we have a mass of hydrogen with a very large number of atoms in orbits, corresponding to very high numbers, and that the revolutional frequency can practically be set equal to the same quantity ω. There may take place transitions between orbits with the difference 1, 2, 3 ... in number, and as the result of these different transitions we shall find, by spectrum analysis, in the emitted radiation frequencies which are practically ω, 2ω, 3ω, etc. According to the radiation theory of electrodynamics we should also get these frequencies and the spectral lines corresponding to them. It must, however, be assumed that they are produced by the simultaneous emission from every individual radiating atom of a fundamental and a series of overtones. According to the Bohr theory, on the other hand, each individual radiating atom at a given time emits only one definite line corresponding to a definite frequency (monochromatic radiation).

We can now realize that the Bohr theory takes us into unknown regions, that it points towards fundamental laws of nature about which we previously had no ideas. The fundamental postulates of electrodynamics, which for a long time seemed to be the fundamental laws of the physical world itself, by which there was hope of explaining the laws of mechanics and of light and of everything else, were disclosed by the Bohr theory as merely superficial and only applicable to large-scale phenomena. The apparently exact account of the activities of nature, obtained by the formulæ of electrodynamics, often veiled processes of a nature entirely different from those the formulæ were supposed to describe.

One might then express some surprise that the laws of electrodynamics could have been obtained at all and interpreted as the most fundamental of all laws. It must, however, be remembered that the Bohr theory for large wave-lengths, i.e., the slow oscillations, leads to a formal agreement with electrodynamics. It must, moreover, be remembered that the laws of electrodynamics are established on the basis of large-scale electric and magnetic processes which do not refer to the activities of separate atoms, but in which very great numbers of electrons are carried in a certain direction in the electric conductors or vibrate in oscillations which are extremely slow compared with light oscillations. Moreover, the observed laws, even if they can account for many phenomena in light, early showed their inability to explain the nature of the spectrum and many other problems connected with the detailed structure of matter. Indeed the more this structure was studied, the greater became the difficulties, the stronger the evidence that the solution cannot be obtained in the classical way.

If we ask whether Bohr has succeeded in setting up new fundamental laws, which can be quantitatively formulated, to replace the laws of electrodynamics and to be used in the derivation of everything that happens in the atom and so in all nature, this question must receive a negative answer. The motion of the electron in a given stationary state may, at any rate to a considerable extent, be calculated by the laws of mechanics. We do not know, however, why certain orbits are, in this way, preferred over others, nor why the electrons jump from outer to inner orbits, nor why they sometimes go from one stationary orbit to the next and sometimes jump over one or more orbits, nor why they cannot come any closer to the nucleus than the innermost orbit, nor why, in these transitions, they emit radiation of a frequency determined according to the rules mentioned.