We can now see very clearly the similarity between the formula derived from the spectrum investigations and that derived from the two Bohr postulates. In both formulæ the frequency appears as the difference between two terms which are characterized in both cases by two integral numbers, in the first formula, numbers denoting two stationary orbits in the Bohr model for hydrogen, and in the second the two numbers which in the Balmer-Ritz formula for the hydrogen spectrum characterize, respectively, a series and one of the lines of the series. To obtain complete agreement we have merely to equate the corresponding terms in the two formulæ. Thus we have for any arbitrary integer n

Aₙ = K or Aₙ = hK
hn ²n ²

For the innermost stationary orbit, for which n = 1, the ionizing work A₁ will accordingly be equal to the product of the constants h and K of Planck and Balmer respectively; and for the orbits No. 2, No. 3, No. 4, etc., the values will be respectively ¼, ¹/₉, ¹/₁₆, etc., of this product. From the charges on the nucleus and the electron, which are both equal to the elementary quantum e of electricity ([see p. 90]), and from the ionizing energy for a given orbit we can now find by the use of simple mechanical considerations the radius of the orbit. If we denote the radii of the orbits 1, 2, 3 ... by a₁, a₂, a₃ ..., we then obtain for the diameters 2a₁, 2a₂, 2a₃ ... the values 2a₁ = 1·056 × 10⁻⁸ cm. (or approximately 2a₁ = 10⁻⁸ cm.), 2a₂ = 4 × 10⁻⁸ cm., 2a₃ = 9 × 10⁻⁸ cm., etc. It is seen that the radii of the orbits are in the proportion 1, 4, 9 ..., or in other words the squares of the integers which determine the orbit numbers. It is in this proportion that the circles in [Fig. 25] are drawn. We must remember, however, that we have here for the moment been thinking of the orbits as circles, while in reality they must in general be assumed to be ellipses. The foregoing considerations will, however, still hold with the single change that 2aₙ will now mean, instead of the diameter of a circle, the major axis of an ellipse.

Let us return to the formulæ

ν = Aₙ″ - Aₙ′ and ν = K - K
hhn″ ²n′ ²

Here n″ denotes in the first formula the index number for the inner of the two orbits between which the transition is supposed to take place, while in the second formula n″ denotes a definite series in the hydrogen spectrum. If n″ is 2 while n′ takes on the values 3, 4, 5 ... ∞ then in the Bohr model of the hydrogen atom this corresponds to a series of transitions to the orbit No. 2 from the orbits 3, 4, 5 ..., while in the hydrogen spectrum this corresponds to the lines in the Balmer series, namely, the red line (Hα) corresponding to the transition 3-2, the blue-green line (Hβ) to 4-2, the violet line (Hγ) to 5-2 and so on. If we now put n″ = 1 while n′ takes the values 2, 3, 4 ..., we get in the atom transitions to the orbit No. 1 from the orbits No. 2, 3, 4 ..., corresponding in the spectrum to what is called the Lyman series in the ultra-violet (named after the American physicist Lyman, who has carried on extensive researches in the ultra-violet region of the spectrum). Thus every line in the hydrogen spectrum is represented by a transition between two definite stationary states in the hydrogen atom, since this transition will give the frequency corresponding to the line in question.

At first sight this would seem perhaps to be such an extraordinary satisfactory result that it would prove an overwhelming witness in favour of the Bohr theory. A little more careful thought, on the other hand, would perhaps cause a complete reversion from enthusiasm and lead some to say that the whole thing has not the slightest value, because the stationary states were so chosen that agreement might be made with the Balmer-Ritz formula. This last consideration, indeed, states the truth in so far that the agreement between the formula and the theory, at least as developed here up to this point, is of a purely formal nature. In the Bohr postulates the frequencies of the emitted radiation are determined by a difference between two of a series of energy quantities, characterizing the stationary states, just as in the Balmer-Ritz formula they appear as a difference between two of a series of terms (K, K/4, K/9, ...) each characterized by its integer. Now by characterizing the quantities of energy in the stationary states by a series of integers (in itself a wholly arbitrary procedure) complete agreement between the Bohr stationary states idea and the spectral formulæ can be attained. It is not even necessary to introduce the Rutherford atomic model to attain this end. By bringing in this specific model, one might join the new theory to the knowledge already gained of the atomic structure, and, so to speak, crystallize the hitherto undefined or only vaguely defined stationary states into more definite form as revolution in certain concrete orbits. This would then lead to a more comprehensive conception of atomic structure. But the theory unfortunately would still be rather arbitrary, since there would seem to be no justification for picking out certain fixed orbits with definite diameters or major axes to play a special rôle. One cannot wonder then that many scientists considered the Bohr theory unacceptable, or at any rate were inclined to look upon it simply as an arbitrary, unreasonable conception which really explained nothing.

Naturally, Bohr himself clearly recognized the formal nature of the agreement between the Balmer-Ritz formula and his postulates. But Bohr was the first to see that the quantum theory afforded the possibility of bringing about such an agreement, and he saw, moreover, that the agreement was not merely fortuitous, but contained within it something really fundamental, on which one could build further. That atomic processes on his theory took on an unreasonable character (compared with the classical theory) was nothing to worry about, for Bohr had come to the clear recognition that it was completely impossible to understand from known laws the Planck-Einstein “quantum radiation,” or to deduce the properties of the spectrum from the Rutherford atom alone. He therefore saw that his theory was really not introducing new improbabilities, but was only causing the fundamental nature of the contradictions which had previously hindered development in this field to appear in a clearer light.

But in addition to this the choice of the dimensions of the stationary states was by no means so arbitrary as might appear in the foregoing. In his first presentation of the theory of the hydrogen spectrum, Bohr had derived his results from certain considerations connected with the quantum theory—considerations of a purely formal nature, indeed, just as those developed in the preceding, but leading to agreement with the spectral formulæ. He, moreover, called attention to the fact that the values obtained for the orbital dimensions were of the same order of magnitude as those which could be expected on wholly different grounds. The diameter of the innermost orbit, i.e., that which defines the outer limit of the atom in the normal state, was found to be, as has been noted above, about 10⁻⁸ cm., i.e., of the same order of magnitude as the values obtained for the diameters of molecules on the kinetic theory of gases ([see p. 27]). The stationary states corresponding to very high quantum numbers one could expect to meet only when hydrogen was very attenuated, for otherwise there could be no room for the large orbits. We note that the 32nd orbit must have a diameter 32² (or over 1000 times) as great as the innermost orbit. Since, now, lines with high number in a hydrogen series correspond on the Bohr theory to transitions from orbits of high number to an inner orbit, it became understandable why only comparatively few lines of the Balmer series are ordinarily observed in the discharge tube, while many more lines are observed in the spectra of certain stars. For in such stars the possibility is left open for hydrogen to exist in a very attenuated state, and yet in such large masses that the lines in question can become strong enough for observation. In fact, one must assume that in a great mass of hydrogen a very large number of atoms send out simultaneously light of the wave-length corresponding to one line. For the ionizing work, i.e., the work necessary to eject the electron completely from the normal state and thus make the atom into a positive ion, the Bohr theory gives a value of the same order of magnitude as the so-called “ionization potentials” which have been found by experiment for various gases. An exact correspondence between theory and experiment could for hydrogen not be attained with certainty, because the hydrogen atoms in hydrogen gas under ordinary conditions always appear united in molecules.

In his very first paper, however, Bohr had studied Balmer’s formula also from another point of view, and had derived in this way an expression for the Rydberg constant K which agreed with experiment. These considerations have reference to the above-mentioned connection of the theory with the classical theory of electrodynamics.