where and 'assume values out of the series:

0, 1, 2,..., ±n, ±(n+1), ... etc.
±1,

and

0, +1, ±2,..., ±(2n−1), ±2n, ... etc. respectively.

From (51) we have as a first approximation for $1, neglecting terms in a。:

Or, if we take account of terms in ao to the first order only in (51), we may write

where the coefficients 1, 2, 3 correspond to simple lines, triplets, or doublets respectively, but cannot be determined. by our analysis, and the notation [], denotes values of the quantities involved calculated for c1, 0=C2, 0= ... =0.

The two functions, and p', introduced in (51) must be understood to involve "hidden" quantum numbers arising from a quantization of the general motion of the whole atom. The intensities and types of polarization of the various Zeeman components will depend on the nature of these two functions, and may be determinable by the application of Bohr's Principle of Correlation to the internal mechanism of the atom. Such an application would be equivalent to a form of Choice Principle relating to the "hidden" quantum numbers, and should lead to an explanation not only of the questions of intensity and polarization, but also of the way in which and 'assume different groups of values for the different components of a doublet or a triplet.

With regard to the effect of the relativity correction on the appearance of the lines, we see from (45), (46), (47), and (48) that if we assume n to be fixed for each series term, then, since (n+n) is also fixed, it follows that n, is also fixed, and hence 8W is a one-valued quantity. Thus the effect of the

relativity correction on any spectral line would on Sommerfeld's theory consist in a slight modification in the position of the line given by

where ♪ W (m) and 8W (n) are given by (45) for the quantum

numbers (m, m1) and (n, n1) respectively. This leads to no fine structure whatever, and an explanation of the doublet and triplet structure of the lines must be sought for elsewhere.

§ 5. Conclusion.

In his remarks on the array of figures given in Table A of the last article, Prof. Sommerfeld † writes: "only so much appears to be certain: that the harmony of integral numbers represented by our Runge Denominators has its ultimate foundation in the rules of hidden quantum numbers, and possesses a quantum derivation." We have attempted in this paper to inquire into the nature of this derivation, and have shown how it may be possible to ascribe the most complex and general types of Zeeman decomposition to small variations in the atomic field arising from the establishment of the external magnetic field. Our theory is, however, limited by our very deficient knowledge of the exact nature of the atomic field itself. The quantities Eo, C1, 0, C2, 0, ..., etc. and the way in which they arise for the different types of atomic motions being only partly comprehended at present, it does not seem yet promising to attempt an evaluation of their variations with a magnetic field from first principles. Thus we have to satisfy ourselves in § 4 with the assumption expressed by equations (51), which may be regarded as empirical. This assumption may, on the other hand, be looked upon as as a guide to the nature of the terms E, €1,0,..., etc. themselves, and may eventually serve as a fresh test for possible theories which claim to assign exact values to these quantities from first principles. Thus the phenomenon of the Zeeman Effect may on the Quantum Theory, as it did on the classical theory, throw light on the problems of atomic structure and radiation.

* Loc. cit.

† Loc. cit. p. 542; this is a free translation from the original, which is in German.

This differs from the similar integral evaluated by Sommerfeld (Atombau, u.s.w.' p. 476, under (e)) by the appearance of the term in D. We have to our degree of approximation

where

J2 =

J1 =J2+ D1J3+ DJ1+ D2J5—§D12J ̧‚·

2

2B
A+ +

2B
A+ +

4

+ G ] dr = −2xi (√C-BA),

+

-2πi

-1/2 dr

(ii.)

J5

доб

the values of all integrals except J being obtained from Sommerfeld's work, already referred to. To evaluate J we note that it is regular at infinity, so that

+ CNC (D,B+ D2 - CD)}, (iii.) (D2B+

✓C being (as pointed out by Sommerfeld) taken as the negative (imaginary) root of C. This is the result quoted in the text.

XIX. The Auroral Spectrum and the Upper Strata of the Atmosphere. Preliminary Communication. By L. VEGARD, Doctor of Science, Professor of Physics at the University of Christiania".

DURING recent years I have been investigating the

auroral spectrum. In the winter 1912-13, I undertook an expedition to Bossekop in Finmarken †, the main object of which was to study the auroral spectrum. With a spectrograph which combined a considerable dispersion with a great power of light, I succeeded in photographing a few of the strongest lines in the blue part of the auroral spectrum, and it was proved that these lines belong to the so-called negative band spectrum of nitrogen.

During the winter of 1921, I continued investigations in Christiania, and here I made determinations of the green auroral line. As the result of a number of measurements, I found the auroral line to have the following wave-length:

λ=55780 (intern. units).

Although the auroral line was determined with such an accuracy that the error is only a fraction of an Å unit, the origin of the line remained as mysterious as ever.

It was to be hoped that a more complete investigation of the whole auroral spectrum might give us some valuable information also with regard to the origin of the green line. But apart from this question, the determination of the auroral spectrum is a problem of the very greatest importance, on account of its bearing on the question regarding the constitution of the upper strata of the atmosphere and the nature of the cosmic electric rays producing the aurora borealis.

In the year 1921 the "Government Fund for Scientific Research" furnished me with the necessary means for taking up this work in a more systematic way. A more complete description of experimental arrangement will be given in a later work. Presently I shall only mention that during the last winter (1922-23) three spectrographs, which were put up last summer, have been at work at the Geophysical Institute of Tromsø, where the top roof of the building has *Communicated by the Author.

L. Vegard, Phys. Z. S. xiv. p. 677 (1913); Ann. d. Phys. 1. p. 853 (1916); Bericht über eine Expedition nach Finmarke, Christiania Vid. selsk. skr. Mat. kl. 1916, No. 7.

L. Vegard, Geofysiske Publikationer, vol. ii. No. 5. Phil. Mag. Ser. 6. Vol. 46. No. 271. July 1923.

kindly been put at my disposal by the Director, O. Krogness. For the sake of convenience we shall give the spectrographs the following notation, viz. I., II, and III. (Ronian numerals), where

I. is a quartz spectrograph with a fairly large dispersion and of high light power, for studying the ultraviolet part of the spectrum;

II. is a fairly big glass spectrograph with a considerable dispersion and a fairly high light power;

III. is a small glass spectrograph with the largest possible light power, but with a much smaller dispersion.

The spectrographs I. and II. were specially designed for an accurate determination of the wave-length of the lines that might appear in the auroral spectrum during the time of exposure, which had to be very long.

The small glass spectrograph, III., was constructed for the study of possible variations of the auroral spectrum, and also to learn how many lines could be observed in the visible part of the spectrum.

The big spectrographs I. and II. were mounted in a wooden box where the temperature could be regulated automatically. The whole box could be turned about a horizontal and a vertical axis.

These spectrographs have been in operation during the last winter (1922-23), and in this work I have been very ably assisted by Mr. Einar Tønsberg. We have already obtained a number of plates showing a considerable number of lines. The plates have been measured at the Physical Institute of Christiania, and in this work I have been ably assisted by Mr. Jonathan Aars.

With the quartz spectrograph we have obtained three spectrograms taken on "Imperial Eclipse" plates. The time of exposure and the lines obtained are given in Table I.

It appears that with a time of exposure of 15-20 hours of northlight a considerable number of lines can be obtained on the plate by means of the quartz spectrograph. It should be noticed that in the spectra 1 and 2 the strongest lines, such as 4278 and 3914, are over-exposed.

To give an idea of the relative strength of the lines, I have given them intensities from 1-10. Later on, I intend to give more accurate intensity values based on quantitative measurements with a registering micro-photometer.

On the spectrograms from the quartz spectrograph 21 lines have been measured.