Combining the above equations with (13), (11), and (12), we obtain

This result is similar in form to Tank's and the coefficients of the two lowest powers of H are the same as in his series, but the coefficient of H3 is different.

In order that the result may be valid, it is necessary that the series (4) converge and represent f(q) throughout the interval of integration, and that the derivatives of shall all be finite throughout a circle whose radius is greater than +H2 drawn about the origin on the complex u plane. From (17) it is evident that the derivatives of will be finite up to the point where w vanishes. It follows from (14') that ƒ′(q)/(q−d) must have no zeros on that part of the complex q plane which is mapped on the abovementioned circle on the u plane. It is necessary, in particular, that ƒ'(q) shall vanish not more than once for real values of q between a and b.

It does not seem worth while to attempt an exact discussion of the boundaries of the region on the complex q plane from which the zeros of f'(q)/(q-d) must be excluded, but we can say qualitatively that there is little chance that the series (7) will not converge throughout the interval of integration if the series

converges rapidly.

ω = 2α + 3βξ + 4γξ2 + 5δξο+ ...

(21)

In the applications of this analysis an expression for H as a function of J will generally be desired. The power series may be reversed to advantage by the following scheme which resembles that of Lagrange. Let H = F(J). Then by Taylor's theorem

Let b denote the coefficient of H" in (20). F'(O), F" (0), etc., can be calculated in terms of the b's. Let us first compute the derivatives of F in terms of H. Let

Since vanishes when H does, the values of F'(O), F"(0), etc., are obtained by setting H equal to zero in the right-hand members of (23) and (24). Thus

F'(0) = 1/b1:

F"(0) = -
= -2b2/b;

(25)

F'" (0)

=-6b3/b1 + 12b/bi;

If these formulas for F'(O), F" (0), etc., are evaluated in terms of the expressions for the b's given by (20), equation (22) becomes.

As a check on the series development here suggested, the writer has derived Sommerfeld's formula,

(12).

by means of the series

(27)

The derivatives of when evaluated by

equations (16) are in this case particularly simple and as a result the series can be summed.

It is of interest to note that the method of development in series here suggested is applicable to a variety of problems. It may be used to evaluate indefinite as well as definite integrals.

One simple application is in the determination of the periods of oscillation of a vibrating system. Iu the case of a conditionally periodic system with orthogonal coördinates, the periods are given by expressions of the form

where a and b are again roots of f(q). Introducing the quantities H, &, u defined as in the preceding work, we obtain

If de/du is developed into a power series of the type (7), we obtain

The coefficients K, vanish when 7 is odd, and the final expression for the period is

1 This is in effect the method used by Sommerfeld, Atombau und Spektrallinien,

2nd edition, Braunschweig, 1921, pp. 476–482.

2 F. Tank, Mitteilungen Physik. Ges. Zurich, 1919, p. 87.

Cf. G. Hettner, Zs. Physik., 1, 4, 1920 (350).

THE FURTHER EXTENSION OF THE ULTRA-VIOLET SPECTRUM AND THE PROGRESSION WITH ATOMIC NUMBER OF THE SPECTRA OF LIGHT ELEMENTS

By R. A. MILLIKAN

RYERSON PHYSICAL LABORATORY, UNIVERSITY OF CHICAGO

Read before the Academy April 26, 1920

The chief purpose of this investigation, outlined and begun in 19161 and briefly reported upon last year,2 has been to explore in the extreme ultra-violet the radiations which can be emitted by the second ring or shell of electrons in the atoms of atomic number from 2 to 13 (helium to aluminium).3

The results obtained to date may be very briefly summarized as follows:3 1. The ultra-violet spectrum has been photographed and its wavelengths determined down to λ = 136.6 Ångstroms in the case of aluminium and down to λ = 149.5 Ångstroms in the case of copper. There is thus a gap represented by a factor of but 10 between the shortest measured ultra-violet waves and the longest X-rays measured by the method of crystal-spectrometry which stops at 13.3 Ångstroms. Fortunately, however, in the ultra-violet region, which has been already opened up and explored, the most interesting and the most important of the hitherto inaccessible radiations are found; thus

2. The La lines of Al, Mg, and Na have been photographed and located at 144.3 Å, 232.2 Å, and 372.2 Å, respectively. These wave-lengths are all fairly accurately on the Mosely line connecting La frequencies and atomic number (see fig. 1). It has thus been definitely proved that the L series continues with its main characteristics unchanged throughout the whole range of atomic numbers from Ur (92) to Ne (10). The linear progression thus revealed clear down to neon could be very roughly inferred from the beautiful measurements of Hjalmar, which gave the Ka and KB lines of the elements down to sodium. For the Kossel relation between the frequencies of the K and L series, namely, KB-Ka La, although known to be quite inexact, was presumably sufficiently near to the truth to enable the order of magnitude of the La wave-lengths to be predicted. Kossel has already made this use of Hjalmar's data. His values of La computed for Al, Mg and Na in this way are actually about 20% too low.

=

3. It has been found that the aluminium atom (atomic number 13) when excited by these condensed sparks in vacuo emits no radiations whatever of wave-length between 144.3 Å and about 1200 Å where its M spectrum, that due to its three outer electrons, begins and extends with considerable complexity into the visible. This shows that optical spectra are quite like

X-ray spectra in that large gaps occur between the frequencies due to the electrons in successive rings or shells. If we could neglect the influence of the negative electrons upon one another we could compute the relative diameters of these shells, for they would be inversely proportional to the limiting frequencies, i.e., in this case in about the ratio 1 to 9; but such computations are of little value save as the roughest sort of indices. The chief lines below 2000 Å, due to the three outer electrons of the Al atom have the wave-lengths 1379.7 Å, 1384.5 A, 1605.9 Å, 1612.0 Å, 1671.0 Å, 1854.7 Å and 1862.7 Å.

Magnesium shows a behavior quite like that of aluminium in that we find a complete blank between its La line at 232.2 Å and the lines due to its two outer electrons whose radiations begin on our plates on the short wave-length side at about 1700 Å. The chief radiations below 2000 Å, arising from these two outer electrons of the magnesium atom, have the wave-lengths 1735.2 Å, 1737.9 Å, 1751.0 Å, and 1753. 7 Å.

Also, sodium, quite consistently with the foregoing, is found to emit no radiations whatever between its L lines, the longest of which is at 376.5 Å and the lines due to its single M electron which have their convergence wave-length at 2412.63 Å, and reach their maximum intensity in the familiar sodium doublet at 5890 Å and 5896 Å.

4. Coming now to the group of atoms below neon, atomic number 10, the spectra due to the electrons in the incomplete second or L-ring in the case of these atoms are completely unpredictable from any theory that we now have but they have been experimentally obtained. For reasons which will appear they will not be considered in the order of atomic number. a. The spectrum due to the six L-ring electrons of the oxygen atom (atomic number 8) begins upon our plates at about 230 Å and extends with much complexity and strength up to 834.0 Å where the strongest oxygen line, which will be arbitrarily called its La line, is found. Above 834.0 Å the oxygen lines are few in number and relatively faint.

α

Since the Ka line of O can be computed with much certainty to be at 23.68 Å, the ratio of the K frequency to the L frequency for the oxygen atom is about 35. It will be recalled that this ratio is only about 7 in the case of atoms of high atomic number and that it slowly increases with decreasing atomic number, reaching the values 17.2, 23.4 and 31.1 in the cases of Al, Mg, and Na, respectively (see above). The strongest of the oxygen lines have the following wave-lengths: 321.2, 374.3, 507.8, 525.7, 554.2, 599.5, 610.1, 616.7, 625.2, 629.6, 644.0, 703.1, 718.5 and 834.0.

The oxygen lines have been identified because they appear as impurities in all easily oxidizable metals, NrLi, Mg, Al, Zn, Fe, etc., The key to the oxygen spectrum was furnished by the discovery that chemically pure aluminium and magnesium showed the extraordinary property