(op. cit. liv. ii. ch. 4), and which he has employed with effect—in a manner probably admitting of wide applicationto discover laws masked by fortuitous happenings, such as the average difference in the height of the barometer at different times of day (op. cit. liv. ii. ch. 5). The case is that in which the crude observations are represented by a rectangle; one value being as likely as another between finite limits-say, ±a.

Altogether, we may conclude with Laplace that, in certain cases the Method of Situation is preferable to the Method of Least Squares, the Method called par excellence "the most advantageous method" (op. cit. Supplement 2, ed. 1847, p. 619); that (in general) it is natural to consider the Median as "a very good approximation" (op. cit., Introduction, p. lxix). "Mais le résultat donné par la methode la plus avantageuse est encore préférable."

CXIV. The Motions of Electrons in Gases and the Formation of Negative Ions in Air. By LEONARD B. LOEB *.

N a recent note in this magazine Dr. V. A. Bailey †, in commenting on the writer's work reported in 1921, appears to have been under an erroneous impression as to the method of measurement employed by the writer. In the paper referred to the writer attempted to verify the J. J. Thomson theory of ion formation by studying the mobility curves in air obtained below 10 cm. pressure, using the Rutherford alternating-current method with alternations of square wave form obtained from a commutator. This method is the method of experimentation termed by Dr. Bailey the "Lattey modification" of the Rutherford method. Thus the writer did not use the method termed by Bailey the "Franck inodification," which involves the use of an alternating current of sinusoidal form, as stated in Bailey's note.

In the work quoted above the equations used for a verification of the theory by the writer were but a rough approximation to the rigorous equations, as owing to the uncertain knowledge of many of the factors a greater precision did not seem warranted. The agreement obtained was merely a qualitative one. Lately much more careful measurements have been made, using the square wave oscillations, with the added advantage that some rough measurements on the mobilities of electrons in air at the pressures *Communicated by the Author.

Bailey, V. A., Phil. Mag. xlvi. p. 213 (July 1923).

Loeb, L. B., Phys Rev. n. s. vol. xvii. p. 89 (Feb. 1921).

employed in this work were also made. The results of these investigations, which are shortly to appear in print, enabled the use to be made of the rigorous theory of J. J. Thomson. The results obtained by including the values of the electron mobilities, as measured directly, in the rigorous attachment equations appear to give a striking verification of the theory. When, however, certain corrections which seem theoretically necessary are made on the values of the electron mobilities the agreement initially observed is completely vitiated. In the application of the corrections to the electron mobilities, certain assumptions are made about the space charge effects due to the accumulation of ions which may not be fulfilled in air. Thus it is still doubtful whether the corrected electron mobilities are more justified than the uncorrected ones. It is therefore, at present, an open question whether the rigorous form of the Thomson theory is correct or not.

Не

The writer agrees with Dr. Bailey that it is possible that "h," the attachment constant of J. J. Thomson, may decrease as the velocity of thermal agitation "u" decreases, though he is unaware of any direct evidence that this is so. desires to point out that for the change in "u" produced by higher fields H. B. Wahlin* found no evidence of a marked change in "h." It is also desired to point out that in order of magnitude "h" is such a distinct function of the chemical nature of the gas that even were the assumption of Bailey correct (viz., that "h" is a function of "u"), the theory of Thomson would be applicable in its broad outlines.

66

The writer has unfortunately long been aware of the fact that the use of sinusoidal alternating potentials of high frequency for the measurement of electron mobilities is an unsatisfactory mode of procedure. Yet up to the present the method has furnished the only means of estimating the electron mobilities in most gases above 100 mm. pressure. It might be pointed out that where the results obtained by the writer using sinusoidal oscillations overlap the very beautiful results of Townsend, Bailey, and their collaborators, they are roughly in agreement with them, as will be seen in a forthcoming paper by K. T. Compton †.

Department of Physics,

University of California,
Berkeley, Calif.

Aug. 25, 1923.

* Wahlin, H. B., Phys. Rev. n. s. vol. xix. p. 173 (Feb. 1922).

+ Compton, K. T., "Mobilities of Electrons in Gases," Phys. Rev. n. s. vol. xxii. (1923) (to appear).

CXV. On the Motion of Electrons in Gases.

To the Editors of the Philosophical Magazine.
GENTLEMEN,-

HAVE received a letter from Dr. L. B. Loeb with which T he encloses a copy of a note he has sent to the Philosophical Magazine, taking exception to a statement I made on his work in connexion with the formation of negative ions and the measurements of the velocities of electrons in gases.

My statement consisted of seven lines, and as Dr. Loeb has written several papers on these subjects, no reader would suppose that I had related all Dr. Loeb's work. The main facts to which I alluded, however, are those published by him in the Philosophical Magazine of January 1922, and in my paper I gave that reference.

In his paper of January 1922 Dr. Loeb gives reasons which led him to suppose that previous determinations of "mobilities were unreliable, and states that in his latest experiments on the determination of the mobility he used "high-frequency oscillations from an audion oscillator." He thus obtained a value of the "mobility mobility" which was 2800 cm./sec. for nitrogen, which he points out is much higher than the value 200 cm./sec. given by other experimenters, and he adopts the higher number as the · fundamental basis of the calculations of the chance of ion formation which he gives in this paper.

Under these conditions I am surprised to find the following sentence in Dr. Loeb's note to the Philosophical Magazine: "Thus the writer did not use the method termed by Bailey the Franck modification,' which involves the use of an alternating current of sinusoidal form, as stated in Bailey's note," as the high-frequency oscillations from an audion oscillator are usually supposed to be of sinusoidal

form.

I have therefore looked up another paper by Dr. Loeb (Physical Review, p. 24, 1922) where he gives a diagram and full details of his experiments with the audion oscillator, and it is clear from this paper that the apparatus is designed to generate oscillations of sinusoidal form, and they are treated as such in the investigation.

Dr. Loeb appears not to have noticed the fact that I referred to his paper in the Philosophical Magazine of January 1922, where he gives the impression that "mobilities" determined by sinusoidal alternating forces are reliable.

The statement in Dr. Loeb's note that the method of determining electron "mobilities" by means of sinusoidal alternating potentials has up to the present "furnished. the only means of estimating electron mobilities in most gases above 100 mm. pressure," is by no means correct. Townsend's method, involving the magnetic deflexion of a stream of electrons moving in a uniform electric field, may be used to determine the velocity of electrons in a gas at any pressure when no ions are present. Most of the determinations, it is true, have been made with gases at pressures below 100 mm., because the smaller the pressure the smaller is the magnetic force required to deflect the stream, and it is easier to work with small magnetic forces than with large forces involving the use of coils with iron cores. From the measurements that have been made it has been found that the velocity in the direction of an electric force Z is a function of the ratio Z/p, so that if the velocities are determined for one value of Z and one value of p, they may be considered as being known for any multiples n'Z and np of this force and pressure.

Electrical Laboratory,

Parks Road, Oxford.

Sept. 13, 1923.

Yours truly,

V. A. BAILEY.

CXVI. On Atomic Structure and the Reflexion of X-Rays by Crystals. By D. R. HARTREE, B.A., St. John's College, Cambridge*.

IN

§ 1. Introduction.

N a recent paper † I have given an account of an attempt to determine approximately the field inside and in the neighbourhood of an atom by quantitative analysis of the terms appearing in the optical and X-ray spectra, following the theory given in a more qualitative form by Bohr. It was pointed out there that the dimensions of the orbits and variation of time along them could be calculated approximately from the results of this analysis and applied to the problem of the intensity of the reflexion of X-rays by crystals, and some preliminary results of this application to sodium were quoted and compared with Bragg's experimental results ‡. * Communicated by R. H. Fowler, M.A.

+ D. R. Hartree. Proc. Camb. Phil. Soc. vol. xxi. part 6 (1923).

↑ W. L. Bragg, R. W. James, and C. II. Bosanquet, Phil. Mag. xli. p. 309; xlii. p. 1 (1921); xliv. p. 433 (1922). These papers will be referred to as B. J. B. i. ii, and iii.

Phil. Mag. S. 6. Vol. 46. No. 276. Dec. 1923.

4 B

The object of the present paper is to consider this application in more detail, and also to inquire whether and to what extent it may be possible to obtain from X-ray reflexion evidence on the orientations of the orbits in the atom and on the relative phases of the electrons in the different orbits

For simplicity we will consider a rocksalt crystal, as it is for rocksalt that the experimental work has been done, and it is quite sufficient for the purpose of this paper. With the arrangement of atoms in the rocksalt crystal, the different possible reflecting faces can be divided into two groups namely, those for which every plane of atoms parallel to the reflecting face is similarly constituted, i. e., contains equal numbers of Na and Cl atoms regularly arranged, and those for which alternate planes are differently constituted, i. e., contain Na atoms only and Cl atoms only. The former are faces for which one index is even, the latter those for which all indexes are odd; they will be referred to as "even" and "odd" faces respectively.

The formula for the intensity of X-ray reflexion need not be quoted here in its entirety. The important part of it for the purposes of this paper is the factor, written F2 in Bragg's papers, which depends on the number and arrangement of electrons in each scattering atom. If the wavelets scattered by the individual electrons are all in phase, then F is equal to the number of electrons concerned in the scattering; this is the case when the electrons are distributed in a space small compared with the wave-length of the incident X-rays, and also in certain other conditions. When the wave-length of the incident X-rays is of the same order as the distances between the various electrons, the scattered wavelets are not in general all in phase, and the amplitude of the wave scattered by the atom as a whole will be less than that scattered when the wavelets are in phase; this is expressed by a smaller value of F. The extent of this interference depends on the electron arrangement and on the angle of scattering.

The scattering by the individual electrons is assumed to be properly deduced from classical electrodynamics; the results so obtained are at any rate of the right order, as pointed out in Bragg's paper. The amplitude of the wavelet scattered by one electron according to classical electrodynamies will be taken as the unit of amplitude; the value of this unit in terms of the mass and charge of the electron etc., and also the summation of the contributions

* B. J. B. iii. p. 446.