discharge showed a striated positive column, but it was impossible to determine whether it was still striated when the final break occurred.

(1) In Geissler tubes fitted with hot cathodes there were two discontinuous changes in the current as the voltage across the tube is increased, these changes being associated with the appearance of two types of luminous discharge. At short distances between the electrodes, however, one of these disappears, and the conditions of the low-voltage arc, stimulated by the emission from a hot cathode, appeared.

(2) The behaviour of the two types of discharge under various changes in conditions was not the same, although the spectra appeared to differ only in intensity.

(3) The least voltage at which it was possible to maintain a discharge was not found to bear any simple relation to the ionizing potential of hydrogen, a reason for which has been offered.

(4) The voltages at which discontinuous changes in the current with mercury vapour occurred were found to be multiples of the ionization potential or these multiples plus the first radiating potential.

In conclusion, the writer wishes to express his appreciation to Professor K. T. Compton, under whose supervision this work was done, for his unfailing interest and many valuable suggestions during the course of this investigation.

Palmer Physical Laboratory,

Princeton University.

LVII. On the Physical Properties of Elements at High
Temperatures. By MEGH NAD SAHA *.

THE investigation of the physical properties of elements at high temperature is at present exciting a considerable amount of interest, on the theoretical as well as on the practical side. The present paper is the outcome of certain investigations undertaken by the author, which were withheld from publication because no positive result was obtained. But in view of the recent works, it appeared advisable to give publicity at least to certain points which appear to have been rather lightly passed over by recent workers.

Let us picture to ourselves a quantity of gas, elementary or compound, which is being raised to higher and higher temperatures. The physical changes occurring in the mass under such an increasing stimulus have been discussed in previous papers †. It has been shown that the gas will become luminous, will emit its characteristic lines-principal lines, sharp and diffuse lines, Bergmann lines-and ultimately will be ionized. The problem before us is: (1) to determine the statistical distribution of the atoms in the various quantum orbits, and from this to deduce the intensity of the different lines of the characteristic spectrum; (2) to determine the electrical and optical properties of such a mass of ionized gas.

The method which the present writer followed was purely thermodynamical. It consisted in the application of a form of the law of reaction-isochore, which was originally developed by Nernst for the study of the dissociation-equilibria of gaseous compounds from their physical properties, to the problem of ionization. The same method was followed in the extension of the method to mixtures of different elements by H. N. Russell. But in this, as in other cases, thermodynamics lead us rather blindfolded to the goal, and do not enable us to see the details of the intervening stages.

A very powerful method has recently been developed by Messrs. Darwin and Fowler § in a number of important papers published in the Phil. Mag. and Proc. of the Camb. Phil. Soc. Probably with the aid of this method the problem is *Communicated by the Author.

+ M. N. Saha, Proc. Roy. Soc. Lond. May 1921. Phil. Mag. vol. xli. p. 267 et seq., see particularly p. 274.

Russell, The Astrophysical Journal,' vol. lv. p. 143. Observatory, vol. xliv. Sept. 1921.

§ Darwin & Fowler, Phil. Mag. vols. xliv. & xlv. Phil. Soc. vol. xxi. parts 3 & 4.

Milne, 'The

Proc. Camb.

brought much nearer to solution, but it seems that there are a number of important points on which the authors have not laid sufficient stress. The first point to which I wish to call attention is that "no theory of dissociation-equilibrium can be said to be complete unless it takes account of the mutual interaction between matter (atoms) and radiant energy, because at high temperatures, exchange of energy takes place mainly by radiation, and only to a slight extent by collision."

Binding of an Electron with a Proton (H+).

To make the above point clear, we shall consider the simplest case conceivable-namely, the binding of an electron with a proton to form an H-atom. This case can rightly be called the simplest, because, thanks to the Bohr theory, all the possible states of combination are known, and the dynamics can be handled with easy mathematics-advantages which are not present in such cases as the reactions

H2H+H, Ca Ca++e.

Let us first treat the dynamical part. An electron starts from infinity with velocity v, and passes past a stationary proton. What are the conditions that this will be captured by the proton? A little consideration will show that as long as the energy of the system remains conserved, the electron will describe an hyperbola with the proton as the inner focus, and thus, after wheeling round the proton, will pass off to infinity. In other words, it can never be captured by the proton and lodged in one of the stationary Bohr orbits having the energy) unless the system loses the

energy

From physical ground, it seems to be fairly well established that such a process actually takes place in nature in all cases of ionization, giving rise to a continuous spectrum beginning from the limit of the series lines. In the case of hydrogen, the continuous spectrum was first detected by Huggins during the observation of the eclipse spectra, and was confirmed by Evershed. The explanation cited above is due to Bohr. (See 'Atombau und Spektralanalyse,' p. 547, 3rd edition.)

But the dynamical interpretation of this process from the standpoint of quantum mechanics is far from satisfactory, as

has been pointed out by Nicholson in a recent paper. Eddington has thrown out the suggestion that during its orbital motion the electron loses energy by radiation just as an accelerated electron would do according to the classical theory. But here we are treading on rather dangerous ground, as the satisfactory working out of the suggestion means nothing less than the discovery of the linkage between the classical theory and the quantum theory.

In the case of hydrogen, the continuous spectrum has been observed at the limiting frequency of the Balmer series (quantum orbit 2, or 22), but this is owing to the fact that observations do not probably extend up to the limit of the Lyman series. In the case of the alkali elements, the continuous spectrum has been observed extending towards the short wave-length side from the limiting frequency of the principal series. These facts are very decisive in favour of the view that combination of an ionized atom with an electron is always accompanied by liberation of energy in the form of continuous waves of light.

The Statistical part; deduction of the Law of Reaction

isochore for the Ionization of the H-atom.

This part has been worked out by Fowler, but before taking up his method of deduction, I shall give, another deduction based on the older methods, because this may serve to bring out the details of the case in a more intelligible manner.

The problem which we are discussing is only a special case of the general problem of association of particles and dissociation of compound particles which was first treated from the standpoint of the kinetic theory by Boltzmann, Natanson, and J. J. Thomson about thirty years ago. A masterly discussion is given in Jeans's Dynamical Theory of Gases,' p. 213 et seq., 3rd edition. This treatment is applicable, of course with some alteration, to the combination of protons and electrons.

Suppose we have a system consisting of 2v, particles of type A, and v particles of type A, formed by the combination of two particles A. Let v=2(v1+v2): i. e., v is the total number of particles if there be no aggregation at all.

=

Then Jeans shows that an encounter between two particles of type A can never result in an association unless the quantity m2+24 (where 24 mutual potential energy of the particles, v relative velocity of the two particles) assumes a negative value. According to Jeans, "this might be effected by collision with a third molecule [it is not at all Nicholson, Phil. Mag. vol. xliv. p. 193 (1922).

† Eddington, Monthly Notices R. A. S. vol. lxxxiii p. 43.

clear how], or possibly, if my2+24 were small at the beginning of the encounter, sufficient energy might be dissipated by radiation for mv2+24 to become negative before the termination of the encounter."

Jeans continues: "we may leave the consideration of this second possibility on one side for the present, with the remark that if this were the primary cause of aggregation, we should no longer be able to use the equations with which we have been working, since they rest upon the assumption of conservation of energy."

The simplest illustration of association is the binding of a proton (H) with an electron. As has been already shown, here it is not possible to leave on one side the action of radiation, for that will be tantamount to staging the play of Hamlet without Hamlet's part; for physical evidences decisively prove that radiation is emitted in all cases of the binding of an ionized atom with the electron. Similarly, absorption of radiation is essential for the splitting up of an atom M into M+ and e.

In spite of rather uncertain knowledge regarding the rôle of radiant energy in these processes, Boltzmann deduced a formula on the assumption that potential energy exists between two molecules (here we should say between H+ and e), when the centre of the second (e) lies within a sensitive region surrounding the first (H). With this assumption, Boltzmann obtains a formula (formula 503 of page 199, 3rd edition, Jeans's' Dynamical Theory of Gases') which, with a slight change in notation, can be put in the form

where volume of the sensitive region.

Comparing this with the formula for reaction - isobar

derived by me, viz.

i. e., the radius of the sensitive layer varies as

2.95 × 10-6

cm.