Expressed in terms of the semi-major axis Ro, eccentricity e and periodic time T, the progress per revolution is

The apses therefore progress or regress according as (k-k) is positive or negative.

We have no knowledge as to the proper forms of m1 and m, for matter in bulk, but the following are results for hypothetical single nuclei.

For the contracted electron using relativity methods,

For spherical conductor which does not change in shape :

This last case is numerically almost the same as that for the contracted electron by relativity methods. This is important, because it shows that so far as inertia enters in the astronomical problem we can get practically the same result

• Proved only for disturbance from a steady state.

as a logical sequence from the fundamental equations as has been obtained by relativity doctrine.

Our results for the apsidal progression per revolution are: Contracted electron by relativity method,

which numerically is in close agreement with observations on "Mercury."

This result is obtained by assuming that the attraction depends on the velocity. It is easily seen from our analysis that if

In order to get the observed value for "Mercury" k would have to be 5/2 if k-k, is 1/4.

It is important to recognize that it is only by introducing either explicitly or implicitly this comparatively large dependence on speed, of the attraction between bodies that Einstein can get the numerical agreement. Such dependence based on the known forces between electrical currents has been recognized before now in the theory of electrodynamics, but is hardly acceptable in gravitational theory. On these lines it appears that orthodox electrodynamics is quite as capable of providing an explanation of this astronomical feature as Einstein's theory. It is, however, important that endeavour should be made to determine, if possible, the numerical value of k1+kз−k, for matter in bulk.

There still remains the question of the effect of transference in space as suggested by Sir Oliver Lodge. Eddington's conclusions on this problem may be modified considerably

when what I hold to be more correct equations of motion are used.

NOTE. At Sir Oliver Lodge's request I have calculated m1 and m2 for Bucherer's electron which has the same form as Lorentz's electron but keeps its volume unchanged. My results are

XXXVII. Molecular Frequency and Molecular Number. By H. STANLEY ALLEN, M.A., D.Sc., University of London, King's College*.

THE

PART I.

§ 1. Molecular Number.

THE work of Moseley on the high-frequency spectra of the elements has established securely the importance of the "atomic number " of an element: that is, the number which determines the place in the periodic classification and fixes the charge carried by the central part of the atom. It is now certain that the atomic number is more fundamental than the atomic weight. Recent investigations of the atomic weight of lead of radioactive origin have shown that the value obtained for this quantity depends upon the source from which the material is derived. An interesting account of these researches has been given by Soddy†, who points out that the atomic weight as ordinarily understood is not the unique quantity hitherto supposed. In the future increasing importance will be attached to the atomic number. It is the conviction of the present author that this will prove true not only in connexion with the properties of the chemical elements but also in dealing with compounds. In the latter case it is convenient to introduce the term "molecular number" to signify the sum of the positive charges carried by the atomic nuclei contained in the molecule. Thus when a molecule contains a atoms of an element A, b atoms of B, c atoms of C, so that its chemical formula is AaBiCe, the molecular number NaNa+bNo+cNe, where Na, Nb, Ne are the atomic numbers of the component elements. For * Communicated by the Author.

+ Royal Institution Lecture, 'Nature,' vol. xcix. p. 414 (1917).

example, the molecular number of water (H2O, hydrol) is for the nuclear charge of hydrogen is 1, and of oxygen

is 8.

It may be remarked that the molecular number is usually even. This arises from the fact that when the valency is odd, the atomic number is usually odd also. But in the case of an element such as copper, which may be either univalent or divalent, or in the case of some of the metals of the eighth group, the molecular number may be odd.

In former papers† it has been shown that simple relations exist between the atomic number of an element and the characteristic frequency deduced from observations of the specific heat in the solid state. In the present communication similar results are found in connexion with the molecular number of a compound and its characteristic frequency. So far as the writer is aware, this is the first attempt to establish a relationship involving molecular number, previous work in different branches of physics having been restricted to considerations of atomic number only.

§ 2. Characteristic Molecular Frequency.

At high temperatures the law as to the specific heat of compounds enunciated by Joule and verified by Kopp§ shows that, as the specific heat is then mainly additive, the heat energy arises for the most part from the vibrations of the individual atoms. At sufficiently high temperatures the vibrational energy of each atom approaches the value 3RT. At low temperatures, on the other hand, Nernst ¶ supposes that the vibrations of the molecules play a more important part than the vibrations of the atoms in the molecule. In the case of regular monatomic solids Debye has deduced an

*This fact is probably at the bottom of the remarkable numerical relations involving powers of 10, pointed out by the author in a paper read before the Physical Society of London (Proceedings, vol. xxvii. p. 425, 1915). It was shown that there must be a numerical connexion between the unit of length and the unit of mass in the C.G.S. system, "and there is no reason why it should not involve the number 10." This negative statement may now be changed to a positive one. There is a reason, in the constitution of water itself, why the number 10 should be introduced.

H. S. Allen, Proc. Roy. Soc. vol. xciv. p. 100 (1917); Phil. Mag. vol. xxxiv. p. 478, p. 488 (1917).

Joule, Phil. Mag. [3] vol. xxv. p. 334 (1844).

§ Kopp, Lieb. Ann. vol. iii. pp. 1 & 289 (1864).

Cf. Sutherland, Phil. Mag. [5] vol. xxxii. p. 550 (1891).

¶ Nernst, Vorträge über die Kinetische Theorie, p. 79 (1914). 'The Theory of the Solid State,' p. 81 (1914).

expression for the specific heat, C, which is reduced, at sufficiently low temperatures, to a simple law of proportionality between C, and T3. That this is also true for certain regular polyatomic substances has been shown experimentally by Eucken and Schwers* in the case of fluorite, CaF, and pyrites, FeS2. Thus it would appear that near the absolute zero the forces uniting the atoms in the molecule are sufficiently great, as compared with the forces uniting the molecules, to compel the individual atoms to follow the movements of the molecule of which each forms a part. At low temperatures the specific heat can be represented by Debye's formula assuming a single characteristic frequency. At higher temperatures Nernst introduces one or Einstein terms, with appropriate characteristic frequencies, to include the vibrations of the atoms in the molecule. It will be shown that the characteristic frequency, v, for the molecular movement conforms to the relation previously found to hold for the elements, viz.

Nv=ny or Nv = (n+1)va,

more

where N is now the molecular instead of the atomic number, n is an integer, and VA is a fundamental frequency having a value very near to 21 x 1012 sec.-1. The term " frequency number" is suggested to denote the numerical factor, n or n+1.

It would, of course, be possible to avoid the introduction of the fraction by introducing a fundamental frequency which is that just quoted, but as the number of cases requiring the fractional value is comparatively small, it seems better to retain for the present the larger value for v.

§ 3. Characteristic Frequency from Specific Heat.

For a small number of compounds low-temperature measurements are available, and the characteristic frequency can be deduced from the specific heat. In 1912 Nernst and Lindemann† published observations on the specific heat of rock-salt and sylvin at temperatures down to 22° K. For NaCl the characteristic frequency, v, was determined by the equation ẞv=287.3, whilst for KC1 Bv=217.6, where B=4.78×10-11. From these results we find the value of Ny for rock-salt to be 8 x 210 x 1012, whilst for sylvin it is nearly the same, 8x 20.5 x 1012. In his address on the Kinetic Theory Nernst gives different values for the Debye

* Eucken and Schwers, Ber. deutsch.phys. Gesell. vol. xv. p. 578 (1913). † Preuss. Akad. Berlin, p. 1160 (1912).