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IN view of the recent discovery of elements with atomic numbers 93–96, it is of interest to recall a calculation based on the assumption of a minimum proper length for the world-line of an electron1. The conception that it is impossible to discriminate in space-time between two positions of the electron when the interval separating them is less than h/m0c (m0 = rest mass) leads to the result that an electron in a Bohr orbit cannot have a velocity greater than c/2. A further consequence of this is that a K-ring cannot exist in an atom for which Z> hc/22pie2and this means that the Bohr-Rutherford model would stop at Z = 96. Although the original argument leading to this result cannot be regarded as free from objection in the light of modern quantum theory, the likelihood of some such limitation remains. A more recent application of the principle of minimum length and time to the Rutherford-Bohr atom sets the upper limit to Z somewhat lower than 962.

It has been the custom for a considerable period, to investigate the dosage in radiation teletherapy, in a plane passing centrally through the site of the lesion to be irradiated, or at a number of points distributed throughout the lesion. The radiation fields employed are usually approximately symmetrical about a central axis, and the dosage-rate measurements have been made chiefly in a plane passing through this axis. A diagram of lines of equal dosage-rate (“isodose curves”) provided with the source of radiation has hitherto been regarded as a quantitative description of the field generated. In many cases, since this central plane can be made to coincide with the central plane of the lesion, the dose applied to the central parts of the tumour can be deduced from such diagrams. In some cases it is not possible, on account of the situation of the lesion, to make the two central planes coincide. In such cases arising in Westminster Hospital, the dose in the central plane (or any other plane) has been foun...

Although there are some changes in method and in some places in the notation in this paper, which make it differ somewhat from the two papers I and II recently published in these ‘Proceedings,’ the same principle and object underlie it, and we may regard it as paper III of the series. It is believed that the alterations made and the manner of introducing the quantum equations make the presentation in the present form more elegant and that the theory is more satisfactory as a unitary theory. The quantum equations are made to emerge from a rather simple conception of a matrix length, of which the square may be identified as the square of the length of the theory of relativity. We begin with a discussion of the geodesics in a five-dimensional continuum and make certain changes both in the constants and in the point of view regarding the nature of the path of the electron in this space. The effect is to remove a difficulty with regard to the Kaluza-Klein cylindrical type of space.

On the view that the theory of the electron is exactly analogous to the theory of the photon, a quantum law is derived which is shown to require the existence of the first-order equation of the quantum theory. The notation required for this purpose is developed, and it is shown to have certain advantages. The limiting character of the fundamental lengths h/m0c and e2/m0c2 follows naturally, and the notation suggests that the latter should be regarded as a limit to the application of present physical theory.

The field theories of the electron and positron and also of the meson are developed by means of a close analogy with the photon. The analogy consists in the representation of the tracks of these particles by means of null-geodesics. The choice of notation is guided by the attempt to arrive at a theory in which the lengths (h/m 0 c) and (e 2 /m 0 c 2 ) occur naturally without reference to the structure of the particles, and in which the concept of quantization of electric charge is included. It is found that these objects can be attained by assuming that an additional degree of freedom is necessary for the description of the particles. If this is regarded as an additional dimension, it is found that an exact analogy can be made with the field theories familiar in the theory of relativity. An important feature is the union, in a single tensor, of energy, momentum and current density. A certain arbitrariness, not unlike that associated with the Poynting vector, is revealed, and it is shown that if this is removed by making a definite choice of the magnitude of the magnetic moment of the electron and positron, the spin angular momefttum is ^hereby fixed at the value 1/2h. In the development of the meson field the analogy shows* that the nuclear sources of the field act as if contributing a current density analogous to a magnetic current density in the electromagnetic case. The use of the additional degreb of freedom in the sinusoidal form indicates that the ratio of the constants g 1 and g 2 introduced into field theories as measures of the strengths of the sources is determined by the mass of the particle emitted in the neutron-proton transition.

This paper is concerned chiefly with the introduction of a co-ordinate operator in the quantum mechanics of the electron. It differs from other work on the same subject in the derivation, of the form of the operator by an analogy with the quantum expression for the angular momentum of the electron. Expressions for velocity and momentum operators are derived, and the relation between them is formally that of the classical theory in the absence of a field of force. By the introduction of reciprocal relations for the co-ordinate and momentum operators a condition is obtained which is applicable when the finite extension of the particle cannot be ignored. This is similar to the condition introduced by Yukawa in the theory of non-local fields. In the course of the development of the work, an explanation is offered of the significance of the fifth co-ordinate introduced in Kaluza’s theory of the gravitational and electromagnetic field and in Klein’s adaptation of it. Further, a new aspect of the question of the quantization of space and time is revealed by the use of co-ordinate operators ( X k ), for it appears that the interest lies, not in a property of space-time, but in the quantization of localization of a particle. It should, however, be emphasized that the co-ordinates ( x k ) are retained with their usual meaning, but it is suggested that they are insufficient as a basis of a mechanical description of the electron. It is by operations upon ( X k ) and not upon ( x k ) that mechanical quantities are obtained. The momentum ( u k ), like ( x k ), keeps its usual meaning, but it also is part of an operator ( M k ) of wider significance. This is analogous to the case of the angular momentum of the electron where Dirac’s introduction of ‘spin’ shows that the orbital angular momentum is a part of a complete angular momentum operator. The application has been limited to the electron, because the derivation of the form of the operator ( X k ) rests upon this analogy. The intention has not been to develop a calculus in which ordinary geometrical co-ordinates are replaced by operators. Finally, in proceeding to a force operator an additional term arises proportional to the derivative of the acceleration with respect to time. This term, which is usually ascribed to the reaction of the field of the electron, occurs here without reference to the self field, and suggests that, in addition to charge and current, a non-Maxwellian quantity must be considered as interacting with the external field.

The relativity theory of gravitation indicates that space-time is a four dimensional continuum in which the line element is measured by the equation ( ds ) 2 = g mn dx m dx n , (1) the notation being that generally adopted. The world-lines or natural tracks of free particles in this space are geodesics. From (1) we have g mn dx m /ds . dx n /ds = 1, (2) the quantity on the left being an expression corresponding to the kinetic energy of ordinary dynamics for a particle of unit mass. This correspondence is readily appreciated if it be noted that dx m /ds is the natural extension of the velocity, dx m /dt .

The works of Kaluza and of O. Klein have shown that it is possible to give to electromagnetic forces a geometrical interpretation similar to that of gravitational forces in the theory of relativity. It is necessary to assume the existence of a five-dimensional continuum defined by d σ 2 = γ μv dx μ dx v , (1) where the summation extends over μ v } = 1, 2, 3, 4, 5. The coefficients, γ μv have the values γ mn = g mn + γ 55 α 2 ϕ m ϕ n , γ 5 m = αγ 55 ϕ m , } (2) where α and γ 55 are constants, the g ’s the Riemannian coefficients and the ϕ ’s the components of the electromagnetic potential.

The vector analysis in use up to the present, as a rule, are concerned with quantities which are represented by straight lines. The straight line AB, in space of three dimensions, is represented by a vector a, and if B has Cartesian co-ordinates ( x, y, z ) with respect to A, we write a = i x + j y + k z , where i, j, k are fundamental vectors. An account will be given of a vector analysis in which the vector is represented by δa = Ʃ n i n δ x n .

An application of a principle proposed by the author and described as the principle of minimum proper time is made to obtain an equation for the ratio of the masses of the proton and electron. The result obtained is in agreement with the experimental value. The possibility of explaining the asymmetry in regard to the masses by means of a metrical consideration is discussed.

A discussion of the nature of the equations of the quantum theory, based upon the theory of relativity, having indicated that difficulties of a fundamental character will arise when dimensions less than the electron-radius are concerned, it is pointed out that the idea that such dimensions are not observable must be included in a geometrical description in physics. The electron-radius is thus the minimum length we discover in our measurements relating to the interaction of particles. But the principle of minimum proper time applied to the track of a heavy particle M0 shows that the particle is in effect a time scale whose finest division is h/M0c2. In the theory of relativity the conception of a minimum length e2/m0c2 becomes equivalent to the conception of a minimum interval of proper time e2/m0c3. The argument shows that these two intervals are of the same order of magnitude and leads to a simple relation between the mass ratio of the proton and electron and the fine-structure constant.

In the early presentation of the wave theory of matter de Broglie considered that every material particle is the seat of a "periodic phenomenon” of a frequency dependent upon the mass of the particle. A consideration of how this periodicity would manifest itself to observers, with respect to whom the particle is in motion, led to the development of some of the basic principles of the wave theory and, in particular, the expression of the mechanical quantities of energy and of momentum in terms of the theory led to the equationv0=m0c2/h. (1)v0denotes the frequency of the periodic phenomenon as it would be measured by an observer at rest with respect to the particle andm0is the rest mass. We have shown in theseProceedings(Flint 1928, pp. 636, 638) that the interval of time τ0= 1/v0can be regarded as representing the smallest interval of time which has any significance in the physical world in relation to a material particle. This result was first developed from the Wilson-Sommerfeld quantum conditions and later from Planck’s principle of the quantum of action.

The vector analyses in use up to the present, as a rule, are concerned with quantities which are represented by straight lines, and the space to which they are applicable is Euclidean in its properties. The straight line, AB, in space of three dimensions, is represented by a vector a, and if B has Cartesian coordinates ( x, y, z ) with respect to A, we write: a = i x + j y + k z , where i, j, k, are fundamental vectors. An account will be given of a vector analysis in which a vector is represented by δa' = Σ n i n δx n . The vector is of infinitesimal length and represents a component measured in any system of co-ordinates.

An account is given of the various physical factors of the new four-gramme radium unit at Westminster Hospital Annexe. The radiation fields available for treatment are described, and it is shown that the protection afforded is adequate, and better than that in the case of the previous two-gramme unit.

In two papers published in these 'Proceedings' the suggestion was made that the wave equation is a metrical equation and that the quantity ψ is characteristic of the metrics of physical space. In developing this view a definition of equality of vectors at different points was given and ψ was interpreted as the natural scale factor. Except for the definition of equality the space was Riemannian. The system was suggested by that of Weyl and Eddington and was, in fact, almost identical with it.

The results are given of a series of water phantom measurements, by Sievert's condenser chamber method, on the intensity distributions of the 2 gm. radium bomb in use at the Westminster Hospital Annexe, London. The best thickness of secondary copper filter is discussed and results are given that show “back-scatter” at the surface to be negligible for this particular unit. The results for the measurement of doses for various radium-skin distances are expressed graphically. The expression of therapeutic doses in r units is discussed, and the skin erythema dose in r, thus found, is compared with values given by other workers.

The first-order equation of the quantum theory is developed by applying a parallel displacement to the matrix length of a vector. The quantum theory reveals itself as a law of parallel displacement from which a modification of Weyl's theory is deduced. In this modification a term containing the mass of a charged particle occurs as well as terms containing the charge and electromagnetic potential. The principle of minimum proper time emerges from this result as a limitation of the application of the quantum equation to a particle in an electromagnetic field.

The problem considered is an example of the propagation of electro-magnetic waves between perfectly conducting parallel plates, where it is possible to see how the conductivity of the dielectric affects the propagation. In addition, the simplicity of the functions involved in the solution of the problem enables its physical aspect to be readily appreciated.

The contents of this paper were suggested by a discussion of the equation:in a paper by Glaisher which appears in the Philosophical Transactions, 1881, Part III.The solutions in series of (1) are:and in the paper referred to it is shewn that the coefficients of hp+1 in the expansions of and of satisfy equation (1) when p is a positive integer.

The wave theory of mechanics substitutes a group of waves extending over a small range of frequencies for the usual conception of the electron, thus leading to the view that the electron is a wave disturbance which travels as a compact group through space, but which does not spread into an increasingly large region like waves from a source of light or sound. An example of this has been considered in detail by Schroedinger (‘Die Naturwiss.’ vol. 28, p. 664), and the group obtained by adding together the phase waves for the linear Planck oscillator.

A generalization of the uncertainty relationships for the case in which an electromagnetic field is present is developed. It is further shown that the atomicity of electric charge is a consequence of the integral relationships which are a feature of the quantum theory.

It is shown how it is possible, by means of geometry and the introduction of a principle of measurement, founded by analogy with the theory of H. Weyl, to discover a unity existing between gravitational, electromagnetic and quantum phenomena. Dirac’s equation and an extension of it are derived from the principle of measurement, and an essential feature of the theory is that it incorporates a theory of the electron in which its mass appears as a geometrical quantity, entering into the equation as a consequence of the existence of a fundamental unit of length. The observed mass is given by this geometrical mass together with interaction terms which also enter naturally into the theory.

§1. In a recent paper by one of us the conclusion was reached that the proper time of a particle had a minimum value h / mc 2 where m is the rest mass. This conclusion was based on a simple assumption about the metrics of space which was found to be in agreement with Schroedinger’s wave equation. The following considerations show that the same conclusion can be reached by a method which is independent of the system of metrics referred to. The expression for the momentum of a particle of rest mass m in relativity mechanics is p α = mg αβ dx 8 / ds , where s denotes an arc of the world line. if p α (= m dx α / ds ) denote the contravariant vector corresponding to p α we have p α p α = m 2 .

Diese Mitteilung ist eine Bemerkung uber die Abhandlung von W.Scherrer unter demselben Titel, die vor kurzem in dieser Zeitschrift erschienen ist. Die Mitteilung zeigt die Verwandtschaft zwischen seinen Vorschlagen und denen von anderen Autoren, insbesondere vonEinstein in seiner Theorie von Fernparallelismus.

A comparison is made between the treatment of hollow wave guides and conducting guides. It is shown how certain problems in the case of the former can be solved by the familiar methods applied to the latter. In conclusion, suggestions are made for the experimental determination of impedances in hollow guides, particularly for the case of the determination of the impedance of an open-ended guide.

A theory of charged particles is developed by an extension ofEinstein’s concept of local axes. A new four vector is added to the geometrical system in addition to the electromagnetic potential vector. The particle is then identified with the geometry and the discussion leads to a relativistic theory of the electron, in which the difficulty concerning the self-energy is avoided.

In Einstein’s theory of gravitation it is assumed that the geometry of space- time is characterised by the following equation for the measurement of displacement:— ds 2 = g mn dx m dx n { m n = 1, 2, 3, 4, the sign of summation being omitted for convenience. It is supposed that the coefficients, of which g mn is the type, are dependent upon the content of space, and the relation existing between them is the law of gravitation.

Mittels einer Geometrie, in der das Linienelement durch eine Matrix dargestellt wird, und der Annahme einer Methode ahnlich der vonWeyl wird gezeigt, das Gleichungen von der Gestalt derDiracschen Gleichung sich als Eichgleichungen auffassen lassen konnen. Ihre Rolle in der Theorie ist vergleichbar mit dem Gravitationsgesetz in der Relativitatstheorie. Diese Theorie wird mit der Feldtheorie vonKaluza zusammengefast. Dabei ergibt sich eine Beziehung zwischen den

The relativity theory of Weyl and Eddington is an extension of what may be described as the geometric principle of relativity to the metrics of space. This extension permits the inclusion of electromagnetic phenomena into a system of geometry and metrics so that we have a consistent and fundamental scheme for the description of gravitational and electromagnetic phenomena. Recently considerable light has been thrown on the relation between classical and quantum theory by the work of de Broglie, Schroedinger and others.

The attempt to unite the first order equations of the quantum theory with the theory of relativity has led to a number of suggestions by various writers, most of whom consider operations in a continuum with four co-ordinates ( x, y, z, t ). Some add a fifth co-ordinate in order to complete the description of a point. For example, Fock and Iwanenko add ζ, a quantity capable of possessing the integral values (1, 2, 3, 4). Einstein in his latest attempt at a unitary theory retains the four dimensional continuum as the fundamental background, but introduces vectors with five components which are related to four dimensional vectors by means of a set of quantities γ, with appropriate affixes. This is a position half way between the four and five dimensional continuum and, in making use of this system, Solomon has found it necessary to introduce a fifth co-ordinate in order to include quantum phenomena into the scheme.

William Wilson was born on 1 March 1875 at Goody Hills, near the village of Mawbray on the Solway shore in Cumberland, where his father was a farmer. His ancestors on the paternal and maternal sides of the family appear to have been farmers for some generations. His education began in the village school of Holme St Cuthbert where, as he has related, he had a master of outstanding ability, named John Routledge. He has recorded that the books he read at this time directed his interest towards scientific study, and that a book by Nesbit owned by his father on mensuration and land surveying gave him his first taste for geometry. It was, however, evidently his intention to take up a career in agriculture, for after gaining the Longcake scholarship of £40 a year for three years when he was not quite fourteen years of age, he went to the Agricultural College at Aspatria as a weekly boarder. In the college he had his first experience of formal scientific teaching and study, especially in chemistry and surveying together with biology, geology and zoology, and became interested in the application of scientific method and knowledge in agriculture.

CERTAIN considerations based on the principles of quantum mechanics lead to the view that the process of localization of a particle in space and time is subject to quantization. Thus if the world line of a particle of rest mass m0 is considered, it has no physical meaning to subdivide it into elements of length less than h/m0c, or, what is the same thing, to associate with the particle intervals of proper time less than h/m0c2 1. This has been expressed as a principle of elementary indeterminacy by Ruark2 and by Furth3.

ONE of the difficulties of modern quantum theory lies in the prediction, in certain circumstances, of high probabilities of transition to negative energy states. This difficulty comes very strikingly into evidence, as O. Klein first pointed out1, in the example of reflection and refraction of electrons at a surface where the electric potential is discontinuous or very rapidly varying.

IT has been suggested that co-ordinates should be represented in the quantum theory by operators analogous to those which represent orbital angular momentum1.

IN attempts to understand the relation between the quantum theory and the theories of relativity and electro-magnetism, a certain limitation, applicable to the motion of a particle of electromagnetic mass m0and of charge, is brought to light. It appears that the expression (m0c2d e/cmdxm) can assume only those values which are multiples of Planck's constant1. d is an element of proper time associated with the track of the particle, dxm is a component of displacement andm is a component of electromagnetic potential. An interesting result is obtained if this be applied to an electron in an electrostatic field of potential Ne/r. This is the case with an electron in an atomic orbit where N is the number of the atom. The above condition then implies (m0c22 Ne2/r)dt ® h where = 12;/c, ½ being the velocity of the particle, and where account is taken of the fact that the nuclear and electronic charges are of opposite sign. If we consider the case of the K-level of an atom and make use of Sommerfeld's value for Ne2/r, which may be regarded as a sufficient approximation, we obtain m0c2 (2 2/=2) dt ® h. In this case = NI±, where is the fine-structure constant 2e2/hc. We note that as approaches the value 1/2 the factor of dt approaches zero very rapidly, and the limitation states that the least possible value of dt is very large. We interpret this as an indication of the breakdown of the description of the charge as a particle in motion. This gives a clue to the nature of the limitation. It provides us at each point of space and time with a criterion for the dynamical description of an electric charge.

THERE are many elementary treatises on wave mechanics. Their multiplicity is due perhaps to the fact that there are many ways of approach to this subject or perhaps because, as in the case of a disease for which many cures are advertised, the right treatment still awaits discovery. Elementary Wave Mechanics Introductory Course of Lectures. By W. Heitler. Notes taken and prepared by W. S. E. Hickson. (Hectographed.) Pp. ii + 88. (Dublin: Dublin Institute for Advanced Studies, 1943.) 5s.

ONE of the consequences of the introduction of a geometry of four dimensions as a basis of physical theory has been to reveal relations between certain quantities which had previously appeared to differ fundamentally in character, the simplest being there relation between space and time. In geometries which introduce new parameters or new dimensions, a similar unexpected unification may appear. This is particularly the case in the theory based upon the use of projective geometry associated with the names of Veblen and Hoffmann, and with the five-dimensional theory introduced by Kaluza. It is necessary to express any results obtained in this way in a form which satisfies the requirements of relativistic invariance; at the development of both these theories has proceeded in such a way as to make this possible.

IN the attempt to discover a place for Dirac's equation in the geometry and metric of the physical world, it has been proposed to make use of Weyl's concept of gauging. It was thus proposed to bring ubout the union of gravitation and electromagnetism by Kaluza's suggestion of the adoption of a five-dimensional continuum leaving the idea of gauging free to bring about the union of the quantum theory with these branches of physics.

THE familiar application of this principle is to a particle, such as an electron, and from it we learn that errors, q in the positional co-ordinate and p in the momentum, must occur such that: "q “p h.