The Quantum Bicycle Society is pleased to share open access to the following selection of articles. *Below you will find:*

**1. The Quantum Bicycle Journal**: papers published exclusively by us.

**2. Papers Published Elsewhere** by Quicycle members and contributors.

**3. Classic Papers** we felt it worthwhile to include.

CLICK HERE for a full list of journal articles.

## 1. The uantum Bicycle Journal

**AUTHORS LISTED:**QJ0027: Vivian N. E. ROBINSON (2023)

QJ0026: Arnie BENN, John G. WILLIAMSON (2022)

QJ0025: John G. WILLIAMSON, Arnie BENN, Michael MERCURY (2022)

QJ0022: Vivian N. E. ROBINSON (2020)

QJ0020: Garnet N. ORD (2020)

QJ0019: Martin B. VAN DER MARK, John G. WILLIAMSON (2016)

## Alternative Path to Solve Einstein’s Field Equations

Vivian N.E. ROBINSON*Quicycle Journal (2023)QJ002*7

**ABSTRACT**:

A solution to Einstein’s gravity field equations is derived directly from his 1916 paper, although it utilizes the conversion from Cartesian to polar co-ordinates, citing Schwarzschild’s paper for its derivation.

* *

Keywords:*Einstein, gravity, relativity, field equations, Schwarzschild*

LINK *to this abstract*

## The Photonic Topology Of Quantum Spin

Arnie BENN, John G. WILLIAMSON

Quicycle Journal (2022)

QJ0026

**ABSTRACT**:

This paper investigates the nature of quantum spin, an internal quantum property of the electron and other subatomic particles and antiparticles. The development of sub-quantum mechanics opened the door to a deeper understanding of the photonic substructure of subatomic particles, and in particular, allowing for the first time the *a priori* calculation of the electron’s charge and anomalous magnetic moment. This represents a significant advance in relativistic quantum mechanics and in our understanding of the substructure of subatomic particles. As a consequence, this theory allows us to investigate the specific components of the electron’s quantum spin, as well as the topological difference between its ‘spin-up’ and ‘spin-down’ states. (As will be clarified, spin up versus down is conceptually different to parallel versus antiparallel because they refer to different spin components.) It is proposed that the electron’s spin is composed of three components, here named intrinsic spin, toroidal spin, and tumble. It is further proposed that the nature of the spin-up or spin-down state is a direct consequence of the direction of polarisation of the circularly-polarised photon constituting the electron (or other particle). Consequently, when two equal-energy photons of the appropriate energy interact to generate a particle/antiparticle pair, one photon polarisation will yield a spin-up electron and a spin-down positron, and the other will yield a spin-down electron and spin-up positron. The nature of quantum spin can then be leveraged to gain insight into the physical reason behind the formation and stability of the *di-electron* boson (electron pair) as a counter-rotating, interwoven, diamagnetic photonic state, a consequence of two superimposed and opposite electron spin configurations, as well as the physical (energy-related) reason behind exclusion.

Keywords:*Quantum spin, electron, positron, circularly polarised photon, intrinsic spin*

LINK *to this abstract*

## Quantum Spin In 4 Derived 3-Spaces

John G. WILLIAMSON, Arnie BENN, Michael MERCURY*Quicycle Journal (2022)QJ0025*

** ABSTRACT**:

A new chemical theory is proposed that integrates sub-quantum mechanics — a development of relativistic quantum mechanics and electromagnetism — in a way that allows new insights into the modeling of electrons in atomic and molecular orbitals. Central to this approach is the mathematical treatment of each aspect of a system’s energy flow in its appropriate space-time components, as well as the fact that the internal wave nature of energy and matter results in quantum systems with a coherent, resonant, harmonic nature. It is this wave nature that lies at the root of quantisation itself. This paper presents an approach to the modeling of a quantum system in a more complete way. It is proposed that derivatives in space-time lead to four, 3-dimensional ‘dynamical spaces.’ This proves significant because these derivative spaces describe physical observables such as spin, electric field, and magnetic field. These have different properties, and this approach allows each to be considered either separately or together. The four 3-component derivative spaces are: energy (or frequency) space, electric field space, magnetic field space, and spin (or angular momentum) space. At the root of each of these are the properties of the

*one*underlying 4-dimensional space-time. In particular, emerging from this new theory is a new way to model quantum spin and its contributions to energy, harmonic stability, and quantisation. This approach may also account for several phenomena that relate to elemental magnetic properties and solid state crystalline properties.

Keywords:*quantum spin, harmonic resonance, field cancellation, relativistic quantum mechanics, quantum chemistry*

LINK *to this abstract*

## Moving** Clocks And Special Relativity Rest Reference Frames**

Vivian N. E. ROBINSON*Quicycle Journal (2020)QJ0022*

** ABSTRACT**:

Einstein’s special relativity theory predicts that time will go slower on moving clocks than stationary clocks. Without an absolute rest reference frame, how do we know which clock is moving and which is stationary? Although there is no absolute rest reference frame there are some practical rest reference frames. When the appropriate rest reference frame is chosen, clocks moving with respect to that rest reference pass time slower than those at rest in that reference frame.

*Below, the second version was released following a video peer review session, viewable HERE .*

Keywords:*Relativity, Special Relativity, Special Relativity corrections, General Relativity, inertial reference frame*

LINK *to this abstract*

**Perspective, Images, And Quantum Mechanics As A Relativistic Effect**

Garnet N. ORD*Quicycle Journal (2020)QJ0020*

**ABSTRACT**:

Much of human cognition is devoted to image analysis. From infancy we learn to process the two dimensional images we receive on our retinas, decoding what we believe to be a representation of a world of objects located in a three dimensional space, viewed from a particular perspective. The consistency of that decoding is very strong: ‘seeing is believing.’ However, it is not infallible, and visual clues can give rise to mistaken perspectives. There is an analog of mistaken perspective in a ‘received view’ on the relation between quantum mechanics and special relativity. We explore a counter-argument, that quantum mechanics is in fact a relativistic effect, primarily through the use of images.

Keywords:*special relativity, light cone, quantum mechanics, perspective*, *Twins Paradox*

LINK *to this abstract*

**Inversion And General Invariance In Spacetime**

Martin B. VAN DER MARK, John G. WILLIAMSON*Quicycle Journal (2016)QJ0019*

** ABSTRACT**:

A general formula for division in a relativistic Clifford-Dirac algebra is derived. Where division is undefined turns out, in many cases, to correspond to areas of physical interest, such as the light cone, invariant quantities in electromagnetism, and the basis set of quantities in the Dirac equation. Apart from such areas, where there has already been significant development in science, new sets of inter-related quantities, involving the spin and the total energy for example, are suggested as possibilities for further investigation and development.

Keywords:*non-division algebra, invariants, inverses, scaling, Dirac algebra, Clifford algebra, geometric algebra, special relativity, generalized covariance, rest mass, light cone, diamond conjugate*

LINK *to this abstract*

## 2. Papers Published Elsewhere

** AUTHORS LISTED:**QJ0024: Martin B. VAN DER MARK, John G. WILLIAMSON (2021)

QJ0023: Vivian N. E. ROBINSON (2021)

QJ0018: John G. WILLIAMSON (2019)

QJ0017: Martin B. VAN DER MARK (2019)

QJ0016: Martin B. VAN DER MARK (2015)

QJ0015: John G. WILLIAMSON (2015)

QJ0014: John G. WILLIAMSON, S. J. LEARY (2015)

QJ0013: W. F. HAGEN (2015)

QJ0012: Martin B. VAN DER MARK (2015)

QJ0011: John G. WILLIAMSON (2014)

QJ0010: John G. WILLIAMSON (2012)

QJ0009: P. H. BUTLER, N. G. GRESNIGT, Martin B. VAN DER MARK, P. F. RENAUD (2012)

QJ0008: Martin B. VAN DER MARK, G. W. ‘t HOOFT (2000)

QJ0007: John G. WILLIAMSON, Martin B. VAN DER MARK (1997)

QJ0006: John G. WILLIAMSON, C. E. TIMMERING (1990)

**Relativistic Inversion, Invariance And Inter-Action**

Martin B. VAN DER MARK, John G. WILLIAMSON *Symmetry (2021)QJ0024*

** ABSTRACT:**A general formula for inversion in a relativistic Clifford-Dirac algebra has been derived. Identifying the base elements of the algebra as those of space and time, the first order differential equations over all quantities proves to encompass the Maxwell equations, and leads to a natural extension incorporating rest mass and spin. Despite the fact that this algebra is not a division algebra it seems to parallel reality well: where division is undefined turns out to correspond to physical limits, such as that of the light cone. The divisor corresponds to invariants of dynamical significance, such as the invariant interval, the general invariant quantities in electromagnetism, and the basis set of quantities in the Dirac equation. The study suggests other Lorentz invariants that may prove of interest, including one relating the spin and total energy. It is speculated that the apparent 3-dimensionality of nature arises from a beautiful symmetry between the three-vector algebra and each of four sets of three derived spaces in the full 4-dimensional algebra. It is conjectured that elements of inversion may play a role in the interaction of fields and matter.

**Keywords**: invariants ; inversion; division; non-division algebra ; Dirac algebra ; Clifford algebra ; geometric algebra ; special relativity, photon interaction

LINK *to this abstract*

**Physical Explanation Of Einstein’s Gravity**

Vivian N E. ROBINSON *J. Phys. Commun. (2021)QJ0023*

** ABSTRACT:**Einstein’s gravitational field equations from his general theory of relativity have formed the foundations of gravitational studies since their publication. His work is widely acknowledged as an example of a theoretical study that made a great contribution to our understanding of gravity. Einstein’s mathematical approach has made the topic complex and open to misinterpretation. This study evaluates the physics upon which his mathematics operates. It shows that mass distorts space–time by the redshift of photons. It derives two alternative metrics to the Schwarzschild metric. One was derived directly from Einstein’s early work on gravity. The other was derived from Einstein’s field equations by removing the approximations introduced in the Schwarzschild metric derivation. Both match observation better than the Schwarzschild metric and show why Einstein did not believe in black holes. The metric derived directly from Einstein’s early gravitational study, predicts the torus shape shown in the Event Horizon Telescope collaboration image. In showing the physics involved, this study suggests it is easier to understand the complexities of his work.

LINK *to this abstract*

## A New **Linear Theory Of Light And Matter**

John G. WILLIAMSON *J. Phys. (2019)QJ0018*

** ABSTRACT:**A new theory is proposed with similar form and mechanics to the Dirac equation, but where the rest-mass is introduced in a more fundamental way. The new theory proves to encompass and extend classical electromagnetism, and reduce to it exactly in the field only case. A sharpening of the principle of relativity allows fully relativistic field-only solutions, which take on some of the properties more usually associated with quantised states. The extended theory has four new solutions which exhibit a non-zero electric field divergence and a double-covering symmetry. These are necessarily charged, essentially fermionic states, two with positive and two with negative charge, differing by an inner spin symmetry. The new configurations are identified with the spin “up” and spin “down” electron and positron.

LINK *to this abstract*

## Quantum Particle, Light Clock Or Heavy Beat Box?

Martin B. VAN DER MARK *J. Phys. (2019)QJ0017*

**ABSTRACT**:

A standing wave of light inside a cavity, while observed from a moving frame, is seen to have a spatial beat of precisely the same nature as that of the de Broglie wave of a quantum particle. The structural forces that hold together the cavity against the internal radiation pressure are compared to the so-called Poincaré stresses inside an elementary particle, as are required for the particle’s stability. This seems to work best if the cavity is non-rigid and hence is itself oscillating, coupled to the radiation inside.

LINK *to this abstract*

## On The Nature Of “Stuff” And The Hierarchy Of Forces

Martin B. VAN DER MARK *SPIE Optics & Photonics (2015)QJ0016*

**ABSTRACT**

From super clusters of galaxies down to the quarks in the proton, at all length scales the structure of matter is the result of a balance of forces. In this paper it is argued that with decreasing size there must necessarily be an increase of the fraction of kinetic and binding energy with respect to the total energy. Smaller sizes require stronger forces which represent more of the energy available. The smallest possible size of granularity is found to be where the internal kinetic energy and total energy become comparable, which occurs at the size of the proton. We infer that the proton is the smallest stable particle, being a light speed circulation of energy.

Keywords: primordial stuff, hierarchy of forces, rest mass, energy balance, proton structure, light speed knot of energy.

LINK *to this abstract*

## On The Nature Of The Photon And The Electron

John G. WILLIAMSON *SPIE Optics & Photonics* *(2015)QJ0015*

** ABSTRACT:**A new theory, describing both light and material particles, is proposed. The experimentally-observed nature of space and time are brought into the theory at the most fundamental level. An equation encompassing the usual free-space Maxwell equations but similar in form to the Dirac equation is proposed. This equation has new kinds of solutions. Propagating, pure-field solutions may have any energy, but the energy transferred must be proportional to the frequency. These are identified with the physical photon. Solutions with a rest-mass term allow any incoming propagating field to merge into re-circulating vortex-like solutions. The minimum energy configuration “rectifies” the oscillating electric field of light into a uni-directional, radial (inward or outward directed) configuration. The resulting apparent external charge may be readily estimated and is found to be of the order of the elementary charge. The spin may, likewise, be calculated, and is found to be half integral, exhibiting a double-covering internal symmetry. Charge is then not a fundamental quantity in the theory – but is a result of the way field folds from a rest-massless bosonic to a rest-massive fermionic configuration. The simplest such charged, fermionic particles are identified with the electron and positron.

Keywords: Electromagnetic electron positron photon.

LINK *to this abstract*

## Absolute Relativity In Classical Electromagnetism: The Quantisatoin Of Light

John G. WILLIAMSON, S. J. LEARY*SPIE Optics & Photonics* *(2015)*

QJ0014

** ABSTRACT:**A rigorous introduction of the underlying nature of space and time forces qualitatively new kinds of solutions in the classical theory of electromagnetism. A class of relativistic wave-functions are derived which are solutions to the first-order, free-space Maxwell equations. These describe all photons from radio to gamma waves and are governed by a single parameter: the exchange frequency. Though the theory remains that of classical, continuous electromagnetism, allowed traveling-wave solutions are quantised in that they come in “lumps” and their characteristic energy is proportional to frequency.

Keywords: light quantisation.

LINK *to this abstract*

## Matter In The Form Of Toroidal Electromagnetic Vortices

W. F. HAGEN*SPIE Optical Eng. & Appl*. *(2015)*

QJ0013

** ABSTRACT:**The creation of charged elementary particles from neutral photons is explained as a conversion process of electromagnetic (EM) energy from linear to circular motion at the speed of light into two localized, toroidal shaped vortices of trapped EM energy that resist change of motion, perceptible as particles with inertia and hence mass. The photon can be represented as a superposition of left and right circular polarized transverse electric fields of opposite polarity originating from a common zero potential axis, the optical axis of the photon. If these components are separated by interaction with a strong field (nucleon) they would curl up into two electromagnetic vortices (EMV) due to longitudinal magnetic field components forming toroids. These vortices are perceptible as opposite charged elementary particles. These spinning toroids generate extended oscillating fields that interact with stationary field oscillations. The velocity-dependent frequency differences cause beat signals equivalent to matter waves, leading to interference. The extended fields entangled with every particle explain wave particle duality issues. Spin and magnetic moment are the natural outcome of these gyrating particles. As the energy and hence mass of the electron increases with acceleration so does its size shrink proportional to its reduced wavelength. The artificial weak and strong nuclear forces can be easily explained as different manifestations of the intermediate EM forces. The unstable neutron consists of a proton surrounded by a contracted and captured electron. The associated radial EM forces represent the weak nuclear force. The deuteron consists of two axially separated protons held together by a centrally captured electron. The axial EM forces represent the strong nuclear force, providing stability for “neutrons” only within nucleons.The same principles were applied to determine the geometries of force-balanced nuclei. The alpha-particle emerges as a very compact symmetric cuboid that provides a unique building block to assemble the isotopic chart. Exotic neutron-4 appears viable which may explain dark matter. The recognition that all heavy particles, including the protons, are related to electrons via muons and pions explains the identity of all charges to within 10

^{–36}. Greater deviations would overpower gravitation. Gravitation can be traced to EM vacuum fluctuations generated by standing EM waves between interacting particles. On that basis, gravity can be correlated via microscopic quantities to the age of the universe of 13.5 billion years. All forces and particles and potentially dark matter and dark energy are different manifestations of EM energy.

Keywords: photon, pair-creation, electron, waves, electromagnetic vortices, matter, elementary particles, neutron, deuteron, gravitation.

LINK *to this abstract*

## Quantum Mechanical Probability Current As Electromagnetic 4-Current From Topological EM Fields

Martin B. VAN DER MARK *SPIE Proceedings (2015)QJ0012*

**ABSTRACT**:

Starting from a complex 4-potential A=αdβ we show that the 4-current density in electromagnetism and the probability current density in relativistic quantum mechanics are of identical form. With the Dirac-Clifford algebra Cl_{1,3} as mathematical basis, the given 4-potential allows topological solutions of the fields, quite similar to Bateman’s construction, but with a double field solution that was overlooked previously. A more general null-vector condition is found and wave-functions of charged and neutral particles appear as topological configurations of the electromagnetic fields.

Keywords: topological fields, electromagnetism, Clifford algebra, Dirac equation, 4-current, knots.

LINK *to this abstract*

## A New Theory of Light And Matter

John G. WILLIAMSON*Journal of Physics: Conference Series (2014)QJ0011*

** ABSTRACT:**Within a new mathematical framework, an extension to the free-space Maxwell equations is pro-posed with four further linear coupled differential equations. Within the formalism, a sharpening of the principle of relativity leads to new solutions of the Maxwell equations corresponding to uncharged rest-massless propagating states identified with the photon. The simplest extension, including a rest-mass density, allows for a self-confinement of the resulting system. The resulting new solutions are necessarily charged and essentially fermionic. These are identified with the electron and positron.

LINK *to this abstract*

## Fermions From Bosons And The Origin Of The Exclusion Principle

John G. WILLIAMSON *Mendel (2012)QJ0010*

** ABSTRACT:**Based on a model of the electron as a localised photon, a possible physical basis for the Pauli exclusion principle is proposed. It arises from the harmonic interference of the underlying fields. One possible theory leading to the self-confinement of electromagnetism is explored: an extension of the Maxwell equations.

Keywords: Electron, electromagnetism, exclusion principle, electromagnetic vortex.

LINK *to this abstract*

## A Fields Only Version Of Lorentz Force Law: Particles Replaced By Their Fields

P. H. BUTLER, N. G. GRESNIGT, Martin B. VAN DER MARK, P. F. RENAUD*arXiv (2012)QJ0009*

**ABSTRACT**:

We show that the Lorentz force law, F^{L}_{1}=q_{1}(E+v_{1}×B) being the charge on particle 1interacting with the electromagnetic fields due to all other particles, can be written in a pure field form F^{L}_{1}=−∇_{1}U^{EM}. In this expression U^{EM} is the total electromagnetic energy of the system of particle1 and all other particles. In deriving this result we review the old but not widely known results thatMaxwell’s equations follow uniquely from Special Relativity, and that the Lorentz force law follows from applying Hamilton’s variational principle to this result.For a two particle system, the standard view is that the electromagnetic force on particle 1 is the result of the charge of particle 1 interacting locally with the field of particle 2, and conversely. Both charges 1 and 2, and fields 1 and 2 are needed. In our approach, the fields of all particles contribute to the electromagnetic interaction everywhere, over all of space. The charges of the particles do not enter the theory except incidentally, via Gauss’s law. This has novel interpretational consequences. In particular, it allows a charged particle to be replaced by its electric and magnetic fields, much as a particle in quantum mechanics is replaced by its complex valued wave function.

Keywords: Lorentz force, Maxwell’s equations, Coulomb force. Hamilton’s principle, energy density.

LINK *to this abstract*

## Light Is Heavy

Martin B. VAN DER MARK, G. W. ‘t HOOFT*arXiv (2000)*

QJ0008

**ABSTRACT**:

Einstein’s relativity theory appears to be very accurate, but at times equally puzzling.On the one hand, electromagnetic radiation must have zero rest mass in order to propagate at the speed of light, but on the other hand, since it definitely carries momentum and energy, it has non-zero inertial mass. Hence, by the principle of equivalence, it must have non-zero gravitational mass, and so, light must be heavy.In this paper, no new results will be derived, but a possibly surprising perspective on the above paradox is given.

LINK *to this abstract*

## Is The Electron A Photon With Toroidal Topology?

John G. WILLIAMSON, Martin B. VAN DER MARK*Annales de la Fondation Louis de Broglie* *(1997)QJ0007*

** ABSTRACT:**We study the properties of a simple semi-classical model of a photon confined in periodic boundary conditions of one wavelength. the topology of this object, together with the photon electric field, give rise to a charge of the order of 10

^{-19}Coulombs and a half-integral spin, independent of its size. The ratio of the electromagnetic energy inside and outside the object leads to an anomalous spin

*g*factor which is close to that of the electron. Although a finite size of order 10

^{-12}meter arises in a natural way, the apparent size of the object will be much smaller iin energetic scattering events.

LINK *to this abstract*

## Quantum Point Contact As A Local Probe Of The Electrostatic Potential Contours

John G. WILLIAMSON, C. E. TIMMERING*Phys. Rev. (1990)QJ0006*

** ABSTRACT:**Quantum point contacts, shifted laterally by +-50nm have been used to study the local electrostatic potential in the two-dimensional electron gas of a GaAs-Al

_{x}Ga

_{1-x}As heterostructure. The resistance of the quantised plateaux is observed to vary with this lateral position. In one device, an alternate suppression of the quantised resistance plateaux is observed, which we interpret as being due to an overlap of strong local potential fluctuations with the transverse profile of the wave function in the quantum point contact.

LINK *to this abstract*

## 3. Classic Papers

**AUTHORS LISTED**:

QJ0005: Louis DE BROGLIE (1924)

QJ0004: Albert EINSTEIN (1920)

QJ0003: William K. CLIFFORD (1878)

## Recherches Sur La Théorie Des Quanta

Louis DE BROGLIE *(1924)QJ0005*

**FROM INTRODUCTION**:

(*Translation*) History shows that there long has been dispute over two viewpoints on the nature of light: corpuscular and undulatory; perhaps however, these two are less at odds with each other than heretofore thought, which is a development that quantum theory is beginning to support. Based on an understanding of the relationship between frequency and energy, we proceed in this work from the assumption of existence of a certain periodic phenomenon of a yet to be determined character, which is to be attributed to each and every isolated energy parcel, and from the PLANCK-EINSTEIN notion of proper mass, to a new theory. In addition, Relativity Theory requires that uniform motion of a material particle be associated with propagation of a certain wave for which the phase velocity is greater than that of light (Chapter 1). For the purpose of generalising this result to nonuniform motion, we posit a proportionality between the momentum world vector of a particle and a propagation vector of a wave, for which the fourth component is its frequency. Application of FERMAT’s Principle for this wave then is identical to the principle of least action applied to a material particle. Rays of this wave are identical to trajectories of a particle (CHAPTER 2).

The application of these ideas to the periodic motion of an electron in a BOHR atom leads then, to the stability conditions of a BOHR orbit being identical to the resonance condition of the associated wave (Chapter 3). This can then be applied to mutually interaction electrons and protons of the hydrogen atoms (CHAPTER 4). The further application of these general ideas to EINSTEIN’s notion of light quanta leads to several very interesting conclusions. In spite of remaining difficulties, there is good reason to hope that this approach can lead further to a quantum and undulatory theory of Optics that can be the basis for a statistical understanding of a relationship between light-quanta waves and MAXWELL’s formulation of Electrodynamics (CHAPTER 5). In particular, study of scattering of X and γ-rays by amorphous materials, reveals just how advantageous such a reformulation of electrodynamics would be (Chapter 6). Finally, we see how introduction of phase waves into Statistical Mechanics justifies the concept of existence of light quanta in the theory of gases and establishes, given the laws of black body radiation, how energy parcellation between atoms of a gas and light quanta follows.

LINK *to this abstract*

## Relativity: The Special And General Theory

Albert EINSTEIN*(1920)QJ0004*

**FROM PREFACE**:

The present book is intended, as far as possible, to give an exact insight into the theory of Relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. The work presumes a standard of education corresponding to that of a university matriculation examination, and, despite the shortness of the book, a fair amount of patience and force of will on the part of the reader. The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they originated.

LINK *to this abstract*

## Applications Of Grassmann’s Extensive Algebra

William K. CLIFFORD

American Journal of Mathematics *(1878)QJ0003*

**FROM INTRODUCTION**:

The present communication endeavors to determine the place of Quaternions and of what I have elsewhere called Biquaternions in the more extended system, thereby explaining the laws of those algebras in terms of simpler laws. It contains, next, a generalization of them, applicable to any number of dimensions; and a demonstration that the algebra thus obtained is always a compound of quaternion algebras which do not interfere with one another.

LINK *to this abstract*