In the process of teaching, repetition is an acceptable procedure. In my own classes, I started out by explaining what I was going to tell them. After this preliminary, I told them. When this was done, I closed by telling them what I had told them. Invariably this was the time when the guy in the back of the room popped up with the question: "Would you mind repeating that?"

In scientific terminology, orders of magnitude are given in powers of ten. As a matter of comparison, if a quantity is one order of magnitude greater than another, it is ten times the other.

In terms of the Newtonian gravitational force, it is much too weak to be the determining factor in the stability of atomic nuclei. In the case of electrons and positrons, the discrepancy is much more extreme.

Since the electron is a mass entity with an internal motion of circulation, a centrifugal force tending to disrupt the particle is present. Then there must exist an internal stabilizing force of sufficient magnitude to prevent the disruptive action. It is found by calculation that the stabilizing mass force is about 45 orders of magnitude greater than the Newtonian gravitational force. This translates into 10⁴⁵ in terms of powers of ten. Since the energy of mass depends on the square of the velocity, we conclude that the internal circulation velocity for an electron at rest is the uncontracted velocity of light.

The electron cannot be a sphere. In the case of a spinning sphere under the action of a central force, the equatorial plane is stabilized, but the poles collapse into the plane. Then there must exist another rotation at right angles to the first to prevent the collapse.

In the case of a sphere, the two rotations at right angles combine to give a single rotation about a new axis. In this case the collapse occurs as before. The only geometrical shape which will provide stability is that of a toroid. In this case, the motion of rotation around the axis of the ring prevents a collapse into the ring. The collapse of the ring is prevented by the motion of particle spin.

In mechanics, pressure is described as force per unit area. The numerical value is equal to the energy per unit volume. Since one-half the energy of the electron is internal, with the other half forming the space of the particle, the internal pressure can be calculated. In the case of the electron, the internal pressure is in excess of 10¹⁸ pounds per square inch.

The electron is a toroid, but there must exist a spherical equivalent. The inverse square law applies at points beyond the surface, but the effect must be modified as the surface is approached. The concept of the particle as a singularity in space doesn't tell us much. In the study of potentials associated with the electron, we find in addition to the static potential, one associated with spin and another related to dipole moment.

In the case of the stabilizing mass force, a condition approaching that of internal saturation must apply. It appears that the external manifestation is affected by motion of the particle so as to provide a mechanism for emission and absorption of photons by the electrons. This residue in excess of saturation appears as the Newtonian gravitational force. A good analogy to the condition of saturation is that of the electrical capacitor. Internal stresses may be quite large, but very little external effect, if any, appears.

The double motion may be described as a sink-source flow through the vortex with a return and continuation very much like that of a smoke ring. As viewed from the rear of the vortex, two spins are possible. The particle may spin to the right or to the left with respect to the vortex axis. It follows that two primary particles exist. As a matter of terminology, we identify the electron as having the left hand spin. The positron then spins to the right.

The nature of charge does not appear, but it is obvious that it is a function of the double flow. In terms of Bernouli's law, parallel flow lines attract and antiparallel flow lines repel. In this case it appears that the simple concept that like charges repel and unlike charges attract is not sufficient to provide an analysis in the case of very close proximity of the units.

The toroidal shape of the unit requires that the radius of the ring axis should equal the radius of the cross section of the ring. The volume of the unit is easily found by the application of the theorem of Pappus in the study of calculus. The radius of the unit can be calculated by quantum theory in terms of Planck's constant, mass, and the velocity of light. The value is found to be in excess of two hundred times the radius of the proton or neutron.

There can exist no other primary particles than positrons and electrons. It follows that the relative infinity of unstable particles known to exist are no more than clumps of positrons and electrons. Proximity results in mutual contraction of units with the size of the group depending directly on the quantum number applying and inversely proportional to the number of units in the group. It follows that the group can be smaller than any one of its units with no limit implied.

One failure to be found in Stephen Hawking's book, "A short History of Time," is the absolute acceptance of Hubble's contention that the universe is expanding. Let us develop a few consequences of that contention. If the universe is expanding, the energy density is reduced with time. Since the solar system is a part of the universe, this system expands. As the sun and the moon and the planets recede, they too get larger since no change in aperture is measured. The measured density is reduced unless the mass is increased to compensate for the increase in volume. Since the earth must enter into the expansion, the earth would increase in size relative to the size of a man unless the man expanded in the proper proportion. Now we get to the essence of the matter: If the expansion extends to the individual, it also applies to his head. If the expansion does not take place, we can visualize Dr. Hubble scratching his head six inches from the scalp and not even knowing the difference.

We offer no proof that the universe is not expanding. The observation is made that it is not necessary to invoke the doppler effect to explain the red shift. Einstein's claim that radiation in a gravitational field followed a straight line through a curved space made it impossible to predict the energy loss from photons in the field. The only added comment that needs to be made is the observation that a toroidal configuration for the universe could very well assure stability against the collapsing effect of gravitational forces.

The analysis of the neutron requires a toroidal configuration. The ring is composed of equal numbers of positrons and electrons alternating in position. Electron and positron spins combine in parallel, but the contraction produced by mutual proximity is such that the spin of the neutron as a unit is still one-half integral. The region of stability is fairly sharply defined, depending on the square of the number of units in the ring. The condition of stability requires 930 electrons and 930 positrons in the ring with a mass defect equivalent to 20 electron masses. The possibility of neutron isotopes within a limited range of values is not rejected. The proton is a toroid formed by the loss of the triplet electron-positron-electron. As explained by contraction theory, the spin of the proton is still one-half integral.

Now we get to the Pauli neutrino. This was invoked to provide a conservation of spin in the case of ß-decay. The question is: Who ever said that spin had to be conserved in the first place? In the case of an electron emitting a photon, the photon has spin. There is no assumption made that the spin of the emitted photon plus the new spin of the electron must equal the original spin of the electron. In fact there is no assumption made that the spin of the electron was changed by the emission of the photon. At least the laws ought to be consistent.

Incidentally, the presently assumed integral spin for the photon is in error. Only one-half of the photon energy applies to spin. Then we find the photon spin to be one-half integral. We also find that particle spin is contracted by the velocity of the particle in motion.

Since we now have electrons, positrons, protons, and neutrons, we can put them together to form atoms of stable matter. Protons and neutrons form the nucleus with electrons taking certain specified orbits called shells. The positron is a leftover having no place outside the interior of proton and neutron.

The presence of antimatter is speculated in other regions of the universe to preserve the concept of symmetry. If we can assume the existence of negative protons, the nucleus of an atom of antimatter would be composed of antineutrons and antiprotons with positrons making up the orbital shells. There can be no proof that such matter exists or that it would remain stable if it did. Whether it exists or not, it is the basis of the propulsion system used on the Starship Enterprise. Since there are no negative masses, the name antimatter is not the proper description.

Hawking's dependence on general relativity theory is most unfortunate. At least he admits that the theory is limited. His comment on the limitations of quantum theory is well-taken. In the formation of neutrons and protons from electrons and positrons, it is found that quantum states lower than unity must apply. Then we extend quantum theory into the region of inverse quantum numbers. If this is the case, quantum state unity as applied to the atom is not the lowest possible state. This explains not only the possibility of K-capture, but in the limit the ingestion of all the planetary electrons into the nucleus. Since electrons cannot exist in the free state within the nucleus, this action would result in breeding the nuclear protons into neutrons. The resulting disruption of the group occasioned by the removal of the stabilizing effect of proton-proton repulsion would produce mesons in unlimited numbers.

In the formation of nuclei of atoms, it is found that the stabilizing force is that of particle spin. The limit of the number of nuclear particles to form a stable group is found by calculation to be about 246. This is in general agreement with experiment. Since spin forces are associated with viscosity, the nucleus is a solid structure in the sense that the particles retain their relative positions. The kinetic energy of motion applying is that of a linear oscillator to provide the condition of a virtual orbit.

Orbital electrons form interlocking systems call shells. The first electron shell is described by energy quantum number unity. It is found to be complete with two electrons in a circular orbit. The second shell is described by quantum energy number two and is complete with eight electrons. In this shell, circular orbits apply to six electrons with elliptical orbits applying to the other two. The analysis can be continued for higher orbits, but it appears that no more than eight electrons can exist in any outer shell. Valence of the atom in chemical reactions is determined by the number of electrons in the outer shell.

As a matter of theory, the electron spin number is . In terms of energy, the positive value is taken. Then it appears that orbital angular momentum numbers cannot be integers as applied to atomic states. Instead, it is found that the corrected orbital angular momentum numbers are given by a sequence of odd half integers in such a manner as to yield integer quantum energy numbers. The angular momentum number zero is reserved for the description of virtual orbits. The correction removes the insanity of electrons oscillating through the nucleus.

Photons are emitted from orbiting electrons in the atom only when a transition to a lower orbit is made. Photons in a certain range of frequency are termed thermal photons. The indication is that a photon has a temperature manifestation. The photon is the basic unit to be considered in the transfer of heat from the sun to the earth. Those photons considered thermal in nature have frequencies of the order of 10¹³ cycles per second.

It must be pointed out that all photons have a temperature manifestation. The association between temperature and frequency can be made by the equality of Planck's energy form to that of Boltzmann. A photon with a frequency of 10¹³ cycles per second has a temperature of about 480 degrees on the Kelvin scale. This is about 180 degrees above room temperature on the Centigrade temperature scale.

Such a photon is a relatively large unit. In the case of a photon of visible light, the mid-range of about 6000 angstroms is found to apply. Since the diameter of an atom in the solid state is calculated to be about 2 angstroms, a single photon would encompass about 3000 atoms in the distance of one wavelength. Since the resolving power of a microscope is about one-half wavelength of the radiation used, the impossibility of viewing atoms or molecules with an optical microscope appears.

Hawking devotes a chapter in his book to the Heisenberg uncertainty principle. One statement of the principle has to do with the simultaneous measurement of momentum and location. If the location can be measured with any degree of exactness, the momentum is uncertain. If the momentum is known, the location of the object is uncertain.

Let us consider the nature of a photon. It has a length given by a single wavelength and a radius given by the wavelength divided by 2. We must consider the unit to be uniform in the assumption of a constant energy density in the interior. If we take a point in the interior of the volume as the location, no momentum can be measured to exist at this point. If the total momentum is measured, the whole volume must be included so that no one point can be called the location. Since the unit has both length and aperture, no point of location applies. If we choose such a point, the remainder of the volume is not considered. The Heisenberg uncertainty principle is simply a recognition of the fact that the entire unit must be considered and that the unit has a physical size.

The mathematics of chaos is related to the uncertainty principle. Predictions can be made only within certain limits. In the case of the weather, it is said that the effect of disturbance as much as that caused by the wing of a butterfly may modify the outcome of a prediction.

The kinetic theory of gases relates temperature to the agitation of molecules. The analysis is valid, but the effect as a temperature source must be considered secondary. Any prediction of the path of a single molecule is impossible. If we associate accelerations and decelerations of molecules with photon emission and absorption, the total effect must consider the effect of the photon as well. This is the nature of the temperature continuum invoked in the study of thermodynamics.

The molecule described by kinetic theory is a non-existent idealization. It is assumed to be completely elastic and spherical in the simplest form. In this case there are no provisions for the absorption and re-emission of photon energy. No internal structure is assumed to exist and no internal temperature applies. The absolute zero temperature is defined for the condition of all of the molecules at rest.

Thermodynamics is limited by the fact that thermal effects associated with accelerations and decelerations of matter are not considered. All compressions and expansions are quasi-static in the sense that they take place so slowly that temperature changes are uniform over the volume. Temperature is a gross property of matter not depending on individual atoms, molecules or photons. The lack of an internal structure requires that the only measure of temperature is that external to the matter itself and applies only to the space of the atom or molecule.

One glaring error of relativity theory is that of the assignment of a zero rest mass to the photon. Since the photon requires no velocity contraction, the effect does not occur. The error was introduced by the mistaken idea that mass increases with velocity to become infinite at the velocity of light. Since the photon does not have infinite mass, a zero rest mass was required in order to explain this fact. No consideration was given to the fact that there are an infinity of different masses applying to the photon.

The first law of thermodynamics is encompassed in the mechanical equivalent of heat. This implies an energy law based on temperature. It follows that if the law of conservation of energy applies, there must exist an equivalent law in terms of heat. Since heat is described by Boltzman's constant times absolute temperature, there also exists a temperature conservation law. Since matter can emit and absorb radiation, the law of conservation of temperature must consider an external temperature to describe photon emission and a remaining temperature within the emitting particle. Somewhere in this mess must appear a temperature effect associated with a particle in motion.

In view of the analysis presented, we are forced to conclude that the definitions of two separate absolute zeros are necessary. The absolute zero as applied to space requires the absence of photon activity. This can be true only for matter at rest, but the implication in this case is that the maximum temperature possible exists within the matter itself.

The other side of the coin requires that the absolute zero of temperature applies in the interior of matter only in the event that the total radiation has been emitted. In view of the mass- energy form, a zero mass results. This is known to occur as the end result of the formation of the neutrino in the decay of the positron-electron pair.

In theory a temperature continuum can be established in the event of a sufficient photon overlap. A thermal photon in the range of 10¹³ cycles per second has a volume of sufficient size to encompass as many as 10⁹ molecules of a gas at atmospheric pressure. If thermal photons are emitted and absorbed as motions occur at this particle density, it is certain that a required photon overlap will establish a temperature continuum.

The second law of thermodynamics is an entropy law. To this point the word "entropy" has not appeared, and no definition has been given. We make the point that the simplicity of a definition does not necessarily make the meaning clear. Applications may range far beyond those presently known to exist. We comment that there has been no advance in understanding in the field of thermodynamics in the two hundred years of its existence.