It is shown that when a point charge is present in an electromagnetic field, the conservation of energy and momentum does not in general lead to conservation of angular momentum for the system as a whole. The conservation laws impose stringent restrictions on the possible equations which may describe the motion of the point charge.If it is required that higher derivatives of the velocity than the second should not appear explicitly in these equations, then the choice is unique and the only possible equations are those originally derived by Lorentz. If the third derivative is allowed to appear explicitly in the equations, but no thigher ones, then it is possible to give one other system of equations for describing the behaviour of a point singularity which can be used without entirely artificial initial and final conditions.

The exact relativistic classical equations taking radiation reaction into account for the rotation and translation of apoint dipole are given for the case where the dipole is always a pure magnetic dipole in the rest system. These equations are entirely free from any singularities. It is shown that the mass M, angular momentum of the spin I and magnetic moment g₂ are three entirely independent constants with no connection between them. The cross-section for the scattering of light by a dipole is given by formula (56). This formula shows that due to radiation reaction the scattering actually decreases as ω⁻² for very high frequenciesω, instead of increasing as ω² when radiation reaction is neglected. The quantum mechanical formula for the scattering of neutral mesons by neutrons is shown to go wrong at energiesħ ω ≳ 3 μ due to neglect of the effects of radiation damping. The classical formula (56) can still be correctly applied in the range 3μ <ħ ω < M, where the quantum formula is wrong, M being the neutron mass. Finally reasons are given for thinking that the quantum theory of the electron fails at energies above about √3 × 137m due to neglect of the effect of radiation damping on the spin, and the quantum theory of the meson and its inter-action with the electromagnetic field at √6 × 137μ.

The previous paper having shown that all divergences and large crosssections for neutral mesons being due entirely to neglect of radiation reaction, an attempt is made in this paper to remove those difficulties in the theory ofcharged mesons which do not occur in the theory of neutral mesons by following up an idea put forward tentatively by the present author some time ago on the ground that it would diminish the excessive scattering of charged mesons. It is assumed that the heavy elementary particles can exist in states of all integral charge, positive, negative or zero, the different states having different rest masses, of which the states with charge 0 ande (neutron and proton) must be assumed to have the lowest rest masses, while the proton states of charge −e and 2e are assumed to have the next lowest. The cross-sections for the creation and annihilation of protons of charge 2e and −e by several processes are calculated. The collision of a fast proton with another stationary proton is the most effective process for creating protons of charge 2e, the cross-section being of the order 10⁻²⁷ cm.² The colliding proton must have a kinetic energy of at least 35 M.e-V. Neutrons of the same energy would produce protons of charge −e on colliding with neutrons. The cross-sections for the production of protons of charge 2e and −e by mesons or photons are of the order 10⁻²⁷ cm.² The life time for spontaneous decay of these particles is of the order of 1/6 seconds, while the life time in air for reconversion into ordinary protons or neutrons by collision with a nucleus is of the order Z 10⁻⁷ secs. for low velocities. These particles have an interaction with the proton or neutron which is the same as the proton-neutron interaction with small additional terms. The energy-range relationship is calculated. The mean ionisation along a tract of a proton of charge 2e is nearly twice that of a proton or half that of an α-particle of the same range. If the theory is correct these particles are expected to occur in the nuclear explosions produced by cosmic rays, though less frequently than ordinary protons. Study of Wilson chamber photographs and photographs of nuclear explosions in the emulsions of photographic plates especially at high altitudes might be expected to reveal or disprove the existence of these particles.

It is shown that the scattering of neutral mesonsby the spin of the heavy particles (g₂ interaction) on the quantum theory agrees completely in its dependence on energy, scattering angle, and polarisation of the incident and scattered meson with the scattering on the classical theory (neglecting radiation reaction), except for being larger by a constant factor 3, which is due to differences in the averaging over the initial directions of spin of the heavy particle.

On the basis of the assumption put forward by Bhabha that the heavy particles can exist in states of all integral charge, it is shown that the scattering of charged mesons by the spin of the heavy particles only differs from the scattering of neutral mesons by factors (1−ΔM₂c²/E) and (1−ΔM₋₁c²/E) for scattering by protons and neutrons respectively, ΔM₂ and ΔM₋₁ being the mass excesses of protons of charge 2e and −e over an ordinary proton, and E being the meson energy.Thus on this assumption the scattering of charged mesons shows complete correspondence with the classical theory, and hence the previously given classical formula (36d) taking radiation reaction into account multiplied by a factor 3 will give the scattering of charged mesons up to energies of roughly 10⁹ e.v., correctly to within about 20%.

A new formula (36e) is given tentatively which should be more accurate.

The scattering of charged mesons on the usual quantum theory shows no correspondence with the classical scattering. The scattering on the basis of Heitler's idea of allowing the heavy particles to exist in higher spin states also shows no correspondence with the classical scattering.

The classical theory of mesons only contains the fundamental constant ξ. The rest mass μ of the meson is introduced only when the theory is quantized by the relation μ=ℏχ. As a result, although the quantum theory goes over strictly into the classical theory when ℏ → 0, the classical theory corresponds not only to the limit in which the momentum properties of mesons can be neglected, but also to the limit μ=0 (but ξ a finite constant). Due however to the fortunate circumstance that in nature M≫μ (M being the neutron mass) the classical theory can be used extensively to calculate processes involving mesons and heavy particles, but is entirely inadequate for calculating processes involving mesons and electrons or neutrinos, just because here μ≫m (m being the electron mass).

It is shown that the original Dirac theory in which a particle of spin half ħ is described by the Dirac equation with all the negative energy states empty, and the hole theory in which all the negative energy states are filled each with one electron, donot in general lead to the same probabilities for second order processes, contrary to what is generally believed, due to the energy denominators in the quantum formula for a second order process being different in the two theories, although the matrix elements for each first order process are the same.

The formula for the scattering of radiation by a free electron on the hole theory is calculated. This isnot the same as the Klein-Nishina formula which gives the scattering on the original Dirac theory. The new formula agrees with the Klein-Nishina formula only when the energy of the light quantum is small compared with the rest mass of the electron. For high energies it gives a greater scattered intensity at small angles and less at large angles than the Klein-Nishina formula and at extremely relativistic energies it gives a total scattering which is five times greater. The new formula at high energies is in definite disagreement with experiment, which agrees very well with the Klein-Nishina formula. Thus the hole theory is in definite disagreement with experiments on scattering although it is in at least qualitative agreement with nature in describing the existence of the positron and the process of pair creation, and is the only form in which the Dirac theory is logically tenable.

A rigorous solution of the equations of the cascade theory taking the radiation and pair creation cross-sections to be those for complete screening and the collision loss to be a constant is given which exactly satisfies the given boundary conditions at the surface of the layer. The solution is in the form of an infinite series but it is not a series in powers of the collision loss β since this enters essentially into the expression for each term. The first term of the series alone gives the whole energy spectrum of cascade electrons both above and below the critical energy with very considerable accuracy. The total number of particles produced at a deptht by an electron of initial energy E₀= β exp. y₀ is given in Table III fort varying from 2 to 30 and y₀ from 3 to 16. It is shown that at the maximum of the cascade the total number of particles above the critical energy is 0.8 times the number below the critical energy.

A formula for the depth of the penetration of a cascade as a function of the energy of the primary electron is given based on the calculations of Bhabha and Chakrabarty. A simple formula for the end of a shower is also given. It is shown that fluctuation plays an important part in determining the ability of an electron to operate two counters separated by a given thickness of absorber and increases the number of such electrons tenfold for thick absorbers. Formulæ are given for the number of electrons which enter an absorber of thicknesst and result in one or more particles emerging from the other side of the absorber.

A new experimental arrangement is described which makes a much more effective use of the cascade process for separating the electrons from the penetrating particles. This arrangement is suitable for measuring the penetrating component in high altitude balloon flights, and for studying the range spectrum of cosmic ray mesons.

Photographic plates exposed at high altitudes show a population of stars and isolated single tracks. Statistics are given covering 288 stars and 655 isolated tracks found in an area of 17 sq. cm. Some tracks are closely associated with the stars, and these have a mean range of 4·1 cm. air. The remainder, which show no association with stars, have a mean range of 3·4cm. air. Some stars have more than one associated single track. Alternative explanations are discussed. The most probable hypothesis appears to be that the single tracks are due to the spontaneous disintegration of unstable neutral particles emitted from the stars.

On the basis of the quantum mechanical cross-sections for radiation emission by electrons and pair creation by quanta when screening is complete, the fluctuation in the mean number of electrons and photons in any given energy interval in a cascade is calculated. The expression for the mean square deviation of this number is given explicitly in the form of a double integral which can be evaluated by the saddle point method.

The present note extends to the actual cascade process a result already established by Scott and Uhlenbeck for a model in which there is only one type of physical entity involved. It is shown that with the sole approximation of neglecting the angular divergence of showers complete information about the stochastic process of cascade generation can be calculated. Theqth order correlation functions are expressed as integrals involving the (q-1)th order correlation functions and the two independent elementary solutions of the cascade equations (8). The latter are the usual functions giving the mean number of electrons and photons in cascades excited respectively by a single electron or photon of some definite energy. If the latter are known exactly, then all the higher order correlation functions can be calculated by repeated integration. One thus obtains explicit expressions for the mean of theqth power of the number of electrons or photons in some finite energy interval, and from this again, through a use of equation (5), of the probability of finding N particles in this energy interval at a deptht in a cascade started by an electron or photon of some given energy.

The entire class of relativistic wave equations derivable from the Lagrange functionψ^°D(α^kp_k+β_χ)ψ and in which the wave function transforms according to the representation ℛ (3/2, 1/2)+ℛ(1/2, 1/2) of the Lorentz group is studied. It is shown that there is only one equation in this class describing particles of finite mass in which the charge density is positive definite, namely, the equation equivalent to the set proposed by Dirac, Fierz and Pauli for a particle of spin 3/2.

It is shown that there is no equation in this class which describes a particle of spin 3/2 and zero rest mass.

There is an equation in this class in which the particle has a state of finite rest mass and spin 3/2 and another state of zero mass and spin 1/2. In the former state the free charge density is positive definite in the latter zero.