Two usual methods of proving the Law of Conservation of Angular Momentum lead to discrepant results. It is shown that the two methods deal with different definitions of angular momentum. When the definitions agree, the discrepancy disappears.

Although a proper derivation of Ohm's law for a particular material should be based on a quantum mechanical treatment of that material, it is interesting to see how far one can proceed on a purely macroscopic basis. It is shown here that for homogeneous, isotropic materials, one can expect an odd voltage-current relation, and that ordinarily Ohm's law will be obeyed as accurately as one wishes over some range of voltage and current. Only the ideas of causality, symmetry, and analyticity are used. Further, if Ohm's law is followed exactly by a material at constant temperature but the resistance is a function of the temperature, and the departure from Ohm's law is due entirely to Joule heating, then it is shown that the odd voltage-current relation still obtains.

If the original data {F¿} have a different scale and origin, so that F¿ = aY{ + b, the new solution of (4) should be y = ay + b, and since / is the same function of y that it was of y, (1) becomes S fnf(y,y,y")dx = 0,

An investigation has been carried out of the current and electric field distributions in the end region of a crossed field MHD device (accelerator or generator), with skewed electrodes, for the case in which the plasma exhibits the Hall effect. In order to simplify the situation enough to permit analytic solutions, a two-dimensional problem was considered wherein the plasma velocity and properties, and magnetic field intensity were taken to be constant and uniform. Solutions were obtained for several values of ωτ (ω=electron cyclotron frequency, τ=electron mean free time); they show that current flows a considerable distance upstream of the beginning of the electrodes, that some current concentration exists at one of the electrode corners, and that displacing this electrode upstream reduces this current concentration somewhat.

It is usual in the analysis of one-dimensional channel flows to study the behaviour of the analogous isentropic flow since, first, it retains the essential features of flows of practical interest and, secondly, it is simpler to describe. Although in conventional channel flows it is sufficient to neglect heat addition and friction to ensure isentropicity, in the MHD case it is in addition necessary to neglect Joule heating. This is accomplished by considering the fluid as having infinite electrical conductivity. However, this procedure does not necessarily imply infinite currents, since the external resistance will limit current flow. In the conventional problem, if we assume an isentropic flow, we are able to obtain a once integrated form of the governing equations. Such once integrated solutions are not possible in the present isentropic MHD channel flow, but equally simple solutions can be found and are presented. Examples of application of these results to the crossed field MHD generator and accelerator are also given.

When a glass capillary tube filled with alternating drops of mercury and electrolyte is made to vibrate mechanically, an alternating voltage is available across terminals inserted in its ends. This effect has been attributed to changes in the surface area of the interfaces [E. Yeager and F. Hovorka, J. Acoust. Soc. Am. 25, 447 (1953)]. New experimental evidence indicates that at low frequencies the output voltage depends on motion of the liquid drops with respect to the capillary tube. A first-order approximation theory which takes into account the tube compliance and the relative motion of the liquid drops with respect to the capillary tube has been obtained and is found to agree qualitatively with the experimental data relating: (1) tube output voltage to frequency at fixed amplitude of vibration; (2) tube output voltage to relative motion of the drops at fixed frequency; and (3) short-circuit current output to relative motion for fixed capillary radius, independent of mercury-electrolyte proportions for a given electrolyte. Theory indicates that the voltage is a function of the length of mercury in a tube rather than of the number of interfaces; this result seems to be in agreement with the data.

This paper presents a review of five papers in which a generalized electrodynamics has been developed. The purpose of the review is to present the results obtained so far, leaving out duplications, false starts, and detailed calculations. The emphasis is on the sequence of ideas, the difficulties encountered, and the methods of procedure.

The components of $\frac{\ensuremath{\rho}\mathrm{v}}{c}$ and $i\ensuremath{\rho}$ must transform as components of a four-vector, so that if measured in one coordinate system they are known in all coordinate systems. On the other hand, any operational definition of $\ensuremath{\rho}(\mathbf{r},t)$ must take account of the positions of all particles at the same time $t$, that of the $s$-th particle being ${\mathbf{r}}_{s}(t)$. Upon performing the Lorentz transformation these will be ${{\mathbf{r}}_{s}}^{\ensuremath{'}}({{t}_{s}}^{\ensuremath{'}})$, and the transformed time ${{t}_{s}}^{\ensuremath{'}}$ will be different for each particle. Another observer, in measuring ${\ensuremath{\rho}}^{\ensuremath{'}}$, would use ${{\mathbf{r}}_{s}}^{\ensuremath{'}}({t}^{\ensuremath{'}})$, ${t}^{\ensuremath{'}}$ being the same for all particles. As particles are in motion ${{\mathbf{r}}_{s}}^{\ensuremath{'}}({{t}_{s}}^{\ensuremath{'}})\ensuremath{\ne}{{\mathbf{r}}_{s}}^{\ensuremath{'}}({t}^{\ensuremath{'}})$, and there appears to be no necessary relation between $\ensuremath{\rho}(\mathbf{r},t)$ and ${\ensuremath{\rho}}^{\ensuremath{'}}({\mathbf{r}}^{\ensuremath{'}},{t}^{\ensuremath{'}})$, operationally defined in each coordinate system. It turns out, however, that if in each coordinate system the charge density is defined by $\ensuremath{\rho}(\mathbf{r},t)=\ensuremath{\Sigma}{s}^{}{e}_{s}\ensuremath{\delta}(\mathbf{r}\ensuremath{-}{\mathbf{r}}_{s}(t))$, then relativistic equations of transformation hold.

Paralleling a work of Fock we are able to eliminate the auxiliary conditions in our generalized quantum electrodynamics. As in the work of Fock this leads to a determination of both the electrostatic self-energy and electrostatic particle-particle interaction. Both turn out to be finite and in agreement with results obtained classically.

If one wishes to derive generalized field equations from a Lagrangian, at the same time preserving the linear character of the equations, one must admit terms involving derivatives of the field quantities. It turns out that the only non-trivial generalization of this kind, leading to differential equations of order below eighth, is obtained by taking ${L}_{f}=(\frac{1}{8\ensuremath{\pi}}){\frac{1}{2}{F}_{\ensuremath{\alpha}{\ensuremath{\beta}}^{2}}+{a}^{2}{(\frac{\ensuremath{\partial}{F}_{\ensuremath{\alpha}\ensuremath{\beta}}}{\ensuremath{\partial}{x}_{\ensuremath{\beta}}})}^{2}}$. This leads to a theory that contains the Land\'e-Thomas theory and accounts for the choice of sign required when one wishes to consider the total field as consisting of the Maxwell-Lorentz and the Yukawa fields.

The Dirac equation may be written so as to give an eigenvalue problem whose eigenvalues would be values of the electron mass. The equation is solved in several cosmological spaces, none requiring any quantization of $m$. A space-time suggested by Eddington leads to wave equations that have solutions with quantized $m$, $m$ depending upon the radius of the universe, constants $c$ and $\ensuremath{\hbar}$, and a quantum number. If the radius of the universe is taken as ${10}^{28}$ cm, the lowest mass state in this group of solutions is of the order of ${10}^{\ensuremath{-}65}$ g. Conversely if the usual electron mass is considered as the lowest state, the radius of the universe is of the order of ${10}^{\ensuremath{-}10}$ cm. Since the only constants occurring in the theory are $\ensuremath{\hbar}$, $m$, and $c$, one would expect that the radius of the universe would come out in terms of $\frac{h}{\mathrm{mc}}$. Only an occurrence, as the result of quantization, of a large dimensionless number could lead to a reasonable result; but Dirac's equation evidently does not provide such a number, and is therefore unsuited to account for the electron mass in terms of the radius of the universe.

Eddington, in his book Relativity Theory of Protons and Electrons, makes statements to the effect that the invariance of Dirac's equation is an elementary consequence of the tensor form which it acquires in the new wave tensor calculus, in which it is merely an identity derivable on epistemological principles and not on physical hypothesis. It is shown in this article that the invariance of Dirac's equation referred to by Eddington is a purely formal one, while the invariance which is of particular interest in quantum mechanics depends upon the physical interpretation of the wave function. The generalization of physical interpretation suggested by Eddington's work, however, has the desirable quality of leaving Dirac's equation invariant in the physical sense. It is further shown that in the derivation of Dirac's equation Eddington makes use of the usual physical assumptions, but in somewhat disguised form.

In a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. In quantummechanics in the case of two physical quantities described by noncommuting operators, the knowledge of one precludes the knowledge of the other. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) these two quantities cannot have simultaneous reality. Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that had previously interacted with it leads to the result that if (1) is false then (2) is also false. One is thus led to conclude that the description of reality as given by a wave function is not complete.

In this paper a point of view is presented according to which the quantities $m$ and $e$, the mass and the charge of elementary particles, need not enter into the electrodynamics and the quantum mechanics of electrons, positrons and photons. The only constants entering into the equations, rewritten in this way, are the velocity of light and two independent lengths, namely: $a=\frac{{e}^{2}}{m{c}^{2}}$, and $b=\frac{h}{\mathrm{mc}}$. The first of these lengths determines the scale of electrodynamic phenomena, and has no especial relation to electronic radius. The second determines the scale of quantum-mechanical phenomena. In the absence of particles electromagnetic phenomena have no definite scale. This fact, together with the possibility of creation of electron-positron pairs, leads to the belief that a theory of interactions of electrons, positrons, and photons, giving as a by-product a derivation of the ratio $\frac{a}{b}=\ensuremath{\alpha}=\frac{{e}^{2}}{\mathrm{hc}}$, could be formulated without introducing the two other pure numbers $\ensuremath{\beta}=\frac{m}{M}$ and $\ensuremath{\gamma}=\frac{G{m}^{2}}{{e}^{2}}$. This theory is envisaged as a limiting theory, obtainable from the future general theory by putting $\ensuremath{\beta}=\ensuremath{\gamma}=0$. It is then considered from the point of view of the necessity of giving up space-time framework for the description of physical phenomena. It is concluded that the first limiting theory should not necessitate abolition of space and time.

Expressions are obtained, in accordance with Einstein's approximate solution of the equations of general relativity valid in weak fields, for the effect of steady pencils and passing pulses of light on the line element in their neighborhood. The gravitational fields implied by these line elements are then studied by examining the velocity of test rays of light and the acceleration of test particles in such fields. Test rays moving parallel to the pencil or pulse do so with uniform unit velocity the same as that in the pencil or pulse itself. Test rays moving in other directions experience a gravitational action. A test particle placed at a point equally distant from the two ends of a pencil experiences no acceleration parallel to the pencil, but is accelerated towards the pencil by twice the amount which would be calculated from a simple application of the Newtonian theory. The result is satisfactory from the point of view of the conservation of momentum. A test particle placed at a point equally distant from the two ends of the track of a pulse experiences no net integrated acceleration parallel to the track, but experiences a net acceleration towards the track which is satisfactory from the point of view of the conservation of momentum.