A conservative system composed of a collection of interacting ions plus externally applied forces is considered. The mechanical problem of motion is defined for such a system, and the equilibrium and the translational and rotational invariance conditions discussed. It is shown that the second equilibrium condition of Born and Huang, namely that the stresses must vanish in the equilibrium configuration for the infinite lattice model, is not a requirement of the theory. The dynamical matrices, whose eigenvalues are simply related to the phonon frequencies, are shown to be of the same form for a homogeneously strained crystal as for an ideal unstrained crystal. The elastic constants of a homogeneously strained crystal are calculated by the method of homogeneous deformation, and also by the method of long waves. The quantities which are observed in the measurement of sound velocities in strained crystals are the effective elastic constants, which differ from the elastic constants by terms involving the stress components. These effective elastic constants are also calculated in the lattice theory, and their properties are discussed.

This summary of much of the author's work stresses the unity of approach to the understanding of the optical properties of a crystalline medium for visible light as well as for x rays. The scattering elements of the crystal are supposed to act as dipoles, and for the interior of the medium the problem is that of finding a balanced, or self-consistent state of optical field and dipole vibration. This is an eigenvalue problem, with the phase velocity as the eigenvalue. Its solution leads on to finding the forced vibrations in a crystal with a plane surface on which an external plane wave falls, and gives the theory of reflection and refraction. The physical interpretation of the various mathematical steps of the theory is stressed. It leads, as an example, to an immediate qualitative understanding of the anomalous absorption occurring for x rays near the Bragg angles (Borrmann effect). A mechanical model of this effect is discussed.

A general discussion of the treatment of the asymmorphic space groups is given, with special reference to the hexagonal close-packed lattice ($\frac{P{6}_3}{\mathrm{mmc}}$), for which the irreducible representations are given in full and the lattice harmonics listed for all $l$. The expansions possess the same properties as those given for the cubic groups in the preceding paper.

The bootstrap yields the strong-interactions symmetry provided the hadron currents participating in nonstrong interactions are first fed into it; the alternative course, in which the bootstrap is required to generate the entire symmetry uniquely, with no a priori information from the nonstrong interactions, seems to lie beyond the present techniques and would also leave unexplained the subsequent emergence of the weaker couplings. We suggest that "internal" symmetries may have a geometrical origin, corresponding to transformations in the global embedding space of the four-dimensional physical Riemann universe. These would be the unitary or orthogonal transformations of the normal subspace, since the latter do transform a small region of curved space-time into itself. This picture fits in with the short ranges of the strong interactions. Cosmological and astrophysical implications are noted.

The embedding of space-time in a higher-dimensional pseudo-Euclidean space is considered as a means of studying the representations of coordinate transformations in curved space-time. To the extent that this approach is valid, the existence of particles belonging to various representations of some internal symmetry group is "predicted" in much the same sense that particles of spin 0, ½, 1, etc. are "predicted" by the structure of representations of the Lorentz group.

We report a beginning in a project to determine the consequences of the following assumption, "that a physical theory in flat space is obtainable as the limit of a physical theory in a curved space." Because of the absence of groups of motion in general curved spaces, we discuss only the case of constant curvature. Then operators of angular and linear momentum exist, and we show that the interesting irreducible unitary representations of the group of motions reduce very simply to those of the inhomogeneous Lorentz group in the limit of zero curvature.

The complete general-relativistic plane-wave space-times (gravitational, electromagnetic, or both) are examined in relation to the focusing effect they exert on null cones. The following remarkable property is then obtained. No spacelike hypersurface exists in the space-time which is adequate for the global specification of Cauchy data. As a consequence, it is not possible to imbed a plane wave globally in any hyperbolic normal pseudo-Euclidean space.

The embedding of various relativistic Riemannian spaces is performed. In each case the original Riemannian line element, the embedding pseudo-Euclidean line element, and the embedding transformation are shown.

This is a review of differential geometry results pertaining to the problem of embedding curved space-time in a pseudo-Euclidean space.

A review is given of some conventions and definitions required for the derivation of the irreducible representations of the space groups, and of a method to obtain lattice harmonics. These are given for all the irreducible representations of the simple cubic ($\mathrm{Pm}3m$), face-centered cubic ($\mathrm{Fm}3m$) and body-centered cubic ($\mathrm{Im}3m$) space groups for $l\le 12$. The expansions are given in polar coordinates and care has been taken that different bases corresponding to the same representation span identical, rather than equivalent, representations, which are given in full. Moreover, all the different expansions listed in the tables are fully orthogonal.

The word "fermion" here means "spin-½ fermion." Discussion is restricted to processes in which two arbitrary incoming fermions are scattered into two arbitrary fermions (not necessarily the same) in the final state. Invariance under space inversion ($P_{\sigma }$), time-reversal ($P_T$) invariance, and charge-conjugation ($P_C$) invariance is not assumed to hold necessarily. Formulas are derived which can be applied to any polarization experiment, including correlation experiments. Necessary conditions are obtained which can be tested experimentally to check whether a certain invariance principle holds in nature. In addition to $P_{\sigma }$, $P_T$, $P_C$ invariance and their combinations, certain helicity invariance principles (related to ${\gamma }^5$ invariance principles) are discussed. The treatment is relativistic. By a suitable choice of rest frames for each particle, the appearance of some trigonometric functions of the scattering angle is avoided, and $\sigma$ matrices can be used instead of $\gamma$ matrices. Experiments in which the two incoming fermions are uncorrelated and one of them is unpolarized, are treated in great detail. It is shown to what extent such experiments can determine the $S$ matrix. A graphical method is introduced which can often be used to simplify discussion of multiple scattering experiments. Fermion—spin-zero-boson scattering is treated as a special case of fermion-fermion scattering. Those consequences of invariance principles which can be checked most easily in general polarization experiments are discussed.

Neutron capture cross sections are averaged over Maxwell-Boltzmann neutron distributions for temperatures pertinent to slow nucleosynthesis ($s$ process) in red giant stars. Such cross sections, together with isotopic abundances, provide the major quantitative experimental tests of the theory. The cross sections are tabulated at convenient intervals from $kT=5\mathrm{}$ keV to $kT=90\mathrm{}$ keV where data are available. In many cases it has been possible to supplement the often fragmentary (as of June 1964) data with evidence from nuclear systematics.

The effective quantum-mechanical Hamiltonian for a system of electrons in static electromagnetic field is derived from quantum electrodynamics for electron velocities small compared with the light velocity. The Hamiltonian obtained is almost identical with the one obtained by Bethe and Salpeter, but there is a small difference, of which an explanation is given.

A general survey of phenomenological, collective-model theories, especially those based upon the hydrodynamic model of rotations and vibrations of a deformed liquid drop, is presented. Details of models of even-even, odd-odd and odd-$A$ nuclei are given as well as a discussion of electromagnetic moments and transitions in these models. The influence the vibrating and rotating nuclear surface has on alpha and beta decay of deformed nuclei is outlined. The discussion is extended to collective model calculations of the photonuclear process from which information can be obtained on the shape of nuclei in their ground states. Comparison of theory with the results of some recent experiments is given where this has been considered useful.

The availability of intense electromagnetic fields produced by lasers has stimulated experimental and theoretical investigations of nonlinear optical phenomena such as second-harmonic generation. Coefficients descriptive of such processes in a medium may be calculated in terms of the matrix elements and characteristic frequencies of the medium by conventional quantum-mechanical perturbation theory of sufficient order. Calculations of this sort for several specific cases have been presented by Armstrong et al., and others. The present paper (i) applies diagrammatic techniques borrowed from field theory to the perturbation calculations, which simplifies the algebra and offers some insight into the structure of the nonlinear processes, and (ii) provides a convenient collection of results for many of the nonlinear optical processes of present and potential interest.