occurs between the onset of the edipse and its midpoint. This agrees qualitatively with the work of Allais' with a paraconical pendulum, where the change of azimuth increased substantially in the 6rst half of the eclipse of 30 June 1954. Both these e6ects would seem to have a gI'RvltRt1onRl bRsls which cannot-bc explained by accepted classical theory.Both our experimental 6ndings and those of Allais cause one to question whether the classical laws of gravitation hold without modi6cation. CONCLUSIONQuantitative observations made with a precise torsion pendulum show, in agreement with many earlier, less precise recordings made at Harvard since 3.953, that ' Maurice F. C. Allais, Aerospace Kng. 18, 46 (1959). the times required to traverse a 6xed. fraction of its total angular path vary markedly during the hours before the eclipse and during its 6rst half, i.e. , up to its midpoint. Also the signihcant changes in these times do not coincide exactly with the astronomically determined. onset, midpoint, and endpoint of the eclipse.These variations are too great to be explained, on the basis of classical gravitational theory, by the relative change in position of the moon with respect to the earth and sun. This leads to the same conclusion arrived. at by Allaisthat classical gravitational theory needs to be modiaed to interpret his (and our) experimental results.Moreover, the 6ndings with the torsion pendulum, the sig~~~~~n™~~~~~~~~~~0~~~perpendicularly to the geogravitic vector, seem to indicate the possibility of a 6ne structure in these observations neither predicted nor recorded using the orthodox methods of quasistationary gravitational investigations.Upon de6ning vector spherical partial waves {Q ) as a basis, a matrix equation is derived describing scattering for general incidence on objects of arbitrary shape. With no losses present, the scattering matrix is then obtained in the symmetric, unitary form 8= -Q'~P, where (perfect conductor) Q is the Schmidt orthogonalisation of Q""= (k/v) J'de p(vXReg")Xg".g, integration extending over the object surface.For quadric (separable) surfaces, Q itself becomes symmetric, e fecting considerable simplihcation. A secular equation is given for constructing eigenfunctions of general objects. Finally, numerical results are presented and compared with experimental measurements.
Multiple scattering effects due to a random array of obstacles are considered. Employing a ``configurational averaging'' procedure, a criterion is obtained for the validity of approximate integral equations describing the various field quantities of interest. The extinction theorem is obtained and shown to give rise to the forward-amplitude theorem of multiple scattering. In the limit of vanishing correlations in position, the complex propagation constant κ of the scattering medium is obtained. Under appropriate restrictions, the expression for κ is shown to include both the square-root law of isotropic scatterers and the additive rule for cross sections valid for sufficiently low densities of anisotropic obstacles. Some specific examples from acoustics and electromagnetic theory then indicate that at least in the simplest cases the results remain valid for physically allowable densities of obstacles.
Upon introducing the outgoing spherical (or circular cylinder) partial waves {ψn} as a basis, the equation QT = − Re (Q) is obtained for the transition matrix T describing scattering for general incidence on a smooth object of arbitrary shape. Elements of Q involve integrals over the object surface, e.g. Qmn = ±(i2)δmn+(k8π)∫dσ⋅∇[Re(ψm)ψn]. where the −, + apply for Dirichlet and Neumann conditions, respectively. For quadric (separable) surfaces, Q is symmetric. Symmetry and unitarity lead to a secular equation defining eigenfunctions for general bodies. Some apparently new closed-form results are obtained in the low frequency limit, and the transition matrix is computed numerically for the infinite strip.
Scalar multiple scattering effects due to a random distribution of spheres are considered in detail. Transformation from a volume to a surface integral allows one to take account of the ``hole corrections'' involved in the equation of multiple scattering, and yields a secular equation for the propagation constant K of the composite medium. In the low-frequency limit a result is given which appears to be exact over the entire range 0 ≤ δ ≤ 1, where δ is the fractional volume occupied by scatterers. Also in this limit, the boundary conditions appropriate to the boundary of the composite medium are established from examination of the total transmitted and reflected fields.
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