Report that "discusses the magnetic vector and scalar potentials, magnetic field components and their derivatives, and flux linkage for single current systems, and the mutual inductance, forces and torques between two such systems, whose axes are coplanar but not necessarily coincident" (p. 1).

A summary of the dynamic behavior of the proton bunches in the Brookhaven Alternating Gradient Synchrotron (AGS) has been given previously. In these reports, the usual linearization of the differential equations involved has been made and the theory was restricted to well bunched beams. The linearized approach is no longer valid at transition where the actual phase angle of the bunch can differ appreciably for a short time from the stable phase angle Φ₀. In this report the non-linearity of the differential equations for phase oscillations will no longer be neglected. At transition the beam is slow enough so that the electronics of the bootstrap system can be considered as being ideal and the radius servo loop can be characterized by one time constant. Under these assumptions the analysis can be carried out in a two-dimensional phase plane. The essential new result will be the short existence of a stable equilibrium point for the bunch motion not coinciding with Φ₀. The results here derived have been tested experimentally and at least a qualitative agreement was found. However, the conclusions are no more valid if debunching takes place since we have still neglected the finite bunch width.

A generalization of Wigner's simple model for the distribution of nuclear level spacings is studied. It is based on a stochastic process which reproduces the correct joing probability distribution of N enerfy levels for small spacings. The case N=3, which includes the effect of the correlation between adjacent spacings, is discussed in detail. The resulting distribution and the correlation coefficient are compared with experimental data. No definite conclusion can be drawn except to say that the effect of the correlations on the spacing distributions on the spacing distribution is very small.