In Part One they formulate in a general way the problem of analyzing and evaluating the aberrations of quadrupole magnet beam systems, and of characterizing the shapes and other properties of the beam envelopes in the neighborhood of foci. They consider all aberrations, including those due to misalignments and faulty construction, through third order in small parameters, for quadrupole beam systems. One result of this study is the development of analytic and numerical techniques for treating these aberrations, yielding useful expressions for the comparison of the aberrations of different beam systems. A second result of this study is a comprehensive digital computer program that determines the magnitude and nature of the aberrations of such beam systems. The code, using linear programming techniques, will adjust the parameters of a beam system to obtain specified optical properties and to reduce the magnitude of aberrations that limit the performance of that system. They examine numerically, in detail, the aberrations of two typical beam systems. In Part Two, they examine the problem of extracting the proton beam from a synchrotron of 'H' type magnet construction. They describe the optical studies that resulted in the design of an external beam from the Bevatron that is …