Many problems in physics and engineering ultimately involve systems of linear differential equations with periodic coefficients. Till quite recently, research engineers, frequently employing nonrigorous methods, merely reduced these problems to Hill's equation or even to Mathieu's equation. [ Modern engineering problems often deman'd he investigation 0" systems with several degrees of freedom (dynamic stability of elastic systems, parametric resonance in high-power transmission lines and particle accelerators, numerous problems of celestial mechanics). In this connection one is often most interested in a quantitative description of phenomena for which there are no analogs in the simple system with one degree of freedom described by Hill's equation. I The old techniques prove inadequate.

It is also noteworthy that, since Lyapunov and Poincar, practical methods for investigating the stability of periodic motions described by nonlinear differential equations have centered increasingly around systems of linear differential equations with periodic coefficients.

Thanks to the efforts of many authors, the past two decades have seen significant progress made in the mathematical theory of such systems (particularly Hamiltontan systems). Practical methods are now available which frequently furnish solutions, quite satisfactory from the engineering standpoint, to many problems which previously seemed quite forbidding. It is characteristic of these methods that the computational difficulties increase only slowly with increase in the order of the system, so that they are particularly fruitful in regard to problems with manydegrees of freedom. Many new and profound results have been obtained for HillTs equation as well. This book aims at a systematic exposition of these results, previously available only in the periodical literature in a brief form, not readily understood by the nonspecialist. Most of the book is intended for the mathematically mature engineer, equipped with the basic mathematical tools of linear algebra and Lyapunov stability theory at the level of a technical college education. The exceptions are Chapters III and VIII, which demand more sophisticated mathematical tools and require an acquaintance with advanced university courses in mechanics, physics and mathematics.