  This paper describes a general investigation of stationary oscillations of
galaxies. It begins with a linear analysis of modes of oscillation with
continuous spectra of real frequencies. Such modes are gravitational analogues
of the van Kampen modes of oscillation in plasmas. The characteristic value
problem governing modes of the van Kampen type in a galaxy is solved with the
aid of a modified version of the matrix method of Kalnajs in which the
perturbation of the distribution function is expressed in terms of generalized
functions. In general, there is no characteristic equation governing the
frequencies in the continuous spectrum. However, isolated frequencies in the
continuous spectrum do satisfy a characteristic equation which, for stellar
systems, is a counterpart of the dispersion relation proposed by Vlasov for
plasma oscillations. The linear analysis also provides a characteristic
equation for modes with a discrete spectrum of real and/or complex frequencies.
The second part of the paper describes a perturbation theory for a stationary
oscillation of a galaxy with a small but finite amplitude. Integrals of the
stellar motion are constructed with the aid of canonical perturbation theory
and used in conjunction with the theorem of Jeans in order to specify the
density of stars in the six-dimensional phase space. These oscillations are
slightly nonlinear counterparts of the modes of the van Kampen type, and they
are stellar-dynamical counterparts of the nonlinear plasma waves described by
Bernstein, Greene, and Kruskal. Fully nonlinear models of stationary
oscillations of galaxies can be constructed with the aid of Schwarzschild's
numerical method for the solution of the fundamental integral equation
describing the self-consistency of a stellar system.
