Integrals of two-dimensional motions of a perfect incompressible fluid of nonuniform density

Integrals of motion are obtained which remain the same by virtue of the linearized equations of two-dimensional motions of a perfect incompressible fluid stratified with respect to the density and which are quadratic forms of the perturbation fields. The properties of these integrals and the variational principles associated with them are discussed. A number of conclusions concerning the problem of stability of stratified flows are drawn.