The chapter discusses the application of statistical methods to hydrodynamics, focusing on the ergodic motion of parallel vortices. The author, L. Onsager, highlights the importance of understanding the stability of unsteady flows, particularly in the context of turbulent dissipation. He mentions the contributions of C. C. Lin and Kolmogoroff in advancing the theory of turbulence. The main focus is on the statistical mechanics of $n$ parallel vortices in an incompressible, frictionless fluid. The system is Hamiltonian and has a finite number of degrees of freedom, allowing for the application of standard statistical mechanics methods. The equations of motion are derived, and the Hamiltonian is defined, considering both unbounded and bounded fluid domains. The chapter also explores the properties of the phase space, noting that the phase-space volume is finite and that the energy can assume all values from $+\infty$ to $-\infty$. The function $\Phi(E)$, which represents the phase-volume corresponding to energies less than a given value, is analyzed, and it is shown that $\Phi'(E)$ is positive and reaches its maximum at a finite energy $E_m$. The temperature $\Theta$ is defined and shown to be positive for energies below $E_m$.The chapter discusses the application of statistical methods to hydrodynamics, focusing on the ergodic motion of parallel vortices. The author, L. Onsager, highlights the importance of understanding the stability of unsteady flows, particularly in the context of turbulent dissipation. He mentions the contributions of C. C. Lin and Kolmogoroff in advancing the theory of turbulence. The main focus is on the statistical mechanics of $n$ parallel vortices in an incompressible, frictionless fluid. The system is Hamiltonian and has a finite number of degrees of freedom, allowing for the application of standard statistical mechanics methods. The equations of motion are derived, and the Hamiltonian is defined, considering both unbounded and bounded fluid domains. The chapter also explores the properties of the phase space, noting that the phase-space volume is finite and that the energy can assume all values from $+\infty$ to $-\infty$. The function $\Phi(E)$, which represents the phase-volume corresponding to energies less than a given value, is analyzed, and it is shown that $\Phi'(E)$ is positive and reaches its maximum at a finite energy $E_m$. The temperature $\Theta$ is defined and shown to be positive for energies below $E_m$.