The derivation of the hydrodynamic equations for a gaseous mixture from the system of kinetic Boltzmann equations is analyzed. The form of the hydrodynamic equations is a unique consequence of necessary and sufficient conditions for the solvability of systems of linear integral equations with symmetrical kernels, which define the terms in the expansion of the distribution functions in a series with respect to a parameter of spatial non-homogeneity (actually, the Knudsen number). The transport laws are presented in a form for which the Onsager symmetry relations hold. In deriving the Onsager relations use is made of symmetry properties of integral operators, which are a consequence of the invariance of the equations of mechanics with respect to a transformation involving changing the sign of the time and the impulses of the particles. The Onsager relations are also derived from expressions for the kinetic coefficients in terms of correlation functions.