In this paper we consider the slow motion of a incompressible, inviscid fluid, that has a high electric conductivity, plunged in a uniform magnetic field, within which an induced magnetic field is generated. Moreover it is subjected to forces of general kind, functions of place and of time. We establish opportune equations for the field of the velocity vectors and for that of the pressure. Finally, making use of the generalized formula of Kirchhoff, we give, by means of integral formulas, the performance of the components of the velocity, which are perpendicular to the applied magnetic field, and of the axial components of the vortex.