Recently, various methods have been proposed to assess the risk of Rolling Contact Fatigue failure by Ekberg, Kabo and Andersson, and in particular, the Dang Van multiaxial fatigue criterion has been suggested in a simple approximate formulation. In a recent note by Ciavarella and Maitournam, it was found that the approximation implied can be very significant. Here, a much larger range of conditions is given, partial slip conditions are considered and analytical formulae are also derived.