Link between the retrograde prograde nutations and nutations in obliquity and longitude

The nutations of the Earth can be seen as variations in longitude and in obliquity of the position of the Celestial Ephemeris Pole (CEP); these variations are given for each nutation frequency, and correspond to an elliptical motion of the CEP in the inertial space. Therefore the nutations can also be expressed as the sum of two circular motions of opposite direction (one prograde and one retrograde) with the same frequency. In the literature, the nutations are given in one or the other representation. Because the conventions are not always the same, we review here the mathematical expressions of both descriptions and we give the way to get one from the other.