In this paper the Author considers the motion of an asymmetric rigid body about a fixed point in a newtonian field of forces, in the only hypothesis that the centre of mass belongs to a principal axis of inertia relative to the fixed point. He gives new particular solutions of the equations of the motion in which the squares of the equatorial components p, q of the angular velocity are functions of second degree in the component r about the baricentral axis. The discovered solution depends on three arbitrary constants and this defines a system of $\infty^{3}$ possible motions.