An infinite family of axially symmetric thin discs of finite radius is presented. The family of discs is obtained by means of a method developed by Hunter and contains, as its first member, the Kalnajs disc. The surface densities of the discs present a maximum at the centre of the disc and then decrease smoothly to zero at the edge, in such a way that the mass distribution of the higher members of the family is more concentrated at the centre. The first member of the family has a circular velocity proportional to the radius, thus representing a uniformly rotating disc. On the other hand, the circular velocities of the other members of the family increase from a value of zero at the centre of the discs to a maximum and then decrease smoothly to a finite value at the edge of the discs, in such a way that, for the higher members of the family, the maximum value of the circular velocity is attained nearest the centre of the discs.