In [3] Miyakawa proved the existence of 418 classes of three-valued logical functions. In this paper it is proved that the number of classes of functions in $P_3$ is 406 (and not 418). This paper contains data on the number of bases and pivotal incomplete sets of any rank and an algorithm for enumeration which is different from that in [4] (The data given in [4] is incorrect because the number of classes is incorrect).