Reverting now to the generation of series by the relation between consecutive terms, we see that, if this method is to be possible, the relation "proper -ancestor" must be an aliorelative, transitive, and connected. Under what circumstances will this occur? It will always be transitive: no matter what sort of relation may be, "-ancestor" and "proper -ancestor" are always both transitive. But it is only under certain circumstances that it will be an aliorelative or connected. Consider, for example, the relation to one's left-hand neighbour at a round dinner-table at which there are twelve people. If we call this relation , the proper -posterity of a person consists of all who can be reached by going round the table from right to left. This includes everybody at the table, including the person himself, since [Pg 36] twelve steps bring us back to our starting-point. Thus in such a case, though the relation "proper -ancestor" is connected, and though itself is an aliorelative, we do not get a series because "proper -ancestor" is not an aliorelative. It is for this reason that we cannot say that one person comes before another with respect to the relation "right of" or to its ancestral derivative. The above was an instance in which the ancestral relation was connected but not contained in diversity. An instance where it is contained in diversity but not connected is derived from the ordinary sense of the word "ancestor." If is a proper ancestor of , and cannot be the same person; but it is not true that of any two persons one must be an ancestor of the other. The question of the circumstances under which series can be generated by ancestral relations derived from relations of consecutiveness is often important. Some of the most important cases are the following: Let be a many-one relation, and let us confine our attention to the posterity of some term . When so confined, the relation "proper -ancestor" must be connected; therefore all that remains to ensure its being serial is that it shall be contained in diversity. This is a generalisation of the instance of the dinner-table. Another generalisation consists in taking to be a one-one relation, and including the ancestry of as well as the posterity. Here again, the one condition required to secure the generation of a series is that the relation "proper -ancestor" shall be contained in diversity.