The generation of series by means of relations more or less resembling that of
to
is very common. The series of the Kings of England, for example, is generated by relations of each to his successor. This is probably the easiest way, where it is applicable, of conceiving the generation of a series. In this method we pass on from each term to the next, as long as there is a next, or back to the one before, as long as there is one before. This method always requires the generalised form of mathematical induction in order to enable us to define "earlier" and "later" in a series so generated. On the analogy of "proper fractions," let us give the name "proper posterity of
with respect to
" to the class of those terms that belong to the