65

120

expressible by a rational fraction. It is less obvious
that the ultimate frequency can have an irrational
value without there being any law of succession;
but there seems to be no doubt that this also is
possible. For although it is impossible that
the ratio of frequency should, at any stage of the endless
succession of occurrences, be other than a vulgar
fraction, yet given any two values,
one greater and tbe other less than the irrational
value, there is no reason why as the succession
proceeds the ratio for all that part of it that has
occurred should not at length forever cease to be
as great as the greater of the two values while ceasing to be as small