Only in this way is the progress from term to term
in a formal series possible (from type to type in the hierarchy
of Russell and Whitehead). (Russell and Whitehead have not
admitted the possibility of this progress but have made use of
it all the same.)
The repeated application of an operation to its
own result I call its successive application ("O' O' O' a"
is the result
of the threefold successive application of
"O'" to "a").
In a similar sense I speak of the successive
application of several operations to a number of
propositions.
The general term of the formal series a,
O' a, O' O' a, . . . I write thus:
"[a, x, O' x]". This
expression in brackets is a variable. The first term of the
expression is the beginning of the formal series, the second
the form of an arbitrary term x of the series, and the
third the form of that term of the series which immediately
follws x.