Thence we obtain
Since this is an identity we may multiply out the left-hand side and
equate the coefficients of the successive powers of to zero; obtaining
relations which enable us to express any function in terms of
members of the series .
7. In the applications to combinatory analysis it usually happens that
we may regard as being indefinitely great and then the relations are
simply continued indefinitely.
The before-written identity now becomes
and herein writing for and transposing the factors we find
an identity which is derivable from the former by interchange of the
symbols and .
There is thus perfect symmetry between the symbols and it follows
as a matter of course that in any relation connecting the quantities
with the quantities we are at liberty to
interchange the symbols , . This interesting fact can be at once verified in
the case of the relations , etc.
Solving these equations we find