Colloquium publications.

FUNCTIONALS AND THEIR APPLICATIONS.

119

72. The Algebra of Permutable and Non-Permutable Functions. In order to facilitate the inverse operation corresponding
to division, we abandon slightly Volterra's notation and introduce complex quantities
K = k + jK(rs),
= u + jU(rs),
etc., where k and u are constants, and K and U are functions of
r, s as indicated. We say that K = t if k = u and K(rs) = U(rs),
and define the zero element and the addition and subtraction of
elements in the usual way. In the quantity K we speak of k as
the ordinary coefficient and K(rs) as the functional coefficient;
and if the ordinary coefficient vanishes we speak of K as a function
of nullity. Both k and K may contain extra parameters x, y, * * *,
which do not enter into the functional operation, now to be
described.
We define the product
(34) Kt = ku + j{kU(rs) + uK(rs) + fK(rsi)U(s1s)dsi},
where the integral sign denotes the combination of the first or
second kind according to the algebra we are treating; if necessary to distinguish between these two products we may denote
them by [Ki]1 and [Kg2 respectively, or make use of the symbols
KU, KU, f Un, etc. Generally, we shall not need to im~ 2 *A
pose a condition of permutability on the functions involved, and
so we shall be using an algebra which does not contain the commutative property. The associative law for multiplication is
however always satisfied, as well as the distributive.
We shall have both left-handed and right-handed division.
In the algebra of the first kind, division by a function K, not of
nullity, is equivalent, by the definition of equality, to first an
ordinary division (to satisfy the relation among the ordinary
coefficients), and second, the resolution of an integral equation
of the second kind of Volterra type. This resolution is always
possible and unique, provided that the functions are finite