The Theory of Permutable Functions

A fundamental property of permutability is expressed in the following theorem:
Two functions permutable with a third are permutable with each other.

A group of permutable functions is characterized by a function of the first order of which the first and second partial derivatives exist and are finite. Consequently when we consider a group of permutable functions, we shall always assume that there is known to us a function of the first order which has finite derivatives of the first and second orders and belongs to the group. This function shall be spoken of as the fundamental function of the group. When a fundamental function of the group has the canonical form, we shall speak of the group as a canonical group.
A remarkable group of permutable functions is the so-called closed-cycle group, which is made up of functions of the form
f(y-x).
Unity belongs to this group, and it is deduced immediately.