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 seco...