Symmetric elliptic integrals of the third kind

Legendre’s incomplete elliptic integral of the third kind can be replaced by an integral which possesses permutation symmetry instead of a set of linear transformations. Two such symmetric integrals are discussed, and direct proofs are given of properties corresponding to the following parts of the Legendre theory: change of parameter, Landen and Gauss transformations, interchange of argument and parameter, relation of the complete integral to integrals of the first and second kinds, and addition theorem. The theory of the symmetric integrals offers gains in simplicity and unity, as well as some new generalizations and some inequalities.