There are many problems where the function under study in symmetric in several variables and its maximum or minimum occurs when the variables are equal. The author of this paper calls this fact the Purkiss principle, in honor of one of the authors who (independently with Terquem) noticed this principle in the middle of the XIXth century. Bunyakovskij gave a counterexample to this ”principle” and the question is to find conditions under which this ”principle” is true. This is the object of this paper where the following result is proved. Theorem. Let f and g be symmetric functions of class \(C^ 2\) on a neighbourhood of a point \(P=(r,...,r)\in R^ n\). On the set where \(g\) equals \(g(P)\), the function \(f\) will have a local maximum or minimum at \(P\), except in degenerate cases, which are described in the paper. Some generalizations of the Purkiss principle are also proved when the group of permutations is replaced by other symmetry groups. The paper ends with a biographical appendix about Terquem, Bunyakovskij, Purkiss and Chrystal.