1. The theorem. Let gik gik(U, v) be the elements of a 2 by 2, symmetric, positive definite (function) matrix, defined in a neighborhood of (un, v) (0, 0). The problem to be dealt with concerns the existence of a 2-dimensional surface, in a 3-dimensional Euclidean space, for which ds2 = g,dU2 + 2g12du dv + g22dV2; in other words, with the existence of funictions x xi(u,iv), where j 1,2, 3, which satisfy the three conditions