1. The smoothness of H and K. A point set 8 in the (x, y, z) -space is said to be a (small piece of a) surface of class Gn, where n ^ 1, if it is the locus of the endpoints of a vector function X = X(u,v) with three com? ponents which is defined on a two-dimensional (u, v) -domain, possesses con? tinuous partial derivatives of n-th order, and is such that the vector product (Xu, Xv) of the first order derivatives does not vanish. If 8 is a surface of class C1, then, after a suitable choice of the coordinate axes, 8 can be repre? sented as the locus of the endpoints of X = (x, y, z(x, y)), where z = z(x, y) is a function of class O1 on some (x, y)-domain. The function z = z(x,y) is of class On if and only if the surface 8 is of class Cn. The first part of this paper will be concerned with relationships between assumptions of smoothness (that is, degree of differentiability) for a surface 8: X = X(u,v) and for its curvatures, either its Gaussian curvature K = K(u,v) or its mean curvature H = H(u,v). These curvatures are defined when 8: X = X(u,v) is of class Cn, where n^.2, and their definitions show that H(u,v) and K(u,v) are then of class Cn~2. Standard theorems in the theory of elliptic differential equations lead to partial converses of this statement.