This paper concerns the application of a finite element method to the numerical solution of a nonrestricted form of the Plateau problem, as well as to a free boundary problem of Plateau type. The solutions obtained here are examined for several examples and are considered to be sufficiently accurate. It is also observed that the hysteresis effect, which is a feature of a nonlinear problem, appears in this problem. 1. Introduction. Methods for the numerical solution of the Plateau problem have so far been examined by D. Greenspan [3], [4], using the combination technique of difference and variational methods, and by P. Conçus [5], using a finite difference method. These two methods can be applied only to the so-called restricted form of the Plateau problem described by Forsythe and Wasow [2, Section 18.9], that is, to the problems where the boundary condition is represented by a single-valued function. Thus, they cannot be applied to the problem where the boundary condition is represented by a multi-valued function, such as Courant's example described later. This paper shows that such multi-valued boundary-value problems can be solved numerically by a finite element method. In this case, two solution methods, one for a free boundary problem and the other in a cyclindrical coordinate system, are presented. 2. Application of a Finite Element Method to the Plateau Problem. The Plateau problem involves finding a twice continuously differentiable function uix, y) in a region D satisfying "(•*> y) = fix, y), or (2.1) duix, y)/dx = gix, y) duix, y)/dy = hix, y) and minimizing the surface area functional (2.2) Ju = Jf (1 + ul + u2y)U2 dx dy, where fix, y), gix, y) and hix, y) are single-valued functions [2, Section 18.9].