In the case of a circular toroidal membrane under internal pressure there is no solution of the linear membrane equations for which the displacements are continuous. One possible resolution of the difficulty depends on introduction of an internal bending boundary layer. Another possibility more appropriate for very thin shells is to resort to the nonlinear membrane theory. The present paper is concerned with the second possibility. The equations are reduced to a linear second-order differential equation for the rotation of shell elements in which the coefficient of the undifferentiated term contains a large parameter. The equation is amenable to asymptotic methods of integration, and the solution is obtained in terms of two analytic functions of a single variable. These two functions, together with their first derivatives and certain integrals pertinent to the problem, are tabulated in the paper. With the help of these tables the title problem can be solved for practical ranges of the parameters without resort to further machine calculations.