The parameters of a system of demand equations are estimated subject to the prior information of classical demand theory. The equations are estimated as a system using a variant of generalized least squares, the parametric restrictions being imposed by Lagrange multipliers. Tests of significance are given, both for individual restrictions and for the restrictions applied collectively. The method is applied to Barten's sixteen commodity consumer expenditure data for Holland. The work was done independently of R. H. Court's [6] similar treatment; however, there are significant differences in the method which warrant further discussion and the application is itself of some interest.