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198 Short Communications AEQ. MATH.for all x~ [0, l]. These assumptions imply that if ~p (x) is a classical solution of (1)-(2), it is unique. Approximate solutions to (1)-(2) can be obtained from applications of the classical Rayleigh-Ritz method to finite-dimensional subspaces of S. Upper bounds for the rates of convergence of the approximations have recently been theoretically determined for certain sequences of subspaces of S, such as the piecewise-polynomial Hermite and spline subspaces. These approximate solutions are not precisely obtainable in practical computation on a digital computer, since certain integrals arising in the Rayleigh-Ritz formulation are replaced necessarily by quadrature formulas.The purpose of this paper is to investigate the errors introduced in the approximate solutions by such quadrature formulas. In particular, we obtain bounds for these errors, as they apply to finite-dimensional subspaces of S of piecewise-polynomial functions, and we determine when these quadrature errors are consistent with (i.e., = o]) [o o Lg(x[o oO3) g(x[o o])j est une fonction multiplicative, c'est-~t-dire F(xy)=F(x)F(y).