Meir and Sharma [1] have suggested a modification of Romberg quadrature using Newton-Cotes and, in particular, Simpson sums in place of trapezoidal sums. By comparing the error term with that obtained by Bulirsch [3] for trapezoidal sums, they concluded that the use of Simpson sums would lead to an improvement of the results. The procedure adopted by Meir and Sharma [1] permits them to obtain an expression for the error in the numerical quadrature. However, for the purpose of numerical computation, this procedure appears to be less suitable. In section 3, we give an alternative formulation which would enable us to carry out the computation, using Simpson sums, in the same wasy as is done in the case of Romberg quadrature with trapezoidal sums. Some numerical results are discussed in section 4.