In a recent work (Roy and Saha, 2000) analytical expression for weight function has been deduced for an elliptic crack under normal loading. The integral equation method developed earlier was used in conjunction with Jacobi's polynomials in solving the problem. In the present work the same method is used to solve the title problem. New results of weight function have been given for the first time for the case of an elliptic crack under shear loading. The analytical expressions for mode II and mode III weight functions given, can be used to evaluate mode II and mode III stress intensity factors for an elliptic crack under polynomial as well as non-polynomial loadings. Some examples have been cited as illustrations.