Viscoelastic and viscoinelastic flows through straight ducts of different cross-sectional shapes have engaged the attention of a number of workers. These results may be found in all standard texts on the subject. The viscoelastic flows present a more complex analysis than the viscous flows, and for this reason a viscoelastic solution can rarely be obtained in an explicit and usable form. In this note, we obtain a simple formula relating the drag per unit length of the duct, the pressure gradient, the cross-sectional area, and the rate of flux across the section for a duct of arbitrary cross-section without explicitly solving for the physical components of velocity. This result may be of interest to workers experimenting with the non-Newtonian fluids. The recent work of Nigam and Arunachalam [1], which gives a similar result for the drag in viscous flows, can be deduced very easily from the present analysis.