The Hour Glass Theory of Davidson and Nedderman was one of the first theoretical predictions of the discharge rate from conical hoppers. Because of the assumptions in the theory, it was applicable only to narrow-angled, smooth-walled hoppers and even here it over-predicted the mass flow rate. Subsequent workers have incorporated the effects of wall friction with minor improvements to the accuracy. In this paper allowance is made of the non-uniform velocity distribution in the hopper. This enables the theory to be extended to wider angles and into the core flow regime. The resulting predictions are in good agreement with established correlations in the mass flow regime but there is less satisfactory agreement in core flow.