A two-dimensional numerical model for compressible branching flow through a slot is described for the purpose of predicting the discharge coefficients of film cooling holes in gas turbine blades. The method employs free-streamline theory and the hodograph transformation. It calculates the area ratio of hole to duct and the contraction coefficient from a set of prescribed boundary conditions. An approximate method for calculating the compressible contraction coefficients is also discussed in the thesis. It employs the incompressible theory previously developed by McNown and Hsu (1951) for the free efflux, the 'compressibility factor' and the flow parameter (P_o-P_j)/(P_o-P₁), where P_o, P_j, P₁ represent the stagnation pressure, the static pressure of the jet and the static pressure of the approach flow, respectively. The advantages of using this method are the direct input of the area ratio of hole to duct and its speed of calculation. Experimental tests were performed using a specially designed rig in a supersonic wind tunnel. The investigations included sharp-edged slots with three different widths, a single hole and a row of two holes. The approach velocity in terms of the characteristic Mach number ranged from 0.18 to 0.58 and the pressure ratio P_o/P_j, ranged from 1.10 to 1.97. Agreement between the experimental data and the theoretical values was good to within the experimental accuracy (typically around +/- 5%) for the slots and the 2-hole configuration. For the 1-hole configuration, less bleed flow than predicted was observed, with the discrepancy varying from 7% to 18%. The latter case is a very severe test of a purely two-dimensional theory. The results for the 2-hole plate suggest that the slot theory can in fact be used to predict the flow through a row of holes with small pitch to diameter ratios.