A review is given of the methods previously adopted of calculating the discharge of a gas through a convergent-divergent nozzle and of a liquid over a broad-crested weir, through a Venturi flume with a horizontal bottom, and in a swirling state through a vertical nozzle. The assumption is made throughout that conditions are uniform over each cross-section of the constriction. Hugoniot's method is shown to be preferable, and it is used to examine the discharge of a liquid through a Venturi flume of any shape and of a swirling gas through a nozzle. It is pointed out that, if an external force is operating in the axial direction, the local velocity of sound or of a long wave, as the case may be, is not attained by the fluid at the exact geometrical throat of the constriction. For the general case of liquid flow, whether swirling or not, the axial velocity of streaming in the absence of external forces is shown to be equal at the throat to the velocity of a long wave.