The relationship for combined increasing and decreasing free surface flow is presented in difference form, and then particularized to spatially increasing and spatially decreasing flows. For spatially decreasing flow, a theoretical difference is noted between the writer’s version and that in a widely-used reference source. By investigating an application for uniformly decreasing, full-pipe flow, it is shown by physical reasoning and comparison of an available analytical solution to the alternate versions of the difference relationship that the writer’s difference relationship is the correct one. Free surface spatially varied flow relationships are specialized to pressure conduits to demonstrate conditions of slope invariance, which has important and somewhat counter-intuitive ramifications for certain practical applications encountered in environmental engineering.

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