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This series of papers investigates the dynamic interior of a quiescentprominence revealed by recent {\it Hinode} and {\it SDO/AIA} high-resolutionobservations. This first paper is a study of the static equilibrium of theKippenhahn-Schl\"{u}ter diffuse plasma slab, suspended vertically in a bowedmagnetic field, under the frozen-in condition and subject to a theoreticalthermal balance among an optically-thin radiation, heating, and field-alignedthermal conduction. The everywhere-analytical solutions to this nonlinearproblem are an extremely restricted subset of the physically admissible statesof the system. For most values of the total mass frozen into a given bowedfield, force-balance and steady energy-transport cannot both be met without afinite fraction of the total mass having collapsed into a cold sheet of zerothickness, within which the frozen-in condition must break down. An exact,resistive hydromagnetic extension of the Kippenhahn-Schl\"{u}ter slab is alsopresented, resolving the mass-sheet singularity into a finite-thickness layerof steadily-falling dense fluid. Our hydromagnetic result suggests that thenarrow, vertical prominence $H_{\alpha}$ threads may be falling across magneticfields, with optically-thick cores much denser and ionized to much lowerdegrees than conventionally considered. This implication is discussed inrelation to (i) the recent {\it SDO/AIA} observations of quiescent prominencesthat are massive and yet draining mass everywhere in their interiors, (ii) thecanonical range of $5-60 G$ determined from spectral-polarimetric observationsof prominence magnetic fields over the years and (iii) the need for a morerealistic multi-fluid treatment.

The Hydromagnetic Interior of a Solar Quiescent Prominence. I. Coupling between Force-balance and Steady Energy-transport