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An atomic hydrogen maser has been operated that confines atoms in a volume whose linear dimensions are approximately ten times larger than those of previous masers of this type. The uncertainties associated with the wall frequency shift, presently the principal limiting factor in the absolute accuracy of frequency measurements with the hydrogen maser, should be reduced with this device, since the fraction of time an atom spends on the storage box wall is inversely proportional to the diameter of the box. Oscillations are achieved by the use of two resonant cavities coupled by a high-gain amplifier. The strong field maintained in one cavity prestimulates the atoms to radiate at an enhanced rate in the weak field of the other. With enough gain, self-sustained oscillation can be achieved at normal hydrogen fluxes and normal cavity -factor values. The theory of the two-cavity large storage box maser is considered and preliminary results are discussed.