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A formula is given for the mean rate at which a point moving randomly in an -dimensional space crosses an arbitrary hypersurface. It is applied to evaluating false-alarm rates of systems of sensors arranged in coincidence or in parallel. Detection probabilities to permit a comparison of these arrangements are calculated for systems with identical sensors emitting independent Gaussian noise voltages and receiving signals of equal strength. A system that adds properly weighted outputs of the sensors is then best. At extreme SNR the parallel system is next best, but because of the high thresholds it requires to maintain a preassigned false-alarm rate, it is inferior to coincidence systems at moderate SNRs.