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This paper is the first in a series on a systematic analysis of a number of catalogs of extragalactic objects. The method is based on the "analysis of power spectra," as elucidated by Tukey. As workers in other fields have abundantly shown, this method, which combines the virtues of the power spectrum with the utility of its transform, the covariance function, can be a powerful and discriminating statistical test. Presented here are a number of theoretical results necessary for the analysis of data distributed on a sphere. Some results are close analogs of corresponding formulae for a distribution in flat space. Others are directed to the problem of relating the observed twodimensional distribution to the three-dimensional spatial distribution. Subject headings: galaxies - galaxies, clusters of
P. J. E. Peebles (Mon,) studied this question.