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The following work is founded on that conception of evolution, the most recent and precise formulation of which is due to Dr. J. C. Willis, and represents an attempt to develop the quantitative consequences of the conception. By his statistical studies of distribution Dr. Willis was led to two conclusions:— (1) Species occupying large areas are, on the whole , older than those occupying small areas, provided that allied forms are compared.

II.—A mathematical theory of evolution, based on the conclusions of Dr. J. C. Willis, F. R. S