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I. By the k-partitions of an r-gon, I mean the number of ways in which it can be divided by k — 1 diagonals, of which none crosses another; two ways being different only when no cyclical permutation or reversion of the numbers 1 2 3. .r at the angles can make them alike: and by the k-partitions of an r-ace (a pencil of r rays in space or a plane), I mean the number of ways in which it can be divided into k smaller pencils, by the introduction of k — 1 connecting lines, of which none enclose a space; two ways being different only when by no cyclical permutation or reversion of the numbers 1 2 3.. r in the angular spaces of the r-ace they can be made identical. The polygon here considered is the section of a pyramid, and its discussion includes that of the polyace. The enumeration of the partitions of the polygon and polyace is indispensable in the theory of the polyedra. In a memoir “On the x-edra which have an (x — 1)-gonal base, and all their Summits Triedral,” in the Transactions of the Royal Society, 1856, page 399, I have investigated the (r — 2)-partitions of the r-ace, or the r-gon; for the number of x-edra there determined is exactly that of these (r — 2)-partitions. What follows may be considered as a completion of the investigation in that memoir begun; yet not properly a continuation, inasmuch as the results there obtained are here deduced by a different and more general method.

XII. On the K-partitions of the R-gon and R-ace