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I. As the partitions of anr -gon are made by drawingdiagonals , so the partitions of anr -ace are made by drawingdiapeds , each a line in two faces non-contiguous about ther -ace. A partitionedr -ace standing on a partitionedr -gon is a partitioned pyramid ofr -gonal base and vertex. I am about to determine the number of such partitions of thisr -pyramid, that can be made with K diapeds andk diagonals, so that no two partitions shall be syntypous;i. e . one the repetition or the reflected image o f the other. I have proved in a memoir “On Autopolar Polyedra” in the Transactions of the Royal Society for 1857, that the problem of the polyedra reduces itself to the determination of thex -edra generable from ther -pyramid. Such anx -edron isr -gonous.

VII. On the partitions of the R -pyramid, being the first class of R -gonous X -edra