311 



have not succeeded in obtaining a satisfactory geometrical definition ; 

 but the search after it led to a variety of theorems, relating chiefly 

 to the first-mentioned curve, and the results of the investigation are 

 contained in the present memoir. Some of these results are due to 

 Mr. Salmon, with whom I was in correspondence on the subject. 

 The character of the results makes it difficult to develope them in a 

 systematic order ; but the results are given in such connexion one 

 with another, as I have been able to present them in. Considering 

 the object of the memoir to be the establishment of a distinct geo- 

 metrical theory of the Pippian, the leading results will be found 

 summed up in the nine definitions or modes of generation of the 

 Pippian, given in the concluding number. In the course of the 

 memoir I give some further developments relating to the theory in 

 the memoirs in Liouville above referred to, showing its relation to 

 the Pippian, and the analogy with theorems of Hesse in relation to 

 the Hessian. 



VIII. On the ^-partitions of a Polygon and Polyace." By 

 the Rev. T. P. KIRKMAN, M.A. Communicated by 

 ARTHUR CAYLEY, Esq. Received November 13, 1856. 



(Abstract.) 



The problem relating to the polyace is the reciprocal of that 

 relating to the polygon, and is not separately discussed. By the 

 ^-partitions of a polygon, the author means the number of ways in 

 which the polygon can be divided by (k 1) diagonals, no one of 

 which crosses another ; two ways being different only when no 

 cyclical permutation or reversion of the numbers at the angles of 

 the polygon can make them alike : it is assumed that the polygon 

 is of the ordinary convex form, so that all the diagonals lie within 

 its area. The author remarks, that the enumeration of the partitions 

 of the polygon and polyace is indispensable in the theory of polyedra, 

 and that in his former memoir " On the Enumeration of a>edra having 

 Triedral Summits and an (x l)-gonal Base," Phil. Trans. 1856, 

 p. 399, he has, in fact, investigated the (; 2)-partitions of the r-ace 



