68 Royal Society :— 
Eigenschaften der algebraischen Curven,” Crelle, t. xlvii. pp. 1-6, 
in the particular case of a basis-curve of the third order; and I also 
found that the Pippian might be considered as occurring implicitly 
in my “Mémoire sur les Courbes du Troisisme Ordre,” Liouville, 
t. ix. p. 285, and “ Nouvelles Remarques sur les Courbes du 
Troisiéme Ordre,” Liouv. t. x. p, 102. As regards the Quippian, I 
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. 
“On the k-partitions of a Polygon and Polyace.” By the Rey. 
T. P. Kirkman, M.A. 
The problem relating to the polyace is the reciprocal of that 
relating to the polygon, and is not separately discussed. By the 
k-partitions of a polygon, the author means the number of ways in 
which the polygon can be divided by (A—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 w-edra having 
Triedral Summits and an (w—1)-gonal Base,” Phil. Trans. 1856, 
p- 399, he has, in fact, investigated the (7 —2)-partitions of the 7-ace 
or 7-gon: so that the present memoir may be considered as a com- 
pletion, or rather an extension and completion, of the investigations 
in his former memoir. The number of distinctions to be made in the 
problem of the present memoir is very great; thus, a partition of the 
polygon may be either reversible or irreversible; and if reversible, 
then the axis of reversion may be either agonal, monogonal, or dia- 
gonal, that is, it may pass through no angle, one angle only, or two 
angles of the polygon; and in the last case it may be either drawn or 
undrawn. Again, there may be a single axis or a greater number of 
axes of reversion: in the case of m such axes, the partition is said to 
be m-ly reversible ; and in like manner an irreversible partition may 
‘consist of a single irreversible sequence of configurations, or it may 
