I 



The Rev. T. P. Kirkman on Autopolar Polyedra. 461 



ntrally autopolar may always be so signed, that two pairs of gamics 

 shall exhibit in each quadruplet a duad of the form aa. rln the 

 above type it is observable that every duad, as 15, occurs four times. 

 The same thing is to be seen in every autopolar type of edges. 



If we make use of the closed 10-"gon 1239804567, as directed in 

 a paper "On the Representation of Polyedra," in the 14()th volume 

 of the Transactions of the Royal Society, a paradigm of this lO-edron 

 can be written out, exhibiting to the eye all the faces, summits, 

 angles, and edges of the figure. 



The problems following are next proposed and solved. 



To find the number of autopolar (r + 2)-edra generable from the 

 (r + \)-edral pyramid. 



The answer is, (r>3), 



i|(r-3r)4, + (r'-3r + 2)4,_, + (r-2;-3).2,_,|, 



where the circulator «^=1 or =0 as ?• is or is not =:«?«. 



To determine the number of autopolar (r + 3)-edra generable from 

 the (r + 1 )-edral pyramid. 



The solution is, (r>3), 



i./;.4_(j;.3+n^_36,.+ 24 + 9r.2, + (?-3 + 29r + 60)2_,]' 

 24 L J 



Hence it appears that there is one autopolar 6-edron, not a pyramid, 

 and five autopolar 7-edra besides the 7-edral pyramid, viz. three 

 generable from the 6-edral and two from the 5-edral pyramid. 



The problem of enumeration of the a;-edra may, by a slight ex- 

 tension of the meaning of partition, be stated thus : to determine 

 the k-paj-titions of a iryramid ; and this depends on the problem, 

 to find the 'k-partitions of a polygon, and on this, which is nearly 

 the same question, to find the k-partitions of a ij}encil. 



By the k-partitions of a ^>-gon is meant the number of ways in 

 which h lines can be drawn not one to cross another, and ter- 

 minated either by the angles of the polygon, or by points assumed 

 upon its sides or within its areas so as to break up the system of one 

 face and^ summits into a system of \-\-h faces and p-\-i summits, 

 where h-\-i=^k; it being understood that if a point be assumed 

 within the area, three lines at least shall meet in it, and if on a side, 

 one segment of it shall be counted among the /• lines. The number 

 of k-partitions proper, for which i=0, or of ways in which /-dia- 

 gonals can be drawn none crossing another, is — 



ik^-X .(1+2 ' 



which is also the number of ways in which a pencil of ^j rays can be 

 broken up into p-\-/c pencils, by the addition of k lines, each one con- 

 necting two pencils. 



