220 

 The answer is, (r>3), 



where the circulator s r = 1 or =0 as r is or is not =sm. 



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

 the (r + 1 )-edral pyramid. 



The solution is, 0>3), 





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 #-edra may, by a slight ex- 

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

 the ^-partitions of a pyramid ; and this depends on the problem, 

 to find the ^.-partitions of a polygon, and on this, which is nearly 

 the same question, to find the ^.-partitions of a pencil. 



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

 which Jc 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 p summits into a system of 1 + 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 k lines. The number 

 of ^.-partitions proper, for which z'=0, or of ways in which A-dia- 

 gonals can be drawn none crossing another, is 



fic+l .fk+2 ' 



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

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

 necting two pencils. 



