BEING  THE  EIEST  CLASS  OF  E-GONOIJS  X-EDEA. 
161 
Here  the  30  edges  abc...za^yh  are  all  3535,  carrying  the  subindex  quadruplets 
112  2, 
7  4 8 11, 
13  3 14  6, 
19  1 20  11, 
15  9 11  6, 
2  8 3 2, 
8  5 9 11, 
14  7 15  6, 
20  1 1 12, 
1 2 5 12, 
3  7 4 2, 
9  5 10  10, 
15  7 16  9, 
18  8 2 1, 
20  12  7 11, 
4  3 5 2, 
10  5 11  9, 
16  8 17  9, 
16  7 3 8, 
19  11  9 10, 
5  3 6 12, 
11  5 12  6, 
17  8 18  10, 
12  5 8 4, 
17  10  10  9, 
6  4 7 12, 
12  4 13  6, 
18  1 19  10, 
13  4 6 3, 
4 7 14  3. 
The  closed  polygon  123...  20  is  di-awn  through  the  20  summits,  and  the  closed 
polygon  12 3..  12  through  the  12  faces.  If  any  five  continuous  edges,  of  which  no  three 
are  in  a pentagon,  be  made  to  convanesce,  as  the  five  asrpq,  the  8-ace  is  restored, 
6 triaces  thus  uniting  to  form  it.  The  disappearance  of  the  remaining  vanescibles  will 
restore  the  8-gon, 
Every  .r-edron  of  the  first  class,  i.  e.  every  partition  of  a pyramid,  can  be  thus  exhibited 
in  a paradigm,  and  the  greater  number  of  those  of  the  higher  classes. 
The  partitions  of  the  r-pyramid  have  all  this  property,  that  each  contains  a discrete 
r-gony,  i.  e.  an  r-gonous  system  of  vanescibles  of  which  no  diaped  meets  a diagonal ; or 
each  contains  an  r-gony  of  the  first  class.  Some  of  them,  however,  contain  also  a mixed 
r-gony,  on  which  are  one  or  more  angles  made  by  a diaped  and  a diagonal.  If  the  figure 
contain  a discrete  r-gony,  it  is  an  r-gonous  polyedron  of  the  first  class ; if  not,  it  is  a 
polyedron  of  a higher  class.  The  diapeds  of  a discrete  r-gony  form  a continuous  line  of 
convanescibles ; if  a diagonal  be  drawn  in  a face  about  one  of  these,  the  r-gony  is  no 
longer  discrete,  but  mixed. 
The  problems  that  are  next  to  be  solved  towards  the  completion  of  the  theory  of  the 
polyedra,  are  the  following ; and  I have  little  hope  of  their  solution,  in  terms  of  r. 
How  many  partitions  can  be  made  of  the  summits  of  a partitioned  r-gon,  so  that  no 
(/’-|-l)-gon  shall  be  introduced? 
How  many  partitions  can  be  made  of  the  faces  of  a partitioned  r-ace,  so  that  no 
(r+l)-ace  shall  be  introduced? 
In  how  many  ways  can  a partitioned  partition  of  the  r-ace  be  laid  on  a partitioned 
partition  of  the  r-gon,  so  that  no  (r+ej-gony,  nor  (r-|-e)-gon,  or  (r-f-e)-ace  shall  be 
introduced  ? 
MDCCCLVIII. 
z 
