AND ENUMERATION OF POLYEDRA. 69 



abcde-fg--'- --'dhij'k-- — 



l^-^.^.ahijk a-bc'-'def-g 



-cw-------^*w -/cAw------- 



-_---o^A/>--- -'---'ken- — 



--dq-----jn- rm-'io-'-'pa 



.--erp-'tq-' j n - f q o - 



The rank of the faces is stated thus, in the order of rank, 

 7744443 3, 7555443 3, 



7654443 3, 7655433 3. 



The two last are the 8-edra (a) and (A) of (16) and (17). 



In the above arrangements every duad of adjoining letters, 

 as dh and kd in the pentagon dh?jk of (A), which is read 

 horizontally, is adjoined also vertically : dh is an angle in that 

 pentagon, and the same edges d and h meet at the summit 

 dch. The first and final letters of a multiplet are considered 

 to be adjoining letters. 



The total number of octaedra, whose summits are all 

 triedral, is thus shown to be 14, viz., four on a heptagonal, 

 eight on a hexagonal, and two on a pentagonal base; and 

 the total number of 8-peaks having only triangular faces is 

 14, viz., four having a 7-edral, eight having a 6-edral, and 

 two having a 5-edral summit. 



The reader will gladly forego the consideration of the cases 

 in which ny8, to say nothing of the polyedra whose summits 

 are not all triedral. 



22. The geometrical problem — how many n-edrons are 

 there ? — reduces itself to the combinatorial problem : — In how 

 many ways can multiplets, «. c, triplets, quadruplets, &c., be 

 made with n symbols, under these two conditions : first, that 

 every contiguous pair of symbols in any multiplet shall be a 

 contiguous pair in some one other, the first and last of a mul- 



