70 REV. T. P. KIRKMAN ON THE REPRESENTATION, &c. 



tiplet bein^ reckoned a contiguous pair; secondly, that no 

 three symbols of any multiplet shall occur in any other? 



Every system of such multiplets made with n symbols 

 represents a distinct ?z-edron, of which the faces, the summits, 

 and the edges, can be assigned. If the symbols stand for 

 faces, the multiplets are summits ; and if the symbols con- 

 tiguous to any symbol A be written out in order, as 



aAb bAc cAd dAe eAf, &c., 

 the edges about the face A are in order 



aA bA cA dA, eA /A, &c. 



If the symbols stand for summits, the multiplets are faces, 

 and the edges in order about the summits are determined in 

 exactly the same manner. 



It is easy to deduce from Theo. I., that, if ct^ be the num- 

 ber of e-gonal faces, and Se that of the e-edral summits, of any 

 jo-edron, am and Sn being the number of the most-angled, 

 a3+*3— 8 — S5—Se— . . . — Sn~i—Sn=a5-h2a6+da'j+ . . . + 

 (7/2 — \—4)am-i+(m~4)am> 



This affirms, that the total number of jo-edrons, having ag 

 triangular faces, and ^3 triedral, S5 5-edral, . , . Sn n-edra\, 

 summits, and their most-angled face m-gona\, and differing in 

 the numerical values of 035 &c., and 53, &c., cannot exceed 

 that of the (m — 4)-partitions of the number M, =03+^3 — 8 

 — S5 — Se — . . . — Sn ; i. e., of the ways in which M can be 

 made up of m — 4 numbers written in order; or, which is the 

 same thing, if myS, of the ways in which M can be divided 

 into X parcels, none of which contains more than m — 4, while 

 one parcel at least contains m — 4. The subindex 4 does not 

 appear in the above equation : this is curious ; but a geome- 

 trical reason can easily be assigned for its disappearance. 



