AND RETICULATIONS OF THE R-GON, PP # 
its circuit; for if any face has three, sections along them 
will cut off three polygons having each at least one angle 
outside its section unoccupied by a diagonal, because 
every polygon has at least one such angle above its base. 
But F and F’ have each only two such angles ; wherefore 
no face of either has more than two diagonals, and conse- 
quently no face of F’ is ampler than a quadrilateral. 
3. If we pare away all the margin of I’, all its faces, 
except the two marginal triangles, will become quadri- 
laterals. F’ is now F”, a (2k+4)-gon, having / quadri- 
laterals, two triangles, and k+1 diagonals. Let all the & 
consecutive edges ab, bc, cd---- of quadrilaterals vanish 
except y, and in the y remaining quadrilaterals let all the 
opposite edges a’b’, U’c’, c'd’.-++ vanish except z; there 
are now only z quadrilaterals. This can be done in 
Ba is different wa 
yit % yelr ye. 
In order that F’ should be one of the resulting figures, 
it is necessary that the spared edges, together with the 
four in the marginal triangles, should be 7’. That is, 
k+z=r —4, or 
z=r —k-A, 
Wherefore F’ is one among 
kia yi 
Y qyll ” [r—k-4)1 figures, (y>09). 
4. To find this sum, let 7’ —-k-4=0@; then all terms 
vanish in which y <0, and the series is 
Kol—-2 Geil orti-1_ (9 41)81-2 KeHAI (94 2) 
-—— + 
yet. ei yoraia eta yoeeiaz yet 
“Fase Ovi. 94] eel (+2)?! 
TO aes Ce eee yer2ii pu 
Ko+3i-l (9 4.8)314 
yo+3i1 pot et 
BIA k= 8 O41, (K-O)(R-O-1) +1042) 
ea a+) yk (9+ 1)(0+2) 1-2 
