Let A+2 be the number of the diagonals ; and let 
R-{r,k-\-2), R'«{r,h-\-2), /c+2), /^+2) P(r,k+2\ 
/(?•, /c+2). 
be the numbers of the five classes of (A+3)-partitioned 
r-gons, having only two marginal faces, both triangles, having 
every other face either a triangle or a quadrilateral, and having 
every angle except two, viz., one in each marginal triangle, 
occupied by a diagonal. The first class has two axes of 
reversion, and has no triangles except the marginals. The 
next three have one axis of reversion, agonal, diagonal, or 
monogonal. The fourth has a repeated irreversible sequence ; 
the fifth is simply irreversible, that is, has its upper surface 
different from its lower, and has no symmetry of repetition. 
The expression of these numbers, in which no figure is 
enumerated which is the repetition or the reflected image of 
another, is the following : 
(,._6— 2A-)- (r—^—2ky 
72+r, 7c+2)=0 +0 ; 
^+2)=S; 
1 
y I -1 i(r-6-A) | -1 
m . 1/ 
12/ 1 1 
lj(r— 6— A)1 
where r and k are even ; y = or -7 0 ; y not 7^(r — 6) ; 
y 1-1 
jR 
(^5 /'^ + 2) ]^_pQi(r-4)— ) jy I 1 
(■|(/c+l)) y 
i(r-k-S) I -1 
li(r-k-6) I 1 
where r is even and k is odd ; y = or 7 0 ; y not 7 > 
y I —1 1 —1 
/c+2) = S (4^) y 
y 12/|1 * li(r-*-5)|l ’ 
where k is even and r is odd ; y = or 7 0, not 7|(?‘ — 5) ; 
I\r, A-+2)=72"^(r, k\2) + ^+2) ; 
/(r,/t+2) = i [s- ^ y 
r—h-& I —1 
1 _j_04(»'— y / 12/ 1 1 ' if— A— 5 1 1 I 
(i?-+i2"‘'+i2''‘+i2"'‘’+/^) (f, /:+2)J, y — OX -7 0,not 7 \{r — 4). 
