cw) 
~ 
ie) 
AND RETICULATIONS OF THE R-GON. 
{R4+2(R+1P) +41} (7,441) 
kr—-*-k-4 {—1 (r est 3) r—«—3 | —1 
® r-a—k-4 |1 4 yr-73 2 
» Qurtats , 
where z has every value that gives a result, namely, 
at0O, etr—-2k-4, wpr—-k—-A. 
For each one of the doubly reversible will be once only 
constructed, having but one possible posture about either 
axis; each one of the R will be twice made, once with 
either extremity of its axis uppermost; each of I? will also 
be twice constructed, to show both surfaces; and each of 
the I will be four times made, with either marginal face 
on the right hand, and with either surface uppermost. 
I cannot find that this sum 2, has any simpler expres- 
sion than the finite series which it represents. 
By this formula, when R*(r,k+1), R(r,k+1) and 
IP’(r,k+1) have been otherwise determined, we easily 
obtain the number of the largest class, the class I(r, +1) 
of these (k+2)-partitions of the r-gon which have two 
marginal faces. 
6. Let the r-gon E be one of the number R?(r,k+1). 
This E has a certain number 2+2 of angles unoccupied 
by a diagonal; and, by the vanishing of x of these, E 
becomes the 7’-gon E’ (7’=r-—wz), having two marginal 
triangles. As the 2 points were symmetrically placed 
with respect to two axes of symmetry, their disappearance 
has not disturbed the symmetry, and E’ has the same two 
axes of reversion. All the k+1 diagonals of E’ are pa- 
rallel to one and perpendicular to the other of these axes, 
and EH’ is made up of & quadrilaterals and two marginal 
triangles, and 7’ =2k+4. 
When £ is odd, 7’ =4i +6, and there is a central quadri- 
lateral, on each side of which stand }(k—1) others. We 
turn E’ into E by adding @ points in the circuit so as 
to preserve the symmetry about the two axes. When 
v=4j+2, two of these will be placed at opposite mid- 
