pin yr |-l Ark 1 
ee ee Oe ee 
Yqyll © yr-k-4i1 yr—k-4 11 ? 
is the number of figures F’, which all differ at least in 
their position, all being placed so as to have their two 
marginal triangles on either hand. 
5. The r-gon F is made from the 7”-gon F” by depositing 
z points (7’+2=r) in any one or more of 7” —2 positions, 
viz., about the vertices of the two marginal triangles of 
F’, and on the »” —4 other edges of I’. This can be done 
in 
(w+1)"-311 at 7-3" 3314 
rsa poe different ways ; 
and the product of this number into &, just found, taken 
for every value of r=r-—7", is the entire number of r-gons 
F. That is,* if R?(r,4+1) denote the number of these 
r-gons which have k+1 diagonals and two axes of rever- 
sion, R(r,*+1) that of the 7-gons F which have one axis 
of reversion, and [?(r,k+1) that of those which are doubly 
irreversible, that is, which have no axis of reversion, but 
have a sequence of configuration repeated in the circuit; . 
and if I(r, +1) denote the number of these F which have 
no symmetry, but are irreversible, i.e. have their lower 
face different from their upper, we obtain, by writing r-—wx 
for 7’, 
* An example of the class R?(7,2) is formed by placing two triangles on 
opposite sides either of a square or of a hexagon. An example of R%(r,2) 
is got by placing two triangles about a hexagon, so that one side of it shall 
stand betweenthem. One of R% (7,2) is made by placing two triangles about 
a hexagon, so that no side shall be between them. Two triangles placed 
on any two sides of a pentagon form an example of R™°(r,2); and ifa 
diameter be drawn which is not one of the two axes in any of the class 
R%(r,2), it becomes one of I?(r,3). — Vid. Art. (8, 9,10, 11). The sym- 
metry of these figures is fully discussed in the memoir referred to in the 
first article, in the Philosophical Transactions. 
