AND RETICULATIONS OF THE R-GON. 231 
r' and x into the preceding sum 3, taken for all values of 
y and of e=r-—r', is the number 
22 (r,k£+1)+R*(7,4+1), 
when £ is odd. 
The sum 3, by the reasoning of Art. 4 is seen to be 
A(k-1)YI2 yk IAR-1)) IA 
=e SS C=», Dir $2 
y yyvit [2@—k—-5) | 1 tr" —k—5) [1 
wherefore, when & is odd, by Art. 6, 
P(r, 4+1) 
=1|3, Se nena ee el) haneeilaaer ere 
aie jie) |1 ]k0—#—k-5) | 
me) k+1)5 
gate onze Bok z | 
4. A ) 
he ee wee sp 
in? bof 
where #<{0, x<tr—2k-4, xpr-k-5, and all fractions 
irreducible are zeros, i.e. r and w are even and f is odd. 
14, When & is even, there is a central diagonal in L”. 
Of the 34% edges of quadrilaterals between this and the 
marginal triangle let all but y vanish, and of these y 
quadrilaterals let all the opposite edges but z vanish. 
Repeating these operations in order in the circuit in the 
other half of the figure, we complete 
lfjylo yell 
a . = figures, 
among which L’ is twice made, if the spared edges 
tk+z=4(r'-4), or 
z=}(r'-k-4), 
In each of these figures we can add 4 points in 4(r'—2) 
positions, viz., about the vertex of a marginal triangle, and 
in k=}(r'—4) consecutive sides of quadrilaterals. We 
repeat this distribution of 4a points in the other half of 
the circuit in order, and thus complete on each r’-gon 
(hat 1) br + —4)ir-4 1 
yr =O) ine Tarver tret! aeeitels figures, 
