362 
Proceedings of the Royal Society 
By making these two covertical with a 2-gon we construct n N, whose 
circle is (A), and thus demonstrate 
Theorem AA. — If at any crossing r of an unifilar knot of n - 1 
crossings we make the two even angles of r covertical with a 2-gon, 
thus adding an edge to each collateral of the 2-gon, we construct an 
unifilar of n crossings by one of its even 2-gons. 
This is the constructing converse of the Theorem A, art. 3. 
8. Let 
v . . . fcdr . . . bam . . s 
be the circle of an unifilar of h crossings, in which cd and bci are 
edges of the mesh H, both dotted inside H at their mid-points a in 
cd and /3 in ba. The circle is read 
v . . fc{a)dr . . . b(/3)am . . s, 
say from c above to d below with the dot (a) on the right in H, till 
we come to the crossing b, and proceed from b below to a above past 
the dot (/?) on the right in the same H. In H draw the 2-gon pq from 
p between (a) and d to q between b and (ft). We now read from 
c above, not to d but to p below, having the dot (a) on our right in 
H • then crossing the upper edge pq of the 2-gon, we proceed along 
the lower pq to q, planting a dot (e) inside the 2-gon on that pq ; at 
q, again crossing the upper pq , we proceed to a past (/3) on our right 
in II, next to m, &c., completing the smaller circle 
v . . . fcpqam ... 5, 
which contains neither the upper edge of the 2-gon pq, nor any of 
the crossings dr ... b. We have constructed a bifilar knot of h + 2 
crossings, and demonstrated 
Theorem D. — If in any mesh H of an unifilar knot n T we con- 
nect by a 2-gon two mid-edges that are dotted either both inside or 
both outside of H, we complete a bifilar n+2 U. 
Observe that the dotting of art. 1 may be done either on the right 
or on the left of every edge. 
9. If in the unifilar n _ 2 Q of art. 5 we replace in the (2 + r)-gonal 
face F the 2-gon 12 effaced from n P, art. 4, we reconstruct the uni- 
filar n P. It follows that of the mid-points 1 and 2 of that face 
F on M _ 2 Q one, and only one, is dotted inside F. For if otherwise 
