of Edinburgh, Session 1884-85. 
363 
we should construct (Theo. D) by the 2-gon 12, not n P unifilar, but 
a bifilar „U. Thus we have proof of 
Theorem BB. — If in the projection of any (2 + r)-gonal mesh 
F of an unifilar of n - 2 crossings we connect by a 2-gon the mid- 
points of two edges of which one, and only one, is dotted (art. 2) 
inside F, we construct an unifilar of n crossings by one of its odd 
2-gons. 
This is the constructing converse of Theorem B, art. 5, and is 
true when r>0. 
10. The constructing converse of Theorem C is 
Theorem CC. — If in any unifilar knot of n - 2 crossings we make, 
at any projected crossing r, either pair of opposed angles co vertical 
with a double flap, adding two edges to each of the other pair of 
opposed meshes about r, we construct an unifilar of n crossings by 
one of its plural flaps. 
Ho base which has a plural flap, not fixed, can be operated upon 
by Theorem AA or BB, unless the operation abolishes the plural 
flap. And every flap, single or plural, is fixed, if its deletion lays 
bare a section through two edges only. Such a fixed flap cannot 
compete for the lead, nor hinder an operation by AA or BB, which 
does not abolish the fixture. Every construction is by a leading 
flap ( vide my paper, xvii.), and the leader has the most 2-gons. 
2, On the Twists of Listing and Tait. By the Bev. Thomas 
P. Kirkman, M.A., F.R.S. 
In the figure 1 following, the knot 9 Aj has a triangular section 
P rr cutting away on each side of it a (3 + r)-gonal mesh, and 9 Ar 2 * 4 
has such a section Ry?p, through one crossing only. Make in these 
knots creases at rr that approach to meet at R in 2, and creases pp 
that meet at P in 2. In 3, 2 is prepared for a rotation through 
two right angles about the fixed axis PR, through the crossing P 
and the crease-kiss R, or, as a more learned man would say, through 
the decussation P and the plicatorial osculation R — taking 2 for 
9 A j : exchange here P and R if 2 is 9 A r 2 . In 4 the rotation is 
effected, undoing the crossing P of 3, which has become a kiss P' in 
4, while the kiss R of 3 has become the crossing R' in 4, if 3 is 
9 A j : exchange here P and R, if 3 is 9 Ar 2 . In 5 the crease kiss is 
VOL. XIII. 2 B 
