361 
of Edinburgh, Session 1884-85. 
in each be in an even place ; then the second 7 is in an odd place, 
art. 1. Let the thread cross over itself in the even places. In both 
circles (F) then the thread, passing under itself at d, 6, and r, goes 
onwards till it passes over itself at p, 6, and s, proceeding till it 
comes under itself again at d. Let now the double flap 765 in the 
projection of each circle he shrunk to a point 7, the four edges of 
its two 2-gonal loops 65 and 67 disappearing. These circles thus 
become the two unifllars of n - 2 crossings, 
. . . dir . . . pis . . . 
. . . dir . . . pis ... . . (G), 
which of course differ exactly as do the circles (F) in the portions 
omitted. The thread in each goes under at d, over at 7, under at r, 
and so on its course till it passes over at p, under at 7, over at s . . . 
and finally comes again under at d , as before the shrinking. And 
this is true whatever be the crossings d , r, p, s, in 2-gons or not, 
and whether these four be or not crossings in like meshes on the 
knots. Thus is proved 
Theorem C. — If any unifilar knot n K of n crossings has a double 
flap, the two contiguous terminal 2-gons of a (2 + i)-ple flap, (*>°)» 
it is reduced, by the shrinking up of the two 2-gons to a point, to 
an unifilar S of n - 2 crossings. 
7. Eeturning to the even 2-gon 12 of art. 1, it is plain that at its 
crossings it is covertical with two meshes whose edges, meeting at 1 
and at 2, will he dotted both of each pair inside or outside its mesh, 
according as the 2-gon 12 has its dots both outside or both inside 
of it. And it is equally plain that every crossing r of a knot has 
two covertical angles about it so evenly dotted, and another cover- 
tical pair which have both one, and only one, dot inside those 
angles about r. We may call the latter pair the odd angles, and 
the former the even angles about r. 
In the unifilar w _xM whose circle is (B), art. 1, we can reverse the 
process by which w _xM is obtained from i.e ., we can, by restoring 
in the projection of , t _iM the deleted 2-gon 12, construct upon it n ~N. 
In the circle (A) a28 and 315 are even angles about the crossings 2 
and 1. In (B), the circle of ^M, we read along 32 and next along 
28, dotting both on the right, and later along a 2 and 25, dotting 
both on the right ; i.e., a28 and 325 are even coverticals about 2. 
