385 
oj Edinburgh , Session 1885-86. 
bend the part thus cut oat round so that its axis becomes the core 
of the anchor-ring; or we may cut the cylinder, and then widen out 
the two ends, and bend them over so that they may unite and form 
a seam not about the core, as in the last mentioned case, but about 
the axis of the anchor-ring. An anchor-ring has thus two seams— 
one a circle with its centre in the axis, and one a circle with its 
centre in the core — -and it can be reduced to a cylinder, either by 
cutting the first, and, if we may coin the word, “unflyping,” or by 
cutting the second, and unbending. 
We see then that just as a cylinder can be represented by two 
parallel lines, so an anchor-ring can be represented by a parallelo- 
gram. The condition here is, that the parallel sides of the paral- 
lelogram cut the complex in precisely the same way. Such a 
parallelogram will in general represent two anchor-rings ; we must 
therefore indicate which of the two pairs of parallel lines represents 
the seam about the axis, and which the seam about the core, To 
transfer from this plane plan to an actual anchor-ring — that is, to 
make a model such as those shown to the Society— we have only 
to remember that the four corners of the parallelogram represent 
the single point in which the two seams intersect ; that the one pair 
of parallel sides represent the one seam, the other the other ; and 
that lines parallel to these pairs of sides are to be measured on the 
anchor-ring, in the one case along the circumference of a circle 
about the axis, in the other case along the circumference of a circle 
about the core. 
As there are two genera of cylinders, one knotted and one locked, 
so there are four genera of anchor-rings. 1st, locked about the 
axis and locked about the core ; 2nd, locked about the axis 
and knotted about the core * 3rd, knotted about the axis and 
locked about the core ; 4th, knotted about the axis and knotted 
about the core. Of these only the first, which is not knotted at all, 
consists of three distinct sheets; the second is reduced to a locked 
Cylinder by cutting it along a seam about the axis, to a knotted 
cylinder by cutting it along a seam about the core ; in the third 
these relations are reversed; in whichever way the fourth is re- 
duced to a cylinder, a knotted cylinder is produced. 
It is worthy of notice that anchor-rings of the fourth kind have 
VOL. XIII. 2 D 
