384 
Proceedings of the Royal Society 
There are, of course, an infinite number of ways in which such a 
strip may be rolled up into a cylinder. But in whatever way the 
cylinder is formed, if we cut it along a generating line, and unroll 
it, we may take the parallel edges of the flat strip as the two lines 
defining, on the plane complex, the particular cylinder. We may 
move these two parallel lines, parallel to themselves, retaining their 
distance from one another, in any way, and they will still represent 
the same cylinder, because we may form this flat strip by cutting 
the cylinder through any generating line. Two parallel lines will 
therefore represent a cylinder if the points in which they inter- 
sect a line at right angles to them are always similarly situated in 
reference to the complex. 
The number of cylinders is obviously infinite, but they may all 
be grouped under two genera. For a part of a hole, cut off by one of 
the parallel lines, may, at the seam, find its continuation in a part 
of a hole, either first, in the same sheet, or second, in one of the 
other sheets. 
In the first case we have three distinct sheets locked together. 
In the second we have only one sheet wound three times round the 
cylinder, and knotted. When we have three independent sheets 
we can colour or shade them independently, each having its own 
colour or shading, but when there is only one sheet this is not 
possible. In this case the only way of distinguishing the layers is 
by varying the colour, or shading, continuously as we go round the 
cylinder, so that after three turns we come back to the colour or 
shading with which we started. This has been done in the models 
exhibited. 
We have assumed that the complex is flexible, we shall now 
assume that it is also extensible, so that we can draw it out in any 
particular direction, and make the circular holes into ellipses. We 
shall assume that any deformation may be produced without affect- 
ing the character of the complex as long as the topological relation 
of the layers is preserved. This extension is not of any use if we 
confine ourselves to cylinders, for there is no topological change 
produced by twisting a cylinder. The meaning of the extension will 
be seen when we come to apply the complex to an anchor-ring. 
An anchor-ring can be made out of a cylinder in two ways. We 
may cut the cylinder by two planes at right angles to the axis, and 
