of Edinburgh, Session 1885 - 86 . 
383 
Each sheet is perforated by equal circular holes so arranged that 
any three neighbouring holes in the same sheet have their centres at 
the apices of an equilateral triangle. The radius of the holes must 
not be greater than half the distance between the centres of two 
neighbouring holes, otherwise the sheet would be cut into separate 
pieces, and must not be less than one-third of the said distance, 
otherwise there would not be room for the neck between two holes 
in one sheet to pass without crumpling through the chink caused by 
the overlapping of the holes in the other two sheets. In the figure 
the radius of the holes is about two-fifths of the distance between 
the centres. 
The complex of three sheets is, as will be seen by inspecting the 
figure, a case of what Professor Tait calls locking. No two sheets are 
linked together ; if any one sheet be abolished the other two come 
apart. Each sheet lies wholly above one of the other two, and 
wholly below the other. 
The analogy of this complex to what we may call the Borromean * 
rings will be seen at once. In the Borromean rings figured below 
(fig. 2), each ring lies wholly above one of the other two, and wholly 
below the other, so that while all are in- 
extricably locked together, no two are linked, 
and if any one is abolished the other two 
come apart. 
The complex of sheets may be applied 
to other surfaces besides the plane. Two 
other surfaces, viz., the cylinder and the 
anchor-ring, will be considered here. 1% 2 - 
To apply the complex to a cylinder or to clothe a cylinder 
with the interlaced sheets, we must cut the complex by two parallel 
lines, and roll up the strip thus cut out so that the two edges 
shall join and form what may be called the seam. But there 
must not be any peculiarity at the seam; the pattern must run 
through the seam without any discontinuity; therefore the two 
parallel lines must cut the complex in the same manner, so that 
each part of a hole divided by one line may find its exact con- 
tinuation at the seam when the strip is rolled up. 
* The interlocked rings shown above occur in the armorial bearings of 
the Italian family Borromeo. 
