
170 PROFESSOR TAIT ON KNOTS. 
not being cut through in the process of splitting, remains a closed curve. It is, 
in fact, a clear coil of two turns, which, having only one intersection, may be 
opened out into a single turn. But in this form it has two whole twists, half a 
twist for each half of the original strip, and a whole twist additional, due to 
the bending into a closed circuit. 
That with one whole twist splits, of course, into two interlinking single 
coils, each having one whole twist. 
That with three half twists gives, when split, the trefoil knot, and when 
flattened out it has three whole twists. 
From two whole twists we get two single coils twice linked, each with two 
whole twists. This result may be obviously obtained from a continuous strip, 
with only half a twist. One continued cut, which takes off a strip constantly 
equal to one quarter of the original breadth of the slip, gives a half twist ring 
of half breadth, intersecting once a double twist ring of quarter breadth. <A 
second cut splits the wider ring into one similar to the narrow one, but there is 
now double linking. 
§ 25. A good many of these relations may be exhibited by dipping a wire, 
forming a two-coil knot, into PLATEAU’s glycerine soap solution, and destroying 
the film which fills up the clear interior of the coil. Neglecting the surface 
curvature of the remaining film, it has twists similar to those of the paper 
strips above treated, and the integral amounts of twist show how far the wire- 
knot is, if at all, reducible. 
This mode of regarding a clear coil of two turns, as, in certain cases, the 
continuous edge of a strip of paper whose ends are pasted together after any 
odd number of half twists, is one of many ways in which we are led to study 
all clear coils as specimens of more or less perfect plaiting, the number of 
threads plaited together being the same as the number of turns of the coil. 
Another mode in which we are led to the same way of regarding them is by 
supposing a cylinder to be passed through the middle of the (flattened) clear 
coil, and then to expand so as to draw all the turns tight. As there can be 
only a finite number of intersections, we have always an infinite choice of gene- 
rating lines of the cylinder on which no intersection lies. ‘Suppose the whole 
to be cut along such a line and rolled out flat. It would, of course, be a more 
or less perfect plait, but with a special characteristic, depending upon the fach 
that 7t is formed from one continuous cord or wire. 
Call the several laps of the cut cord a, B, y, &c. Then we may arrange 
the cut ends anyhow as follows :—a to y, y to «, « to B, B to 4, 6 to a if there be 
but five; and similarly for any other number, exhausting all before repeating 
any one oftener than once. We may now, after having settled their order, 
change their designations, so as to name them, as they occur, in the natural 
order of the alphabet. Thus any such plait may be represented by a diagram 
