PROFESSOR TAIT ON KNOTS. 169 
Let us confine our attention for a moment to clear coils. It is easy to see 
that 
Tf the number of windings is even the number of crossings is odd, ht vice 
versa. 
Various proofs of this may be given, all depending on the fundamental 
theorem of §1, but the following one is simple enough, and will be useful in 
some other applications. 
First, in a clear coil of two turns there must be an odd number of intersec- 
tions. For there must be one intersection, and the two loops thus formed 
must have their other intersections (if any) in pairs. 
Now begin with any point in a clear coil, where the curve intersects itself 
for the first time. The loop so formed intersects the rest in an even number of 
points. Hence every turn we take off removes an odd number of intersections. 
Thus, as two turns give an odd number (or, more simply, as one turn gives 
none), the proposition is proved. 
Thus, to form the symmetrical clear coil of two turns and of any (odd) 
number of intersections, make the wire into a helix, and bring one end through 
the axis in the same direction as the helix (not in the opposite direction, as in 
Ampére’s Solenoids), then join the ends. [The solenoidal arrangement, re- 
garded from any point of view, has only nugatory intersections. | 
§ 23. A very curious illustration of the irreducible clear coils which have 
two turns only is given by the edges of a long narrow strip of paper. Bend it, 
without twisting, till the ends meet, and then paste them together. The two 
edges will form separate non-linked closed curves without crossings. 
Give the slip one half twist (1.¢., through 180°) before pasting the ends 
together. The edges now form one continuous curve—a clear coil of two 
turns with one (nugatory) crossing. 
Give one full twist before pasting. Each edge forms a closed curve, 
but there are two crossings. The curves are, in fact, once linked into one 
another. (See Plate XV. fig. 13.) 
Give three half twists before joming. The edges now form one continuous 
clear coil with three intersections. 
Two full twists give two separate closed curves with four crossings, 7.¢., 
twice linked together. (See Plate XVI. fig. 12.) 
Five half twists give the pentacle of § 7 above. And soon. In all these 
examples, from the very nature oF the case, the crossings are ueutaie 
+ and — 
§ 24. Now. suppose that, in any of the above examples, after the pasting, we 
cut the slip of paper up the middle throughout its whole length. 
The first, with no twist, splits of course into two separate simple circuits. 
That which has half a twist, having originally only one edge, and that edge 
VOL. XXVIII. PART I. PD 4 
