150 PROFESSOR TAIT ON KNOTS. 
— in itself, and the two knots may be put together, so that this condition is satis- 
fied in the partial schemes, but not in the whole. As there cannot be a knot with 
fewer than three crossings, we do not meet with this difficulty till we come to 
knots with six crossings. And as there can be no linking without at least two 
crossings, we do not meet with linked knots on the same string till we come to 
eight crossings at least. 
§ 5. We are now prepared to attack our main question. 
Given the number of its double points, to find all the essentially different 
Sorms which a closed curve can assume. 
Going round the curve continuously, call the first, third, &c., intersections 
A, B,C, &c. In this category we evidently exhaust all the intersections. The 
complete scheme is then to be formed by properly interpolating the same letters 
in the even places ; and the form of the curve depends solely upon the way in 
which this is done. 
It cannot, however, be done at random. For, jirst, incited A nor B can 
occur in the second place, B nor C in the fourth, and so on, else we should have 
necessarily nugatory intersections, as shown in § 2. Thus the number of pos- — 
sible arrangements of z letters (viz.,.—1.... 2.1) is immensely greater than 
the number which need here be tried. But, secondly, even when this is attended 
to, the scheme may be an impossible one. Thus, the scheme 
ADBECADBEC|A 
is lawful, but © 
ADBACEDCEB|A 
is not. 
The former, in fact, may be treated as s the result of superposing two cloned 
(and not self-intersecting) curves, both denoted by the letters A D BEC A, so 
as to make them cross one another at the points marked B, C, D, E, then cut-— 
ting them open at A, and joining the free ends so as to make a continuous 
circuit with a crossing at A. 
But in the latter scheme above we have to deal with the curves ADB A 
and C EC E, and in the last of these we cannot have junctions alternately + 
and —:as required by our fundamental principle. In fact, the scheme would — 
require the point C to lie sommes inside and outside the closed circuit 
ADBA. 
Or we may treat ADBA and CEDCas closed curves infersecuae one 
another and yet having only one point, D, in common. . 
Thus, to test any arrangement, we may strike out from the Fale sched q 
all the letters of any one closed part as A——A, and the remaining letters 
must satisfy the fundamental principle, z.¢., that they can be taken with suffixes — 
+ and — alternately, or what comes to the same thing) that an even number — 

