238 
Proceedings of the Iioyal Society 
which the intersections are either wholly or partially nugatory, i.e ., 
not in reality contributing to the knot, whether on account of the 
order of their arrangement or their signs. All nugatory intersec- 
tions can be detected at once by the scheme itself, and may be 
struck out. As will be understood from what follows, the schemes 
AABBCC and ACBBCA 
are wholly nugatory, while in 
ACBDCBDAEGFEGF 
only the intersection A is necessarily nugatory. In fact a group 
like C B D C B D, when not itself nugatory by reason of its signs, 
is self-contained, and forms a special knot which may be drawn tight 
so as to present only a roughness in the string. The following 
sketches illustrate these essentially nugatory crossings: — 
I. Given the number of its double points, to find all the essen- 
tially different forms which a closed curve can assume. 
(a.) 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. 
(6) It cannot, however, be done at random. For instance, the 
scheme 
ADBECADBEC | A 
is lawful, but 
ADBACEDCEB | A 
is not. 
The former, in fact, may be treated as the result of superposing 
two closed (and not self-intersecting) curves, both denoted by the 
letters A D B E C A, so as to make them cross one another at the 
points marked B, C, D, E, then cutting them open at A, and joining 
the free ends so as to make a continuous circuit with a crossing at A. 
