330 
Proceedings of the Royal Society 
into links, by the removal of an intersection. For, to take off an 
intersection is easily seen to be equivalent simply to rubbing out 
one connecting line in the figure, and simultaneously diminishing 
by unity each of the exponents at its ends. If it be the only line 
connecting these exponents, they are (after reduction by unit 
each), to be added together. And this consideration enables us 
to obtain, even more simply than before, the rules for distinguish- 
ing a knot from a link. I propose, when I have sufficient leisure, 
to re-investigate the whole subject from this point of view. 
Meanwhile I may notice that it is exceedingly easy to draw the 
outline of any knot or link by this method. All that is necessary is 
to select a point in each of the lines in the figure, and join (two and 
two) all these points which are in the boundary of each closed area. 
The four lines which will thus be drawn to each of the chosen 
points must be treated as pairs of continuous lines intersecting at 
these points, and at these only. When there are only two sides — 
and, therefore, only two points — in an area, two separate lines must 
be drawn between them, and these must cross one another at each 
of the two points. 
The annexed diagram shows the result of this process as applied 
to the following symbol 
5 r— rf 5 
fy» 5 /v»5 
This method also clears up in a remarkable manner the whole 
