311 
of Edinburgh , Session 1876-77. 
Listing’s notation the double trefoil knot which has appeared in 
each of my papers ; for, although irreducible (at least so far as I 
am aware), it contains several meshes which have angles of essen- 
tially different characters. Listing’s avowed object was to simplify 
notation as far as possible. My impression is that, in one respect 
at least, he has carried simplification a little too far; for it cost 
me some little time and trouble to draw, from his type-symbol, 
the one knot which he speaks of, but leaves undrawn, viz., as 
above, — 
2r 4 + 3r 2 ) 
2Z 5 + 2l 2 J 
Here is one of its forms: transformation will give the four 
others. 
In fact the type-symbol, even in this specially simple and symme- 
trical case, where it is much condensed, contains just as many sepa- 
rate typographical characters as the scheme ; and I think there can 
be no doubt whatever that it is almost incomparably more easy to 
draw the figure from the scheme than from the symbol. Given the 
scheme, the symbol can be formed from it in a moment; while the 
finding of the scheme from the symbol is very troublesome. But in 
such a matter experience is the only guide, and I have had almost no 
practice in trying to draw from the symbol. Listing’s type-symbol 
leads directly, however, to an inquiry not even suggested by the 
scheme; (for the latter, as I have given it, is essentially confined to 
a single closed curve), — viz., the forms of more than one closed 
curve, intersecting one another or not, which jointly divide an 
unlimited plane into given numbers of meshes with given numbers 
of sides. 
