326 Proceedings of the Royal Society 
The common type-symbol is 
r 5 + r 4 + r 3 + 2 r 2 I 
P+l* + P + 2P / 
But the schemes are 
A,E 6 B 4 G 8 C 6 H 6 D 6 B 4 E o A 0 F ! C e a 6 F 2 H li D 8 
and 
A 6 D 4 B 2 H 4 C 6 F 4 D 4 A 8 E 4 G 2 F 4 C 6 G 2 E 4 A ,b. 
Now no change in lettering can affect the suffixes, so that the 
two schemes are essentially different. In fact the sum of the suf- 
fixes is 84 in the first scheme, but only 64 in the second. The 
first has only one degree of beknottedness, the second has two. 
The first is not amphicheiral, the second is. 
There is no connection between the type-symbol, as Listing gives 
it, and the singleness or complexity of the curve represented, but it 
is possible to make analogous symbols capable of expressing every- 
thing of this kind. Only we must now adopt something very much 
resembling Crum Brown’s Graphical Formulae for chemical compo- 
sition. Some very remarkable relations follow from this process, but 
I can only allude to a few of the simpler of them in this abstract. 
The only necessary relations among the numbers forming the 
right or left part of a symbol are satisfied if no one is greater than 
the sum of the others, and if the sum of all is even. With any set 
of numbers subject to these conditions, we can form the right or 
left-hand side of a symbol-— and from that we can form the other 
when we know the grouping. 
An example or two will make this clear. Take, for instance, the 
symbol 
2r 4 + r 2 ) 
W + P J 
which represents the five-crossing knot of p. 242 above. 
A glance at the figure shows that the following is the arrange- 
ment of the right-handed meshes. 
2 
/ \ 
yA — — p4 
the single mesh with two corners having one of these corners in 
common with each of the two four-sided meshes, which again 
