T11.—On Kwors, with a Census For OrpER TEN. by ONG 
Lirttx, Lincontn, N rs. 
1, Gauss in 1833* called attention to the importance of the study 
of the ways in which cords might be linked. Nothing, however, 
appears to have been written upon the subject until in 1847 Listing 
published his Vorstudien zur Topologie.t In this he briefly but in a 
masterly way touched upon the subject of knots, established some of 
the fundamental propositions, and proposed a notation which, as 
slightly modified by Prof. Tait, furnishes the point of view for the 
present paper. In a communication} to. Prof. Tait in 1877, Listing 
points out the fragmentary character of bis own contribution to the 
. subject, and says that the type-symbol used by him is “nichts weiter 
als ein derartiger Fingerzeig.” 
It is to Professor Tait, however, that the greater part of our pres- 
ent knowledge of the subject is due. He, independently of Listing, 
obtained the fundamental propositions and found the knots and their 
forms for orders from three to seven inclusive.§ 
In 1884 Kirkman || published the forms of knots of orders eight and 
nine, and imniediately Tait, making use of Kirkman’s work, extended 
his census of knots to these orders. 4] 
2. Professor Tait has shown that any closed plane curve of 7 cross- 
ings divides its plane into +2 compartments; that these compart- 
‘ments are in two groups; that, at the crossings, like compartments 
are vertically opposite. We shall call these compartments of the 
plane parts. A part is represented by the number equal to the num- 
ber of double points on its perimeter. The sum of the numbers 
representing the parts of either group is 2n, that is, these numbers 
together constitute a partition of 2n. The partitions for the two 
groups together make up Listing’s type-symbol. As it can lead to 
* Hine Hauptaufgabe aus dem Grenzgebiet der Geometria Situs und der Geo- 
metria Magnitudinis wird die sein, die Umschlingungen zweier geschlossener oder 
unendlicher Linien zu zihlen.”—Werke. Gottingen. 1867, vol. v, p. 605. 
+ Gottingen Studien, 1847. I have been able to see only Tait’s apparently full 
abstract in Proc. Roy. Soc. Edin., vol. ix, pp. 306-309. 
¢ Proc. Roy. Soc. Edin., vol. ix, p. 316. 
§ On Knots, Trans. Roy. Soc. Edin., xxviii, 145-191, 1876-77 
| Trans. Roy. Soc. Edin., xxxii, 281-309, 
4] Trans. Roy. Soc. Edin., xxxii, 327-342 
