190 PROFESSOR TAIT ON KNOTS. 
return to the consideration of this very curious part of the subject, and at the 
same time to develop a method of treating knots altogether different from 
anything here given, which I recently described to the Society.* 
After the papers, of which the foregoing is a digest, had been read, I 
obtained from Professors Listinc and KLEIN a few references to the literature 
of the subject of knots. It is very scanty, and has scarcely any bearing upon 
the main question which I have treated above. Considering that Listine’s 
Essay was published thirty years ago, and that it seems to be pretty well 
known in Germany, this is acurious fact. From List1ne’s letter (Proc. R. S. E. 
1877, p. 316), it is clear that he has published only a small part of the results of 
his investigations. KLeInt himself has made the very singular discovery that 
in space of four dimensions there cannot be knots. 
The value of Gauss’ integral has: been discussed at considerable length by 
BoEDDICKER (by the help of the usual co-ordinates for potentials) in an Inaugural 
Dissertation, with the title, Bectrag zur Theorie des Winkels, Gottingen, 1876. 
An Inaugural Dissertation by WEITH, Topologische Untersuchung der Kurven- 
Verschlingung, Ziirich 1876, is professedly based on Listine’s Essay. It con- 
tains a proof that there is an infinite number of different forms of knots! The 
author points out what he (erroneously) supposes to be mistakes in LisTING’s 
Essay; and, in consequence, gives as something quite new an illustration of the 
obvious fact that there can be irreducible knots in which the crossings are 
not alternately over and under. The rest of this paper is devoted to the 
relations of knots to R1EMANN’s surfaces. 
* Proceedings, R. 8S. E., May 7th, 1877. 
+ Mathematische Annalen, IX. 478. 
+ Professor FiscHpr has just shown me an enlarged copy of Borppicker’s pamphlet above 
mentioned. Twenty pages are now added, mainly referring to the connection of knots with Rizmann’s 
surfaces, and the title is altered to Erweiterung der Gauss’schen Theorie der Verschlingungen. 
Stuttgart, 1876. 

ERRATA. 
Page 167. In the lower group of figures the numbers II. and IV. are to be interchanged. 
Also the cut marked III. is merely a form of V. The correct symbol for 
III. differs from these by having the 2 between the two lines joining 5 and 4. 
», 185. Interchange the words “first” and ‘‘second ” in lines 20 and 25 respectively. 
rot 
