* 146 PROFESSOR TAIT ON KNOTS. 
kinetic stability, and thus to get the sixty or seventy permanent forms required 
for the known elements, we may have to go to a very high order of complexity. 
This leads to a physical question of excessive difficulty. THomson has briefly 
treated the subject in his recent paper on “ Vortex Statics,”* but he cannot be 
said to have as yet even crossed the threshold. But secondly, stable or not, are 
there after all very many different forms of knots with any given small number — 
of crossings? This is the main question treated in the following paper, and it 
seems, so far as as I can ascertain, to be an entirely novel one. 
When I commenced my investigations I was altogether unaware that any- 
thing had been written (from a scientific point of view) about knots. No one 
in Section A at the British Association meeting of 1876, when I read a little 
paper on the subject, could give me any reference; and it was not till after I 
had sent my second paper to this Society that I obtained, in consequence of a 
hint from Professor CLERK-MAXWELL, a copy of the very remarkable Essay by 
Listine, Vorstudien zur Topologie,t of which (so far as it bears upon my present 
subject) I have given a full abstract in the Proceedings of the Society for Feb. 
3, 1877. Here, as was to be expected, I found many of my results anticipated, 
but I also obtained one or two hints which, though of the briefest, have since 
been very useful to me. Listine does not enter upon the determination of the — 
number of distinct forms of knots with a given number of intersections, in fact 
he gives only a very few forms as examples, and they are curiously enough 
confined to three, five, and seven crossings only; but he makes several very 
suggestive remarks about the representation of knots in general, and gives a 
special notation for the representation of a particular class of “ reduced ” knots. 
Though this has absolutely no resemblance to the notation employed by me for 
the purpose of finding the number of distinct forms of knots, I have found a — 
slight modification of it to be very useful for various purposes of illustration 
and transformation. This work of Listine’s, and an acute remark made by 
Gauss (which, with some comments on it by CLERK-MAXwELL, will be referred 
to later), seem to be all of any consequence that has been as yet written on the © 
subject. I have acknowledged in the text all the hints I have got from these 
writers ; and the abstract of Listina’s work above referred to will show wherein 
he has: anticipated me. 
Parr i 
The Scheme of a Knot, and the number of distinct Schemes for each degree 
of Knottiness. 
§ 1. My investigations commenced with a recognition of the fact that in ~ 
any knot or linkage whatever the crossings may be taken throughout alternately 
* Proc. R. S. E. 1875-6 (p. 59). + Gottinger Studien, 1847. 

