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when designing Vortex- Atoms of various forms, and which I gave 
to Section A at the late meeting of the British Association. The 
second of these theorems (as will be seen by reference to my British 
Association paper, Messenger of Mathematics , Jan. 1877), was virtually 
the same as Listing’s division of the meshes of a reduced knot into 
right- and left-handed, — only I called them black and white , but. 
as I did not see how to connect this theorem directly with the measure 
of beknottedness, I did not formally introduce it into my papers read 
to the Society. It is, of course, virtually included in the state- 
ments regarding coins thrown into the corners of cells, — for, taking 
the case of the silver and copper coins, the pair of left-handed 
vertical angles are those in which, or in one of them, there is 
silver — the right-handed, copper. 
Nothing can be clearer than Listing’s statements on several 
parts of the subject : it is greatly to be desired that he had made 
many more. Still, with a cordial recognition of the great value of 
all that is to be found in Listing’s paper, I adhere to what I said 
in my last communication, to the effect that the full character of 
a knot cannot be learned except from its “ scheme,” or some- 
thing equivalent to it. That the type- symbol (when such a repre- 
sentation is possible) is ultimately equivalent to the scheme may 
possibly be true,* especially when we consider that it virtually 
contains two independent descriptions of a knot ( i.e . in terms of its 
right-handed and its left-handed meshes separately) ; that for pur- 
poses of classification it is superior is, I think, obvious, but I think 
it equally obvious that for the purpose of drawing the knot it is 
inferior. And the scheme for a reducible knot is quite as simple 
as that for a reduced one, while it is not easy to see what would be 
called the symbol of a reducible knot. Nor can I represent by 
* (Added Feb. 7.) — I have just found symbols for which this is not the case. 
The following single instance is sufficient, for the present, to show that the 
type-symbol is not always equivalent to the scheme. The symbol 
r 4 + 2 r 3 4- 2 r 2 ) 
P+P+P+P ] 
may represent either a continuous curve with 7 intersections, or a complex 
system consisting of a circle intersected at six points by a skewed figure of 
8. I shall discuss the subject fully in a paper “ On Links,” which I have 
in preparation. 
