312 
Proceedings of the Royal Society 
The first idea of this was suggested to me when I endeavoured to 
draw the curve with six intersections, whose type symbol is 
This symbol obviously satisfies the three numerical conditions; 
but, on trying to draw the corresponding figure, I found that it 
always came out as a species of endless chain of three separate 
links. One of its forms, from which the others can be found by 
transformation, is three circles, every two of which intersect. 
Two figures of 8, linked together at each end. give the symbol 
And by shifting the twist from one to the other, as explained in 
I have not as yet studied the theory of type-symbols, as it differs 
so much from my own method; but it is obviously desirable to find 
the criterion by which to distinguish from one another the type- 
symbols necessarily denoting one closed curve, and those neces- 
sarily denoting two or more intersecting curves. It is probable that 
there are symbols which may represent either kind of figure. The 
inquiry will no doubt be found very simple, if only approached from 
the proper side. 
I now pass to the sole point of Listing’s paper which (so far as 
knots are concerned) was thoroughly new to me, though not unex- 
pected ; and I shall lead up gradually to the special case which he 
gives, using for the purpose the properties of the amphicheiral knots 
mentioned in my last paper. 
To apply the amphicheiral property to the production of new 
forms, we may begin by studying under what conditions the internal 
arrangements of a knot can be altered while four points of its con- 
tour, and two of the parts of the cord or wire joining pairs of these, 
are fixed. [The reason for the number four is, that when two only 
3r 4 
2*3 + 3* I 2 
the latter part of this paper, the symbol may be changed to 
r 4 + 2r 3 + r 2 
2Z 4 + 2 1 2 
}■ 
