296 Proceedings of the Royal Society 
simple loop passed unsymmetrically through a simple knot of 
three intersections (figures 1 and 3 of my former paper), and that 
the knot and loop are interchangeable between two groups of 
three intersections. The knot is right-handed when transferred 
to the second group ; left-handed in the first. But the figures 
plainly show that they may also be regarded as a right nnd a 
left-handed simple knot having a part common to each, so that 
neither can be pulled tight, subject to our convention that there 
shall be nothing higher than a double point. And here the peculiar 
difficulty associated with the amphicheiral forms comes in ; for, in 
estimating the electromagnetic work, we find we must leave out one 
copper and one silver junction — the result being + 877- - 877-. This 
is to be treated as ± 167 t (or two degrees of beknottedness) because 
the portions with different signs belong to what are, virtually at 
least, two separate knots.* 
The possibility of such amphicheiral forms is obvious from one 
of the first illustrations in my former paper; where we have 
only to suppose the irrelevant crossing removed, and one of the 
separate simple knots (which are both right-handed in the cut) 
made left-handed. But I was not at first prepared to find this 
property in any knot not separable into detached, self-contained 
portions ; so that it is possible that some of my former statements 
may require modification. 
It may be well to notice that when, in a slight variation of the 
* Feb. 19. — This is not correct. There is but one degree of beknottedness, 
for the two knots are not “ virtually separate,” as they have a part in 
common, while one is right-handed and the other left-handed. In fact, the 
figures above are mere transformations of the last cut in my former paper 
— which is shown to be capable of being opened up by a single change of 
sign. This can be done in the figures above, at either end of the lower 
coil where it forms part of the external boundary. But if, without altering 
the outline of the figure, we change all the signs in either of the two com- 
ponent knots, so as to make them both right-handed, or both left-handed, the 
whole acquires the double degree of beknottedness wrongly assigned to it in 
the text. But it has now continuations of sign, and in virtue of these it hap- 
pens to be reducible. In fact, when we make it into a clear coil after these 
changes of sign it becomes the pentacle (fig. 1 above), the only knot with 
fewer than six crossings which possesses, as we have seen, two degrees of be- 
knottedness. I stated in my first paper, that when the signs in any non- 
nugatory arrangement are alternately + and - the cord “ is obviously as 
completely knotted as its scheme will admit of.” This completeness must 
