315 
of Edinburgh, Session 1876-77. 
Thus, it will easily be seen that the figures below are mere distor- 
tions of fig. 2 above ; and that, the dotted parts being fixed, they 
can be changed on an actual cord into one another, and even 
reversed (as from left to right), thus giving four distinguishable 
forms, of which I figure only two,-— 
Each of these may be changed without undoing the fixtures to its 
reverse (from left to right) by the first simple process just described, 
the loop, in fact, being transferred from one branch of the (undotted) 
cord to the other. 
Of course the number of ways in which the dotted part can be 
varied is infinite. I therefore give here only that which reproduces 
the forms already quoted from Listing as equivalent. 
r 5 + 3r 3 ) f 2r 4 + 2 r 3 
Z 4 + 2Z 3 -f 2P j an tZ 4 + 2t 3 + 2Z* 
which are those already quoted from Listing. 
In fact, looking at Listing’s figures above, we see that in each 
there is a part of the curve, containing four crossings, exactly the 
same as one or other of the two (partly dotted) distortions of the 
4-crossing knot above. 
The paper contains many other instances of these applications of 
amphicheiral forms. 
In conclusion, it appears that the problem of finding all the abso- 
lutely distinct forms of knots, with a given number of intersections, is 
a much more difficult one than I at first thought; and it is so because 
the number of really distinct species of each order is very much less 
than I was prepared to find it. The question now belongs more to 
VOL. ix. 2 T 
