307 
of Edinburgh, Session 1876 - 77 . 
He then applies the notion of lines winding or screwing round 
one another to the projection of a knot on a surface, and shows that 
we can thus obtain a knowledge of the relative situation of the 
various coils. At each crossing portions of the two branches may 
be regarded as small parts of lines twisted round one another. Of 
the four angles so formed, two vertical or opposite ones are bounded 
towards the right , the other two towards the left , by that line which 
passes over the other. We thus distinguish these pairs of vertical 
angles into right- and left-handed. [Listing uses right-handed for 
what we should call left-handed in screwing, but the difference is 
of no importance, so far as his results are concerned.] 
Next he shows that perversion, but not inversion, changes right- 
handed into left-handed angles. 
He then gives the complete knot with three intersections, and 
shows that when it is in a reduced (as distinguished by him from 
a reducible ) form, all the angles in each separate mesh have a 
common character; but that, when it is reducible, some of the 
meshes have angles of both kinds. He distinguishes between the 
right- and left-handed forms of the reduced knot, and shows that 
they are not convertible into one another; also that (including 
external space, or the Amplexum ) there are three meshes with two 
corners each, and two with three corners; one class being right- 
handed, the other left. And he states that the Amplex may be 
made to change its character from right to left by being changed 
from a three-cornered to a two-cornered mesh, or vice versa. 
He points out that a loop (i.e., a mesh with only one corner) does 
not appear in the reduced form, and then writes, as the type- 
symbol of the reduced right-handed knot of three intersections, the 
expression 
3 r 2 'i 
2P ) ’ 
denoting three right-handed meshes with two corners each ( Oesen ), 
and two left-handed meshes with three corners each ( Mascheri ). 
[The perverted, or left-handed, form is of course represented by the 
same symbols, with the interchange of r and l .] 
The sum of the numerical coefficients in the symbol is the 
number of meshes (the Amplex included), and is greater by two 
2 s 
VOL. IX. 
