327 
of Edinburgh, Session 1876 - 77 . 
have three corners in common. Hence in this notation the joining 
lines represent the crossings. Hence also the characters of the left- 
hand meshes are obvious from the figure. Outer space has the 
three external lines for corners — inside there is one triangle and 
two spaces bounded by two lines each (i.e. 9 with two corners). Thus 
we reproduce the left-hand part of Listing’s symbol. But the 
figure also shows us which lines (corners) each pair of these has in 
common, and enables us at once to draw the annexed figure 
which gives us exactly the same information as the first, only from 
a different point of view. 
The connections in the former figure cannot be varied, so that, 
in this particular case, Listing’s symbol for the right-handed meshes 
alone suffices to draw the figure; at least if nugatory crossings be 
rejected. Such would arise, for instance, if we tried to draw the 
symbol in the form 
r 2 
II 
7*4 
11 
r 4 
O 
which would give three ovals joined like the links of a chain— the 
last having an internal nugatory loop. In this case the second part 
of the symbol would be 
P + 2P + 1 
where the nugatory character of one intersection is clearly ex- 
hibited. 
But, if we had merely the left-hand part of the symbol given us, 
we might adjust it thus 
1 2 
