329 
of Edinburgh, Session 1876 - 77 . 
ate closed curve. In fact, the upper line represents a crossing on 
the boundary , at which there is (internally) only a left-handed 
mesh, which is impossible if the boundary were a closed curve. 
And the lowest line in the figure is a point in the boundary 
which forms- a common vertex of three (internal) meshes, two right 
and one left-handed. This, also, is inconsistent with the boundaiy’s 
being a closed curve. 
There is only one other case which may cause a little trouble. 
It can easily be seen by the fig. of last page. For we may take out 
the following part of the symbol 
P 
which must obviously represent the lemniscate in the figure. Its 
exponents and lines do not satisfy our condition : but they will 
do so if we remove the diagonal line— which corresponds to what 
is (in the lemniscate when alone) a nugatory intersection. 
I conclude by giving the representations, according to the 
method just explained, of some of the preceding figures. Thus 
the three first figs, of p. 325 are, respectively, 
/ 
P — P = P 
\ / 
W 
7 3 
II \ 
l 4 _P 
while the pair of common-symbol knots on the same page are 
P—P r 2 
Z 5 =Z 4 
and 
It may be observed that the present method gives great facilities 
for the study of cases in which knots are reduced, or are changed 
