324 
Proceedings of the Royal Society 
figures which will be obtained when they are made in turn the 
point of section. In the scheme above written the suffixes express 
the numbers of letters which intervene, in the scheme, between the 
two appearances of the same letter. If n be the whole number of 
letters, the suffix may of course be either 2 r or 2n - 2r - 2. It is 
convenient to write always that one of these two numbers which is 
not greater than the other. When a particular suffix occurs only 
once, the corresponding crossing has evidently different properties 
from the others ; if twice, we find in general that the corresponding 
crossings have similar properties. If three times, two of them have 
usually like properties, but the third not — and so on. This method 
is useful, but it is in certain cases misleading. In fact, we must 
look not only at the suffix itself, but at the place which it occupies 
relatively to the whole group of suffixes, in order to obtain absolutely 
definite information. Something similar to this is obviously hinted 
at in Listing’s paper, where he seems to determine the number of 
possible transformations of the figure representing a symbol, by 
treating the numerical coefficients much as I have here treated the 
suffixes. But this is mere conjecture on my part.] 
By this method then, or by examining the diagram, we see that 
A and D are similar, so are B and C, while E may possibly possess 
distinct properties of its own. We need, therefore, take only three 
cases, A, B, and E, 
a.) Divide at A. Then we have either 
DBEC | D DCEB | D 
two ovals crossing one another, one taken right handed, the other 
left ; or 
DBECBECD | D = BECBECIB 
2r 3 ) 
3 1 2 I 
the trefoil knot ; for D becomes nugatory. 
b.) Divide at B. We have either 
A D | A ECADCE | E = DA | 
2r 2 ) 
2 1 2 1 
D 
