323 
of Edinburgh , Session 1876 - 77 . 
The process presents no difficulties, so that I shall give only two 
simple examples. Thus the scheme of the pentacle, viz. : — 
ADBEC ADBEC | A 
is divided at A (in this case it does not matter which junction we 
take) into the two superposed non-autotomic ovals 
DB EC | D, DBEC | D, 
by the first mode : — , and is simplified into 
DBECCEBD | D 
(• i.e ., a wholly nugatory scheme) by the second. 
The type-symbols in the original state, and in the two altered 
states, are, respectively, 
2r 5 I 
5 l 2 1 
2 r 4 1 
4 l 2 ) 
r ' ' \ 
2>l 2 + 21 ) 
The last of these is virtually nothing. In fact, terms in 
r or l to the first power are rejected by Listing. And, when 
these loops are taken off by untwisting or by opening up, the scheme 
becomes 
r 4 ) 
l 2 + 21 ) 
and a second application of the process removes the whole. 
Operating in a similar way upon the only other figure with five 
non-nugatory intersections — viz. : — 
A 4 D 4 B 2 E 2 C 2 A 4 D 4 C 2 E 2 B 2 I A 
2 v*4 + r 2 ) 
2P + 1 2 i 
we find three classes of cases, according to the particular intersec- 
tion operated on. 
[I may here introduce, though it involves a slight digression, a 
method which I have found very convenient as an assistance in 
finding which intersections have similar properties as regards the 
