522 Proceedings of the Royal Society 
Under Odd on even of unifilars, 5 on 7, are 32 unifilars, con- 
structed on 
7 E 7 C 4, 7 E 7 L 2, 7 L 7 E 4, 7 U 7 D 1, 
7 E 7 D 4, 7 L 7 C 2, 7 L 7 L 1, 7 U 7 E2, 
7 E 7 E 8, 7 L 7 D 2, 7 U 7 C 1, 7 U 7 L 1. 
Under Odd on odd of unifilars, 5 on 7, are 8 bifilars, constructed 
on 
7 E 7 E 4, 7 E 7 L 2, 7 E 7 U 1, 7 L 7 U 1. 
Every one m 12 of these 577 knots gives by unkissing at P and 
R of its linear section PR two unifilar 11-folds which, if the knot 
m 12 has no complementary about PR, can be twisted either one into 
the other, and which, if m 12 has a complementary, are two knots in 
a group of mutually convertibles, if I rightly conceive the matter. 
Bat not every two in a group can be twisted each into the other by 
a single twist. When m 12 has a complementary, two pairs of 
unifilar 1 1-folds are thereby given, each pair being two knots con- 
vertible into each other by a single twist, and both pairs, if I am 
not in this point mistaken, belonging to the same group. 
When m 12 has a complementary about PR, it stands drawn next 
to m 12 among the 577 figures herewith presented. 
I have had no difficulty in the cases 1 and 2 of art. 1, because 
I have the bifilars of 8 and 9 crossings. As the bifilars of ten 
crossings are not before the student, he may find it useful to know 
how to lay 2 of the unifilar K 2+2 on the bifilar L 12 _ 2 , without 
drawing the bifilar 10-folds, and by inspection only of the unifilar 
9-folds. The rule is this: — Make every odd angle at the crossing 
P of an unifilar 9-fold covertical with the figure P ab, «5R; i.e., 
with two triangles collateral with the 2-gon ab and having their 
third angles at P and R in the constructed linear section PR. 
Every result of laying an even 2-gon on a bifilar 2-gon of a 10-fold 
is thus obtained. The proof of this is easy by my theorems A and 
C given recently in these Proceedings, applied to the figure P ab 
ab R on the 12-fold completed by it. By similar devices the use 
of bifilar bases and charges can always be evaded in our construc- 
tions. 
