520 
Proceedings of the Eoyal Society 
mentary about the section PR, has P and R alike ; and that such 
drawn M n will always give a pair of mutually convertibles. 
11. Every K e+2 imposed on L n _ e in the (M w ) above constructed 
is supposed to be either a subsolid defined in page 283 of Trans. 
R.S.E., vol. xxxii. part 2 ; or such a subsolid carrying a plural 
flap, of which neither P nor R is a crossing. 
In laying 4 on 8, I have never laid 2 ‘2 on 8 ; nor have I laid 
2*3 nor 3 '2 on 6 (vide, for this kind of operation, the memoir 
above cited) ; and this for the reason given in art. 9. 
12. The number of unsolid knots (M 12 ), a portion of U 12 (see my 
paper “On the Twists, &c.”), which I herewith offer, is 577, of 
which 199 are bifilars and 378 unifilars. They are few in com- 
parison of the entire U 12 . As they will be useful, if ever the 12-fold 
knots are wanted, it is worth the while to give a summary of 
them, which will, to a student working on this subject, be of 
almost as much service as the 577 engraved figures. The references 
to fA, &c. ; 8 A, &c., are to knots nearly all engraved in my plates in 
the Trans. R.S.E., and clearly given in the text when not 
engraved. 
Under the heading, Bifilar on bifilar, 4 on 8, are 47 bifilars, 
constructed on 
8 A2, 
8 P2, 
8 V2, 
g BZ 6, 
8 C 4, 
8 R 4, 
8 A w 6, 
gBm 2, 
gD 2, 
s S3, 
s Ba 1, 
8 Bx 3, of 
8 1 4, 
8 U 4, 
g B& 2, 
the 47. 
Under Even 
of unifilar on 
bifilar, 4 on 8, 
are 67 unifilars, 
constructed on 
8 A 3, 
8 P3, 
8 V3, 
8 b; 7, 
gC6, 
8 R5, 
g A w 9, 
8 B m 2, 
3, 
8 S 3, 
g B& 2, 
g B^ 6, of 
8 E 6, 
8 U 4, 
g B lc 3, 
the 67. 
Under Bifilar on even of unifilar, 4 on 8, 
are 47 unifilars, 
constructed on 
00 
sQ 1, 
8 Av 4, 
g B w 5, 
gFl, 
sYl, 
gAic 6, 
8 B^ 3. 
,»4, 
■rH' 
GO 
8 B n 4, 
8 L4, 
8 A u 4, 
8 B» 2, 
