514 
Proceedings of the Royal Society 
circle, and inclined at an angle of — to its plane. 
The surface there- 
fore intersects itself in this straight line. It is obvious that the 
surface has “helicoid asymmetry” ; as for each sense in which the 
point of intersection may rotate, there are two senses in which the 
generating line may rotate. This gives four forms, which obviously 
coincide in pairs. 
4. On the Linear Section PE of a Knot M n , which passes 
through two Crossings P and R, which meets no Edge, 
and which cuts away a (3 + r)-gonal Mesh of M n . By 
Rev. Thomas P. Kirkman, M.A., E.RS. 
1. The purpose of this paper is to give rules for the construction 
of knots M b of n crossings, having each one or more linear sections 
PR, such that by unkissing at P and at R, shall be obtained two 
unifilar knots of n - 1 crossings. See my paper “ On the Twists of 
Listing and Tait,” page 363 of the Proc. R.S.E. The knot M„ is 
always made by a section ffc, i.e., by uniting the P’s and R’s of two 
2-gons (PR) on K e+2 and L„_ e , after cutting away the four edges 
PR. These 2-gons can only be (a bifilar 2-gon has edges in two 
circles) — 
(a) Bifilars found on bifilars K and L ; 
(i b ) Bifilar found on bifilar K, and even on unifilar L ; 
(c) Odd laid on even of unifilars K and L ; 
( d ) Odd laid on odd of unifilars K and L ; 
(e) Even laid on even of unifilars K and L ; or 
(f) Bifilar of bifilar K laid on odd of unifilar L. 
2. (a) Bifilar on Bifilar . — Let the pairs (A) and (B) 
. . . aPR5 . . . . . . ePR f . . . 
. . . cFRd . . . ^ ' ... gFFJi . . . 
be the circles of the bifilar knots K e+2 and L re _ e . In (A) the 
angles aFc and RPR are co vertical, as are 5R d and PRP. In 
(B) eF g and RPR are co vertical, and also fFUi and PRP. If we 
