516 Proceedings of the Royal Society 
4. (b) Bifilar on Even of Unifilar . — Let the pair 
. . . aPR& . . . and cPRc? . . . (A) 
be the circles of a bifilar knot K e+2 , having the bifilar 2-gon 
PR, and let 
...ePR/...fcRP$r... (B) 
be the circle of an unifilar L n _ e which has the even 2-gon PR ; 
and let the knots be joined by the section ffc. We have a knot M„ 
whose crossings P and R show cVa covertical over eP g, c and e on 
the left ; and bRd covertical over hRfi b and k on the left. Call 
this figure (C). Reading from c in (C) we get the circle 
. . cPg . . . eP a . . . 5R/ . . . hRd . . . , 
i.e.j M n is unifilar. Unkissing in (C) at R, we find the circle 
. . hb . . aVe . . gVc . . . df. . . 
of an unifilar of n- 1 crossings ; and another such unifilar by 
unkissing at P, thus proving 
Theorem F. — If the bifilar knot K e+2 , at its bifilar 2-gon PR, 
be joined by section ffc to the unifilar L n _ e at its even 2-gon 
PR, an unifilar M n is contracted which gives, by unkissing at P 
and at R, two unifilars of n - 1 crossings. 
5. (c) Odd on Even of Unifilars . — Let the circles 
. . mPR/r . . . sPR{ . . . (A) 
.../RPe...aPR&... (B) 
be those containing the odd and even 2-gons PR, PR. Joined by 
section ffc they form the knot M B , whose crossings P and R show 
eP a covertical over sPm, e and s on the left, and KRf covertical 
over IcRi , h and h on the left. Call this configuration (C). 
Reading from e in (C), we get the circle 
. . eP m . . . iRh . . . /Rr . . . sPa . . , 
proving M n unifilar. 
Unkissing at R in (C) we get 
if...hh... sVa . . . ePm . . . 
