518 Proceedings of the Royal Society 
left. Call this figure (C). Heading from e and h in (C) we get 
two circles 
. . . eP m . . . sPa . . . , and hPh . . . iPf . . . , 
of the bifilar M w . 
Unkissing at E we get the circle 
... hi ... kf . . . 
which contains neither of the sequences e ... a and m ... s, so that 
its knot of n - 1 crossings is no unifilar. 
Unkissing at P, we get the circle 
... me ... as . . . 
which contains neither i ... k, nor f ... h, and is not the circle of an 
unifilar. Both the knots of n- 1 crossings are plurifils. This 
proves 
Theorem J. — If at their even 2-gons PE, PE, we unite two 
unifilars K e+2 , and L n _ e by the section fic, we construct a bifilar 
knot M b , which gives by unkissing at P and E two plurifil knots 
each of n - 1 crossings. 
8. (/) Bifilar on Odd of Unifilar . — Let 
. . . aPPb . . . and . . cPE<i . . . (A) 
he the circles of a bifilar knot having the bifilar 2-gon PE. 
let 
. . . ePPf . . . hPBg ... (B) 
And 
be the circle of an unifilar, on which is the odd 2-gon PE. Draw- 
ing the angle aPc covertical over h~Pe, a and h on the left, and bPd 
cover tical over gBfi b and g on the left, we have the crossings P 
and E of M n constructed by the union of the knots at their 2-gons 
PE. Call this figure (C). Heading from a in (C) we find the 
circle 
. . aPe . . . gBd . . . cPh . . . fBb . . . 
of this M„, showing that it is unifilar. Unkiss now at E in (C); we 
get, 
. .Pd ...cPh.. 
a circle of the knot so formed of 1 crossings, which contains 
neither d ... a, nor g ... e\ so that the knot is not unifilar; and 
