515 
of Edinburgh, Session 1885-86. 
efface the four edges PR and make aPe covertical over eP g, c and e 
on the left, and bRd covertical over hPf, b and h on the left, cPg 
crossing aPe , and bRf crossing dRh, we have before c a ft <g 
us the two summits P and R of M n . Draw this figure, 
and call it (C). No change has been made at any- 
crossing besides P and R. Beginning to read in(C) eg hf 
along cPg, we find the two circles 
cPg . . . hRd . . . , and aPe . . .fPlb . . . , 
of the bifilar M n ; for P g in (B) brings us to AR, i.e., in (C) to ARc?; 
and RcZ in (A) brings us to cP, i.e., in (C) to cP^, repeating the 
round. Also along aPe in (C) we reach by (B) /R, i.e., in (C) 
fRb ; and R6 in (A) brings us to aP, i.e., in (C) to aPe, repeating 
the circle. 
3. Let us now unkiss at R in (C), so as to make the section PR 
into Prr'. We are to read not fb and lid crossing at R, but fd and 
lib kissing at R ; and evidently this can make no difference in the 
course of the thread from R through b, d, h, or /. We find the 
circle of an unifilar of n - 1 crossings, 
frd . . . cPg . . . hr'b . . . aPe . . . , 
or omitting the creases r and P, 
fd . . . cPg . . . hb . . . aPe . . ., 
containing all the sequences d ... c, g ... h, b ... a, e . . . f ; for Rd 
in (A) brings us to cP, i.e., by (C) to cPg ; and Pg in (B) leads to 
AR, i.e., in C to hr'b, whence R b in (A) brings us to aP, i.e., by (C) 
to aPe ; and Pe in (B) brings us to our start in frd. 
Unkissing next at P we get the circle 
ga . . . bPf . . . ec . . . dPh . . . , 
of an unifilar on which lies the triangular section R :pp. Thus we 
have proved 
Theorem E, — If we unite by the section ffc two bifilar knots 
K e+2 and L n _ e by bifilar 2-gons PR and PR, we construct a 
bifilar M w , which gives by unkissing at P and R two unifilar knots 
each of n - 1 crossings. 
