of Edinburgh, Session 1885-85. 517 
the circle of an unifilar of n - 1 crossings ; and by unkissing at P 
we find another such unifilar, thus establishing 
Theorem G. — If at its odd 2-gon PR, we join by section ffc the 
unifilar K e+2 and the unifilar L n _ e at its even 2-gon PR, we con- 
struct an unifilar M*, from which by unkissing at P and R we 
obtain two unifilars each of n - 1 crossings. 
6. (d) Odd on Odd of Unifilars. — Let 
. . . «PR7i . . . ePR /. . . (A) 
. . . mPPk . . . sPRz . . . (B) 
be the circles of the unifilars K e+2 and L, t _ e which have each an 
odd 2-gon PR. Making the angle eP a covertical over sPm, e and s 
on the left, and hPff over kPi, h and k on the left, we have the 
crossings P and R of M n , the result of joining (A) and (B) by the 
section ffc. Call this figure (C). Reading in (C) from e , we get 
the two circles 
. . . eP m . . . iPh . . . and . . . aPs . . . kPf . . . , 
proving that M n is bifilar. 
Unkissing at R in (C) we obtain 
K h . . . ePm ...if... aPs . . . 
an unifilar of n - 1 crossings, which has every summit of (A) and 
(B) except R, and every edge of them but the four PR’s. And by 
unkissing at P in (C) we get another such unifilar. This proves 
Theorem H. — If at their odd 2-gons PR, PR, we unite the 
unifilars K e+2 and L n _ e by the section ffc , we construct a bifilar 
M„, which yields by unkissing at P and at R, two unifilars each of 
n - 1 crossings. 
7. (e) Even on Even of Unifilars. — Let 
...aPR7*.../RPe... (A) 
. . . sPR^ . . . hRPm . . . (B) 
be the circles of two unifilars having each an even 2-gon PR. 
Uniting them at those 2-gons by the section ffc , we get the knot 
M w whose R linear section PR shows at P the angle ePa over sPm , 
e and s on the left, and at R the angle 7*R/ over iPk, h and i on the 
