of Edinburgh, Session 1884-85. 
359 
APPENDIX. 
1. Demonstration of Theorems A, B, C, &c., page 484 of Trans- 
actions, Royal Society of Edinburgh, vol. xxxii. By Rev. 
T. P. Kirkman, M.A., F.R.S. 
1. In the circle of an unifilar knot every crossing a is read twice, 
once in an odd and once in an even place ; and the thread is sup- 
posed to pass under and over itself alternately at successive crossings 
Every contiguous duad of the circle is a different edge of the 
knot. Let every mid-edge round the circle be dotted on the right. 
Let 128 .. . a213 . . . 5, . . . (A) 
of 2 n terms 1, 2, &c., where 51 is a contiguous duad, be the circle 
of an unifilar of n crossings. We see that 1 and 2 are the 
crossings of a 2-gon ; for no mesh of n N but a 2-gon can have two 
summits joined by two different edges of the mesh. (A) is the 
circle of any unifilar wdiich has the duads 12 and 21. 
Let us write the above thus, omitting only the crossing 1, and 
simply reversing one of the sequences between 1 and 1 — 
28 . . . a25 ... 3, . . . (B) 
where 32 is a contiguous duad. What does this mean, when in the 
projection of ra N the 2-gon is shrunk up to a point 2, the edges 12 
and 21 disappearing? 
It is the circle of an unifilar n _iM of n— 1 crossings, which has 
every duad of (A) except 12 and 21; for it has 51 and 13 because 
it has 52 and 23, 2 and 1 being now the same point. 
2. In (A) as we w T alk from 1 to 2, and later from 2 to 1, we 
make the circuit of the 2-gon 12 in the same direction round it; 
therefore our two dots will be both inside or both outside of it. 
This 12 is an even 2-gon; and every even 2-gon mn of an uni- 
filar is known by a glance at the circle, by its exhibition of mn 
and nm. 
3. We have demonstrated the following 
Theorem A. — Every unifilar knot n N of n crossings, which has 
