of Edinburgh, Session 1884-85. 
367 
The reader, having before him in the plates of vol. xxxii. Trans. 
R.S.E. nearly all the knots unifilar or not of fewer than ten 
crossings, can amuse himself by considering the question above 
proposed. He can begin with S 5 = 5 B + 5 C. Compounds like G E 
are rejected all through. But the student will not be able to con- 
fine his attention to unifilars only. 
Observe that in the linear sections PR neither P nor E can be a 
non-terminal crossing of a plural flap. Also, if in PR R is a crossing 
of the 2-gon RR', PR and PR' are for our purpose the same linear 
section, because we get the same w-fold knot whether we unkiss at 
R or at R' upon the (?z+l)-fold. Hence it follows that P rr and 
P r'r' are the same triangular section when rRr and r'Rr' are 
covertical angles at the crossing R. 
It is not difficult to give simple rules whereby S n is found without 
error or repetition from S n _! and S„_ 2 by the leading flap of each 
knot of S B . But a definition of a fixed flap must be made and 
stuck to. 
I conclude with two useful little theorems. 
a. On every unsolid unifilar V in U n+1 each of its linear sections 
PR (P and R unlike) lies through four angles at P and R, which 
are all odd or all even. 
b. The couples of w-folds obtained by unkissing on Y, or on Y 
and its complementary, are all unifilars or not, according as these 
angles (in a) are odd or not. 
My objection to the twistings, that they put a twist upon the 
tape, has been answered by Tait. In the case of Listing’s twist, I 
have satisfied myself that his answer is sufficient. It appears to me 
that it ought to be formally demonstrated as sufficient in all cases. 
After all, as it is certainly not on record who invented kissing, it 
may come to be forgotten who invented unkissing. 
P.S. Nov. 7. — I have learned how to form readily all the uni- 
filars only of U n+1 , required for unifilar couples, by operating on 
unifilars only of n and of fewer crossings. I shall soon present to 
Professor Tait the requisite unifilars in U 12 , and thus I hope to save 
him much time and trouble in grouping the unifilar convertibles of 
eleven crossings. 
O 
