of Edinburgh, Session 1885-86. 519 
by unkissing at P, we form another plurifil knot of n - 1 crossings. 
This proves 
Theorem K. — If we unite the bifilar knot K e+2 at its bifilar 
2-gon PR to the unifilar L n _ e by its odd 2-gon PR, we construct 
an unifilar M w of n crossings, which yields by unkissing at P and 
at R two plurifil knots each of n - 1 crossings. 
9. It is evident from the above reasoning that in the construction 
of M„ which has a linear section PR, at which by unkissing two uni- 
filars of n - 1 crossings can be got, we are in every case to use one 
of the Theorems E, F, G, H, in articles 3, 4, 5, 6 ; and that these are 
both necessary and sufficient rules. By them we can form all the 
required unsolid knots M ra without omission or repetition. 
We have only to lay e upon n - e, i.e. K e+2 upon L n _ e , where 
e + 2 > n - e, beginning at e = 2. 
In forming the unsolid knots (M 12 ) which I herewith present to 
Professor Tait, I have begun with e = 4 ; i.e., I have laid 4 upon 8, 
and 5 upon 7. For not only is the number of figures for e= 2 and 
e = 3 enormous, but I am sure that they would be of no service to 
Professor Tait, in his rapid handling of his twists. Nor am I at all 
sanguine in the hope that these results for e = 4 and e = 5 will assist 
him except by furnishing a ready mode of verifying the most com- 
plex work of his grouping of the 11 folds with which I have had 
the honour to supply him. I have for myself never attempted this 
task of grouping the twisted knots of eight or more crossings, as this 
problem appears to me of less consequence than an accurate census 
of the knots with a description of their symmetry. 
10. I should have remarked in my paper “On the Twists, &c.,” 
that a complementary pair of w-fold unsolids will give only one 
couple of convertible (n— 1) folds, if each of the pair has P and R 
alike, which can be only when a zonal trace passes between P and 
R. And I was in error in saying that “ every 9-fold so got from P 
and R alike is a 9-fold having a triangular section at which it can 
be twisted into its reflected image.” The truth is that every n— 1-fold 
got from P and R alike on the knot M n , whether M m has or has 
not a complementary, is a knot which has a triangular section 
at which it can be twisted either into itself or into its reflected 
image. 
I believe that no M 12 that I have drawn, which has no comple- 
