of Edinburgh, Session 1884-85. 
385 
This is the twist of Listing, who found it in his study of knots of 
seven crossings, under what seems to me the needless disguise of 
gracefully flowing curves. It was sharp wit in Listing so to 
observe it ; it was still more acute in Tait to detect under a more 
complete disguise the manoeuvre of his twist of greater complexity. 
The figures 3 and 4, considered as unsolids of ten crossings, are 
the complementary pair C71 and C74. If we had these two before 
we had obtained 9 A /, 9 A q, 9 A r 2 , and 9 A b, all of which are subsolids 
to be regularly constructed by their leading flaps, we could, without 
the minute comparisons of that process, at once draw these sub- 
solids — the first pair by unkissing at It and P', the second by 
unkissing at P and E\ And if we had all the unsolids of ten 
crossings, having each a linear section PE, which cuts away on both 
hands a (3 + r)-gonal mesh, we could with the same ease draw every 
knot of nine crossings that has a triangular section P rr. All we 
have to do is to take first every complementary pair of 10 -fold 
unsolids, and by unkissing, as just shown, write down from it two 
couples of 9-fold convertibles j next to take every unsolid 10-fold 
without complementary, which has a linear section PE, above 
described, of which the two crossings P and E are not identical, 
and by unkissing first at P and then at E, to obtain from each such 
unsolid one couple of convertibles. The remaining unsolids in 
which those crossings P and E are identical will give, by unkissing 
at either in each, every 9-fold not already found which has a linear 
section P rr ; where the words just used, “ remaining unsolids,” 
include every unsolid before handled as having a section PE with 
P and E unlike, to be handled again once, twice, &c., according as 
it has one, two, &c., different linear sections PE with P and E 
alike. And every 9-fold so got from P and E alike is a 9-fold 
having a triangular section at which it can be twisted into its 
reflected image. It will sometimes happen that amongst our 
couples obtained of convertibles, AB, BC, &c., the couple CC will 
be found. This means that the knot C has two different sections, 
P rr and S tt, at which it can be twisted into itself, besides one or 
more at which it can be twisted into B or B' of the couples BC and 
B'C. Por the truth is, that the number of different crossings P in 
the linear sections PE on all the knots of n crossings is exactly the 
number of different triangular sections P rr, above described, on all 
