366 
Proceedings of the Royal Society 
the knots of n - 1 crossings. We shall obtain repetitions of some 
nine-folds, but no vain repetition ; for when the unkissing is finished 
we shall have grouped all the convertibles in twos, threes, sixes, &c., 
without ever attempting to perform a twist, as well as have given 
an accurate account of the number of different triangular sections on 
the grouped ones and on the uniques, without ever trying to count 
these sections, or to distinguish them by their symmetry. 
In all that precedes, no distinction is made or supposed between 
unifilar and plurifil knots; nor do I know any reason why unifilars 
only should be considered. 
The crossing P in fig. 1 may stand for any tessarace in the pro- 
jection of any knot, or of any w-acron whatever, through which lies 
the triangular section P rr there described. The twisting can take 
place in them all, no matter what the faces and the other n - 1 
summits may be, and the groups can be formed of which every 
figure can be so twisted into one or more of the others. 
The question here presents itself — Will it be profitable, supposing 
that the census of all the knots of n crossings is wanted, to employ 
the method of unkissing above opened ? I am of opinion that it will. 
Let the subsolids (which have no linear section PE) be divided 
into S n , all that have no triangular section P rr cutting away on 
both hands a (3 + r)-gonal mesh, and T n , all those which admit one 
or more such sections P rr, and let U n comprise all the unsolids, 
which have each one or more linear sections PE, cutting away on 
each hand a (3 + r)-gonal mesh. 
Suppose that S n and U„ are found, and that the subsolids T n , in 
general more numerous than S n , are wanting. If we can obtain 
U n+ i, we shall readily get by the simple process of unkissing, not 
only the missing T M , but every pair of convertibles possible out of 
T m and the unsolids U M that admit a triangular section P rr, and 
every fact required for our table of uniques and grouped con- 
vertibles. 
I believe that even when U M+1 is more numerous than T n , it can 
be more easily found than T„, which comprises only subsolids to be 
obtained by the minute and often many comparisons that determine 
the leading flap ; whereas U n+1 is rapidly put together without 
minute comparisons by laying 2 upon n- 1,3 upon n - 2, &c., by 
the simplest marginal sections ffc only. 
