364 Proceedings of the Royal Society 
For not only are the great majority of possible knots not stable forms 
for vortices ; but altogether independently of the question of kinetic 
stability, the number of distinct forms with each degree of knotti- 
ness is exceedingly small, — very much smaller than I was prepared 
to find it. I have already stated that for three, four, five, and sixfold 
knottiness, the numbers are only 1, 1, 2, 4. For a reason given 
in my first paper, knots whose number of crossings is a multiple of 6 
form an exceptional class : so I thought it might be useful to dis- 
cover and to figure all the distinct forms with seven -fold knottiness. 
Eight and higher numbers are not likely to be attacked by a 
rigorous process until the methods are immensely simplified. The 
method of partitions, supplemented by the graphic formulse of my 
last paper, is to some extent tentative. I have verified the present 
results by means of it, and have extended it to 8-fold knottiness, 
but I am not certain that I have got all the possible forms of the 
latter. 
As I did not see how to abridge the process, I wrote out all the 
admissible permutations of the seven letters in the even places of 
the scheme. These I found to be 579, five of which were, of course, 
unique. The others (as 7 is a prime number) were divisible into 
82 groups-— those of each group being mutually equivalent. On 
examination, it was found that only 22 of the 87 selected arrange- 
ments satisfied the criterion for possible knots (see I §(£>) of my paper, 
ante p. 238), and several even of these were repetitions. These 
repetitions were of two kinds — 1st, the mere inversion of the order 
of the scheme ; 2d, the relative positions of a 3-fold and a 4-fold knot 
which in certain cases wero found combined as a 7-fold form. 
Clearing off these repetitions, and along with them a form really 
belonging to 6-fold knots (because consisting of two trefoil knots 
and one nugatory intersection), there remain only eleven distinct 
forms of the 7th order. These are as follows : — 
1 . 
3 
This has a great many forms, with correspondingly different 
