C. N. Little—Knots, with a Census for Order Ten. 39 
bonds in the connections of pairs of parts. Therefore in these orders 
all forms from any clutch are forms of ‘the same knot. The sub- 
ordinate partitions of these forms are then to be examined and all 
forms added which are obtained from them by changing the positions 
of connections of their parts, retaining the given clutch of the subor- 
dinate partition. These forms in turn are treated in the same way, 
but it will usually happen that no new form of the knot is obtained 
and the complete determination of the knot in all of its forms is 
finished. 
23. We take for illustration Knot I of Class IV, shewn on Plate I. 
Ba, becomes Bu, by twisting about a vertical axis the 2-gon connect- 
ing the 8-gon and 7-gon. The first crossing below is opened and the 
strings above are crossed, the rest of the knot remaining fixed. 
Twisting the 2-gon again Ba, is obtained and nothing new is gotten 
by further changes of the forms. Since the negative partition in 
every case consists of two parts joined by three symmetrical connec- 
tions, which have only one circular arrangement, there are no other 
forms of Knot I. 
24, The knots of Class III, order n, are unique; since three things 
have but one circular arrangement. 
25. The knots of Classes III, [V, V will be found with their forms 
grouped together in Plates I-V. On Plates V—VII are figured the 
forms of Class VI grouped as they come from the clutches, except 
that no form is repeated. The knots of this class will be found in 
Table VI. Every knot-form is the projection of two knots, one of 
which is the perverted image of the other, and consequently each 
group of knot-forms belongs to two knots which are in general 
different. If a series of knot-forms contains any amphicheiral form 
then it will also contain the perversion of every form of the series 
not amphicheiral. The series consists of the forms of one knot and 
not of two. . 
26. In order 10 I find, counting a knot and its irreconcilable perver- 
sion as two: 
Class. Forms. Knots. Knots. 
III, 6 12 6 
EV 25 30 15 
V, 200 128 64 
VI, 133 64 39 
Totals, 364 234 124 
