CO. N. Little—Knots, with a Census for Order Ten. 37 
b) e 
In Table V, clutch No. (1) of 53° is thrown out by I. In (2) the 
two 3-gons E and F are joined together by a single bond and to A 
in each case by two bonds, and VI applies; or we might use I. 
Theorem VI throws out (4), since the two 3-gons B and F are joined 
to each other by a single bond and to other parts in each case by 
two bonds. In (5), drop F, by VI; B becomes a 2-gon joining D 
and K, and they are 3-gons connecting A and C. An obvious exten- 
sion of Theorem VII throws out the clutch. Since in (7) B and C 
are joined together and to A, I applies. 
This process is continued until all clutches which do not give 
knot-forms have been thrown out, as well as those clutches which 
are repetitions of clutches previously given. The samples given of 
Tables IV and V constitute about one-twentieth of the complete 
tables. 
19. We are now ready to draw the forms which the remaining 
clutches furnish. 
Derinition.—The number of Circular arrangements of n things is 
the number of distinct ways in which m things can be arranged in 
a circle, the order whether direct or retrograde being of no con- 
sequence. 
For illustration of the general method we consider in detail the 
productive clutches of partition 6°42*. In clutch (6) the two 6-gons 
are connected, by three bonds, by a 2 gon, and by a 4-gon which is 
connected with one 6-gon by two bonds and with the other by a 
2-gon and one bond. There will be as many forms from this clutch 
as there are circular arrangements of aaa b (cd), where c and d 
may not be separated. These are 
aaa b (cd) 
aaa b (de) 
aab a (cd) 
aab a (de). 
The four forms O’c,, C%c,, C’c,, C*e,, can at once be drawn. 
See Plate V, Knot LXIV. 
In (19), the only other clutch of this partition that affords forms, a 
6-gon is connected with a 4-gon directly, by a 2-gon, and by a 6-gon 
which is connected with the 4-gon by a single bond and by a 2-gon. 
There are two circular arrangements of a b (cd), namely a b (cd) and 
b a (cd); these furnish the two forms C*d,, C*d, Knot XXX, 
Plate III. In practice the operations described in this section and 
in the preceding, are performed simultaneously. 
20. In Class VI every productive partition is used as a leading 
