36 CU, N. Little—Knots, with a Census for Order Ten. 
and all the possible solutions of this set of equations here as 
throughout the work are written down by inspection. The rest of 
this table is made out in the same way. In Table V we must, in 
every possible way, add the partitions of 3% into four parts or fewer, 
none greater than x—1. In the general case we add the partitions 
of (p—3)x into p—1 parts none greater than x—1. In the com- 
plete Table IV there are 400 clutches, and in Table V 1000. It has 
not been thought necessary to publish more than a sample of either 
table. 
18. Having completed the tables of clutches we next cast out in 
Tables IV and V all clutches unproductive of knot-forms. The 
theorems already established are sufficient for this purpose. The 
routine followed will be readily understood from considering its use 
in the samples given of Tables IV and V. 
Partition 7°2*: in clutch (1) A and B are connected by two 
2-gons, and V applies ; (4) becomes (3) by interchanging D and E; 
(7) becomes (6) by interchanging C and D; and in (2) and (5) we 
have a 2-gon solely bound to a single part, and I applies. 
Partition 6? 42°: clutch (1) is thrown out by IV since a 6-gon B is 
joined to A by four bonds and to C by two; Theorem IV casts out 
(4) because of C, (14) because of CO, (17) because of B, and (20) be- 
cause of A; in (2), (3), (8), (9), (11), (12), (15) and (18) a 2-gon is solely 
joined to a single part, and I applies; in (5) B and C form a combi- 
nation solely bound to A, and I applies; I throws out also (10) where 
D and E are bound together and to C, and (13) where the same 
parts are bound to B; interchanging D and E (7) becomes (6); 
interchanging A and B (16) becomes (7). 
Partition 5°32: I throws out (2), (3), (6), (8), (14), (16), (19), (20), 
(26), (27), (30), (31), (33), (87) and (88) ; by interchanges of parts 
()=(4, COD=(5), @=(9), (82)=00), 
(N=), (08) 8), — (3E( 4), (= (22), 
(13)=(10), (21)=(10), (28)=(23), (35)=( 1), 
(15)=(12), (22)=(21), (29)=(28), and (36)=( 9). 
In (4) D is dropped by VI, and B thus is changed to a 4-gon; 
then the application of §12 leaves the two 3-gons A and C joined 
by two 2-gons, E and the still farther reduced B; V, there- 
fore, shows that the clutch gives links only. In (9), drop C by VI 
and B becomes a 2-gon and A a 3-gon; the two 3-gons A and D are 
connected by the two 2-gons E and the reduced B; therefore the 
clutch gives links, by V. In (11) drop C by VI and the two 3-gons 
_ A and B are connected by the two 2-gons E and the reduced D; V 
applies. 
