34 C. N. Little—Knots, with a Census for Order Ten. 
ns LL Soren OR Props. or 
n, « | Partition.| Add c= eel Bee eS Form Sunn 
@ ‘e7/ ee eve eee eee eee eee Link. IV. 
e€ 00 e€ 0 0 ooe a §12, V 
eeoo eee e€0oo eoo eee fs IV. 
e0oo eee ooe ey EVe 
o0e€0 00° 0e€0 Knot. |§12, VI§11. 
G4} 101.0) OF ON NOZORO eee 000 000 Link. §12, VIL. 
oee oee eeo 7; §12, VI. 
ooee|] 000 eoo oee 000 Knot. |§12, VI, 11. 
oee 000 ereno ts §12, VI, 11. 
eoe : eeo eoe Link. Vi. 
0 e€| ooee eee oee oee eee a IV. 
| € 00 000 ooe s §12, V. 
oeo eeo oe 0 Knot. 812. 
e|0000/] eee 000 000 ee e ne $12, 
eoo oee ooe * §12. 
o|eeece 000 000 eee 000 Link. IV. 
oee eoo eeo IV. 
ee0o0| 000 oee eoo 000 : §12, V 
oee eee eeo Z: LVe 
eoe ooe eoe Knot. §12. 
The first line of this scheme says that when 7 is even and % even, 
and 2 divided into four even parts, the minimum values of (AB), 
(AC), (AD), will be even, and that if a partition of % into three 
even parts be added to 6—x, y—u, O—x the numbers constituting 
the clutch will be even, and the form a link by Prop. IV. 
One of the propositions proved in this scheme is worthy of separate 
enunciation. 
IX. Turorrem.—In even orders partitions of 2” into four even or 
four odd parts give link-forms only. In odd orders partitions into 
four even parts give link-forms only. 
16. X. THeorem.—In all orders partitions of six even parts give 
link-forms only. 
For, an even number may be divided into five parts, of which four, 
two or none shall be odd. After the application of §12 the only 
parts to be found in the leading partition will be 4-gons and 2-gons. 
Every form under consideration will be reduced so as to have as lead- 
ing partition one of the following, with a clutch of which every 
number is 1. 
Ae Ase" AR ee a 
In the lower orders 4°2°, 472‘, 42°, and 2° have been found to give 
no knots. 
If the form given by 4° be drawn it is found in any particular 
case to consist of four closed curves, and therefore by Theorem III is 
always a link-form. On drawing 4*2° it also proves to be a linkage. 
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