Q. N. Littleh—Knots, with a Census for Order Ten. 35 
The partition 4°2 with the given clutch can not exist in a plane. 
This happens in two cases in order 10. On drawing the forms in 
such cases additional crossings will be found to be necessary. It is 
therefore true that in all orders, partitions of six even parts give link- 
forms only. 
A synthetic proof of this theorem is possible. 
17. Instead of continuing the consideration of the general subject 
we shall now illustrate the method of determining the knot-forms of 
any order by a particular consideration of order 10. 
The number of partitions of 20 into parts none greater than 10 or 
less than 2 is 107.* These, arranged in dictionary order, are given 
in full in Table I. 
Of these the single partition of Class H gives a link, §11; the 
thirty-seven marked II are cut out by Theorem II. In Class III, 
Theorem VIII throws out 8°4 and 86°; in Class IV, Theorem IX the 
eight partitions marked IX; and in Class X, Theorem X the three 
marked X. The partitions remaining in Classes ILI to VI inclusive 
are alone taken as leading partitions; for, all remaining partitions 
appear in every possible way as subordinate partitions (§ 2), and 
can, therefore, furnish no additional knot-forms. 
We have then to tabulate the clutches of those partitions still 
remaining in Classes III to VI. We at once write down Tables II 
and III in which are omitted all clutches that are thrown out by 
Sections 14 and 15. In Class V, a table with headings as shown 
in Table IV is used. We take for illustration the first partition, 
72°, Here x=3,and 6—x, y—u, O—u, e—x are 4, —1, —1, —1. 
In this class we must add in every possible way to the minimum 
values of (AB), (AC), (AD), (AE) the partitions of 2x into four 
parts, or fewer, none greater than x—1. The only partitions of 6 
meeting these conditions are 2° and 2°1°, which may be added in 
seven different ways to 4, —1, —1, —1, giving the different clutches 
of the table. Thus adding 1, 2, 2,1, to G—x, y—x, O—u, e+ we 
have (AB), (AC), (AD), (AE), equal to 5, 1, 1, 0, respectively. Sub- 
tracting (AB) from {, ete. 
='B =(BC) + (BD) + (BE) =2 
2C =(BC) + (CD) + (CE) =1 
2D =(BD) + (CD)+(DE)=1 
2K=(BE) + (CE) + (DE)=2. 
These quantities are put in the columns headed >B, SC, SD, SE, 
*See Tait, Trans. Roy. Soc. Edin., vol. xxxii, p. 342. 
