O. N. Little—Knots, with a Census for Order Ten. 33 
> 
For, HCM (fig. 2, Plate I) under we will say at C is over at H and 
M. LCN is then over at C and under at L and N. LHN is over 
at L, under at H, and over at N. Its continuation is therefore NML 
which is over at N, under at M, and over at L. 
14. Crass III, OnpER v.—In this class we have 
(AB) + (AC) + (BC)=n 
and an unique solution of equations (a), § 9. Therefore (BC)=x, 
(AB)=6—x and (AC)=y—x. If two or three of these quantities 
be even, the clutch to which they belong will give a linkage, by 
Theorem IV. In other cases by § 12 there is a knot. 
Suppose 2 to be odd and a@ even; then x is odd, and f and y are 
both odd, or both even. In the first case the clutch gives a link, in 
the second a knot. 
Suppose 7 to be odd, and @ odd; then % is even, and of f and y 
one must be odd and the other even. The clutch gives a link. 
Suppose 7 to be even and @ even; then x is even and /, y are 
both odd or both even. In the first case a knot, in the second a link. 
Suppose z to be even and @ odd; then x is odd, and of f and y 
one must be odd and one even, and there is a knot. This proves the 
following : 
VIII. THrorEem.—In odd orders only partitions of 2 into three 
even parts, give knots, while in even orders, only these partitions 
give links. 
15. Crass 1V, OrpER n. Here 
—(AB)+(CD)=x—£f 
—(AC) +(BD)=x—y 
—(AD) + (BC)=x—6 
(AB) +(AC)+(AD) o> 9, 
The minimum values of (AB), (AC), (AD), (BC), (BD), (CD), are 
respectively 6—x, y—u, 0—x,0,0,0. To get every clutch we must 
add in every possible way to the minimum values of (AB), (AC), 
(AD), all the partitions of into not more than three parts, none 
greater than 2—1. But evidently we must increase the minimum 
values of any quantity of the second set (BC), (BD), (CD), by the 
same number that we increase the corresponding quantity of the first 
set. The following scheme which considers in detail every possible 
case, expresses clearly the propositions for determining whether 
clutches of partitions of this class furnish knots or links. Let e or o 
indicate whether a number be even or odd. 
or 
Trans, Conn. Acap., Vou. VII. SEPT., 1885, 
