30 C. N. Littlh— Knots, with a Census for Order Ten. 
8. I, Tororem.—If a part be solely bound to a second part, or if 
any g parts (¢ “%p—2) be bound mutually in any way and all free 
bonds of these parts go to a single part, then this portion of the form 
constitutes a separate knot (unless there be linkage) and the string 
concerned in it may be drawn tight without affecting the remainder 
of the knot-form. Such knots are not considered as belonging to 
order 7. 
In particular a 2-gon so bound throws out from consideration a 
clutch. 
9. Il. Tororem.—No knot-form of the nth order has as leading par- 
tition one whose class exceeds + 2. 
Adding equations (a) above, dividing by two, and subtracting the 
first and any other, say the second, we find 
—(AB)+(CD)+(CE)+ ... (CP)+ ... (OP)=n—a—A£, 
=xu— fi. 
Therefore, (AB) x 6— x. 
In a similar way 
(AC) xy—x 
(AD) x 6— x 
(AP) x az— 
Adding (AB)+(AC)+ ... (AP) xf+y+ ... m—(p—1)x 
; Kn+u—(p—1)u 
XKn— (p—2)x, 
we have then the two conditions 
(AB)+(AC)+ ... (AP) xn—(p—2)x () 
=n— Nn. 
Now suppose, if possible, p= +3 
(AB) + (AC) + pt. (AP)=n=x — i — 
Kn—("%+1)x Xn—u—H. 
To the minimum values of (AB), (AC), etce., (that is, to 6—x, 
VY, -.- ) must be added x in all, and to no one more than x—1, 
by I. The +2 smallest parts of ~ square are evidently (%—1)* 
(~—2)«. By adding these ~+2 parts in any way to the mini- 
mum values, a clutch will be given in which each of four parts will 
have a single bond not going to A, and each of —2 parts will have 
two. A part of the latter kind cannot have its two free bonds car- 
ried to a second part of the same kind, by I. If two parts be joined 
by a single bond there will be left two free bonds. Ultimately it 
will be necessary to join two parts of the first kind to a combina- 
tion of parts having but two free bonds, and I will apply. If any of 
the x+2 parts of x° be diminished by s then will s parts of 22% be 
OO es 
