CO. N. Littl—Knots, with a Census for Order Ten. 31 
added to those of the first kind, and however the free bonds may be 
arranged, ultimately the same result as before will be reached. 
Therefore p cannot equal ~+3 and still give knot-forms. 
Much less can p be greater than %+3. 
10. A given clutch of a leading partition does not uniquely deter- 
mine a form. The following proposition however holds. 
IJ. Tatorem.—All or none of the forms determined by any 
given clutch of a partition are knot-forms of the order considered. 
For, all forms to be had from any clutch of a given partition may 
be obtained by taking all the possible different changes (consistent 
with the given clutch) of relative position of the various parts. 
But these can all be effected by successive interchanges of the con- 
nections of two parts, whether such connections are direct (by a 
single bond), by a 2-gon, by a 3-gon, or are more complicated. We 
may therefore confine the attention to a definite portion of the knot 
and keep the remainder fixed. Let A and B, (Fig. 1, Plate I), be two 
parts connected as shown. ‘Two strings, or two parts of a single 
string, are involved. If there were more all but two would be 
closed. Let the ends of these strings, or parts of a single string, 
leave the portion of the form under consideration at @ and ¢ on the 
perimeter of A, and 6 and d on that of B. Cut at these points and 
call the ends a and a’, band 6',c and ¢’,d and d'; a, 6, c and d remain 
fixed. Now revolve through 180° about the axis AB, and join the 
free ends. 
Before the change there may be three cases. The strings may be 
oe ae or i ee After the change a’ is joined to ¢, and c’ to 
a; b' tod, and d' to b. 
He c 4 Heconare ke gee 
ac’ d'b 
\" ’ ee bd bd 
Mee dee 
i bd'ae 
wa ac’ b'd 
Therefore coming up to this part of the knot on any string, we 
must leave on the same string before and after the change. If then 
the form was a knot before the change it will be one after, and 
if a link before it will be a link after. 
11. Cotls.—A succession of n 2-gons constitutes an n-coil, which 
may be open or closed. Since at the 3rd or 2n+ 1st crossing of a 
coil the strings have the relative position of the first crossing, if the 
coil be closed by carrying around the ends to the beginning and 
