38 C. N. Littleh—Knots, with a Census for Order Ten. 
partition. But since the subordinate partitions here belong to the 
same class, every form, with certain exceptions, will be found 
twice; this affords a check on the completeness of this part of the 
work. The exceptions are the amphicheiral knot-forms. These have 
the same partitions similarly connected both as leading and as sub- 
ordinate partitions, These therefore appear but once. 
In the partition 53°, which was given as the shortest possible 
illustration of Table V, clutch (3) furnishes a single form D’s,, which 
already had been obtained under 54°32*; clutch (6) gives two which 
had been obtained, Du, under 64372’ and D*x, under 54°32*. Clutch 
(8) gives the only new knot-form from this partition, viz: the 
amphicheiral D*o. 
From the clutches of Classes III, IV, V and VI 364 ten-fold knot 
forms are obtained. 
21. The Derivation of Knots from Knot-Forms.—Prof. Tait has 
not described the methods which he used in his derivation of the 
knots of lower orders from the knot-forms. In 1884 he says :* “ the 
treatment to which I have subjected Kirkman’s collection of forms, 
in order to group together mere varieties or transformations of one 
special form, is undoubtedly still more tentative in its nature; and 
thus, though I have grouped together many widely different forms, I 
cannot be absolutely certain that all those groups are essentially dif- 
ferent from one another.” 
If a ten-fold knot be placed upon a plane in such a way as to have 
but ten crossings the eye will project it upon the plane in a form 
which will be found among the 364 above obtained. If the knot 
gives more than one form it will be possible to obtain any other of 
its forms by one or more turnings over of restricted portions of the 
knot while the remainder is held fixed. Now the string cannot issue 
from the portion of the knot that is turned at more than four points, 
for in that case the turning would introduce consecutive overs, and 
one or more additional crossings; the portion of the knot that is 
turned must therefore be wholly between two parts of the given 
knot form and in turning it we untwist two of the strings at one 
point and twist two at another, the result being simply to change the 
position of a single bond from one end of the connection to the 
other. The class of the form is therefore not changed, and all the 
forms of any knot belong to the same class. 
22. Moreover in order 10 and lower orders all the forms coming 
from any clutch are obtained by changing the position of single 
* Trans. Roy. Soc. Edin., vol. xxxii, p. 327. 
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