334 PROFESSOR TAIT ON KNOTS. 



XXXII. XXXIII. XXXIV. XXXV. 



44442 64422 7632 6633 7542 5544 44433 

 552222 or 552222 4422222 or 4422222 4422222 or 4422222 333333 



.XXXVI. XXXVII. XXXVIII. XXXIX. XL. XLI. 



666 864 882 66222 7722 99 



33222222 33222222 33222222 552222 4422222 222222222 



It will be seen that the above list suggests many curious remarks. Thus, 

 in the eightfolds, we have two different amphicheirals, each having the parti- 



54322 



tions 44332. Again, we have p^o^o f° r a knot which is not amphicheiral, 



as well as 54322 for one which is amphicheiral. (See § 47 of my former paper.) 



54322 

 And we have 44000 standing for two quite distinct knots. All these apparent 



difficulties, however, are due to the incompleteness of the definition by parti- 

 tions merely {i.e., as by Listing's Type-Symbol). For, in addition to this, it is 

 requisite that we should know the relative grouping of the right-handed or of 

 the left-handed partitions. 



In the Plate I have inserted the designations given in my former paper to 

 the various forms of 6-fold and 7-fold knottiness : — and I have also appended to 

 each form the designation of the corresponding figure in Kirkman's drawings. 



The Plate contains a great deal of information of a kind not yet alluded to 

 in this paper. It gives, for instance, an excellent set of examples of Knot- 

 fulness. This term implies (§ 35 of my former paper) " the number of knots of 

 lower orders [whether interlinked or not) of which a given knot is built up." It is 

 to be understood as applied to simple forms only ; for we have set aside, as 

 composite knots, all such as have any one component separable, so that it may 

 be drawn tight without fastening together two laps belonging to one or two of 

 the other components. 



Thus, as a few of the examples of 2-fold knotfulness among the 8-folds, we 

 have 



vi. and xi. (3-fold and once-beknotted 5-fold) ; 

 and 11. and v. (each two 4-folds) ; while 



in., ix., and xiv. are different forms of two (linked) 3-folds. 



Among the 9-folds we have, for instance, 



xxx. and xxxin. (4-fold and clear-coiled 5-fold), 

 xvi. and xxvi. (3-fold and 8 6-fold), 



xiv., xv., xvin., and xxv. (4-fold and once-beknotted 5-fold). 

 But we have also 



iv., xiii., xxiii., and xxiv. (linked 3-fold and 4-fold), 

 xx., xxvii. (two 3-folds, linked, and with one kink). 



