PROFESSOR TAIT ON KNOTS. 329 



55332 64332 



443322 an 433332. 



These numbers would necessarily be identical if the forms could be repre- 

 sented by the same scheme. As will be seen by the list below, § 6, these are 

 respectively the second, and the sixth, of the group of equivalent forms of 

 number vin of the ninefold knots. (See Plate XLIV.) 



The characters of the various faces of the representative polyhedra (so far at 

 least as the number of their sides is concerned) are widely different in the two 

 cases. [Mr Kirkman objects to this process that it introduces twisting of the 

 cord or tape itself. No doubt it does, or at least seems to do so, but the 

 algebraic sum of all the twists thus introduced is always zero; i.e., by "iron- 

 ing out " the tape in its new form, all this twist will be removed. I have often 

 used a comparison very analogous to this, to give to students a notion of the 

 nature of the kinematical explanation of the equal quantities of + and — elec- 

 tricity, which are always produced by electrification. If the two ends of a 

 stretched rope, along whose cylindrical surface a generating line is drawn, be 

 fixed, and torsion be applied to the middle by means of a marlinspike passed 

 through it at right angles, one-half of the generating line becomes a right- 

 handed, the other an equal left-handed cork-screw. Thus the algebraic sum 

 of the distortions is zero. And, in consequence, if the rope be untwistable 

 (the Universal Flexure Joint of § 109 of Thomson and Tait's Natural Philosophy) 

 and endless, the turning of the spike merely^ gives it rotation like that of a 

 vortex-ring. Such considerations are of weighty import in many modern 

 physical theories.] 



As will be seen, by an examination of the latter part of Plate XLIV., even 

 among the forms of 9-fold knottiness there are several which are capable of 

 more than one different changes of this kind. Some of these I may have failed 

 to notice. But it is worthy of remark that the 8-folds seem, with two excep- 

 tions, to resemble the 7-folds in having at most two distinct polyhedral forms 

 for any one knot. 



4. Kirkman's results for knottiness 3, 4, 5, 6, 7, when bifilars and composites 

 are excluded, agree exactly with those given in my former paper. I have 

 figured these afresh in Plate XLIV., in the forms suggested by Kirkman's 

 drawings, omitting only the single 6-fold, and the single 7-fold, which are com- 

 posite knots. 



As will be seen in the Plate, where they are figured in groups, there are but 

 18 simple forms of 8-fold knottiness. Besides these there are 3 not properly 

 8-fold, being composite {i.e., made up of two separate knots on the same string) ; 

 either two of the unique 4-fold, or a trefoil with one or other of the two 5-folds. 

 These it was not thought necessary to figure, especially as they may present 

 themselves in a variety of forms. 



