CONSTRUCTION OF KNOTS OF FEWER THAN TEN CROSSINGS. 289 



every possible set of e charges. These, as well as their reflected images when 

 required (although such images are neither registered or figured by us), have to 

 be imposed in every different posture, by every kind of possible section, ffc, ff, 

 &c, on every different set of e flaps or edges that can be selected on the subject 

 knot, and in every different order that symmetry permits without repetition of 

 results (art. 4), so that when the work is done not one of the unsolids Q of n 

 crossings shall have a least marginal subsolid besides the e that we have just 

 imposed, nor have a concurrence upon it. 



26. The linear sections by which the charges are imposed may be any of the 

 five of art. 15. But, observe, when we use the section^, we are to select the 

 charge from our list of subsolids of k + 2 crossings; because {art. 16) two will 

 be lost. For other sections our selection of the charges will be from those of 

 k crossings. 



27. The number of different postures in which a charge can be imposed on 

 Q' depends on the symmetry of the united portions of the Q completed by the 

 union. Let e denote the edge or flap of the subject and e' that of the charge in 

 all the five sections ffc, ff, fe, ef, ee. The rules are three — 



(1) If one or both of e and e be zoned polar, only one configuration is pos- 

 sible by the union ; no second and different (art. 4) can be formed by turning 

 the charge C through two right angles about e , nor by using C, the reflected 

 image of C, when C is not C. Every knot on which is a zone is its own 

 mirrored image. 



(2) If neither e nor e be zoned polar, and if they are not both asymmetric, 

 two and two only different configurations can be made by the above variation 

 of posture of the charge. 



(3) If both e and e are asymmetric, four different configurations can and 

 must be made and registered, due to such variation. 



No more results can be obtained by putting for Q' the reflected image of 

 Q' : nothing is so attainable but repetitions or reflected images of knots already 

 registered. 



On almost every subject Q' and charge C, though having any symmetry, 

 may be found asymmetric, i.e., zoneless and non-polar, flaps and edges, which 

 are to be dealt with by the above rules. 



28. The subject Q' to be charged with a set of least marginal subsolids may 

 have or not have concurrences. All that is required in order that the com- 

 pleted Q shall have no concurrence, is that our number e of charges of k 

 crossings shall be large enough to spoil all concurrences on Q', as well as to 

 cover at least once every marginal subsolid on Q' which has fewer than k + 1 

 crossings. 



In the constructions of this paper, Q and C are one or both symmetric. 

 When asymmetries come to be handled both as subject and charge, the number 



VOL. XXXII. PART. II. 3 B 



