490 PROFESSOR TAIT ON KNOTS. 



in the congruent figure correspond to A, B, 0, &c, in the original, they will be 

 adjusted to one another as follows. (The case of five is taken as being suffi- 

 cient to show the general principle.) 



ABODE 

 E' D' C B' A'. 



Now if B be joined with D, however the joining line be linked with the others, 

 B' will be joined to D' ; and these parts will form, together, one closed circuit, 

 so that the figure is not a knot. Similarly if A and E be joined. Similarly 

 if A be joined to B, and also D to E. If C be the terminal of the free 

 lap, so will C ; and again we have a figure consisting of more than one 

 string. 



It will be observed that the common characteristic of these excepted cases 

 is that each possesses at least partial symmetry in the mode in which points 

 to be joined are selected from the group. Hence the rule for selection is 

 simply to avoid every trace of symmetry. 



Even when this is done the final result may be a composite knot, i.e., two 

 or more separate knots on the same string. These can be detected and 

 removed at once, so that it is not necessary to lay clown rules for preventing 

 their occurrence. 



Repetitions of the same form from different points of view form the only 

 really troublesome part of this process. These are inevitable, as we see at 

 once from the fact that there may be several essentially different ways of 

 cutting the complete quasi-symmetrical figure into congruent halves by lines 

 meeting it in the same odd number of points. But it may also often be cut by 

 one such line in one odd number of points and by another in a different odd 

 number. 



Still, with all these inherent drawbacks, the method is applicable without 

 much labour to the tenfold amphicheirals ; and it fully answered my purpose. 



0. I had proceeded but a short way with the application of this method 

 when I found that there may be more than one distinct amjmicheiral belonging 

 to the same type. 



One example of this had been already given in § 48 of Part I. while I was 

 dealing with amphicheirals, and again in Part II. in my census of eightfolds 

 (Type V.), but I had carelessly passed it over as a special peculiarity probably 

 due to the fact that the knot in question, though not composite, was constructed 

 of portions each of which possessed, all but complete, the outline of the four- 

 fold amphicheiral. From the point of view taken in § 4 above, however, the 

 reason of the property is evident. For if the half knot, when the extremities 

 of the strings are all held fixed, be capable of a distortion which shall change 

 the relative positions of some of its meshes or the numbers of their corners, 



