500 PROFESSOR TAIT ON KNOTS. 



(indicated by heavier lines, one continuous, the other dotted) in which we can 

 choose the laps which are to form the circle with regard to which it is its own 

 inverse. 



Fig. 08 of the 10-folds, which by reason of its symmetry belongs to both 

 orders of amphicheirals, can have its circles shown as in figs. (E) and (F). 



15. But if we take a non-symmetrical knot of the kind B(a), such as fig. A 

 above treated, we obtain some still more striking results as to the number of 

 ways in which we can choose the laps which form the circular portion. In 

 this figure corresponding right and left handed meshes are marked with the 

 same letter. 



Thus, if we throw out the right hand mesh, d, from the contents of the 

 circle and take in the left hand d instead, the figure (drawn to show the new 

 circle) becomes fig. G. 



If we throw out/, and take the amplexum instead, we obtain fig. H. 



But, if we throw out from the circle g, c, and e, and take instead of them 

 the corresponding external meshes, the figure takes the curious form K. Here 

 the full line is the new boundary between the two halves of the figure. This 

 new boundary, as well as the entire figure, is easily seen to be its own inverse 

 with regard to the part bounded by the heavier portion of the full line. This, 

 however, is only one of four ways in which it might be selected from the full 

 line alone. Such modifications are very curious as well as numerous, but we 

 cannot pursue them here. 



16. In the upper rows of Plate LXXIX. I have given the amphicheirals 

 of the first class, up to the tenfolds inclusive. They are drawn on the principle 

 of § 4 above, and the first form in which each presented itself has been 

 preserved. A comparison of these, with the corresponding figures as drawn 

 in Plate LXXX. directly from Kirkman's results, is very instructive. 



[Added, Oct. 19, 1885. — Though the general statement in § 11 above is 

 true from the point of view there taken, there is a possibility of evading it. 

 Thus, if we draw a figure like E, Plate LXXIX., but with a four-pointed star 

 inside, we get vii. of the 8-folds; which is thus shown to be an amphicheiral 

 of the Second, as well as of the First, Order. But, if we try a three-pointed 

 star, we get the simp 1 est trifilar locking; as in Part I. § 42 (1), and Part II. § 8.] 



