330 PROFESSOR TAIT ON KNOTS. 



And the Plate also shows that there are .41 simple forms of 9-fold knotti- 

 ness. Besides these, and not figured, there are 5 made up of two mere separate 

 knots of lower orders, and one which is made up of three separate trefoils. 



5. Thus the distinct forms of each order, from the 3rd to the 9th inclusive, 

 are in number 



1, 1, 2, 4, 8, 21, 47 ; 



or, if we exclude combinations of separate knots, 



1, 1, 2, 3, 7, 18, 41. 



The later and larger of the numbers in these series, however, would be con- 

 siderably increased if we were to take account of arrangements of sign at the 

 crossings, other than the alternate over and under which has been tacitly 

 assumed; and which are, in certain cases, compatible with non-degradation of 

 the order of knottiness. This raises a question of considerable difficulty, upon 

 which I do not enter at present. Applications to one of the 8-folds and to one 

 of the 9-folds will be found in my former paper, § 42 (1). 



Another interesting fact which appears from Plate XLIV. is, that there are 

 six distinct amphicheiral forms of 8-fold knottiness : at least if we include one, 

 not figured, which consists of two separate 4-folds ; in which case we must 

 consider that there are two six-fold amphicheirals, the second being the com- 

 bination of right and left handed trefoils, described in § 13 of my former paper. 

 Thus the number of amphicheirals is, in the 4-fold, 6-fold, and 8-fold knots 

 respectively, either 1, 2, 6, or (if we exclude composites), 1, 1, 5. All but two 

 of these 8-fold amphicheirals were treated in my former paper, two having been 

 separately figured, and the other being a mere common case of the general 

 forms of § 47. 



Finally, as a curious addition to the paragraphs on the genesis of amphicheiral 

 knots, given in my first paper, I mention the following, which is at once suggested 

 by the amphicheiral 6 -fold : — Keeping one end of a string fixed, make a loop on 

 the other ; pass the free end through it and across the fixed end ; pass the free 

 end again through the external loop last made, then across the fixed end, 

 and so on indefinitely. The second time the fixed end is reached we have 

 the trefoil (if the alternate over and under be adhered to), the third time we 

 have the amphicheiral 6-fold; and, generally, the nth time, a knot of 3(w-l) 

 fold knottiness, which is amphicheiral if n is odd. Three of these were, inci- 

 dentally, given in my former paper. 



But, reverting to the main object of my former paper, we now see that the 

 distinctive forms of less than 10-fold knottiness are together more than sufficient 

 (with their perversions, &c.) for the known elements, as on the Vortex Atom 

 Theory. 



6. From the point of view of theory, as suggested in §§ 12, 21, of my 





