296 REV. T. P. KIRKMAN ON THE ENUMERATION, DESCRIPTION, AND 

 We have on 6 E, 



By 



by 



(««) = 64>(23) or (45) = 63. 

 (fl«) = 55>(12) or (45) = 53. 



(<m)=64 we get 8 Pi, and 

 (aa) ~55 we get 8 S. 



Upon 6 F we see that (ab), (ad), (ac), are the only flaps that can block linear 

 section and exclude concurrence, 



not 



(«5) = 54,33>(43) = 54,33? 

 (ac)=54 >(43)=44, 

 (ad) = 63 >(43) = 64. 



Drawing the flaps (ab) and (ac) we get 8 T and 8 U. 

 Upon 6 G only (ab) can exclude concurrence. Here (34) is fixed (art. 22). 



(a&)=54>(56)^54? (12) = 34. 



Drawing (ab) we obtain 8 V symmetric as we expected, 

 having three different flaps, one zonal, one epizonal, and 

 one asymmetric. 



One more subsolid, 8 W, is built on 6 J, which has only 

 one edge and one angle. 



We have constructed twenty-two subsolids of eight crossings, 8 A . . . 8 W, 

 whose symmetry and circles are seen on their figures. Seven of them are 

 unifilar. 



In naming these knots I use an alphabet of 25 letters, omitting the letter 0. 



Thus, 



A, B, . . . z ; Aa, Ab, ... Az; Ba, Bb, . . . ~Bz ; 



are each a set of 25. 



45. For the unsolids 8 X, &c, of eight crossings which have no concurrence, 

 we have to lay 2 upon 6, 3 upon 5, 4 and 2 - 2 upon 4. 



For 2 upon 6 we can impose 4 Ajfc on G A once, on G B twice, on 6 C once, on 

 6 E once, and on fi F once, but on no other of six crossings, without violating the 

 rules in arts. 25, 28. 



i A ff c on e A s ives s x ; 



„ 6 B „ 8 Y and 8 Z ; 

 F Ac- 



» 6 » 8 ' 



wliose descriptions are read on their figures. The monozone 8 A& has four 

 different flaps, one epizonal upon the marginal charge, and three zonals, in the 

 zonal plane. 



