CONSTRUCTION OF KNOTS OF FEWER THAN TEN CROSSINGS. 293 



(af) gives us 7 E, asymmetric ; 

 (hf) „ 7 F, monozone. 



Thus there are six subsolids, 7 A 7 B . . . 7 F, of which we read on their figures 

 that 7 C, 7 D, and 7 E are unifilar. 



38. To obtain the unsolids 7 G, &c., without concurrences, we have to lay 2 

 upon 5, 3 upon 4, and 2 . 2 upon 3, 



±Affc on 5 A gives 7 G, monozone, 



This is 2 upon 5. We cannot lay the same charge on 5 B destroying the con- 

 currence, without completing an unsolid having two marginal charges of which 

 we have just imposed only one ; which is forbidden (art. 25). 

 For 3 upon 4, 



3 Aff on 4 A gives 7 H : 



z Afe on 4 A gives 7 I; vide the figures. 



Observe that we can impose 3 A only by the sections ff&ndfe; for it has no 

 edge to lose at a section ef or ee ; and if we attempt to lay it on a flap h by ffc, 

 we merely turn h into a concurrence of two, which is not permitted as a result 

 of any charging operation. 



In 7 H and 7 I the 2-zoned and the zoneless 2-ple axes of 4 A, after being 

 loaded symmetrically by 3 A, retain their repeating polarity, but from being 

 , janal have become heteroid, not janal. 



For 2 . 2 upon 3, we lay on 3 A the two charges iAffc, which stands for 

 twice tAffc. The result is 7 J, in which one flap is zoned polar, and two are 

 epizonal. Thus there are four unsolids, 7 G, 7 H, 7 I, 7 J, without concurrence. 



39. For unsolids, 7 K, &c, having concurrences, we obtain on 6 A, 7 K ; on 

 6 B, 7 L and 7 M ; on 6 C, 7 N ; on 6 D, 7 P ; on 6 E, 7 Q ; on 5 A 7 R and 7 S ; on 4 A, 7 T 

 and 7 U ; on 3 A, 7 V ; eleven of them. 



Thus we have 21 knots of seven crossings, of which 6 are subsolids and 15 

 are unsolids. Their symmetry and circles are to be read on their figures. 



Twelve of them, 7 C, 7 D, 7 E, 7 G, 7 H, 7 I, 7 L, 7 P, 7 S, 7 T, 7 U, 7 V, are unifilar, of 

 which all but 7 I have been found and figured by Tait. See Plate XV., Trans. 

 R.S.E., 1876-77, for his eleven figured unifilars, and his reduction of them to 

 eight. 



40. The meaning of the symbols ffc, ff, and fe is clear from the figures 

 7 G, 7 H, 7 I. In reduction of 7 G, after making the linear section, two flaps have 

 to be restored. Also after section of the two parallels in 7 H, the cut portions 

 have to be united to make two flaps on the severed knots ; and after section 

 in 7 I they have to be united to restore the flap on the charge 3 A and the edge 

 on 4 A. 



We shall see an example of the section ef in 9 D^, and of ee in 8 Ak and 9 Di ; 

 vide the figures. 



