498 PROFESSOR TAIT ON KNOTS. 



choose, be taken so as together to form one closed curve ; and this, along with 

 a great circle of the sphere, forms a link of two cords possessing the required 

 property. On the plane we can carry out this construction by describing any 

 figure within a circle, along with its inverse as regards the circle but on the 

 opposite side of the centre ; and arranging so that these may join into a con- 

 tinuous curve linked with the circle. But this arrangement remains a link 

 when we unite the new curve with the circle by so introducing new meshes as 

 to leave the whole possessed of the required property. 



Or, we may trace any curve on a hemisphere, and its image (in the common 

 base) on the other hemisphere. These, together with the great circle separat- 

 ing the hemispheres, give another link solution. 



It is clear, from the essentially limited nature of the spherical surface, that 

 these two methods give the only possible solutions of the problem : — i.e., when 

 the corresponding right and left handed meshes required by the conditions are 

 made equal in pairs, the lines joining similarly situated points in them must 

 either meet in one point (which, of course, must be the centre of the sphere), 

 or they must be parallel. 



10. As I did not at once see how to obtain solutions corresponding to 

 unifilar knots by means of either of these methods, I asked Mr Kihkman 

 whether he knew of a polyhedron which possessed the requisite property. The 

 first he suggested to me corresponded, as I easily found, to a trifilar which 

 belongs to the results of the first method above : — i.e., one of its cords being 

 taken as the circle, the other two were inverses of one another with regard to 

 it. But, as soon as he mentioned to me that the polyhedron, corresponding to 

 a composite knot consisting of two separate once-beknotted 5-folds on the 

 same string, satisfies the special conditions of the present question (though 

 inadmissible on other grounds), I saw why I had failed in obtaining unifilars 

 by the first of the two methods above. For the purpose of avoiding trifilars 

 from the first I had always made the curve traced by either end of the moving 

 diameter (in the process of § 9 above) cross the great circle wherever it met it, 

 so as to join that traced by the other end. No insertions of new meshes could 

 then reduce the whole to a unifilar without depriving it of the property for 

 which it was sought, 



11. But if we make the closed curve traced by one end of the moving 

 diameter touch the great circle in one point, the point of contact must of course 

 be regarded as a crossing, while the circle and the closed curve necessarily fuse 

 into one continuous line. The same happens with the curve traced by the 

 opposite end of the diameter. Thus we may obtain with the greatest ease any 

 number of unifilars satisfying the conditions. And it is clear that, by a slight 

 extension of the definition above, all such knots will be brought under the 

 general term amphicheiral. To make them true knots, i.e., not composites, the 



