776 PROFESSOR C. N. LITTLE ON 



14. For the latter, tables of all possible crossings were made out, as described in 

 [A] § G. To secure accuracy these tables were written out twice, once by myself and 

 again later under my direction.* The limitation of § 7 keeps these tables in bounds. 

 Nevertheless a considerable number of the forms given are reducible, and these must 

 in the after work be detected and excluded. This was invariably done by so distorting 

 the form that a sequence of three overs occurred. 



1 5. Upon these tables also the twist of each form was computed. 



1 6. From the tables the crossings were marked upon the tracings, giving all possible 

 forms of non-alternate knots. 



Forms are regarded as distinct only when compartments are dissimilar, or when the 

 direct connections of the compartments are dissimilar in the number or character of the 

 crossings. The tables lead to a certain number of forms equivalent to others already 

 given, and differing only in the circumstance that the form had been rotated through an 

 angle -n about an axis of symmetry in the plane of the projection. These were excluded 

 from the plates. In the case of forms which as alternates are amphicheiral, two 

 equivalent forms occur which differ in that the amplexum partition of one is made the 

 non-amplexum partition of the other. Such duplicates are also omitted. 



17. The task of finding the knots from the knot forms was exceedingly laborious, 

 and one that I should not have been able to accomplish except for the constant check 

 given by the twist. 



The process was as follows : — The twenty-four knots of which the forms obtained in 

 (a) § 13 are projections were easily found. Every other form was examined to see if 

 it could be distorted so as to have two parts with three alternate connections. If this 

 could be done, its proper place in these twenty-four knots was at once known. Where 

 this was found to be impossible, the form was distorted into a form of which the knot 

 was known. In the end it was necessary to see that all permissible distortions of the 

 forms of each knot gave only forms of the same knot. The conditions attendant upon 

 this process are very numerous ; I must repeat the caution previously given that until 

 a simple test is found which shall distinguish the forms of one knot from those of every 

 other in the same class, it may happen that non-alternate knots regarded as distinct may 

 be in reality the same. 



1 8. The identification of a given tenfold non-alternate by means of the census may, 

 in a particular instance, require some labour. In the first place it must be ascertained 

 that the given knot is reduced. For this, unfortunately, no simple test is as yet known. 

 Next observe the twist, to determine the class in which the knot will be found. Then 

 find the number of the corresponding alternate knot and compare with the list of § 19. 

 If this number is to be found in more than one knot of the class, a closer comparison 

 with the forms of the plate must be made. In case of apparent failure to find the form, 

 the limitations of § 16 must be kept in mind. 



* By Dr H. F. Blichfeldt, now instructor, at the time a student in Stanford University, for whose care my 

 thanks are due. 



