502 PROFESSOR TAIT ON KNOTS. 



Kirkman's lists. This had the disadvantage of mixing up together types 

 with very different relative numbers of right and left handed meshes. On the 

 present occasion I have taken in the first rank the knots which have an equal 

 number of meshes (six) of each kind, next those which have respectively 5 and 

 7, 4 and 8, &c. This will considerably simplify the process of seeking for any 

 particular ten-fold in so long a list. The arrangement of the various types in 

 each rank, however, follows somewhat closely the order of their earliest 

 appearance in the first list which I got from Mr Kirkman, that upon which I 

 commenced the present work. 



To identify any 10 fold, all that is necessary is to count the numbers of 

 corners in the respective right and left handed meshes, look out the contracted 

 expressions for the corresponding partitions of 20 in § 18, and then search below 

 for the symbol, or pair of letters so obtained. Their order, of course, is imma- 

 terial, as it can be altered by a mere change of mode of projection. If the 

 symbol occur more than once, a closer examination must be made, account 

 being now taken of the way in which the right, or the left handed, meshes 

 are coupled together. This is easily done as in § 20 of my first paper. 



20. The number of distinct forms which I detected as not contained in Mr 

 Kirkman's first list of 10 folds bears a far smaller ratio to the whole than was 

 the case with the ninefolds. I consider that this is due not to my remissness, 

 but to Mr Kirkman's improvements in his methods, i.e., rather to the non- 

 existence than to the non-detection of omissions ; and I think it is improbable 

 that any distinct variety of a recognised type has escaped detection. Thus in 

 the present census some types may be omitted (this is more likely to be true 

 of unique types than of others) ; and I may have, as already indicated, 

 grouped in two or more smaller detachments the varieties of one and the 

 same type. But the possibility of either defect is due to the somewhat 

 tentative nature of the methods employed. 



The guarded way in which I spoke (Part II., § 1) of the completeness of the 

 Census has been justified by a recent observation made by Mr Kirkman, viz., 

 a 9 fold not included either in his list or in mine. Fortunately this knot, 

 figured as fig. L, PI. LXXIX., is not a new type but a distinct form of type 

 VI. of the 9 folds as shown in the Plate attached to Part II. My methods 

 ought to have supplied this additional member of a group, of which some 

 forms had been furnished by Kirkman ; but I had not, at the time, much 

 readiness in applying them. The labour of the 10 folds has made me much 

 more skilful than before in this matter. 



21. In the following list, the order is the same as in the plates. The symbols 

 for each knot are so written that the second, in all cases, corresponds to the 

 group of meshes to which (as the figure happens to be drawn) the amplex 

 belongs. 



