328 PROFESSOR TAIT ON KNOTS. 



so far as to make the assigned numbers of essentially different forms accurate, 

 cannot in any other sense compensate. In other words, there may still be 

 some fundamental forms omitted, while others may be retained in more than 

 one group of their possible transformations. Both difficulties grow at a fear- 

 fully rapid rate as we pass from one order of knottiness to the next above ; and 

 thus I have thought it well to make the most I could of the valuable materials 

 placed before me ; for the full study of 10-fold and 11-fold knottiness seems to 

 be relegated to the somewhat distant future. 



2. The problem which Kirkman has attacked may, from the point of view 

 which I adopt, be thus stated : — •" Form all the essentially distinct polgehdra * 

 {whether solids, quasi-solids, or unsolids) which have three, four, &c, eight, or 

 nine, four-edged solid angles." Thus, in his results, there is no fear of 

 encountering two different projections of the same polyhedron; or, in the 

 language of my former paper, no two of his results will give the same scheme. 

 Thus there is no one which can be formed from another by the processes of § 5 

 of my former paper. 



3. But, when a projection of a knot is viewed as a polyhedron, we necessarily 

 lose sight of the changes which may be produced, by twisting, in the knot itself 

 when formed of cord or wire ; a process which (without introducing nugatory 

 crossings) may alter, often in many ways, the character of the corresponding 

 polyhedron. This subject was treated in §§ 4, 11, 14, &c, of my former paper. 

 But it is so essential in the present application that it is necessary to say some- 

 thing more about it here. It would lead to great detail were I to discuss each 

 example which has presented itself, especially in the 9-folds ; but they can all 

 be seen in PI. XLIV., by comparing together two and two the various members 

 of each of the groups. 



The following example, however, though one only of several possible trans- 

 formations is given, is sufficiently general to show the whole bearing of the 

 remark, so far at least as we at present require it. 



It is obvious that either figure may be converted into the other, by merely 

 rotating through two right angles the part drawn in full lines, the dotted part 

 of the cord being held fixed. Also, the numbers of corners or edges in the 

 right and left handed meshes in these two figures are respectively as below :— 



* This word is objectionable, on many grounds, in the present connection. But a more suitable 

 one docs not occur to me ; and the qualification (given in brackets) will prevent any misconception. 

 Of course no projection of a true polyhedron can bo cut by a straight line in two points only. 



