PROFESSOR TAIT ON KNOTS. 497 



the same can of course be done with the congruent half. The whole pre- 

 serves its type, and is still amphicheiral, but it becomes an essentially distinct 

 form. 



It will be seen that there is one type of tenfolds which has four different 

 amphicheiral forms ; another contains three ; while there are four types each 

 with two forms. The remaining seven amphicheiral types are either unique 

 forms or have no amphicheiral distortion. 



7. We are now prepared for one extension of the definition of an amphi- 

 cheiral given in § 1 above. But we prefer to establish a new and independent 

 definition : — thus 



An amphicheiral knot of the First Order and Second Class is one which can 

 be distorted into its own perversion. 



Under this definition every distortion of an amphicheiral knot is included, 

 even although it be such that its right and left handed meshes do not corre- 

 spond to one another in pairs. For, whatever be the distortion, and whatever 

 parts of the knot be affected by it, an exactly similar distortion might have 

 been applied to the congruent parts of the original amphicheiral. These two 

 distorted forms are, of course, capable of being distorted one into the other : — 

 and that other is its perversion. 



Every amphicheiral knot of the first order and second class corresponds to, 

 and can be distorted into, at least one of the first class : — but the converse is 

 not necessarily true. 



8. Whether there are other classes of amphicheirals of the first order besides 

 these I do not yet know. I have made attempts to construct a specimen of a 

 supposed Third Class which should have the property of being changed into its 

 own perversion by the twisting of a single, limited, portion, while the result could 

 not be obtained by any simpler method. Such forms, if they exist, must in 

 general be incapable of distortion into amphicheirals of either the first or the 

 second class. This search has been fruitless. Among the requirements which 

 it introduces, is the necessity for an ordinary amphicheiral in which two pairs 

 of corresponding right and left hand meshes shall have one common corner ; a 

 condition which does not seem to be satisfied except by the simplest (amphi- 

 cheiral) link, in which indeed it must be satisfied, as there are but four com- 

 partments in all. But this gives no satisfactory solution. 



9. We may now take up the curious question raised in the last paragraph 

 of § 3 above. 



A simple method of producing arrangements in which the group of left- 

 handed meshes is similar to, but not congruent with, that of the right-handed 

 follows at once from the fact that, if one end of a diameter of a sphere trace a 

 figure of any kind, the other end traces a similar and equal but (except in 

 special cases of symmetry) non-congruent figure. These figures can, if we 



