PROFESSOR TAIT ON KNOTS. 495 



Hence, for a true knot, both ends of the axis must be the middle points 

 of laps ; and therefore — 



There must be two laps, at least, in every amphicheiral knot, each of which is 

 common to a pair of corresponding right and left handed meshes ; and when the 

 whole is symmetrically stretched oxer a sphere the middle points of these laps are 

 at opposite ends of a diameter. 



3. With regard to the middle point of either of these laps, the various 

 pairs of corresponding right and left handed meshes are situated at equal 

 arcual distances measured in opposite directions on the same great circle. 

 Hence if the whole be opened up at the middle point of either of these laps 

 and projected on a plane symmetrically about the middle point of the other, 

 the halves into which the plane figure is divided by any straight line passing 

 through the latter point are congruent figures applied on opposite sides of 

 that line as base ; the point being, as it were, a centre. There are, thus, at 

 least two ways of opening up any amphicheiral knot so as to exhibit this 

 species of quasi-symmetry. 



What precedes is on the supposition that the system of right, or of left, 

 handed meshes can be applied to itself in one way only. If there be, as 

 happens in some specially symmetrical cases, more than one way of doing this, 

 there is a corresponding increase in the number of pairs of common laps, as 

 defined in the preceding section. 



It has also been assumed above that, on the sphere, the systems of right 

 and left handed meshes are not only similar but congruent. The question of 

 the possible existence of knots in which the system of right hand meshes 

 shall be the reversion of the left hand system will be considered later. 



4. We now obtain a perfectly general, though of course in one sense tentative, 

 method of constructing amphicheirals of any order. Think of the result of § 3 

 as to the congruency of the halves of the knot when opened at either of the pair 

 of corresponding laps. As a continuous line necessarily cuts the projection of a 

 complete knot in an even number of points, the half figure which is to be 

 drawn on one side of the common base must meet it in an odd number of 

 points because one lap has been opened. Let these be called, in order, A, B, 

 C, &c. Then, to form the half figure, these points must be joined in pairs, 

 the odd one forming one end of the line whose other terminal is at the broken 

 lap. These joining lines, and that with the free end, must be made to inter- 

 sect one another in a number of points equal to half the knottiness of the 

 amphicheirals sought. Every mode of doing this gives a figure which, when 

 its congruent has been applied on the other side of the base, possesses the 

 amphicheiral quasi-symmetry above described. 



5. To ensure that the figure shall be a knot, and not a link or a set of 

 detached figures, the following precautions are necessary. If A', B'. C, &c, 



