WITH A CENSUS OF THE AM PHICHEIR ALS WITH TWELVE CROSSINGS. 245 
There exist four reversible tangles R 1? Ri, V, R x ' of the type 
— 4 — 
1 ; 
3 
II . 
- 6 — 
/ \ 
R x , R/ being conjugate to R 1; R/ respectively. The distortions DiD x , D/D/ 
transform the knot into the amphicheiral form 
as shown in fig. 4\ 
The knot shown in fig. 4" is obtained from the original by means of the distortion 
DjD/. Its compartment symbol is 
but the knot is not an amphicheiral knot of the first class. 
The distortion DiD/ reproduces the original amphicheiral with amphicheiral 
centres 0, O', as shown in fig 4'". In this case the distortion DiD/ amounts to a 
deformation of the knot. Possibly the effect of the above distortion may be accounted 
for by the peculiar symmetry of the knot. 
§ 4. Amphicheirals of the First Order. 
A census of the twelvefold amphicheiral knots of the first and second orders 
is given on pp. 253-255 (shown also on the Plate) ; in the construction of these 
the methods of Tait # have been used. 
Tait’s Method of Construction . — When an amphicheiral knot is fitted on the 
surface of a sphere, as stated on p. 244, the part of the knot on one hemisphere is 
congruent to the part on the other. This congruence persists when the knot is 
subjected to symmetrical deformations by shortening or lengthening corresponding 
* Trans. Boy. Soc. Edin., xxxii, pp. 494-497 ; or Scientific Papers , i, 336-340. 
