WITH A CENSUS OF THE AMPHICHEIRA LS WITH TWELVE CROSSINGS. 247 
the complementary form of the knot. For example, the complementary form given 
by fig. 5 is fig. 5' ; the identity of this with fig. 5 may be recognised either from the 
compartment symbol or from the intrinsic symbol (p. 240). 
Tait points out how, in the structure of the framework, m-filar knots can be 
avoided, and that composite knots, which are at once detected, must be discarded. 
By the index of a knot, with respect to the pair of amphicheiral centres 0, O', is 
meant the number p which has been defined earlier. For a knot of order n, it may 
be shown that n is not less than 4 p\ hence, for n = 12, the only values of p to 
consider are 1, 2, 3. That is, in the construction of the amphicheiral knots of 
order 12, it is unnecessary to consider frameworks with more than four bends, 
including the infinite bend. 
If a knot has a second pair of amphicheiral centres, not necessarily on the line 
00', the corresponding index may be the same as or different from p, and hence 
the knot may be constructed on a different framework. For example, knot No. 31 on 
the Plate, shown also in fig. 6, is of index 2 with respect to the pair of amphicheiral 
centres 0, O', but of index 3 with respect to the pair 0 l5 0\. Also Nos. 34, 35, for 
which p = 2, belong to the set p = 3 (see the Plate). 
§ 5. Amphicheirals op the Second Order. 
Tait’s Method of Construction . — When an amphicheiral knot of the second order 
is fitted on the surface of a sphere, a compartment A must either be diametrically 
opposite to its corresponding compartment or it must be the image of the com- 
partment \ in a diametral plane. 
In the first case the desired arrangement of compartments may be obtained by 
means of a closed curve on a sphere and its diametrically opposite curve, together 
with a great circle. Such an arrangement can only lead to a trifilar knot. Further, 
the knot is bifilar if the curve is taken as its own opposite. 
As may be seen by a projection from one pole of the great circle on to the 
tangent plane to the sphere at the other, the proper correspondence of compartments 
in a plane is secured by means of a circle, a closed curve, and its inverse as to the 
circle, but reflected in the origin. If now the closed curve is made to touch the 
