WITH A CENSUS OF THE AMPHICHEIRALS WITH TWELVE CROSSINGS. 251 
compartment ^ is joined once to each of the compartments <J 3 , any two adjacent 
compartments S 3 being joined singly. 
In the construction of these amphicheirals it is not, however, necessary that the 
curve used be non-singular ; it is possible to obtain a figure which exhibits the 
amphicheiral property by means of two singular curves arranged as above. For 
example, fig. 7 represents the amphicheiral knot whose compartment symbol is 
Corresponding compartments are not opposite on the sphere ; nevertheless the 
primary and secondary symbols exhibit the identity as to the number and arrange- 
ment of the joins, differentiated only by the right- and left-handed property peculiar 
to an amphicheiral of the second order. By reversal of the one set of compartments 
the amphicheiralism is undisturbed, and this knot is found to be No. 22 (p = 2) of 
the amphicheirals of the first order. However, the skew amphicheiral constructed 
as shown in No. 61 in the plate of knots is not an amphicheiral of the first class 
as defined by Tait. It is equivalent to the knot shown in fig. 4 W , p. 244, which is 
the result of applying to the amphicheiral knot shown in fig. 4, p. 244, two non- 
conjugate distortions. Consequently Tait would call it an amphicheiral of the first 
order and second class. But its compartment symbol 
4 2 4—2 
shows the particular character that belongs to an amphicheiral of the second order. 
It is prevented from being classed as an amphicheiral of the second order by the 
TRANS. ROY. SOC. EDIN., VOL. LII, PART I (NO. 11). 40 
