of Edinburgh, Session 1885-86. 
697 
12 circles. Each 15-gon is of zoneless 5-ple repetition, whose 
collaterals are the meshes 246 .. . five times written, showing a 
zoneless repetition. 
The knot must be the zoneless hexarchaxine G = 15 12 6 20 4 30 3 60 2 60 , 
i.e ., of twelve 15-gons, twenty 6-gons, &c. It has six principal 
5-ple axes, ten secondary 3-ple axes, and fifteen 2-ple tertiary axes, 
all the axes zoneless-janal. The triangles and 2-gons are all 
asymmetrical and all alike. 
To construct the knot G. Cut away the summits of the hexarch- 
axine 5 12 , and make what remains of the edges into thirty 2-gons. 
You have the zoned hexarchaxine knot F = 10 12 3 20 2 30 . Each 
triangle aho of F is collateral with three 10-gons, A, B, and C. 
At a in A on the left of b, complete by the 2-gon j3y the small 
triangle a/3y, and at b in B and c in C complete by 2-gons the 
triangles by a and c/3a. Do the like at each of the twenty triangles 
of E, operating in the same direction round each. Thus G is con- 
structed. 
It is easy in like manner to form upon the regular 20-edron 
hexarchaxine knots, both zoned and zoneless, on the 4-edron. such 
tetrarchaxine, and on the cube or its reciprocal such triarchaxine 
knots. 
The following examples of knot-symmetry, perhaps not yet 
noticed, may be found useful : — 
1. 6 2 5 4 3 8 2 4 , 2-ple monaxine contrajanal, a bifilar of 16 crossings, 
whose circles are 
l/ec7y54234896790c5ac; 
lWoa&876532. 
The contrajanal poles are the 6-gons 12486c, and 90/56, whose 
zoneless axis is the only contrajanal diameter. 
2. 9 2 5 6 3 12 2 6 , 3-ple monaxine contrajanal, a bifilar of 24 crossings, 
with the circles 
1 2ij7cif3hg3Adecd05ba5GS978llmn ; 
2hgfec4:ba097 QmnljJc. 
The 3-ple contrajanal poles are the 9-gons 
127234556m, and 890 defigl. 
