241 
of Edinburgh, Session 1876-77. 
1 and 3 below, where it will be seen that one turn of the coil may 
be regarded as wound round the other — the screw being right- 
handed in 3 and left-handed in 1. 
3 . 
It is easy to show that these methods give merely different 
views of the same knot. The simplest way of doing this is to 
suppose the knot projected on a sphere, the eye being at the centre. 
Arrange so that one closed branch, e.g., A— A , forms nearly 
a great circle. Shifting the eye to opposite sides of the plane 
of this great circle the coil presents exactly the two appearances 
related to one another by the deformation processes given above. 
What was inside the closed branch from the one point of view 
is outside it from the other, and vice versa. 
Thus 1 and 2 above are the only forms with three non-nugatory 
intersections. 2 may be formed from 1 by putting 0 in either of the 
three border areas, each of which has two sides only. If 0 be placed 
in external space, or in the inner three-sided area, 1 is reproduced. 
Similar remarks apply to the deformation of 2. 
Figures 4 and 5 are the only forms with four valid intersections. 
Like 1 and 2, the first of them is a clear coil (see below), the second 
not clear. And, of course, any deformation of either produces the 
other, or reproduces itself. 6, 7, 8, 9, are forms of an essentially 
not-clear arrangement, with five intersections. The numbers 
inserted in 6 show which form is produced by placing 0 in the 
corresponding area. The only other forms having five intersec- 
