246 
Proceedings of the Royal Society 
Whatever be the number of intersections a postponement of no 
places leads to nugatory results. 
A postponement of one place is possible for three and for four 
intersections only. 
Postponement of two places is possible only for ( four ), five, and 
eight — three for seven and ten — four for nine and fourteen — five 
for (eight), eleven and sixteen, — six for (ten), thirteen, and twenty, 
&c. Generally there are in all cases n postponements for 2n + 1 
intersections: — and for 3n + 2 , or 3w + 1 intersections, according 
as n is even or odd. The numbers which are italicised and put in 
brackets above, arise from the fact that a postponement of r places, 
when there are n intersections, gives the same result as a postpone- 
ment of n-r-1 places. [It will be observed that this cyclical 
order of the letters in the even places is possible for any number 
of intersections which is not 6 or a multiple of 6 .] 
When there are n postponements with 2^4-1 intersections the 
curve is the symmetrical double coil — i.e., the plait is a simple 
twist. 
The case with 3rc + 2 or 3n+ 1 intersections is a clear coil of 
three turns, corresponding to a regular plait of three strands. 
VI. Numerous examples are given of the application of various 
methods of reduction. For instance, the scheme 
A^BTCG ! D4EIFI(ji)H5KCLi7| A 
- + + + -P- + - + - + - + - 4- - 4- 
which is rendered irreducible by changing the sign of B, is reduced 
by successive stages as follows : — 
ABC(?D4AIfii)HKBC'LF| A, 
- + - + -+ + 4---+- 4- - 4- 
BC A GD AGcDYLBC H\ B, 
+ __ + _ + _ + __ ++ 
CA(rDdGBC| C; 
1 1 1 - 4 . 
and, finally, 
AGVAGrD\A, 
-+-+-+ 
which is the simple irreducible knot of figures 1 and 3 above. 
