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Proceedings of the Royal Society 
tions, are the clear coil of two turns, whose scheme is the first 
given in I ( b ) above, and its solitary deformation. 
(c.) Hence to draw a scheme, select in it any closed circuit, e.g., 
A ... . A — the more extensive the better, provided it do not 
include any less extensive one. Draw this, and build upon it the 
rest of the scheme ; commencing always with the common point A, 
and passing each way from this to the next occurring of the junctions 
named in the closed circuit. [It is better to construct both parts 
of the rest of the scheme inside , and then invert one of them, as we 
thus avoid some puzzling ambiguities.] Inversions with respect 
to various origins will now give all possible forms of the scheme. 
III. Thus the scheme is. perfectly definite as to the general 
shape of the curve, if we take the possible deformations into account. 
And the spherical projection, already mentioned, will in general 
allow us to regard and exhibit the knot as a more or less perfect plait. 
It does so always when the coil is clear , i.e., when all the turns 
of the cord maybe regarded as passing in the same direction round 
a common axis thrust through the knot. When the coil is not 
clear some of the cords of the plait are doubled back on them- 
selves. Thus by drawing the plait from a given scheme we can 
tell at once whether one of its forms is a clear coil or not. 
From this point of view another notation for clear coils is given 
in the form 
a y a 
P ay /? " " 
