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Proceedings of the Royal Society 
figure. Hence the scheme is a complete and definite statement 
of the nature of the knot. 
(a.) One illustration depends upon the fact that all deforma- 
tions of such a cord or wire may be considered as being effected by 
bending at a time only a limited portion of the wire, the rest being 
held fixed. This corresponds to changing the point of view finitely 
with regard to the part altered, and yet infinitesimally with regard 
to all the rest. This, it is clear, can always be done, as the relative 
dimensions of the various coils may be altered to any extent without 
altering the character of the knot All such deformations may be 
obtained by altering the position of a luminous point, and the plane 
on which it casts a shadow of the knot. Any addition to the 
normal number of intersections which may be produced by this 
process is essentially nugatory. 
Another mode, really depending on the same principles, consists 
in fixing temporarily one or more of the crossings, and considering 
the impossibility of unlocking in any way what is now virtually 
two or more separate interlacing closed curves, or a single closed 
curve with full knottings, but with fewer intersections than the 
original one. 
Another depends upon the study of cases of knots in which one 
or more crossings can be got rid of. Here, it is proved that con- 
tinuations of sign are in general lost when an intersection is lost; 
so that, as our system has no continuations of sign, it can lose no 
intersections. 
(6.) Practical processes for producing all such deformations gra- 
phically are given at once by various simple mechanisms. Thus, 
taking 0 any fixed point whatever, let p } a point in the deformed 
curve, be found from its corresponding point, P, by joining PO 
and producing it so that 
PO * Op = a 2 , 
or so that 
PO -pOp — a, Ac., Ac. 
The essential thing is that points near 0 should have images 
distant from 0, and vice versa. And p must be taken in OP produced, 
else the distorted knot is altered from a right-handed to a left- 
handed one, and vice versd. This distinction is shown in the cuts 
