294 
Proceedings of the Royal Society 
considerations. And it is probable after all that the true measure 
of beknottedness is the smallest number of signs in a scheme which 
must be altered in order that the wire may cease to be knotted. 
It will be found that the alteration of five signs is sufficient to 
remove the knotting from the annexed figure, and the stages of 
operation of the various modes of reduction show that this form can 
be regarded as made up of simpler knots intersecting one another 
on the same string. In such a case it is not easy to give a strict 
definition of the beknottedness in any other way than by defining 
it as the smallest number of changes of sign which will take off all 
the knotting. For the separate knots are virtually independent, 
and to change all the signs in any one of them does not in every 
case necessarily simplify the knot. Uncorrected the work is 
- 13 x 47 t. Corrected it is - 10 x 47r, which agrees with the 
removal of the beknottedness by change of five signs only. 
If the sign of the one unsymmetrical crossing be altered, four 
changes of sign will suffice; for the uncorrected work is - 11 x 47r; 
corrected it is - 8 x 47r, corresponding to four changes of sign. 
The various modes alluded to in my paper of adjusting the 
(lower) signs so that there shall be no beknottedness, follow at once 
from these remarks. For we may make all the free letters in 
each circuit +, save those which we have taken, in pairs, in some 
previous circuit. 
This test, though extremely useful as above explained in classi- 
fying knots with the same number of intersections, is not fully 
descriptive of a knot, being ambiguous whenever there is more 
than one class of knots with the same number of intersections and 
with the same excess or defect of silver as regards copper crossings. 
This consideration, which promises to clear up some obscure and 
difficult parts of the subject, has led me to some very curious re- 
sults. The most important of these is when a knot, whatever be 
its number of intersections, has equal numbers of silver and of 
