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Proceedings of the Royal Society 
but the number of such will usually depend upon the particular 
intersection with which we commence the scheme. 
Additional changes of sign, still without any knotting, may be 
introduced by various processes, of which the following is the 
simplest: — When two letters appear together twice, not necessarily 
in the same order, but with like signs, these signs may be changed. 
Thus, the following parts of a scheme 
may be changed to 
PQ....QP 
+ + 
PQ . . . . QP 
+ + 
and the statements already made about nugatory intersections can 
be applied to these and other combinations even when they occur 
separately once only in each of two separate knots on the same 
cord. This, and a great number of similar theorems, allow of a great 
special extension of the nugatory test already given — but an exten- 
sion which cannot be made in any case until the signs of the 
intersections are given as well as the order of their occurrence. 
Again, though, as has been said above, continuations of sign 
disappear when an intersection is lost, it does 
not follow that if a scheme have continuations 
of sign it must necessarily be reducible. The 
annexed diagram is an excellent instance. Its 
scheme contains fourteen continuations, and 
only twelve changes, of sign, and yet the knot 
is irreducible. But if we suppose it cut across 
twice at the single unsymmetrieally placed crossing, and the ends 
joined so as still to preserve continuity in the string, the scheme 
has still fourteen continuations, but only ten 
changes, of sign ; and it does not involve any 
real beknottedness. 
The remaining figure illustrates a fully 
knotted scheme, where there are no continu- 
ations of sign, but in which the mere change 
of sign of one of the intersections produces four continuations of 
sign, and the whole beknottedness disappears. Similar remarks 
apply to most of the preceding figures. 
