322 Proceedings of the Boyal Society 
Of course such figures can be transformed or deformed according 
to the methods given in my first paper — the scheme and the type- 
symbol alike remaining unaltered. And alterations of both scheme 
and symbol are, in various classes of cases, producible by the pro- 
cesses of my last paper without any change of links or linking. 
The genesis of the scheme of a link may be most easily studied 
by forming a knot into a link. This is done by cutting both turns 
of the wire at any junction, and joining them again so as to make 
two closed curves instead of one. No intersections are lost by this 
process, except that which was cut across, provided, of course, that 
the original knot had no nugatory intersections, and that none are 
rendered nugatory by the operation of cutting the whole across. 
Any crossing with four adjacent crossings when the turns of the 
coil pass alternately over and under one another will appear in a 
scheme as follows : — 
AXB C X D . . . . 
— I — + - + 
implying that from X through B and C back to X forms one con- 
tinuous circuit; similarly from X through D and A back to X. 
There are but two ways in which continuity can be kept up if 
we cut the cord twice at X, and reunite the ends in a different 
arrangement from the original one. 
It is obvious that if we pass from C to B, by way of X (abolished), 
and similarly with the rest, we divide the continuous closed curve 
into two separate (but generally inter-linked) closed curves. If 
we pass from A, by way of X (abolished) to C, we pass thence in 
time to B, and finally by way of D to A. Thus the curve remains 
continuous, but with one intersection less than at first. And, in 
either case, the alternate order of the signs of the crossings will be 
maintained throughout. 
In the former of these modes we take the part containing C and B 
(and we may, if we please, also take the rest) in the same order as 
before the change. The scheme is therefore, without any other 
change, simply divided into two parts, which are separated from 
one another by the (abolished) junction X in its two positions. 
In the second mode, it is obvious that the letters in one of the 
two parts separated from one another by the mark X in its two 
places are simply to be inverted in order without change. 
