*148 PROFESSOR TAIT ON KNOTS. 
though not always diminished—by a change of position of the luminous point, 
or by a distortion of the wire or cord, which we may suppose to form the knot. 
This wire or cord must be supposed capable of being bent, extended, or con- 
tracted to any extent whatever, subject to the sole condition that no lap of it 
can be pulled through another, 7.¢., that its continuity cannot be interrupted. 
There are, therefore, projections of every knot which give a minimum number 
of intersections, and it is to these that our attention must mainly be confined. 
Later we will consider the question how to determine this minimum number, 
which we will call AKnottiness, for any particular knot ; but for our present pur- 
pose it is sufficient to get rid of what are necessarily nugatory intersections, 1.é., 
intersections which no alteration of the mode of crossing can render permanent. 
These crossings are essentially such that if both branches of the string were cut 
across at one of them, and their ends reunited crosswise, so as to form two 
separate closed curves, these separate curves shall not be linked together, how- 
ever they may individually be knotted, 7.¢., that if they are knots they are 
separate from one another, so that one of them may be drawn tight so as to 
present only a roughness in the string. For in this case the nugatory crossing 
will thus be made to bound a mere Joop. 
{We may define a necessarily nugatory crossing as one through which a 
closed, or an infinitely extended, surface may pass without meeting the string 
anywhere but at the crossing. Or, as will be seen later (§ 20), we may recognise 
a necessarily nugatory crossing as a point where a compartment meets itself. | 
In the first two of the sketches subjoined all the crossings are necessarily 
nugatory ; in the third, only the middle one is so. 
Now these diagrams, when lettered in the manner forthwith to be explained 
(see, for instance, Plate X VI. fig. 1), present respectively the following schemes :— 
AABB|A 
ACBBCA|A 
ACBDCBDAEGFEGF|A 
These and similar examples show that in a scheme a crossing is necessarily 
nugatory, if between the two appearances of the letter denoting that crossing 
there is a group consisting of any set of letters each occurring twice. The set 
may consist of any number whatever, including zero. For our present purpose 
it will be found sufficient to consider this last special case alone, 7.¢., the same 
letter twice in succession denotes a necessarily nugatory crossing. 

