PROFESSOR TAIT ON KNOTS. 161 
Part II. 
The number of Forms for each Scheme. 
. 
§14. A possible scheme being made according to the methods just described, 
with the requisite number of intersections, let it be constructed in cord, with the 
intersections alternately + and —. Then [since all schemes involving essentially 
nugatory crossings, like those mentioned in § 2, must be got rid of, as they 
do not really possess the requisite number of intersections] no deformation 
which the cord can suffer will reduce, though it may incréase, the number of 
double points. Ifit do increase the number, the added terms will be of the 
- nugatory character presently to be explained. If it do not increase that 
_ number, the scheme will in general still represent the altered figure. For, as 
we have seen, the scheme is a complete and definite statement of the nature of 
the knot. But, as already stated, in certain cases the knot can be distorted so 
as no longer to be represented by the same scheme. 
All deformations of such a knotted 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 jinitely 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. In general © 
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. As is easily seen, it generally occurs in the form of the 
avoidable overlapping of two branches, giving continuations of sign. 
The process pointed out in § 11 gives a species of deformation which it. 
is perhaps hardly fair to class with those just described, though by a slight 
extension of mathematical language such a classification maybe made strictly 
‘accurate. It may be well to present, in passing, a somewhat different view of 
the application of this method. Thus, it is obvious at a glance that the two 
following figures are mere distortions of the second form of the 4-fold knot 
figured in § 17 below :— 
Also it will be seen that by twisting, the dotted pete being held fixed, 
VOL. XXVIII. PART I. ; OT 
