


162 © PROFESSOR TAIT ON KNOTS. 
either of these may be changed into the other, or changed to its own reverse 
(as from left to right). 
‘We may now substitute what we please for the dotted parts. I give only 
the particular mode which reproduces the two forms Spine by LIsTING to be 
equivalent :— 
Another mode of viewing the subject, really depending on the same principles, 
consists in fixing temporarily one or more of the crossings, and considering the 
impossibility of unlocking in any way what is now virtually two or more 
separate interlacing closed curves, or a single closed curve with full knotting, 
but with fewer intersections than the original one. 
Another depends upon the study of cases of knots in which one or more 
crossings can be got rid of. Here, as will be seen in § 33 below, it is proved 
that continuations of sign are in general lost when an intersection is lost ; so 
that, as our system has no continuations of sign, it can lose no intersections. 
§ 15. Practical processes for producing graphically all such deformations 
as are represented by the same scheme are given at once by various simple 
mechanisms. ‘Thus, taking O any fixed point whatever, let p, a poimt in the 
‘deformed curve, be found from its corresponding point, P, by joining PO and 
producing it according to any rule such as | 
PO" Op=a, 
or 
PO + Op=a, &c., &e. 
The essential thing is that points near O should have images distant from 
O, and vice versd. And p must be taken in PO produced, else the distorted 
knot is altered from a right-handed to a left-handed one, and vice versd, as will 
be seen at once by taking the image of the crossing figured in § 1 above. 
It is obvious, from the mode of formation, that these figures are all repre- 
sented by the same scheme,—for the scheme tells the order in which the 
various crossings occur,—and it is easy to show that they give merely different 
views of the same knot. The simplest way of doing this is to suppose the knot 
projected on a sphere, and ¢here constructed in cord, the eye being at the centre. 
Arrange so that one closed branch, ¢.g., A———A, forms nearly a great circle. 
Looking towards the centre of the sphere from opposite sides of the plane of 
this great circle, the coil presents exactly the two appearances related to one © 
another by the deformation processes given above. What was inside the closed 

