PROFESSOR TAIT ON KNOTS. 181 
mating the electro-magnetic work. To clear this away we require a definite 
statement of how the pole moves along the curve itself. For if its path screw 
round the curve +47 must be added to the work for each complete turn. As 
an illustration, suppose we bend, as in the figure, an 
india-rubber band coloured black on one side, so 
that the black is always the concave surface, and so 
that one loop is the perversion of the other, we find 
on pulling it out straight that it has no twist. If both loops be made by 
overlaying, when pulled out it becomes twisted through two whole turns. This 
illustrates the kinematical principle that spiral springs act by torsion. An 
excellent instance of its connection with knots is to be seen in the process 
employed in §11. For if we have portions of a cord, as in the diagram 
(Plate XVI. fig. 7), the pulling out of the loop in the upper cord changes the 
arrangement, as shown in the second figure. 
A practical rule, which completely meets the GAussIAn problem, may easily 
be given from the consideration of the cylindrical magnetized surface above men- 
tioned. Go round the curve, marking an arrow-head after each crossing to 
show the direction in which you passed it. Then a junction 
like the following gives +47 for the upper branch, and 
nothing for the lower (which, on this supposition, does not ie <a 
pass through the magnetic sheet). Change the crossing from 
over to under, and this quantity changes sign. The junction figured above 
would, in our first illustration, be a silver one. But a still simpler process is 
to go round, as in § 36 (y), putting a dot to the rzght after each crossing over, 
and vice versd. 
§ 41. Now, in order that our rule when applied to knots may give no work 
where there is no beknottedness, we must make the required expression such as 
to vanish whenever all the intersections are nugatory. Those which are nugatory 
only in consequence of their signs are in pairs, silver and copper, and will take 
care of themselves, as we see by special examples like the 
following. Hence the only part to correct for is that de- eG 
pending on the number of whole turns, and the sketch of | 
the india-rubber band above shows that the work at the vertex of each such 
partial closed circuit is simply not to be counted, 2.¢., that the 42, which 
would be reckoned for each such crossing by our rule (positively or negatively 
as the case may be), is to be considered as made up for by the corresponding 
screwing of the pole round the curve. 
§ 42. There must be some very simple method of determining the amount of 
beknottedness for any given knot; but I have not hit upon it. I shall there- 
fore content myself with a few remarks on the subject, some of which are 
general, others applicable only to certain classes of forms. There seems to be 
VOL. XXVIII. PART I. Sua 

