290 
Proceedings of the Royal Society 
going round the curve and pitching a coin into each field or cell as 
it is reached. To make the required distinction between crossing 
over and crossing under , we may suppose the two coins to be of dif- 
ferent kinds, — silver and copper for instance. Let the rule be : — 
silver to the right when crossing over, to the left when crossing under. 
Then, however the path be arranged, of the four angles at each 
crossing, one will have no coins, th'e vertical or opposite corner will 
have two silver or two copper coins, the others one copper or one 
silver coin each. 
It is easily seen that a reversal of the direction of going round 
leaves the single coins as they were, but shifts the pair of coins 
into the angle formerly vacant; also that in the deformed figures 
the circumstances are exactly the same as in the original. Hence 
we may divide the crossings into silver and copper ones, according 
as two silver or two copper coins come together. And the excess 
of the silver over the copper crossings, or vice versa, furnishes an 
exceedingly simple and readily applied test (not however, as will 
soon he seen, in itself absolutely conclusive of identity, though 
absolutely conclusive against it), which is of great value in arrang- 
ing in family groups (those of each family having the same number 
of silver crossings), the various knots having a given number of 
intersections. 
I soon saw that this process, so limited, was intimately connected 
with that required for the estimation of the work necessary to carry 
a magnetic pole along the curve, the curve being supposed to be 
traversed by an electric current, and it occurred to me that we 
might possibly obtain a definite measurement of beknottedness in 
terms of such a physical quantity: as it obviously must be always 
the same for the same knot, and must vanish when there is no 
beknottedness. The measure may be made more complete by 
recording the numbers of non-nugatory silver and copper cross- 
ings separately, with the number to be deducted as due merely to 
the coiling of the figure. I shall recur to this point later. 
When unit current circulates in a circuit, the work required to 
carry unit pole once round any closed curve once linked with 
the circuit is ± 47 t. Instead of the current we may substitute a 
uniformly and normally magnetised surface bounded by the circuit. 
The potential energy of the pole in any position is always measured 
