


178 PROFESSOR TAIT ON KNOTS. 
(6). To make this process give the distinction between crossing over and 
crossing wander, we may suppose the two coins to be of different 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, the vertical or opposite 
corner will have two silver or two copper coins, the others one copper or one 
suver 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 all 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 versd, furnishes an 
exceedingly simple and readily applied test (not, however, as will soon be seen, 
in itself absolutely conclusive of identity, though absolutely conclusive against 
it), which is of great value in arranging in family groups (those of each family 
having the same number of silver crossings), the various knots having a given 
number of intersections. 
(y). Or, still more simply, we may dispense altogether with the copper coins, 
so that, going round, we pitch a coin into the field to the s7ght at each crossing 
over, to the eft at each crossing wader. When the coins are in the same angle 
the crossing is a silver one, when in two vertical angles it is copper. Each of 
these three processes has its special uses. 
§ 37. This process, thus limited, is obviously 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. 
Hence 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 beknot- 
tedness. To make the measure complete, we must record the numbers of non- 
nugatory silver and copper crossings separately, with the number to be deducted 
as due merely to the coding of the figure. This last is a very important matter, 
and will be dealt with later. 
§ 38. When unit current circulates in a simple circuit, it is known that 
the work required to carry unit magnetic pole once round any closed curve 
once linked with the circuit is +47. Instead of the current we may substitute 
a uniformly and normally magnetized surface bounded by the circuit. The 
potential energy of the pole in any position is measured by the spherical aperture 
subtended at the pole by the circuit ; but its sign depends upon whether the north 
or south polar side is turned to the pole. Hence the pole has no potential 
energy when it is situated in the plane of the circuit but external to it, and the 
