PROFESSOR TAIT ON KNOTS. 179 
potential energy is +2a when the pole just reaches the plane of the circuit 
internally. 
In fact the electro-magnetic force exerted by an element da of a unit 
current, on a unit north pole placed at the origin of a, is 
Vada 
Ta? 
or, aS We may write it, 
1 
V.daV To: 
This is identical in form with the expression for the differential whose 
integral, taken round a closed circuit, is AMpERE’s Directrice.* 
Hence the element of work done by the closed circuit while the pole de- 
scribes a vector Sa, is 
ll 
SW=—S8.8af Mt = —S.8a/daV Fe - 

But, if dQ be the spherical angle subtended at a by a little plane area ds, 
whose unit normal vector (drawn towards the origin of a) is Uv, obviously 
S.UvUa 
1 
dQ =F = —S.UvV 74s. 
Now, in the general formula (Trans. R. 8. E. 1870, p. 76) 
JS Voda=fdsV.(VUvV )ao, 
put 
it 
c= Vi 
and we have 
Vad 1 1 
— Te = flis( Ur Vx, — VSUrV a2) 
= fdsUvv’ at Oe 
Now the double integral always vanishes while Ta is finite, and we have there- 
fore 

sW=— et __S.3ayO=80.. 
That is, the work done during any infinitesimal displacement of the pole is 
numerically equal to the change in the value of the spherical angle subtended 
by the circuit. The angle is, of course, a discontinuous function, its values 
differing by 47 at points indefinitely near to one another, but lying on opposite 
* Electrodynamics and Magnetism, §§ 5-8, Quarterly Math. Journal, 1859. 
