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of Edinburgh, Session 1876 - 77 . 
by the spherical opening subtended by the circuit ; but its sign 
depends upon whether the north or south polar side is turned to the 
pole. Hence there is no potential energy when the pole is situated 
in the plane of the circuit but external to it, and the value is 
± 27r when the pole just reaches the plane of the circuit internally. 
G-auss gave from these results the value of a remarkable double 
integral extending over each of any two closed curves linked 
together in space. Clerk-Maxwell ( Electricity , § 422) has shown 
that this integral may vanish even for a complex linking of the 
two circuits; and a similar difficulty is met with in the single 
circuits with which we are now dealing, so that a special set of rules 
must be made for determining the beknottedness in terms of the 
silver and copper junctions. But the difficulty just mentioned 
leads, as will be seen, to a very curious result. 
To construct the magnetised surface which shall exert the same 
external action on a pole as a current in any given closed circuit 
does, we may either suppose a surface extending 
to infinity in one direction (say, for definiteness, 
upwards from the plane of the paper), and having 
the circuit for its edge ; or we may form, as in 
the figure, a finite autotomic surface of one sheet, 
having the circuit for its edge. The only diffi- 
culty in estimating the work lies in the definite statement of how 
the pole is to move along the curve itself. For, if its path screw 
round the curve, ± 47r must be added to the work for each such com- 
plete turn. As an illustration, 
suppose we bend an india-rubber 
band coloured black on one side, as 
in the figure, so that the black is 
always the concave surface, 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 is an instance 
of the kinematic principle that spiral springs act by torsion. 
Perhaps the most simple definite condition is that which I first 
employed, viz., to make the pole move along the curve, keeping always 
in the osculating plane and on the convex side. But we have then 
to arrange beforehand what is to be done at a point of inflexion. 
A practical rule, however, may easily be given from the con- 
vol. ix. 2 Q 
