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It is now well known as Stokes's theorem connecting a certain surface integral with a line integral, and has many applications both in physics and in geometry.

The work done in carrying a unit magnetic pole once round a circuit conveying a current is called the “line integral of magnetic force” along that path.

If, for instance, we carry a unit pole in a circular path of radius r once round an infinitely long straight filamentary current I, the line integral is 4πI.

It is easy to prove that this is a general law, and that if we have any currents flowing in a conductor the line integral of magnetic force taken once round a path linked with the current circuit is 4π times the total current flowing through the circuit.

If it is made endless, the magnetic force H is the same everywhere along the central axis and the line integral along the axis is Hl.