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In Maxwell's treatise, he discusses the potential of two closed curves:

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I have a lengthy derivation up to equation 15. In equation 16, $\dfrac{dx}{ds} \dfrac{dx}{ds'} + \dfrac{dy}{ds} \dfrac{dy}{ds'} + \dfrac{dz}{ds} \dfrac{dz}{ds'}$ is replaced by cos $\epsilon$. That is fine. Also since the strength ($\phi$ and $\phi'$) of each shell is unity, it disappears in equation 16. But what I don't understand is the reversing of sign in equation 16.

It is written that "equation 15 gives the potential energy due to the mutual action of the two shells". It is also written that "equation 15 with its sign reversed, when the strength of each shell is unity, is called the potential of two closed curves $s$ and $s'$"

Since a thin magnetic shell is equivalent to a closed circuit, therefore the potential energy of one shell due to another should be equivalent to the potential energy of one closed circuit due to another closed circuit.

Then how is it that the sign got reversed in case of closed curves (circuits).

My derivation:

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Neumann's potential energy existed in positive and negative forms.

Here is an article about Neumann's potential energy taken from the book "Newtonian Electrodynamics"

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Maxwell however treated Neumann's potential energy as kinetic energy which has to be positive.

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