Confusion in Maxwell's derivation of Ampere's Force Law

Question

I am reading Maxwell's "a treatise on electricity and magnetism, Volume 2, page 155" about "Ampere's Force Law". I have some confusion in the following pages:

On the top of the page, Maxwell says: (The coordinates of points on either current are functions of s or of s')

Does he mean we parametrize the path of one circuit with variable s and another circuit with variable s'?

Then he says: (If F is any function of the position of a point, then we shall use the subscript$_{(s,0)}$ to denote the excess of its value at P over that at A, thus $$F_{(s,0)}=F_{P}-F_{A}$$ Such functions necessarily disappear when the circuit is closed.)

Does he mean$$\oint F.ds = 0$$ or something else?

If he means something else by "Such functions necessarily disappear when the circuit is closed" what is it?

Answer 1

\begin{align} \int_{0}^{s^{\prime}}\dfrac{dP}{ds^{\prime}} \xi^{2}ds^{\prime} & = \biggl[P\xi^{2}\biggr]_{0}^{s^{\prime}} -\int_{0}^{s^{\prime}}P d\xi^{2}\\ & = \biggl[P\xi^{2}\biggr]_{0}^{s^{\prime}}-\int_{0}^{s^{\prime}}2P \xi d\xi\\ & = \biggl[P\xi^{2}\biggr]_{0}^{s^{\prime}}-\int_{0}^{s^{\prime}}2P \xi \underbrace{\dfrac{d\xi}{ds^{\prime}}}_{=l ^{\prime}}ds^{\prime}\\ & =\biggl[P\xi^{2}\biggr]_{0}^{s^{\prime}} -\int_{0}^{s^{\prime}}2P \xi l ^{\prime}ds^{\prime} \end{align}