My question refers to the following problem in electromagnetic theory which Gauss was concerned with (see Zur theorie der electrodynamik, volume 5 of Gauss's werke): Determine the work done on a magnetic monopole of strengh m when it is moved along a closed loop L_1 in the magnetic field produced by a closed current loop L_2 with current I. This problem led Gauss to the introduction of the Gauss's linking integral for computing the linking number of two interwinged loops. According to several books, this integral is constant under derformations of the loops as long as their number of interwingings is constant (so it's a measure of the degree of entaglement of the two loops). I'm trying to understand what was his result (what was his topological constant) and also what is his meaning, because according to several books is has an advanced physical meaning (the Gauss's linking integral is an advanced analytic tool).
I would suggest the following paper: http://www.maths.ed.ac.uk/~aar/papers/ricca.pdf
it contains a quite detailed analysis of Gauss original works in modern terms. The linking number appears in computation of so-called helicity of vector fields, and in this form it is often applied to magnetic vector field.