Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/458

 CHAPTER XXIII. THEORIES OF ACTION AT A DISTANCE.

On the Explanation of Ampère's Formula given by Gauss and Weber.

846.] The attraction between the elements $$ds$$ and $$ds'$$ of two circuits, carrying electric currents of intensity $$i$$ and $$i'$$ is, by, Ampère's formula,

the currents being estimated in electromagnetic units. See Art. 526.

The quantities, whose meaning as they appear in these expressions we have now to interpret, are

and the most obvious phenomenon in which to seek for an interpretation founded on a direct relation between the currents is the relative velocity of the electricity in the two elements.

847.] Let us therefore consider the relative motion of two particles, moving with constant velocities $$v$$ and $$v'$$ along the elements $$ds$$ and $$ds'$$ respectively. The square of the relative velocity of these particles is

and if we denote by $$r$$ the distance between the particles,