Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/189

519.] In the language of quaternions, the resultant force on $$ds$$ is the vector part of the product of the directrix multiplied by $$ds$$.

Since we already know that the directrix is the same thing as the magnetic force due to a unit current in the circuit $$s^\prime$$, we shall henceforth speak of the directrix as the magnetic force due to the circuit.

518.] We shall now complete the calculation of the components of the force acting between two finite currents, whether closed or open.

Let $$\rho$$ be a new function of $$r$$, such that Rhthen by (17) and (20) Rh and equations (11) become

With these values of the component forces, equation (13) becomes

519.] Let

These quantities have definite values for any given point of space. When the circuits are closed, they correspond to the components of the vector-potentials of the circuits.

Let $$L$$ be a new function of $$r$$, such that Rhand let $$M$$ be the double integral Rh