Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/267

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616.]
VECTOR-POTENTIAL OF CURRENTS.
235

According to our hypothesis a, b, c are identical with μα, μβ, μγ respectively. We therefore obtain


(1)
If we write (2)
and[1] (3)

we may write equation (1),

Similarly, (4)


If we write


(5)

(6)

where r is the distance of the given point from the element x y z, and the integrations are to be extended over all space, then


(7)


The quantity χ disappears from the equations (A), and it is not related to any physical phenomenon. If we suppose it to be zero everywhere, J will also be zero everywhere, and equations (5), omitting the accents, will give the true values of the components of .

  1. The negative sign is employed here in order to make our expressions consistent with those in which Quaternions are employed.