Page:Electromagnetic phenomena.djvu/8

 $$\mathfrak{G}=\frac{1}{c}\int\left[\mathfrak{d.h}\right]dS$$,

shows that

$$\mathfrak{G}_{x}=\frac{1}{c}\int\left(\mathfrak{d}_{y}\mathfrak{h}_{z}-\mathfrak{d}_{z}\mathfrak{h}_{y}\right)dS$$

Therefore, by (6), since $$\mathfrak{h}'=0$$

§ 7. Our second special case is that of a particle having an electric moment, i.e. a small space S, with a total charge $$\int\varrho\ dS=0$$, but with such a distribution of density, that the integrals $$\int\varrho\ x\ dS,\ \int\varrho\ y\ dS,\ \int\varrho\ z\ dS$$ have values differing from 0.

Let x, y, z be the coordinates, taken relatively to a fixed point A of a particle, which may be called its centre, and let the electric moment be defined as a vector $$\mathfrak{p}$$ whose components are

Then

Of course, if x, y, z are treated as infinitely small, $$\mathfrak{u}_{x},\ \mathfrak{u}_{y},\ \mathfrak{u}_{z}$$ must be so likewise. We shall neglect squares and products of these six quantities.

We shall now apply the equations (17) to the determination of the scalar potential φ' for an exterior point P(x, y, z), at finite distance from the polarized particle, and for the instant at which the local time of this point has some definite value t' . In doing so, we shall give the symbol $$\left[\varrho\right]$$, which, in (17), relates to the instant at which local time in dS is $$t'-\frac{r'}{c}$$, a slightly different meaning. Distinguishing by $$r_{0}^{'}$$ the value of r'  for the centre A, we shall understand by $$\left[\varrho\right]$$ the value of the density existing in the element dS at the point (x, y, z), at the instant t0 at which the local time of A is $$t'-\frac{r_{0}^{'}}{c}$$.

It may be seen from (5) that this instant precedes that for which we have to take the numerator in (17) by