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( 816 ) shows that Therefore, by (6), since 1/ = 0 • (22) $ 7. Our second special case is that of a particle having an electric moment, i. e. a small space S, with a total charge, but with such a distribution of density, that the integrals have values differing from 0. Let x,y, z be the coordinates, taken relatively to a fixed point of the particle, which may be called its centre, and lei the electric moment be defined as a vector p whose components are . . (23) Then • (24) Of course, if x, y, z are treated as infinitely small, u We shall now apply the equation (17) to the determination of the scalar potential V for an exterior point P(x, y, z), at finite distance from the polarized particle, and for the instant at which the Jocal time of this point has some definite value $. In doing so, we shall give the symbol [9], which, in (17), relates to the instant at which I the local time in d S isa slightly different meaning. Distinguishing by r\ the value of r' for the centre A, we shall understand by the value of the density existing in the element dS at the point (X, y, z), at the instant t It may be seen from (5) that this instant precedes that tor which we have to take the numerator in (17; by | (817) units of time. In this last expression we may put for the differential coefficients their values at the point A. In (17) we have now to replace [j>] by . • (25) whererelates again to the time t which the calculations are to be performed having been chosen, this time t etc., by which (25) becomes Again, if henceforth we understand by r‘ what has above been called r\, the factoimust be replaced by so that after all, in the integral (17), the element cl S is multiplied by This is simpler than the primitive form, because neither r', nor the time for which the quantities enclosed in brackets are to be taken, depend on x, y, z. Using (23) and remembering thai we get a formula in which all the enclosed quantities are to be taken for the instant at which the local time of the centre of the particle is We shall conclude these calculations by introducing a new vector V', whose components are 54 Proceedings Royal Acad, Amsterdam. Vol. VI. |