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 with similar expressions for the components of the force parallel to the axes of y and z.

From this we see that if q1 be the acceleration of the second sphere, the forces on the first sphere are an attraction $$\tfrac{\mu ee'}{3R^{2}}qq'\cos\epsilon$$ along the line joining the centres of the spheres, a force $$\tfrac{\mu ee'}{3R}q_{1}$$ in the direction opposite to the acceleration of the second sphere, and a force $$\tfrac{\mu ee'}{3R}q_{1}\tfrac{d}{dt}\left(\tfrac{1}{R}\right)$$ in the direction opposite to the direction of motion of the second sphere. There are, of course, corresponding forces on the second sphere; and we see that, unless both spheres move with equal uniform velocities in the same direction, the forces on the two spheres are not equal and opposite. If we suppose that the two spheres are moving with uniform velocities q in the same direction, the repulsion between them is $$\tfrac{ee'}{K\cdot R^{2}}\left(1-\tfrac{\mu Kq^{2}}{3}\right)$$; or if c be the velocity of light in the medium through which they are moving, the repulsion $$=\tfrac{ee'}{KR^{2}}\left(1-\tfrac{q^{2}}{3c^{2}}\right)$$. Hence, if the repulsion between two electrified particles is to be changed into an attraction by means of their motion, their velocities must exceed $$\sqrt{3c}$$; hence we should expect the molecular streams in a vacuum-tube to repel each other, as we could not suppose that the velocity of the particles forming these streams is as great as that of light; and Mr. Crookes has, in fact (see Phil. Trans. 1879, part ii.), experimentally determined that they do repel each other.

It is remarkable that the law of force between two moving charged particles, which we have deduced from Maxwell's theory, agrees with that assumed by Clausius, in his recent researches on Electrodynamics (see Phil. Mag. Oct. 1880); but it differs from Weber's well-known law materially. According to Weber's law, the force does not depend on the actual velocities of the particles, but only on their velocity relative to each other, whereas, according to the laws we have investigated, the forces depend on the actual velocities of the particles as well as on their relative velocities: thus there is a force between two charged particles moving with equal velocities in the same direction, in which case, of course, the relative velocity is nothing. It must be remarked that what we have for convenience called the actual velocity of the particle is, in fact, the velocity of the particle relative to the medium through which it is moving : thus, in equation (6), q, q' are the velocities of the first and second particles respectively relative to the medium whose magnetic permeability is μ.