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to the direction of motion of the sphere and to the magnetic induction; and if ω be the resultant velocity of the sphere, and θ the angle between the direction of motion of the sphere and the direction of magnetic induction, the magnitude of the force

$={\frac {\mu e}{2}}\omega {\sqrt {a^{2}+b^{2}+c^{2}}}\sin \theta$.

It will be useful to endeavour to calculate the magnitude of this force on a particle of air moving in a vacuum-tube; although our knowledge of the magnitude of several of the quantities involved is so vague that our result must only be looked upon as showing that the force is of an order great enough to produce appreciable effects, and must not be looked upon as having any quantitative value.

Let us suppose that the mass of a molecule of air is 10^-22 (C.G.S. system); that a, the radius of the molecule, =10^-7; that, as before, $e=K\times 3\times 10^{12}a^{2}=K\times 3\times 10^{-2}$ (this quantity is probably enormously underrated); and as we know nothing about the velocity of the charged particles, let us assume it to be the mean velocity of the air-molecules, viz. $4\times 10^{-5}$. We shall suppose the vacuum-tube placed in a magnetic field whose strength is 10³. Then, by the formula, the acceleration of the particle of air when the magnetic force is at right angles to its path is about 10⁷; this acceleration would produce a deflection of about 2 millims. per decimetre of path, a deflection which could easily be observed. We know from the experiments of Mr. Crookes and others that a magnet produces very decided deflections of the molecular streams; and the direction of the deflections (see Phil. Trans. 1879, part 1, pp. 154 & 156) agrees with that given by formulae (5), if we suppose that the particles projected from the negative pole are negatively charged.

§ 6. Let us now calculate the expression given by Maxwell's theory for the force between two charged moving particles.

Let u, v, w be the components of the velocity of the centre of one of the particles, u', v', w' those of the other; let R denote the distance between the particles, e the charge on one of the particles, e' the charge on the other; let r denote the distance of a point from the centre of the first particle, r' the distance of the same point from the centre of the second particle. We shall suppose, for the sake of simplicity, that the particles are very small; we shall calculate the kinetic energy of the system and deduce the forces between the particles by means of Lagrange's equations.