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which these electromotive forces diminish with the distance, we will take the case of a particle stopped by a screen at a distance of ${\tfrac {1}{100}}$ of a millimetre from the glass, and compare the electromotive force at the glass with the electromotive force which would be produced at the glass if there were no screen. By substituting in the formula giving the electromotive force, we find that the electromotive force at the glass when the screen is present is only about ${\tfrac {1}{10000}}$ of what it is when the screen is away; and as the intensity of the phosphorescence will vary as the square of the electromotive force, we see that when the screen is present the phosphorescence is quite imperceptible. This explains an experiment of Goldstein's, in which he coated the glass with a layer of collodion whose thickness he estimated at a few hundredths of a millimetre, and found the glass behind quite black.

§ 5. To find the effect produced by a magnet on a moving electrified sphere. To do this we shall calculate the kinetic energy of the system; we can then, by means of Lagrange's equations, calculate the force on the sphere.

Let α, β, γ be the components of magnetic induction, α₁, β₁, γ₁ the components of magnetic force; if A, B, C be the components of magnetization,

$\alpha =\alpha _{1}+4\pi A,\quad \beta =\beta _{1}+4\pi B,\quad \gamma =\gamma _{1}+4\pi C$.

The kinetic energy of the system

$={\frac {1}{8\pi }}\int \int \int \left(\alpha \alpha _{1}+\beta \beta _{2}+\gamma \gamma _{1}\right)dx\ dy\ dz$.

To get the force on the sphere due to the magnet, we only want that part of the kinetic energy which involves both the coordinates of the sphere and the coordinates of the magnet. We may write the kinetic energy as

${\frac {1}{8\pi }}\int \int \int \left(\alpha ^{2}+\beta ^{2}+\gamma ^{2}-4\pi \alpha A-4\pi \beta B-4\pi \gamma C\right)dx\ dy\ dz$.

Let F', G', H' be the components of the vector-potential due to the magnet alone; then, by equation (4),

$\left.{\begin{array}{ll}\alpha &=\mu e\left(r{\frac {d}{dy}}{\frac {1}{R}}-q{\frac {d}{dz}}{\frac {1}{R}}\right)+{\frac {dH'}{dy}}-{\frac {dG'}{dz}},\\\\\beta &=\mu e\left(p{\frac {d}{dz}}{\frac {1}{R}}-r{\frac {d}{dx}}{\frac {1}{R}}\right)+{\frac {dF'}{dz}}-{\frac {dH'}{dx}},\\\\\gamma &=\mu e\left(q{\frac {d}{dx}}{\frac {1}{R}}-p{\frac {d}{dy}}{\frac {1}{R}}\right)+{\frac {dG'}{dx}}-{\frac {dF'}{dy}}.\end{array}}\right\}$