Page:Thomson1881.djvu/19

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by the preceding work, that the coefficient of uu' is zero; the coefficient of vv', 3σπ; and the coefficient of ww', 5σπ. Adding, we get the whole kinetic energy due to the vector-potential arising from e and the electric displacement arising from e'

.

We can get that part of the kinetic energy due to the vector-potential arising from e' and the electric displacement from e by writing e' for e, and u', v', w' for u, v, w respectively. Hence, that part of the kinetic energy which is multiplied by ee'

;

or, substituting for σ its value,

.

Or if q and q' be the velocities of the spheres, and ε the angle between their directions of motion, this part of the kinetic energy

,

and the whole kinetic energy due to the electrification

(6)

If x, y, z be the coordinates of the centre of one sphere, x', y', z' those of the other, we may write the last part of the kinetic energy in the form

.

By Lagrange's equations, the force parallel to the axis of x acting on the first sphere

,