Page:Thomson1881.djvu/19

 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' 

$$=\frac{8\pi\sigma}{R}(uu'+vv'+ww')$$;

or, substituting for σ its value,

$$=\frac{\pi ee'}{3R}(uu'+vv'+ww')$$.

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

$$=\frac{\mu ee'}{3R}qq'\cos\epsilon$$,

and the whole kinetic energy due to the electrification

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

$$\frac{\mu ee'}{3R}\left(\frac{dx}{dt}\frac{dx'}{dt}+\frac{dy}{dt}\frac{dy'}{dt}+\frac{dz}{dt}\frac{dz'}{dt}\right)$$.

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