Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/122

82 by the first of equations (12).

Multiplying by $$Rl$$, and remembering that at the second surface

where $$\sigma$$ is the surface-density; or, multiplying the equations (12) by $$l, m, n$$ respectively, and adding,

This equation is called the characteristic equation of $$V$$ at a surface. This equation may also be written

where $$v_1, v_2$$ are the normals to the surface drawn towards the first and the second medium respectively, and $$V_1, V_2$$ the potentials  at points on these normals. We may also write it where $$R_1, R_2$$ are the resultant forces, and $$\epsilon_1, \epsilon_2$$ the angles which they make with the normals drawn from the surface on either side.

79.] Let us next determine the total mechanical force acting on an element of the electrified surface.

The general expression for the force parallel to $$x$$ on an element whose volume is $$dx\,dy\,dz$$, and volume-density $$\rho$$, is