Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/227

in the space beyond the surface $\alpha _{2}$, we shall have obtained the complete solution of this particular case.

The resultant force at any point of either sheet is

$R_{1}=-{\frac {dV}{ds_{1}}}=-{\frac {dV}{d\alpha }}{\frac {d\alpha }{ds_{1}}},$

(19)

or

$R_{1}={\frac {V_{1}-V_{2}}{\alpha _{1}-\alpha _{2}}}{\frac {c}{D_{2}D_{3}}}$

(20)

If $p_{1}$ be the perpendicular from the centre on the tangent plane at any point, and $P_{1}$ the product of the semi-axes of the surface, then $p_{1}D_{2}D_{3}=P_{1}$.

Hence we find

$R_{1}={\frac {V_{1}-V_{2}}{\alpha _{1}-\alpha _{2}}}{\frac {cp_{1}}{P_{1}}},$

(21)

or the force at any point of the surface is proportional to the perpendicular from the centre on the tangent plane.

The surface-density $\sigma$ may be found from the equation

$4\pi \sigma =R_{1}.$

(22)

)

The total quantity of electricity on a segment cut off by a plane whose equation is $x=a$ from one sheet of the hyperboloid is

$Q={\frac {c}{2}}{\frac {V_{1}-V_{2}}{\alpha _{1}-\alpha _{2}}}\left({\frac {a}{\lambda _{1}}}-1\right).$

(23)

The quantity on the whole infinite sheet is therefore infinite. The limiting forms of the surface are :—

(1) When $\alpha =F_{(k)}$ the surface is the part of the plane of $xz$ on the positive side of the positive branch of the hyperbola whose equation is

${\frac {x^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=1.$

(24)

(2) When $\alpha =0$ the surface is the plane of $yz$.

(3) When $\alpha =-F_{(k)}$ the surface is the part of the plane of $xz$ on the negative side of the negative branch of the same hyperbola.

The Hyperloloids of One Sheet.

By making $\beta$ constant we obtain the equation of the hyperboloid of one sheet. The two surfaces which form the boundaries of the electric field must therefore belong to two different hyperboloids. The investigation will in other respects be the same as for the hyperboloids of two sheets, and when the difference of potentials is given the density at any point of the surface will be proportional to the perpendicular from the centre on the tangent plane, and the whole quantity on the infinite sheet will be infinite.