Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/279

When the angle is a salient one $\alpha$ is less than $\pi$, and the surface-density varies according to some inverse power of the distance from the edge, so that at the edge itself the density becomes infinite, although the whole charge reckoned from the edge to any finite distance from it is always finite.

Thus, when $\alpha =0$ the edge is infinitely sharp, like the edge of a mathematical plane. In this case the density varies inversely as the square root of the distance from the edge.

When $\alpha ={\tfrac {\pi }{3}}$ the edge is like that of an equilateral prism, and the density varies inversely as the ${\tfrac {2}{5}}$ power of the distance.

When $\alpha ={\tfrac {\pi }{2}}$ the edge is a right angle, and the density is inversely as the cube root of the distance.

When $\alpha ={\tfrac {2\pi }{3}}$ the edge is like that of a regular hexagonal prism, and the density is inversely as the fourth root of the distance.

When $\alpha =\pi$ the edge is obliterated, and the density is constant.

When $\alpha ={\tfrac {4}{3}}\pi$ the edge is like that in the inside of the hexagonal prism, and the density is directly as the square root of the distance from the edge.

When $\alpha ={\tfrac {3}{2}}\pi$ the edge is a re-entrant right angle, and the density is directly as the distance from the edge.

When $\alpha ={\tfrac {5}{3}}\pi$ the edge is a re-entrant angle of 60°, and the density is directly as the square of the distance from the edge.

In reality, in all cases in which the density becomes infinite at any point, there is a discharge of electricity into the dielectric at that point, as is explained in Art. 55.

EXAMPLE V. Ellipses and Hyperbolas. Fig. X.

192.] We have seen that if

$x_{1}=e^{\phi }\cos \psi ,\ y_{1}=e^{\phi }\sin \psi$.

(1)

$x$ and $y$ will be conjugate functions of $\phi$ and $\psi$.

Also, if

$x_{2}=e^{-\phi }\cos \psi ,\ y_{2}=-e^{-\phi }\sin \psi$

(2)

$x_{2}$ and $y_{2}$ will be conjugate functions. Hence, if

$2x=x_{1}+x_{2}=\left(e^{\phi }+e^{-\phi }\right)\cos \psi ,\ 2y=y_{1}+y_{2}=\left(e^{\phi }-e^{-\phi }\right)\sin \psi ,$

(3)

$x$ and $y$ will also be conjugate functions of $\phi$ and $\psi$.

In this case the points for which $\phi$ is constant lie in the ellipse whose axes are $e^{\phi }+e^{-\phi }$ and $e^{\phi }-e^{-\phi }$.