Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/231

 If the equatorial radius is very small compared to the polar radius, as in a wire with rounded ends,

When both $$b$$ and $$c$$ become zero, their ratio remaining finite, the system of surfaces becomes two systems of confocal cones, and a system of spherical surfaces of which the radius is inversely proportional to $$\gamma$$.

If the ratio of $$b$$ to $$c$$ is zero or unity, the system of surfaces becomes one system of meridian planes, one system of right cones having a common axis, and a system of concentric spherical surfaces of which the radius is inversely proportional to $$\gamma$$. This is the ordinary system of spherical polar coordinates.

153.] When $$c$$ is infinite the surfaces are cylindric, the generating lines being parallel to $$z$$. One system of cylinders is elliptic, with the equation

The other is hyperbolic, with the equation

This system is represented in Fig. X, at the end of this volume.



154.] If in the general equations we transfer the origin of co ordinates to a point on the axis of $$x$$ distant $$t$$ from the centre of the system, and if we substitute for $$x, \lambda, d,$$ and $$c, t + x, t + \lambda, t + b$$, and $$t + c$$ respectively, and then make $$t$$ increase indefinitely, we obtain, in the limit, the equation of a system of paraboloids whose foci are at the points $$x = b$$ and $$x = c$$,

If the variable parameter is $$\lambda$$ for the first system of elliptic paraboloids, $$\mu$$ for the hyperbolic paraboloids, and $$\nu$$ for the second system of elliptic paraboloids, we have $$\lambda, b,\mu, c,\nu$$ in ascending order of magnitude, and