An introduction to the mathematical theory of attraction ...

194 Surfaces and Curves of the Second Degree.
10. If the equipotential surfaces of a field of force be paraboloids of revolution, show that the force component at any point P in the direction joining P
to the focus F varies inversely as PF.
This follows from the expression for the potential, since
(À - h) sec2 Wm = PF = À - v.
11. If the boundaries of the field in the last Example be the surfaces of
conductors in electric equilibrium, show that the density of the distribution of
mass on one of these surfaces varies inversely as the focal perpendicular on the
tangent plane.
106. Uniplanar Distribution.-Results analogous to
those obtained in Arts. 99-105 hold good for a uniplanar
distribution of mass acting inversely as the distance.
The equation
4(x - À) + = 0 (65)
represents a system of confocal parabolas containing two
families such that the curves belonging to the one and
those belonging to the other are turned in opposite directions.
In fact if x and y be given in (65), the quadratic equation
for determining X has two roots, one between + co and h
which may be called À, and one between h and - oc which
may be called y. Then we have
x = x +, - h, y2 = 4(À - A) (h - 6). (66)
Corresponding points whose coordinates are x, y, and x' y',
on confocal parabolas of the same family, whose parameters
are À and X', are defined by the equations
x - = x' - - = À -h_ (67)
If ds and dt denote the elements of the curves whose parameters are X and fi at the point x, y, since these curves cut
orthogonally, by (66) we have
dX dx
dt dt os