An introduction to the mathematical theory of attraction ...

170 SurJaces and Czures of the Second Degree.
Here the equation of the hyperboloid referred to its centre and axes is;supposed to be
x2 2 z2
a2` b2 C2
12. Find the total mass on the portion of a hyperboloidal homoeoid of one
sheet intercepted between the plane of xy and a parallel plane.
Here if x, y, z be the coordinates of a point on the surface, we may
assume
x = a cosh 0 cos p, y = b cosh 0 sin, z= c sinh O,
and we find
Q = 27rebc J = 27rbf,
wheref is the distance of the given plane from the centre,
and the equation of the hyperboloid is
x2 / \
a4 - - = 1.
a2 b2 C2
It is to be observed that the total mass of the entire hom~eoid is infinite for
the hyperboloid of one sheet, and for each sheet of the hyperboloid of two sheets.
In the case of the hyperboloid of two sheets, if one of these sheets be composed
of positive mass and the other of negative, the total mass is zero.
13. If one sheet of a hyperboloidal homeoid of two sheets be composed of
positive mass and the other of negative, show that the potential of the homoeoid
at its centre is zero.
14. Show that a hyperboloidal +
homeoid of two sheets, one positive,
the other negative, exercises no attraction at points on the sides of the two
sheets remote from the centre.
This is proved by a method similar
to that employed in Art. 18.
15. If the equipotential surfaces of a field of force be confocal quadrics of
the same family, prove that corresponding points lie on the same line of force.
If a, a', a" be the primary semiaxes of the three confocals passing through
the point x, y, z, by a well-known theorem, Salmon, "Geometry of Three
Dimensions," Art. 160,
a2 a'2 a"2
(x 2 - b2)(a2 - e2)'
whence if a' and a" remain unaltered - is constant, and similar results hold good
a
for y and z. Hence, corresponding points on ellipsoids confocal with a lie on
the intersection of the quadrics a' and a", which is perpendicular to the ellipsoids, and is therefore a line of force when the ellipsoids are equipotentials.