The numbers
,
and
connected with Cartesian rectangular coordinates
,
and
by the formulas
where
.
The coordinate surfaces are (see
Fig.): ellipses
,
one-sheet hyperbolas
(
),
and two-sheet hyperbolas
(
),
with centres at the coordinate origin.
Figure: e035420a
The system of ellipsoidal coordinates is orthogonal. To every triple of numbers
,
and
correspond 8 points (one in each octant), which are symmetric to
each other relative to the coordinate planes of the system
.
The
Lamé coefficients
are
If one of the conditions
in the definition of ellipsoidal coordinates is replaced by
an equality, then degenerate ellipsoidal coordinate systems are obtained.