The equation of an ellipsoid, referred to its principal axes, is
| x2 | + | y2 | + | z2 | = 1. |
| a2 | b2 | c2 |
Here a, b, c, are the lengths of the principal semi-axes. A plane section is an ellipse, but there are two particular directions of the section for which the ellipse reduces to a circle.
The ellipsoid has important applications in dynamics, as a means of interpreting algebraic formulæ for physical quantities; examples are the ellipsoids of strain and of gyration, and Poinsot's momental ellipsoid.
Two ellipsoids in which the principal sections lie in the same planes, and have the same foci, are called confocal.
Ellipsoidal Harmonics are mathematical functions by means of which certain physical problems in heat and electricity relating to ellipsoidal surfaces can be solved. For ellipsoids of revolution, see Spheroid.
Elliptic Functions are generalizations of the circular functions sine, cosine, &c. If
we have x = sin u. Similarly, if
