The true catenary is that assumed by a chain of uniform weight per unit of length, but the form generally adopted for suspension bridges is that assumed by a chain under a weight uniformly distributed relatively to a horizontal line.
Finally, we may refer to the catenary of uniform strength, where the cross-section of the wire (or cable) is supposed to vary as the tension.
The only surface of revolution having this property is the catenoid formed by the revolution of a catenary about its directrix.
This catenoid, however, is in stable equilibrium only when the portion considered is such that the tangents to the catenary at its extremities intersect before they reach the directrix.
Every catenary lying between them has its directrix higher, and every catenary lying beyond them has its directrix lower than that of the two catenaries.