An elastic string of uniform section and density in its natural state is suspended from two points. Find its equation of equilibrium.
Let the element stretch into ; then, by Hooke’s law, , where is the elastic constant of the string; hence the weight of the stretched element . Accordingly, as before,
which is the intrinsic equation of the curve, and reduces to that of the common catenary when . The coordinates , may be expressed in terms of the single parameter by putting
These equations are more convenient than the result of eliminating , which is somewhat complicated.