A soap film is formed between two parallel rings of radius

separated by a distance

. To minimize the surface-tension energy of the soap film, its total area

seeks a minimum value. The derivation of the shape of the film involves a problem in the calculus of variations. Let

represent the functional form of the film in cylindrical coordinates. The area is then given by

. The integrand

is determined by the Euler–Lagrange equation

, which can be reduced to its first integral

, a constant. The solution works out to

, a catenary of revolution, with the boundary condition

. When

, the film collapses to disks within the two rings.