  Elliptic equation $(y')^2=a_0+a_2y^2+a_4y^4$ is the foundation of the
elliptic function expansion method of finding exact solutions to nonlinear
differential equation. In some references, some new form solutions to the
elliptic equation have been claimed. In the paper, we discuss its solutions in
detail. By detailed computation, we prove that those new form solutions can be
derived from a very few known solutions. This means that those new form
solutions are just new representations of old solutions. From our discussion,
some new identities of the elliptic function can be obtained. In the course of
discussion, we give an example of this kind of formula.
